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Introduction
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James H. Stock
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Mark W. Watson
Princeton University
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Library of Congress Cataloging-in-Publication Data
Stock, James H.
Introduction to econometrics/James H. Stock, Harvard University, Mark W. Watson, Princeton University.—
Third edition update.
pages cm.—(The Pearson series in economics)
Includes bibliographical references and index.
ISBN 978-0-13-348687-2—ISBN 0-13-348687-7 1. Econometrics. I. Watson, Mark W. II. Title.
HB139.S765 2015 330.01’5195––dc23
2014018465
www.pearsonhighered.com
ISBN-10: 0-13-348687-7 ISBN-13: 978-0-13-348687-2
Brief Contents
PART ONE
CHaPter 1 CHaPter 2 CHaPter 3
PART TWO
CHaPter 4 CHaPter 5
CHaPter 6 CHaPter 7 CHaPter 8 CHaPter 9
PART THREE
CHaPter 10 CHaPter 11 CHaPter 12 CHaPter 13
PART FOuR
CHaPter 14 CHaPter 15 CHaPter 16
PART FIvE
CHaPter 17 CHaPter 18
Introduction and Review
economic Questions and Data 1 review of Probability 14
review of Statistics 65
Fundamentals of Regression Analysis
Linear regression with One regressor 109
regression with a Single regressor: Hypothesis tests and Confidence Intervals 146
Linear regression with Multiple regressors 182
Hypothesis tests and Confidence Intervals in Multiple regression 217 Nonlinear regression Functions 256
assessing Studies Based on Multiple regression 315
Further Topics in Regression Analysis
regression with Panel Data 350
regression with a Binary Dependent Variable 385 Instrumental Variables regression 424 experiments and Quasi-experiments 475
Regression Analysis of Economic Time Series Data
Introduction to time Series regression and Forecasting 522 estimation of Dynamic Causal effects 589
additional topics in time Series regression 638
The Econometric Theory of Regression Analysis
the theory of Linear regression with One regressor 676 the theory of Multiple regression 705
v
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Contents
Preface xxix
PART ONE
CHAPTER 1
1.1
1.2 1.3
CHAPTER 2
2.1
2.2
2.3
Introduction and Review
Economic Questions and Data 1
economic Questions We examine 1
Question #1: Does reducing Class Size Improve elementary School education? 2 Question #2: Is there racial Discrimination in the Market for Home Loans? 3 Question #3: How Much Do Cigarette taxes reduce Smoking? 3
Question #4: By How Much Will U.S. GDP Grow Next Year? 4
Quantitative Questions, Quantitative answers 5 Causal effects and Idealized experiments 5
estimation of Causal effects Forecasting and Causality 7
Data: Sources and types
6
experimental Versus Observational Data 7 Cross-Sectional Data 8
time Series Data 9
Panel Data 11
Review of Probability 14
random Variables and Probability Distributions
15
7
Probabilities, the Sample Space, and random Variables Probability Distribution of a Discrete random Variable 16 Probability Distribution of a Continuous random Variable 19
expected Values, Mean, and Variance 19
the expected Value of a random Variable 19
the Standard Deviation and Variance 21
Mean and Variance of a Linear Function of a random Variable 22 Other Measures of the Shape of a Distribution 23
two random Variables 26 Joint and Marginal Distributions 26
15
vii
viii Contents
2.4
2.5 2.6
CHAPTER 3
3.1
3.2
Conditional Distributions 27
Independence 31
Covariance and Correlation 31
the Mean and Variance of Sums of random Variables 32
the Normal, Chi-Squared, Student t, and F Distributions 36 the Normal Distribution 36
the Chi-Squared Distribution 41 the Student t Distribution 41 the F Distribution 42
random Sampling and the Distribution of the Sample average 43
random Sampling 43
the Sampling Distribution of the Sample average 44
Large-Sample approximations to Sampling Distributions 47
3.3 3.4
the Law of Large Numbers and Consistency 48 the Central Limit theorem 50
aPPeNDIx 2.1 Derivation of results in Key Concept 2.3 63 Review of Statistics 65
estimation of the Population Mean 66 estimators and their Properties 66
Properties of Y 68
the Importance of random Sampling 70
Hypothesis tests Concerning the Population Mean 71 Null and alternative Hypotheses 71
the p-Value 72
Calculating the p-Value When sY Is Known 73
the Sample Variance, Sample Standard Deviation, and Standard error Calculating the p-Value When sY Is Unknown 76
the t-Statistic 76
Hypothesis testing with a Prespecified Significance Level 77 One-Sided alternatives 79
Confidence Intervals for the Population Mean 80
Comparing Means from Different Populations 82
74
Hypothesis tests for the Difference Between two Means 82
Confidence Intervals for the Difference Between two Population Means 84
PART TWO
CHAPTER 4
4.1 4.2
4.3
4.4
Scatterplots, the Sample Covariance, and the Sample Correlation 91
Scatterplots 91
Sample Covariance and Correlation 92
aPPeNDIx 3.1 the U.S. Current Population Survey 106
aPPeNDIx 3.2 two Proofs that Y Is the Least Squares estimator of μY 107 aPPeNDIx 3.3 a Proof that the Sample Variance Is Consistent 108
Fundamentals of Regression Analysis
Linear Regression with One Regressor 109
the Linear regression Model 109
estimating the Coefficients of the Linear regression
Model 114
the Ordinary Least Squares estimator 116
OLS estimates of the relationship Between test Scores and the Student– teacher ratio 118
Why Use the OLS estimator? 119 Measures of Fit 121
the R2 121
the Standard error of the regression 122 application to the test Score Data 123
the Least Squares assumptions 124
assumption #1: the Conditional Distribution of ui Given Xi Has a Mean of Zero 124
assumption #2: (Xi, Yi), i = 1,…, n, are Independently and Identically Distributed 126
assumption #3: Large Outliers are Unlikely 127 Use of the Least Squares assumptions 128
3.5
3.6 3.7
Differences-of-Means estimation of Causal effects Using experimental Data 84
the Causal effect as a Difference of Conditional expectations estimation of the Causal effect Using Differences of Means
Using the t-Statistic When the Sample Size Is Small the t-Statistic and the Student t Distribution 87
Use of the Student t Distribution in Practice 89
85 85
87
Contents ix
x Contents 4.5
4.6
CHAPTER 5
5.1
5.2 5.3
5.4
5.5
5.6
5.7
Sampling Distribution of the OLS estimators 129 the Sampling Distribution of the OLS estimators 130
Conclusion 133
aPPeNDIx 4.1 the California test Score Data Set 141 aPPeNDIx 4.2 Derivation of the OLS estimators 141
aPPeNDIx 4.3 Sampling Distribution of the OLS estimator 142
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals 146
testing Hypotheses about One of the regression Coefficients 146
two-Sided Hypotheses Concerning β1 147
One-Sided Hypotheses Concerning β1 150
testing Hypotheses about the Intercept β0 152
Confidence Intervals for a regression Coefficient 153
regression When X Is a Binary Variable 155 Interpretation of the regression Coefficients 155
Heteroskedasticity and Homoskedasticity 157
What are Heteroskedasticity and Homoskedasticity? Mathematical Implications of Homoskedasticity 160 What Does this Mean in Practice? 161
158
the theoretical Foundations of Ordinary Least Squares 163
Linear Conditionally Unbiased estimators and the Gauss–Markov theorem 164
regression estimators Other than OLS 165
Using the t-Statistic in regression When the Sample Size Is Small 166
the t-Statistic and the Student t Distribution 166
Use of the Student t Distribution in Practice 167
Conclusion 168
aPPeNDIx 5.1 Formulas for OLS Standard errors 177
aPPeNDIx 5.2 the Gauss–Markov Conditions and a Proof of the Gauss–Markov theorem 178
CHAPTER 6
6.1
6.2 6.3 6.4
6.5
6.6 6.7
6.8
Contents xi Linear Regression with Multiple Regressors 182
Omitted Variable Bias 182
Definition of Omitted Variable Bias 183
a Formula for Omitted Variable Bias 185
addressing Omitted Variable Bias by Dividing the Data into
Groups 187
the Multiple regression Model 189
the Population regression Line 189
the Population Multiple regression Model 190
the OLS estimator in Multiple regression 192
the OLS estimator 193
application to test Scores and the Student–teacher ratio 194
Measures of Fit in Multiple regression 196 the Standard error of the regression (SER) 196
the R2 196
the “adjusted R2” 197 application to test Scores 198
the Least Squares assumptions in Multiple
regression 199
assumption #1: the Conditional Distribution of ui Given X1i, X2i, c, Xki Has a Mean of Zero 199
assumption #2: (X1i, X2i, c, Xki, Yi), i = 1, c, n, are i.i.d. 199 assumption #3: Large Outliers are Unlikely 199
assumption #4: No Perfect Multicollinearity 200
the Distribution of the OLS estimators in Multiple regression 201
Multicollinearity 202
examples of Perfect Multicollinearity 203 Imperfect Multicollinearity 205
Conclusion 206
aPPeNDIx 6.1 Derivation of equation (6.1) 214
aPPeNDIx 6.2 Distribution of the OLS estimators When there are two regressors and Homoskedastic errors 214
aPPeNDIx 6.3 the Frisch–Waugh theorem 215
xii Contents CHAPTER 7
7.1
7.2
7.3 7.4 7.5
7.6 7.7
CHAPTER 8
8.1
8.2
Hypothesis Tests and Confidence Intervals in Multiple Regression 217
Hypothesis tests and Confidence Intervals for a Single Coefficient 217
Standard errors for the OLS estimators 217
Hypothesis tests for a Single Coefficient 218
Confidence Intervals for a Single Coefficient 219
application to test Scores and the Student–teacher ratio 220
tests of Joint Hypotheses 222
testing Hypotheses on two or More Coefficients 222
the F-Statistic 224
application to test Scores and the Student–teacher ratio 226 the Homoskedasticity-Only F-Statistic 227
testing Single restrictions Involving Multiple Coefficients 229
Confidence Sets for Multiple Coefficients 231
Model Specification for Multiple regression 232
Omitted Variable Bias in Multiple regression 233
the role of Control Variables in Multiple regression 234 Model Specification in theory and in Practice 236 Interpreting the R2 and the adjusted R2 in Practice 237
analysis of the test Score Data Set 238
Conclusion 243
aPPeNDIx 7.1 the Bonferroni test of a Joint Hypothesis 251 aPPeNDIx 7.2 Conditional Mean Independence 253
Nonlinear Regression Functions 256
a General Strategy for Modeling Nonlinear regression Functions 258 test Scores and District Income 258
the effect on Y of a Change in X in Nonlinear Specifications 261
a General approach to Modeling Nonlinearities Using Multiple regression 266
Nonlinear Functions of a Single Independent Variable 266
Polynomials 267
Logarithms 269
Polynomial and Logarithmic Models of test Scores and District Income 277
8.3
8.4 8.5
CHAPTER 9
9.1 9.2
9.3
9.4
9.5
Interactions Between Independent Variables 278
Interactions Between two Binary Variables 279
Interactions Between a Continuous and a Binary Variable 282 Interactions Between two Continuous Variables 286
Nonlinear effects on test Scores of the Student–teacher ratio 293
Discussion of regression results 293 Summary of Findings 297
Conclusion 298
aPPeNDIx 8.1 regression Functions that are Nonlinear in the Parameters 309
aPPeNDIx 8.2 Slopes and elasticities for Nonlinear regression Functions 313
Assessing Studies Based on Multiple Regression 315
Internal and external Validity 315
threats to Internal Validity 316 threats to external Validity 317
threats to Internal Validity of Multiple regression analysis 319
Omitted Variable Bias 319
Misspecification of the Functional Form of the regression Function 321 Measurement error and errors-in-Variables Bias 322
Missing Data and Sample Selection 325
Simultaneous Causality 326
Sources of Inconsistency of OLS Standard errors 329
Internal and external Validity When the regression Is Used for Forecasting 331
Using regression Models for Forecasting 331
assessing the Validity of regression Models for Forecasting 332
example: test Scores and Class Size 332
external Validity 332
Internal Validity 339
Discussion and Implications 341
Conclusion 342
aPPeNDIx 9.1 the Massachusetts elementary School testing Data 349
Contents xiii
xiv Contents PART THREE
CHAPTER 10
10.1 10.2 10.3
10.4 10.5
10.6 10.7
CHAPTER 11
11.1 11.2
11.3
Further Topics in Regression Analysis
Regression with Panel Data 350
Panel Data 351
example: traffic Deaths and alcohol taxes 352
Panel Data with two time Periods: “Before and after” Comparisons 354
Fixed effects regression 357
the Fixed effects regression Model 357 estimation and Inference 359 application to traffic Deaths 361
regression with time Fixed effects 361
time effects Only 362
Both entity and time Fixed effects 363
the Fixed effects regression assumptions and Standard errors for Fixed effects regression 365
the Fixed effects regression assumptions 365
Standard errors for Fixed effects regression 367
Drunk Driving Laws and traffic Deaths 368
Conclusion 372
aPPeNDIx 10.1 the State traffic Fatality Data Set 380
aPPeNDIx 10.2 Standard errors for Fixed effects regression 380
Regression with a Binary Dependent variable 385
Binary Dependent Variables and the Linear Probability Model 386
Binary Dependent Variables 386 the Linear Probability Model 388
Probit and Logit regression 391
Probit regression 391
Logit regression 396
Comparing the Linear Probability, Probit, and Logit Models 398
estimation and Inference in the Logit and Probit Models 398 Nonlinear Least Squares estimation 399
11.4 11.5
CHAPTER 12
12.1
12.2
12.3
12.4 12.5
12.6
Maximum Likelihood estimation 400 Measures of Fit 401
application to the Boston HMDa Data 402
Conclusion 409
aPPeNDIx 11.1 the Boston HMDa Data Set 418
aPPeNDIx 11.2 Maximum Likelihood estimation 418 aPPeNDIx 11.3 Other Limited Dependent Variable Models 421
Instrumental variables Regression 424
the IV estimator with a Single regressor and a Single Instrument 425
the IV Model and assumptions 425
the two Stage Least Squares estimator 426
Why Does IV regression Work? 427
the Sampling Distribution of the tSLS estimator 431 application to the Demand for Cigarettes 433
the General IV regression Model 435
tSLS in the General IV Model 437
Instrument relevance and exogeneity in the General IV Model 438 the IV regression assumptions and Sampling Distribution of the
tSLS estimator 439
Inference Using the tSLS estimator 440 application to the Demand for Cigarettes 441
Checking Instrument Validity 442
assumption #1: Instrument relevance 443 assumption #2: Instrument exogeneity 445
application to the Demand for Cigarettes 448
Where Do Valid Instruments Come From? 453 three examples 454
Conclusion 458
aPPeNDIx 12.1 the Cigarette Consumption Panel Data Set 467
aPPeNDIx 12.2 Derivation of the Formula for the tSLS estimator in equation (12.4) 467
Contents xv
xvi Contents
CHAPTER 13
13.1
13.2 13.3
13.4
13.5 13.6
aPPeNDIx 12.3 Large-Sample Distribution of the tSLS estimator 468 aPPeNDIx 12.4 Large-Sample Distribution of the tSLS estimator When
the Instrument Is Not Valid 469
aPPeNDIx 12.5 Instrumental Variables analysis with Weak Instruments 471
aPPeNDIx 12.6 tSLS with Control Variables 473 Experiments and Quasi-Experiments 475
Potential Outcomes, Causal effects, and Idealized experiments 476
Potential Outcomes and the average Causal effect 476 econometric Methods for analyzing experimental Data 478
threats to Validity of experiments 479
threats to Internal Validity 479 threats to external Validity 483
experimental estimates of the effect of Class Size reductions 484
experimental Design 485
analysis of the Star Data 486
Comparison of the Observational and experimental estimates of Class Size effects 491
Quasi-experiments 493
examples 494
the Differences-in-Differences estimator 496 Instrumental Variables estimators 499 regression Discontinuity estimators 500
Potential Problems with Quasi-experiments 502
threats to Internal Validity 502 threats to external Validity 504
experimental and Quasi-experimental estimates in Heterogeneous Populations 504
OLS with Heterogeneous Causal effects 505
IV regression with Heterogeneous Causal effects 506
PART FOuR
CHAPTER 14
14.1 14.2
14.3 14.4
14.5 14.6
Forecasting GDP Growth Using the term Spread Stationarity 540
time Series regression with Multiple Predictors Forecast Uncertainty and Forecast Intervals 544
537 541
13.7
Conclusion 509
aPPeNDIx 13.1 the Project Star Data Set 518
aPPeNDIx 13.2 IV estimation When the Causal effect Varies across Individuals 518
aPPeNDIx 13.3 the Potential Outcomes Framework for analyzing Data from experiments 520
Regression Analysis of Economic Time Series Data
Introduction to Time Series Regression and Forecasting 522
Using regression Models for Forecasting 523
Introduction to time Series Data and Serial Correlation 524
real GDP in the United States 524
Lags, First Differences, Logarithms, and Growth rates 525 autocorrelation 528
Other examples of economic time Series 529
autoregressions 531
the First-Order autoregressive Model 531
the pth-Order autoregressive Model 534
time Series regression with additional Predictors and the autoregressive Distributed Lag Model 537
Contents xvii
Lag Length Selection Using Information Criteria 547
Determining the Order of an autoregression 547
Lag Length Selection in time Series regression with Multiple Predictors 550
Nonstationarity I: trends 551
What Is a trend? 551
Problems Caused by Stochastic trends 554
Detecting Stochastic trends: testing for a Unit ar root 556 avoiding the Problems Caused by Stochastic trends 561
xviii
Contents 14.7
14.8
CHAPTER 15
15.1 15.2
15.3
15.4
15.5
15.6
Nonstationarity II: Breaks 561
What Is a Break? 562
testing for Breaks 562
Pseudo Out-of-Sample Forecasting 567 avoiding the Problems Caused by Breaks 573
Conclusion 573
aPPeNDIx 14.1 time Series Data Used in Chapter 14 583 aPPeNDIx 14.2 Stationarity in the ar(1) Model 584
aPPeNDIx 14.3 Lag Operator Notation 585
aPPeNDIx 14.4 arMa Models 586
aPPeNDIx 14.5 Consistency of the BIC Lag Length estimator 587
Estimation of Dynamic Causal Effects 589
an Initial taste of the Orange Juice Data 590
Dynamic Causal effects 593
Causal effects and time Series Data 593 two types of exogeneity 596
estimation of Dynamic Causal effects with exogenous regressors 597
the Distributed Lag Model assumptions 598
autocorrelated ut, Standard errors, and Inference 599 Dynamic Multipliers and Cumulative Dynamic Multipliers 600
Heteroskedasticity- and autocorrelation-Consistent Standard errors 601
Distribution of the OLS estimator with autocorrelated errors 602
HaC Standard errors 604
estimation of Dynamic Causal effects with Strictly exogenous regressors 606
the Distributed Lag Model with ar(1) errors 607
OLS estimation of the aDL Model 610
GLS estimation 611
the Distributed Lag Model with additional Lags and ar(p) errors 613
Orange Juice Prices and Cold Weather 616
15.7
15.8
CHAPTER 16
16.1 16.2
16.3
16.4
16.5
16.6
Is exogeneity Plausible? Some examples 624
U.S. Income and australian exports 624
Oil Prices and Inflation 625
Monetary Policy and Inflation 626
the Growth rate of GDP and the term Spread
626 aPPeNDIx 15.1 the Orange Juice Data Set 634
Conclusion 627
aPPeNDIx 15.2 the aDL Model and Generalized Least Squares in Lag Operator Notation 634
Additional Topics in Time Series Regression 638
Vector autoregressions 638
the Var Model 639
a Var Model of the Growth rate of GDP and the term Spread 642
Multiperiod Forecasts 643
Iterated Multiperiod Forecasts 643 Direct Multiperiod Forecasts 645 Which Method Should You Use? 648
Orders of Integration and the DF-GLS Unit root test 649
Other Models of trends and Orders of Integration 649
the DF-GLS test for a Unit root 651
Why Do Unit root tests Have Nonnormal Distributions? 654
Cointegration 656
Cointegration and error Correction 656
How Can You tell Whether two Variables are Cointegrated? 658 estimation of Cointegrating Coefficients 659
extension to Multiple Cointegrated Variables 661
application to Interest rates 662
Volatility Clustering and autoregressive Conditional Heteroskedasticity 664
Volatility Clustering 664
autoregressive Conditional Heteroskedasticity 666 application to Stock Price Volatility 667
Conclusion 670
Contents xix
xx Contents PART FIvE
CHAPTER 17
17.1 17.2
17.3
17.4
17.5
The Econometric Theory of Regression Analysis
The Theory of Linear Regression with One Regressor 676
the extended Least Squares assumptions and the OLS estimator 677
the extended Least Squares assumptions 677 the OLS estimator 679
Fundamentals of asymptotic Distribution theory 679
Convergence in Probability and the Law of Large Numbers 680 the Central Limit theorem and Convergence in Distribution 682
Slutsky’s theorem and the Continuous Mapping theorem application to the t-Statistic Based on the Sample Mean 684
asymptotic Distribution of the OLS estimator and
t-Statistic 685
Consistency and asymptotic Normality of the OLS estimators 685 Consistency of Heteroskedasticity-robust Standard errors 685 asymptotic Normality of the Heteroskedasticity-robust t-Statistic 687
exact Sampling Distributions When the errors are Normally Distributed 687
n
Weighted Least Squares 690
WLS with Known Heteroskedasticity 690
WLS with Heteroskedasticity of Known Functional Form 691 Heteroskedasticity-robust Standard errors or WLS? 694
aPPeNDIx 17.1 the Normal and related Distributions and Moments of
Continuous random Variables 700 aPPeNDIx 17.2 two Inequalities 703
The Theory of Multiple Regression 705
the Linear Multiple regression Model and OLS estimator in Matrix Form 706
the Multiple regression Model in Matrix Notation 706
the extended Least Squares assumptions 708 the OLS estimator 709
CHAPTER 18
18.1
Distribution of β1 with Normal errors 687
Distribution of the Homoskedasticity-Only t-Statistic 689
683
Contents xxi
18.2 asymptotic Distribution of the OLS estimator and t-Statistic 710
the Multivariate Central Limit theorem 710 asymptotic Normality of bn 711 Heteroskedasticity-robust Standard errors 712 Confidence Intervals for Predicted effects 713 asymptotic Distribution of the t-Statistic 713
18.3 tests of Joint Hypotheses 713 Joint Hypotheses in Matrix Notation 714
asymptotic Distribution of the F-Statistic 714 Confidence Sets for Multiple Coefficients 715
18.4 Distribution of regression Statistics with Normal errors 716 Matrix representations of OLS regression Statistics 716
Distribution of bn with Normal errors 717 Distribution of s2uN 718 Homoskedasticity-Only Standard errors 718 Distribution of the t-Statistic 719 Distribution of the F-Statistic 719
18.5 efficiency of the OLS estimator with Homoskedastic errors 720
the Gauss–Markov Conditions for Multiple regression 720 Linear Conditionally Unbiased estimators 720
the Gauss–Markov theorem for Multiple regression 721
18.6 Generalized Least Squares 722
the GLS assumptions 723
GLS When Ω Is Known 725
GLS When Ω Contains Unknown Parameters 726
the Zero Conditional Mean assumption and GLS 726
18.7 Instrumental Variables and Generalized Method of Moments estimation 728
the IV estimator in Matrix Form 729
asymptotic Distribution of the tSLS estimator 730
Properties of tSLS When the errors are Homoskedastic 731 Generalized Method of Moments estimation in Linear Models 734 aPPeNDIx 18.1 Summary of Matrix algebra 746
aPPeNDIx 18.2 Multivariate Distributions 749
aPPeNDIx 18.3 Derivation of the asymptotic Distribution of βn 751
xxii Contents
Appendix 757 References 765 Glossary 771 Index 779
aPPeNDIx 18.4 Derivations of exact Distributions of OLS test Statistics with Normal errors 752
aPPeNDIx 18.5 Proof of the Gauss–Markov theorem for Multiple regression 753
aPPeNDIx 18.6 Proof of Selected results for IV and GMM estimation 754
Key Concepts
PART ONE
1.1 2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.1 3.2 3.3 3.4 3.5 3.6 3.7
PART TWO
4.1 4.2 4.3 4.4 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3
6.4 6.5 7.1 7.2
Introduction and Review
Cross-Sectional, time Series, and Panel Data 12
expected Value and the Mean 20
Variance and Standard Deviation 21
Means, Variances, and Covariances of Sums of random Variables 35 Computing Probabilities Involving Normal random Variables 37
Simple random Sampling and i.i.d. random Variables 44
Convergence in Probability, Consistency, and the Law of Large Numbers 48 the Central Limit theorem 52
estimators and estimates 67
Bias, Consistency, and efficiency 68
efficiency of Y : Y Is BLUe 69
the Standard error of Y 75
the terminology of Hypothesis testing 78
testing the Hypothesis E(Y) = μY,0 against the alternative E(Y) ≠ μY,0 79 Confidence Intervals for the Population Mean 81
Fundamentals of Regression Analysis
terminology for the Linear regression Model with a Single regressor 113 the OLS estimator, Predicted Values, and residuals 117
the Least Squares assumptions 129
Large-Sample Distributions of bn0 and bn1 131
General Form of the t-Statistic 147
testing the Hypothesis b1 = b1,0 against the alternative b1 ≠ b1,0 149
Confidence Interval for β1 154
Heteroskedasticity and Homoskedasticity 159
the Gauss–Markov theorem for bn1 165
Omitted Variable Bias in regression with a Single regressor 185
the Multiple regression Model 192
the OLS estimators, Predicted Values, and residuals in the Multiple regression
Model 194
the Least Squares assumptions in the Multiple regression Model 201
Large-Sample Distribution of b , b , c, b 202 n0 n1 nk
testing the Hypothesis bj = bj,0 against the alternative bj ≠ bj,0 219 Confidence Intervals for a Single Coefficient in Multiple regression 220
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xxiv Key Concepts
7.3 7.4 8.1
8.2 8.3
8.4 8.5 9.1 9.2
9.3 9.4 9.5 9.6 9.7
PART THREE
10.1 10.2 10.3 11.1 11.2 11.3 12.1
12.2 12.3 12.4 12.5 12.6
PART FOuR
14.1 14.2 14.3 14.4
Omitted Variable Bias in Multiple regression 233
R2 and R 2: What they tell You—and What they Don’t 238
the expected Change on Y of a Change in X1 in the Nonlinear regression Model (8.3) 263
Logarithms in regression: three Cases 276
a Method for Interpreting Coefficients in regressions with Binary Variables 281
Interactions Between Binary and Continuous Variables 284 Interactions in Multiple regression 289
Internal and external Validity 316
Omitted Variable Bias: Should I Include More Variables in
My regression? 321
Functional Form Misspecification 322
errors-in-Variables Bias 324
Sample Selection Bias 326
Simultaneous Causality Bias 329
threats to the Internal Validity of a Multiple regression Study 330
Further Topics in Regression Analysis
Notation for Panel Data 351
the Fixed effects regression Model 359
the Fixed effects regression assumptions 366
the Linear Probability Model 389
the Probit Model, Predicted Probabilities, and estimated effects 394 Logit regression 396
the General Instrumental Variables regression Model and terminology 436
two Stage Least Squares 438
the two Conditions for Valid Instruments 439
the IV regression assumptions 440
a rule of thumb for Checking for Weak Instruments 444
the Overidentifying restrictions test (the J-Statistic) 448
Regression Analysis of Economic Time Series Data
Lags, First Differences, Logarithms, and Growth rates 527 autocorrelation (Serial Correlation) and autocovariance 528 autoregressions 535
the autoregressive Distributed Lag Model 540
14.5 14.6 14.7 14.8 14.9 14.10 15.1 15.2 15.3 15.4 16.1 16.2 16.3 16.4 16.5
PART FIvE
17.1 18.1
18.2 18.3 18.4
Stationarity 541
time Series regression with Multiple Predictors 542
Granger Causality tests (tests of Predictive Content) 543
the augmented Dickey–Fuller test for a Unit autoregressive root 559 the QLr test for Coefficient Stability 566
Pseudo Out-of-Sample Forecasts 568
the Distributed Lag Model and exogeneity 598
the Distributed Lag Model assumptions 599
HaC Standard errors 607
estimation of Dynamic Multipliers Under Strict exogeneity 616
Vector autoregressions 639
Iterated Multiperiod Forecasts 646
Direct Multiperiod Forecasts 648
Orders of Integration, Differencing, and Stationarity 650 Cointegration 657
Regression Analysis of Economic Time Series Data
the extended Least Squares assumptions for regression with a Single regressor 678
the extended Least Squares assumptions in the Multiple regression Model 707
the Multivariate Central Limit theorem 711 Gauss–Markov theorem for Multiple regression 722 the GLS assumptions 724
Key Concepts xxv
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General Interest Boxes
the Distribution of earnings in the United States in 2012 33 a Bad Day on Wall Street 39
Financial Diversification and Portfolios 46
Landon Wins! 70
the Gender Gap of earnings of College Graduates in the United States 86 a Novel Way to Boost retirement Savings 90
the “Beta” of a Stock 120
the economic Value of a Year of education: Homoskedasticity or Heteroskedasticity? 162
the Mozart effect: Omitted Variable Bias? 186
the return to education and the Gender Gap 287
the Demand for economics Journals 290
Do Stock Mutual Funds Outperform the Market? 327
James Heckman and Daniel McFadden, Nobel Laureates 410
Who Invented Instrumental Variables regression? 428
a Scary regression 446
the externalities of Smoking 450
the Hawthorne effect 482
What Is the effect on employment of the Minimum Wage? 497
Can You Beat the Market? Part I 536
the river of Blood 546
Can You Beat the Market? Part II 570
Orange trees on the March 623
NeWS FLaSH: Commodity traders Send Shivers through Disney World 625 Nobel Laureates in time Series econometrics 669
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Preface
econometrics can be a fun course for both teacher and student. The real world of economics, business, and government is a complicated and messy place, full of competing ideas and questions that demand answers. Is it more effective to tackle drunk driving by passing tough laws or by increasing the tax on alcohol? Can you make money in the stock market by buying when prices are historically low, relative to earnings, or should you just sit tight, as the random walk theory of stock prices suggests? Can we improve elementary education by reducing class sizes, or should we simply have our children listen to Mozart for 10 minutes a day? Econometrics helps us sort out sound ideas from crazy ones and find quantitative answers to important quantitative questions. Econometrics opens a window on our complicated world that lets us see the relationships on which people, busi- nesses, and governments base their decisions.
Introduction to Econometrics is designed for a first course in undergradu- ate econometrics. It is our experience that to make econometrics relevant in an introductory course, interesting applications must motivate the theory and the theory must match the applications. This simple principle represents a sig- nificant departure from the older generation of econometrics books, in which theoretical models and assumptions do not match the applications. It is no won- der that some students question the relevance of econometrics after they spend much of their time learning assumptions that they subsequently realize are unre- alistic so that they must then learn “solutions” to “problems” that arise when the applications do not match the assumptions. We believe that it is far better to motivate the need for tools with a concrete application and then to provide a few simple assumptions that match the application. Because the theory is imme- diately relevant to the applications, this approach can make econometrics come alive.
New to the third edition
• Updated treatment of standard errors for panel data regression
• Discussion of when and why missing data can present a problem for regression analysis
• The use of regression discontinuity design as a method for analyzing quasi- experiments
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• Updated discussion of weak instruments
• Discussion of the use and interpretation of control variables integrated into the core development of regression analysis
• Introduction of the “potential outcomes” framework for experimental data
• Additional general interest boxes
• Additional exercises, both pencil-and-paper and empirical
This third edition builds on the philosophy of the first and second editions that applications should drive the theory, not the other way around.
One substantial change in this edition concerns inference in regression with panel data (Chapter 10). In panel data, the data within an entity typically are correlated over time. For inference to be valid, standard errors must be com- puted using a method that is robust to this correlation. The chapter on panel data now uses one such method, clustered standard errors, from the outset. Clustered standard errors are the natural extension to panel data of the heteroskedasticity- robust standard errors introduced in the initial treatment of regression analysis in Part II. Recent research has shown that clustered standard errors have a number of desirable properties, which are now discussed in Chapter 10 and in a revised appendix to Chapter 10.
Another substantial set of changes concerns the treatment of experiments and quasi-experiments in Chapter 13. The discussion of differences-in-differences regression has been streamlined and draws directly on the multiple regression principles introduced in Part II. Chapter 13 now discusses regression discontinuity design, which is an intuitive and important framework for the analysis of quasi- experimental data. In addition, Chapter 13 now introduces the potential outcomes framework and relates this increasingly commonplace terminology to concepts that were introduced in Parts I and II.
This edition has a number of other significant changes. One is that it incor- porates a precise but accessible treatment of control variables into the initial discussion of multiple regression. Chapter 7 now discusses conditions for con- trol variables being successful in the sense that the coefficient on the variable of interest is unbiased even though the coefficients on the control variables generally are not. Other changes include a new discussion of missing data in Chapter 9, a new optional calculus-based appendix to Chapter 8 on slopes and elasticities of nonlinear regression functions, and an updated discussion in Chapter 12 of what to do if you have weak instruments. This edition also includes new general interest boxes, updated empirical examples, and additional exercises.
the Updated third edition
• The time series data used in Chapters 14–16 have been extended through the beginning of 2013 and now include the Great Recession.
• The empirical analysis in Chapter 14 now focuses on forecasting the growth rate of real GDP using the term spread, replacing the Phillips curve forecasts from earlier editions.
• Several new empirical exercises have been added to each chapter. Rather than include all of the empirical exercises in the text, we have moved many of them to the Companion Website, www.pearsonhighered.com/stock_watson. This has two main advantages: first, we can offer more and more in-depth exercises, and second, we can add and update exercises between editions. We encourage you to browse the empirical exercises available on the Companion Website.
Features of this Book
Introduction to Econometrics differs from other textbooks in three main ways. First, we integrate real-world questions and data into the development of the theory, and we take seriously the substantive findings of the resulting empirical analysis. Second, our choice of topics reflects modern theory and practice. Third, we provide theory and assumptions that match the applications. Our aim is to teach students to become sophisticated consumers of econometrics and to do so at a level of mathematics appropriate for an introductory course.
real-World Questions and Data
We organize each methodological topic around an important real-world question that demands a specific numerical answer. For example, we teach single-variable regression, multiple regression, and functional form analysis in the context of estimating the effect of school inputs on school outputs. (Do smaller elementary school class sizes produce higher test scores?) We teach panel data methods in the context of analyzing the effect of drunk driving laws on traffic fatalities. We use possible racial discrimination in the market for home loans as the empirical appli- cation for teaching regression with a binary dependent variable (logit and probit). We teach instrumental variable estimation in the context of estimating the demand elasticity for cigarettes. Although these examples involve economic reasoning, all
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can be understood with only a single introductory course in economics, and many can be understood without any previous economics coursework. Thus the instruc- tor can focus on teaching econometrics, not microeconomics or macroeconomics.
We treat all our empirical applications seriously and in a way that shows students how they can learn from data but at the same time be self-critical and aware of the limitations of empirical analyses. Through each application, we teach students to explore alternative specifications and thereby to assess whether their substantive findings are robust. The questions asked in the empirical applica- tions are important, and we provide serious and, we think, credible answers. We encourage students and instructors to disagree, however, and invite them to rean- alyze the data, which are provided on the textbook’s Companion Website (www .pearsonhighered.com/stock_watson).
Contemporary Choice of topics
Econometrics has come a long way since the 1980s. The topics we cover reflect the best of contemporary applied econometrics. One can only do so much in an introductory course, so we focus on procedures and tests that are commonly used in practice. For example:
• Instrumental variables regression. We present instrumental variables regres- sion as a general method for handling correlation between the error term and a regressor, which can arise for many reasons, including omitted variables and simultaneous causality. The two assumptions for a valid instrument— exogeneity and relevance—are given equal billing. We follow that presenta- tion with an extended discussion of where instruments come from and with tests of overidentifying restrictions and diagnostics for weak instruments, and we explain what to do if these diagnostics suggest problems.
• Program evaluation. An increasing number of econometric studies analyze either randomized controlled experiments or quasi-experiments, also known as natural experiments. We address these topics, often collectively referred to as program evaluation, in Chapter 13. We present this research strategy as an alternative approach to the problems of omitted variables, simultaneous causality, and selection, and we assess both the strengths and the weaknesses of studies using experimental or quasi-experimental data.
• Forecasting. The chapter on forecasting (Chapter 14) considers univariate (autoregressive) and multivariate forecasts using time series regression, not large simultaneous equation structural models. We focus on simple and reli- able tools, such as autoregressions and model selection via an information
criterion, that work well in practice. This chapter also features a practically oriented treatment of stochastic trends (unit roots), unit root tests, tests for structural breaks (at known and unknown dates), and pseudo out-of-sample forecasting, all in the context of developing stable and reliable time series forecasting models.
• Time series regression. We make a clear distinction between two very dif- ferent applications of time series regression: forecasting and estimation of dynamic causal effects. The chapter on causal inference using time series data (Chapter 15) pays careful attention to when different estimation meth- ods, including generalized least squares, will or will not lead to valid causal inferences and when it is advisable to estimate dynamic regressions using OLS with heteroskedasticity- and autocorrelation-consistent standard errors.
theory that Matches applications
Although econometric tools are best motivated by empirical applications, stu- dents need to learn enough econometric theory to understand the strengths and limitations of those tools. We provide a modern treatment in which the fit between theory and applications is as tight as possible, while keeping the mathematics at a level that requires only algebra.
Modern empirical applications share some common characteristics: The data sets typically are large (hundreds of observations, often more); regressors are not fixed over repeated samples but rather are collected by random sampling (or some other mechanism that makes them random); the data are not normally dis- tributed; and there is no a priori reason to think that the errors are homoskedastic (although often there are reasons to think that they are heteroskedastic).
These observations lead to important differences between the theoretical development in this textbook and other textbooks:
• Large-sample approach. Because data sets are large, from the outset we use large-sample normal approximations to sampling distributions for hypothesis testing and confidence intervals. In our experience, it takes less time to teach the rudiments of large-sample approximations than to teach the Student t and exact F distributions, degrees-of-freedom corrections, and so forth. This large-sample approach also saves students the frustration of discover- ing that, because of nonnormal errors, the exact distribution theory they just mastered is irrelevant. Once taught in the context of the sample mean, the large-sample approach to hypothesis testing and confidence intervals carries directly through multiple regression analysis, logit and probit, instrumental variables estimation, and time series methods.
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• Random sampling. Because regressors are rarely fixed in econometric appli- cations, from the outset we treat data on all variables (dependent and inde- pendent) as the result of random sampling. This assumption matches our initial applications to cross-sectional data, it extends readily to panel and time series data, and because of our large-sample approach, it poses no additional conceptual or mathematical difficulties.
• Heteroskedasticity. Applied econometricians routinely use heteroskedasticity- robust standard errors to eliminate worries about whether heteroskedasticity is present or not. In this book, we move beyond treating heteroskedasticity as an exception or a “problem” to be “solved”; instead, we allow for heteroskedasticity from the outset and simply use heteroskedasticity-robust standard errors. We present homoskedasticity as a special case that provides a theoretical motivation for OLS.
Skilled Producers, Sophisticated Consumers
We hope that students using this book will become sophisticated consumers of empir- ical analysis. To do so, they must learn not only how to use the tools of regression analysis but also how to assess the validity of empirical analyses presented to them.
Our approach to teaching how to assess an empirical study is threefold. First, immediately after introducing the main tools of regression analysis, we devote Chapter 9 to the threats to internal and external validity of an empirical study. This chapter discusses data problems and issues of generalizing findings to other settings. It also examines the main threats to regression analysis, including omit- ted variables, functional form misspecification, errors-in-variables, selection, and simultaneity—and ways to recognize these threats in practice.
Second, we apply these methods for assessing empirical studies to the empiri- cal analysis of the ongoing examples in the book. We do so by considering alterna- tive specifications and by systematically addressing the various threats to validity of the analyses presented in the book.
Third, to become sophisticated consumers, students need firsthand experi- ence as producers. Active learning beats passive learning, and econometrics is an ideal course for active learning. For this reason, the textbook website features data sets, software, and suggestions for empirical exercises of different scopes.
approach to Mathematics and Level of rigor
Our aim is for students to develop a sophisticated understanding of the tools of modern regression analysis, whether the course is taught at a “high” or a “low” level of mathematics. Parts I through IV of the text (which cover the substantive
material) are accessible to students with only precalculus mathematics. Parts I through IV have fewer equations and more applications than many introductory econometrics books and far fewer equations than books aimed at mathemati- cal sections of undergraduate courses. But more equations do not imply a more sophisticated treatment. In our experience, a more mathematical treatment does not lead to a deeper understanding for most students.
That said, different students learn differently, and for mathematically well- prepared students, learning can be enhanced by a more explicitly mathematical treatment. Part V therefore contains an introduction to econometric theory that is appropriate for students with a stronger mathematical background. When the mathematical chapters in Part V are used in conjunction with the material in Parts I through IV, this book is suitable for advanced undergraduate or master’s level econometrics courses.
Contents and Organization
There are five parts to Introduction to Econometrics. This textbook assumes that the student has had a course in probability and statistics, although we review that material in Part I. We cover the core material of regression analysis in Part II. Parts III, IV, and V present additional topics that build on the core treatment in Part II.
Part I
Chapter 1 introduces econometrics and stresses the importance of providing quantitative answers to quantitative questions. It discusses the concept of cau- sality in statistical studies and surveys the different types of data encountered in econometrics. Material from probability and statistics is reviewed in Chapters 2 and 3, respectively; whether these chapters are taught in a given course or are simply provided as a reference depends on the background of the students.
Part II
Chapter 4 introduces regression with a single regressor and ordinary least squares (OLS) estimation, and Chapter 5 discusses hypothesis tests and confidence inter- vals in the regression model with a single regressor. In Chapter 6, students learn how they can address omitted variable bias using multiple regression, thereby esti- mating the effect of one independent variable while holding other independent variables constant. Chapter 7 covers hypothesis tests, including F-tests, and confi- dence intervals in multiple regression. In Chapter 8, the linear regression model is
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extended to models with nonlinear population regression functions, with a focus on regression functions that are linear in the parameters (so that the parameters can be estimated by OLS). In Chapter 9, students step back and learn how to identify the strengths and limitations of regression studies, seeing in the process how to apply the concepts of internal and external validity.
Part III
Part III presents extensions of regression methods. In Chapter 10, students learn how to use panel data to control for unobserved variables that are constant over time. Chapter 11 covers regression with a binary dependent variable. Chapter 12 shows how instrumental variables regression can be used to address a variety of problems that produce correlation between the error term and the regressor, and examines how one might find and evaluate valid instruments. Chapter 13 intro- duces students to the analysis of data from experiments and quasi-, or natural, experiments, topics often referred to as “program evaluation.”
Part IV
Part IV takes up regression with time series data. Chapter 14 focuses on forecast- ing and introduces various modern tools for analyzing time series regressions, such as unit root tests and tests for stability. Chapter 15 discusses the use of time series data to estimate causal relations. Chapter 16 presents some more advanced tools for time series analysis, including models of conditional heteroskedasticity.
Part V
Part V is an introduction to econometric theory. This part is more than an appendix that fills in mathematical details omitted from the text. Rather, it is a self-contained treatment of the econometric theory of estimation and inference in the linear regression model. Chapter 17 develops the theory of regression analysis for a single regressor; the exposition does not use matrix algebra, although it does demand a higher level of mathematical sophistication than the rest of the text. Chapter 18 presents and studies the multiple regression model, instrumental variables regression, and generalized method of moments estimation of the linear model, all in matrix form.
Prerequisites Within the Book
Because different instructors like to emphasize different material, we wrote this book with diverse teaching preferences in mind. To the maximum extent possible,
the chapters in Parts III, IV, and V are “stand-alone” in the sense that they do not require first teaching all the preceding chapters. The specific prerequisites for each chapter are described in Table I. Although we have found that the sequence of topics adopted in the textbook works well in our own courses, the chapters are written in a way that allows instructors to present topics in a different order if they so desire.
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TaBLE i
Chapter
10
11
12.1, 12.2
12.3–12.6
13
14
15
16
17
Sample Courses
This book accommodates several different course structures.
Guide to Prerequisites for Special-Topic Chapters in Parts III, Iv, and v
prerequisite parts or chapters
part ii
part iii
part iV
4–7, 9 8
10.1, 12.1, 10.2 12.2
14.1–14.4 14.5–14.8 15
Xa X
Xa X
Xa X
Xa X
XX
Xa X
XX
Xa b
Xa b
X
Xa b
XXX
XX
XX
X
part i
1–3
Xa
Xa
Xa
Xa
Xa
Xa
Xa
Xa
X
part V 17
18X X
This table shows the minimum prerequisites needed to cover the material in a given chapter. For example, estimation of dynamic causal effects with time series data (Chapter 15) first requires Part I (as needed, depending on student preparation, and except as noted in footnote a), Part II (except for Chapter 8; see footnote b), and Sections 14.1 through 14.4.
aChapters 10 through 16 use exclusively large-sample approximations to sampling distributions, so the optional Sections 3.6 (the Student t distribution for testing means) and 5.6 (the Student t distribution for testing regression coefficients) can be skipped. bChapters 14 through 16 (the time series chapters) can be taught without first teaching Chapter 8 (nonlinear regression functions) if the instructor pauses to explain the use of logarithmic transformations to approximate percentage changes.
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Standard Introductory econometrics
This course introduces econometrics (Chapter 1) and reviews probability and sta- tistics as needed (Chapters 2 and 3). It then moves on to regression with a single regressor, multiple regression, the basics of functional form analysis, and the evaluation of regression studies (all Part II). The course proceeds to cover regres- sion with panel data (Chapter 10), regression with a limited dependent variable (Chapter 11), and instrumental variables regression (Chapter 12), as time permits. The course concludes with experiments and quasi-experiments in Chapter 13, topics that provide an opportunity to return to the questions of estimating causal effects raised at the beginning of the semester and to recapitulate core regression methods. Prerequisites: Algebra II and introductory statistics.
Introductory econometrics with time Series and
Forecasting applications
Like a standard introductory course, this course covers all of Part I (as needed) and Part II. Optionally, the course next provides a brief introduction to panel data (Sections 10.1 and 10.2) and takes up instrumental variables regression (Chapter 12, or just Sections 12.1 and 12.2). The course then proceeds to Part IV, covering forecasting (Chapter 14) and estimation of dynamic causal effects (Chapter 15). If time permits, the course can include some advanced topics in time series analysis such as volatility clustering and conditional heteroskedasticity (Section 16.5). Prerequisites: Algebra II and introductory statistics.
applied time Series analysis and Forecasting
This book also can be used for a short course on applied time series and forecast- ing, for which a course on regression analysis is a prerequisite. Some time is spent reviewing the tools of basic regression analysis in Part II, depending on student preparation. The course then moves directly to Part IV and works through forecast- ing (Chapter 14), estimation of dynamic causal effects (Chapter 15), and advanced topics in time series analysis (Chapter 16), including vector autoregressions and conditional heteroskedasticity. An important component of this course is hands-on forecasting exercises, available to instructors on the book’s accompanying website. Prerequisites: Algebra II and basic introductory econometrics or the equivalent.
Introduction to econometric theory
This book is also suitable for an advanced undergraduate course in which the students have a strong mathematical preparation or for a master’s level course in
econometrics. The course briefly reviews the theory of statistics and probability as necessary (Part I). The course introduces regression analysis using the nonmath- ematical, applications-based treatment of Part II. This introduction is followed by the theoretical development in Chapters 17 and 18 (through Section 18.5). The course then takes up regression with a limited dependent variable (Chapter 11) and maximum likelihood estimation (Appendix 11.2). Next, the course optionally turns to instrumental variables regression and generalized method of moments (Chapter 12 and Section 18.7), time series methods (Chapter 14), and the estima- tion of causal effects using time series data and generalized least squares (Chapter 15 and Section 18.6). Prerequisites: Calculus and introductory statistics. Chapter 18 assumes previous exposure to matrix algebra.
Pedagogical Features
This textbook has a variety of pedagogical features aimed at helping students understand, retain, and apply the essential ideas. Chapter introductions provide real-world grounding and motivation, as well as brief road maps highlighting the sequence of the discussion. Key terms are boldfaced and defined in context throughout each chapter, and Key Concept boxes at regular intervals recap the central ideas. General interest boxes provide interesting excursions into related topics and highlight real-world studies that use the methods or concepts being discussed in the text. A Summary concluding each chapter serves as a helpful framework for reviewing the main points of coverage. The questions in the Review the Concepts section check students’ understanding of the core content, Exercises give more intensive practice working with the concepts and techniques introduced in the chapter, and Empirical Exercises allow students to apply what they have learned to answer real-world empirical questions. At the end of the textbook, the Appendix provides statistical tables, the References section lists sources for further reading, and a Glossary conveniently defines many key terms in the book.
Supplements to accompany the textbook
The online supplements accompanying the third edition update of Introduction to Econometrics include the Instructor’s Resource Manual, Test Bank, and Power- Point® slides with text figures, tables, and Key Concepts. The Instructor’s Resource Manual includes solutions to all the end-of-chapter exercises, while the Test Bank, offered in Testgen, provides a rich supply of easily edited test problems and
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questions of various types to meet specific course needs. These resources are avail- able for download from the Instructor’s Resource Center at www.pearsonhighered .com/stock_watson.
Companion Website
The Companion Website, found at www.pearsonhighered.com/stock_watson, provides a wide range of additional resources for students and faculty. These resources include more and more in depth empirical exercises, data sets for the empirical exercises, replication files for empirical results reported in the text, practice quizzes, answers to end-of-chapter Review the Concepts questions and Exercises, and EViews tutorials.
MyEconLab
The third edition update is accompanied by a robust MyEconLab course. The MyEconLab course includes all the Review the Concepts questions as well as some Exercises and Empirical Exercises. In addition, the enhanced eText avail- able in MyEconLab for the third edition update includes URL links from the Exercises and Empirical Exercises to questions in the MyEconLab course and to the data that accompanies them. To register for MyEconLab and to learn more, log on to www.myeconlab.com.
acknowledgments
A great many people contributed to the first edition of this book. Our biggest debts of gratitude are to our colleagues at Harvard and Princeton who used early drafts of this book in their classrooms. At Harvard’s Kennedy School of Govern- ment, Suzanne Cooper provided invaluable suggestions and detailed comments on multiple drafts. As a coteacher with one of the authors (Stock), she also helped vet much of the material in this book while it was being developed for a required course for master’s students at the Kennedy School. We are also indebted to two other Kennedy School colleagues, Alberto Abadie and Sue Dynarski, for their patient explanations of quasi-experiments and the field of program evaluation and for their detailed comments on early drafts of the text. At Princeton, Eli Tamer taught from an early draft and also provided helpful comments on the penultimate draft of the book.
We also owe much to many of our friends and colleagues in econometrics who spent time talking with us about the substance of this book and who collec- tively made so many helpful suggestions. Bruce Hansen (University of Wisconsin– Madison) and Bo Honore (Princeton) provided helpful feedback on very early outlines and preliminary versions of the core material in Part II. Joshua Angrist (MIT) and Guido Imbens (University of California, Berkeley) provided thought- ful suggestions about our treatment of materials on program evaluation. Our presentation of the material on time series has benefited from discussions with Yacine Ait-Sahalia (Princeton), Graham Elliott (University of California, San Diego), Andrew Harvey (Cambridge University), and Christopher Sims (Princeton). Finally, many people made helpful suggestions on parts of the manuscript close to their area of expertise: Don Andrews (Yale), John Bound (University of Michigan), Gregory Chow (Princeton), Thomas Downes (Tufts), David Drukker (StataCorp.), Jean Baldwin Grossman (Princeton), Eric Hanushek (Hoover Institution), James Heckman (University of Chicago), Han Hong (Princeton), Caroline Hoxby (Harvard), Alan Krueger (Princeton), Steven Levitt (University of Chicago), Richard Light (Harvard), David Neumark (Michigan State University), Joseph Newhouse (Harvard), Pierre Perron (Boston University), Kenneth Warner (University of Michigan), and Richard Zeckhauser (Harvard).
Many people were very generous in providing us with data. The Califor- nia test score data were constructed with the assistance of Les Axelrod of the Standards and Assessments Division, California Department of Education. We are grateful to Charlie DePascale, Student Assessment Services, Massachusetts Department of Education, for his help with aspects of the Massachusetts test score data set. Christopher Ruhm (University of North Carolina, Greensboro) graciously provided us with his data set on drunk driving laws and traffic fatali- ties. The research department at the Federal Reserve Bank of Boston deserves thanks for putting together its data on racial discrimination in mortgage lending; we particularly thank Geoffrey Tootell for providing us with the updated version of the data set we use in Chapter 9 and Lynn Browne for explaining its policy context. We thank Jonathan Gruber (MIT) for sharing his data on cigarette sales, which we analyze in Chapter 12, and Alan Krueger (Princeton) for his help with the Tennessee STAR data that we analyze in Chapter 13.
We thank several people for carefully checking the page proof for errors. Kerry Griffin and Yair Listokin read the entire manuscript, and Andrew Fraker, Ori Heffetz, Amber Henry, Hong Li, Alessandro Tarozzi, and Matt Watson worked through several chapters.
In the first edition, we benefited from the help of an exceptional development editor, Jane Tufts, whose creativity, hard work, and attention to detail improved
Preface xli
xlii Preface
the book in many ways, large and small. Pearson provided us with first-rate sup- port, starting with our excellent editor, Sylvia Mallory, and extending through the entire publishing team. Jane and Sylvia patiently taught us a lot about writing, organization, and presentation, and their efforts are evident on every page of this book. We extend our thanks to the superb Pearson team, who worked with us on the second edition: Adrienne D’Ambrosio (senior acquisitions editor), Bridget Page (associate media producer), Charles Spaulding (senior designer), Nancy Fenton (managing editor) and her selection of Nancy Freihofer and Thompson Steele Inc. who handled the entire production process, Heather McNally (sup- plements coordinator), and Denise Clinton (editor-in-chief). Finally, we had the benefit of Kay Ueno’s skilled editing in the second edition. We are also grate- ful to the excellent third edition Pearson team of Adrienne D’Ambrosio, Nancy Fenton, and Jill Kolongowski, as well as Mary Sanger, the project manager with Nesbitt Graphics. We also wish to thank the Pearson team who worked on the third edition update: Christina Masturzo, Carolyn Philips, Liz Napolitano, and Heidi Allgair, project manager with Cenveo® Publisher Services.
We also received a great deal of help and suggestions from faculty, students, and researchers as we prepared the third edition and its update. The changes made in the third edition incorporate or reflect suggestions, corrections, com- ments, data, and help provided by a number of researchers and instructors: Don- ald Andrews (Yale University), Jushan Bai (Columbia), James Cobbe (Florida State University), Susan Dynarski (University of Michigan), Nicole Eichelberger (Texas Tech University), Boyd Fjeldsted (University of Utah), Martina Grunow, Daniel Hamermesh (University of Texas–Austin), Keisuke Hirano (University of Arizona), Bo Honore (Princeton University), Guido Imbens (Harvard Uni- versity), Manfred Keil (Claremont McKenna College), David Laibson (Harvard University), David Lee (Princeton University), Brigitte Madrian (Harvard Uni- versity), Jorge Marquez (University of Maryland), Karen Bennett Mathis (Flor- ida Department of Citrus), Alan Mehlenbacher (University of Victoria), Ulrich Müller (Princeton University), Serena Ng (Columbia University), Harry Patrinos (World Bank), Zhuan Pei (Brandeis University), Peter Summers (Texas Tech University), Andrey Vasnov (University of Sydney), and Douglas Young (Mon- tana State University). We also benefited from student input from F. Hoces dela Guardia and Carrie Wilson.
Thoughtful reviews for the third edition were prepared for Addison-Wesley by Steve DeLoach (Elon University), Jeffrey DeSimone (University of Texas at Arlington), Gary V. Engelhardt (Syracuse University), Luca Flabbi (Georgetown University), Steffen Habermalz (Northwestern University), Carolyn J. Heinrich (University of Wisconsin–Madison), Emma M. Iglesias-Vazquez (Michigan State
University), Carlos Lamarche (University of Oklahoma), Vicki A. McCracken (Washington State University), Claudiney M. Pereira (Tulane University), and John T. Warner (Clemson University). We also received very helpful input on draft revisions of Chapters 7 and 10 from John Berdell (DePaul University), Janet Kohlhase (University of Houston), Aprajit Mahajan (Stanford University), Xia Meng (Brandeis University), and Chan Shen (Georgetown University).
Above all, we are indebted to our families for their endurance throughout this project. Writing this book took a long time, and for them, the project must have seemed endless. They, more than anyone else, bore the burden of this commit- ment, and for their help and support we are deeply grateful.
Preface xliii
Introduction
to Econometrics
C h1a p t e r
Economic Questions and Data
Ask a half dozen econometricians what econometrics is, and you could get a half dozen different answers. One might tell you that econometrics is the science of testing economic theories. A second might tell you that econometrics is the set of tools used for forecasting future values of economic variables, such as a firm’s sales, the overall growth of the economy, or stock prices. Another might say that econo- metrics is the process of fitting mathematical economic models to real-world data. A fourth might tell you that it is the science and art of using historical data to make numerical, or quantitative, policy recommendations in government and business.
In fact, all these answers are right. At a broad level, econometrics is the science and art of using economic theory and statistical techniques to analyze economic data. Econometric methods are used in many branches of economics, including finance, labor economics, macroeconomics, microeconomics, marketing, and eco- nomic policy. Econometric methods are also commonly used in other social sci- ences, including political science and sociology.
This book introduces you to the core set of methods used by econometricians. We will use these methods to answer a variety of specific, quantitative questions from the worlds of business and government policy. This chapter poses four of those questions and discusses, in general terms, the econometric approach to answering them. The chapter concludes with a survey of the main types of data available to econometricians for answering these and other quantitative economic questions.
1.1
Economic Questions We Examine
Many decisions in economics, business, and government hinge on understanding relationships among variables in the world around us. These decisions require quantitative answers to quantitative questions.
This book examines several quantitative questions taken from current issues in economics. Four of these questions concern education policy, racial bias in mortgage lending, cigarette consumption, and macroeconomic forecasting.
1
2 ChaptEr 1 Economic Questions and Data
Question #1: Does Reducing Class Size Improve
Elementary School Education?
Proposals for reform of the U.S. public education system generate heated debate. Many of the proposals concern the youngest students, those in elementary schools. Elementary school education has various objectives, such as developing social skills, but for many parents and educators, the most important objective is basic academic learning: reading, writing, and basic mathematics. One prominent pro- posal for improving basic learning is to reduce class sizes at elementary schools. With fewer students in the classroom, the argument goes, each student gets more of the teacher’s attention, there are fewer class disruptions, learning is enhanced, and grades improve.
But what, precisely, is the effect on elementary school education of reducing class size? Reducing class size costs money: It requires hiring more teachers and, if the school is already at capacity, building more classrooms. A decision maker contemplating hiring more teachers must weigh these costs against the benefits. To weigh costs and benefits, however, the decision maker must have a precise quantitative understanding of the likely benefits. Is the beneficial effect on basic learning of smaller classes large or small? Is it possible that smaller class size actu- ally has no effect on basic learning?
Although common sense and everyday experience may suggest that more learning occurs when there are fewer students, common sense cannot provide a quantitative answer to the question of what exactly is the effect on basic learning of reducing class size. To provide such an answer, we must examine empirical evidence—that is, evidence based on data—relating class size to basic learning in elementary schools.
In this book, we examine the relationship between class size and basic learn- ing, using data gathered from 420 California school districts in 1999. In the Cali- fornia data, students in districts with small class sizes tend to perform better on standardized tests than students in districts with larger classes. While this fact is consistent with the idea that smaller classes produce better test scores, it might simply reflect many other advantages that students in districts with small classes have over their counterparts in districts with large classes. For example, districts with small class sizes tend to have wealthier residents than districts with large classes, so students in small-class districts could have more opportunities for learning outside the classroom. It could be these extra learning opportunities that lead to higher test scores, not smaller class sizes. In Part II, we use multiple regres- sion analysis to isolate the effect of changes in class size from changes in other factors, such as the economic background of the students.
1.1 Economic Questions We Examine 3
Question #2: Is There Racial Discrimination
in the Market for Home Loans?
Most people buy their homes with the help of a mortgage, a large loan secured by the value of the home. By law, U.S. lending institutions cannot take race into account when deciding to grant or deny a request for a mortgage: Applicants who are identical in all ways except their race should be equally likely to have their mortgage applications approved. In theory, then, there should be no racial bias in mortgage lending.
In contrast to this theoretical conclusion, researchers at the Federal Reserve Bank of Boston found (using data from the early 1990s) that 28% of black appli- cants are denied mortgages, while only 9% of white applicants are denied. Do these data indicate that, in practice, there is racial bias in mortgage lending? If so, how large is it?
The fact that more black than white applicants are denied in the Boston Fed data does not by itself provide evidence of discrimination by mortgage lenders because the black and white applicants differ in many ways other than their race. Before concluding that there is bias in the mortgage market, these data must be examined more closely to see if there is a difference in the probability of being denied for otherwise identical applicants and, if so, whether this difference is large or small. To do so, in Chapter 11 we introduce econometric methods that make it possible to quantify the effect of race on the chance of obtaining a mort- gage, holding constant other applicant characteristics, notably their ability to repay the loan.
Question #3: How Much Do Cigarette Taxes
Reduce Smoking?
Cigarette smoking is a major public health concern worldwide. Many of the costs of smoking, such as the medical expenses of caring for those made sick by smoking and the less quantifiable costs to nonsmokers who prefer not to breathe secondhand cigarette smoke, are borne by other members of society. Because these costs are borne by people other than the smoker, there is a role for government intervention in reducing cigarette consumption. One of the most flexible tools for cutting consumption is to increase taxes on cigarettes.
Basic economics says that if cigarette prices go up, consumption will go down. But by how much? If the sales price goes up by 1%, by what percentage will the quantity of cigarettes sold decrease? The percentage change in the quantity demanded resulting from a 1% increase in price is the price elasticity of demand.
4 ChaptEr 1 Economic Questions and Data
If we want to reduce smoking by a certain amount, say 20%, by raising taxes, then we need to know the price elasticity of demand to calculate the price increase necessary to achieve this reduction in consumption. But what is the price elasticity of demand for cigarettes?
Although economic theory provides us with the concepts that help us answer this question, it does not tell us the numerical value of the price elasticity of demand. To learn the elasticity, we must examine empirical evidence about the behavior of smokers and potential smokers; in other words, we need to analyze data on cigarette consumption and prices.
The data we examine are cigarette sales, prices, taxes, and personal income for U.S. states in the 1980s and 1990s. In these data, states with low taxes, and thus low cigarette prices, have high smoking rates, and states with high prices have low smoking rates. However, the analysis of these data is complicated because causal- ity runs both ways: Low taxes lead to high demand, but if there are many smokers in the state, then local politicians might try to keep cigarette taxes low to satisfy their smoking constituents. In Chapter 12, we study methods for handling this “simultaneous causality” and use those methods to estimate the price elasticity of cigarette demand.
Question #4: By How Much Will U.S. GDP
Grow Next Year?
It seems that people always want a sneak preview of the future. What will sales be next year at a firm that is considering investing in new equipment? Will the stock market go up next month, and, if it does, by how much? Will city tax receipts next year cover planned expenditures on city services? Will your microeconomics exam next week focus on externalities or monopolies? Will Saturday be a nice day to go to the beach?
One aspect of the future in which macroeconomists are particularly interested is the growth of real economic activity, as measured by real gross domestic product (GDP), during the next year. A management consulting firm might advise a man- ufacturing client to expand its capacity based on an upbeat forecast of economic growth. Economists at the Federal Reserve Board in Washington, D.C., are man- dated to set policy to keep real GDP near its potential in order to maximize employment. If they forecast anemic GDP growth over the next year, they might expand liquidity in the economy by reducing interest rates or other measures, in an attempt to boost economic activity.
Professional economists who rely on precise numerical forecasts use econo- metric models to make those forecasts. A forecaster’s job is to predict the future
1.2 Causal Effects and Idealized Experiments 5
by using the past, and econometricians do this by using economic theory and statistical techniques to quantify relationships in historical data.
The data we use to forecast the growth rate of GDP are past values of GDP and the “term spread” in the United States. The term spread is the difference between long-term and short-term interest rates. It measures, among other things, whether investors expect short-term interest rates to rise or fall in the future. The term spread is usually positive, but it tends to fall sharply before the onset of a recession. One of the GDP growth rate forecasts we develop and evaluate in Chapter 14 is based on the term spread.
Quantitative Questions, Quantitative Answers
Each of these four questions requires a numerical answer. Economic theory pro- vides clues about that answer—for example, cigarette consumption ought to go down when the price goes up—but the actual value of the number must be learned empirically, that is, by analyzing data. Because we use data to answer quantitative questions, our answers always have some uncertainty: A different set of data would produce a different numerical answer. Therefore, the conceptual frame- work for the analysis needs to provide both a numerical answer to the question and a measure of how precise the answer is.
The conceptual framework used in this book is the multiple regression model, the mainstay of econometrics. This model, introduced in Part II, provides a math- ematical way to quantify how a change in one variable affects another variable, holding other things constant. For example, what effect does a change in class size have on test scores, holding constant or controlling for student characteristics (such as family income) that a school district administrator cannot control? What effect does your race have on your chances of having a mortgage application granted, holding constant other factors such as your ability to repay the loan? What effect does a 1% increase in the price of cigarettes have on cigarette consumption, hold- ing constant the income of smokers and potential smokers? The multiple regres- sion model and its extensions provide a framework for answering these questions using data and for quantifying the uncertainty associated with those answers.
1.2
Causal Effects and Idealized Experiments
Like many other questions encountered in econometrics, the first three questions in Section 1.1 concern causal relationships among variables. In common usage, an action is said to cause an outcome if the outcome is the direct result, or consequence,
6 ChaptEr 1 Economic Questions and Data
of that action. Touching a hot stove causes you to get burned; drinking water causes you to be less thirsty; putting air in your tires causes them to inflate; putting fertilizer on your tomato plants causes them to produce more tomatoes. Causality means that a specific action (applying fertilizer) leads to a specific, measurable consequence (more tomatoes).
Estimation of Causal Effects
How best might we measure the causal effect on tomato yield (measured in kilo- grams) of applying a certain amount of fertilizer, say 100 grams of fertilizer per square meter?
One way to measure this causal effect is to conduct an experiment. In that experiment, a horticultural researcher plants many plots of tomatoes. Each plot is tended identically, with one exception: Some plots get 100 grams of fertilizer per square meter, while the rest get none. Moreover, whether a plot is fertilized or not is determined randomly by a computer, ensuring that any other differences between the plots are unrelated to whether they receive fertilizer. At the end of the growing season, the horticulturalist weighs the harvest from each plot. The difference between the average yield per square meter of the treated and untreated plots is the effect on tomato production of the fertilizer treatment.
This is an example of a randomized controlled experiment. It is controlled in the sense that there are both a control group that receives no treatment (no fertil- izer) and a treatment group that receives the treatment (100 g/m2 of fertilizer). It is randomized in the sense that the treatment is assigned randomly. This random assignment eliminates the possibility of a systematic relationship between, for example, how sunny the plot is and whether it receives fertilizer so that the only systematic difference between the treatment and control groups is the treatment. If this experiment is properly implemented on a large enough scale, then it will yield an estimate of the causal effect on the outcome of interest (tomato produc- tion) of the treatment (applying 100 g/m2 of fertilizer).
In this book, the causal effect is defined to be the effect on an outcome of a given action or treatment, as measured in an ideal randomized controlled experi- ment. In such an experiment, the only systematic reason for differences in out- comes between the treatment and control groups is the treatment itself.
It is possible to imagine an ideal randomized controlled experiment to answer each of the first three questions in Section 1.1. For example, to study class size, one can imagine randomly assigning “treatments” of different class sizes to differ- ent groups of students. If the experiment is designed and executed so that the only systematic difference between the groups of students is their class size, then in
theory this experiment would estimate the effect on test scores of reducing class size, holding all else constant.
The concept of an ideal randomized controlled experiment is useful because it gives a definition of a causal effect. In practice, however, it is not possible to perform ideal experiments. In fact, experiments are relatively rare in economet- rics because often they are unethical, impossible to execute satisfactorily, or pro- hibitively expensive. The concept of the ideal randomized controlled experiment does, however, provide a theoretical benchmark for an econometric analysis of causal effects using actual data.
Forecasting and Causality
Although the first three questions in Section 1.1 concern causal effects, the fourth—forecasting the growth rate of GDP—does not. You do not need to know a causal relationship to make a good forecast. A good way to “forecast” whether it is raining is to observe whether pedestrians are using umbrellas, but the act of using an umbrella does not cause it to rain.
Even though forecasting need not involve causal relationships, economic theory suggests patterns and relationships that might be useful for forecasting. As we see in Chapter 14, multiple regression analysis allows us to quantify historical relationships suggested by economic theory, to check whether those relationships have been stable over time, to make quantitative forecasts about the future, and to assess the accuracy of those forecasts.
1.3
Data: Sources and Types
In econometrics, data come from one of two sources: experiments or nonexperi- mental observations of the world. This book examines both experimental and nonexperimental data sets.
Experimental Versus Observational Data
Experimental data come from experiments designed to evaluate a treatment or policy or to investigate a causal effect. For example, the state of Tennessee financed a large randomized controlled experiment examining class size in the 1980s. In that experiment, which we examine in Chapter 13, thousands of students were randomly assigned to classes of different sizes for several years and were given standardized tests annually.
1.3 Data: Sources and Types 7
8 ChaptEr 1 Economic Questions and Data
The Tennessee class size experiment cost millions of dollars and required the ongoing cooperation of many administrators, parents, and teachers over several years. Because real-world experiments with human subjects are difficult to admin- ister and to control, they have flaws relative to ideal randomized controlled exper- iments. Moreover, in some circumstances, experiments are not only expensive and difficult to administer but also unethical. (Would it be ethical to offer randomly selected teenagers inexpensive cigarettes to see how many they buy?) Because of these financial, practical, and ethical problems, experiments in economics are relatively rare. Instead, most economic data are obtained by observing real-world behavior.
Data obtained by observing actual behavior outside an experimental setting are called observational data. Observational data are collected using surveys, such as telephone surveys of consumers, and administrative records, such as historical records on mortgage applications maintained by lending institutions.
Observational data pose major challenges to econometric attempts to esti- mate causal effects, and the tools of econometrics are designed to tackle these challenges. In the real world, levels of “treatment” (the amount of fertilizer in the tomato example, the student–teacher ratio in the class size example) are not assigned at random, so it is difficult to sort out the effect of the “treatment” from other relevant factors. Much of econometrics, and much of this book, is devoted to methods for meeting the challenges encountered when real-world data are used to estimate causal effects.
Whether the data are experimental or observational, data sets come in three main types: cross-sectional data, time series data, and panel data. In this book, you will encounter all three types.
Cross-Sectional Data
Data on different entities—workers, consumers, firms, governmental units, and so forth—for a single time period are called cross-sectional data. For example, the data on test scores in California school districts are cross sectional. Those data are for 420 entities (school districts) for a single time period (1999). In general, the number of entities on which we have observations is denoted n; so, for example, in the California data set, n = 420.
The California test score data set contains measurements of several different variables for each district. Some of these data are tabulated in Table 1.1. Each row lists data for a different district. For example, the average test score for the first district (“district #1”) is 690.8; this is the average of the math and science test scores for all fifth graders in that district in 1999 on a standardized test (the Stanford
1.3 Data: Sources and Types 9
taBLe 1.1 Selected Observations on test Scores and Other Variables for California School Districts in 1999
Observation (District) Number
1
2
3
4
5
District average test Score (fifth grade)
690.8
661.2
643.6
647.7
640.8
Student–teacher ratio
17.89
21.52
18.70
17.36
18.67
expenditure per pupil ($)
$6385
5099
5502
7102
5236
percentage of Students Learning english
0.0%
4.6
30.0
0.0
13.9
….. ….. …..
418 645.0
419 672.2
420 655.8
21.89 4403 24.3
20.20 4776 3.0
19.04 5993 5.0
Note: The California test score data set is described in Appendix 4.1.
Achievement Test). The average student–teacher ratio in that district is 17.89; that is, the number of students in district #1 divided by the number of classroom teachers in district #1 is 17.89. Average expenditure per pupil in district #1 is $6385. The percentage of students in that district still learning English—that is, the percentage of students for whom English is a second language and who are not yet proficient in English—is 0%.
The remaining rows present data for other districts. The order of the rows is arbitrary, and the number of the district, which is called the observation number, is an arbitrarily assigned number that organizes the data. As you can see in the table, all the variables listed vary considerably.
With cross-sectional data, we can learn about relationships among variables by studying differences across people, firms, or other economic entities during a single time period.
Time Series Data
Time series data are data for a single entity (person, firm, country) collected at multiple time periods. Our data set on the growth rate of GDP and the term spread in the United States is an example of a time series data set. The data set
10
ChaptEr 1 Economic Questions and Data
taBLe 1.2 Selected Observations on the Growth rate of GDp and the term Spread in the United States: Quarterly Data, 1960:Q1–2013:Q1
Observation Number
Date (year:quarter)
GDp Growth rate (% at an annual rate)
8.8%
−1.5
1.0
−4.9
2.7
term Spread (% per year)
0.6%
1.3
1.5
1.6
1.4
1 1960:Q1
2 1960:Q2
3 1960:Q3
4 1960:Q4
5 1961:Q1
…. …. ….
211 2012:Q3 2.7 1.5
212 2012:Q4 0.1 1.6
213 2013:Q1 1.1 1.9
Note: The United States GDP and term spread data set is described in Appendix 14.1.
contains observations on two variables (the growth rate of GDP and the term spread) for a single entity (the United States) for 213 time periods. Each time period in this data set is a quarter of a year (the first quarter is January, Febru- ary, and March; the second quarter is April, May, and June; and so forth). The observations in this data set begin in the first quarter of 1960, which is denoted 1960:Q1, and end in the first quarter of 2013 (2013:Q1). The number of observa- tions (that is, time periods) in a time series data set is denoted T. Because there are 213 quarters from 1960:Q1 to 2013:Q1, this data set contains T = 213 observations.
Some observations in this data set are listed in Table 1.2. The data in each row correspond to a different time period (year and quarter). In the first quarter of 1960, for example, GDP grew 8.8% at an annual rate. In other words, if GDP had continued growing for four quarters at its rate during the first quarter of 1960, the level of GDP would have increased by 8.8%. In the first quarter of 1960, the long-term interest rate was 4.5%, the short-term interest rate was 3.9%, so their difference, the term spread, was 0.6%.
By tracking a single entity over time, time series data can be used to study the evolution of variables over time and to forecast future values of those variables.
State
average price Cigarette Sales per pack
Year (packs per capita) (including taxes)
total taxes (cigarette excise tax + sales tax)
1.3 Data: Sources and Types 11
taBLe 1.3
Observation Number
Selected Observations on Cigarette Sales, prices, and taxes, by State and Year for U.S. States, 1985–1995
1 Alabama
2 Arkansas
3 Arizona
1985 116.5 $1.022 $0.333
1985 128.5 1.015 0.370
1985 104.5 1.086 0.362
…… …… ……
47 West Virginia 1985
48 Wyoming 1985
49 Alabama 1986
112.8 1.089
129.4 0.935
117.2 1.080
0.382
0.240
0.334
…… …… ……
96 Wyoming 1986 127.8 1.007 0.240
97 Alabama 1987 115.8 1.135 0.335
…… …… ……
528 Wyoming 1995 112.2 1.585 0.360
Note: The cigarette consumption data set is described in Appendix 12.1.
Panel Data
Panel data, also called longitudinal data, are data for multiple entities in which each entity is observed at two or more time periods. Our data on cigarette con- sumption and prices are an example of a panel data set, and selected variables and observations in that data set are listed in Table 1.3. The number of entities in a panel data set is denoted n, and the number of time periods is denoted T. In the cigarette data set, we have observations on n = 48 continental U.S. states (entities) for T = 11 years (time periods) from 1985 to 1995. Thus there is a total ofn * T = 48 * 11 = 528observations.
12 ChaptEr 1 Economic Questions and Data
Cross-Sectional, time Series, and panel Data
1.1
KeY CONCept
• Cross-sectional data consist of multiple entities observed at a single time period.
• Time series data consist of a single entity observed at multiple time periods.
• Panel data (also known as longitudinal data) consist of multiple entities, where each entity is observed at two or more time periods.
Some data from the cigarette consumption data set are listed in Table 1.3. The first block of 48 observations lists the data for each state in 1985, organized alpha- betically from Alabama to Wyoming. The next block of 48 observations lists the data for 1986, and so forth, through 1995. For example, in 1985, cigarette sales in Arkansas were 128.5 packs per capita (the total number of packs of cigarettes sold in Arkansas in 1985 divided by the total population of Arkansas in 1985 equals 128.5). The average price of a pack of cigarettes in Arkansas in 1985, including tax, was $1.015, of which 37¢ went to federal, state, and local taxes.
Panel data can be used to learn about economic relationships from the expe- riences of the many different entities in the data set and from the evolution over time of the variables for each entity.
The definitions of cross-sectional data, time series data, and panel data are summarized in Key Concept 1.1.
Summary
1. Many decisions in business and economics require quantitative estimates of how a change in one variable affects another variable.
2. Conceptually, the way to estimate a causal effect is in an ideal randomized controlled experiment, but performing such experiments in economic appli- cations is usually unethical, impractical, or too expensive.
3. Econometrics provides tools for estimating causal effects using either observa- tional (nonexperimental) data or data from real-world, imperfect experiments.
4. Cross-sectional data are gathered by observing multiple entities at a single point in time; time series data are gathered by observing a single entity at multiple points in time; and panel data are gathered by observing multiple
entities, each of which is observed at multiple points in time.
Key Terms
randomized controlled experiment (6)
control group (6) treatment group (6) causal effect (6) experimental data (7)
observational data (8) cross-sectional data (8) observation number (9) time series data (9) panel data (11) longitudinal data (11)
Review the Concepts 13
MyEconLab Can Help You Get a Better Grade
MyEconLab If your exam were tomorrow, would you be ready? For each chapter, MyEconLab Practice Tests and Study Plan help you prepare for your exams. You can also find the Exercises and all Review the Concepts Questions available now in MyEconLab.
To see how it works, turn to the MyEconLab spread on the inside front cover of this book and then go to www.myeconlab.com.
For additional Empirical Exercises and Data Sets, log on to the Companion Website at www.pearsonhighered.com/stock_watson.
Review the Concepts
1.1 Design a hypothetical ideal randomized controlled experiment to study the effect of hours spent studying on performance on microeconomics exams. Suggest some impediments to implementing this experiment in practice.
1.2 Design a hypothetical ideal randomized controlled experiment to study the effect on highway traffic deaths of wearing seat belts. Suggest some impediments to implementing this experiment in practice.
1.3 You are asked to study the casual effect of hours spent on employee train- ing (measured in hours per worker per week) in a manufacturing plant on the productivity of its workers (output per worker per hour). Describe:
a. an ideal randomized controlled experiment to measure this causal effect;
b. an observational cross-sectional data set with which you could study this effect;
c. an observational time series data set for studying this effect; and
d. an observational panel data set for studying this effect.
14
C h2a p t e r
Review of Probability
This chapter reviews the core ideas of the theory of probability that are needed to understand regression analysis and econometrics. We assume that you have taken an introductory course in probability and statistics. If your knowledge of probability is stale, you should refresh it by reading this chapter. If you feel confident with the material, you still should skim the chapter and the terms and concepts at the end to make sure you are familiar with the ideas and notation.
Most aspects of the world around us have an element of randomness. The theory of probability provides mathematical tools for quantifying and describing this randomness. Section 2.1 reviews probability distributions for a single random variable, and Section 2.2 covers the mathematical expectation, mean, and variance of a single random variable. Most of the interesting problems in economics involve more than one variable, and Section 2.3 introduces the basic elements of probability theory for two random variables. Section 2.4 discusses three special probability distributions that play a central role in statistics and econometrics: the normal, chi- squared, and F distributions.
The final two sections of this chapter focus on a specific source of randomness of central importance in econometrics: the randomness that arises by randomly drawing a sample of data from a larger population. For example, suppose you survey ten recent college graduates selected at random, record (or “observe”) their earnings, and compute the average earnings using these ten data points (or “observations”). Because you chose the sample at random, you could have chosen ten different graduates by pure random chance; had you done so, you would have observed ten different earnings and you would have computed a different sample average. Because the average earnings vary from one randomly chosen sample to the next, the sample average is itself a random variable. Therefore, the sample average has a probability distribution, which is referred to as its sampling distribution because this distribution describes the different possible values of the sample average that might have occurred had a different sample been drawn.
Section 2.5 discusses random sampling and the sampling distribution of the sample average. This sampling distribution is, in general, complicated. When the
2.1 Random Variables and Probability Distributions 15
sample size is sufficiently large, however, the sampling distribution of the sample average is approximately normal, a result known as the central limit theorem, which is discussed in Section 2.6.
2.1
Random Variables and Probability Distributions
Probabilities, the Sample Space, and Random Variables
Probabilities and outcomes. The gender of the next new person you meet, your grade on an exam, and the number of times your computer will crash while you are writing a term paper all have an element of chance or randomness. In each of these examples, there is something not yet known that is eventually revealed.
The mutually exclusive potential results of a random process are called the outcomes. For example, your computer might never crash, it might crash once, it might crash twice, and so on. Only one of these outcomes will actually occur (the outcomes are mutually exclusive), and the outcomes need not be equally likely.
The probability of an outcome is the proportion of the time that the outcome occurs in the long run. If the probability of your computer not crashing while you are writing a term paper is 80%, then over the course of writing many term papers you will complete 80% without a crash.
Thesamplespaceandevents. Thesetofallpossibleoutcomesiscalledthesample space. An event is a subset of the sample space, that is, an event is a set of one or more outcomes. The event “my computer will crash no more than once” is the set consisting of two outcomes: “no crashes” and “one crash.”
Random variables. A random variable is a numerical summary of a random outcome. The number of times your computer crashes while you are writing a term paper is random and takes on a numerical value, so it is a random variable.
Some random variables are discrete and some are continuous. As their names suggest, a discrete random variable takes on only a discrete set of values, like 0, 1, 2, c, whereas a continuous random variable takes on a continuum of possible values.
16 ChaPteR 2 Review of Probability
Probability Distribution of a Discrete
Random Variable
Probability distribution. The probability distribution of a discrete random vari- able is the list of all possible values of the variable and the probability that each value will occur. These probabilities sum to 1.
For example, let M be the number of times your computer crashes while you are writing a term paper. The probability distribution of the random variable M is the list of probabilities of each possible outcome: The probability that M = 0, denoted Pr(M = 0), is the probability of no computer crashes; Pr(M = 1) is the probability of a single computer crash; and so forth. An example of a probability distribution for M is given in the second row of Table 2.1; in this distribution, if your computer crashes four times, you will quit and write the paper by hand. According to this distribution, the probability of no crashes is 80%; the probabil- ity of one crash is 10%; and the probability of two, three, or four crashes is, respectively, 6%, 3%, and 1%. These probabilities sum to 100%. This probability distribution is plotted in Figure 2.1.
Probabilities of events. The probability of an event can be computed from the probability distribution. For example, the probability of the event of one or two crashes is the sum of the probabilities of the constituent outcomes. That is, Pr(M = 1 or M = 2) = Pr(M = 1) + Pr(M = 2) = 0.10 + 0.06 = 0.16, or 16%.
Cumulative probability distribution. The cumulative probability distribution is the probability that the random variable is less than or equal to a particular value. The last row of Table 2.1 gives the cumulative probability distribution of the random variable M. For example, the probability of at most one crash, Pr(M … 1), is 90%, which is the sum of the probabilities of no crashes (80%) and of one crash (10%).
taBLe 2.1 Probability of Your Computer Crashing M times
Outcome (number of crashes)
01234
Probability distribution 0.80 0.10 0.06 0.03 0.01
Cumulative probability
distribution 0.80 0.90 0.96 0.99 1.00
2.1 Random Variables and Probability Distributions 17 Figure 2.1 Probability Distribution of the Number of Computer
Crashes
The height of each bar is the probability that the computer crashes the indi- cated number of times. The height of the first bar is 0.8, so the probability
of 0 computer crashes is 80%. The height of the second bar is 0.1, so the probability of 1 computer crash is 10%, and so forth for the other bars.
Probability
0.8
0.7
0.6
0.5
0.4 0.3 0.2 0.1 0.0
01234
Number of crashes
A cumulative probability distribution is also referred to as a cumulative distribution function, a c.d.f., or a cumulative distribution.
The Bernoulli distribution. An important special case of a discrete random vari- able is when the random variable is binary, that is, the outcomes are 0 or 1. A binary random variable is called a Bernoulli random variable (in honor of the seventeenth-century Swiss mathematician and scientist Jacob Bernoulli), and its probability distribution is called the Bernoulli distribution.
For example, let G be the gender of the next new person you meet, where G = 0 indicates that the person is male and G = 1 indicates that she is female. The outcomes of G and their probabilities thus are
G = e 1 with probability p (2.1) 0 with probability 1 – p,
where p is the probability of the next new person you meet being a woman. The probability distribution in Equation (2.1) is the Bernoulli distribution.
Electronic Publishing Services Inc.
18
ChaPteR 2 Review of Probability
Figure 2.2
Cumulative Distribution and Probability Density Functions of Commuting time
Probability
Pr (Commuting time£20) = 0.78
Pr (Commuting time£15) = 0.20
1.0
0.8
0.6
0.4
0.2
0.0
(a) Cumulative distribution function of commuting time Probability density
0.15
0.12
0.09
0.06
0.03
0.00
10 15 20 25 30 35 40
Commuting time (minutes)
Pr (Commuting time£15) = 0.20
Pr (15 < Commuting time£20) = 0.58
Pr (Commuting time>20) = 0.22
0.22
10 15 20 25 30 35 40
Commuting time (minutes)
(b) Probability density function of commuting time
Figure 2.2a shows the cumulative probability distribution (or c.d.f.) of commuting times. The probability that a commuting time is less than 15 minutes is 0.20 (or 20%), and the probability that it is less than 20 minutes
is 0.78 (78%). Figure 2.2b shows the probability density function (or p.d.f.) of commuting times. Probabilities are given by areas under the p.d.f. The probability that a commuting time is between 15 and 20 minutes is 0.58 (58%) and is given by the area under the curve between 15 and 20 minutes.
0.58
0.20
Probability Distribution of a Continuous
Random Variable
Cumulative probability distribution. The cumulative probability distribution for a continuous variable is defined just as it is for a discrete random variable. That is, the cumulative probability distribution of a continuous random variable is the probability that the random variable is less than or equal to a particular value.
For example, consider a student who drives from home to school. This student’s commuting time can take on a continuum of values and, because it depends on random factors such as the weather and traffic conditions, it is natural to treat it as a continuous random variable. Figure 2.2a plots a hypothetical cumulative distribu- tion of commuting times. For example, the probability that the commute takes less than 15 minutes is 20% and the probability that it takes less than 20 minutes is 78%.
Probability density function. Because a continuous random variable can take on a continuum of possible values, the probability distribution used for discrete variables, which lists the probability of each possible value of the random variable, is not suitable for continuous variables. Instead, the probability is summarized by the probability density function. The area under the probability density function between any two points is the probability that the random variable falls between those two points. A probability density function is also called a p.d.f., a density function, or simply a density.
Figure 2.2b plots the probability density function of commuting times corre- sponding to the cumulative distribution in Figure 2.2a. The probability that the com- mute takes between 15 and 20 minutes is given by the area under the p.d.f. between 15 minutes and 20 minutes, which is 0.58, or 58%. Equivalently, this probability can be seen on the cumulative distribution in Figure 2.2a as the difference between the probability that the commute is less than 20 minutes (78%) and the probability that it is less than 15 minutes (20%). Thus the probability density function and the cumu- lative probability distribution show the same information in different formats.
2.2
2.2 Expected Values, Mean, and Variance 19
Expected Values, Mean, and Variance The Expected Value of a Random Variable
Expectedvalue. TheexpectedvalueofarandomvariableY,denotedE(Y),isthe long-run average value of the random variable over many repeated trials or occur- rences. The expected value of a discrete random variable is computed as a weighted average of the possible outcomes of that random variable, where the weights are the probabilities of that outcome. The expected value of Y is also called the expectation of Y or the mean of Y and is denoted mY.
20 ChaPteR 2 Review of Probability
For example, suppose you loan a friend $100 at 10% interest. If the loan is repaid, you get $110 (the principal of $100 plus interest of $10), but there is a risk of 1% that your friend will default and you will get nothing at all. Thus the amount you are repaid is a random variable that equals $110 with probability 0.99 and equals $0 with probability 0.01. Over many such loans, 99% of the time you would be paid back $110, but 1% of the time you would get nothing, so on average you would be repaid $110 * 0.99 + $0 * 0.01 = $108.90. Thus the expected value of your repayment (or the “mean repayment”) is $108.90.
As a second example, consider the number of computer crashes M with the probability distribution given in Table 2.1. The expected value of M is the average number of crashes over many term papers, weighted by the frequency with which a crash of a given size occurs. Accordingly,
E(M)=0*0.80+1*0.10+2*0.06+3*0.03+4*0.01=0.35. (2.2)
That is, the expected number of computer crashes while writing a term paper is 0.35. Of course, the actual number of crashes must always be an integer; it makes no sense to say that the computer crashed 0.35 times while writing a particular term paper! Rather, the calculation in Equation (2.2) means that the average number of crashes over many such term papers is 0.35.
The formula for the expected value of a discrete random variable Y that can take on k different values is given as Key Concept 2.1. (Key Concept 2.1 uses “summation notation,” which is reviewed in Exercise 2.25.)
expected Value and the Mean
2.1
Key COnCept
Suppose the random variable Y takes on k possible values, y1, c, yk, where y1 denotes the first value, y2 denotes the second value, and so forth, and that the probability that Y takes on y1 is p1, the probability that Y takes on y2 is p2, and so forth. The expected value of Y, denoted E(Y), is
ak i=1
E(Y) = y1p1 + y2p2 + g+ ykpk =
where the notation gki = 1yi pi means “the sum of yi pi for i running from 1 to k.” The expected value of Y is also called the mean of Y or the expectation of Y and is denoted mY.
yipi, (2.3)
2.2 Expected Values, Mean, and Variance 21
Expected value of a Bernoulli random variable. An important special case of the general formula in Key Concept 2.1 is the mean of a Bernoulli random variable. Let G be the Bernoulli random variable with the probability distribution in Equation (2.1). The expected value of G is
E(G) = 1 * p + 0 * (1 – p) = p. (2.4) Thus the expected value of a Bernoulli random variable is p, the probability that
it takes on the value “1.”
Expected value of a continuous random variable. The expected value of a con- tinuous random variable is also the probability-weighted average of the possible outcomes of the random variable. Because a continuous random variable can take on a continuum of possible values, the formal mathematical definition of its expectation involves calculus and its definition is given in Appendix 17.1.
The Standard Deviation and Variance
The variance and standard deviation measure the dispersion or the “spread” of a probability distribution. The variance of a random variable Y, denoted var(Y), is the expected value of the square of the deviation of Y from its mean:
Y2 var(Y) = E3(Y – m ) 4.
Because the variance involves the square of Y, the units of the variance are the units of the square of Y, which makes the variance awkward to interpret. It is therefore common to measure the spread by the standard deviation, which is the square root of the variance and is denoted sY. The standard deviation has the same units as Y. These definitions are summarized in Key Concept 2.2.
Variance and Standard Deviation
The variance of the discrete random variable Y, denoted s2Y, is s2Y = var(Y) = E3(Y – mY)24 = ak (yi – mY)2pi.
i=1
Key COnCept
2.2
The standard deviation of Y is sY, the square root of the variance. The units of the standard deviation are the same as the units of Y.
(2.5)
22 ChaPteR 2 Review of Probability
For example, the variance of the number of computer crashes M is the probability-weighted average of the squared difference between M and its mean, 0.35:
var(M)=(0-0.35)2 *0.80+(1-0.35)2 *0.10+(2-0.35)2 *0.06 +(3-0.35) *0.03+(4-0.35) *0.01=0.6475. (2.6)
The standard deviation of M is the square root of the variance, so sM = 20.64750 ≅ 0.80. 2 2
VarianceofaBernoullirandomvariable. ThemeanoftheBernoullirandomvari- able G with probability distribution in Equation (2.1) is mG = p [Equation (2.4)], so its variance is
var(G)=s2G =(0-p)2 *(1-p)+(1-p)2 *p=p(1-p). (2.7) Thus the standard deviation of a Bernoulli random variable is s = 2p(1 – p).
Mean and Variance of a Linear Function
of a Random Variable
This section discusses random variables (say, X and Y) that are related by a linear function. For example, consider an income tax scheme under which a worker is taxed at a rate of 20% on his or her earnings and then given a (tax-free) grant of $2000. Under this tax scheme, after-tax earnings Y are related to pre-tax earnings X by the equation
Y = 2000 + 0.8X. (2.8)
That is, after-tax earnings Y is 80% of pre-tax earnings X, plus $2000.
Suppose an individual’s pre-tax earnings next year are a random variable with mean mX and variance s2X. Because pre-tax earnings are random, so are after-tax earnings. What are the mean and standard deviations of her after-tax earnings under this tax? After taxes, her earnings are 80% of the original pre-tax earnings,
plus $2000. Thus the expected value of her after-tax earnings is
E(Y) = mY = 2000 + 0.8mX. (2.9)
The variance of after-tax earnings is the expected value of (Y – mY)2. Because Y=2000+0.8X,Y-mY =2000+0.8X – (2000+0.8mX)=0.8(X-mX).
G
2.2 Expected Values, Mean, and Variance 23
Thus E3(Y – mY)24 = E530.8(X – mX)426 = 0.64E3(X – mX)24. It follows that var(Y) = 0.64var(X), so, taking the square root of the variance, the standard deviation of Y is
sY = 0.8sX. (2.10)
That is, the standard deviation of the distribution of her after-tax earnings is 80% of the standard deviation of the distribution of pre-tax earnings.
This analysis can be generalized so that Y depends on X with an intercept a (instead of $2000) and a slope b (instead of 0.8) so that
Y = a + bX. Then the mean and variance of Y are
mY = a + bmX and
s2Y = b2s2X,
(2.11)
(2.12) (2.13)
and the standard deviation of Y is sY = bsX. The expressions in Equations (2.9)
and (2.10) are applications of the more general formulas in Equations (2.12) and (2.13) with a = 2000 and b = 0.8.
Other Measures of the Shape of a Distribution
The mean and standard deviation measure two important features of a distribu- tion: its center (the mean) and its spread (the standard deviation). This section discusses measures of two other features of a distribution: the skewness, which measures the lack of symmetry of a distribution, and the kurtosis, which measures how thick, or “heavy,” are its tails. The mean, variance, skewness, and kurtosis are all based on what are called the moments of a distribution.
Skewness. Figure 2.3 plots four distributions, two which are symmetric (Figures 2.3a and 2.3b) and two which are not (Figures 2.3c and 2.3d). Visually, the distri- bution in Figure 2.3d appears to deviate more from symmetry than does the dis- tribution in Figure 2.3c. The skewness of a distribution provides a mathematical way to describe how much a distribution deviates from symmetry.
s3 Y
The skewness of the distribution of a random variable Y is
Skewness = E3(Y – mY)34, (2.14)
24
ChaPteR 2 Review of Probability
Figure 2.3
0.5
0.4
0.3
0.2
0.1
0.0
–4 –3 –2 –1 0 1 2 3 4
(a) Skewness = 0, kurtosis = 3 0.5
0.4
0.3
0.2
0.1
0.0
–4 –3 –2 –1 0 1 2 3 4
(c) Skewness = –0.1, kurtosis = 5
Four Distributions with Different Skewness and Kurtosis
0.6
0.5
0.4
0.3
0.2
0.1
0.0
–4 –3 –2 –1 0 1 2 3 4
(b) Skewness = 0, kurtosis = 20
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
–4 –3 –2 –1 0 1 2 3 4
(d) Skewness = 0.6, kurtosis = 5
All of these distributions have a mean of 0 and a variance of 1. The distributions with skewness of 0 (a and b) are symmetric; the distributions with nonzero skewness (c and d) are not symmetric. The distributions with kurtosis exceeding 3 (b–d) have heavy tails.
where sY is the standard deviation of Y. For a symmetric distribution, a value of Y a given amount above its mean is just as likely as a value of Y the same amount below its mean. If so, then positive values of (Y – mY)3 will be offset on average (in expectation) by equally likely negative values. Thus, for a symmetric distribu-
Y3
tion, E3(Y – m ) 4 = 0; the skewness of a symmetric distribution is zero. If a
The kurtosis of the distribution of Y is
2.2 Expected Values, Mean, and Variance 25
distribution is not symmetric, then a positive value of (Y – mY)3 generally is not offset on average by an equally likely negative value, so the skewness is nonzero for a distribution that is not symmetric. Dividing by s3Y in the denom- inator of Equation (2.14) cancels the units of Y3 in the numerator, so the skewness is unit free; in other words, changing the units of Y does not change its skewness.
Below each of the four distributions in Figure 2.3 is its skewness. If a distribu- tion has a long right tail, positive values of (Y – mY)3 are not fully offset by nega- tive values, and the skewness is positive. If a distribution has a long left tail, its skewness is negative.
Kurtosis. The kurtosis of a distribution is a measure of how much mass is in its tails and, therefore, is a measure of how much of the variance of Y arises from extreme values. An extreme value of Y is called an outlier. The greater the kur- tosis of a distribution, the more likely are outliers.
Kurtosis = E3(Y – mY)44. (2.15) s4
If a distribution has a large amount of mass in its tails, then some extreme depar- tures of Y from its mean are likely, and these departures will lead to large values, on average (in expectation), of (Y – mY)4. Thus, for a distribution with a large amount of mass in its tails, the kurtosis will be large. Because (Y – mY)4 cannot be negative, the kurtosis cannot be negative.
The kurtosis of a normally distributed random variable is 3, so a random vari- able with kurtosis exceeding 3 has more mass in its tails than a normal random variable. A distribution with kurtosis exceeding 3 is called leptokurtic or, more simply, heavy-tailed. Like skewness, the kurtosis is unit free, so changing the units of Y does not change its kurtosis.
Below each of the four distributions in Figure 2.3 is its kurtosis. The distribu- tions in Figures 2.3b–d are heavy-tailed.
Moments. The mean of Y, E(Y), is also called the first moment of Y, and the expected value of the square of Y, E(Y2), is called the second moment of Y. In general, the expected value of Yr is called the rth moment of the random variable Y. That is, the rth moment of Y is E(Yr). The skewness is a function of the first, second, and third moments of Y, and the kurtosis is a function of the first through fourth moments of Y.
Y
26 ChaPteR 2 Review of Probability
2.3
Two Random Variables
Most of the interesting questions in economics involve two or more variables. Are college graduates more likely to have a job than nongraduates? How does the distribution of income for women compare to that for men? These questions con- cern the distribution of two random variables, considered together (education and employment status in the first example, income and gender in the second). Answering such questions requires an understanding of the concepts of joint, marginal, and conditional probability distributions.
Joint and Marginal Distributions
Joint distribution. The joint probability distribution of two discrete random vari- ables, say X and Y, is the probability that the random variables simultaneously take on certain values, say x and y. The probabilities of all possible (x, y) combina- tions sum to 1. The joint probability distribution can be written as the function Pr(X = x, Y = y).
For example, weather conditions—whether or not it is raining—affect the commuting time of the student commuter in Section 2.1. Let Y be a binary ran- dom variable that equals 1 if the commute is short (less than 20 minutes) and equals 0 otherwise and let X be a binary random variable that equals 0 if it is rain- ing and 1 if not. Between these two random variables, there are four possible outcomes: it rains and the commute is long (X = 0, Y = 0); rain and short com- mute(X = 0,Y = 1);norainandlongcommute(X = 1,Y = 0);andnorainand short commute (X = 1, Y = 1). The joint probability distribution is the frequency with which each of these four outcomes occurs over many repeated commutes.
An example of a joint distribution of these two variables is given in Table 2.2. According to this distribution, over many commutes, 15% of the days have rain and a long commute (X = 0, Y = 0); that is, the probability of a long, rainy com- mute is 15%, or Pr(X = 0,Y = 0) = 0.15. Also, Pr(X = 0,Y = 1) = 0.15,
taBLe 2.2 Joint Distribution of Weather Conditions and Commuting times
Long commute (Y = 0)
Short commute (Y = 1)
Total
rain (X = 0)
0.15
0.15
0.30
no rain (X = 1) total
0.07 0.22
0.63 0.78
0.70 1.00
Pr(X = 1,Y = 0) = 0.07, and Pr(X = 1,Y = 1) = 0.63. These four possible outcomes are mutually exclusive and constitute the sample space so the four prob- abilities sum to 1.
Marginalprobabilitydistribution. Themarginalprobabilitydistributionofaran- dom variable Y is just another name for its probability distribution. This term is used to distinguish the distribution of Y alone (the marginal distribution) from the joint distribution of Y and another random variable.
The marginal distribution of Y can be computed from the joint distribution of X and Y by adding up the probabilities of all possible outcomes for which Y takes on a specified value. If X can take on l different values x1, c, xl, then the mar- ginal probability that Y takes on the value y is
al i=1
For example, in Table 2.2, the probability of a long rainy commute is 15% and the probability of a long commute with no rain is 7%, so the probability of a long commute (rainy or not) is 22%. The marginal distribution of commuting times is given in the final column of Table 2.2. Similarly, the marginal probability that it will rain is 30%, as shown in the final row of Table 2.2.
Conditional Distributions
Conditional distribution. The distribution of a random variable Y conditional on another random variable X taking on a specific value is called the conditional distribution of Y given X. The conditional probability that Y takes on the value y when X takes on the value x is written Pr(Y = y X = x).
For example, what is the probability of a long commute (Y = 0) if you know it is raining (X = 0)? From Table 2.2, the joint probability of a rainy short com- mute is 15% and the joint probability of a rainy long commute is 15%, so if it is raining a long commute and a short commute are equally likely. Thus the proba- bility of a long commute (Y = 0), conditional on it being rainy (X = 0), is 50%, or Pr(Y = 0 X = 0) = 0.50. Equivalently, the marginal probability of rain is 30%; that is, over many commutes it rains 30% of the time. Of this 30% of com- mutes, 50% of the time the commute is long (0.15>0.30).
In general, the conditional distribution of Y given X = x is
Pr(Y = y X = x) = Pr(X = x, Y = y). (2.17) Pr(X = x)
Pr(Y = y) =
Pr(X = xi,Y = y). (2.16)
2.3 Two Random Variables 27
28 ChaPteR 2 Review of Probability
taBLe 2.3 Joint and Conditional Distributions of Computer Crashes (M) and
Computer age (A) a. Joint Distribution
Old computer (A = 0)
New computer (A = 1)
Total
M=0 M=1 M=2 M=3 M=4 total
0.35 0.065 0.05 0.025 0.01 0.50
0.45 0.035 0.01 0.005 0.00 0.50
0.80 0.10 0.06 0.03 0.01 1.00
B. Conditional Distributions of M given A
M=0 M=1 M=2 M=3 M=4 total
Pr(MA = 0)
Pr(MA = 1)
0.70 0.13 0.10 0.05 0.02 1.00
0.90 0.07 0.02 0.01 0.00 1.00
For example, the conditional probability of a long commute given that it is rainy isPr(Y = 0X = 0) = Pr(X = 0,Y = 0)>Pr(X = 0) = 0.15>0.30 = 0.50.
As a second example, consider a modification of the crashing computer exam- ple. Suppose you use a computer in the library to type your term paper and the librarian randomly assigns you a computer from those available, half of which are new and half of which are old. Because you are randomly assigned to a computer, the age of the computer you use, A (= 1 if the computer is new, = 0 if it is old), is a random variable. Suppose the joint distribution of the random variables M and A is given in Part A of Table 2.3. Then the conditional distribution of computer crashes, given the age of the computer, is given in Part B of the table. For example, the joint probability M = 0 and A = 0 is 0.35; because half the computers are old, the conditional probability of no crashes, given that you are using an old computer, is Pr(M = 0 A = 0) = Pr(M = 0, A = 0)>Pr(A = 0) = 0.35>0.50 = 0.70, or 70%. In contrast, the conditional probability of no crashes given that you are assigned a new computer is 90%. According to the conditional distributions in Part B of Table 2.3, the newer computers are less likely to crash than the old ones; for example, the probability of three crashes is 5% with an old computer but 1% with a new computer.
Conditionalexpectation. TheconditionalexpectationofYgivenX,alsocalledthe conditional mean of Y given X, is the mean of the conditional distribution of Y given X. That is, the conditional expectation is the expected value of Y, computed
2.3 Two Random Variables 29 using the conditional distribution of Y given X. If Y takes on k values y1, c, yk,
then the conditional mean of Y given X = x is
ak i=1
For example, based on the conditional distributions in Table 2.3, the expected number of computer crashes, given that the computer is old, is E(M 0 A = 0) = 0 * 0.70 + 1 * 0.13 + 2 * 0.10 + 3 * 0.05 + 4 * 0.02 = 0.56. The expected number of computer crashes, given that the computer is new, is E(M 0 A = 1) = 0.14, less than for the old computers.
The conditional expectation of Y given X = x is just the mean value of Y when X = x. In the example of Table 2.3, the mean number of crashes is 0.56 for old computers, so the conditional expectation of Y given that the computer is old is 0.56. Similarly, among new computers, the mean number of crashes is 0.14, that is, the conditional expectation of Y given that the computer is new is 0.14.
The law of iterated expectations. The mean of Y is the weighted average of the conditional expectation of Y given X, weighted by the probability distribution of X. For example, the mean height of adults is the weighted average of the mean height of men and the mean height of women, weighted by the propor- tions of men and women. Stated mathematically, if X takes on the l values x1, c, xl, then
al i=1
Equation (2.19) follows from Equations (2.18) and (2.17) (see Exercise 2.19). Stated differently, the expectation of Y is the expectation of the conditional
expectation of Y given X,
E(Y) = E[E(YX)], (2.20)
where the inner expectation on the right-hand side of Equation (2.20) is computed using the conditional distribution of Y given X and the outer expectation is com- puted using the marginal distribution of X. Equation (2.20) is known as the law of iterated expectations.
For example, the mean number of crashes M is the weighted average of the conditional expectation of M given that it is old and the conditional expectation of
E(YX = x) =
yiPr(Y = yi X = x). (2.18)
E(Y) =
E(YX = xi)Pr(X = xi). (2.19)
30 ChaPteR 2 Review of Probability
Mgiventhatitisnew,soE(M) = E(MA = 0) * Pr(A = 0) + E(MA = 1) * Pr(A = 1) = 0.56 * 0.50 + 0.14 * 0.50 = 0.35.Thisisthemeanofthemarginal distribution of M, as calculated in Equation (2.2).
The law of iterated expectations implies that if the conditional mean of Y given X is zero, then the mean of Y is zero. This is an immediate consequence of Equation (2.20): if E(Y X) = 0, then E(Y) = E[E(YX)] = E[0] = 0. Said differently, if the mean of Y given X is zero, then it must be that the probability-weighted average of these conditional means is zero, that is, the mean of Y must be zero.
The law of iterated expectations also applies to expectations that are condi- tional on multiple random variables. For example, let X, Y, and Z be random variables that are jointly distributed. Then the law of iterated expectations says that E(Y ) = E[E(Y X, Z )], where E(Y X, Z ) is the conditional expectation of Y given both X and Z. For example, in the computer crash illustration of Table 2.3, let P denote the number of programs installed on the computer; then E(M A, P) is the expected number of crashes for a computer with age A that has P programs installed. The expected number of crashes overall, E(M), is the weighted average of the expected number of crashes for a computer with age A and number of pro- grams P, weighted by the proportion of computers with that value of both A and P.
Exercise 2.20 provides some additional properties of conditional expectations with multiple variables.
Conditional variance. The variance of Y conditional on X is the variance of the conditional distribution of Y given X. Stated mathematically, the conditional variance of Y given X is
ak 2
var(YX=x)= [yi -E(YX=x)] Pr(Y=yiX=x). (2.21)
i=1
For example, the conditional variance of the number of crashes given that the computer is old is var(M A = 0) = (0 – 0.56)2 * 0.70 + (1 – 0.56)2 * 0.13 + (2 – 0.56)2 * 0.10 + (3 – 0.56)2 * 0.05 + (4 – 0.56)2 * 0.02 ≅ 0.99. The standard deviation of the conditional distribution of M given that A = 0 is thus
10.99 = 0.99. The conditional variance of M given that A = 1 is the variance of the distribution in the second row of Panel B of Table 2.3, which is 0.22, so the standard deviation of M for new computers is 10.22 = 0.47. For the conditional distributions in Table 2.3, the expected number of crashes for new computers (0.14) is less than that for old computers (0.56), and the spread of the distribution of the number of crashes, as measured by the conditional standard deviation, is smaller for new computers (0.47) than for old (0.99).
Independence
Two random variables X and Y are independently distributed, or independent, if knowing the value of one of the variables provides no information about the other. Specifically, X and Y are independent if the conditional distribution of Y given X equals the marginal distribution of Y. That is, X and Y are independently distributed if, for all values of x and y,
Pr(Y = yX = x) = Pr(Y = y) (independenceofXandY). (2.22)
Substituting Equation (2.22) into Equation (2.17) gives an alternative expression for independent random variables in terms of their joint distribution. If X and Y are independent, then
Pr(X = x,Y = y) = Pr(X = x)Pr(Y = y). (2.23) That is, the joint distribution of two independent random variables is the product
of their marginal distributions.
Covariance and Correlation
Covariance. One measure of the extent to which two random variables move together is their covariance. The covariance between X and Y is the expected value E[(X – mX)(Y – mY)], where mX, where mX is the mean of X and mY is the mean of Y. The covariance is denoted cov(X, Y) or sXY. If X can take on l values and Y can take on k values, then the covariance is given by the formula
cov(X, Y) = sXY = E[(X – mX)(Y – mY)]
ak al i=1j=1
To interpret this formula, suppose that when X is greater than its mean (so that X – mX is positive), then Y tends be greater than its mean (so that Y – mY is positive), and when X is less than its mean (so that X – mX 6 0), then Y tends to be less than its mean (so that Y – mY 6 0). In both cases, the product (X – mX) * (Y – mY) tends to be positive, so the covariance is positive. In con- trast, if X and Y tend to move in opposite directions (so that X is large when Y is small, and vice versa), then the covariance is negative. Finally, if X and Y are independent, then the covariance is zero (see Exercise 2.19).
=
(xj – mX)(yi – mY)Pr(X = xj, Y = yi). (2.24)
2.3 Two Random Variables 31
32 ChaPteR 2 Review of Probability
Correlation. BecausethecovarianceistheproductofXandY,deviatedfromtheir means, its units are, awkwardly, the units of X multiplied by the units of Y. This “units” problem can make numerical values of the covariance difficult to interpret.
The correlation is an alternative measure of dependence between X and Y that solves the “units” problem of the covariance. Specifically, the correlation between X and Y is the covariance between X and Y divided by their standard deviations:
2var(X ) var(Y )
corr(X,Y) = cov(X,Y) = sXY . (2.25)
sX sY
Because the units of the numerator in Equation (2.25) are the same as those of the denominator, the units cancel and the correlation is unitless. The random variables X and Y are said to be uncorrelated if corr(X, Y ) = 0.
The correlation always is between −1 and 1; that is, as proven in Appendix 2.1,
-1 … corr(X, Y) … 1 (correlation inequality). (2.26)
Correlationandconditionalmean. IftheconditionalmeanofYdoesnotdepend on X, then Y and X are uncorrelated. That is,
if E(Y X) = mY, then cov(Y, X) = 0 and corr(Y, X) = 0. (2.27)
We now show this result. First suppose that Y and X have mean zero so that cov(Y, X ) = E[(Y – mY)(X – mX)] = E(YX). By the law of iterated expecta- tions [Equation (2.20)], E(YX ) = E[E(YX X )] = E[E(Y X )X ] = 0 because E(Y X ) = 0, so cov(Y, X ) = 0. Equation (2.27) follows by substituting cov(Y, X ) = 0 into the definition of correlation in Equation (2.25). If Y and X do not have mean zero, first subtract off their means, then the preceding proof applies.
It is not necessarily true, however, that if X and Y are uncorrelated, then the conditional mean of Y given X does not depend on X. Said differently, it is pos- sible for the conditional mean of Y to be a function of X but for Y and X nonethe- less to be uncorrelated. An example is given in Exercise 2.23.
The Mean and Variance of Sums
of Random Variables
The mean of the sum of two random variables, X and Y, is the sum of their means:
E(X+Y)=E(X)+E(Y)=mX +mY. (2.28)
2.3 Two Random Variables 33 the Distribution of earnings in the united States in 2012
Some parents tell their children that they will be able to get a better, higher-paying job if they get a college degree than if they skip higher education. Are these parents right? Does the dis- tribution of earnings differ between workers who are college graduates and workers who have only a high school diploma, and, if so, how? Among workers with a similar education, does the dis- tribution of earnings for men and women differ?
For example, do the best-paid college-educated women earn as much as the best-paid college- educated men?
One way to answer these questions is to examine the distribution of earnings of full-time workers, con- ditional on the highest educational degree achieved (high school diploma or bachelor’s degree) and on gender. These four conditional distributions are shown in Figure 2.4, and the mean, standard deviation, and
Figure 2.4 Conditional Distribution of average hourly earnings of U.S. Full-time Workers in 2012, Given education Level and Gender
The four distributions of earnings are for women
and men, for those with only a high school diploma (a and c) and those whose highest degree is from a four-year college (b and d).
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
0.00
0 102030405060708090
Dollars
(a) Women with a high school diploma
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
0.00
0 102030405060708090
Dollars
(c) Men with a high school diploma
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00
0 102030405060708090
Dollars
(b) Women with a college degree
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
0.00
0 102030405060708090
Dollars
(d) Men with a college degree
continued on next page
Density Density
Density Density
34
ChaPteR 2 Review of Probability
taBLe 2.4 Summaries of the Conditional Distribution of average hourly earnings of U.S.
Full-time Workers in 2012 Given education Level and Gender
percentile
(a) Women with high school diploma
(b) Women with four-year college degree
(c) Men with high school diploma
(d) Men with four-year college degree
Mean
$15.49
25.42
20.25
32.73
Standard Deviation
$8.42
13.81
11.00
18.11
25%
$10.10
16.15
12.92
19.61
50% (median)
$14.03
22.44
17.86
28.85
75%
$18.75
31.34
24.83
41.68
90%
$24.52
43.27
33.78
57.30
Average hourly earnings are the sum of annual pretax wages, salaries, tips, and bonuses divided by the number of hours worked annually.
some percentiles of the conditional distributions are presented in Table 2.4.1 For example, the conditional mean of earnings for women whose highest degree is a high school diploma—that is, E(Earnings|Highest degree = high school diploma, Gender = female)—is $15.49 per hour.
The distribution of average hourly earnings for female college graduates (Figure 2.4b) is shifted to the right of the distribution for women with only a high school degree (Figure 2.4a); the same shift can be seen for the two groups of men (Figure 2.4d and Figure 2.4c). For both men and women, mean earnings are higher for those with a college degree (Table 2.4, first numeric column). Interestingly, the spread of the distribution of earnings, as measured by the standard deviation, is greater for those with a college degree than for those with a high school diploma. In addition, for both men and women, the
90th percentile of earnings is much higher for work- ers with a college degree than for workers with only a high school diploma. This final comparison is con- sistent with the parental admonition that a college degree opens doors that remain closed to individuals with only a high school diploma.
Another feature of these distributions is that the distribution of earnings for men is shifted to the right of the distribution of earnings for women. This “gender gap” in earnings is an important— and, to many, troubling—aspect of the distribu- tion of earnings. We return to this topic in later chapters.
1The distributions were estimated using data from the March 2013 Current Population Survey, which is discussed in more detail in Appendix 3.1.
2.3 Two Random Variables 35
Means, Variances, and Covariances of Sums of Random Variables
Key COnCept
2.3
Let X, Y, and V be random variables, let mX and s2X be the mean and variance of X, let sXY be the covariance between X and Y (and so forth for the other vari- ables), and let a, b, and c be constants. Equations (2.29) through (2.35) follow from the definitions of the mean, variance, and covariance:
E(a+bX+cY)=a+bmX +cmY, var(a + bY) = b2s2Y,
var(aX + bY) = a2s2X + 2absXY + b2s2Y,
E(Y2) = s2Y + m2Y,
(2.29) (2.30) (2.31) (2.32) (2.33) (2.34) (2.35)
cov(a + bX + cV,Y) = bsXY + csVY,
corr(X, Y)| … 1 and |s | … 2s s (correlation inequality). XY XY
E(XY) = sXY + mX mY, 22
The variance of the sum of X and Y is the sum of their variances plus two times their covariance:
var(X + Y) = var(X) + var(Y) + 2cov(X,Y) = s2X + s2Y + 2sXY. (2.36) If X and Y are independent, then the covariance is zero and the variance of their
sum is the sum of their variances:
v a r ( X + Y ) = v a r ( X ) + v a r ( Y ) = s 2X + s 2Y
(if X and Y are independent). (2.37)
Useful expressions for means, variances, and covariances involving weighted sums of random variables are collected in Key Concept 2.3. The results in Key Concept 2.3 are derived in Appendix 2.1.
36 ChaPteR 2 Review of Probability
2.4
The Normal, Chi-Squared, Student t, and F Distributions
The probability distributions most often encountered in econometrics are the nor- mal, chi-squared, Student t, and F distributions.
The Normal Distribution
A continuous random variable with a normal distribution has the familiar bell- shaped probability density shown in Figure 2.5. The function defining the normal probability density is given in Appendix 17.1. As Figure 2.5 shows, the normal density with mean m and variance s2 is symmetric around its mean and has 95% of its probability between m – 1.96s and m + 1.96s.
Some special notation and terminology have been developed for the normal distribution. The normal distribution with mean m and variance s2 is expressed concisely as “N(m, s2).” The standard normal distribution is the normal distribu- tion with mean m = 0 and variance s2 = 1 and is denoted N(0, 1). Random vari- ables that have a N(0, 1) distribution are often denoted Z, and the standard normal cumulative distribution function is denoted by the Greek letter Φ; accord- ingly, Pr(Z … c) = Φ(c), where c is a constant. Values of the standard normal cumulative distribution function are tabulated in Appendix Table 1.
To look up probabilities for a normal variable with a general mean and variance, we must standardize the variable by first subtracting the mean, then by dividing
Figure 2.5 the Normal Probability Density
The normal probability
95%
density function with mean m and variance s2 is a bell-shaped curve, centered at m. The area under the normal p.d.f. between m – 1.96s and
m + 1.96s is 0.95.
The normal distribution is denoted N(m, s2).
m – 1.96s
m m + 1.96s y
2.4 The Normal, Chi-Squared, Student t, and F Distributions 37
Computing Probabilities Involving Normal Random Variables
Key COnCept
2.4
Suppose Y is normally distributed with mean m and variance s2; in other words, Y is distributed N(m, s2). Then Y is standardized by subtracting its mean and dividing by its standard deviation, that is, by computing Z = (Y – m)/s.
Let c1 and c2 denote two numbers with c1 6 c2 and let d1 = (c1 – m)/s and d2 = (c2 – m)/s. Then
Pr(Y … c2) = Pr(Z … d2) = Φ(d2),
Pr(Y Ú c1) = Pr(Z Ú d1) = 1 – Φ(d1),
Pr(c1 …Y…c2)=Pr(d1 …Z…d2)=Φ(d2)-Φ(d1).
(2.38) (2.39) (2.40)
The normal cumulative distribution function Φ is tabulated in Appendix Table 1.
the result by the standard deviation. For example, suppose Y is distributed
11
tion, that is, (Y – 1) > 14 = (Y – 1). Accordingly, the random variable (Y – 1)
the standard normal distribution shown in Figure 2.6b. Now Y … 2 is equivalent to 1(Y – 1) … 1(2 – 1)—that is, 1(Y – 1) … 1. Thus,
N(1, 4)—that is, Y is normally distributed with a mean of 1 and a variance of 4.
What is the probability that Y … 2—that is, what is the shaded area in Figure 2.6a?
The standardized version of Y is Y minus its mean, divided by its standard devia-
22
is normally distributed with mean zero and variance one (see Exercise 2.8); it has
2222
Pr(Y…2)=Pr[1(Y-1)…1]=Pr(Z…1)=Φ(0.5)=0.691, (2.41) 222
where the value 0.691 is taken from Appendix Table 1.
The same approach can be applied to compute the probability that a normally
distributed random variable exceeds some value or that it falls in a certain range. These steps are summarized in Key Concept 2.4. The box “A Bad Day on Wall Street” presents an unusual application of the cumulative normal distribution.
The normal distribution is symmetric, so its skewness is zero. The kurtosis of the normal distribution is 3.
38 ChaPteR 2 Review of Probability
Figure 2.6 Calculating the Probability that YÅ2 When Y Is Distributed N(1, 4)
To calculate Pr(Y … 2), standardize Y, then use the standard normal distribution table. Y is standardized by subtracting its mean (m = 1) and dividing by its standard deviation (s = 2). The probability that Y … 2 is shown in
Figure 2.6a, and the corresponding probability
Pr(Y < 2)
N(1, 4) distribution
after standardizing Y is shown in Figure 2.6b.
(Y - 1)>2, is a standard normal (Z) random
Pr(Z … 0.5). From Appendix Table 1, Pr(Z … 0.5) = Φ(0.5) = 0.691.
Because the standardized random variable,
variable, Pr(Y … 2) = Pr1Y – 1 … 2 – 1 2 = 22
1.0 2.0 y
(a) N(1, 4)
Pr(Z < 0.5)
0.691
N(0, 1) distribution
(b) N(0, 1)
Themultivariatenormaldistribution. Thenormaldistributioncanbegeneralized to describe the joint distribution of a set of random variables. In this case, the distribution is called the multivariate normal distribution, or, if only two variables are being considered, the bivariate normal distribution. The formula for the bivar- iate normal p.d.f. is given in Appendix 17.1, and the formula for the general mul- tivariate normal p.d.f. is given in Appendix 18.1.
The multivariate normal distribution has four important properties. If X and Y have a bivariate normal distribution with covariance sXY and if a and b are two
(2.42)
Final
Electronic Publishing Services Inc.
constants, then aX + bY has the normal distribution: Stock/Watson, Econometrics 1e
aX + bY is distrSiTbOutCe.dITNE(Mam.000+4 bm , a2s2 + b2s2 + 2 abs ) X Y X Y XY
Fig. (0X2.,0Y4 bivariate normal).
1st Proof 2nd Proof 3rd Proof
0.0 0.5 z
2.4 The Normal, Chi-Squared, Student t, and F Distributions 39
a Bad Day on Wall Street
On a typical day the overall value of stocks traded on the U.S. stock market can rise or fall by 1% or even more. This is a lot—but nothing com- pared to what happened on Monday, October 19, 1987. On “Black Monday,” the Dow Jones Industrial Average (an average of 30 large industrial stocks) fell by 22.6%! From January 1, 1980, to December 31, 2012, the standard deviation of daily percentage price changes on the Dow was 1.12%, so the drop of 22.6% was a negative return of 20(= 22.6>1.12)
standard deviations. The enormity of this drop can be seen in Figure 2.7, a plot of the daily returns on the Dow during the 1980s.
If daily percentage price changes are normally dis- tributed, then the probability of a change of at least 20 standard deviations is Pr(|Z| Ú 20) = 2 * Φ(-20). You will not find this value in Appendix Table 1, but you can calculate it using a computer (try it!). This probability is 5.5 * 10-89, that is, 0.000 . . . 00055, where there are a total of 88 zeros!
Figure 2.7
From 1980
through 2012,
the average
percentage daily
change of “the
Dow” index was
0.04% and its
standard deviation
was 1.12%. On
October 19, 1987—
”Black Monday”—
the Dow fell 22.6%,
or more than 20 –10 standard deviations.
–15
–20
–25
Daily Percentage Changes in the Dow Jones Industrial average in the 1980s
Percent change
10 5 0 –5
October 19, 1987
1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990
Year
continued on next page
40 ChaPteR 2 Review of Probability
How small is 5.5 * 10-89? Consider the following:
• The world population is about 7 billion, so the prob- ability of winning a random lottery among all living people is about one in 7 billion, or 1.4 * 10-10.
• The universe is believed to have existed for 14 bil- lion years, or about 5 * 1017 seconds, so the prob- ability of choosing a particular second at random from all the seconds since the beginning of time is 2 * 10-18.
• There are approximately 1043 molecules of gas in the first kilometer above the earth’s surface. The probability of choosing one at random is 10-43.
Although Wall Street did have a bad day, the fact that it happened at all suggests its probabil- ity was more than 5.5 * 10-89. In fact, there have been many days—good and bad—with stock price changes too large to be consistent with a normal distribution with a constant variance. Table 2.5 lists the ten largest daily percentage price changes in the
Dow Jones Industrial Average in the 8325 trading days between January 1, 1980, and December 31, 2012, along with the standardized change using the mean and variance over this period. All ten changes exceed 6.4 standard deviations, an extremely rare event if stock prices are normally distributed.
Clearly, stock price percentage changes have a distribution with heavier tails than the normal dis- tribution. For this reason, finance professionals use other models of stock price changes. One such model treats stock price changes as normally distributed with a variance that evolves over time, so periods like October 1987 and the financial crisis in the fall of 2008 have higher volatility than others (models with time- varying variances are discussed in Chapter 16). Other models abandon the normal distribution in favor of distributions with heavier tails, an idea popularized in Nassim Taleb’s 2007 book, The Black Swan. These models are more consistent with the very bad—and very good—days we actually see on Wall Street.
taBLe 2.5
Date
October 19, 1987
October 13, 2008
October 28, 2008
October 21, 1987
October 26, 1987
October 15, 2008
December 01, 2008
October 09, 2008
October 27, 1997
September 17, 2001
the ten Largest Daily Percentage Changes in the Dow Jones Industrial Index, 1980–2012, and the Normal Probability of a Change at Least as Large
percentage Change (x)
-22.6
11.1
10.9
10.1
-8.0
-7.9
-7.7
-7.3
-7.2
-7.1
Standardized Change
Z = (x − M),S
-20.2
9.9
9.7
9.0
-7.2
-7.1
-6.9
-6.6
-6.4
-6.4
normal probability of a Change at Least this Large Pr(∣Z∣ # z) = 2𝚽(−z)
5.5 * 10-89
6.4 * 10-23
3.8 * 10-22
1.8 * 10-19
5.6 * 10-13
1.6 * 10-12
4.9 * 10-12
4.7 * 10-11
1.2 * 10-10
1.6 * 10-10
2.4 The Normal, Chi-Squared, Student t, and F Distributions 41
More generally, if n random variables have a multivariate normal distribution, then any linear combination of these variables (such as their sum) is normally distributed.
Second, if a set of variables has a multivariate normal distribution, then the marginal distribution of each of the variables is normal [this follows from Equa- tion (2.42) by setting a = 1 and b = 0].
Third, if variables with a multivariate normal distribution have covariances that equal zero, then the variables are independent. Thus, if X and Y have a bivariate normal distribution and sXY = 0, then X and Y are independent. In Section 2.3 it was shown that if X and Y are independent, then, regardless of their joint distribution, sXY = 0. If X and Y are jointly normally distributed, then the converse is also true. This result—that zero covariance implies independence—is a special property of the multivariate normal distribution that is not true in general.
Fourth, if X and Y have a bivariate normal distribution, then the conditional expectation of Y given X is linear in X; that is, E(YX = x) = a + bx, where a and b are constants (Exercise 17.11). Joint normality implies linearity of conditional expectations, but linearity of conditional expectations does not imply joint normality.
The Chi-Squared Distribution
The chi-squared distribution is used when testing certain types of hypotheses in statistics and econometrics.
The chi-squared distribution is the distribution of the sum of m squared inde- pendent standard normal random variables. This distribution depends on m, which is called the degrees of freedom of the chi-squared distribution. For exam- ple, let Z1, Z2, and Z3 be independent standard normal random variables. Then Z21 + Z2 + Z23 has a chi-squared distribution with 3 degrees of freedom. The name for this distribution derives from the Greek letter used to denote it: A chi- squared distribution with m degrees of freedom is denoted x2m.
Selected percentiles of the x2m distribution are given in Appendix Table 3. For example, Appendix Table 3 shows that the 95th percentile of the x2m distribution is 7.81, so Pr(Z21 + Z2 + Z3 … 7.81) = 0.95.
The Student t Distribution
The Student t distribution with m degrees of freedom is defined to be the distribu- tion of the ratio of a standard normal random variable, divided by the square root of an independently distributed chi-squared random variable with m degrees of freedom divided by m. That is, let Z be a standard normal random variable, let W be a random variable with a chi-squared distribution with m degrees of freedom,
42 ChaPteR 2 Review of Probability
and let Z and W be independently distributed. Then the random variable Z>2W/m has a Student t distribution (also called the t distribution) with m degrees of freedom. This distribution is denoted tm. Selected percentiles of the Student t distribution are given in Appendix Table 2.
The Student t distribution depends on the degrees of freedom m. Thus the 95th percentile of the tm distribution depends on the degrees of freedom m. The Student t distribution has a bell shape similar to that of the normal distribution, but when m is small (20 or less), it has more mass in the tails—that is, it is a “fat- ter” bell shape than the normal. When m is 30 or more, the Student t distribution is well approximated by the standard normal distribution and the t ∞ distribution equals the standard normal distribution.
The F Distribution
The F distribution with m and n degrees of freedom, denoted Fm,n, is defined to be the distribution of the ratio of a chi-squared random variable with degrees of freedom m, divided by m, to an independently distributed chi-squared random variable with degrees of freedom n, divided by n. To state this mathematically, let W be a chi-squared random variable with m degrees of freedom and let V be a
chi-squared random variable with n degrees of freedom, where W and V are W>m
independently distributed. Then V>n has an Fm,n distribution—that is, an F dis- tribution with numerator degrees of freedom m and denominator degrees of freedom n.
In statistics and econometrics, an important special case of the F distribution
arises when the denominator degrees of freedom is large enough that the Fm,n
the denominator random variable V>n is the mean of infinitely many squared
distribution can be approximated by the Fm,∞ distribution. In this limiting case,
standard normal random variables, and that mean is 1 because the mean of a
squared standard normal random variable is 1 (see Exercise 2.24). Thus the Fm, ∞
of freedom, divided by m: W>m is distributed F . For example, from Appendix m,∞
Table 4, the 95th percentile of the F3,∞ distribution is 2.60, which is the same as the 95th percentile of the x23 distribution, 7.81 (from Appendix Table 2), divided by the degrees of freedom, which is 3 (7.81>3 = 2.60).
The 90th, 95th, and 99th percentiles of the Fm,n distribution are given in Appendix Table 5 for selected values of m and n. For example, the 95th percentile of the F3,30 distribution is 2.92, and the 95th percentile of the F3,90 distribution is 2.71. As the denominator degrees of freedom n increases, the 95th percentile of the F3,n distribution tends to the F3, ∞ limit of 2.60.
distribution is the distribution of a chi-squared random variable with m degrees
2.5
2.5 Random Sampling and the Distribution of the Sample Average 43 Random Sampling and the Distribution
of the Sample Average
Almost all the statistical and econometric procedures used in this book involve averages or weighted averages of a sample of data. Characterizing the distribu- tions of sample averages therefore is an essential step toward understanding the performance of econometric procedures.
This section introduces some basic concepts about random sampling and the distributions of averages that are used throughout the book. We begin by dis- cussing random sampling. The act of random sampling—that is, randomly draw- ing a sample from a larger population—has the effect of making the sample average itself a random variable. Because the sample average is a random vari- able, it has a probability distribution, which is called its sampling distribution. This section concludes with some properties of the sampling distribution of the sample average.
Random Sampling
Simple random sampling. Suppose our commuting student from Section 2.1 aspires to be a statistician and decides to record her commuting times on various days. She selects these days at random from the school year, and her daily com- muting time has the cumulative distribution function in Figure 2.2a. Because these days were selected at random, knowing the value of the commuting time on one of these randomly selected days provides no information about the commuting time on another of the days; that is, because the days were selected at random, the values of the commuting time on each of the different days are independently distributed random variables.
The situation described in the previous paragraph is an example of the sim- plest sampling scheme used in statistics, called simple random sampling, in which n objects are selected at random from a population (the population of commuting days) and each member of the population (each day) is equally likely to be included in the sample.
The n observations in the sample are denoted Y1, c, Yn, where Y1 is the first observation, Y2 is the second observation, and so forth. In the commuting exam- ple, Y1 is the commuting time on the first of her n randomly selected days and Yi is the commuting time on the ith of her randomly selected days.
Because the members of the population included in the sample are selected at random, the values of the observations Y1, c, Yn are themselves random. If
44 ChaPteR 2 Review of Probability
Simple Random Sampling and i.i.d. Random Variables
2.5
Key COnCept
In a simple random sample, n objects are drawn at random from a population and each object is equally likely to be drawn. The value of the random variable Y for the ith randomly drawn object is denoted Yi. Because each object is equally likely to be drawn and the distribution of Yi is the same for all i, the random variables Y1, c, Yn are independently and identically distributed (i.i.d.); that is, the distri- bution of Yi is the same for all i = 1, c, n and Y1 is distributed independently of Y2, c, Yn and so forth.
different members of the population are chosen, their values of Y will differ. Thus the act of random sampling means that Y1, c, Yn can be treated as random vari- ables. Before they are sampled, Y1, c, Yn can take on many possible values; after they are sampled, a specific value is recorded for each observation.
i.i.d. draws. Because Y1, c, Yn are randomly drawn from the same population, the marginal distribution of Yi is the same for each i = 1, c, n; this marginal distribution is the distribution of Y in the population being sampled. When Yi has the same marginal distribution for i = 1, c, n, then Y1, c, Yn are said to be identically distributed.
Under simple random sampling, knowing the value of Y1 provides no infor- mation about Y2, so the conditional distribution of Y2 given Y1 is the same as the marginal distribution of Y2. In other words, under simple random sampling, Y1 is distributed independently of Y2, c, Yn.
When Y1, c, Yn are drawn from the same distribution and are indepen- dently distributed, they are said to be independently and identically distributed (or i.i.d.).
Simple random sampling and i.i.d. draws are summarized in Key Concept 2.5.
The Sampling Distribution of the Sample Average
The sample average or sample mean, Y, of the n observations Y1, c, Yn is 1 1 an
Y=n(Y1 +Y2 + g+Yn)=n Yi. (2.43) i=1
An essential concept is that the act of drawing a random sample has the effect of making the sample average Y a random variable. Because the sample was drawn
2.5 Random Sampling and the Distribution of the Sample Average 45
at random, the value of each Yi is random. Because Y1, c, Yn are random, their average is random. Had a different sample been drawn, then the observations and their sample average would have been different: The value of Y differs from one randomly drawn sample to the next.
For example, suppose our student commuter selected five days at random to record her commute times, then computed the average of those five times. Had she chosen five different days, she would have recorded five different times—and thus would have computed a different value of the sample average.
Because Y is random, it has a probability distribution. The distribution of Y is called the sampling distribution of Y because it is the probability distribution associated with possible values of Y that could be computed for different possible samples Y1, c, Yn.
The sampling distribution of averages and weighted averages plays a central role in statistics and econometrics. We start our discussion of the sampling distri- bution of Y by computing its mean and variance under general conditions on the population distribution of Y.
Mean and variance of Y_. Suppose that the observations Y1, c, Yn are i.i.d., and
let mY and s2Y denote the mean and variance of Yi (because the observations are i.i.d.
the mean and variance is the same for all i = 1, c,n). When n = 2, the mean
of the sum Y1 + Y2 is given by applying Equation (2.28): E(Y1 + Y2) = mY + 11
m = 2m . Thus the mean of the sample average is E3 (Y + Y )4 = * 2m = YY 2122Y
mY. In general,
122Y1ni and Yj are independently distributed for i ≠ j, so cov(Yi, Yj) = 0. Thus,
1 an
i=1
E(Yi ) = mY.
n = 2, var(Y1 + Y2) = 2s2Y, so [by applying Equation (2.31) with a = b = 12 and
E(Y) = n
The variance of Y is found by applying Equation (2.37). For example, for
12
cov(Y , Y ) = 04, var(Y) = as . For general n, because Y , c, Y are i.i.d., Y
var(Y) = vara1 n Yb ni=1 i
= n1 var(Yi ) + n1 cov(Yi,Yj) 2an 2an an
The standard deviation of Y is the square root of the variance, s 2n. Y
i = 1 i = 1 j = 1, j ≠ i = s2Y.
(2.45)
n
(2.44)
46 ChaPteR 2 Review of Probability
Financial Diversification and portfolios
T he principle of diversification says that you can reduce your risk by holding small investments in multiple assets, compared to putting all your money into one asset. That is, you shouldn’t put all
your eggs in one basket.
The math of diversification follows from Equa-
dollars in each asset, the actual payoff of your port- folioafter1yearis(Y + Y + g + Y )>n = Y.
E(Y) = mY,and,forlargen,thevarianceoftheport- folio payout is var(Y) = rs2 (Exercise 2.26). Putting all your money into one asset or spreading it equally across all n assets has the same expected payout, but diversifying reduces the variance from s2 to rs2.
The math of diversification has led to financial products such as stock mutual funds, in which the fund holds many stocks and an individual owns a share of the fund, thereby owning a small amount of many stocks. But diversification has its limits: For many assets, payouts are positively correlated, so var(Y) remains positive even if n is large. In the case of stocks, risk is reduced by holding a portfolio, but that portfolio remains subject to the unpredictable fluctuations of the overall stock market.
tion (2.45). Suppose you divide $1 equally among n assets. Let Yi represent the payout in 1 year of $1
th
invested in the i asset. Because you invested 1>n
12n
To keep things simple, suppose that each asset has
the same expected payout, mY, the same variance, s2,
that cov(Y , Y ) = rs 4. Then the expected payout is ij2
and the same positive correlation r across assets [so
In summary, the mean, the variance, and the standard deviation of Y are
E(Y) = mY.
2 s2Y2n
(2.46)
(2.47) (2.48)
var(Y) = sY = n , and
std.dev(Y) = sY = sY .
These results hold whatever the distribution of Yi is; that is, the distribution of Yi
does not need to take on a specific form, such as the normal distribution, for Equations (2.46) through (2.48) to hold.
The notation s2Y denotes the variance of the sampling distribution of the sample average Y. In contrast, s2Y is the variance of each individual Yi, that is, the variance of the population distribution from which the observation is drawn. Sim- ilarly, sY denotes the standard deviation of the sampling distribution of Y.
Sampling distribution of Y_ when Y is normally distributed. Suppose that Y1, c, Yn are i.i.d. draws from the N(mY, s2Y) distribution. As stated following Equation (2.42), the sum of n normally distributed random variables is itself
2.6
Large-Sample Approximations to Sampling Distributions
Sampling distributions play a central role in the development of statistical and econometric procedures, so it is important to know, in a mathematical sense, what the sampling distribution of Y is. There are two approaches to characterizing sampling distributions: an “exact” approach and an “approximate” approach.
The “exact” approach entails deriving a formula for the sampling distribution
that holds exactly for any value of n. The sampling distribution that exactly
describes the distribution of Y for any n is called the exact distribution or finite-
2.6 Large-Sample Approximations to Sampling Distributions 47
normally distributed. Because the mean of Y is mY and the variance of Y is s2Y>n,
tributed N(m , s >n). Y 2Y
this means that, if Y1, c, Yn are i.i.d. draws from the N(mY, s2Y), then Y is dis-
sample distribution of Y. For example, if Y is normally distributed and Y1, c, Yn
with mean m and variance s >n. Unfortunately, if the distribution of Y is not Y 2Y
are i.i.d., then (as discussed in Section 2.5) the exact distribution of Y is normal
normal, then in general the exact sampling distribution of Y is very complicated and depends on the distribution of Y.
The “approximate” approach uses approximations to the sampling distribution that rely on the sample size being large. The large-sample approximation to the sam- pling distribution is often called the asymptotic distribution—“asymptotic” because the approximations become exact in the limit that n S ∞. As we see in this section, these approximations can be very accurate even if the sample size is only n = 30 observations. Because sample sizes used in practice in econometrics typically number in the hundreds or thousands, these asymptotic distributions can be counted on to provide very good approximations to the exact sampling distribution.
This section presents the two key tools used to approximate sampling distri-
butions when the sample size is large: the law of large numbers and the central
limit theorem. The law of large numbers says that, when the sample size is large,
Y will be close to mY with very high probability. The central limit theorem says
sample average, (Y – m )>s , is approximately normal. YY
that, when the sample size is large, the sampling distribution of the standardized
Although exact sampling distributions are complicated and depend on the dis-
the asymptotic normal distribution of (Y – m )>s does not depend on the YY
distribution of Y. This normal approximate distribution provides enormous sim- plifications and underlies the theory of regression used throughout this book.
tribution of Y, the asymptotic distributions are simple. Moreover—remarkably—
48 ChaPteR 2 Review of Probability
Key COnCept
2.6
Convergence in Probability, Consistency, and the Law of Large Numbers
The sample average Y converges in probability to mY (or, equivalently, Y is con- sistentformY)iftheprobabilitythatYisintherange(mY – c)to(mY + c)becomes arbitrarily close to 1 as n increases for any constant c 7 0. The convergence of Y to mY in probability is written, Y ¡p mY.
The law of large numbers says that if Yi, i = 1, c, n are independently and identically distributed with E(Yi) = mY and if large outliers are unlikely (techni- cally if var(Yi) = s2Y 6 ∞), then Y ¡p mY.
The Law of Large Numbers and Consistency
The law of large numbers states that, under general conditions, Y will be near mY with very high probability when n is large. This is sometimes called the “law of averages.” When a large number of random variables with the same mean are averaged together, the large values balance the small values and their sample average is close to their common mean.
For example, consider a simplified version of our student commuter’s exper- iment in which she simply records whether her commute was short (less than 20 minutes) or long. Let Yi = 1 if her commute was short on the ith randomly selected day and Yi = 0 if it was long. Because she used simple random sampling, Y1, c, Yn are i.i.d. Thus Yi, i = 1, c, n are i.i.d. draws of a Bernoulli random variable, where (from Table 2.2) the probability that Yi = 1 is 0.78. Because the expectation of a Bernoulli random variable is its success probability, E(Yi ) = mY = 0.78. The sample average Y is the fraction of days in her sample in which her commute was short.
Figure 2.8 shows the sampling distribution of Y for various sample sizes n. When n = 2 (Figure 2.8a), Y can take on only three values: 0, 12, and 1 (neither commute was short, one was short, and both were short), none of which is par- ticularly close to the true proportion in the population, 0.78. As n increases, how- ever (Figures 2.8b–d), Y takes on more values and the sampling distribution becomes tightly centered on mY.
The property that Y is near mY with increasing probability as n increases is called convergence in probability or, more concisely, consistency (see Key Con- cept 2.6). The law of large numbers states that, under certain conditions, Y con- verges in probability to mY or, equivalently, that Y is consistent for mY.
2.6 Large-Sample Approximations to Sampling Distributions 49 Figure 2.8 Sampling Distribution of the Sample average of n Bernoulli
Probability
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0.0 (a) n=2
Probability
0.25 0.20 0.15 0.10 0.05 0.00
0.0 (c) n=25
0.25
0.50
0.75
1.00
Probability
0.5 0.4 0.3 0.2 0.1 0.0
0.0 0.25 (b) n=5
Probability
0.125 0.100 0.075 0.050 0.025
0.00
0.0 0.25
(d) n=100
0.50 0.75 1.00
Value of sample average
Random Variables
m = 0.78
m = 0.78
Value of sample average
m = 0.78
m = 0.78
0.25
0.50
0.75 1.00
0.50 0.75 1.00
Value of sample average
Value of sample average
The distributions are the sampling distributions of Y, the sample average of n independent Bernoulli random variables with p = Pr(Yi = 1) = 0.78 (the probability of a short commute is 78%). The variance of the sampling distribution of Y decreases as n gets larger, so the sampling distribution becomes more tightly concentrated around its mean m = 0.78 as the sample size n increases.
50 ChaPteR 2 Review of Probability
The conditions for the law of large numbers that we will use in this book are that Yi, i = 1, c, n are i.i.d. and that the variance of Yi, s2Y, is finite. The math- ematical role of these conditions is made clear in Section 17.2, where the law of large numbers is proven. If the data are collected by simple random sampling, then the i.i.d. assumption holds. The assumption that the variance is finite says that extremely large values of Yi—that is, outliers—are unlikely and observed infrequently; otherwise, these large values could dominate Y and the sample average would be unreliable. This assumption is plausible for the applications in this book. For example, because there is an upper limit to our student’s commuting time (she could park and walk if the traffic is dreadful), the variance of the distribution of commuting times is finite.
The Central Limit Theorem
The central limit theorem says that, under general conditions, the distribution of
mean of Y is m and its variance is s = s >n. According to the central limit Y Y22Y
Y is well approximated by a normal distribution when n is large. Recall that the
theorem, when n is large, the distribution of Y is approximately N(mY, s2). As 2Y
discussed at the end of Section 2.5, the distribution of Y is exactly N(mY, sY) when the sample is drawn from a population with the normal distribution N(mY, s2Y). The central limit theorem says that this same result is approximately true when n is large even if Y1, c, Yn are not themselves normally distributed.
The convergence of the distribution of Y to the bell-shaped, normal approxi- mation can be seen (a bit) in Figure 2.8. However, because the distribution gets quite tight for large n, this requires some squinting. It would be easier to see the shape of the distribution of Y if you used a magnifying glass or had some other way to zoom in or to expand the horizontal axis of the figure.
One way to do this is to standardize Y by subtracting its mean and dividing
by its standard deviation so that it has a mean of 0 and a variance of 1. This
Y, (Y – m )>s . According to the central limit theorem, this distribution should YY
YY
process leads to examining the distribution of the standardized version of
be well approximated by a N(0, 1) distribution when n is large.
The distribution of the standardized average (Y – m )>s is plotted in Fig-
ure 2.9 for the distributions in Figure 2.8; the distributions in Figure 2.9 are exactly the same as in Figure 2.8, except that the scale of the horizontal axis is changed so that the standardized variable has a mean of 0 and a variance of 1. After this change of scale, it is easy to see that, if n is large enough, the distribution of Y is well approximated by a normal distribution.
One might ask, how large is “large enough”? That is, how large must n be for the distribution of Y to be approximately normal? The answer is, “It depends.” The
2.6 Large-Sample Approximations to Sampling Distributions 51 Figure 2.9 Distribution of the Standardized Sample average of n Bernoulli
Random Variables with p = 0.78 Probability
Probability
0.5 0.4 0.3 0.2 0.1 0.0
–3.0 –2.0 –1.0 0.0 1.0 2.0 3.0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
–3.0 –2.0 –1.0 0.0 1.0 2.0 3.0
(a) n=2 Probability
0.25 0.20 0.15 0.10 0.05 0.00
(b) n=5 Probability
0.12
0.09
0.06
0.03
0.00
Standardized value of sample average
Standardized value of sample average
–3.0 –2.0 –1.0 0.0 1.0 2.0 3.0
Standardized value of sample average
–3.0 –2.0 –1.0 0.0 1.0 2.0 3.0
Standardized value of sample average
The sampling distribution of Y in Figure 2.8 is plotted here after standardizing Y. This plot centers the distributions in Figure 2.8 and magnifies the scale on the horizontal axis by a factor of 2n. When the sample size is large, the sam- pling distributions are increasingly well approximated by the normal distribution (the solid line), as predicted by the central limit theorem. The normal distribution is scaled so that the height of the distributions is approximately the same in all figures.
(c) n=25
(d) n=100
52 ChaPteR 2 Review of Probability
the Central Limit theorem
2.7
Key COnCept
06s 6∞.AsnS∞,thedistributionof(Y-m)>s (wheres =s >n) 2Y YY Y22Y
2 Suppose that Y1, c, Yn are i.i.d. with E(Yi) = mY and var(Yi) = sY, where
becomes arbitrarily well approximated by the standard normal distribution.
quality of the normal approximation depends on the distribution of the underly- ing Yi that make up the average. At one extreme, if the Yi are themselves nor- mally distributed, then Y is exactly normally distributed for all n. In contrast, when the underlying Yi themselves have a distribution that is far from normal, then this approximation can require n = 30 or even more.
This point is illustrated in Figure 2.10 for a population distribution, shown in Figure 2.10a, that is quite different from the Bernoulli distribution. This distribu- tion has a long right tail (it is “skewed” to the right). The sampling distribution of Y, after centering and scaling, is shown in Figures 2.10b–d for n = 5, 25, and 100, respectively. Although the sampling distribution is approaching the bell shape for n = 25, the normal approximation still has noticeable imperfections. By n = 100, however, the normal approximation is quite good. In fact, for n Ú 100, the normal approximation to the distribution of Y typically is very good for a wide variety of population distributions.
The central limit theorem is a remarkable result. While the “small n” distribu- tions of Y in parts b and c of Figures 2.9 and 2.10 are complicated and quite different from each other, the “large n” distributions in Figures 2.9d and 2.10d are simple and, amazingly, have a similar shape. Because the distribution of Y approaches the normal as n grows large, Y is said to have an asymptotic normal distribution.
The convenience of the normal approximation, combined with its wide appli- cability because of the central limit theorem, makes it a key underpinning of mod- ern applied econometrics. The central limit theorem is summarized in Key Concept 2.7.
Summary
1. The probabilities with which a random variable takes on different values are summarized by the cumulative distribution function, the probability distri- bution function (for discrete random variables), and the probability density function (for continuous random variables).
Probability
0.50 0.40 0.30 0.20 0.10 0.00
–3.0 –2.0 –1.0 0.0 1.0 2.0 3.0
Probability
0.12 0.09 0.06 0.03 0.00
–3.0 –2.0 –1.0 0.0 1.0 2.0 3.0
Summary 53
Figure 2.10 Distribution of the Standardized Sample average of n Draws from a Skewed Distribution
(a) n = 1 Probability
0.12 0.09 0.06 0.03 0.00
(b) n = 5 Probability
0.12 0.09 0.06 0.03 0.00
Standardized value of sample average
Standardized value of sample average
–3.0 –2.0 –1.0 0.0 1.0 2.0 3.0
Standardized value of sample average
(d) n=100
The figures show the sampling distribution of the standardized sample average of n draws from the skewed (asymmetric) population distribution shown in Figure 2.10a. When n is small (n = 5), the sampling distribution, like the population distribution, is skewed. But when n is large (n = 100), the sampling distribution is well approximated by a standard normal distribution (solid line), as predicted by the central limit theorem. The normal distribution is scaled so that the height of the distributions is approximately the same in all figures.
–3.0 –2.0 –1.0 0.0 1.0 2.0 3.0
Standardized value of sample average
(c) n=25
54 ChaPteR 2 Review of Probability
2. The expected value of a random variable Y (also called its mean, mY),
is s = E3(Y – m ) 4, and the standard deviation of Y is the square root 2Y Y2
denoted E(Y), is its probability-weighted average value. The variance of Y
of its variance.
3. The joint probabilities for two random variables X and Y are summarized
by their joint probability distribution. The conditional probability distribu- tion of Y given X = x is the probability distribution of Y, conditional on X taking on the value x.
4. A normally distributed random variable has the bell-shaped probability density in Figure 2.5. To calculate a probability associated with a normal random variable, first standardize the variable and then use the standard normal cumulative distribution tabulated in Appendix Table 1.
5. Simple random sampling produces n random observations Y1, c, Yn that are independently and identically distributed (i.i.d.).
6. The sample average, Y, varies from one randomly chosen sample to the next and thus is a random variable with a sampling distribution. If Y1, c, Yn are i.i.d., then:
a. the sampling distribution of Y has mean m and variance s = s >n; Y Y22Y
b. the law of large numbers says that Y converges in probability to mY; and
c. the central limit theorem says that the standardized version of Y,
(Y – m )>s , has a standard normal distribution 3N(0, 1) distribution]
Y
when n is large.
Key Terms
outcomes (15)
probability (15)
sample space (15)
event (15)
discrete random variable (15) continuous random variable (15) probability distribution (16) cumulative probability
distribution (16)
cumulative distribution function
(c.d.f.) (17)
Bernoulli random variable (17) Bernoulli distribution (17)
probability density function (p.d.f.) (19)
density function (19) density (19) expected value (19) expectation (19) mean (19)
variance (21)
standard deviation (21) moments of a distribution (23) skewness (23)
kurtosis (25)
outlier (25)
Y
leptokurtic (25)
rth moment (25)
joint probability distribution (26) marginal probability distribution (27) conditional distribution (27) conditional expectation (28) conditional mean (28)
law of iterated expectations (29) conditional variance (30) independently distributed (31) independent (31)
covariance (31)
correlation (32)
uncorrelated (32)
normal distribution (36)
standard normal distribution (36) standardize a variable (36) multivariate normal distribution (38) bivariate normal distribution (38)
chi-squared distribution (41) Student t distribution (41)
t distribution (42)
F distribution (42)
simple random sampling (43) population (43)
identically distributed (44) independently and identically
distributed (i.i.d.) (44)
sample average (44)
sample mean (44)
sampling distribution (45)
exact (finite-sample) distribution (47) asymptotic distribution (47)
law of large numbers (48) convergence in probability (48) consistency (48)
central limit theorem (50) asymptotic normal distribution (52)
Review the Concepts 55
MyEconLab Can Help You Get a Better Grade
MyEconLab If your exam were tomorrow, would you be ready? For each chapter, MyEconLab Practice Tests and Study Plan help you prepare for your exams. You can also find the Exercises and all Review the Concepts Questions available now in MyEconLab.
To see how it works, turn to the MyEconLab spread on the inside front cover of this book and then go to www.myeconlab.com.
For additional Empirical Exercises and Data Sets, log on to the Companion Website at www.pearsonhighered.com/stock_watson.
Review the Concepts
2.1. Examples of random variables used in this chapter included (a) the gender of the next person you meet, (b) the number of times a computer crashes, (c) the time it takes to commute to school, (d) whether the computer you are assigned in the library is new or old, and (e) whether it is raining or not. Explain why each can be thought of as random.
2.2. Suppose that the random variables X and Y are independent and you know their distributions. Explain why knowing the value of X tells you nothing about the value of Y.
56 ChaPteR 2 Review of Probability
2.3. Suppose that X denotes the amount of rainfall in your hometown during a randomly selected month and Y denotes the number of children born in Los Angeles during the same month. Are X and Y independent? Explain.
2.4. An econometrics class has 80 students, and the mean student weight is 145 lb. A random sample of 4 students is selected from the class, and their average weight is calculated. Will the average weight of the students in the sample equal 145 lb? Why or why not? Use this example to explain why the sample average, Y, is a random variable.
2.5. Suppose that Y1, c, Yn are i.i.d. random variables with a N(1, 4) distri- bution. Sketch the probability density of Y when n = 2. Repeat this for n = 10 and n = 100. In words, describe how the densities differ. What is the relationship between your answer and the law of large numbers?
2.6. Suppose that Y1, c, Yn are i.i.d. random variables with the probability distribution given in Figure 2.10a. You want to calculate Pr( Y … 0.1). Would it be reasonable to use the normal approximation if n = 5? What about n = 25 or n = 100? Explain.
2.7. Y is a random variable with mY = 0, sY = 1, skewness = 0, and kurtosis = 100. Sketch a hypothetical probability distribution of Y. Explain why n random variables drawn from this distribution might have some large outliers.
Exercises
2.1 Let Y denote the number of “heads” that occur when two coins are tossed.
a. Derive the probability distribution of Y.
b. Derive the cumulative probability distribution of Y.
c. Derive the mean and variance of Y.
2.2 Use the probability distribution given in Table 2.2 to compute (a) E(Y) and E(X); (b) s2X and s2Y; and (c) sXY and corr(X, Y).
2.3 Using the random variables X and Y from Table 2.2, consider two new random variables W = 3 + 6X and V = 20 – 7Y. Compute (a) E(W) and E(V); (b) s2W and s2V; and (c) sWV and corr(W, V).
2.4 Suppose X is a Bernoulli random variable with P(X = 1) = p.
a. Show E(X3) = p.
b. ShowE(Xk) = pfork 7 0.
c. Suppose that p = 0.3. Compute the mean, variance, skewness, and kurtosis of X. (Hint: You might find it helpful to use the formulas given in Exercise 2.21.)
2.5 In September, Seattle’s daily high temperature has a mean of 70°F and a standard deviation of 7°F. What are the mean, standard deviation, and variance in °C?
2.6 The following table gives the joint probability distribution between employ- ment status and college graduation among those either employed or looking for work (unemployed) in the working-age U.S. population for 2012.
Joint Distribution of employment Status and College Graduation in the U.S. Population aged 25 and Older, 2012
Exercises 57
Non–college grads (X = 0)
College grads (X = 1)
Total
unemployed (Y = 0)
0.053
0.015
0.068
employed (Y = 1)
0.586
0.346
0.932
total
0.639
0.361
1.000
a. Compute E(Y).
b. The unemployment rate is the fraction of the labor force that is
unemployed. Show that the unemployment rate is given by 1 − E(Y).
c. Calculate E(YX = 1) and E(YX = 0).
d. Calculate the unemployment rate for (i) college graduates and (ii) non–college graduates.
e. A randomly selected member of this population reports being unem- ployed. What is the probability that this worker is a college graduate? A non–college graduate?
f. Are educational achievement and employment status independent? Explain.
2.7 In a given population of two-earner male-female couples, male earnings have a mean of $40,000 per year and a standard deviation of $12,000. Female earnings have a mean of $45,000 per year and a standard deviation of $18,000. The correlation between male and female earnings for a couple is 0.80. Let C denote the combined earnings for a randomly selected couple.
a. What is the mean of C?
b. What is the covariance between male and female earnings?
58 ChaPteR 2 Review of Probability
c. What is the standard deviation of C?
d. Convert the answers to (a) through (c) from U.S. dollars ($) to euros (:).
2.8 The random variable Y has a mean of 1 and a variance of 4. Let Z = 12(Y – 1). Show that mZ = 0 and s2Z = 1.
2.9 X and Y are discrete random variables with the following joint distribution:
Value of Y
14 22 30 40 65
Value of X
1 0.02
5 0.17
0.05 0.10 0.03 0.01
0.15 0.05 0.02 0.01
8 0.02
That is, Pr(X = 1, Y = 14) = 0.02, and so forth.
a. Calculate the probability distribution, mean, and variance of Y.
b. Calculate the probability distribution, mean, and variance of Y given
X = 8.
c. Calculate the covariance and correlation between X and Y.
2.10 Compute the following probabilities:
a. If Y is distributed N(1, 4), find Pr(Y … 3). b. If Y is distributed N(3, 9), find Pr(Y 7 0).
c. If Y is distributed N(50, 25), find Pr(40 … Y … 52). d. If Y is distributed N(5, 2), find Pr(6 … Y … 8).
2.11 Compute the following probabilities:
a. If Y is distributed x24, find Pr(Y … 7.78).
b. If Y is distributed x210, find Pr(Y 7 18.31).
c. If Y is distributed F10,∞, find Pr(Y 7 1.83).
d. Why are the answers to (b) and (c) the same?
e. If Y is distributed x21, find Pr(Y … 1.0). (Hint: Use the definition of
the x21 distribution.)
2.12 Compute the following probabilities:
a. If Y is distributed t15, find Pr(Y 7 1.75).
0.03 0.15 0.10 0.09
b. If Y is distributed t90, find Pr(-1.99 … Y … 1.99).
c. If Y is distributed N(0, 1), find Pr(-1.99 … Y … 1.99).
d. Why are the answers to (b) and (c) approximately the same?
e. If Y is distributed F7,4, find Pr(Y 7 4.12).
f. If Y is distributed F7,120, find Pr(Y 7 2.79).
2.13 X is a Bernoulli random variable with Pr(X = 1) = 0.99, Y is distributed N(0, 1), W is distributed N(0, 100), and X, Y, and W are independent. Let S = XY + (1 – X)W. (That is, S = Y when X = 1, and S = W when X = 0.)
a. Show that E(Y 2) = 1 and E(W 2) = 100.
b. Show that E(Y3) = 0 and E(W3) = 0. (Hint: What is the skewness
for a symmetric distribution?)
c. Show that E(Y4) = 3 and E(W4) = 3 * 1002. (Hint: Use the fact that the kurtosis is 3 for a normal distribution.)
d. Derive E(S), E(S2), E(S3) and E(S4). (Hint: Use the law of iterated expectations conditioning on X = 0 and X = 1.)
e. Derive the skewness and kurtosis for S.
2.14 In a population mY = 100 and s2Y = 43. Use the central limit theorem to
answer the following questions:
a. In a random sample of size n = 100, find Pr(Y … 101). b. In a random sample of size n = 165, find Pr(Y 7 98).
c. In a random sample of size n = 64, find Pr(101 … Y … 103).
2.15 Suppose Yi, i = 1, 2, c, n, are i.i.d. random variables, each distributed
N(10, 4).
a. Compute Pr(9.6 … Y … 10.4) when (i) n = 20, (ii) n = 100, and
(iii) n = 1000.
b. Suppose c is a positive number. Show that Pr(10 – c … Y … 10 + c)
becomes close to 1.0 as n grows large.
c. Use your answer in (b) to argue that Y converges in probability
to 10.
2.16 Y is distributed N(5, 100) and you want to calculate Pr(Y 6 3.6). Unfor- tunately, you do not have your textbook, and do not have access to a nor- mal probability table like Appendix Table 1. However, you do have your
Exercises 59
60 ChaPteR 2 Review of Probability
computer and a computer program that can generate i.i.d. draws from the N(5, 100) distribution. Explain how you can use your computer to compute an accurate approximation for Pr(Y 6 3.6).
2.17 Yi, i = 1, c, n, are i.i.d. Bernoulli random variables with p = 0.4. Let Y denote the sample mean.
a. Use the central limit to compute approximations for i. Pr(Y Ú 0.43) when n = 100.
ii. Pr(Y … 0.37) when n = 400.
b. How large would n need to be to ensure that Pr(0.39 … Y … 0.41) Ú 0.95? (Use the central limit theorem to compute an approximate answer.)
2.18 In any year, the weather can inflict storm damage to a home. From year to year, the damage is random. Let Y denote the dollar value of damage in any given year. Suppose that in 95% of the years Y = $0, but in 5% of the years Y = $20,000.
a. What are the mean and standard deviation of the damage in any year?
b. Consider an “insurance pool” of 100 people whose homes are suffi- ciently dispersed so that, in any year, the damage to different homes can be viewed as independently distributed random variables. Let Y denote the average damage to these 100 homes in a year. (i) What is the expected value of the average damage Y? (ii) What is the prob- ability that Y exceeds $2000?
2.19 Consider two random variables X and Y. Suppose that Y takes on k values y , c, y and that X takes on l values x , c, x .
c. Suppose that X and Y are independent. Show that sXY = 0 and corr(X, Y) = 0.
2.20 Consider three random variables X, Y, and Z. Suppose that Y takes on k values y1, c, yk, that X takes on l values x1, c, xl, and that Z takes on m values z1, c, zm. The joint probability distribution of X, Y, Z is Pr(X = x, Y = y, Z = z), and the conditional probability distribution of
YgivenXandZisPr(Y = yX = x,Z = z) = Pr(Y = y,X = x,Z = z). Pr(X = x,Z = z)
a. Show that Pr(Y = y ) = g
Use the definition of Pr(Y = yj X = xi).]
1k 1l
l ji=1jii
Pr(Y = y X = x ) Pr(X = x ). [Hint: b. Use your answer to (a) to verify Equation (2.19).
a. Explain how the marginal probability that Y = y can be calculated from the joint probability distribution. [Hint: This is a generalization of Equation (2.16).]
b. Show that E(Y) = E[E(Y 0 X, Z)]. [Hint: This is a generalization of Equations (2.19) and (2.20).]
2.21 X is a random variable with moments E(X), E(X2), E(X3), and so forth.
a. Show E(X – m)3 = E(X3) – 3[E(X2)][E(X)] + 2[E(X)]3.
b. Show E(X – m)4 = E(X4) – 4[E(X)][E(X3)] + 6[E(X)]2[E(X2)] – 3[E(X)]4.
2.22 Suppose you have some money to invest—for simplicity, $1—and you are planning to put a fraction w into a stock market mutual fund and the rest, 1 – w, into a bond mutual fund. Suppose that $1 invested in a stock fund yields Rs after 1 year and that $1 invested in a bond fund yields Rb, suppose that Rs is random with mean 0.08 (8%) and standard deviation 0.07, and suppose that Rb is random with mean 0.05 (5%) and standard deviation 0.04. The correlation between Rs and Rb is 0.25. If you place a fraction w of your money in the stock fund and the rest, 1 – w, in the bond fund, then the return on your investment is R = wRs + (1 – w)Rb.
a. Suppose that w = 0.5. Compute the mean and standard deviation of R.
b. Suppose that w = 0.75. Compute the mean and standard deviation of R.
c. What value of w makes the mean of R as large as possible? What is the standard deviation of R for this value of w?
d. (Harder) What is the value of w that minimizes the standard deviation of R? (Show using a graph, algebra, or calculus.)
2.23 This exercise provides an example of a pair of random variables X and Y for which the conditional mean of Y given X depends on X but corr(X, Y) = 0. Let X and Z be two independently distributed standard normal random variables, and let Y = X + Z.
2
a. Show that E(Y 0 X ) = X
b. Show that mY = 1.
. 2
c. Show that E(XY ) = 0. (Hint: Use the fact that the odd moments of a standard normal random variable are all zero.)
d. Show that cov(X, Y) = 0 and thus corr(X, Y) = 0.
Exercises 61
62 ChaPteR 2 Review of Probability
2.24 Suppose Y is distributed i.i.d. N(0, s2) for i = 1, 2, c, n.
22
a. Show that E(Y > s ) = 1.
i
i
b. Show that W = (1>s )g Y is distributed x .
2n2 2 i=1i n
Bn-1
c. Show that E(W) = n. [Hint: Use your answer to (a).] d. Show that V = Y1n gni=2Yi2 is distributed tn-1.
2.25 (Review of summation notation) Let x1, c, xn denote a sequence of numbers, y1, c, yn denote another sequence of numbers, and a, b, and c denote three constants. Show that
a. b. c. d.
axi=a xi an an
i=1 i=1
(xi +yi)= xi + yi an an an
i a= 1 n
i=1 an
i = 1 i = 1
a = na
2 2 2an2 2an2 an (a+bxi +cyi) =na +b xi +c yi +2ab xi +
i=1a a i=1 i=1 i=1 nn
2ac yi + 2bc xiyi i=1 i=1
2.26 SupposethatY1,Y2,c,YnarerandomvariableswithacommonmeanmY, a common variance sY2, and the same correlation r (so that the correlation between Yi and Yj is equal to r for all pairs i and j, where i ≠ j).
a. Show that cov(Yi, Yj) = rsY2 for i ≠ j.
b. Suppose that n = 2. Show that E(Y) = mY and var(Y) = 2sY + 2rsY.
c. For n Ú 2, show that E(Y) = m and var(Y) = s >n + 1 2 1 2
[(n – 1)>n]rs2Y.
2.27 X and Z are two jointly distributed random variables. Suppose you know
Y 2Y d. When n is very large, show that var(Y) ≈ rs2Y.
∼ thevalueofZ,butnotthevalueofX.LetX = E(XZ)denoteaguess
of the value of X using the information on Z, and let W = X – X∼ denote the error associated with this guess.
a. Show that E(W ) = 0. (Hint: Use the law of iterated expectations.) b. Show that E(WZ) = 0.
Derivation of Results in Key Concept 2.3 63 nn
c. Let X = g(Z) denote another guess of X using Z, and V = X – X denote its error. Show that E(V2) Ú E(W 2). [Hint: Let h(Z ) = g(Z) – E(X Z), so that V = 3X – E(X Z)4 – h(Z). Derive E(V2).]
Empirical Exercise
E2.1 On the text website, http://www.pearsonhighered.com/stock_watson/, you will find the spreadsheet Age_HourlyEarnings, which contains the joint distribution of age (Age) and average hourly earnings (AHE) for 25- to 34-year-old full-time workers in 2012 with an education level that exceeds a high school diploma. Use this joint distribution to carry out the follow- ing exercises. (Note: For these exercises, you need to be able to carry out calculations and construct charts using a spreadsheet.)
a. Compute the marginal distribution of Age.
b. Compute the mean of AHE for each value of Age; that is, compute,
E(AHE|Age = 25), and so forth.
c. Compute and plot the mean of AHE versus Age. Are average hourly
earnings and age related? Explain.
d. Use the law of iterated expectations to compute the mean of AHE;
that is, compute E(AHE).
e. Compute the variance of AHE.
f. Compute the covariance between AHE and Age.
g. Compute the correlation between AHE and Age.
h. Relate your answers in parts (f) and (g) to the plot you constructed in (c).
2.1
appenDix
Derivation of Results in Key Concept 2.3
This appendix derives the equations in Key Concept 2.3.
Equation (2.29) follows from the definition of the expectation.
To derive Equation (2.30), use the definition of the variance to write var(a + bY) =
E{[a + bY – E(a + bY)]2} = E{[b(Y – mY)]2} = b2E[(Y – mY)2] = b2s2Y.
64 ChaPteR 2 Review of Probability
To derive Equation (2.31), use the definition of the variance to write
var(aX+bY)=E53(aX+bY)-(am +bm)46 XY
= E53a(X – m ) + b(Y – m )4 62 XY2
= E3a (X – m ) 4 + 2E3ab(X – m )(Y – m )4 2X2XY
2Y2 + E3b (Y – m ) 4
= a2var(X) + 2abcov(X, Y) + b2 var(Y) = a2s2X + 2absXY + b2s2Y,
(2.49)
where the second equality follows by collecting terms, the third equality follows by expanding
ToderiveEquation(2.32),writeE(Y ) = E53(Y – m ) + m ] 6 = E[(Y – m ) 4 + 2YY2Y2
Y
the quadratic, and the fourth equality follows by the definition of the variance and covariance.
2 mYE(Y – mY) + mY2 = sY2 + mY2 because E(Y – mY) = 0.
To derive Equation (2.33), use the definition of the covariance to write
cov(a + bX + cV,Y) = E5[a + bX + cV – E(a + bX + cV)43Y – m ]6
= E5[b(X – m ) + c(V – m )43Y – m ]6 XVY
= E53b(X – m )43Y – m ]6 + E53c(V – m )43Y – m 46 XY VY
= bsXY + csVY, (2.50) ToderiveEquation(2.34),writeE(XY) = E53(X – m ) + m 43(Y – m ) + m ]6 =
which is Equation (2.33). E3(X-m)(Y-m)4+mE(Y-m)+mE(X-m)+m m =s +m m.
XXYY X Y X Y Y X XY XY XY
We now prove the correlation inequality in Equation (2.35); that is, 0 corr (X, Y ) 0 … 1. Let a = -sXY>s2X and b = 1. Applying Equation (2.31), we have that
var(aX + Y) = a2s2X + s2Y + 2asXY
=(-s >s)s +s +2(-s >s)s
XY2X22X2Y XY2XXY =s -s >s.
2Y 2XY 2X
tion (2.51), it must be that s – s > s Ú 0. Rearranging this inequality yields
(2.51) Because var(aX + Y) is a variance, it cannot be negative, so from the final line of Equa-
2Y 2XY 2X
s2 … s2 s2 (covariance inequality). (2.52)
The covariance inequality implies that s >(s s ) … 1 or, equivalently, 2XY 2X2Y
XY XY
0 s > (s s ) 0 … 1, which (using the definition of the correlation) proves the correlation XY XY
inequality, 0corr(XY)0 … 1.
C h3a p t e r
Review of Statistics
Statistics is the science of using data to learn about the world around us. Statisti- cal tools help us answer questions about unknown characteristics of distribu- tions in populations of interest. For example, what is the mean of the distribution of earnings of recent college graduates? Do mean earnings differ for men and women, and, if so, by how much?
These questions relate to the distribution of earnings in the population of workers. One way to answer these questions would be to perform an exhaustive survey of the population of workers, measuring the earnings of each worker and thus finding the population distribution of earnings. In practice, however, such a comprehensive survey would be extremely expensive. The only comprehensive sur- vey of the U.S. population is the decennial census, which cost $13 billion to carry out in 2010. The process of designing the census forms, managing and conducting the surveys, and compiling and analyzing the data takes ten years. Despite this extraordinary commitment, many members of the population slip through the cracks and are not surveyed. Thus a different, more practical approach is needed.
The key insight of statistics is that one can learn about a population distribution by selecting a random sample from that population. Rather than survey the entire U.S. population, we might survey, say, 1000 members of the population, selected at random by simple random sampling. Using statistical methods, we can use this sample to reach tentative conclusions—to draw statistical inferences—about char- acteristics of the full population.
Three types of statistical methods are used throughout econometrics: estima- tion, hypothesis testing, and confidence intervals. Estimation entails computing a “best guess” numerical value for an unknown characteristic of a population distri- bution, such as its mean, from a sample of data. Hypothesis testing entails formulat- ing a specific hypothesis about the population, then using sample evidence to decide whether it is true. Confidence intervals use a set of data to estimate an inter- val or range for an unknown population characteristic. Sections 3.1, 3.2, and 3.3 review estimation, hypothesis testing, and confidence intervals in the context of statistical inference about an unknown population mean.
Most of the interesting questions in economics involve relationships between two or more variables or comparisons between different populations. For example,
65
66 ChapteR 3 Review of Statistics
3.1
Estimation of the Population Mean
Suppose you want to know the mean value of Y (that is, mY) in a population, such as the mean earnings of women recently graduated from college. A natural way to estimate this mean is to compute the sample average Y from a sample of n independently and identically distributed (i.i.d.) observations, Y1, c, Yn (recall that Y1, c, Yn are i.i.d. if they are collected by simple random sam- pling). This section discusses estimation of mY and the properties of Y as an estimator of mY.
Estimators and Their Properties
Estimators. The sample average Y is a natural way to estimate mY, but it is not the only way. For example, another way to estimate mY is simply to use the first observation, Y1. Both Y and Y1 are functions of the data that are designed to estimate mY; using the terminology in Key Concept 3.1, both are estimators of mY. When evaluated in repeated samples, Y and Y1 take on different values (they produce different estimates) from one sample to the next. Thus the estimators Y and Y1 both have sampling distributions. There are, in fact, many estimators of mY, of which Y and Y1 are two examples.
There are many possible estimators, so what makes one estimator “better” than another? Because estimators are random variables, this question can be phrased more precisely: What are desirable characteristics of the sampling distri- bution of an estimator? In general, we would like an estimator that gets as close as possible to the unknown true value, at least in some average sense; in other words, we would like the sampling distribution of an estimator to be as tightly
is there a gap between the mean earnings for male and female recent college grad- uates? In Section 3.4, the methods for learning about the mean of a single popula- tion in Sections 3.1 through 3.3 are extended to compare means in two different populations. Section 3.5 discusses how the methods for comparing the means of two populations can be used to estimate causal effects in experiments. Sections 3.2 through 3.5 focus on the use of the normal distribution for performing hypothesis tests and for constructing confidence intervals when the sample size is large. In some special circumstances, hypothesis tests and confidence intervals can be based on the Student t distribution instead of the normal distribution; these special cir- cumstances are discussed in Section 3.6. The chapter concludes with a discussion of the sample correlation and scatterplots in Section 3.7.
3.1 Estimation of the Population Mean 67
estimators and estimates
Key ConCept
3.1
An estimator is a function of a sample of data to be drawn randomly from a population. An estimate is the numerical value of the estimator when it is actually computed using data from a specific sample. An estimator is a random variable because of randomness in selecting the sample, while an estimate is a nonrandom number.
centered on the unknown value as possible. This observation leads to three specific desirable characteristics of an estimator: unbiasedness (a lack of bias), consis- tency, and efficiency.
Unbiasedness. Supposeyouevaluateanestimatormanytimesoverrepeatedran- domly drawn samples. It is reasonable to hope that, on average, you would get the right answer. Thus a desirable property of an estimator is that the mean of its sampling distribution equals mY; if so, the estimator is said to be unbiased.
To state this concept mathematically, let mnY denote some estimator of mY, such as Y or Y1. The estimator mnY is unbiased if E(mnY) = mY, where E(mnY) is the mean of the sampling distribution of mnY; otherwise, mnY is biased.
Consistency. Another desirable property of an estimator mY is that, when the sample size is large, the uncertainty about the value of mY arising from random variations in the sample is very small. Stated more precisely, a desirable property of mnY is that the probability that it is within a small interval of the true value mY approaches 1 as the sample size increases, that is, mnY is consistent for mY (Key Concept 2.6).
Variance and efficiency. Suppose you have two candidate estimators, mnY and m∼Y, both of which are unbiased. How might you choose between them? One way to do so is to choose the estimator with the tightest sampling distribution. This suggests choosing between mnY and m∼Y by picking the estimator with the smallest variance. If mnY has a smaller variance than m∼Y, then mnY is said to be more efficient than m∼Y. The terminology “efficiency” stems from the notion that if mnY has a smaller variance than m∼Y, then it uses the information in the data more efficiently than does m∼Y.
68 ChapteR 3 Review of Statistics
Bias, Consistency, and efficiency
3.2
Key ConCept
Let mnY be an estimator of mY. Then:
• The bias of mnY is E(mnY) – mY.
• mnY is an unbiased estimator of mY if E(mnY) = mY.
• mnY is a consistent estimator of mY if mnY ¡p mY.
• Let m∼Y be another estimator of mY and suppose that both mn Y and m∼Y are unbiased.
∼
Then mnY is said to be more efficient than mnY if var(mnY) 6 var(mY).
Bias, consistency, and efficiency are summarized in Key Concept 3.2. Properties of Y
How does Y fare as an estimator of mY when judged by the three criteria of bias, consistency, and efficiency?
Bias and consistency. The sampling distribution of Y has already been examined in Sections 2.5 and 2.6. As shown in Section 2.5, E(Y ) = mY, so Y is an unbiased estimator of mY. Similarly, the law of large numbers (Key Concept 2.6) states that Y ¡p mY; that is, Y is consistent.
Efficiency. What can be said about the efficiency of Y? Because efficiency entails a comparison of estimators, we need to specify the estimator or estimators to which Y is to be compared.
We start by comparing the efficiency of Y to the estimator Y1. Because Y1, c, Yn are i.i.d., the mean of the sampling distribution of Y1 is E(Y1) = mY; thus Y1 is an unbiased estimator of mY. Its variance is var(Y1) = s2Y. From Section
2Y
2.5, the variance of Y is s >n. Thus, for n Ú 2, the variance of Y is less than the
variance of Y1; that is, Y is a more efficient estimator than Y1, so, according to the criterion of efficiency, Y should be used instead of Y1. The estimator Y1 might strike you as an obviously poor estimator—why would you go to the trouble of collecting a sample of n observations only to throw away all but the first?—and the concept of efficiency provides a formal way to show that Y is a more desirable estimator than Y1.
3.1 Estimation of the Population Mean 69
efficiency of Y: Y Is BLUe
Let mnY be an estimator of mY that is a weighted average of Y1, c, Yn, that is,
Key ConCept
3.3
n
mn = (1>n)g a Y , where a , c, a are nonrandom constants. If mn is un-
Yi=1ii1n Y biased, then var(Y) 6 var(mnY) unless mnY = Y. Thus Y is the Best Linear Unbiased
Estimator (BLUE); that is, Y is the most efficient estimator of mY among all unbiased estimators that are weighted averages of Y1, c, Yn.
What about a less obviously poor estimator? Consider the weighted average in which the observations are alternately weighted by 12 and 32:
Y∼=1a1Y1 +3Y2 +1Y3 +3Y4 + g +1Yn-1 +3Ynb, (3.1) n222222
mean of Y is m and its variance is var(Y) = 1.25s >n (Exercise 3.11). Thus Y is ∼∼∼
where the number of observations n is assumed to be even for convenience. The ∼Y ∼2Y ∼
unbiased and, because var(Y) S 0 as n S ∞, Y is consistent. However, Y has a larger variance than Y. Thus Y is more efficient than Y∼.
The estimators Y, Y1, and Y∼ have a common mathematical structure: They are weighted averages of Y1, c, Yn. The comparisons in the previous two para- graphs show that the weighted averages Y1 and Y∼ have larger variances than Y. In fact, these conclusions reflect a more general result: Y is the most efficient estimator of all unbiased estimators that are weighted averages of Y1, c, Yn. Said differently, Y is the Best Linear Unbiased Estimator (BLUE); that is, it is the most efficient (best) estimator among all estimators that are unbiased and are linear functions of Y1, c, Yn. This result is stated in Key Concept 3.3 and is proved in Chapter 5.
Y is the least squares estimator of mY. The sample average Y provides the best fit to the data in the sense that the average squared differences between the observa- tions and Y are the smallest of all possible estimators.
Consider the problem of finding the estimator m that minimizes an 2
(Yi – m) , (3.2) i=1
which is a measure of the total squared gap or distance between the estimator m and the sample points. Because m is an estimator of E(Y), you can think of it as a
70 ChapteR 3 Review of Statistics Landon Wins!
S hortly before the 1936 U.S. presidential election, the Literary Gazette published a poll indicating that Alf M. Landon would defeat the incumbent, Franklin D. Roosevelt, by a landslide—57% to 43%. The Gazette was right that the election was a land- slide, but it was wrong about the winner: Roosevelt
won by 59% to 41%!
How could the Gazette have made such a big
mistake? The Gazette’s sample was chosen from telephone records and automobile registration
files. But in 1936 many households did not have cars or telephones, and those that did tended to be richer—and were also more likely to be Republican. Because the telephone survey did not sample randomly from the population but instead undersampled Democrats, the estimator was biased and the Gazette made an embarrass- ing mistake.
Do you think surveys conducted using social media might have a similar problem with bias?
prediction of the value of Yi, so the gap Yi – m can be thought of as a prediction mistake. The sum of squared gaps in Expression (3.2) can be thought of as the sum of squared prediction mistakes.
The estimator m that minimizes the sum of squared gaps Yi – m in Expres- sion (3.2) is called the least squares estimator. One can imagine using trial and error to solve the least squares problem: Try many values of m until you are satis- fied that you have the value that makes Expression (3.2) as small as possible. Alternatively, as is done in Appendix 3.2, you can use algebra or calculus to show that choosing m = Y minimizes the sum of squared gaps in Expression (3.2) so that Y is the least squares estimator of mY.
The Importance of Random Sampling
We have assumed that Y1, c, Yn are i.i.d. draws, such as those that would be obtained from simple random sampling. This assumption is important because nonrandom sampling can result in Y being biased. Suppose that, to estimate the monthly national unemployment rate, a statistical agency adopts a sampling scheme in which interviewers survey working-age adults sitting in city parks at 10 a.m. on the second Wednesday of the month. Because most employed people are at work at that hour (not sitting in the park!), the unemployed are overly represented in the sample, and an estimate of the unemployment rate based on this sampling plan would be biased. This bias arises because this sampling scheme overrepresents, or oversamples, the unemployed members of the population. This example is fictitious, but the “Landon Wins!” box gives a real-world example of biases introduced by sampling that is not entirely random.
3.2 Hypothesis Tests Concerning the Population Mean 71
It is important to design sample selection schemes in a way that minimizes bias. Appendix 3.1 includes a discussion of what the Bureau of Labor Statistics actually does when it conducts the U.S. Current Population Survey (CPS), the survey it uses to estimate the monthly U.S. unemployment rate.
3.2
Hypothesis Tests Concerning the Population Mean
Many hypotheses about the world around us can be phrased as yes/no questions. Do the mean hourly earnings of recent U.S. college graduates equal $20 per hour? Are mean earnings the same for male and female college graduates? Both these questions embody specific hypotheses about the population distribution of earn- ings. The statistical challenge is to answer these questions based on a sample of evidence. This section describes hypothesis tests concerning the population mean (Does the population mean of hourly earnings equal $20?). Hypothesis tests involving two populations (Are mean earnings the same for men and women?) are taken up in Section 3.4.
Null and Alternative Hypotheses
The starting point of statistical hypotheses testing is specifying the hypothesis to be tested, called the null hypothesis. Hypothesis testing entails using data to com- pare the null hypothesis to a second hypothesis, called the alternative hypothesis, that holds if the null does not.
The null hypothesis is that the population mean, E(Y), takes on a specific value, denoted mY,0. The null hypothesis is denoted H0 and thus is
H0: E(Y) = mY,0. (3.3)
For example, the conjecture that, on average in the population, college graduates earn $20 per hour constitutes a null hypothesis about the population distribution of hourly earnings. Stated mathematically, if Y is the hourly earning of a randomly selected recent college graduate, then the null hypothesis is that E(Y) = 20; that is, mY,0 = 20 in Equation (3.3).
The alternative hypothesis specifies what is true if the null hypothesis is not. The most general alternative hypothesis is that E(Y) ≠ mY,0, which is called a two-sided alternative hypothesis because it allows E(Y) to be either less than or greater than mY,0. The two-sided alternative is written as
H1: E(Y) ≠ mY,0 (two@sided alternative). (3.4)
72 ChapteR 3 Review of Statistics
One-sided alternatives are also possible, and these are discussed later in this section.
The problem facing the statistician is to use the evidence in a randomly selected sample of data to decide whether to accept the null hypothesis H0 or to reject it in favor of the alternative hypothesis H1. If the null hypothesis is “accepted,” this does not mean that the statistician declares it to be true; rather, it is accepted tentatively with the recognition that it might be rejected later based on additional evidence. For this reason, statistical hypothesis testing can be posed as either rejecting the null hypothesis or failing to do so.
The p-Value
In any given sample, the sample average Y will rarely be exactly equal to the hypothesized value mY,0. Differences between Y and mY,0 can arise because the true mean in fact does not equal mY,0 (the null hypothesis is false) or because the true mean equals mY,0 (the null hypothesis is true) but Y differs from mY,0 because of random sampling. It is impossible to distinguish between these two possibilities with certainty. Although a sample of data cannot provide conclusive evidence about the null hypothesis, it is possible to do a probabilistic calculation that permits testing the null hypothesis in a way that accounts for sampling uncertainty. This calculation involves using the data to compute the p-value of the null hypothesis.
The p-value, also called the significance probability, is the probability of draw- ing a statistic at least as adverse to the null hypothesis as the one you actually com- puted in your sample, assuming the null hypothesis is correct. In the case at hand, the p-value is the probability of drawing Y at least as far in the tails of its distribu- tion under the null hypothesis as the sample average you actually computed.
For example, suppose that, in your sample of recent college graduates, the average wage is $22.64. The p-value is the probability of observing a value of Y at least as different from $20 (the population mean under the null) as the observed value of $22.64 by pure random sampling variation, assuming that the null hypoth- esis is true. If this p-value is small, say 0.5%, then it is very unlikely that this sample would have been drawn if the null hypothesis is true; thus it is reasonable to conclude that the null hypothesis is not true. By contrast, if this p-value is large, say 40%, then it is quite likely that the observed sample average of $22.64 could have arisen just by random sampling variation if the null hypothesis is true; accordingly, the evidence against the null hypothesis is weak in this probabilistic sense, and it is reasonable not to reject the null hypothesis.
To state the definition of the p-value mathematically, let Yact denote the value of the sample average actually computed in the data set at hand and let PrH0
3.2 Hypothesis Tests Concerning the Population Mean 73 denote the probability computed under the null hypothesis (that is, computed
assuming that E(Yi) = mY,0). The p-value is
p@value=Pr 30Y-m 070Y -m 04. (3.5)
That is, the p-value is the area in the tails of the distribution of Y under the null hypothesis beyond mY,0 { Yact – mY,0 . If the p-value is large, then the observed value Yact is consistent with the null hypothesis, but if the p-value is small, it is not.
To compute the p-value, it is necessary to know the sampling distribution of
Y under the null hypothesis. As discussed in Section 2.6, when the sample size is
small this distribution is complicated. However, according to the central limit
theorem, when the sample size is large, the sampling distribution of Y is well
H0 Y,0 act Y,0
approximated by a normal distribution. Under the null hypothesis the mean of
N(m , s ), where s = s >n. This large-sample normal approximation makes Y,0Y2 Y22Y
it possible to compute the p-value without needing to know the population distri- bution of Y, as long as the sample size is large. The details of the calculation, however, depend on whether s2Y is known.
Calculating the p-Value When sY Is Known
this normal distribution is mY,0, so under the null hypothesis Y is distributed
The calculation of the p-value when sY is known is summarized in Figure 3.1. If
of Y is N(m , s ), where s s >n. Thus, under the null hypothesis, the stan- Y,0 Y2 Y2 = 2Y
the sample size is large, then under the null hypothesis the sampling distribution
dardized version of Y, (Y – m )>s , has a standard normal distribution. The Y,0 Y act
p-value is the probability of obtaining a value of Y farther from mY,0 than Y
(Y – m ) > s greater than (Y – m ) > s in absolute value. This probability Y,0 Y act Y,0 Y
under the null hypothesis or, equivalently, is the probability of obtaining
222222
is the shaded area shown in Figure 3.1. Written mathematically, the shaded tail probability in Figure 3.1 (that is, the p-value) is
p@value=Pr aY-mY,0 7 Yact-mY,0 b=2Φa-Yact-mY,0 b,(3.6)
H0
where Φ is the standard normal cumulative distribution function. That is, the
{0Y -m 0>s Y,0 Y.
sss
YYY
p-value is the area in the tails of a standard normal distribution outside
act
The formula for the p-value in Equation (3.6) depends on the variance of the population distribution, s2Y. In practice, this variance is typically unknown. [An exception is when Yi is binary so that its distribution is Bernoulli, in which case
74 ChapteR 3 Review of Statistics
Figure 3.1 Calculating a p-value
The p-value is the
probability of drawing
a value of Y that differs
from mY,0 by at least as
much as Y act. In large
samples, Y is distrib-
–act 0 – Y –mY,0
s Y–
–act z Y –mY,0
s Y–
The p-value is the shaded area in the graph
N(0, 1)
uted N(mY,0, s2Y), under
(Y – m )>s is distrib- Y,0 Y
the null hypothesis, so
uted N(0, 1). Thus the
p-value is the shaded
{(Y -m)>s. Y,0 Y
the variance is determined by the null hypothesis; see Equation (2.7) and Exer- cise 3.2.] Because in general s2Y must be estimated before the p-value can be computed, we now turn to the problem of estimating s2Y.
The Sample Variance, Sample Standard Deviation,
and Standard Error
The sample variance s2Y is an estimator of the population variance s2Y, the sample standard deviation sY is an estimator of the population standard deviation sY, and the standard error of the sample average Y is an estimator of the standard devia- tion of the sampling distribution of Y.
standard normal tail
probability outside
act
The sample variance and standard deviation. The sample variance, s2Y, is 21an 2
sY = n – 1 i=1(Yi – Y) . (3.7) The sample standard deviation, sY, is the square root of the sample variance.
Electronic Publishing Services Inc.
The formula for the sample variance is much like the formula for the popula- Stock/Watson, Econometrics 12e
tion variance. The population variance, E(Y – mY) , is the average value of 2 STOC.ITEM.0009
(Y – mY) in the population distribution. Similarly, the sample variance is the Fig. 032.01
sample average of (Yi – mY) , i = 1, c, n, with two modifications: First, mY is replaced by Y, and second, t1hsetaPvreoroafge uses t2hnediPvriosorf n – 1 in3srtdeaPdrooffn. Final
3.2 Hypothesis Tests Concerning the Population Mean 75
the Standard error of Y
The standard error of Y is an estimator of the standard deviation of Y. The stan-
Key ConCept
3.4
SE(Y) = sn = s >2n. (3.8) YY
dard error of Y is denoted SE(Y) or snY. When Y1, c,Yn are i.i.d.,
The reason for the first modification—replacing mY by Y—is that mY is
unknown and thus must be estimated; the natural estimator of mY is Y. The reason
for the second modification—dividing by n – 1 instead of by n—is that estimating
inExercise3.18,E3(Y -Y)4=3(n-1)>n4s .ThusEg (Y -Y) = i2 2Yni=1i2
mY by Y introduces a small downward bias in (Yi – Y )2. Specifically, as is shown
nE3(Y – Y) 4 = (n – 1)s .Dividingbyn – 1inEquation(3.7)insteadofn i2 2Y
corrects for this small downward bias, and as a result s2Y is unbiased.
Dividing by n – 1 in Equation (3.7) instead of n is called a degrees of freedom correction: Estimating the mean uses up some of the information—that is, uses up 1 “degree of freedom”—in the data, so that only n – 1 degrees of freedom remain.
Consistencyofthesamplevariance. Thesamplevarianceisaconsistentestimator of the population variance:
s2Y ¡ s2Y. (3.9)
In other words, the sample variance is close to the population variance with high probability when n is large.
The result in Equation (3.9) is proven in Appendix 3.3 under the assumptions that Y1, c, Yn are i.i.d. and Yi has a finite fourth moment; that is, E(Y4i ) 6 ∞ . Intuitively, the reason that s2Y is consistent is that it is a sample average, so s2Y obeys the law of large numbers. But for s2Y to obey the law of large numbers in Key Concept 2.6, (Yi – mY)2 must have finite variance, which in turn means that E(Y4i ) must be finite; in other words, Yi must have a finite fourth moment.
ThestandarderrorofY. Becausethestandarddeviationofthesamplingdistribu- tion of Y is s = s > 1n, Equation (3.9) justifies using sY > 1n as an estimator of
YY
s . The estimator of s , s > 1n, is called the standard error of Y and is denoted
YYY
SE(Y) or snY (the caret “^” over the symbol means that it is an estimator of sY).
The standard error of Y is summarized as in Key Concept 3.4.
76 ChapteR 3 Review of Statistics
When Y1, c, Yn are i.i.d. draws from a Bernoulli distribution with success probability p, the formula for the variance of Y simplifies to p(1 – p)>n (see Exercise 3.2). The formula for the standard error also takes on a simple form that depends only on Y and n: SE(Y ) = 2Y(1 – Y)>n.
Calculating the p-Value When sY Is Unknown
Because s2Y is a consistent estimator of s2Y, the p-value can be computed by replac- ing sY in Equation (3.6) by the standard error, SE(Y ) = sn Y . That is, when sY is unknown and Y1, c, Yn are i.i.d., the p-value is calculated using the formula
22 p@value = 2Φa- Yact – mY,0 b.
(3.10)
The t-Statistic
The standardized sample average (Y – m )>SE(Y) plays a central role in testing
Y,0
statistical hypotheses and has a special name, the t-statistic or t-ratio:
t = Y – mY,0. (3.11) SE(Y)
In general, a test statistic is a statistic used to perform a hypothesis test. The t-statistic is an important example of a test statistic.
SE(Y)
Large-sampledistributionofthet-statistic. Whennislarge,s2Yisclosetos2Ywith
as the distribution of (Y – m ) > s , which in turn is well approximated by the
high probability. Thus the distribution of the t-statistic is approximately the same
Y,0
standard normal distribution when n is large because of the central limit theorem
Y
(Key Concept 2.7). Accordingly, under the null hypothesis,
t is approximately distributed N(0,1) for large n. (3.12)
The formula for the p-value in Equation (3.10) can be rewritten in terms of the t-statistic. Let t act denote the value of the t-statistic actually computed:
tact = Y act – mY,0. (3.13) SE(Y)
3.2 Hypothesis Tests Concerning the Population Mean 77 Accordingly, when n is large, the p-value can be calculated using
p@value = 2Φ(-tact ). (3.14)
s = $18.14. Then the standard error of Y is s > 2n = 18.14 > 2200 = 1.28. The YY
As a hypothetical example, suppose that a sample of n = 200 recent college grad-
uates is used to test the null hypothesis that the mean wage, E(Y), is $20 per hour.
The sample average wage is Yact = $22.64, and the sample standard deviation is
act
value of the t-statistic is t = (22.64 – 20)>1.28 = 2.06. From Appendix Table 1,
the p-value is 2Φ(-2.06) = 0.039, or 3.9%. That is, assuming the null hypothesis to be true, the probability of obtaining a sample average at least as different from the null as the one actually computed is 3.9%.
Hypothesis Testing with a Prespecified
Significance Level
When you undertake a statistical hypothesis test, you can make two types of mistakes: You can incorrectly reject the null hypothesis when it is true, or you can fail to reject the null hypothesis when it is false. Hypothesis tests can be performed without computing the p-value if you are willing to specify in advance the probability you are willing to tolerate of making the first kind of mistake—that is, of incorrectly rejecting the null hypothesis when it is true. If you choose a prespecified probability of rejecting the null hypothesis when it is true (for example, 5%), then you will reject the null hypothesis if and only if the p-value is less than 0.05. This approach gives preferential treatment to the null hypothesis, but in many practical situations this preferential treatment is appropriate.
Hypothesistestsusingafixedsignificancelevel. Supposeithasbeendecidedthat the hypothesis will be rejected if the p-value is less than 5%. Because the area under the tails of the standard normal distribution outside { 1.96 is 5%, this gives a simple rule:
Reject H0 if t act 7 1.96. (3.15)
That is, reject if the absolute value of the t-statistic computed from the sample is greater than 1.96. If n is large enough, then under the null hypothesis the t-statistic has a N(0, 1) distribution. Thus the probability of erroneously rejecting the null hypothesis (rejecting the null hypothesis when it is in fact true) is 5%.
78 ChapteR 3 Review of Statistics
the terminology of hypothesis testing
3.5
Key ConCept
A statistical hypothesis test can make two types of mistakes: a type I error, in which the null hypothesis is rejected when in fact it is true, and a type II error, in which the null hypothesis is not rejected when in fact it is false. The prespecified rejection probability of a statistical hypothesis test when the null hypothesis is true—that is, the prespecified probability of a type I error—is the significance level of the test. The critical value of the test statistic is the value of the statistic for which the test just rejects the null hypothesis at the given significance level. The set of values of the test statistic for which the test rejects the null hypothesis is the rejection region, and the values of the test statistic for which it does not reject the null hypothesis is the acceptance region. The probability that the test actually incorrectly rejects the null hypothesis when it is true is the size of the test, and the probability that the test correctly rejects the null hypothesis when the alternative is true is the power of the test.
The p-value is the probability of obtaining a test statistic, by random sampling variation, at least as adverse to the null hypothesis value as is the statistic actually observed, assuming that the null hypothesis is correct. Equivalently, the p-value is the smallest significance level at which you can reject the null hypothesis.
This framework for testing statistical hypotheses has some specialized termi- nology, summarized in Key Concept 3.5. The significance level of the test in Equa- tion (3.15) is 5%, the critical value of this two-sided test is 1.96, and the rejection region is the values of the t-statistic outside ±1.96. If the test rejects at the 5% significance level, the population mean mY is said to be statistically significantly different from mY,0 at the 5% significance level.
Testing hypotheses using a prespecified significance level does not require computing p-values. In the previous example of testing the hypothesis that the mean earnings of recent college graduates is $20 per hour, the t-statistic was 2.06. This value exceeds 1.96, so the hypothesis is rejected at the 5% level. Although performing the test with a 5% significance level is easy, reporting only whether the null hypothesis is rejected at a prespecified significance level conveys less information than reporting the p-value.
What significance level should you use in practice? In many cases, statisticians and econometricians use a 5% significance level. If you were to test many statistical
3.2 Hypothesis Tests Concerning the Population Mean 79
testing the hypothesis E(Y) = mY,0 against the alternative E(Y) ≠ mY,0
1. Compute the standard error of Y, SE(Y) [Equation (3.8)].
2. Compute the t-statistic [Equation (3.13)].
3. Compute the p-value [Equation (3.14)]. Reject the hypothesis at the 5% sig- nificance level if the p-value is less than 0.05 (equivalently, if t act 7 1.96).
Key ConCept
3.6
hypotheses at the 5% level, you would incorrectly reject the null on average once in 20 cases. Sometimes a more conservative significance level might be in order. For example, legal cases sometimes involve statistical evidence, and the null hypothesis could be that the defendant is not guilty; then one would want to be quite sure that a rejection of the null (conclusion of guilt) is not just a result of random sample variation. In some legal settings, the significance level used is 1%, or even 0.1%, to avoid this sort of mistake. Similarly, if a government agency is considering permitting the sale of a new drug, a very conservative standard might be in order so that consumers can be sure that the drugs available in the market actually work.
Being conservative, in the sense of using a very low significance level, has a cost: The smaller the significance level, the larger the critical value and the more difficult it becomes to reject the null when the null is false. In fact, the most con- servative thing to do is never to reject the null hypothesis—but if that is your view, then you never need to look at any statistical evidence for you will never change your mind! The lower the significance level, the lower the power of the test. Many economic and policy applications can call for less conservatism than a legal case, so a 5% significance level is often considered to be a reasonable compromise.
Key Concept 3.6 summarizes hypothesis tests for the population mean against the two-sided alternative.
One-Sided Alternatives
In some circumstances, the alternative hypothesis might be that the mean exceeds mY,0. For example, one hopes that education helps in the labor market, so the relevant alternative to the null hypothesis that earnings are the same for college graduates and non–college graduates is not just that their earnings differ, but
80 ChapteR 3 Review of Statistics
rather that graduates earn more than nongraduates. This is called a one-sided
alternative hypothesis and can be written
H1 : E(Y) 7 mY,0 (one@sided alternative). (3.16)
The general approach to computing p-values and to hypothesis testing is the same for one-sided alternatives as it is for two-sided alternatives, with the modification that only large positive values of the t-statistic reject the null hypothesis rather than values that are large in absolute value. Specifically, to test the one-sided hypothesis in Equation (3.16), construct the t-statistic in Equation (3.13). The p-value is the area under the standard normal distribution to the right of the cal- culated t-statistic. That is, the p-value, based on the N(0, 1) approximation to the distribution of the t-statistic, is
p@value = PrH0(Z 7 tact) = 1 – Φ(tact). (3.17)
The N(0, 1) critical value for a one-sided test with a 5% significance level is 1.64. The rejection region for this test is all values of the t-statistic exceeding 1.64.
The one-sided hypothesis in Equation (3.16) concerns values of mY exceeding mY,0. If instead the alternative hypothesis is that E(Y) 6 mY,0, then the discussion of the previous paragraph applies except that the signs are switched; for example, the 5% rejection region consists of values of the t-statistic less than −1.64.
3.3
Confidence Intervals
for the Population Mean
Because of random sampling error, it is impossible to learn the exact value of the population mean of Y using only the information in a sample. However, it is pos- sible to use data from a random sample to construct a set of values that contains the true population mean mY with a certain prespecified probability. Such a set is called a confidence set, and the prespecified probability that mY is contained in this set is called the confidence level. The confidence set for mY turns out to be all the possible values of the mean between a lower and an upper limit, so that the confidence set is an interval, called a confidence interval.
Here is one way to construct a 95% confidence set for the population mean. Begin by picking some arbitrary value for the mean; call it mY,0. Test the null hypoth- esisthatmY = mY,0againstthealternativethatmY ≠ mY,0bycomputingthet-statistic; if its absolute value is less than 1.96, this hypothesized value mY,0 is not rejected at the 5% level, and write down this nonrejected value mY,0. Now pick another arbitrary value of mY,0 and test it; if you cannot reject it, write down this value on your list.
3.3 Confidence Intervals for the Population Mean 81
Confidence Intervals for the population Mean
Key ConCept
3.7
A 95% two-sided confidence interval for mY is an interval constructed so that it contains the true value of mY in 95% of all possible random samples. When the sample size n is large, 95%, 90%, and 99% confidence intervals for mY are
95% confidence interval for m = 5Y { 1.96SE(Y )6.
90% confidence interval for mY = 5Y { 1.64SE(Y )6.
99% confidence interval for mY = 5Y { 2.58SE(Y )6. Y
Do this again and again; indeed, do so for all possible values of the population mean. Continuing this process yields the set of all values of the population mean that cannot be rejected at the 5% level by a two-sided hypothesis test.
This list is useful because it summarizes the set of hypotheses you can and cannot reject (at the 5% level) based on your data: If someone walks up to you with a specific number in mind, you can tell him whether his hypothesis is rejected or not simply by looking up his number on your handy list. A bit of clever reason- ing shows that this set of values has a remarkable property: The probability that it contains the true value of the population mean is 95%.
The clever reasoning goes like this: Suppose the true value of mY is 21.5 (although we do not know this). Then Y has a normal distribution centered on 21.5, and the t-statistic testing the null hypothesis mY = 21.5 has a N(0, 1) distribu- tion. Thus, if n is large, the probability of rejecting the null hypothesis mY = 21.5 at the 5% level is 5%. But because you tested all possible values of the population mean in constructing your set, in particular you tested the true value, mY = 21.5. In 95% of all samples, you will correctly accept 21.5; this means that in 95% of all samples, your list will contain the true value of mY. Thus the values on your list constitute a 95% confidence set for mY.
This method of constructing a confidence set is impractical, for it requires you to test all possible values of mY as null hypotheses. Fortunately, there is a much easier approach. According to the formula for the t-statistic in Equation (3.13), a trial value of mY,0 is rejected at the 5% level if it is more than 1.96 standard errors away from Y. Thus the set of values of mY that are not rejected at the 5% level consists of those values within { 1.96SE(Y) of Y; that is, a 95% confidence interval for mY is Y – 1.96SE(Y) … mY … Y + 1.96SE(Y). Key Concept 3.7 sum- marizes this approach.
82 ChapteR 3 Review of Statistics
3.4
Comparing Means from Different Populations
Do recent male and female college graduates earn the same amount on average? This question involves comparing the means of two different population distribu- tions. This section summarizes how to test hypotheses and how to construct con- fidence intervals for the difference in the means from two different populations.
Hypothesis Tests for the Difference
Between Two Means
To illustrate a test for the difference between two means, let mw be the mean hourly earning in the population of women recently graduated from college and let mm be the population mean for recently graduated men. Consider the null hypothesis that mean earnings for these two populations differ by a certain amount, say d0. Then the null hypothesis and the two-sided alternative hypothesis are
H0:mm – mw = d0 vs.H1:mm – mw ≠ d0. (3.18)
The null hypothesis that men and women in these populations have the same mean earnings corresponds to H0 in Equation (3.18) with d0 = 0.
As an example, consider the problem of constructing a 95% confidence inter- val for the mean hourly earnings of recent college graduates using a hypothetical random sample of 200 recent college graduates where Y = $22.64 and SE(Y) = 1.28. The 95% confidence interval for mean hourly earnings is 22.64 { 1.96 * 1.28 = 22.64 { 2.51 = 3$20.13, $25.154.
This discussion so far has focused on two-sided confidence intervals. One could instead construct a one-sided confidence interval as the set of values of mY that cannot be rejected by a one-sided hypothesis test. Although one-sided confi- dence intervals have applications in some branches of statistics, they are uncom- mon in applied econometric analysis.
Coverageprobabilities. Thecoverageprobabilityofaconfidenceintervalforthe population mean is the probability, computed over all possible random samples, that it contains the true population mean.
3.4 Comparing Means from Different Populations 83
Because these population means are unknown, they must be estimated from samples of men and women. Suppose we have samples of nm men and nw women drawn at random from their populations. Let the sample average annual earnings be Ym for men and Yw for women. Then an estimator of mm – mw is Ym – Yw.
To test the null hypothesis that mm – mw = d0 using Ym – Yw, we need to
theorem, approximately distributed N(m , s >n ), where s is the population m2mm 2m
recall from Section 2.4 that a weighted average of two normal random variables
know the distribution of Ym – Yw. Recall that Ym is, according to the central limit
variance of earnings for men. Similarly, Yw is approximately distributed N(m , s >n ) where s is the population variance of earnings for women. Also,
w2ww 2w
is itself normally distributed. Because Ym and Yw are constructed from different
Y – Y is distributed N3m – m , (s >n ) + (s >n )4. m w m w 2m m 2w w
randomly selected samples, they are independent random variables. Thus
If s2m and s2w are known, then this approximate normal distribution can be used to compute p-values for the test of the null hypothesis that mm – mw = d0. In practice, however, these population variances are typically unknown so they must be estimated. As before, they can be estimated using the sample variances, s2m and s2w where s2m is defined as in Equation (3.7), except that the statistic is com- puted only for the men in the sample, and s2w is defined similarly for the women. Thus the standard error of Ym – Yw is
SE(Ym – Yw) = s2m + s2w. (3.19) Cn n
For a simplified version of Equation (3.19) when Y is a Bernoulli random variable, see Exercise 3.15.
The t-statistic for testing the null hypothesis is constructed analogously to the t-statistic for testing a hypothesis about a single population mean, by subtracting the null hypothesized value of mm – mw from the estimator Ym – Yw and dividing the result by the standard error of Ym – Yw:
t = (Ym – Yw) – d0 (t@statistic for comparing two means). (3.20) SE(Ym – Yw)
If both nm and nw are large, then this t-statistic has a standard normal distribution when the null hypothesis is true.
Because the t-statistic in Equation (3.20) has a standard normal distribution under the null hypothesis when nm and nw are large, the p-value of the two-sided
mw
84 ChapteR 3 Review of Statistics
3.5
Differences-of-Means Estimation of Causal Effects Using Experimental Data
Recall from Section 1.2 that a randomized controlled experiment randomly selects subjects (individuals or, more generally, entities) from a population of interest, then randomly assigns them either to a treatment group, which receives the exper- imental treatment, or to a control group, which does not receive the treatment. The difference between the sample means of the treatment and control groups is an estimator of the causal effect of the treatment.
test is computed exactly as it was in the case of a single population. That is, the p-value is computed using Equation (3.14).
To conduct a test with a prespecified significance level, simply calculate the t-statistic in Equation (3.20) and compare it to the appropriate critical value. For example, the null hypothesis is rejected at the 5% significance level if the absolute value of the t-statistic exceeds 1.96.
If the alternative is one-sided rather than two-sided (that is, if the alternative is that mm – mw 7 d0), then the test is modified as outlined in Section 3.2. The p-value is computed using Equation (3.17), and a test with a 5% significance level rejects when t 7 1.64.
Confidence Intervals for the Difference
Between Two Population Means
The method for constructing confidence intervals summarized in Section 3.3 extends to constructing a confidence interval for the difference between the means, d = mm – mw. Because the hypothesized value d0 is rejected at the 5% level if t 7 1.96, d0 will be in the confidence set if t … 1.96. But t … 1.96 means that the estimated difference, Ym – Yw, is less than 1.96 standard errors away from d0. Thus the 95% two-sided confidence interval for d consists of those values of d within { 1.96 standard errors of Ym – Yw:
95% confidence interval for d = mm – mw is
(Ym – Yw) { 1.96SE(Ym – Yw). (3.21)
With these formulas in hand, the box “The Gender Gap of Earnings of College Graduates in the United States” contains an empirical investigation of gender differences in earnings of U.S. college graduates.
3.5 Differences-Of-Means Estimation of Causal Effects Using Experimental Data 85
The Causal Effect as a Difference
of Conditional Expectations
The causal effect of a treatment is the expected effect on the outcome of interest of the treatment as measured in an ideal randomized controlled experiment. This effect can be expressed as the difference of two conditional expectations. Spe- cifically, the causal effect on Y of treatment level x is the difference in the condi- tional expectations, E(Y 0X = x) – E(Y 0X = 0), where E(Y 0X = x) is the expected value of Y for the treatment group (which receives treatment level X = x) in an ideal randomized controlled experiment and E(Y 0 X = 0) is the expected value of Y for the control group (which receives treatment level X = 0). In the context of experiments, the causal effect is also called the treatment effect. If there are only two treatment levels (that is, if the treatment is binary), then we can let X = 0 denote the control group and X = 1 denote the treatment group. If the treatment is binary treatment, then the causal effect (that is, the treatment effect) is E(Y 0 X = 1) – E(Y 0 X = 0) in an ideal randomized con- trolled experiment.
Estimation of the Causal Effect Using
Differences of Means
If the treatment in a randomized controlled experiment is binary, then the causal effect can be estimated by the difference in the sample average outcomes between the treatment and control groups. The hypothesis that the treatment is ineffective is equivalent to the hypothesis that the two means are the same, which can be tested using the t-statistic for comparing two means, given in Equation (3.20). A 95% confidence interval for the difference in the means of the two groups is a 95% confidence interval for the causal effect, so a 95% confidence interval for the causal effect can be constructed using Equation (3.21).
A well-designed, well-run experiment can provide a compelling estimate of a causal effect. For this reason, randomized controlled experiments are commonly conducted in some fields, such as medicine. In economics, however, experiments tend to be expensive, difficult to administer, and, in some cases, ethically ques- tionable, so they are used less often. For this reason, econometricians sometimes study “natural experiments,” also called quasi-experiments, in which some event unrelated to the treatment or subject characteristics has the effect of assigning different treatments to different subjects as if they had been part of a randomized controlled experiment. The box “A Novel Way to Boost Retirement Savings” provides an example of such a quasi-experiment that yielded some surprising conclusions.
86 ChapteR 3 Review of Statistics
the gender gap of earnings of College graduates in the united States
T he box in Chapter 2 “The Distribution of Earn- ings in the United States in 2012” shows that, on average, male college graduates earn more than female college graduates. What are the recent trends in this “gender gap” in earnings? Social norms and laws governing gender discrimination in the work- place have changed substantially in the United States. Is the gender gap in earnings of college graduates
stable, or has it diminished over time?
Table 3.1 gives estimates of hourly earnings
for college-educated full-time workers ages 25–34 in the United States in 1992, 1996, 2000, 2004, 2008, and 2012, using data collected by the Cur- rent Population Survey. Earnings for 1992, 1996, 2000, 2004, and 2008 were adjusted for inflation by putting them in 2012 dollars using the Consumer Price Index (CPI).1 In 2012, the average hourly
earnings of the 2004 men surveyed was $25.30,
and the standard deviation of earnings for men
was $12.09. The average hourly earnings in 2012
of the 1951 women surveyed was $21.50, and the
$0.35 (= 212.09 >2004 + 9.99 >1951). The 95% con- fidence interval for the gender gap in earnings in 2012 is 3.80 { 1.96 * 0.35 = ($3.11, $4.49).
The results in Table 3.1 suggest four conclusions. First, the gender gap is large. An hourly gap of $3.80 might not sound like much, but over a year it adds up to $7600, assuming a 40-hour workweek and 50 paid weeks per year. Second, from 1992 to 2012, the estimated gender gap increased by $0.36 per hour in real terms, from $3.44 per hour to $3.80 per hour;
standard deviation of earnings was $9.99. Thus the
estimate of the gender gap in earnings for 2012 is
$3.80 (= $25.30 – $21.50), with a standard error of 22
taBLe 3.1
trends in hourly earnings in the United States of Working College Graduates, ages 25–34, 1992 to 2012, in 2012 Dollars
year Ym sm nm
1992 24.83 10.85 1594
1996 23.97 10.79 1380
2000 26.55 12.38 1303
2004 26.80 12.81 1894
2008 26.63 12.57 1839
2012 25.30 12.09 2004
Yw sw nw
21.39 8.39 1368
20.26 8.48 1230
22.13 9.98 1181
22.43 9.99 1735
22.26 10.30 1871
21.50 9.99 1951
Ym -Yw
3.44**
3.71**
4.42**
4.37**
4.36**
3.80**
SE(Ym -Yw)
0.35
0.38
0.45
0.38
0.38
0.35
95% Confidence interval ford
2.75–4.14
2.97–4.46
3.54–5.30
3.63–5.12
3.62–5.10
3.11–4.49
Men Women
Difference, Men vs. Women
These estimates are computed using data on all full-time workers ages 25–34 surveyed in the Current Population Survey conducted in March of the next year (for example, the data for 2012 were collected in March 2013). The difference is sig- nificantly different from zero at the **1% significance level.
(continued )
however, this increase is not statistically significant at the 5% significance level (Exercise 3.17). Third, the gap is large if it is measured instead in percent- age terms: According to the estimates in Table 3.1, in 2012 women earned 15% less per hour than men did ($3.80>$25.30), slightly more than the gap of 14% seen in 1992 ($3.44>$24.83). Fourth, the gen- der gap is smaller for young college graduates (the group analyzed in Table 3.1) than it is for all college graduates (analyzed in Table 2.4): As reported in Table 2.4, the mean earnings for all college-educated women working full-time in 2012 was $25.42, while for men this mean was $32.73, which corresponds to a gender gap of 22% 3= (32.73 – 25.42)>32.734 among all full-time college-educated workers.
This empirical analysis documents that the “gen- der gap” in hourly earnings is large and has been fairly stable (or perhaps increased slightly) over the recent past. The analysis does not, however, tell us why this
gap exists. Does it arise from gender discrimination in the labor market? Does it reflect differences in skills, experience, or education between men and women? Does it reflect differences in choice of jobs? Or is there some other cause? We return to these questions once we have in hand the tools of multiple regression analysis, the topic of Part II.
1Because of inflation, a dollar in 1992 was worth more than a dollar in 2012, in the sense that a dollar in 1992 could buy more goods and services than a dollar in 2012 could. Thus earnings in 1992 cannot be directly compared to earn- ings in 2012 without adjusting for inflation. One way to make this adjustment is to use the CPI, a measure of the price of a “market basket” of consumer goods and services constructed by the Bureau of Labor Statistics. Over the 20 years from 1992 to 2012, the price of the CPI market basket rose by 63.6%; in other words, the CPI basket of goods and services that cost $100 in 1992 cost $163.64 in 2012. To make earnings in 1992 and 2012 comparable in Table 3.1, 1992 earnings are inflated by the amount of overall CPI price inflation, that is, by multiplying 1992 earnings by 1.636 to put them into “2012 dollars.”
3.6 Using the t-Statistic When the Sample Size Is Small 87
3.6
Using the t-Statistic When the Sample Size Is Small
In Sections 3.2 through 3.5, the t-statistic is used in conjunction with critical values from the standard normal distribution for hypothesis testing and for the construc- tion of confidence intervals. The use of the standard normal distribution is justi- fied by the central limit theorem, which applies when the sample size is large. When the sample size is small, the standard normal distribution can provide a poor approximation to the distribution of the t-statistic. If, however, the popula- tion distribution is itself normally distributed, then the exact distribution (that is, the finite-sample distribution; see Section 2.6) of the t-statistic testing the mean of a single population is the Student t distribution with n – 1 degrees of freedom, and critical values can be taken from the Student t distribution.
The t-Statistic and the Student t Distribution
Thet-statistictestingthemean. Considerthet-statisticusedtotestthehypothesis that the mean of Y is mY,0, using data Y1, c, Yn. The formula for this statistic is
88 ChapteR 3 Review of Statistics
given by Equation (3.10), where the standard error of Y is given by Equation (3.8). Substitution of the latter expression into the former yields the formula for the t-statistic:
(3.22)
where s2Y is given in Equation (3.7).
As discussed in Section 3.2, under general conditions the t-statistic has a stan-
dard normal distribution if the sample size is large and the null hypothesis is true [see Equation (3.12)]. Although the standard normal approximation to the t-sta- tistic is reliable for a wide range of distributions of Y if n is large, it can be unreli- able if n is small. The exact distribution of the t-statistic depends on the distribution of Y, and it can be very complicated. There is, however, one special case in which the exact distribution of the t-statistic is relatively simple: If Y is normally distrib- uted, then the t-statistic in Equation (3.22) has a Student t distribution with n – 1 degrees of freedom. (The mathematics behind this result is provided in Sections 17.4 and 18.4.)
If the population distribution is normally distributed, then critical values from the Student t distribution can be used to perform hypothesis tests and to construct confidence intervals. As an example, consider a hypothetical problem in which tact =2.15andn=20sothatthedegreesoffreedomisn-1=19.From Appendix Table 2, the 5% two-sided critical value for the t19 distribution is 2.09. Because the t-statistic is larger in absolute value than the critical value (2.15 7 2.09), the null hypothesis would be rejected at the 5% significance level against the two-sided alternative. The 95% confidence interval for mY, constructed using the t19 distribution, would be Y { 2.09 SE(Y). This confidence interval is somewhat wider than the confidence interval constructed using the standard nor- mal critical value of 1.96.
The t-statistic testing differences of means. The t-statistic testing the difference of two means, given in Equation (3.20), does not have a Student t distribution, even if the population distribution of Y is normal. (The Student t distribution does not apply here because the variance estimator used to compute the standard error in Equation (3.19) does not produce a denominator in the t-statistic with a chi- squared distribution.)
A modified version of the differences-of-means t-statistic, based on a differ- ent standard error formula—the “pooled” standard error formula—has an exact Student t distribution when Y is normally distributed; however, the pooled
2s >n Y
t = Y – mY,0, 2
3.6 Using the t-Statistic When the Sample Size Is Small 89
standard error formula applies only in the special case that the two groups have the same variance or that each group has the same number of observations (Exer- cise 3.21). Adopt the notation of Equation (3.19) so that the two groups are denoted as m and w. The pooled variance estimator is
2 1 anm (Yi-Ym)2+ anw (Yi-Ym)2
s = C S, (3.23)
pooled nm + nw – 2 i=1 i=1 group m group w
ference in means is SE (Y – Y ) = s * 11>n + 1>n , and the pooled m w pooled m w
pooled t-statistic is computed using Equation (3.20), where the standard error is the pooled standard error, SEpooled(Ym – Yw).
If the population distribution of Y in group m is N(mm, s2m), if the population distribution of Y in group w is N(mw, s2w), and if the two group variances are the same (that is, s2m = s2w), then under the null hypothesis the t-statistic computed using the pooled standard error has a Student t distribution with nm + nw – 2 degrees of freedom.
The drawback of using the pooled variance estimator s2pooled is that it applies only if the two population variances are the same (assuming nm ≠ nw). If the population variances are different, the pooled variance estimator is biased and inconsistent. If the population variances are different but the pooled variance formula is used, the null distribution of the pooled t-statistic is not a Student t distribution, even if the data are normally distributed; in fact, it does not even have a standard normal distribution in large samples. Therefore, the pooled stan- dard error and the pooled t-statistic should not be used unless you have a good reason to believe that the population variances are the same.
Use of the Student t Distribution in Practice
For the problem of testing the mean of Y, the Student t distribution is applicable if the underlying population distribution of Y is normal. For economic variables, however, normal distributions are the exception (for example, see the boxes in Chapter 2 “The Distribution of Earnings in the United States in 2012” and “A Bad Day on Wall Street”). Even if the underlying data are not normally distrib- uted, the normal approximation to the distribution of the t-statistic is valid if the sample size is large. Therefore, inferences—hypothesis tests and confidence intervals—about the mean of a distribution should be based on the large-sample normal approximation.
where the first summation is for the observations in group m and the second sum-
mation is for the observations in group w. The pooled standard error of the dif-
90 ChapteR 3 Review of Statistics
a novel Way to Boost retirement Savings
Many economists think that people do not save enough for retirement. Conventional methods for encouraging retirement savings focus on financial incentives, but there also has been an upsurge in interest in unconventional ways to encourage saving for retirement.
In an important study published in 2001, Brigitte Madrian and Dennis Shea considered one such unconventional method for stimulating retirement savings. Many firms offer retirement savings plans in which the firm matches, in full or in part, savings taken out of the paycheck of participating employ- ees. Enrollment in such plans, called 401(k) plans after the applicable section of the U.S. tax code, is always optional. However, at some firms employees are automatically enrolled in the plan, although they can opt out; at other firms, employees are enrolled only if they choose to opt in. According to conven- tional economic models of behavior, the method of enrollment—opt out or opt in—should not matter: The rational worker computes the optimal action, then takes it. But, Madrian and Shea wondered, could conventional economics be wrong? Does the method of enrollment in a savings plan directly affect its enrollment rate?
To measure the effect of the method of enroll- ment, Madrian and Shea studied a large firm that changed the default option for its 401(k) plan from nonparticipation to participation. They compared two groups of workers: those hired the year before the change and not automatically enrolled (but could opt in) and those hired in the year after the change and automatically enrolled (but could opt out). The financial aspects of the plan remained the same, and Madrian and Shea found no systematic differences
between the workers hired before and after the change. Thus, from an econometrician’s perspec- tive, the change was like a randomly assigned treat- ment and the causal effect of the change could be estimated by the difference in means between the two groups.
Madrian and Shea found that the default enroll- ment rule made a huge difference: The enroll- ment rate for the “opt-in” (control) group was 37.4% (n = 4249), whereas the enrollment rate for the “opt-out” (treatment) group was 85.9% (n = 5801). The estimate of the treatment effect is 48.5% (= 85.9% – 37.4%). Because their sample is large, the 95% confidence (computed in Exer- cise 3.15) for the treatment effect is tight, 46.8% to 50.2%.
How could the default choice matter so much? Maybe workers found these financial choices too confusing, or maybe they just didn’t want to think about growing old. Neither explanation is economi- cally rational—but both are consistent with the predictions of the growing field of “behavioural economics,” and both could lead to accepting the default enrollment option.
This research had an important practical impact. In August 2006, Congress passed the Pension Pro- tection Act that (among other things) encouraged firms to offer 401(k) plans in which enrollment is the default. The econometric findings of Madrian and Shea and others featured prominently in testimony on this part of the legislation.
To learn more about behavioral economics and the design of retirement savings plans, see Benartzi and Thaler (2007) and Beshears, Choi, Laibson, and Madrian (2008).
3.7 Scatterplots, the Sample Covariance, and the Sample Correlation 91
When comparing two means, any economic reason for two groups having different means typically implies that the two groups also could have different variances. Accordingly, the pooled standard error formula is inappropriate, and the correct standard error formula, which allows for different group variances, is as given in Equation (3.19). Even if the population distributions are normal, the t-statistic computed using the standard error formula in Equation (3.19) does not have a Student t distribution. In practice, therefore, inferences about differences in means should be based on Equation (3.19), used in conjunction with the large- sample standard normal approximation.
Even though the Student t distribution is rarely applicable in economics, some software uses the Student t distribution to compute p-values and confidence inter- vals. In practice, this does not pose a problem because the difference between the Student t distribution and the standard normal distribution is negligible if the sample size is large. For n 7 15, the difference in the p-values computed using the Student t and standard normal distributions never exceeds 0.01; for n 7 80, the difference never exceeds 0.002. In most modern applications, and in all appli- cations in this textbook, the sample sizes are in the hundreds or thousands, large enough for the difference between the Student t distribution and the standard normal distribution to be negligible.
3.7
Scatterplots, the Sample Covariance, and the Sample Correlation
What is the relationship between age and earnings? This question, like many oth- ers, relates one variable, X (age), to another, Y (earnings). This section reviews three ways to summarize the relationship between variables: the scatterplot, the sample covariance, and the sample correlation coefficient.
Scatterplots
A scatterplot is a plot of n observations on Xi and Yi, in which each observation is represented by the point (Xi, Yi ). For example, Figure 3.2 is a scatterplot of age (X) and hourly earnings (Y) for a sample of 200 managers in the information industry from the March 2009 CPS. Each dot in Figure 3.2 corresponds to an (X, Y) pair for one of the observations. For example, one of the workers in this sample is 40 years old and earns $35.78 per hour; this worker’s age and earnings are indicated by the highlighted dot in Figure 3.2. The scatterplot shows a positive
92
ChapteR 3 Review of Statistics
Figure 3.2 Scatterplot of average hourly earnings vs. age
Average hourly earnings
100
90
80
70
60
50
40
30
20
10
0
20 25 30 35 40 45 50 55 60 65
Age
Each point in the plot represents the age and average earnings of one of the 200 workers in the sample. The high- lighted dot corresponds to a 40-year-old worker who earns $35.78 per hour. The data are for computer and informa- tion systems managers from the March 2009 CPS.
relationship between age and earnings in this sample: Older workers tend to earn more than younger workers. This relationship is not exact, however, and earnings could not be predicted perfectly using only a person’s age.
Sample Covariance and Correlation
The covariance and correlation were introduced in Section 2.3 as two properties of the joint probability distribution of the random variables X and Y. Because the population distribution is unknown, in practice we do not know the population covariance or correlation. The population covariance and correlation can, however, be estimated by taking a random sample of n members of the population and col- lecting the data (Xi, Yi ), i = 1, c, n.
3.7 Scatterplots, the Sample Covariance, and the Sample Correlation 93
The sample covariance and correlation are estimators of the population covariance and correlation. Like the estimators discussed previously in this chapter, they are computed by replacing a population mean (the expectation) with a sample mean. The sample covariance, denoted sXY, is
1 an
sXY = n – 1i=1(Xi – X)(Yi – Y). (3.24)
Like the sample variance, the average in Equation (3.24) is computed by dividing by n – 1 instead of n; here, too, this difference stems from using X and Y to esti- mate the respective population means. When n is large, it makes little difference whether division is by n or n – 1.
The sample correlation coefficient, or sample correlation, is denoted rXY and is the ratio of the sample covariance to the sample standard deviations:
rXY = sXY . (3.25) sXsY
The sample correlation measures the strength of the linear association between X and Y in a sample of n observations. Like the population correlation, the sample correlation is unitless and lies between −1 and 1: rXY … 1.
The sample correlation equals 1 if Xi = Yi for all i and equals −1 if Xi = – Yi for all i. More generally, the correlation is ±1 if the scatterplot is a straight line. If the line slopes upward, then there is a positive relationship between X and Y and the correlation is 1. If the line slopes down, then there is a negative relationship and the correlation is −1. The closer the scatterplot is to a straight line, the closer is the correlation to ±1. A high correlation coefficient does not necessarily mean that the line has a steep slope; rather, it means that the points in the scatterplot fall very close to a straight line.
Consistencyofthesamplecovarianceandcorrelation. Likethesamplevariance, the sample covariance is consistent. That is,
sXY¡p sXY. (3.26)
In other words, in large samples the sample covariance is close to the population covariance with high probability.
The proof of the result in Equation (3.26) under the assumption that (Xi, Yi) are i.i.d. and that Xi and Yi have finite fourth moments is similar to the proof in Appendix 3.3 that the sample covariance is consistent and is left as an exercise (Exercise 3.20).
94 ChapteR 3 Review of Statistics
Figure 3.3
Scatterplots for Four hypothetical Data Sets
The scatterplots in Figures 3.3a and
3.3b show strong linear relationships between X and Y.
In Figure 3.3c, X is independent of Y and the two variables are uncorrelated. In Figure 3.3d, the two variables also are uncorrelated even though they are related nonlinearly.
y
y
70
60
50
40
30
20
10
70
60
50
40
30
20
10
00
70 80 90 100 110 120 130 70 80 90 100 110 120 130
xx
(a) Correlation = +0.9 (b) Correlation = –0.8 yy
70
60
50
40
30
20
10
70
60
50
40
30
20
10
070 80
(c) Correlation = 0.0 (d) Correlation = 0.0 (quadratic)
Because the sample variance and sample covariance are consistent, the sam- ple correlation coefficient is consistent, that is, rXY ¡p corr(Xi, Yi).
Example. Asanexample,considerthedataonageandearningsinFigure3.2.For
these 200 workers, the sample standard deviation of age is sA = 9.07 years and
the sample standard deviation of earnings is sE = $14.37 per hour. The sample
covariance between age and earnings is sAE = 33.16 (the units are years * dollars Electronic Publishing Services Inc.
90
100 110 120 130
070 80 90
100 110 120 130
xx
Stock/Watson, Econometrics 1e
= 33.16>(9.07 *14.37) = 0.25 or 25%. The correlation of 0.25 means that there
per hour, not readily interpretable). Thus the sample correlation coefficient is
r
AE
STOC.ITEM.0011 Fig. 03.03
1st Proof 2nd Proof 3rd Proof Final
is a positive relationship between age and earnings, but as is evident in the scatterplot, this relationship is far from perfect.
To verify that the correlation does not depend on the units of measurement, suppose that earnings had been reported in cents, in which case the sample stan- dard deviations of earnings is 1437¢ per hour and the covariance between age and earnings is 3316 (units are years * cents per hour); then the correlation is 3316>(9.07 * 1437) = 0.25 or 25%.
Figure 3.3 gives additional examples of scatterplots and correlation. Figure 3.3a shows a strong positive linear relationship between these variables, and the sam- ple correlation is 0.9.
Figure 3.3b shows a strong negative relationship with a sample correlation of −0.8. Figure 3.3c shows a scatterplot with no evident relationship, and the sample correlation is zero. Figure 3.3d shows a clear relationship: As X increases, Y ini- tially increases, but then decreases. Despite this discernable relationship between X and Y, the sample correlation is zero; the reason is that, for these data, small values of Y are associated with both large and small values of X.
This final example emphasizes an important point: The correlation coefficient is a measure of linear association. There is a relationship in Figure 3.3d, but it is not linear.
Summary
1. The sample average, Y, is an estimator of the population mean, mY. When Y1, c, Yn are i.i.d., Y2 Y2 >n a. the sampling distribution of Y has mean mY and variance s = s ; b. Y is unbiased;
c. by the law of large numbers, Y is consistent; and
d. by the central limit theorem, Y has an approximately normal sampling
distribution when the sample size is large.
2. The t-statistic is used to test the null hypothesis that the population mean
takes on a particular value. If n is large, the t-statistic has a standard normal
sampling distribution when the null hypothesis is true.
3. The t-statistic can be used to calculate the p-value associated with the null
hypothesis. A small p-value is evidence that the null hypothesis is false.
4. A 95% confidence interval for mY is an interval constructed so that it con-
tains the true value of mY in 95% of all possible samples.
5. Hypothesis tests and confidence intervals for the difference in the means of two populations are conceptually similar to tests and intervals for the mean
of a single population.
Summary 95
96 Chapter 3 Review of Statistics
6. The sample correlation coefficient is an estimator of the population correlation coefficient and measures the linear relationship between two variables—that is, how well their scatterplot is approximated by a straight line.
Key Terms
estimator (67) estimate (67)
bias, consistency, and
efficiency (68)
BLUE (Best Linear Unbiased
Estimator) (69)
least squares estimator (70) hypothesis tests (71)
null hypothesis (71) alternative hypothesis (71) two-sided alternative
hypothesis (71) p-value (significance probability) (72) sample variance (74)
sample standard deviation (74) degrees of freedom (75) standard error of Y (75) t-statistic (t-ratio) (76)
test statistic (76) type I error (78)
type II error (78) significance level (78) critical value (78) rejection region (78) acceptance region (78) size of a test (78) power of a test (78) one-sided alternative
hypothesis (80)
confidence set (80)
confidence level (80)
confidence interval (80)
coverage probability (82)
test for the difference between two
means (82)
causal effect (85)
treatment effect (85) scatterplot (91)
sample covariance (93) sample correlation coefficient
(sample correlation) (93)
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Review the Concepts
3.1 Explain the difference between the sample average Y and the population mean.
3.2 Explain the difference between an estimator and an estimate. Provide an example of each.
3.3 A population distribution has a mean of 10 and a variance of 16. Determine the mean and variance of Y from an i.i.d. sample from this population for (a) n = 10; (b) n = 100; and (c) n = 1000. Relate your answers to the law of large numbers.
3.4 What role does the central limit theorem play in statistical hypothesis test- ing? In the construction of confidence intervals?
3.5 What is the difference between a null hypothesis and an alternative hypothesis? Among size, significance level, and power? Between a one- sided alternative hypothesis and a two-sided alternative hypothesis?
3.6 Why does a confidence interval contain more information than the result of a single hypothesis test?
3.7 Explain why the differences-of-means estimator, applied to data from a randomized controlled experiment, is an estimator of the treatment effect.
3.8 Sketch a hypothetical scatterplot for a sample of size 10 for two random variables with a population correlation of (a) 1.0; (b) −1.0; (c) 0.9; (d) −0.5; (e) 0.0.
Exercises
3.1 In a population, mY = 100 and s2Y = 43. Use the central limit theorem to answer the following questions:
a. In a random sample of size n = 100, find Pr(Y 6 101).
b. In a random sample of size n = 64, find Pr(101 6 Y 6 103).
c. In a random sample of size n = 165, find Pr( Y 7 98).
3.2 Let Y be a Bernoulli random variable with success probability Pr(Y = 1) = p, and let Y1, c, Yn be i.i.d. draws from this distribution. Let pn be the fraction of successes (1s) in this sample.
Exercises 97
98 ChapteR 3 Review of Statistics
a. Show that pn = Y.
b. Show that pn is an unbiased estimator of p.
c. Show that var(pn) = p(1 – p)>n.
3.3 In a survey of 400 likely voters, 215 responded that they would vote for the incumbent, and 185 responded that they would vote for the challenger. Let p denote the fraction of all likely voters who preferred the incumbent at the time of the survey, and let pn be the fraction of survey respondents who preferred the incumbent.
a. Use the survey results to estimate p.
b. Use the estimator of the variance of pn, np(1 – pn)>n, to calculate the
standard error of your estimator.
c. What is the p-value for the test H0: p = 0.5 vs. H1: p ≠ 0.5?
d. What is the p-value for the test H0: p = 0.5 vs. H1: p 7 0.5?
e. Why do the results from (c) and (d) differ?
f. Did the survey contain statistically significant evidence that the incumbent was ahead of the challenger at the time of the survey? Explain.
3.4 Using the data in Exercise 3.3:
a. Construct a 95% confidence interval for p.
b. Construct a 99% confidence interval for p.
c. Why is the interval in (b) wider than the interval in (a)?
a. Without doing any additional calculations, test the hypothesis
H0: p = 0.50 vs. H1: p ≠ 0.50 at the 5% significance level.
3.5 A survey of 1055 registered voters is conducted, and the voters are asked to choose between candidate A and candidate B. Let p denote the fraction of voters in the population who prefer candidate A, and let pn denote the fraction of voters in the sample who prefer Candidate A.
a. You are interested in the competing hypotheses H0: p = 0.5 vs. H1: p ≠ 0.5. Suppose that you decide to reject H0 if
pn – 0.5 7 0.02.
i. What is the size of this test?
ii. Compute the power of this test if p = 0.53.
b. In i. ii. iii. iv. v.
the survey, pn = 0.54.
Test H0: p = 0.5 vs. H1: p ≠ 0.5 using a 5% significance level. Test H0: p = 0.5 vs. H1: p 7 0.5 using a 5% significance level. Construct a 95% confidence interval for p.
Construct a 99% confidence interval for p.
Construct a 50% confidence interval for p.
c. Suppose that the survey is carried out 20 times, using independently selected voters in each survey. For each of these 20 surveys, a 95% confidence interval for p is constructed.
i. What is the probability that the true value of p is contained in all 20 of these confidence intervals?
ii. How many of these confidence intervals do you expect to contain the true value of p?
d. In survey jargon, the “margin of error” is 1.96 * SE(pn); that is, it
is half the length of 95% confidence interval. Suppose you want to design a survey that has a margin of error of at most 1%. That is, you want Pr(pn – p 7 0.01) … 0.05. How large should n be if the survey uses simple random sampling?
3.6 Let Y1, c, Yn be i.i.d. draws from a distribution with mean m. A test of H0: m = 5 vs. H1: m ≠ 5 using the usual t-statistic yields a p-value of 0.03.
a. Does the 95% confidence interval contain m = 5? Explain.
b. Can you determine if m = 6 is contained in the 95% confidence
interval? Explain.
3.7 In a given population, 11% of the likely voters are African American. A sur- vey using a simple random sample of 600 landline telephone numbers finds 8% African Americans. Is there evidence that the survey is biased? Explain.
3.8 A new version of the SAT is given to 1000 randomly selected high school seniors. The sample mean test score is 1110, and the sample standard devi- ation is 123. Construct a 95% confidence interval for the population mean test score for high school seniors.
3.9 Suppose that a lightbulb manufacturing plant produces bulbs with a mean life of 2000 hours and a standard deviation of 200 hours. An inventor claims to have developed an improved process that produces bulbs with a longer mean life and the same standard deviation. The plant manager randomly selects 100 bulbs produced by the process. She says that she will believe the
Exercises 99
100 ChapteR 3 Review of Statistics
inventor’s claim if the sample mean life of the bulbs is greater than 2100 hours; otherwise, she will conclude that the new process is no better than the old process. Let m denote the mean of the new process. Consider the null and alternative hypotheses H0: m = 2000 vs. H1: m 7 2000.
a. What is the size of the plant manager’s testing procedure?
b. Suppose the new process is in fact better and has a mean bulb life of 2150
hours. What is the power of the plant manager’s testing procedure?
c. What testing procedure should the plant manager use if she wants the size of her test to be 5%?
3.10 Suppose a new standardized test is given to 100 randomly selected third- grade students in New Jersey. The sample average score Y on the test is 58 points, and the sample standard deviation, sY, is 8 points.
a. The authors plan to administer the test to all third-grade students in New Jersey. Construct a 95% confidence interval for the mean score of all New Jersey third graders.
b. Suppose the same test is given to 200 randomly selected third graders from Iowa, producing a sample average of 62 points and sample stan- dard deviation of 11 points. Construct a 90% confidence interval for the difference in mean scores between Iowa and New Jersey.
c. Can you conclude with a high degree of confidence that the popula- tion means for Iowa and New Jersey students are different? (What is the standard error of the difference in the two sample means? What is the p-value of the test of no difference in means versus some differ- ence?)
2
(a) E(Y) = m and (b) var(Y) = 1.25s >n.
3.11 Consider the estimator Y∼, defined in Equation (3.1). Show that
∼∼
YY
3.12 To investigate possible gender discrimination in a firm, a sample of 100 men and 64 women with similar job descriptions are selected at random. A summary of the resulting monthly salaries follows:
average Salary (Y )
Men $3100
Women $2900
Standard Deviation (sY) n
$200 100
$320 64
a. What do these data suggest about wage differences in the firm? Do they represent statistically significant evidence that average wages of
men and women are different? (To answer this question, first state the null and alternative hypotheses; second, compute the relevant t-statistic; third, compute the p-value associated with the t-statistic; and finally, use the p-value to answer the question.)
b. Do these data suggest that the firm is guilty of gender discrimination in its compensation policies? Explain.
3.13 Data on fifth-grade test scores (reading and mathematics) for 420 school districts in California yield Y = 646.2 and standard deviation sY = 19.5.
a. Construct a 95% confidence interval for the mean test score in the population.
b. When the districts were divided into districts with small classes (6 20 students per teacher) and large classes (≥ 20 students per teacher), the following results were found:
Class Size
Small
Large
average Score (Y) 657.4
650.0
Standard Deviation (sY) n
19.4 238
17.9 182
3.15 LetYaandYbdenoteBernoullirandomvariablesfromtwodifferentpopu- lations, denoted a and b. Suppose that E(Ya) = pa and E(Yb) = pb. A random sample of size na is chosen from population a, with sample average denoted pna, and a random sample of size nb is chosen from population b, with sample average denoted pnb. Suppose the sample from population a is independent of the sample from population b.
a. Show that E( pn ) = p and var(pn ) = p (1 – p )>n . Show that aaaaaa
E(pn)=p andvar(pn)=p(1-p)>n. bbbbbb
p (1 – p ) p (1 – p )
b. Showthatvar(pn – pn ) = a n a + b n b .(Hint:Rememberthat
abab the samples are independent.)
Exercises 101
Is there statistically significant evidence that the districts with smaller classes have higher average test scores? Explain.
3.14 Valuesofheightininches(X)andweightinpounds(Y)arerecordedfrom a sample of 300 male college students. The resulting summary statistics are X = 70.5in.,Y = 158 lb., sX = 1.8in.,sY = 14.2 lb., sXY = 21.73 in. * lb., and rXY = 0.85. Convert these statistics to the metric system (meters and kilograms).
pna(1 – pna) pnb(1 – pnb) val for p – p is given by (pn – pn ) { 1.964 n + n .
102 ChapteR 3 Review of Statistics
c. Suppose that na and nb are large. Show that a 95% confidence inter-
ababab How would you construct a 90% confidence interval for pa – pb?
d. Read the box “A Novel Way to Boost Retirement Savings” in Section 3.6. Let population a denote the “opt-out” (treatment) group and popula- tion b denote the “opt-in” (control) group. Construct a 95% confi- dence interval for the treatment effect, pa – pb.
3.16 Grades on a standardized test are known to have a mean of 1000 for students in the United States. The test is administered to 453 randomly selected students in Florida; in this sample, the mean is 1013, and the stan- dard deviation (s) is 108.
a. Construct a 95% confidence interval for the average test score for Florida students.
b. Is there statistically significant evidence that Florida students perform differently than other students in the United States?
c. Another 503 students are selected at random from Florida. They are given a 3-hour preparation course before the test is adminis- tered. Their average test score is 1019, with a standard deviation of 95.
i. Construct a 95% confidence interval for the change in average test score associated with the prep course.
ii. Is there statistically significant evidence that the prep course helped?
d. The original 453 students are given the prep course and then are asked to take the test a second time. The average change in their test scores is 9 points, and the standard deviation of the change is
60 points.
i. Construct a 95% confidence interval for the change in average test scores.
ii. Is there statistically significant evidence that students will perform better on their second attempt, after taking the prep course?
iii. Students may have performed better in their second attempt because of the prep course or because they gained test-taking experience in their first attempt. Describe an experiment that would quantify these two effects.
Empirical Exercises 103
3.17 Read the box “The Gender Gap of Earnings of College Graduates in the
United States” in Section 3.5.
a. Construct a 95% confidence interval for the change in men’s average hourly earnings between 1992 and 2012.
b. Construct a 95% confidence interval for the change in women’s aver- age hourly earnings between 1992 and 2012.
c. Construct a 95% confidence interval for the change in the gender gap in average hourly earnings between 1992 and 2012. (Hint: Ym,1992 – Yw,1992 is independent of Ym,2012 – Yw,2012.)
3.18 This exercise shows that the sample variance is an unbiased estimator of the population variance when Y1, c, Yn are i.i.d. with mean mY and variance s2Y.
a. Use Equation (2.31) to show that
E3(Y – Y) 4 = var(Y ) – 2cov(Y,Y) + var(Y).
i2ii
b. Use Equation (2.33) to show that cov(Y, Y ) = s >n.
i 2Y
c. Use the results in (a) and (b) to show that E(s2Y) = s2Y.
3.19 a. Y is an unbiased estimator of mY. Is Y2 an unbiased estimator of m2Y? b. Y is a consistent estimator of mY. Is Y 2 a consistent estimator of m2Y?
3.20 Suppose that (Xi, Yi ) are i.i.d. with finite fourth moments. Prove that the sample covariance is a consistent estimator of the population covariance, that is, sXY ¡p sXY, where sXY is defined in Equation (3.24). (Hint: Use the strategy of Appendix 3.3.)
3.21 Show that the pooled standard error 3SE ( Y – Y )4 given fol- pooled m w
lowing Equation (3.23) equals the usual standard error for the differ- ence in means in Equation (3.19) when the two group sizes are the same (nm = nw).
Empirical Exercises
E3.1 On the text website, http://www.pearsonhighered.com/stock_watson/, you will find the data file CPS92_12, which contains an extended version of the data set used in Table 3.1 of the text for the years 1992 and 2012. It contains data on full-time workers, ages 25–34, with a high school diploma or B.A./B.S. as their highest degree. A detailed description is given in
104 ChapteR 3 Review of Statistics
CPS92_12_Description, available on the website. Use these data to answer
the following questions.
a. i. Compute the sample mean for average hourly earnings (AHE) in 1992 and 2012.
ii. Compute the sample standard deviation for AHE in 1992 and 2012.
iii. Construct a 95% confidence interval for the population means of AHE in 1992 and 2012.
iv. Construct a 95% confidence interval for the change in the popula- tion mean of AHE between 1992 and 2012.
b. In 2012, the value of the Consumer Price Index (CPI) was 229.6. In 1992, the value of the CPI was 140.3. Repeat (a) but use AHE mea- sured in real 2012 dollars ($2012); that is, adjust the 1992 data for the price inflation that occurred between 1992 and 2012.
c. If you were interested in the change in workers’ purchasing power from 1992 to 2012, would you use the results from (a) or (b)? Explain.
d. Using the data for 2012:
i. Construct a 95% confidence interval for the mean of AHE for
high school graduates.
ii. Construct a 95% confidence interval for the mean of AHE for workers with a college degree.
iii. Construct a 95% confidence interval for the difference between the two means.
e. Repeat (d) using the 1992 data expressed in $2012.
f. Using appropriate estimates, confidence intervals, and test statistics,
answer the following questions:
i. Did real (inflation-adjusted) wages of high school graduates increase from 1992 to 2012?
ii. Did real wages of college graduates increase?
iii. Did the gap between earnings of college and high school gradu- ates increase? Explain.
g. Table 3.1 presents information on the gender gap for college gradu- ates. Prepare a similar table for high school graduates, using the 1992 and 2012 data. Are there any notable differences between the results for high school and college graduates?
E3.2 A consumer is given the chance to buy a baseball card for $1, but he declines the trade. If the consumer is now given the baseball card, will he be willing to sell it for $1? Standard consumer theory suggests yes, but behavioral economists have found that “ownership” tends to increase the value of goods to consumers. That is, the consumer may hold out for some amount more than $1 (for example, $1.20) when selling the card, even though he was willing to pay only some amount less than $1 (for example, $0.88) when buying it. Behavioral economists call this phenomenon the “endowment effect.” John List investigated the endowment effect in a ran- domized experiment involving sports memorabilia traders at a sports-card show. Traders were randomly given one of two sports collectibles, say good A or good B, that had approximately equal market value.1 Those receiv- ing good A were then given the option of trading good A for good B with the experimenter; those receiving good B were given the option of trading good B for good A with the experimenter. Data from the experiment and a detailed description can be found on the textbook website, http://www .pearsonhighered.com/stock_watson/, in the files Sportscards and Sports- cards_Description.2
a. i. Suppose that, absent any endowment effect, all the subjects pre- fer good A to good B. What fraction of the experiment’s subjects would you expect to trade the good that they were given for the other good? (Hint: Because of random assignment of the two treat- ments, approximately 50% of the subjects received good A and 50% received good B.)
ii. Suppose that, absent any endowment effect, 50% of the subjects prefer good A to good B, and the other 50% prefer good B to good A. What fraction of the subjects would you expect to trade the good that they were given for the other good?
iii. Suppose that, absent any endowment effect, X% of the subjects prefer good A to good B, and the other (100 – X)% prefer good B to good A. Show that you would expect 50% of the subjects to trade the good that they were given for the other good.
1Good A was a ticket stub from the game in which Cal Ripken, Jr., set the record for consecutive games played, and good B was a souvenir from the game in which Nolan Ryan won his 300th game.
2These data were provided by Professor John List of the University of Chicago and were used in his paper “Does Market Experience Eliminate Market Anomalies,” Quarterly Journal of Economics, 2003, 118(1): 41–71.
Empirical Exercises 105
106 ChapteR 3 Review of Statistics
appenDix
b. Using the sports-card data, what fraction of the subjects traded the good they were given? Is the fraction significantly different from 50%? Is there evidence of an endowment effect? (Hint: Review Exercises 3.2 and 3.3)
c. Some have argued that the endowment effect may be present, but that it is likely to disappear as traders gain more trading experience. Half of the experimental subjects were dealers, and the other half were nondealers. Dealers have more experience than nondealers. Repeat (b) for dealers and nondealers. Is there a significant differ- ence in their behavior? Is the evidence consistent with the hypothesis that the endowment effect disappears as traders gain more experi- ence? (Hint: Review Exercise 3.15).
3.1
The U.S. Current Population Survey
Each month, the U.S. Census Bureau and the U.S. Bureau of Labor Statistics conduct the Current Population Survey (CPS), which provides data on labor force characteristics of the population, including the levels of employment, unemployment, and earnings. Approxi- mately 60,000 U.S. households are surveyed each month. The sample is chosen by ran- domly selecting addresses from a database of addresses from the most recent decennial census augmented with data on new housing units constructed after the last census. The exact random sampling scheme is rather complicated (first, small geographical areas are randomly selected, then housing units within these areas are randomly selected); details can be found in the Handbook of Labor Statistics and on the Bureau of Labor Statistics website (www.bls.gov).
The survey conducted each March is more detailed than in other months and asks questions about earnings during the previous year. The statistics in Tables 2.4 and 3.1 were computed using the March surveys. The CPS earnings data are for full-time workers, defined to be somebody employed more than 35 hours per week for at least 48 weeks in the previous year.
appenDix
3.2
Two Proofs That Y Is the Least Squares Estimator of mY
This appendix provides two proofs, one using calculus and one not, that Y minimizes the sum of squared prediction mistakes in Equation (3.2)—that is, that Y is the least squares estimator of E(Y).
Calculus Proof
To minimize the sum of squared prediction mistakes, take its derivative and set it to zero:
dan2an an
dm (Yi -m) =-2 (Yi -m)=-2 Yi +2nm=0. (3.27)
i=1 i=1 i=1
Solving for the final equation for m shows that g ni = 1(Yi – m)2 is minimized when
m = Y.
Noncalculus Proof
The strategy is to show that the difference between the least squares estimator and Y must
Two Proofs That Y Is the Least Squares Estimator of mY 107
m=Y-d. Then (Y -m) =(Y -3Y-d4) =(3Y -Y4+d) =(Y -Y) + i2i 2i 2i2
be zero, from which it follows that Y is the least squares estimator. Let d = Y – m, so that
2d(Yi – Y) + d2. Thus the sum of squared prediction mistakes [Equation (3.2)] is
an 2 an 2 an 2 an 2 2
(Yi -m) = (Yi -Y) +2d (Yi -Y)+nd = (Yi -Y) +nd,
i=1 i=1 i=1 i=1
(3.28)
where the second equality uses the fact that g ni = 1(Yi – Y) = 0. Because both terms in the final line of Equation (3.28) are nonnegative and because the first term does not depend on d, g ni = 1(Yi – m)2 is minimized by choosing d to make the second term, nd2, as small as possible. This is done by setting d = 0—that is, by setting m = Y—so that Y is the least squares estimator of E(Y).
108 ChapteR 3 Review of Statistics appenDix
3.3
A Proof That the Sample Variance Is Consistent
This appendix uses the law of large numbers to prove that the sample variance s2Y is a con- sistent estimator of the population variance s2Y, as stated in Equation (3.9), when Y1, c, Yn are i.i.d. and E(Y4i ) 6 ∞.
First, consider a version of the sample variance that uses n instead of n − 1 as a divisor: 1an 21an2 1an 2
n (Yi-Y) =n Yi -2Yn Yi+Y i=1 i=1 i=1
1an2 2 =n Yi-Y
i=1
¡p ( s 2 Y + m 2 Y ) – m 2 Y
p2p22n1n2p2 Y ¡ m so that Y ¡ m . Finally, s = 1 21n g (Y – Y) 2 ¡ s
= s2Y,
wherethefirstequalityuses(Y – Y) = Y – 2YY +Y ,andthesecondusesng Y = Y.
i22i i2 1ni=1i
The convergence in the third line follows from (i) applying the law of large numbers to
1n2p2 2
n g i = 1Yi ¡ E(Y ) (which follows because Yi are i.i.d. and have finite variance because
E(Y4i ) is finite), (ii) recognizing that E(Y2i ) = s2Y + m2Y (Key Concept 2.3), and (iii) noting
from Equation (3.29) and 1
2 S 1.
follows
YYYn-1i=1iY
n
n-1
(3.29)
CHAPTER
4
Linear Regression with One Regressor
Astate implements tough new penalties on drunk drivers: What is the effect
on highway fatalities? A school district cuts the size of its elementary school classes: What is the effect on its students’ standardized test scores? You successfully complete one more year of college classes: What is the effect on your future earnings?
All three of these questions are about the unknown effect of changing one variable, X (X being penalties for drunk driving, class size, or years of schooling), on another variable, Y (Y being highway deaths, student test scores, or earnings).
This chapter introduces the linear regression model relating one variable, X, to another, Y. This model postulates a linear relationship between X and Y; the slope of the line relating X and Y is the effect of a one-unit change in X on Y. Just as the mean of Y is an unknown characteristic of the population distribution of Y, the slope of the line relating X and Y is an unknown characteristic of the population joint distribution of X and Y. The econometric problem is to estimate this slope—that is, to estimate the effect on Y of a unit change in X—using a sample of data on these two variables.
This chapter describes methods for estimating this slope using a random sample of data on X and Y. For instance, using data on class sizes and test scores from different school districts, we show how to estimate the expected effect on test scores of reducing class sizes by, say, one student per class. The slope and the intercept of the line relating X and Y can be estimated by a method called ordinary least squares (OLS).
4.1
The Linear Regression Model
The superintendent of an elementary school district must decide whether to hire additional teachers and she wants your advice. If she hires the teachers, she will reduce the number of students per teacher (the student–teacher ratio) by two. She faces a trade-off. Parents want smaller classes so that their children can receive more individualized attention. But hiring more teachers means spending more money, which is not to the liking of those paying the bill! So she asks you: If she cuts class sizes, what will the effect be on student performance?
109
110 CHAPTER 4
Linear Regression with One Regressor
In many school districts, student performance is measured by standardized tests, and the job status or pay of some administrators can depend in part on how well their students do on these tests. We therefore sharpen the superintendent’s question: If she reduces the average class size by two students, what will the effect be on standardized test scores in her district?
A precise answer to this question requires a quantitative statement about changes. If the superintendent changes the class size by a certain amount, what would she expect the change in standardized test scores to be? We can write this as a math- ematical relationship using the Greek letter beta, bClassSize, where the subscript ClassSize distinguishes the effect of changing the class size from other effects. Thus,
bClassSize = changeinTestScore = ∆TestScore, (4.1) change in ClassSize ∆ClassSize
where the Greek letter ∆ (delta) stands for “change in.” That is, bClassSize is the change in the test score that results from changing the class size divided by the change in the class size.
If you were lucky enough to know bClassSize, you would be able to tell the superintendent that decreasing class size by one student would change district- wide test scores by bClassSize. You could also answer the superintendent’s actual question, which concerned changing class size by two students per class. To do so, rearrange Equation (4.1) so that
∆TestScore = bClassSize * ∆ClassSize. (4.2)
Suppose that bClassSize = -0.6. Then a reduction in class size of two students per class would yield a predicted change in test scores of (-0.6) * (-2) = 1.2; that is, you would predict that test scores would rise by 1.2 points as a result of the reduction in class sizes by two students per class.
Equation (4.1) is the definition of the slope of a straight line relating test scores and class size. This straight line can be written
TestScore = b0 + bClassSize * ClassSize, (4.3)
where b0 is the intercept of this straight line and, as before, bClassSize is the slope. According to Equation (4.3), if you knew b0 and bClassSize, not only would you be able to determine the change in test scores at a district associated with a change in class size, but you also would be able to predict the average test score itself for a given class size.
When you propose Equation (4.3) to the superintendent, she tells you that something is wrong with this formulation. She points out that class size is just one of many facets of elementary education and that two districts with the same class sizes will have different test scores for many reasons. One district might have bet- ter teachers or it might use better textbooks. Two districts with comparable class sizes, teachers, and textbooks still might have very different student populations; perhaps one district has more immigrants (and thus fewer native English speak- ers) or wealthier families. Finally, she points out that even if two districts are the same in all these ways they might have different test scores for essentially random reasons having to do with the performance of the individual students on the day of the test. She is right, of course; for all these reasons, Equation (4.3) will not hold exactly for all districts. Instead, it should be viewed as a statement about a rela- tionship that holds on average across the population of districts.
A version of this linear relationship that holds for each district must incorpo- rate these other factors influencing test scores, including each district’s unique characteristics (for example, quality of their teachers, background of their stu- dents, how lucky the students were on test day). One approach would be to list the most important factors and to introduce them explicitly into Equation (4.3) (an idea we return to in Chapter 6). For now, however, we simply lump all these “other factors” together and write the relationship for a given district as
TestScore = b0 + bClassSize * ClassSize + other factors. (4.4)
Thus the test score for the district is written in terms of one component, b0 + bClassSize * ClassSize, that represents the average effect of class size on scores in the population of school districts and a second component that represents all other factors.
Although this discussion has focused on test scores and class size, the idea expressed in Equation (4.4) is much more general, so it is useful to introduce more general notation. Suppose you have a sample of n districts. Let Yi be the average test score in the ith district, let Xi be the average class size in the ith district, and let ui denote the other factors influencing the test score in the ith district. Then Equa- tion (4.4) can be written more generally as
Yi =b0 +b1Xi +ui, (4.5)
for each district (that is, i = 1, c, n), where b0 is the intercept of this line and b1 is the slope. [The general notation b1 is used for the slope in Equation (4.5) instead of bClassSize because this equation is written in terms of a general variable Xi.]
4.1 The Linear Regression Model 111
112 CHAPTER 4
Linear Regression with One Regressor
Equation (4.5) is the linear regression model with a single regressor, in which Y is the dependent variable and X is the independent variable or the regressor.
The first part of Equation (4.5), b0 + b1Xi, is the population regression line or the population regression function. This is the relationship that holds between Y and X on average over the population. Thus, if you knew the value of X, accord- ing to this population regression line you would predict that the value of the dependent variable, Y, is b0 + b1X.
The intercept b0 and the slope b1 are the coefficients of the population regres- sion line, also known as the parameters of the population regression line. The slope b1 is the change in Y associated with a unit change in X. The intercept is the value of the population regression line when X = 0; it is the point at which the population regression line intersects the Y axis. In some econometric applications, the intercept has a meaningful economic interpretation. In other applications, the intercept has no real-world meaning; for example, when X is the class size, strictly speaking the intercept is the predicted value of test scores when there are no stu- dents in the class! When the real-world meaning of the intercept is nonsensical, it is best to think of it mathematically as the coefficient that determines the level of the regression line.
The term ui in Equation (4.5) is the error term. The error term incorporates all of the factors responsible for the difference between the ith district’s average test score and the value predicted by the population regression line. This error term contains all the other factors besides X that determine the value of the dependent variable, Y, for a specific observation, i. In the class size example, these other factors include all the unique features of the ith district that affect the per- formance of its students on the test, including teacher quality, student economic background, luck, and even any mistakes in grading the test.
The linear regression model and its terminology are summarized in Key Concept 4.1.
Figure 4.1 summarizes the linear regression model with a single regressor for seven hypothetical observations on test scores (Y) and class size (X). The popula- tion regression line is the straight line b0 + b1X. The population regression line slopes down (b1 6 0), which means that districts with lower student–teacher ratios (smaller classes) tend to have higher test scores. The intercept b0 has a math- ematical meaning as the value of the Y axis intersected by the population regression line, but, as mentioned earlier, it has no real-world meaning in this example.
Because of the other factors that determine test performance, the hypotheti- cal observations in Figure 4.1 do not fall exactly on the population regression line. For example, the value of Y for district #1, Y1, is above the population regression line. This means that test scores in district #1 were better than predicted by the
4.1 The Linear Regression Model 113
Terminology for the Linear Regression Model with a Single Regressor
The linear regression model is
Yi =b0 +b1Xi +ui,
where
the subscript i runs over observations, i = 1, c, n;
Yi is the dependent variable, the regressand, or simply the left-hand variable; Xi is the independent variable, the regressor, or simply the right-hand variable; b0 +b1Xisthepopulationregressionlineorthepopulationregressionfunction; b0 is the intercept of the population regression line;
b1 is the slope of the population regression line; and
ui is the error term.
KEY CONCEPT
4.1
FIGURE 4.1 Scatterplot of Test Score vs. Student–Teacher Ratio (Hypothetical Data)
The scatterplot shows hypothetical observations for seven school districts. The population regres- sion line is b0 + b1X. The vertical distance from the ith point to the population regression line is
Yi – (b0 + b1Xi), which
is the population error term ui for the ith observation.
Test score (Y) 700
680
660
640
620
600
10 15 20 25 30
Student–teacher ratio (X)
(X1,Y1) u1
u2
(X2,Y2) b0 +b1X
114 CHAPTER 4
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4.2
Estimating the Coefficients
of the Linear Regression Model
In a practical situation such as the application to class size and test scores, the intercept b0 and slope b1 of the population regression line are unknown. There- fore, we must use data to estimate the unknown slope and intercept of the popu- lation regression line.
This estimation problem is similar to others you have faced in statistics. For example, suppose you want to compare the mean earnings of men and women who recently graduated from college. Although the population mean earnings are unknown, we can estimate the population means using a random sample of male and female college graduates. Then the natural estimator of the unknown popula- tion mean earnings for women, for example, is the average earnings of the female college graduates in the sample.
The same idea extends to the linear regression model. We do not know the population value of bClassSize, the slope of the unknown population regression line relating X (class size) and Y (test scores). But just as it was possible to learn about the population mean using a sample of data drawn from that population, so is it possible to learn about the population slope bClassSize using a sample of data.
The data we analyze here consist of test scores and class sizes in 1999 in 420 California school districts that serve kindergarten through eighth grade. The test score is the districtwide average of reading and math scores for fifth graders. Class size can be measured in various ways. The measure used here is one of the broadest, which is the number of students in the district divided by the number of teachers— that is, the districtwide student–teacher ratio. These data are described in more detail in Appendix 4.1.
Table 4.1 summarizes the distributions of test scores and class sizes for this sam- ple. The average student–teacher ratio is 19.6 students per teacher, and the standard deviation is 1.9 students per teacher. The 10th percentile of the distribution of the
population regression line, so the error term for that district, u1, is positive. In contrast, Y2 is below the population regression line, so test scores for that district were worse than predicted, and u2 6 0.
Now return to your problem as advisor to the superintendent: What is the expected effect on test scores of reducing the student–teacher ratio by two students per teacher? The answer is easy: The expected change is (-2) * bClassSize. But what is the value of bClassSize?
4.2 Estimating the Coefficients of the Linear Regression Model 115
TABLE 4.1
Summary of the Distribution of Student–Teacher Ratios and Fifth-Grade Test Scores for 420 K–8 Districts in California in 1999
Standard Average Deviation
10% 25%
17.3 18.6 630.4 640.0
Percentile
40% 50% (median)
19.3 19.7 649.1 654.5
60% 75% 90%
20.1 20.9 21.9 659.4 666.7 679.1
Student–teacher ratio 19.6 1.9
Test score
654.2 19.1
FIGURE 4.2
student–teacher ratio is 17.3 (that is, only 10% of districts have student–teacher ratios below 17.3), while the district at the 90th percentile has a student–teacher ratio of 21.9.
A scatterplot of these 420 observations on test scores and the student–teacher ratio is shown in Figure 4.2. The sample correlation is -0.23, indicating a weak negative relationship between the two variables. Although larger classes in this sample tend to have lower test scores, there are other determinants of test scores that keep the observations from falling perfectly along a straight line.
Despite this low correlation, if one could somehow draw a straight line through these data, then the slope of this line would be an estimate of bClassSize
Scatterplot of Test Score vs. Student–Teacher Ratio (California School District Data)
Data from 420
California school dis- 720 tricts. There is a weak negative relationship between the student– teacher ratio and test
scores: The sample correlation is -0.23.
Test score
700
680
660
640
620
600 10
15 20 25 30
Student–teacher ratio
116 CHAPTER 4
Linear Regression with One Regressor
based on these data. One way to draw the line would be to take out a pencil and a ruler and to “eyeball” the best line you could. While this method is easy, it is very unscientific, and different people will create different estimated lines.
How, then, should you choose among the many possible lines? By far the most common way is to choose the line that produces the “least squares” fit to these data—that is, to use the ordinary least squares (OLS) estimator.
The Ordinary Least Squares Estimator
The OLS estimator chooses the regression coefficients so that the estimated regression line is as close as possible to the observed data, where closeness is measured by the sum of the squared mistakes made in predicting Y given X.
As discussed in Section 3.1, the sample average, Y, is the least squares estimator of the population mean, E(Y); that is, Y minimizes the total squared estimation mistakes
g ni = 1(Yi – m)2 among all possible estimators m [see Expression (3.2)].
The OLS estimator extends this idea to the linear regression model. Let b0 and b1 be some estimators of b0 and b1. The regression line based on these estimators is b0 +b1X,sothevalueofYipredictedusingthislineisb0 +b1Xi.Thusthemistake made in predicting the ith observation is Yi – (b0 + b1Xi) = Yi – b0 – b1Xi.
The sum of these squared prediction mistakes over all n observations is n
a(Yi – b0 – b1Xi)2. (4.6) i=1
The sum of the squared mistakes for the linear regression model in Expression (4.6) is the extension of the sum of the squared mistakes for the problem of estimating the mean in Expression (3.2). In fact, if there is no regressor, then b1 does not enter Expression (4.6) and the two problems are identical except for the different notation [m in Expression (3.2), b0 in Expression (4.6)]. Just as there is a unique estimator, Y, that minimizes the Expression (3.2), so is there a unique pair of estimators of b0 and b1 that minimize Expression (4.6).
The estimators of the intercept and slope that minimize the sum of squared mistakes in Expression (4.6) are called the ordinary least squares (OLS) estima- tors of b0 and b1.
OLS has its own special notation and terminology. The OLS estimator of b0 is denoted bn0, and the OLS estimator of b1 is denoted bn1. The OLS regression line, also called the sample regression line or sample regression function, is the straight line constructed using the OLS estimators: bn0 + bn1X. The predicted value of Yi
4.2 Estimating the Coefficients of the Linear Regression Model 117
The OLS Estimator, Predicted Values, and Residuals
The OLS estimators of the slope b1 and the intercept b0 are
n
a(Xi – X)(Yi – Y) s
bn = i=1 = XY
1n s2 a(Xi – X)2 X
i=1
bn0 = Y – bn1X.
The OLS predicted values Yni and residuals uni are Yni=bn0+bn1Xi, i=1,c,n
uni=Yi-Yni, i=1,c,n.
KEY CONCEPT
4.2
(4.7)
(4.8)
(4.9) (4.10)
The estimated intercept (bn0), slope (bn1), and residual (uni) are computed from a sample of n observations of Xi and Yi, i = 1, c, n. These are estimates of the unknown true population intercept (b0), slope (b1), and error term (ui).
given Xi, based on the OLS regression line, is Yi = b0 + b1Xi. The residual for the ith observation is the difference between Yi and its predicted value: uni = Yi – Yni.
The OLS estimators, bn0 and bn1, are sample counterparts of the population nn
coefficients, b0 and b1. Similarly, the OLS regression line b0 + b1X is the sample counterpart of the population regression line b0 + b1X, and the OLS residuals uni are sample counterparts of the population errors ui.
You could compute the OLS estimators bn0 and bn1 by trying different values of b0 and b1 repeatedly until you find those that minimize the total squared mis- takes in Expression (4.6); they are the least squares estimates. This method would be quite tedious, however. Fortunately, there are formulas, derived by minimiz- ing Expression (4.6) using calculus, that streamline the calculation of the OLS estimators.
The OLS formulas and terminology are collected in Key Concept 4.2. These formulas are implemented in virtually all statistical and spreadsheet programs. These formulas are derived in Appendix 4.2.
nnn
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FIGURE 4.3
OLS Estimates of the Relationship Between Test
Scores and the Student–Teacher Ratio
When OLS is used to estimate a line relating the student–teacher ratio to test scores using the 420 observations in Figure 4.2, the estimated slope is -2.28 and the estimated intercept is 698.9. Accordingly, the OLS regression line for these 420 observations is
TestScore = 698.9 – 2.28 * STR, (4.11)
where TestScore is the average test score in the district and STR is the student– teacher ratio. The “N” over TestScore in Equation (4.11) indicates that it is the predicted value based on the OLS regression line. Figure 4.3 plots this OLS regression line superimposed over the scatterplot of the data previously shown in Figure 4.2.
The slope of – 2.28 means that an increase in the student–teacher ratio by one student per class is, on average, associated with a decline in districtwide test scores by 2.28 points on the test. A decrease in the student–teacher ratio by two students per class is, on average, associated with an increase in test scores of 4.56 points 3= -2 * (-2.28)4. The negative slope indicates that more students per teacher (larger classes) is associated with poorer performance on the test.
The Estimated Regression Line for the California Data
The estimated regres- sion line shows a negative relationship between test scores and the student– teacher ratio. If class sizes fall by one student, the estimated regression predicts that test scores will increase by 2.28 points.
Test score
720
700
680
660
640
620
TestScore = 698.9 – 2.28¥STR ˆ
600
10
15 20 25 30
Student–teacher ratio
4.2 Estimating the Coefficients of the Linear Regression Model 119
It is now possible to predict the districtwide test score given a value of the student– teacher ratio. For example, for a district with 20 students per teacher, the predicted testscoreis698.9 – 2.28 * 20 = 653.3.Ofcourse,thispredictionwillnotbeexactly right because of the other factors that determine a district’s performance. But the regression line does give a prediction (the OLS prediction) of what test scores would be for that district, based on their student–teacher ratio, absent those other factors.
Is this estimate of the slope large or small? To answer this, we return to the superintendent’s problem. Recall that she is contemplating hiring enough teach- ers to reduce the student–teacher ratio by 2. Suppose her district is at the median of the California districts. From Table 4.1, the median student–teacher ratio is 19.7 and the median test score is 654.5. A reduction of two students per class, from 19.7 to 17.7, would move her student–teacher ratio from the 50th percentile to very near the 10th percentile. This is a big change, and she would need to hire many new teachers. How would it affect test scores?
According to Equation (4.11), cutting the student–teacher ratio by 2 is pre- dicted to increase test scores by approximately 4.6 points; if her district’s test scores are at the median, 654.5, they are predicted to increase to 659.1. Is this improvement large or small? According to Table 4.1, this improvement would move her district from the median to just short of the 60th percentile. Thus a decrease in class size that would place her district close to the 10% with the small- est classes would move her test scores from the 50th to the 60th percentile. According to these estimates, at least, cutting the student–teacher ratio by a large amount (two students per teacher) would help and might be worth doing depend- ing on her budgetary situation, but it would not be a panacea.
What if the superintendent were contemplating a far more radical change, such as reducing the student–teacher ratio from 20 students per teacher to 5? Unfortunately, the estimates in Equation (4.11) would not be very useful to her. This regression was estimated using the data in Figure 4.2, and, as the figure shows, the smallest student–teacher ratio in these data is 14. These data contain no information on how districts with extremely small classes perform, so these data alone are not a reliable basis for predicting the effect of a radical move to such an extremely low student–teacher ratio.
Why Use the OLS Estimator?
There are both practical and theoretical reasons to use the OLS estimators bn0 and bn1. Because OLS is the dominant method used in practice, it has become the com- mon language for regression analysis throughout economics, finance (see “The ‘Beta’ of a Stock” box), and the social sciences more generally. Presenting results
120 CHAPTER 4 Linear Regression with One Regressor The “Beta” of a Stock
Afundamental idea of modern finance is that an investor needs a financial incentive to take a risk. Said differently, the expected return1 on a risky investment, R, must exceed the return on a safe, or risk-free, investment, Rf . Thus the expected excess return, R – Rf , on a risky investment, like owning stock in a company, should be positive.
At first it might seem like the risk of a stock should be measured by its variance. Much of that risk, however, can be reduced by holding other stocks in a “portfolio”—in other words, by diversify- ing your financial holdings. This means that the right way to measure the risk of a stock is not by its vari- ance but rather by its covariance with the market.
The capital asset pricing model (CAPM) formal- izes this idea. According to the CAPM, the expected excess return on an asset is proportional to the expected excess return on a portfolio of all available assets (the “market portfolio”). That is, the CAPM says that
contrast, a stock with a b 7 1 is riskier than the mar- ket portfolio and thus commands a higher expected excess return.
The “beta” of a stock has become a workhorse of the investment industry, and you can obtain esti- mated betas for hundreds of stocks on investment firm websites. Those betas typically are estimated by OLS regression of the actual excess return on the stock against the actual excess return on a broad market index.
The table below gives estimated betas for seven U.S. stocks. Low-risk producers of consumer sta- ples like Kellogg have stocks with low betas; riskier stocks have high betas.
R – Rf = b(Rm – Rf),
(4.12)
Company
Verizon (telecommunications) Wal-Mart (discount retailer) Kellogg (breakfast cereal)
Waste Management (waste disposal) Google (information technology) Ford Motor Company (auto producer) Bank of America (bank)
Source: finance.yahoo.com.
Estimated B
0.0 0.3 0.5 0.6 1.0 1.3 2.2
where Rm is the expected return on the market portfolio and b is the coefficient in the population regression of R – Rf on Rm – Rf . In practice, the risk-free return is often taken to be the rate of inter- est on short-term U.S. government debt. Accord- ing to the CAPM, a stock with a b 6 1 has less risk than the market portfolio and therefore has a lower expected excess return than the market portfolio. In
1The return on an investment is the change in its price plus any payout (dividend) from the investment as a percentage of its initial price. For example, a stock bought on January 1 for $100, which then paid a $2.50 dividend during the year and sold on December 31 for $105, would have a return of R = 3($105 – $100) + $2.504 > $100 = 7.5%.
using OLS (or its variants discussed later in this book) means that you are “speak- ing the same language” as other economists and statisticians. The OLS formulas are built into virtually all spreadsheet and statistical software packages, making OLS easy to use.
The OLS estimators also have desirable theoretical properties. They are anal- ogous to the desirable properties, studied in Section 3.1, of Y as an estimator of the population mean. Under the assumptions introduced in Section 4.4, the OLS estimator is unbiased and consistent. The OLS estimator is also efficient among a certain class of unbiased estimators; however, this efficiency result holds under some additional special conditions, and further discussion of this result is deferred until Section 5.5.
4.3
Measures of Fit
Having estimated a linear regression, you might wonder how well that regression line describes the data. Does the regressor account for much or for little of the variation in the dependent variable? Are the observations tightly clustered around the regression line, or are they spread out?
The R2 and the standard error of the regression measure how well the OLS regression line fits the data. The R2 ranges between 0 and 1 and measures the fraction of the variance of Yi that is explained by Xi. The standard error of the regression measures how far Yi typically is from its predicted value.
The R2
The regression R2 is the fraction of the sample variance of Yi explained by (or predicted by) Xi. The definitions of the predicted value and the residual (see Key Concept 4.2) allow us to write the dependent variable Yi as the sum of the pre- dicted value, Yni, plus the residual uni:
Yi = Yni + uni. (4.13)
In this notation, the R2 is the ratio of the sample variance of Yn i to the sample vari- ance of Yi.
Mathematically, the R2 can be written as the ratio of the explained sum of squares to the total sum of squares. The explained sum of squares (ESS) is the sum of squared deviations of the predicted value,Yni, from its average, and the total sum of squares (TSS) is the sum of squared deviations of Yi from its average:
n
ESS = a(Yni – Y)2
i=1 n
TSS = a(Yi – Y)2. i=1
(4.14) (4.15)
4.3 Measures of Fit 121
122 CHAPTER 4
Linear Regression with One Regressor
Equation (4.14) uses the fact that the sample average OLS predicted value equals Y (proven in Appendix 4.3).
The R2 is the ratio of the explained sum of squares to the total sum of squares: R2 = ESS. (4.16)
TSS
Alternatively, the R2 can be written in terms of the fraction of the variance of Yi not explained by Xi. The sum of squared residuals, or SSR, is the sum of the squared OLS residuals:
n
SSR = aun2i. (4.17)
i=1
It is shown in Appendix 4.3 that TSS = ESS + SSR. Thus the R2 also can be expressed as 1 minus the ratio of the sum of squared residuals to the total sum of squares:
R2 = 1 – SSR. (4.18) TSS
Finally, the R2 of the regression of Y on the single regressor X is the square of the correlation coefficient between Y and X (Exercise 4.12).
The R2 ranges between 0 and 1. If bn1 = 0, then Xi explains none of the varia- tion of Yi and the predicted value of Yi is Yni = bn0 = Y [from Equation (4.8)]. In this case, the explained sum of squares is zero and the sum of squared residuals equals the total sum of squares; thus the R2 is zero. In contrast, if Xi explains all of the variation of Yi, then Yi = Yni for all i and every residual is zero (that is, uni = 0), so that ESS = TSS and R2 = 1. In general, the R2 does not take on the extreme values of 0 or 1 but falls somewhere in between. An R2 near 1 indicates that the regressor is good at predicting Yi, while an R2 near 0 indicates that the regressor is not very good at predicting Yi.
The Standard Error of the Regression
The standard error of the regression (SER) is an estimator of the standard devia- tion of the regression error ui. The units of ui and Yi are the same, so the SER is a measure of the spread of the observations around the regression line, measured in the units of the dependent variable. For example, if the units of the dependent variable are dollars, then the SER measures the magnitude of a typical deviation
from the regression line—that is, the magnitude of a typical regression error—in dollars.
Because the regression errors u1, c, un are unobserved, the SER is com- puted using their sample counterparts, the OLS residuals un1, c, unn. The formula for the SER is
1n SSR SER=suN=2s2uN,wheres2uN=n-2aun2i =n-2, (4.19)
i=1
where the formula for s2uN uses the fact (proven in Appendix 4.3) that the sample average of the OLS residuals is zero.
The formula for the SER in Equation (4.19) is similar to the formula for the sample standard deviation of Y given in Equation (3.7) in Section 3.2, except that Yi – Y in Equation (3.7) is replaced by uni and the divisor in Equation (3.7) is n – 1, whereas here it is n – 2. The reason for using the divisor n – 2 here (instead of n) is the same as the reason for using the divisor n – 1 in Equation (3.7): It corrects for a slight downward bias introduced because two regression coefficients were estimated. This is called a “degrees of freedom” correction because two coefficients were estimated (b0 and b1), two “degrees of freedom” of the data were lost, so the divisor in this factor is n – 2. (The mathematics behind this is discussed in Section 5.6.) When n is large, the difference between dividing by n, by n – 1, or by n – 2 is negligible.
Application to the Test Score Data
Equation (4.11) reports the regression line, estimated using the California test score data, relating the standardized test score (TestScore) to the student–teacher ratio (STR). The R2 of this regression is 0.051, or 5.1%, and the SER is 18.6.
The R2 of 0.051 means that the regressor STR explains 5.1% of the variance of the dependent variable TestScore. Figure 4.3 superimposes this regression line on the scatterplot of the TestScore and STR data. As the scatterplot shows, the student–teacher ratio explains some of the variation in test scores, but much vari- ation remains unaccounted for.
The SER of 18.6 means that standard deviation of the regression residuals is 18.6, where the units are points on the standardized test. Because the standard deviation is a measure of spread, the SER of 18.6 means that there is a large spread of the scatterplot in Figure 4.3 around the regression line as measured in points on the test. This large spread means that predictions of test scores made using only the student–teacher ratio for that district will often be wrong by a large amount.
4.3 Measures of Fit 123
124 CHAPTER 4
Linear Regression with One Regressor
4.4
The Least Squares Assumptions
This section presents a set of three assumptions on the linear regression model and the sampling scheme under which OLS provides an appropriate estimator of the unknown regression coefficients, b0 and b1. Initially, these assumptions might appear abstract. They do, however, have natural interpretations, and understand- ing these assumptions is essential for understanding when OLS will—and will not—give useful estimates of the regression coefficients.
Assumption #1: The Conditional Distribution of ui Given Xi Has a Mean of Zero
The first of the three least squares assumptions is that the conditional distribution of ui given Xi has a mean of zero. This assumption is a formal mathematical state- ment about the “other factors” contained in ui and asserts that these other factors are unrelated to Xi in the sense that, given a value of Xi, the mean of the distribu- tion of these other factors is zero.
This assumption is illustrated in Figure 4.4. The population regression is the relationship that holds on average between class size and test scores in the popu- lation, and the error term ui represents the other factors that lead test scores at a given district to differ from the prediction based on the population regression line. As shown in Figure 4.4, at a given value of class size, say 20 students per class, sometimes these other factors lead to better performance than predicted (ui 7 0) and sometimes to worse performance (ui 6 0), but on average over the popula- tion the prediction is right. In other words, given Xi = 20, the mean of the distri- bution of ui is zero. In Figure 4.4, this is shown as the distribution of ui being centered on the population regression line at Xi = 20 and, more generally, at other values x of Xi as well. Said differently, the distribution of ui, conditional on Xi = x, has a mean of zero; stated mathematically, E(uiXi = x) = 0, or, in somewhat simpler notation, E(ui Xi) = 0.
What should we make of this low R2 and large SER? The fact that the R2 of this regression is low (and the SER is large) does not, by itself, imply that this regression is either “good” or “bad.” What the low R2 does tell us is that other important factors influence test scores. These factors could include differences in the student body across districts, differences in school quality unrelated to the student–teacher ratio, or luck on the test. The low R2 and high SER do not tell us what these factors are, but they do indicate that the student–teacher ratio alone explains only a small part of the variation in test scores in these data.
4.4 The Least Squares Assumptions 125 The Conditional Probability Distributions and the Population
Regression Line
25
15 20 25 30
Student–teacher ratio
The figure shows the conditional probability of test scores for districts with class sizes of 15, 20, and 25 students. The mean of the conditional distribution of test scores, given the student– teacher ratio, E(Y X), is the population regression line. At a given value of X, Y is distributed around the regression line and the error, u = Y – (b0 + b1X), has a conditional mean of zero for all values of X.
As shown in Figure 4.4, the assumption that E(uiXi) = 0 is equivalent to assuming that the population regression line is the conditional mean of Yi given Xi (a mathematical proof of this is left as Exercise 4.6).
The conditional mean of u in a randomized controlled experiment. In a random- ized controlled experiment, subjects are randomly assigned to the treatment group (X = 1) or to the control group (X = 0). The random assignment typically is done using a computer program that uses no information about the subject, ensuring that X is distributed independently of all personal characteristics of the subject. Random assignment makes X and u independent, which in turn implies that the conditional mean of u given X is zero.
In observational data, X is not randomly assigned in an experiment. Instead, the best that can be hoped for is that X is as if randomly assigned, in the precise sense that E(uiXi) = 0. Whether this assumption holds in a given empirical application with observational data requires careful thought and judgment, and we return to this issue repeatedly.
FIGURE 4.4
Test score
720
700
680
660
640
620
Distribution of Y when X = 15
E(YΩX = 15)
Distribution of Y when X = 20
E(YΩX = 20)
Distribution of Y when X =
b0 +b1X
E(YΩX = 25)
600
10
126 CHAPTER 4
Linear Regression with One Regressor
Correlation and conditional mean. Recall from Section 2.3 that if the conditional mean of one random variable given another is zero, then the two random variables have zero covariance and thus are uncorrelated [Equation (2.27)]. Thus the condi- tional mean assumption E(ui Xi) = 0 implies that Xi and ui are uncorrelated, or corr(Xi, ui) = 0. Because correlation is a measure of linear association, this impli- cation does not go the other way; even if Xi and ui are uncorrelated, the conditional mean of ui given Xi might be nonzero. However, if Xi and ui are correlated, then it must be the case that E(ui Xi) is nonzero. It is therefore often convenient to discuss the conditional mean assumption in terms of possible correlation between Xi and ui. If Xi and ui are correlated, then the conditional mean assumption is violated.
Assumption #2: (Xi, Yi), i = 1, . . . , n, Are
Independently and Identically Distributed
The second least squares assumption is that (Xi, Yi ), i = 1, c, n, are indepen- dently and identically distributed (i.i.d.) across observations. As discussed in Sec- tion 2.5 (Key Concept 2.5), this assumption is a statement about how the sample is drawn. If the observations are drawn by simple random sampling from a single large population, then (Xi, Yi ), i = 1, c, n, are i.i.d. For example, let X be the age of a worker and Y be his or her earnings, and imagine drawing a person at random from the population of workers. That randomly drawn person will have a certain age and earnings (that is, X and Y will take on some values). If a sample of n workers is drawn from this population, then (Xi, Yi ), i = 1, c, n, necessar- ily have the same distribution. If they are drawn at random they are also distrib- uted independently from one observation to the next; that is, they are i.i.d.
The i.i.d. assumption is a reasonable one for many data collection schemes. For example, survey data from a randomly chosen subset of the population typi- cally can be treated as i.i.d.
Not all sampling schemes produce i.i.d. observations on (Xi, Yi), however. One example is when the values of X are not drawn from a random sample of the popu- lation but rather are set by a researcher as part of an experiment. For example, suppose a horticulturalist wants to study the effects of different organic weeding methods (X) on tomato production (Y) and accordingly grows different plots of tomatoes using different organic weeding techniques. If she picks the techniques (the level of X) to be used on the ith plot and applies the same technique to the ith plot in all repetitions of the experiment, then the value of Xi does not change from one sample to the next. Thus Xi is nonrandom (although the outcome Yi is random), so the sampling scheme is not i.i.d. The results presented in this chapter developed for i.i.d. regressors are also true if the regressors are nonrandom. The case of a
nonrandom regressor is, however, quite special. For example, modern experimen- tal protocols would have the horticulturalist assign the level of X to the different plots using a computerized random number generator, thereby circumventing any possible bias by the horticulturalist (she might use her favorite weeding method for the tomatoes in the sunniest plot). When this modern experimental protocol is used, the level of X is random and (Xi, Yi) are i.i.d.
Another example of non-i.i.d. sampling is when observations refer to the same unit of observation over time. For example, we might have data on inven- tory levels (Y) at a firm and the interest rate at which the firm can borrow (X), where these data are collected over time from a specific firm; for example, they might be recorded four times a year (quarterly) for 30 years. This is an example of time series data, and a key feature of time series data is that observations falling close to each other in time are not independent but rather tend to be correlated with each other; if interest rates are low now, they are likely to be low next quar- ter. This pattern of correlation violates the “independence” part of the i.i.d. assumption. Time series data introduce a set of complications that are best han- dled after developing the basic tools of regression analysis, so we postpone discus- sion of time series data until Chapter 14.
Assumption #3: Large Outliers Are Unlikely
The third least squares assumption is that large outliers—that is, observations with values of Xi, Yi, or both that are far outside the usual range of the data—are unlikely. Large outliers can make OLS regression results misleading. This potential sensitivity of OLS to extreme outliers is illustrated in Figure 4.5 using hypothetical data.
In this book, the assumption that large outliers are unlikely is made mathe- matically precise by assuming that X and Y have nonzero finite fourth moments: 0 6 E(X4i) 6 ∞ and 0 6 E(Y4i) 6 ∞. Another way to state this assumption is that X and Y have finite kurtosis.
The assumption of finite kurtosis is used in the mathematics that justify the large-sample approximations to the distributions of the OLS test statistics. For example, we encountered this assumption in Chapter 3 when discussing the con- sistency of the sample variance. Specifically, Equation (3.9) states that the sample variance is a consistent estimator of the population variance s2Y (s2Y ¡p s2Y). If Y1, c, Yn are i.i.d. and the fourth moment of Yi is finite, then the law of large numbers in Key Concept 2.6 applies to the average, n1 g ni = 1Yi2, a key step in the proof in Appendix 3.3 showing that s2Y is consistent.
One source of large outliers is data entry errors, such as a typographical error or incorrectly using different units for different observations. Imagine collecting data on the height of students in meters, but inadvertently recording one student’s
4.4 The Least Squares Assumptions 127
128 CHAPTER 4 Linear Regression with One Regressor FIGURE 4.5 The Sensitivity of OLS to Large Outliers
This hypothetical data set has one
outlier. The OLS regression line
estimated with the outlier shows
a strong positive relationship between 1700 X and Y, but the OLS regression line estimated without the outlier shows 1400 no relationship.
1100
800
500
200
Y
2000
OLS regression line including outlier
OLS regression line excluding outlier
0
30 40 50 60 70
X
height in centimeters instead. This would create a large outlier in the sample. One way to find outliers is to plot your data. If you decide that an outlier is due to a data entry error, then you can either correct the error or, if that is impossible, drop the observation from your data set.
Data entry errors aside, the assumption of finite kurtosis is a plausible one in many applications with economic data. Class size is capped by the physical capac- ity of a classroom; the best you can do on a standardized test is to get all the ques- tions right and the worst you can do is to get all the questions wrong. Because class size and test scores have a finite range, they necessarily have finite kurtosis. More generally, commonly used distributions such as the normal distribution have four moments. Still, as a mathematical matter, some distributions have infinite fourth moments, and this assumption rules out those distributions. If the assumption of finite fourth moments holds, then it is unlikely that statistical inferences using OLS will be dominated by a few observations.
Use of the Least Squares Assumptions
The three least squares assumptions for the linear regression model are summa- rized in Key Concept 4.3. The least squares assumptions play twin roles, and we return to them repeatedly throughout this textbook.
4.5 Sampling Distribution of the OLS Estimators 129
The Least Squares Assumptions
Yi = b0 + b1Xi + ui, i = 1, c, n, where
1. The error term ui has conditional mean zero given Xi: E(ui Xi) = 0;
KEY CONCEPT
4.3
2. (Xi, Yi ), i = 1, c, n, are independent and identically distributed (i.i.d.) draws from their joint distribution; and
3. Large outliers are unlikely: Xi and Yi have nonzero finite fourth moments.
Their first role is mathematical: If these assumptions hold, then, as is shown in the next section, in large samples the OLS estimators have sampling distribu- tions that are normal. In turn, this large-sample normal distribution lets us develop methods for hypothesis testing and constructing confidence intervals using the OLS estimators.
Their second role is to organize the circumstances that pose difficulties for OLS regression. As we will see, the first least squares assumption is the most important to consider in practice. One reason why the first least squares assump- tion might not hold in practice is discussed in Chapter 6, and additional reasons are discussed in Section 9.2.
It is also important to consider whether the second assumption holds in an applica- tion. Although it plausibly holds in many cross-sectional data sets, the independence assumption is inappropriate for panel and time series data. Therefore, the regression methods developed under assumption 2 require modification for some applications with time series data. These modifications are developed in Chapters 10 and 14–16.
The third assumption serves as a reminder that OLS, just like the sample mean, can be sensitive to large outliers. If your data set contains large outliers, you should examine those outliers carefully to make sure those observations are correctly recorded and belong in the data set.
4.5
Sampling Distribution of the OLS Estimators
Because the OLS estimators bn0 and bn1 are computed from a randomly drawn sam- ple, the estimators themselves are random variables with a probability distribution— the sampling distribution—that describes the values they could take over different possible random samples. This section presents these sampling distributions.
130 CHAPTER 4
Linear Regression with One Regressor
In small samples, these distributions are complicated, but in large samples, they are approximately normal because of the central limit theorem.
The Sampling Distribution of the OLS Estimators
Review of the sampling distribution of Y . Recall the discussion in Sections 2.5 and 2.6 about the sampling distribution of the sample average, Y, an estimator of the unknown population mean of Y, mY. Because Y is calculated using a randomly drawn sample, Y is a random variable that takes on different values from one sample to the next; the probability of these different values is summarized in its sampling distribution. Although the sampling distribution of Y can be complicated when the sample size is small, it is possible to make certain statements about it that hold for all n. In particular, the mean of the sampling distribution is mY, that is, E(Y) = mY, so Y is an unbiased estimator of mY. If n is large, then more can be said about the sampling distribution. In particular, the central limit theorem (Section 2.6) states that this distribution is approximately normal.
The sampling distribution of bn0 and bn1. These ideas carry over to the OLS estima- tors bn0 and bn1 of the unknown intercept b0 and slope b1 of the population regres- sion line. Because the OLS estimators are calculated using a random sample, bn0 and bn1 are random variables that take on different values from one sample to the next; the probability of these different values is summarized in their sampling distributions.
Although the sampling distribution of bn0 and bn1 can be complicated when the sample size is small, it is possible to make certain statements about it that hold for all n. In particular, the mean of the sampling distributions of bn0 and bn1 are b0 and b1. In other words, under the least squares assumptions in Key Concept 4.3,
E(bn0) = b0 and E(bn1) = b1; (4.20)
that is, bn0 and bn1 are unbiased estimators of b0 and b1. The proof that bn1 is unbiased is given in Appendix 4.3, and the proof that bn0 is unbiased is left as Exercise 4.7.
If the sample is sufficiently large, by the central limit theorem the sampling distribution of bn0 and bn1 is well approximated by the bivariate normal distribution (Section 2.4). This implies that the marginal distributions of bn0 and bn1 are normal in large samples.
This argument invokes the central limit theorem. Technically, the central limit theorem concerns the distribution of averages (like Y). If you examine the numerator in Equation (4.7) for bn1, you will see that it, too, is a type of average—not a simple average, like Y, but an average of the product, (Yi – Y)(Xi – X). As discussed
4.5 Sampling Distribution of the OLS Estimators 131
Large-Sample Distributions of bn0 and bn1
KEY CONCEPT
4.4
If the least squares assumptions in Key Concept 4.3 hold, then in large samples
bn0 and bn1 have a jointly normal sampling distribution. The large-sample normal
distribution of bn is N(b , s2 ), where the variance of this distribution, s2 , is
11bb NN
1
1 var3(Xi – mX)ui4 sbN1 =n 3var(Xi)42 .
The large-sample normal distribution of bn is N(b , s2 ), where
0
s2 =1var(Hiui),whereH=1-c mX dX. bN 0 n 3 E ( H 2i ) 4 2 i E ( X 2i ) i
1
2
(4.21)
(4.22)
00b N
further in Appendix 4.3, the central limit theorem applies to this average so that, like the simpler average Y, it is normally distributed in large samples.
The normal approximation to the distribution of the OLS estimators in large samples is summarized in Key Concept 4.4. (Appendix 4.3 summarizes the deriva- tion of these formulas.) A relevant question in practice is how large n must be for these approximations to be reliable. In Section 2.6, we suggested that n = 100 is sufficiently large for the sampling distribution of Y to be well approximated by a normal distribution, and sometimes smaller n suffices. This criterion carries over to the more complicated averages appearing in regression analysis. In virtually all modern econometric applications, n 7 100, so we will treat the normal approxi- mations to the distributions of the OLS estimators as reliable unless there are good reasons to think otherwise.
The results in Key Concept 4.4 imply that the OLS estimators are consistent— that is, when the sample size is large, bn0 and bn1 will be close to the true population coefficients b0 and b1 with high probability. This is because the variances s2bN0 and s2bN1 of the estimators decrease to zero as n increases (n appears in the denominator of the formulas for the variances), so the distribution of the OLS estimators will be tightly concentrated around their means, b0 and b1, when n is large.
Another implication of the distributions in Key Concept 4.4 is that, in general, the larger is the variance of Xi, the smaller is the variance s2bN1 of bn1. Mathemati- cally, this implication arises because the variance of bn1 in Equation (4.21) is inversely proportional to the square of the variance of Xi: the larger is var(Xi), the larger is the denominator in Equation (4.21) so the smaller is s2bN1. To get a better sense
132 CHAPTER 4 Linear Regression with One Regressor FIGURE 4.6 The Variance of Bn1 and the Variance of X
The colored dots represent
a set of Xi’s with a small 206 variance. The black dots represent a set of Xi’s with 204 a large variance. The
regression line can be
estimated more accurately 202 with the black dots than
with the colored dots.
200 198 196
194
97 98 99 100 101 102 103
X
of why this is so, look at Figure 4.6, which presents a scatterplot of 150 artificial data points on X and Y. The data points indicated by the colored dots are the 75 observa- tions closest to X. Suppose you were asked to draw a line as accurately as possible through either the colored or the black dots—which would you choose? It would be easier to draw a precise line through the black dots, which have a larger variance than the colored dots. Similarly, the larger the variance of X, the more precise is bn1.
The distributions in Key Concept 4.4 also imply that the smaller is the vari- ance of the error ui, the smaller is the variance of bn1. This can be seen mathemat- ically in Equation (4.21) because ui enters the numerator, but not denominator, of s2bN1: If all ui were smaller by a factor of one-half but the X’s did not change, then sbN1 would be smaller by a factor of one-half and s2bN1 would be smaller by a factor of one-fourth (Exercise 4.13). Stated less mathematically, if the errors are smaller (holding the X’s fixed), then the data will have a tighter scatter around the popu- lation regression line so its slope will be estimated more precisely.
The normal approximation to the sampling distribution of bn0 and bn1 is a pow- erful tool. With this approximation in hand, we are able to develop methods for making inferences about the true population values of the regression coefficients using only a sample of data.
Y
4.6
Conclusion
This chapter has focused on the use of ordinary least squares to estimate the intercept and slope of a population regression line using a sample of n observa- tions on a dependent variable, Y, and a single regressor, X. There are many ways to draw a straight line through a scatterplot, but doing so using OLS has several virtues. If the least squares assumptions hold, then the OLS estimators of the slope and intercept are unbiased, are consistent, and have a sampling distribution with a variance that is inversely proportional to the sample size n. Moreover, if n is large, then the sampling distribution of the OLS estimator is normal.
These important properties of the sampling distribution of the OLS estimator hold under the three least squares assumptions.
The first assumption is that the error term in the linear regression model has a conditional mean of zero, given the regressor X. This assumption implies that the OLS estimator is unbiased.
The second assumption is that (Xi, Yi) are i.i.d., as is the case if the data are col- lected by simple random sampling. This assumption yields the formula, presented in Key Concept 4.4, for the variance of the sampling distribution of the OLS estimator.
The third assumption is that large outliers are unlikely. Stated more formally, X and Y have finite fourth moments (finite kurtosis). The reason for this assump- tion is that OLS can be unreliable if there are large outliers. Taken together, the three least squares assumptions imply that the OLS estimator is normally distrib- uted in large samples as described in Key Concept 4.4.
The results in this chapter describe the sampling distribution of the OLS esti- mator. By themselves, however, these results are not sufficient to test a hypoth- esis about the value of b1 or to construct a confidence interval for b1. Doing so requires an estimator of the standard deviation of the sampling distribution—that is, the standard error of the OLS estimator. This step—moving from the sam- pling distribution of bn1 to its standard error, hypothesis tests, and confidence intervals—is taken in the next chapter.
Summary
1. The population regression line, b0 + b1X, is the mean of Y as a function of the value of X. The slope, b1, is the expected change in Y associated with a one-unit change in X. The intercept, b0, determines the level (or height) of the regression line. Key Concept 4.1 summarizes the terminology of the population linear regression model.
Summary 133
134 CHAPTER 4
Linear Regression with One Regressor
2. The population regression line can be estimated using sample observations (Yi, Xi), i = 1, c, n by ordinary least squares (OLS). The OLS estimators of the regression intercept and slope are denoted bn0 and bn1.
3. The R2 and standard error of the regression (SER) are measures of how close the values of Yi are to the estimated regression line. The R2 is between 0 and 1, with a larger value indicating that the Yi’s are closer to the line. The standard error of the regression is an estimator of the standard deviation of the regression error.
4. There are three key assumptions for the linear regression model: (1) The regression errors, ui, have a mean of zero, conditional on the regressors Xi; (2) the sample observations are i.i.d. random draws from the population; and (3) large outliers are unlikely. If these assumptions hold, the OLS estimators bn0 and bn1 are (1) unbiased, (2) consistent, and (3) normally distributed when the sample is large.
Key Terms
linear regression model with a single regressor (112)
dependent variable (112) independent variable (112) regressor (112)
population regression line (112) population regression function (112) population intercept (112) population slope (112)
population coefficients (112) parameters (112)
error term (112)
ordinary least squares (OLS)
estimators (116)
OLS regression line (116) sample regression line (116) sample regression function (116) predicted value (116)
residual (117)
regression R2 (121)
explained sum of squares
(ESS) (121)
total sum of squares (TSS) (121) sum of squared residuals
(SSR) (122)
standard error of the regression
(SER) (122)
least squares assumptions (124)
MyEconLab Can Help You Get a Better Grade
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To see how it works, turn to the MyEconLab spread on the inside front cover of this book and then go to www.myeconlab.com.
For additional Empirical Exercises and Data Sets, log on to the Companion Website at www.pearsonhighered.com/stock_watson.
Review the Concepts
4.1 Explain the difference between bn1 and b1; between the residual uni and the regression error ui; and between the OLS predicted value Yn i and E(Yi Xi).
4.2 For each least squares assumption, provide an example in which the assump- tion is valid and then provide an example in which the assumption fails.
4.3 SER and R2 are “measures of fit” for a regression. Explain how SER mea- sures the fit of a regression. What are the units of SER? Explain how R2 measures the fit of a regression. What are the units of R2?
4.4 Sketch a hypothetical scatterplot of data for an estimated regression with R2 = 0.9. Sketch a hypothetical scatterplot of data for a regression with R2 = 0.5.
Exercises
4.1 Suppose that a researcher, using data on class size (CS) and average test scores from 100 third-grade classes, estimates the OLS regression:
TestScore = 520.4 – 5.82 * CS, R2 = 0.08, SER = 11.5.
a. A classroom has 22 students. What is the regression’s prediction for
that classroom’s average test score?
b. Last year a classroom had 19 students, and this year it has 23 students. What is the regression’s prediction for the change in the classroom average test score?
c. The sample average class size across the 100 classrooms is 21.4. What is the sample average of the test scores across the 100 classrooms? (Hint: Review the formulas for the OLS estimators.)
d. What is the sample standard deviation of test scores across the 100 classrooms? (Hint: Review the formulas for the R2 and SER.)
4.2 Suppose that a random sample of 200 20-year-old men is selected from a population and that these men’s height and weight are recorded. A regres- sion of weight on height yields
Weight = – 99.41 + 3.94 * Height, R2 = 0.81, SER = 10.2, where Weight is measured in pounds and Height is measured in inches.
Exercises 135
136 CHAPTER 4
Linear Regression with One Regressor
a. What is the regression’s weight prediction for someone who is 70 in. tall? 65 in. tall? 74 in. tall?
b. A man has a late growth spurt and grows 1.5 in. over the course of a year. What is the regression’s prediction for the increase in this man’s weight?
c. Suppose that instead of measuring weight and height in pounds and inches, these variables are measured in centimeters and kilograms. What are the regression estimates from this new centimeter–kilogram regression? (Give all results, estimated coefficients, R2, and SER.)
4.3 A regression of average weekly earnings (AWE, measured in dollars) on age (measured in years) using a random sample of college-educated full-time workers aged 25–65 yields the following:
AWE = 696.7 + 9.6 * Age, R2 = 0.023, SER = 624.1.
a. Explain what the coefficient values 696.7 and 9.6 mean.
b. The standard error of the regression (SER) is 624.1. What are the units of measurement for the SER? (Dollars? Years? Or is SER unit-free?)
c. The regression R2 is 0.023. What are the units of measurement for the R2? (Dollars? Years? Or is R2 unit-free?)
d. What does the regression predict will be the earnings for a 25-year-old worker? For a 45-year-old worker?
e. Will the regression give reliable predictions for a 99-year-old worker? Why or why not?
f. Given what you know about the distribution of earnings, do you think it is plausible that the distribution of errors in the regression is normal? (Hint: Do you think that the distribution is symmetric or skewed? What is the smallest value of earnings, and is it consistent with a normal distribution?)
g. The average age in this sample is 41.6 years. What is the average value of AWE in the sample? (Hint: Review Key Concept 4.2.)
4.4 Read the box “The ‘Beta’ of a Stock” in Section 4.2.
a. Suppose that the value of b is greater than 1 for a particular stock. Show that the variance of (R – Rf) for this stock is greater than the variance of (Rm – Rt).
b. Suppose that the value of b is less than 1 for a particular stock. Is it possible that variance of (R – Rf) for this stock is greater than the variance of (Rm – Rt)? (Hint: Don’t forget the regression error.)
c. In a given year, the rate of return on 3-month Treasury bills is 2.0% and the rate of return on a large diversified portfolio of stocks (the S&P 500) is 5.3%. For each company listed in the table in the box, use the estimated value of b to estimate the stock’s expected rate of return.
4.5 A professor decides to run an experiment to measure the effect of time pressure on final exam scores. He gives each of the 400 students in his course the same final exam, but some students have 90 minutes to com- plete the exam, while others have 120 minutes. Each student is randomly assigned one of the examination times, based on the flip of a coin. Let Yi denote the number of points scored on the exam by the ith student (0 … Yi … 100), let Xi denote the amount of time that the student has to complete the exam (Xi = 90 or 120), and consider the regression model Yi =b0 +b1Xi +ui.
a. Explain what the term ui represents. Why will different students have different values of ui?
b. Explain why E(ui Xi) = 0 for this regression model.
c. Are the other assumptions in Key Concept 4.3 satisfied? Explain.
d. The estimated regression is Yni = 49 + 0.24 Xi.
i. Compute the estimated regression’s prediction for the average score of students given 90 minutes to complete the exam. Repeat for 120 minutes and 150 minutes.
ii. Compute the estimated gain in score for a student who is given an additional 10 minutes on the exam.
4.6 Show that the first least squares assumption, E(uiXi) = 0, implies that E(YiXi) = b0 + b1Xi.
4.7 Show that bn0 is an unbiased estimator of b0. (Hint: Use the fact that bn1 is unbiased, which is shown in Appendix 4.3.)
4.8 Suppose that all of the regression assumptions in Key Concept 4.3 are satis- fied except that the first assumption is replaced with E(ui Xi) = 2. Which parts of Key Concept 4.4 continue to hold? Which change? Why? (Is bn1 normally distributed in large samples with mean and variance given in Key Concept 4.4? What about bn0?)
4.9 a. A linear regression yields bn1 = 0. Show that R2 = 0.
b. A linear regression yields R2 = 0. Does this imply that bn1 = 0?
Exercises 137
138 CHAPTER 4
Linear Regression with One Regressor
4.10 Suppose that Yi = b0 + b1Xi + ui, where (Xi, ui) are i.i.d., and Xi is a Bernoulli random variable with Pr(X = 1) = 0.20. When X = 1, ui is N(0, 4); when X = 0, ui is N(0, 1).
a. Show that the regression assumptions in Key Concept 4.3 are satisfied.
b. Derive an expression for the large-sample variance of bn1. [Hint: Evaluate the terms in Equation (4.21).]
4.11 Consider the regression model Yi = b0 + b1Xi + ui.
a. Suppose you know that b0 = 0. Derive a formula for the least squares
estimator of b1.
b. Suppose you know that b0 = 4. Derive a formula for the least squares
4.12 a.
estimator of b1.
Show that the regression R2 in the regression of Y on X is the squared value of the sample correlation between X and Y. That is, show that R 2 = r 2X Y .
b. Show that the R2 from the regression of Y on X is the same as the R2 from the regression of X on Y.
c. Show that bn1 = rXY(sY>sX), where rXY is the sample correlation between X and Y, and sX and sY are the sample standard deviations of X and Y.
4.13 Suppose that Yi = b0 + b1Xi + kui, where k is a nonzero constant and (Yi, Xi) satisfy the three least squares assumptions. Show that the large
n
tion is the variance given in Equation (4.21) multiplied by k2.]
4.14 Show that the sample regression line passes through the point (X, Y).
Empirical Exercises
(Only two empirical exercises for this chapter are given in the text, but you can find more on the text website, http://www.pearsonhighered.com/stock_watson/.)
E4.1 On the text website, http://www.pearsonhighered.com/stock_watson/, you will find the data file Growth, which contains data on average growth rates from 1960 through 1995 for 65 countries, along with variables that are potentially related to growth. A detailed description is given in
2 21var3(Xi – mX)ui4
sample variance of b is given by sN 1 = k n 3var(X ) 4 . [Hint: This equa- 1b2
i
Empirical Exercises 139 Growth_Description, also available on the website. In this exercise, you
will investigate the relationship between growth and trade.1
a. Construct a scatterplot of average annual growth rate (Growth) on the average trade share (TradeShare). Does there appear to be a relationship between the variables?
b. One country, Malta, has a trade share much larger than the other countries. Find Malta on the scatterplot. Does Malta look like an outlier?
c. Using all observations, run a regression of Growth on TradeShare. What is the estimated slope? What is the estimated intercept? Use the regression to predict the growth rate for a country with a trade share of 0.5 and with a trade share equal to 1.0.
d. Estimate the same regression, excluding the data from Malta. Answer the same questions in (c).
e. Plot the estimated regression functions from (c) and (d). Using the scatterplot in (a), explain why the regression function that includes Malta is steeper than the regression function that excludes Malta.
f. Where is Malta? Why is the Malta trade share so large? Should Malta be included or excluded from the analysis?
E4.2 On the text website, http://www.pearsonhighered.com/stock_watson/, you will find the data file Earnings_and_Height, which contains data on earn- ings, height, and other characteristics of a random sample of U.S. workers.2 A detailed description is given in Earnings_and_Height_Description, also available on the website. In this exercise, you will investigate the relation- ship between earnings and height.
a. What is the median value of height in the sample?
b. i. Estimate average earnings for workers whose height is at most
67 inches.
ii. Estimate average earnings for workers whose height is greater than 67 inches.
1These data were provided by Professor Ross Levine of the University of California at Berkeley and were used in his paper with Thorsten Beck and Norman Loayza, “Finance and the Sources of Growth,” Journal of Financial Economics, 2000, 58: 261–300.
2These data were provided by Professors Anne Case (Princeton University) and Christina Paxson (Brown University) and were used in their paper “Stature and Status: Height, Ability, and Labor Market Outcomes,” Journal of Political Economy, 2008, 116(3): 499–532.
140 CHAPTER 4
Linear Regression with One Regressor
iii. On average, do taller workers earn more than shorter workers? How much more? What is a 95% confidence interval for the difference in average earnings?
c. Construct a scatterplot of annual earnings (Earnings) on height (Height). Notice that the points on the plot fall along horizontal lines. (There are only 23 distinct values of Earnings). Why? (Hint: Carefully read the detailed data description.)
d. Run a regression of Earnings on Height.
i. What is the estimated slope?
ii. Use the estimated regression to predict earnings for a worker who is 67 inches tall, for a worker who is 70 inches tall, and for a worker who is 65 inches tall.
e. Suppose height were measured in centimeters instead of inches. Answer the following questions about the Earnings on Height (in cm) regression.
i. What is the estimated slope of the regression?
ii. What is the estimated intercept?
iii. What is the R2?
iv. What is the standard error of the regression?
f. Run a regression of Earnings on Height, using data for female workers only.
i. What is the estimated slope?
ii. A randomly selected woman is 1 inch taller than the average woman in the sample. Would you predict her earnings to be higher or lower than the average earnings for women in the sam- ple? By how much?
g. Repeat (f) for male workers.
h. Do you think that height is uncorrelated with other factors that cause earning? That is, do you think that the regression error term, say ui, has a conditional mean of zero, given Height (Xi)? (You will investigate this more in the Earnings and Height exercises in later chapters.)
APPENDIX
4.1
The California Test Score Data Set
The California Standardized Testing and Reporting data set contains data on test per- formance, school characteristics, and student demographic backgrounds. The data used here are from all 420 K–6 and K–8 districts in California with data available for 1999. Test scores are the average of the reading and math scores on the Stanford 9 Achieve- ment Test, a standardized test administered to fifth-grade students. School characteris- tics (averaged across the district) include enrollment, number of teachers (measured as “full-time equivalents”), number of computers per classroom, and expenditures per stu- dent. The student–teacher ratio used here is the number of students in the district divided by the number of full-time equivalent teachers. Demographic variables for the students also are averaged across the district. The demographic variables include the percentage of students who are in the public assistance program CalWorks (formerly AFDC), the percentage of students who qualify for a reduced-price lunch, and the percentage of students who are English learners (that is, students for whom English is a second lan- guage). All of these data were obtained from the California Department of Education (www.cde.ca.gov).
Derivation of the OLS Estimators 141
APPENDIX
4.2
Derivation of the OLS Estimators
This appendix uses calculus to derive the formulas for the OLS estimators given in Key Concept 4.2. To minimize the sum of squared prediction mistakes g ni = 1(Yi – b0 – b1Xi)2 [Equation (4.6)], first take the partial derivatives with respect to b0 and b1:
0nn
0b a(Yi – b0 – b1Xi)2 = -2a(Yi – b0 – b1Xi) and
0i=1 i=1 0nn
0b a(Yi – b0 – b1Xi)2 = -2a(Yi – b0 – b1Xi)Xi. 1i=1 i=1
(4.23) (4.24)
The OLS estimators, bn0 and bn1, are the values of b0 and b1 that minimize g ni = 1(Yi – b0 – b1Xi)2, or, equivalently, the values of b0 and b1 for which the derivatives in Equations (4.23) and (4.24) equal zero. Accordingly, setting these derivatives equal to
142 CHAPTER 4
Linear Regression with One Regressor
zero, collecting terms, and dividing by n shows that the OLS estimators, bn0 and bn1, must satisfy the two equations
Y – bn0 – bn1X = 0and
1n 1n naXiYi – bn0X – bn1naX2i = 0.
i=1 i=1 Solving this pair of equations for bn0 and bn1 yields
1nn
naXiYi – XY a(Xi – X)(Yi – Y) bn = i = 1 = i = 1
11nn
n a X 2i – ( X ) 2 a ( X i – X ) 2
i=1 i=1 bn0 = Y – bn1X.
a n d
(4.25)
(4.26)
( 4 . 2 7 )
(4.28)
Equations (4.27) and (4.28) are the formulas for bn0 and bn1 given in Key Concept 4.2; the formula bn1 = sXY > s2X is obtained by dividing the numerator and denominator in Equation (4.27) by n – 1.
4.3
APPENDIX
Sampling Distribution of the OLS Estimator
In this appendix, we show that the OLS estimator bn1 is unbiased and, in large samples, has the normal sampling distribution given in Key Concept 4.4.
Representation of bn1 in Terms of the Regressors
and Errors
We start by providing an expression for bn1 in terms of the regressors and errors. Because Yi = b0 + b1Xi + ui,Yi – Y = b1(Xi – X) + ui – u,sothenumeratoroftheformulafor bn1 in Equation (4.27) is
nn
a(Xi -X)(Yi -Y)= a(Xi -X)3b1(Xi -X)+(ui -u)4 i=1 i=1
nn
= b1a(Xi – X)2 + a(Xi – X)(ui – u). i=1 i=1
(4.29)
Sampling Distribution of the OLS Estimator 143
Now gni=1(Xi – X)(ui – u) = gni=1(Xi – X)ui – gni=1(Xi – X)u = gni=1(Xi -X)ui,where
the final equality follows from the definition of X, which implies that gni=1(Xi – X)u =
3gn X – nX4u = 0. Substituting gn (X – X)(u – u) = gn (X – X)u into the i=1i i=1iii=1ii
final expression in Equation (4.29) yields g ni = 1(Xi – X)(Yi – Y) = b1 g ni = 1(Xi – X)2 + g ni = 1(Xi – X)ui. Substituting this expression in turn into the formula for bn1 in Equation (4.27) yields
1n
na(Xi – X)ui
bn=b+ i=1 . 111n
(4.30)
Proof That bn1 Is Unbiased
The expectation of bn1 is obtained by taking the expectation of both sides of Equation (4.30).
Thus,
n i=1
E(b)=b +E≥na(Xi -X)ui¥
1n
na(Xi – X)2 i=1
111n
na(Xi – X)2
i=1 1n
= b + E≥na(Xi – X)E(uiXi,c,Xn)¥ = b, i=1
11n1 na(Xi – X)2
i=1
(4.31)
where the second equality in Equation (4.31) follows by using the law of iterated expecta- tions (Section 2.3). By the second least squares assumption, ui is distributed independently of X for all observations other than i, so E(ui X1, c, Xn) = E(ui Xi). By the first least squares assumption, however, E(ui Xi) = 0. It follows that the conditional expectation in large brackets in the second line of Equation (4.31) is zero, so that E(bn1 – b1 X1, c Xn) = 0. Equivalently, E(bn1X1, c, Xn) = b1; that is, bn1 is conditionally unbiased, given X1, c, Xn. By the law of iterated expectations, E(bn1 – b1) = E3E(bn1 – b1 X1, c, Xn)4 = 0, so that E(bn1) = b1; that is, bn1 is unbiased.
Large-Sample Normal Distribution
of the OLS Estimator
The large-sample normal approximation to the limiting distribution of bn1 (Key Concept 4.4) is obtained by considering the behavior of the final term in Equation (4.30).
144 CHAPTER 4
Linear Regression with One Regressor
First consider the numerator of this term. Because X is consistent, if the sample size is large, X is nearly equal to mX. Thus, to a close approximation, the term in the numerator of Equation (4.30) is the sample average n, where vi = (Xi – mX)ui. By the first least squares assumption, vi has a mean of zero. By the second least squares assumption, vi is i.i.d. The variance of vi is s2v = 3var(Xi – mX)ui4, which, by the third least squares assump- tion, is nonzero and finite. Therefore, v satisfies all the requirements of the central limit theorem (Key Concept 2.7). Thus v>sv is, in large samples, distributed N(0, 1), where s2v = s2v > n. Thus the distribution of v is well approximated by the N(0, s2v > n) distribution.
Next consider the expression in the denominator in Equation (4.30); this is the sample variance of X (except dividing by n rather than n – 1, which is inconsequential if n is large). As discussed in Section 3.2 [Equation (3.8)], the sample variance is a consistent estimator of the population variance, so in large samples it is arbitrarily close to the population variance of X.
Combining these two results, we have that, in large samples, bn1 – b1 ≅ v > var(Xi),
n
2
so that the sampling distribution of b is, in large samples, N(b , sN ), where
222 b1 iiXii
Equation (4.21).
Some Additional Algebraic Facts About OLS
The OLS residuals and predicted values satisfy
1n
n a un i = 0 ,
i=1
1nn naYi = Y,
i=1
auniXi = 0 and suNX = 0, and
TSS = SSR + ESS.
( 4 . 3 2 )
(4.33)
(4.34)
(4.35)
n i=1
11b
1
s N = var(v) > 3var(X )4 = var3(X – m )u 4 > 5n3var(X )4 6, which is the expression in
Equations (4.32) through (4.35) say that the sample average of the OLS residuals is zero; the sample average of the OLS predicted values equals Y; the sample covariance suNX between the OLS residuals and the regressors is zero; and the total sum of squares is the sum of squared residuals and the explained sum of squares. [The ESS, TSS, and SSR are defined in Equations (4.14), (4.15), and (4.17).]
To verify Equation (4.32), note that the definition of bn0 lets us write the OLS residuals asuni =Yi -bn0 -bn1Xi =(Yi -Y)-bn1(Xi -X);thus
nnn
a un i = a ( Y i – Y ) – bn 1 a ( X i – X ) . i=1 i=1 i=1
But the definitions of Y and X imply that gni=1(Yi – Y) = 0 and gni=1(Xi – X) = 0, so g ni = 1 un i = 0 .
Sampling Distribution of the OLS Estimator 145
To verify Equation (4.33), note that Yi = Yni + uni, so gni=1Yi = gni=1 Yni + gni = 1 un1 = gni = 1 Yni, where the second equality is a consequence of Equation (4.32).
To verify Equation (4.34), note that gni = 1 uni = 0 implies gni = 1 uniXi = gni = 1 uni(Xi – X), so
nn
auniXi = a3(Yi – Y) – bn1(Xi – X)4(Xi – X) i=1 i=1
nn
= a(Yi -Y)(Xi -X)-bn1a(Xi -X)2 =0,
i=1 i=1
(4.36)
where the final equality in Equation (4.36) is obtained using the formula for bn1 in Equa- tion (4.27). This result, combined with the preceding results, implies that suNX = 0.
Equation (4.35) follows from the previous results and some algebra:
nn
TSS= a(Yi -Y)2 = a(Yi -Yni +Yni -Y)2
i=1 i=1 nnn
= a ( Y i – Yn i ) 2 + a ( Yn i – Y ) 2 + 2 a ( Y i – Yn i ) ( Yn i – Y ) i=1 i=1 i=1
n
= SSR + ESS + 2auniYni = SSR + ESS, i=1
(4.37)
where the final equality follows from gni=1 uniYni = gni=1 uni(bn0 + bn1Xi) = bn0gni=1 uni + bn1 g ni = 1 un iXi = 0 by the previous results.
146
CHAPTER
5
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
5.1
Testing Hypotheses About
One of the Regression Coefficients
Your client, the superintendent, calls you with a problem. She has an angry tax- payer in her office who asserts that cutting class size will not help boost test scores, so reducing them is a waste of money. Class size, the taxpayer claims, has no effect on test scores.
The taxpayer’s claim can be rephrased in the language of regression analysis. Because the effect on test scores of a unit change in class size is bClassSize, the tax- payer is asserting that the population regression line is flat—that is, the slope bClassSize of the population regression line is zero. Is there, the superintendent asks,
This chapter continues the treatment of linear regression with a single regressor. Chapter 4 explained how the OLS estimator b of the slope coefficient b differs
n1 1 from one sample to the next—that is, how bn1 has a sampling distribution. In this
chapter, we show how knowledge of this sampling distribution can be used to make statements about b1 that accurately summarize the sampling uncertainty. The starting point is the standard error of the OLS estimator, which measures the spread of the sampling distribution of bn1. Section 5.1 provides an expression for this standard error (and for the standard error of the OLS estimator of the intercept), then shows how to use bn1 and its standard error to test hypotheses. Section 5.2 explains how to construct confidence intervals for b1. Section 5.3 takes up the special case of a binary regressor.
Sections 5.1 through 5.3 assume that the three least squares assumptions of Chapter 4 hold. If, in addition, some stronger conditions hold, then some stronger results can be derived regarding the distribution of the OLS estimator. One of these stronger conditions is that the errors are homoskedastic, a concept introduced in Section 5.4. Section 5.5 presents the Gauss–Markov theorem, which states that, under certain conditions, OLS is efficient (has the smallest variance) among a cer- tain class of estimators. Section 5.6 discusses the distribution of the OLS estimator when the population distribution of the regression errors is normal.
5.1 Testing Hypotheses About One of the Regression Coefficients 147
General Form of the t-Statistic In general, the t-statistic has the form
t = estimator – hypothesized value. standard error of the estimator
KEY CONCEPT
5.1
(5.1)
evidence in your sample of 420 observations on California school districts that this slope is nonzero? Can you reject the taxpayer’s hypothesis that bClassSize = 0, or should you accept it, at least tentatively pending further new evidence?
This section discusses tests of hypotheses about the slope b1 or intercept b0 of the population regression line. We start by discussing two-sided tests of the slope b1 in detail, then turn to one-sided tests and to tests of hypotheses regarding the intercept b0.
Two-Sided Hypotheses Concerning b1
The general approach to testing hypotheses about the coefficient b1 is the same as
to testing hypotheses about the population mean, so we begin with a brief review.
Testing hypotheses about the population mean. Recall from Section 3.2 that the null hypothesis that the mean of Y is a specific value mY,0 can be written as H0 : E(Y ) = mY,0, and the two-sided alternative is H1 : E(Y ) ≠ mY,0.
The test of the null hypothesis H0 against the two-sided alternative proceeds as in the three steps summarized in Key Concept 3.6. The first is to compute the standard error of Y, SE(Y ), which is an estimator of the standard deviation of the sampling distribution of Y. The second step is to compute the t-statistic, which has the general formgiveninKeyConcept5.1;appliedhere,thet-statisticist = (Y – mY,0)>SE(Y).
The third step is to compute the p-value, which is the smallest significance level at which the null hypothesis could be rejected, based on the test statistic actually observed; equivalently, the p-value is the probability of obtaining a statistic, by random sampling variation, at least as different from the null hypothesis value as is the statistic actually observed, assuming that the null hypothesis is correct (Key Concept 3.5). Because the t-statistic has a standard normal distribution in large samples under the null hypothesis, the p-value for a two-sided hypothesis test is 2Φ(- t act ), where tact is the value of the t-statistic actually computed and Φ is the cumulative standard normal distribution tabulated in Appendix Table 1. Alternatively,
148 CHAPTER 5
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
the third step can be replaced by simply comparing the t-statistic to the critical value appropriate for the test with the desired significance level. For example, a two-sided test with a 5% significance level would reject the null hypothesis if t act 7 1.96. In this case, the population mean is said to be statistically significantly different from the hypothesized value at the 5% significance level.
Testing hypotheses about the slope b1. At a theoretical level, the critical feature justifying the foregoing testing procedure for the population mean is that, in large samples, the sampling distribution of Y is approximately normal. Because bn1 also has a normal sampling distribution in large samples, hypotheses about the true value of the slope b1 can be tested using the same general approach.
The null and alternative hypotheses need to be stated precisely before they can be tested. The angry taxpayer’s hypothesis is that bClassSize = 0. More gener- ally, under the null hypothesis the true population slope b1 takes on some specific value, b1,0. Under the two-sided alternative, b1 does not equal b1,0. That is, the null hypothesis and the two-sided alternative hypothesis are
H0 : b1 = b1,0 vs. H1 : b1 ≠ b1,0 (two@sided alternative). (5.2)
To test the null hypothesis H0, we follow the same three steps as for the popula- tion mean.
The first step is to compute the standard error of Bn1, SE(bn1). The standard error of bn1 is an estimator of sbn1 the standard deviation of the sampling distribu- tion of bn1. Specifically,
n sn2, SE(b1) = 4 bn1
1n
n – 2 a ( X i – X ) 2 un 2i
1
sn2n=* i=1 .
(5.3)
where
b1
is computed by regression software so that it is easy to use in practice.
The second step is to compute the t-statistic,
t = bn1 – b1,0. (5.5) SE(bn1)
b1nn2 (5.4) c1 (X – X)2d
nai i=1
The estimator of the variance in Equation (5.4) is discussed in Appendix (5.1). Although the formula for sn 2n is complicated, in applications the standard error
5.1 Testing Hypotheses About One of the Regression Coefficients 149
Testing the Hypothesis b1 = b1,0 Against the Alternative b1 ≠ b1,0
1. Compute the standard error of bn1, SE(bn1) [Equation (5.3)].
2. Compute the t-statistic [Equation (5.5)].
3. Compute the p-value [Equation (5.7)]. Reject the hypothesis at the 5% sig- nificance level if the p-value is less than 0.05 or, equivalently, if t act 7 1.96.
The standard error and (typically) the t-statistic and p-value testing b1 = 0 are computed automatically by regression software.
KEY CONCEPT
5.2
The third step is to compute the p-value, the probability of observing a value of bn at least as different from b as the estimate actually computed (bn act), assum-
1 1,0 1 ing that the null hypothesis is correct. Stated mathematically,
p@value=Pr 30bn -b 0 7 0bnact -b 04
H0 1 1,0 1
1 1,0 1 1,0 act
1,0
= P r c ` bn – b ` 7 ` bn a c t – b ` d = P r ( 0 t 0 7 0 t 0 ) , ( 5 . 6 ) H0 SE(bn1) SE(bn1) H0
where PrH0 denotes the probability computed under the null hypothesis, the sec-
standard normal random variable, so in large samples,
p@value = Pr(0Z0 7 0tact0) = 2Φ(-0tact0). (5.7)
A p-value of less than 5% provides evidence against the null hypothesis in the sense that, under the null hypothesis, the probability of obtaining a value of bn1 at least as far from the null as that actually observed is less than 5%. If so, the null hypothesis is rejected at the 5% significance level.
Alternatively, the hypothesis can be tested at the 5% significance level simply by comparing the absolute value of the t-statistic to 1.96, the critical value for a two-sided test, and rejecting the null hypothesis at the 5% level if 0 t act 0 7 1.96.
These steps are summarized in Key Concept 5.2.
n act ond equality follows by dividing by SE(b1), and t
is the value of the t-statistic actually computed. Because bn1 is approximately normally distributed in large samples, under the null hypothesis the t-statistic is approximately distributed as a
150 CHAPTER 5
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
Reporting regression equations and application to test scores. The OLS regres- sion of the test score against the student–teacher ratio, reported in Equation (4.11), yielded bn0 = 698.9 and bn1 = – 2.28. The standard errors of these estimates are SE(bn0) = 10.4 and SE(bn1) = 0.52.
Because of the importance of the standard errors, by convention they are included when reporting the estimated OLS coefficients. One compact way to report the standard errors is to place them in parentheses below the respective coefficients of the OLS regression line:
TestScore = 698.9 – 2.28 * STR, R2 = 0.051, SER = 18.6. (5.8) (10.4) (0.52)
Equation (5.8) also reports the regression R2 and the standard error of the regres- sion (SER) following the estimated regression line. Thus Equation (5.8) provides the estimated regression line, estimates of the sampling uncertainty of the slope and the intercept (the standard errors), and two measures of the fit of this regres- sion line (the R2 and the SER). This is a common format for reporting a single regression equation, and it will be used throughout the rest of this book.
Suppose you wish to test the null hypothesis that the slope b1 is zero in the population counterpart of Equation (5.8) at the 5% significance level. To do so, con- struct the t-statistic and compare its absolute value to 1.96, the 5% (two-sided) critical value taken from the standard normal distribution. The t-statistic is con- structed by substituting the hypothesized value of b1 under the null hypothesis (zero), the estimated slope, and its standard error from Equation (5.8) into the general formula inEquation(5.5);theresultistact = (-2.28 – 0)>0.52 = -4.38.Theabsolutevalue of this t-statistic exceeds the 5% two-sided critical value of 1.96, so the null hypothesis is rejected in favor of the two-sided alternative at the 5% significance level.
Alternatively, we can compute the p-value associated with tact = -4.38. This probability is the area in the tails of standard normal distribution, as shown in Figure 5.1. This probability is extremely small, approximately 0.00001, or 0.001%. That is, if the null hypothesis bClassSize = 0 is true, the probability of obtaining a value of bn1 as far from the null as the value we actually obtained is extremely small, less than 0.001%. Because this event is so unlikely, it is reasonable to con- clude that the null hypothesis is false.
One-Sided Hypotheses Concerning b1
The discussion so far has focused on testing the hypothesis that b1 = b1,0 against thehypothesisthatb1 ≠b1,0.Thisisatwo-sidedhypothesistest,becauseunderthe alternative b1 could be either larger or smaller than b1,0. Sometimes, however, it
5.1 Testing Hypotheses About One of the Regression Coefficients 151 FIGURE 5.1 Calculating the p-Value of a Two-Sided Test When tact = -4.38
The p-value of a two-sided test is the probability that
N(0, 1)
0 Z 0 7 0 t act 0 where Z is a standard normal random variable and tact is the value of the t-statistic calculated from the sample. When
tact = -4.38, the p-value is only 0.00001.
–4.38 0
The p-value is the area
to the left of –4.38
+
the area to the right of +4.38.
4.38 z
is appropriate to use a one-sided hypothesis test. For example, in the student– teacher ratio/test score problem, many people think that smaller classes provide a better learning environment. Under that hypothesis, b1 is negative: Smaller classes lead to higher scores. It might make sense, therefore, to test the null hypothesis that b1 = 0 (no effect) against the one-sided alternative that b1 6 0.
For a one-sided test, the null hypothesis and the one-sided alternative hypoth- esis are
H0:b1 =b1,0vs.H1:b1 6b1,0 (one@sidedalternative). (5.9)
where b1,0 is the value of b1 under the null (0 in the student–teacher ratio example) and the alternative is that b1 is less than b1,0. If the alternative is that b1 is greater than b1,0, the inequality in Equation (5.9) is reversed.
Because the null hypothesis is the same for a one- and a two-sided hypothesis test, the construction of the t-statistic is the same. The only difference between a one- and two-sided hypothesis test is how you interpret the t-statistic. For the one- sided alternative in Equation (5.9), the null hypothesis is rejected against the one- sided alternative for large negative, but not large positive, values of the t-statistic: Instead of rejecting if tact 7 1.96, the hypothesis is rejected at the 5% signifi- cance level if tact 6 -1.64.
152 CHAPTER 5
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
The p-value for a one-sided test is obtained from the cumulative standard normal distribution as
p@value = Pr(Z 6 tact) = Φ(tact) (p@value, one@sided left@tail test). (5.10)
If the alternative hypothesis is that b1 is greater than b1,0, the inequalities in Equa- tions (5.9) and (5.10) are reversed, so the p-value is the right-tail probability, Pr(Z 7 tact).
Whenshouldaone-sidedtestbeused? Inpractice,one-sidedalternativehypoth- eses should be used only when there is a clear reason for doing so. This reason could come from economic theory, prior empirical evidence, or both. However, even if it initially seems that the relevant alternative is one-sided, upon reflection this might not necessarily be so. A newly formulated drug undergoing clinical tri- als actually could prove harmful because of previously unrecognized side effects. In the class size example, we are reminded of the graduation joke that a univer- sity’s secret of success is to admit talented students and then make sure that the faculty stays out of their way and does as little damage as possible. In practice, such ambiguity often leads econometricians to use two-sided tests.
Application to test scores. The t-statistic testing the hypothesis that there is no effect of class size on test scores [so b1,0 = 0 in Equation (5.9)] is tact = -4.38. This value is less than -2.33 (the critical value for a one-sided test with a 1% signifi- cance level), so the null hypothesis is rejected against the one-sided alternative at the 1% level. In fact, the p-value is less than 0.0006%. Based on these data, you can reject the angry taxpayer’s assertion that the negative estimate of the slope arose purely because of random sampling variation at the 1% significance level.
Testing Hypotheses About the Intercept b0
This discussion has focused on testing hypotheses about the slope, b1. Occasion- ally, however, the hypothesis concerns the intercept b0. The null hypothesis con- cerning the intercept and the two-sided alternative are
H0 : b0 = b0,0 vs. H1 : b0 ≠ b0,0 (two@sided alternative). (5.11)
The general approach to testing this null hypothesis consists of the three steps in Key Concept 5.2 applied to b0 (the formula for the standard error of bn0 is given in Appendix 5.1). If the alternative is one-sided, this approach is modified as was discussed in the previous subsection for hypotheses about the slope.
5.2 Confidence Intervals for a Regression Coefficient 153
Hypothesis tests are useful if you have a specific null hypothesis in mind (as did our angry taxpayer). Being able to accept or reject this null hypothesis based on the statistical evidence provides a powerful tool for coping with the uncertainty inherent in using a sample to learn about the population. Yet, there are many times that no single hypothesis about a regression coefficient is dominant, and instead one would like to know a range of values of the coefficient that are con- sistent with the data. This calls for constructing a confidence interval.
5.2
Confidence Intervals
for a Regression Coefficient
Because any statistical estimate of the slope b1 necessarily has sampling uncer- tainty, we cannot determine the true value of b1 exactly from a sample of data. It is possible, however, to use the OLS estimator and its standard error to construct a confidence interval for the slope b1 or for the intercept b0.
Confidence interval for b1. Recall from the discussion of confidence intervals in Section 3.3 that a 95% confidence interval for B1 has two equivalent definitions. First, it is the set of values that cannot be rejected using a two-sided hypothesis test with a 5% significance level. Second, it is an interval that has a 95% probability of contain- ing the true value of b1; that is, in 95% of possible samples that might be drawn, the confidence interval will contain the true value of b1. Because this interval contains the true value in 95% of all samples, it is said to have a confidence level of 95%.
The reason these two definitions are equivalent is as follows. A hypothesis test with a 5% significance level will, by definition, reject the true value of b1 in only 5% of all possible samples; that is, in 95% of all possible samples, the true value of b1 will not be rejected. Because the 95% confidence interval (as defined in the first definition) is the set of all values of b1 that are not rejected at the 5% significance level, it follows that the true value of b1 will be contained in the con- fidence interval in 95% of all possible samples.
As in the case of a confidence interval for the population mean (Section 3.3), in principle a 95% confidence interval can be computed by testing all possible values of b1 (that is, testing the null hypothesis b1 = b1,0 for all values of b1,0) at the 5% significance level using the t-statistic. The 95% confidence interval is then the collection of all the values of b1 that are not rejected. But constructing the t-statistic for all values of b1 would take forever.
An easier way to construct the confidence interval is to note that the t-statistic will reject the hypothesized value b1,0 whenever b1,0 is outside the range
154 CHAPTER 5 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
Confidence Interval for b1
5.3
KEY CONCEPT
A 95% two-sided confidence interval for b1 is an interval that contains the true value of b1 with a 95% probability; that is, it contains the true value of b1 in 95% of all possible randomly drawn samples. Equivalently, it is the set of values of b1 that cannot be rejected by a 5% two-sided hypothesis test. When the sample size is large, it is constructed as
95% confidence interval for b1 = 3bn1 – 1.96SE(bn1), bn1 + 1.96SE(bn1)4. (5.12)
bn1 { 1.96SE(bn1). This implies that the 95% confidence interval for b1 is the inter- val 3bn1 – 1.96SE(bn1), bn1 + 1.96SE(bn1)4. This argument parallels the argument used to develop a confidence interval for the population mean.
The construction of a confidence interval for b1 is summarized as Key Concept 5.3.
Confidence interval for b0. A 95% confidence interval for b0 is constructed as in Key Concept 5.3, with bn0 and SE(bn0) replacing bn1 and SE(bn1).
Applicationtotestscores. TheOLSregressionofthetestscoreagainstthestudent– teacher ratio, reported in Equation (5.8), yielded bn1 = -2.28 and SE(bn1) = 0.52. The 95% two-sided confidence interval for b1 is 5-2.28 { 1.96 * 0.526, or – 3.30 … b1 … – 1.26. The value b1 = 0 is not contained in this confidence interval, so(asweknewalreadyfromSection5.1)thehypothesisb1 = 0canberejectedatthe 5% significance level.
Confidence intervals for predicted effects of changing X. The 95% confidence interval for b1 can be used to construct a 95% confidence interval for the pre- dicted effect of a general change in X.
Consider changing X by a given amount, ∆x. The predicted change in Y asso- ciated with this change in X is b1∆x. The population slope b1 is unknown, but because we can construct a confidence interval for b1, we can construct a confi- dence interval for the predicted effect b1∆x. Because one end of a 95% confidence interval for b1 is bn1 – 1.96SE(bn1), the predicted effect of the change ∆x using this estimate of b1 is 3bn1 – 1.96SE(bn1)4 * ∆x. The other end of the confidence
5.3 Regression When X Is a Binary Variable 155
interval is bn1 + 1.96SE(bn1), and the predicted effect of the change using that esti- mate is 3bn1 + 1.96SE(bn1)4 * ∆x. Thus a 95% confidence interval for the effect of changing x by the amount ∆x can be expressed as
95% confidence interval for b1∆x
= 3(bn1 – 1.96SE(bn1))∆x, (bn1 + 1.96SE(bn1))∆x4. (5.13)
For example, our hypothetical superintendent is contemplating reducing the student–teacher ratio by 2. Because the 95% confidence interval for b1 is 3-3.30, -1.264, the effect of reducing the student–teacher ratio by 2 could be as great as -3.30 * (-2) = 6.60 or as little as -1.26 * (-2) = 2.52. Thus decreas- ing the student–teacher ratio by 2 is predicted to increase test scores by between 2.52 and 6.60 points, with a 95% confidence level.
5.3
Regression When X Is a Binary Variable
The discussion so far has focused on the case that the regressor is a continuous variable. Regression analysis can also be used when the regressor is binary—that is, when it takes on only two values, 0 or 1. For example, X might be a worker’s gender (=1 if female, = 0 if male), whether a school district is urban or rural (= 1 if urban, = 0 if rural), or whether the district’s class size is small or large (= 1 if small, = 0 if large). A binary variable is also called an indicator variable or sometimes a dummy variable.
Interpretation of the Regression Coefficients
The mechanics of regression with a binary regressor are the same as if it is con- tinuous. The interpretation of b1, however, is different, and it turns out that regression with a binary variable is equivalent to performing a difference of means analysis, as described in Section 3.4.
To see this, suppose you have a variable Di that equals either 0 or 1, depend- ing on whether the student–teacher ratio is less than 20:
th
D = e 1 if the student9teacher ratio in i district 6 20.
i 0 if the student9teacher ratio in ith district Ú 20 The population regression model with Di as the regressor is
Yi = b0 + b1Di + ui,i = 1,c,n.
(5.14)
(5.15)
156 CHAPTER 5
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This is the same as the regression model with the continuous regressor Xi except that now the regressor is the binary variable Di. Because Di is not continu- ous, it is not useful to think of b1 as a slope; indeed, because Di can take on only two values, there is no “line,” so it makes no sense to talk about a slope. Thus we will not refer to b1 as the slope in Equation (5.15); instead we will simply refer to b1 as the coefficient multiplying Di in this regression or, more compactly, the coefficient on Di.
If b1 in Equation (5.15) is not a slope, what is it? The best way to interpret b0 and b1 in a regression with a binary regressor is to consider, one at a time, the two possible cases, Di = 0 and Di = 1. If the student–teacher ratio is high, then Di = 0 and Equation (5.15) becomes
Yi =b0 +ui (Di =0). (5.16)
Because E(ui Di) = 0, the conditional expectation of Yi when Di = 0 is E(Yi Di = 0) = b0; that is, b0 is the population mean value of test scores when the student–teacher ratio is high. Similarly, when Di = 1,
Yi =b0 +b1 +ui (Di =1). (5.17)
Thus, when Di = 1,E(Yi Di = 1) = b0 + b1; that is, b0 + b1 is the population mean value of test scores when the student–teacher ratio is low.
Because b0 + b1 is the population mean of Yi when Di = 1 and b0 is the population mean of Yi when Di = 0, the difference (b0 + b1) – b0 = b1 is the difference between these two means. In other words, b1 is the difference between the conditional expectation of Yi when Di = 1 and when Di = 0, or b1 = E(Yi Di = 1) – E(Yi Di = 0). In the test score example, b1 is the differ- ence between mean test score in districts with low student–teacher ratios and the mean test score in districts with high student–teacher ratios.
Because b1 is the difference in the population means, it makes sense that the OLS estimator b1 is the difference between the sample averages of Yi in the two groups, and, in fact, this is the case.
Hypothesis tests and confidence intervals. If the two population means are the same, then b1 in Equation (5.15) is zero. Thus the null hypothesis that the two population means are the same can be tested against the alternative hypothesis that they differ by testing the null hypothesis b1 = 0 against the alternative b1 ≠ 0. This hypothesis can be tested using the procedure outlined in Section 5.1. Specifically, the null hypothesis can be rejected at the 5% level against the two-sided
5.4 Heteroskedasticity and Homoskedasticity 157
alternative when the OLS t-statistic t = bn1 > SE(bn1) exceeds 1.96 in absolute value. Similarly, a 95% confidence interval for b1, constructed as bn1 { 1.96SE(bn1). as described in Section 5.2, provides a 95% confidence interval for the difference between the two population means.
Application to test scores. As an example, a regression of the test score against the student–teacher ratio binary variable D defined in Equation (5.14) estimated by OLS using the 420 observations in Figure 4.2 yields
TestScore = 650.0 + 7.4D, R2 = 0.037, SER = 18.7,
(1.3) (1.8) (5.18)
where the standard errors of the OLS estimates of the coefficients b0 and b1 are given in parentheses below the OLS estimates. Thus the average test score for the subsample with student–teacher ratios greater than or equal to 20 (that is, for which D = 0) is 650.0, and the average test score for the subsample with student– teacher ratios less than 20 (so D = 1) is 650.0 + 7.4 = 657.4. The difference between the sample average test scores for the two groups is 7.4. This is the OLS estimate of b1, the coefficient on the student–teacher ratio binary variable D.
Is the difference in the population mean test scores in the two groups statisti- cally significantly different from zero at the 5% level? To find out, construct the t-statistic on b1 : t = 7.4 > 1.8 = 4.04. This value exceeds 1.96 in absolute value, so the hypothesis that the population mean test scores in districts with high and low student–teacher ratios is the same can be rejected at the 5% significance level.
The OLS estimator and its standard error can be used to construct a 95% con- fidence interval for the true difference in means. This is 7.4 { 1.96 * 1.8 = (3.9, 10.9). This confidence interval excludes b1 = 0, so that (as we know from the previous paragraph) the hypothesis b1 = 0 can be rejected at the 5% significance level.
5.4
Heteroskedasticity and Homoskedasticity
Our only assumption about the distribution of ui conditional on Xi is that it has a mean of zero (the first least squares assumption). If, furthermore, the variance of this conditional distribution does not depend on Xi, then the errors are said to be homoskedastic. This section discusses homoskedasticity, its theoretical implica- tions, the simplified formulas for the standard errors of the OLS estimators that arise if the errors are homoskedastic, and the risks you run if you use these simpli- fied formulas in practice.
158
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FIGURE 5.2
What Are Heteroskedasticity and Homoskedasticity?
Definitions of heteroskedasticity and homoskedasticity. The error term ui is homoskedastic if the variance of the conditional distribution of ui given Xi is con- stant for i = 1, c, n and in particular does not depend on Xi. Otherwise, the error term is heteroskedastic.
As an illustration, return to Figure 4.4. The distribution of the errors ui is shown for various values of x. Because this distribution applies specifically for the indicated value of x, this is the conditional distribution of ui given Xi = x. As drawn in that figure, all these conditional distributions have the same spread; more precisely, the variance of these distributions is the same for the various values of x. That is, in Figure 4.4, the conditional variance of ui given Xi = x does not depend on x, so the errors illustrated in Figure 4.4 are homo- skedastic.
In contrast, Figure 5.2 illustrates a case in which the conditional distribution of ui spreads out as x increases. For small values of x, this distribution is tight, but for larger values of x, it has a greater spread. Thus in Figure 5.2 the variance of ui given Xi = x increases with x, so that the errors in Figure 5.2 are heteroskedastic.
The definitions of heteroskedasticity and homoskedasticity are summarized in Key Concept 5.4.
An Example of Heteroskedasticity
Like Figure 4.4, this shows the conditional distribution of test scores for three differ- ent class sizes. Unlike Figure 4.4, these distributions become more spread out (have a larger variance)
Test score
720
700
680
660
640
620
600
10 15 20 25 30
Student–teacher ratio
Distribution of Y when X = 15
Distribution of Y when X = 20
Distribution of Y when X = 25
b0 +b1X
for larger class sizes. Because the variance of the distribution of u given X, var(u X ), depends on X, u is heteroskedastic.
5.4 Heteroskedasticity and Homoskedasticity 159
Heteroskedasticity and Homoskedasticity
KEY CONCEPT
5.4
The error term ui is homoskedastic if the variance of the conditional distribution of ui given Xi, var(ui Xi = x), is constant for i = 1, c, n and in particular does not depend on x. Otherwise, the error term is heteroskedastic.
Example. These terms are a mouthful, and the definitions might seem abstract. To help clarify them with an example, we digress from the student–teacher ratio/ test score problem and instead return to the example of earnings of male versus female college graduates considered in the box in Chapter 3 “The Gender Gap in Earnings of College Graduates in the United States.” Let MALEi be a binary variable that equals 1 for male college graduates and equals 0 for female gradu- ates. The binary variable regression model relating a college graduate’s earnings to his or her gender is
Earningsi = b0 + b1MALEi + ui (5.19)
for i = 1, c, n. Because the regressor is binary, b1 is the difference in the popu- lation means of the two groups—in this case, the difference in mean earnings between men and women who graduated from college.
The definition of homoskedasticity states that the variance of ui does not depend on the regressor. Here the regressor is MALEi, so at issue is whether the variance of the error term depends on MALEi. In other words, is the variance of the error term the same for men and for women? If so, the error is homoskedastic; if not, it is heteroskedastic.
Deciding whether the variance of ui depends on MALEi requires thinking hard about what the error term actually is. In this regard, it is useful to write Equation (5.19) as two separate equations, one for men and one for women:
Earningsi = b0 + ui (women) and (5.20) Earningsi = b0 + b1 + ui (men). (5.21)
Thus, for women, ui is the deviation of the ith woman’s earnings from the popula- tion mean earnings for women (b0), and for men, ui is the deviation of the ith man’s earnings from the population mean earnings for men (b0 + b1). It follows that the statement “the variance of ui does not depend on MALE” is equivalent to the
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statement “the variance of earnings is the same for men as it is for women.” In other words, in this example, the error term is homoskedastic if the variance of the population distribution of earnings is the same for men and women; if these variances differ, the error term is heteroskedastic.
Mathematical Implications of Homoskedasticity
The OLS estimators remain unbiased and asymptotically normal. Because the least squares assumptions in Key Concept 4.3 place no restrictions on the condi- tional variance, they apply to both the general case of heteroskedasticity and the special case of homoskedasticity. Therefore, the OLS estimators remain unbiased and consistent even if the errors are homoskedastic. In addition, the OLS estima- tors have sampling distributions that are normal in large samples even if the errors are homoskedastic. Whether the errors are homoskedastic or heteroskedastic, the OLS estimator is unbiased, consistent, and asymptotically normal.
Efficiency of the OLS estimator when the errors are homoskedastic. If the least squares assumptions in Key Concept 4.3 hold and the errors are homoskedastic, then the OLS estimators bn0 and bn1 are efficient among all estimators that are linear in Y1, c, Yn and are unbiased, conditional on X1, c, Xn. This result, which is called the Gauss–Markov theorem, is discussed in Section 5.5.
Homoskedasticity-only variance formula. If the error term is homoskedastic, then the formulas for the variances of bn0 and bn1 in Key Concept 4.4 simplify. Con- sequently, if the errors are homoskedastic, then there is a specialized formula that can be used for the standard errors of bn0 and bn1. The homoskedasticity-only stan-
dard error of bn1, derived in Appendix (5.1), is SE(bn1) = 2s∼2n where s∼n2 nb1 b1
homoskedasticity-only estimator of the variance of b1:
s∼ 2n = s 2un ( h o m o s k e d a s t i c i t y @ o n l y ) ,
is the
(5.22)
b1 n
a(Xi – X)2
i=1
where s2un is given in Equation (4.19). The homoskedasticity-only formula for the standard error of bn0 is given in Appendix (5.1). In the special case that X is a binary variable, the estimator of the variance of bn1 under homoskedasticity (that is, the square of the standard error of bn1 under homoskedasticity) is the so-called pooled variance formula for the difference in means, given in Equation (3.23).
Because these alternative formulas are derived for the special case that the errors are homoskedastic and do not apply if the errors are heteroskedastic, they
5.4 Heteroskedasticity and Homoskedasticity 161
will be referred to as the “homoskedasticity-only” formulas for the variance and standard error of the OLS estimators. As the name suggests, if the errors are heteroskedastic, then the homoskedasticity-only standard errors are inappropri- ate. Specifically, if the errors are heteroskedastic, then the t-statistic computed using the homoskedasticity-only standard error does not have a standard normal distribution, even in large samples. In fact, the correct critical values to use for this homoskedasticity-only t-statistic depend on the precise nature of the heteroskedas- ticity, so those critical values cannot be tabulated. Similarly, if the errors are hetero- skedastic but a confidence interval is constructed as { 1.96 homoskedasticity-only standard errors, in general the probability that this interval contains the true value of the coefficient is not 95%, even in large samples.
In contrast, because homoskedasticity is a special case of heteroskedasticity,
the estimators snbn2 and snbn2 of the variances of bn1 and bn0 given in Equations (5.4) 10
and (5.26) produce valid statistical inferences whether the errors are heteroske- dastic or homoskedastic. Thus hypothesis tests and confidence intervals based on those standard errors are valid whether or not the errors are heteroskedastic. Because the standard errors we have used so far [that is, those based on Equations (5.4) and (5.26)] lead to statistical inferences that are valid whether or not the errors are heteroskedastic, they are called heteroskedasticity-robust standard errors. Because such formulas were proposed by Eicker (1967), Huber (1967), and White (1980), they are also referred to as Eicker–Huber–White standard errors.
What Does This Mean in Practice?
Which is more realistic, heteroskedasticity or homoskedasticity? The answer to this question depends on the application. However, the issues can be clarified by returning to the example of the gender gap in earnings among college graduates. Familiarity with how people are paid in the world around us gives some clues as to which assumption is more sensible. For many years—and, to a lesser extent, today— women were not found in the top-paying jobs: There have always been poorly paid men, but there have rarely been highly paid women. This suggests that the distribu- tion of earnings among women is tighter than among men (see the box in Chapter 3 “The Gender Gap in Earnings of College Graduates in the United States”). In other words, the variance of the error term in Equation (5.20) for women is plausi- bly less than the variance of the error term in Equation (5.21) for men. Thus the presence of a “glass ceiling” for women’s jobs and pay suggests that the error term in the binary variable regression model in Equation (5.19) is heteroskedastic. Unless there are compelling reasons to the contrary—and we can think of none—it makes sense to treat the error term in this example as heteroskedastic.
162 CHAPTER 5 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
The Economic Value of a Year of Education: Homoskedasticity or Heteroskedasticity?
On average, workers with more education have higher earnings than workers with less educa- tion. But if the best-paying jobs mainly go to the col- lege educated, it might also be that the spread of the distribution of earnings is greater for workers with more education. Does the distribution of earnings spread out as education increases?
This is an empirical question, so answering it requires analyzing data. Figure 5.3 is a scatterplot of the hourly earnings and the number of years of edu- cation for a sample of 2829 full-time workers in the United States in 2012, ages 29 and 30, with between 6 and 18 years of education. The data come from the March 2013 Current Population Survey, which is described in Appendix 3.1.
Figure 5.3 has two striking features. The first is that the mean of the distribution of earnings increases with the number of years of education. This increase is summarized by the OLS regression line,
Earnings = -7.29 + 1.93YearsEducation, (1.10) (0.08)
R2 = 0.162, SER = 10.29. (5.23) This line is plotted in Figure 5.3. The coefficient
of 1.93 in the OLS regression line means that, on
average, hourly earnings increase by $1.93 for each additional year of education. The 95% confidence interval for this coefficient is 1.93 { 1.96 * 0.08, or 1.77 to 2.09.
The second striking feature of Figure 5.3 is that the spread of the distribution of earnings increases with the years of education. While some workers with many years of education have low-paying jobs, very few workers with low levels of education have high-paying jobs. This can be quantified by looking at the spread of the residuals around the OLS regres- sion line. For workers with ten years of education, the standard deviation of the residuals is $4.32; for workers with a high school diploma, this standard deviation is $7.80; and for workers with a college degree, this standard deviation increases to $12.46. Because these standard deviations differ for differ- ent levels of education, the variance of the residuals in the regression of Equation (5.23) depends on the value of the regressor (the years of education); in other words, the regression errors are heteroskedas- tic. In real-world terms, not all college graduates will be earning $50 per hour by the time they are 29, but some will, and workers with only ten years of educa- tion have no shot at those jobs.
FIGURE 5.3 Scatterplot of Hourly Earnings and Years of Education for 29- to 30-Year-Olds in the United States in 2012
Hourly earnings are plotted against years of education for 2,829 full-time 29- to 30-year-old workers. The spread around the regression line increases with the years of education, indicating that the regression errors are heteroskedastic.
150
ahe Fitted values 100
50
0
Average hourly earnings
5 10 15 20
Years of education
5.5 The Theoretical Foundations of Ordinary Least Squares 163
As this example of modeling earnings illustrates, heteroskedasticity arises in many econometric applications. At a general level, economic theory rarely gives any reason to believe that the errors are homoskedastic. It therefore is prudent to assume that the errors might be heteroskedastic unless you have compelling rea- sons to believe otherwise.
Practical implications. The main issue of practical relevance in this discussion is whether one should use heteroskedasticity-robust or homoskedasticity-only stan- dard errors. In this regard, it is useful to imagine computing both, then choosing between them. If the homoskedasticity-only and heteroskedasticity-robust stan- dard errors are the same, nothing is lost by using the heteroskedasticity-robust standard errors; if they differ, however, then you should use the more reliable ones that allow for heteroskedasticity. The simplest thing, then, is always to use the heteroskedasticity-robust standard errors.
For historical reasons, many software programs report homoskedasticity- only standard errors as their default setting, so it is up to the user to specify the option of heteroskedasticity-robust standard errors. The details of how to imple- ment heteroskedasticity-robust standard errors depend on the software package you use.
All of the empirical examples in this book employ heteroskedasticity-robust standard errors unless explicitly stated otherwise.1
*5.5 The Theoretical Foundations of Ordinary Least Squares
As discussed in Section 4.5, the OLS estimator is unbiased, is consistent, has a variance that is inversely proportional to n, and has a normal sampling distribu- tion when the sample size is large. In addition, under certain conditions the OLS estimator is more efficient than some other candidate estimators. Specifically, if the least squares assumptions hold and if the errors are homoskedastic, then the OLS estimator has the smallest variance of all conditionally unbiased estimators that are linear functions of Y1, c, Yn. This section explains and discusses this result, which is a consequence of the Gauss–Markov theorem. The section concludes
1In case this book is used in conjunction with other texts, it might be helpful to note that some text- books add homoskedasticity to the list of least squares assumptions. As just discussed, however, this additional assumption is not needed for the validity of OLS regression analysis as long as heteroskedasticity-robust standard errors are used.
*This section is optional and is not used in later chapters.
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with a discussion of alternative estimators that are more efficient than OLS when the conditions of the Gauss–Markov theorem do not hold.
Linear Conditionally Unbiased Estimators and
the Gauss–Markov Theorem
If the three least squares assumptions (Key Concept 4.3) hold and if the error is homoskedastic, then the OLS estimator has the smallest variance, conditional on X1, c, Xn, among all estimators in the class of linear conditionally unbiased esti- mators. In other words, the OLS estimator is the Best Linear conditionally Unbi- ased Estimator—that is, it is BLUE. This result is an extension of the result, summarized in Key Concept 3.3, that the sample average Y is the most efficient estimator of the population mean among the class of all estimators that are unbi- ased and are linear functions (weighted averages) of Y1, c, Yn.
Linear conditionally unbiased estimators. The class of linear conditionally unbi-
ased estimators consists of all estimators of b1 that are linear functions of
Y ,c,Y and that are unbiased, conditional on X ,c,X . That is, if b∼ is a 1n 1n1
linear estimator, then it can be written as
∼n∼
b1 = aaiYi (b1 is linear), (5.24)
i=1
where the weights a1, c, an can depend on X1, c, Xn but not on Y1, c, Yn. The estimator ∼b1 is conditionally unbiased if the mean of its conditional sampling distribution, given X1, c, Xn, is b1. That is, the estimator ∼b1 is conditionally unbiased if
E( b∼ X , c, X ) = b ( b∼ is conditionally unbiased). (5.25) 11n11
The estimator b∼ is a linear conditionally unbiased estimator if it can be written 1
in the form of Equation (5.24) (it is linear) and if Equation (5.25) holds (it is con- ditionally unbiased). It is shown in Appendix 5.2 that the OLS estimator is linear and conditionally unbiased.
TheGauss–Markovtheorem. TheGauss–Markovtheoremstatesthat,underaset of conditions known as the Gauss–Markov conditions, the OLS estimator bn1 has the smallest conditional variance, given X1, c, Xn, of all linear conditionally unbiased estimators of b1; that is, the OLS estimator is BLUE. The Gauss–Markov conditions, which are stated in Appendix 5.2, are implied by the three least
5.5 The Theoretical Foundations of Ordinary Least Squares 165
The Gauss–Markov Theorem for bn1
KEY CONCEPT
5.5
If the three least squares assumptions in Key Concept 4.3 hold and if errors are homoskedastic, then the OLS estimator bn1 is the Best (most efficient) Linear conditionally Unbiased Estimator (BLUE).
squares assumptions plus the assumption that the errors are homoskedastic. Con- sequently, if the three least squares assumptions hold and the errors are homo- skedastic, then OLS is BLUE. The Gauss–Markov theorem is stated in Key Concept 5.5 and proven in Appendix 5.2.
LimitationsoftheGauss–Markovtheorem. TheGauss–Markovtheoremprovides a theoretical justification for using OLS. However, the theorem has two important limitations. First, its conditions might not hold in practice. In particular, if the error term is heteroskedastic—as it often is in economic applications—then the OLS estimator is no longer BLUE. As discussed in Section 5.4, the presence of hetero- skedasticity does not pose a threat to inference based on heteroskedasticity-robust standard errors, but it does mean that OLS is no longer the efficient linear condi- tionally unbiased estimator. An alternative to OLS when there is heteroskedasticity of a known form, called the weighted least squares estimator, is discussed below.
The second limitation of the Gauss–Markov theorem is that even if the condi- tions of the theorem hold, there are other candidate estimators that are not linear and conditionally unbiased; under some conditions, these other estimators are more efficient than OLS.
Regression Estimators Other Than OLS
Under certain conditions, some regression estimators are more efficient than OLS.
Theweightedleastsquaresestimator. Iftheerrorsareheteroskedastic,thenOLS is no longer BLUE. If the nature of the heteroskedasticity is known—specifically, if the conditional variance of ui given Xi is known up to a constant factor of proportionality—then it is possible to construct an estimator that has a smaller variance than the OLS estimator. This method, called weighted least squares (WLS), weights the ith observation by the inverse of the square root of the condi- tional variance of ui given Xi. Because of this weighting, the errors in this weighted regression are homoskedastic, so OLS, when applied to the weighted data, is BLUE.
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*5.6
LAD estimator is less sensitive to large outliers in u than is OLS.
In many economic data sets, severe outliers in u are rare, so use of the LAD estimator, or other estimators with reduced sensitivity to outliers, is uncommon in applications. Thus the treatment of linear regression throughout the remainder
of this text focuses exclusively on least squares methods.
Using the t-Statistic in Regression When the Sample Size Is Small
When the sample size is small, the exact distribution of the t-statistic is compli- cated and depends on the unknown population distribution of the data. If, how- ever, the three least squares assumptions hold, the regression errors are homoskedastic, and the regression errors are normally distributed, then the OLS estimator is normally distributed and the homoskedasticity-only t-statistic has a Student t distribution. These five assumptions—the three least squares assump- tions, that the errors are homoskedastic, and that the errors are normally distrib- uted—are collectively called the homoskedastic normal regression assumptions.
The t-Statistic and the Student t Distribution
Recall from Section 2.4 that the Student t distribution with m degrees of freedom is defined to be the distribution of Z>2W>m, where Z is a random variable with a standard normal distribution, W is a random variable with a chi-squared distribution
*This section is optional and is not used in later chapters.
Although theoretically elegant, the practical problem with weighted least squares is that you must know how the conditional variance of ui depends on Xi, some- thing that is rarely known in econometric applications. Weighted least squares is therefore used far less frequently than OLS, and further discussion of WLS is deferred to Chapter 17.
The least absolute deviations estimator. As discussed in Section 4.3, the OLS
estimator can be sensitive to outliers. If extreme outliers are not rare, then other
estimators can be more efficient than OLS and can produce inferences that are
more reliable. One such estimator is the least absolute deviations (LAD) estima-
tor, in which the regression coefficients b0 and b1 are obtained by solving a mini-
mization problem like that in Equation (4.6) except that the absolute value of the
prediction “mistake” is used instead of its square. That is, the LAD estimators of
b andb arethevaluesofb andb thatminimize gn 0Y – b – bX0.The 01 01 i=1i01i
5.6 Using the t-Statistic in Regression When the Sample Size Is Small 167
with m degrees of freedom, and Z and W are independent. Under the null hypoth- esis, the t-statistic computed using the homoskedasticity-only standard error can be written in this form.
The details of the calculation is presented in Sections 17.4 and 18.4, but the main
ideas are as follows. The homoskedasticity-only t-statistic testing b1 = b1,0 is
∼t = (bn1 – b1,0)>s∼bn , where s∼bn2 is defined in Equation (5.22). Under the homoske- 11
dastic normal regression assumptions, Y has a normal distribution, conditional on
X1, c, Xn. As discussed in Section 5.5, the OLS estimator is a weighted average
of Y1, c, Yn, where the weights depend on X1, c, Xn [see Equation (5.32) in
Appendix 5.2]. Because a weighted average of independent normal random variables
is normally distributed, bn1 has a normal distribution, conditional on X1, c, Xn.
Thus (bn1 – b1,0) has a normal distribution under the null hypothesis, conditional
on X1, c, Xn. In addition, sections 17.4 and 18.4 show that the (normalized)
homoskedasticity-only variance estimator has a chi-squared distribution with n – 2
degrees of freedom, divided by n – 2, and s∼2 and bn are independently distributed. bn1 1
Consequently, the homoskedasticity-only t-statistic has a Student t distribution with n – 2 degrees of freedom.
This result is closely related to a result discussed in Section 3.5 in the context of testing for the equality of the means in two samples. In that problem, if the two population distributions are normal with the same variance and if the t-statistic is con- structed using the pooled standard error formula [Equation (3.23)], then the (pooled) t-statistic has a Student t distribution. When X is binary, the homoskedasticity-only standard error for bn1 simplifies to the pooled standard error formula for the difference of means. It follows that the result of Section 3.5 is a special case of the result that if the homoskedastic normal regression assumptions hold, then the homoskedasticity- only regression t-statistic has a Student t distribution (see Exercise 5.10).
Use of the Student t Distribution in Practice
If the regression errors are homoskedastic and normally distributed and if the homoskedasticity-only t-statistic is used, then critical values should be taken from the Student t distribution (Appendix Table 2) instead of the standard normal distribution. Because the difference between the Student t distribution and the normal distribution is negligible if n is moderate or large, this distinction is rele- vant only if the sample size is small.
In econometric applications, there is rarely a reason to believe that the errors are homoskedastic and normally distributed. Because sample sizes typically are large, however, inference can proceed as described in Section 5.1 and 5.2—that is, by first computing heteroskedasticity-robust standard errors and then by using the standard normal distribution to compute p-values, hypothesis tests, and confidence intervals.
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5.7
Conclusion
Return for a moment to the problem that started Chapter 4: the superintendent who is considering hiring additional teachers to cut the student–teacher ratio. What have we learned that she might find useful?
Our regression analysis, based on the 420 observations for 1998 in the Cali- fornia test score data set, showed that there was a negative relationship between the student–teacher ratio and test scores: Districts with smaller classes have higher test scores. The coefficient is moderately large, in a practical sense: Districts with two fewer students per teacher have, on average, test scores that are 4.6 points higher. This corresponds to moving a district at the 50th percentile of the distribu- tion of test scores to approximately the 60th percentile.
The coefficient on the student–teacher ratio is statistically significantly different from 0 at the 5% significance level. The population coefficient might be 0, and we might simply have estimated our negative coefficient by random sampling variation. However, the probability of doing so (and of obtaining a t-statistic on b1 as large as we did) purely by random variation over potential samples is exceedingly small, approximately 0.001%. A 95% confidence interval for b1 is -3.30 … b1 … -1.26.
This result represents considerable progress toward answering the superin- tendent’s question yet a nagging concern remains. There is a negative relation- ship between the student–teacher ratio and test scores, but is this relationship necessarily the causal one that the superintendent needs to make her decision? Districts with lower student–teacher ratios have, on average, higher test scores. But does this mean that reducing the student–teacher ratio will, in fact, increase scores?
There is, in fact, reason to worry that it might not. Hiring more teachers, after all, costs money, so wealthier school districts can better afford smaller classes. But students at wealthier schools also have other advantages over their poorer neigh- bors, including better facilities, newer books, and better-paid teachers. Moreover, students at wealthier schools tend themselves to come from more affluent families and thus have other advantages not directly associated with their school. For example, California has a large immigrant community; these immigrants tend to be poorer than the overall population, and in many cases, their children are not native English speakers. It thus might be that our negative estimated relationship between test scores and the student–teacher ratio is a consequence of large classes being found in conjunction with many other factors that are, in fact, the real cause of the lower test scores.
These other factors, or “omitted variables,” could mean that the OLS analysis done so far has little value to the superintendent. Indeed, it could be misleading:
Changing the student–teacher ratio alone would not change these other factors that determine a child’s performance at school. To address this problem, we need a method that will allow us to isolate the effect on test scores of changing the student–teacher ratio, holding these other factors constant. That method is multi- ple regression analysis, the topic of Chapters 6 and 7.
Summary
1. Hypothesis testing for regression coefficients is analogous to hypothesis test- ing for the population mean: Use the t-statistic to calculate the p-values and either accept or reject the null hypothesis. Like a confidence interval for the population mean, a 95% confidence interval for a regression coefficient is computed as the estimator {1.96 standard errors.
2. When X is binary, the regression model can be used to estimate and test hypotheses about the difference between the population means of the “X = 0” group and the “X = 1” group.
3. In general, the error ui is heteroskedastic—that is, the variance of ui at a given value of Xi, var(ui Xi = x), depends on x. A special case is when the error is homoskedastic—that is, var(ui Xi = x) is constant. Homoskedasticity-only standard errors do not produce valid statistical inferences when the errors are heteroskedastic, but heteroskedasticity-robust standard errors do.
4. If the three least squares assumption hold and if the regression errors are homoskedastic, then, as a result of the Gauss–Markov theorem, the OLS estimator is BLUE.
5. If the three least squares assumptions hold, if the regression errors are homoskedastic, and if the regression errors are normally distributed, then the OLS t-statistic computed using homoskedasticity-only standard errors has a Student t distribution when the null hypothesis is true. The difference between the Student t distribution and the normal distribution is negligible if the sample size is moderate or large.
Key Terms
null hypothesis (148)
two-sided alternative hypothesis
(148)
standard error of bn1 (148)
t-statistic (148)
p-value (149)
confidence interval for b1 (153) confidence level (153)
Key Terms 169
170 CHAPTER 5
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
indicator variable (155) dummy variable (155) coefficient multiplying Di (156) coefficient on Di (156) heteroskedasticity and
homoskedasticity (158) homoskedasticity-only standard
errors (160) heteroskedasticity-robust standard
error (161)
Gauss–Markov theorem (164) best linear unbiased estimator
(BLUE) (165)
weighted least squares (165) homoskedastic normal regression
assumptions (166) Gauss–Markov conditions
(179)
MyEconLab Can Help You Get a Better Grade
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For additional Empirical Exercises and Data Sets, log on to the Companion Website at www.pearsonhighered.com/stock_watson.
Review the Concepts
5.1 Outline the procedures for computing the p-value of a two-sided test of H0 : mY = 0 using an i.i.d. set of observations Yi, i = 1, c, n. Outline the procedures for computing the p-value of a two-sided test of H0 : b1 = 0 in a regression model using an i.i.d. set of observations (Yi, Xi), i = 1, c, n.
5.2 Explain how you could use a regression model to estimate the wage gender gap using the data on earnings of men and women. What are the depen- dent and independent variables?
5.3 Define homoskedasticity and heteroskedasticity. Provide a hypothetical empirical example in which you think the errors would be heteroskedastic and explain your reasoning.
5.4 Consider the regression Yi = b0 + b1Xi + ui, where Yi denotes a worker’s average hourly earnings (measured in dollars) and Xi is a binary (or indi- cator) variable that is equal to 1 if the worker has a college degree and is equal to 0 otherwise. Suppose that b1 = 8.1. Explain what this value means. Include the units of b1 in your answer.
Exercises
5.1 Suppose that a researcher, using data on class size (CS) and average test scores from 100 third-grade classes, estimates the OLS regression
TestScore = 520.4 – 5.82 * CS, R2 = 0.08, SER = 11.5. (20.4) (2.21)
a. Construct a 95% confidence interval for b1, the regression slope coefficient.
b. Calculate the p-value for the two-sided test of the null hypothesis
H0 : b1 = 0. Do you reject the null hypothesis at the 5% level? At the 1% level?
c. Calculate the p-value for the two-sided test of the null hypothesis H0: b1 = -5.6. Without doing any additional calculations, determine whether -5.6 is contained in the 95% confidence interval for b1.
d. Construct a 99% confidence interval for b0.
5.2 Suppose that a researcher, using wage data on 250 randomly selected male
workers and 280 female workers, estimates the OLS regression
Wage = 12.52 + 2.12 * Male, R2 = 0.06, SER = 4.2, (0.23) (0.36)
where Wage is measured in dollars per hour and Male is a binary variable that is equal to 1 if the person is a male and 0 if the person is a female. Define the wage gender gap as the difference in mean earnings between men and women.
a. What is the estimated gender gap?
b. Is the estimated gender gap significantly different from 0? (Compute the
p-value for testing the null hypothesis that there is no gender gap.)
c. Construct a 95% confidence interval for the gender gap.
d. In the sample, what is the mean wage of women? Of men?
e. Another researcher uses these same data but regresses Wages on Female, a variable that is equal to 1 if the person is female and 0 if the person a male. What are the regression estimates calculated from this regression?
Wage= + *Female,R2 = ,SER= .
Exercises 171
172 CHAPTER 5
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
5.3 Suppose that a random sample of 200 20-year-old men is selected from a population, and their heights and weights are recorded. A regression of weight on height yields
Weight = – 99.41 + 3.94 * Height, R2 = 0.81, SER = 10.2, (2.15) (0.31)
where Weight is measured in pounds, and Height is measured in inches. A man has a late growth spurt and grows 1.5 inches over the course of a year. Construct a 99% confidence interval for the person’s weight gain.
5.4 Read the box “The Economic Value of a Year of Education: Homoskedas- ticity or Heteroskedasticity?” in Section 5.4. Use the regression reported in Equation (5.23) to answer the following.
a. A randomly selected 30-year-old worker reports an education level of 16 years. What is the worker’s expected average hourly earnings?
b. A high school graduate (12 years of education) is contemplating going to a community college for a 2-year degree. How much is this worker’s average hourly earnings expected to increase?
c. A high school counselor tells a student that, on average, college grad- uates earn $10 per hour more than high school graduates. Is this state- ment consistent with the regression evidence? What range of values is consistent with the regression evidence?
5.5 In the 1980s, Tennessee conducted an experiment in which kindergarten students were randomly assigned to “regular” and “small” classes and given standardized tests at the end of the year. (Regular classes contained approx- imately 24 students, and small classes contained approximately 15 students.) Suppose that, in the population, the standardized tests have a mean score of 925 points and a standard deviation of 75 points. Let SmallClass denote a binary variable equal to 1 if the student is assigned to a small class and equal to 0 otherwise. A regression of TestScore on SmallClass yields
TestScore = 918.0 + 13.9 * SmallClass, R2 = 0.01, SER = 74.6.
(1.6) (2.5)
a. Do small classes improve test scores? By how much? Is the effect large? Explain.
b. Is the estimated effect of class size on test scores statistically signifi- cant? Carry out a test at the 5% level.
Exercises 173 c. Construct a 99% confidence interval for the effect of SmallClass on
Test Score.
5.6 Refer to the regression described in Exercise 5.5.
a. Do you think that the regression errors are plausibly homoskedastic? Explain.
b. SE(bn1) was computed using Equation (5.3). Suppose that the regression errors were homoskedastic: Would this affect the valid- ity of the confidence interval constructed in Exercise 5.5(c)? Explain.
5.7 Suppose that (Yi, Xi) satisfy the least squares assumptions in Key Concept 4.3. A random sample of size n = 250 is drawn and yields
Yn = 5.4 + 3.2X,R2 = 0.26,SER = 6.2. (3.1) (1.5)
a. Test H0 : b1 = 0 vs. H1 : b1 ≠ 0 at the 5% level.
b. Construct a 95% confidence interval for b1.
c. Suppose you learned that Yi and Xi were independent. Would you be surprised? Explain.
d. Suppose that Yi and Xi are independent and many samples of size
n = 250 are drawn, regressions estimated, and (a) and (b) answered. In what fraction of the samples would H0 from (a) be rejected? In what fraction of samples would the value b1 = 0 be included in the confidence interval from (b)?
5.8 Suppose that (Yi, Xi) satisfy the least squares assumptions in Key Concept 4.3 and, in addition, ui is N(0, s2u) and is independent of Xi. A sample of size n = 30 yields
Yn = 43.2 + 61.5X, R2 = 0.54, SER = 1.52,
(10.2) (7.4)
where the numbers in parentheses are the homoskedastic-only standard errors for the regression coefficients.
a. Construct a 95% confidence interval for b0.
b. TestH0: b1 = 55vs.H1: b1 ≠ 55atthe5%level.
c. TestH0: b1 = 55vs.H1: b1 7 55atthe5%level.
174 CHAPTER 5
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
5.9 Consider the regression model
Yi =bXi +ui,
where ui and Xi satisfy the least squares assumptions in Key Concept 4.3. Let b denote an estimator of b that is constructed as b = Y > X , where Y and X are the sample means of Yi and Xi, respectively.
a. Show that b is a linear function of Y1, Y2, c, Yn.
b. Show that b is conditionally unbiased.
5.10 Let Xi denote a binary variable and consider the regression Yi = b0 + b1Xi + ui. Let Y0 denote the sample mean for observations with X = 0 and let Y1 denote the sample mean for observations with X = 1. Show that bn0 = Y0, bn0 + bn1 = Y1, and bn1 = Y1 – Y0.
5.11 Arandomsampleofworkerscontainsnm = 120menandnw = 131women.
= (1>n )gnm Y
m i=1 m,i
m nm-1i=1m,i m
is $68.10. The corresponding values for women are Yw = $485.10 and sw = $51.10. Let Women denote an indicator variable that is equal to 1 for women and 0 for men and suppose that all 251 observations are used in the regression Yi = b0 + b1 Womeni + ui. Find the OLS estimates of b0 and
b1 and their corresponding standard errors.
5.12 Starting from Equation (4.22), derive the variance of bn0 under homoske-
dasticity given in Equation (5.28) in Appendix 5.1.
5.13 Suppose that (Yi, Xi) satisfy the least squares assumptions in Key Concept
4.3 and, in addition, ui is N(0, s2u) and is independent of Xi.
a. Is bn1 conditionally unbiased?
b. Is bn1 the best linear conditionally unbiased estimator of b1?
c. How would your answers to (a) and (b) change if you assumed only that (Yi, Xi) satisfied the least squares assumptions in Key Concept 4.3 and var(ui Xi = x) is constant?
d. How would your answers to (a) and (b) change if you assumed only that (Yi, Xi) satisfied the least squares assumptions in Key Concept 4.3?
5.14 Suppose that Yi = bXi + ui, where (ui, Xi) satisfy the Gauss–Markov con- ditions given in Equation (5.31).
a. Derive the least squares estimator of b and show that it is a linear function of Y1, c, Yn.
The sample average of men’s weekly earnings 3Y $523.10, and the sample standard deviation 3s = 2
4 is 1 g nm (Y – Y )24
m
b. Show that the estimator is conditionally unbiased.
c. Derive the conditional variance of the estimator.
d. Prove that the estimator is BLUE.
5.15 A researcher has two independent samples of observations on (Yi, Xi). To be specific, suppose that Yi denotes earnings, Xi denotes years of school- ing, and the independent samples are for men and women. Write the regression for men as Ym,i = bm,0 + bm,1Xm,i + um,i and the regression for women as Yw,i = bw,0 + bw,1Xw,i + uw,i. Let bnm,1 denote the OLS estimator constructed using the sample of men, bnw,1 denote the OLS estimator con- structed from the sample of women, and SE(bnm,1) and SE(bnw,1) denote the corresponding standard errors. Show that the standard error of bnm,1 – bnw,1 is given by SE(bnm,1 – bnw,1) = 23SE(bnm,1)42 + 3SE(bnw,1)42.
Empirical Exercises
(Only three empirical exercises for this chapter are given in the text, but you can find more on the text website, http://www.pearsonhighered.com/stock_watson/.)
E5.1 Use the data set Earnings_and_Height described in Empirical Exercise 4.2 to carry out the following exercises.
a. Run a regression of Earnings on Height.
i. Is the estimated slope statistically significant?
ii. Construct a 95% confidence interval for the slope coefficient.
b. Repeat (a) for women.
c. Repeat (a) for men.
d. Test the null hypothesis that the effect of height on earnings is the
same for men and women. (Hint: See Exercise 5.15.)
e. One explanation for the effect on height on earnings is that some professions require strength, which is correlated with height. Does the effect of height on earnings disappear when the sample is restricted to occupations in which strength is unlikely to be important?
E5.2 Using the data set Growth described in Empirical Exercise 4.1, but exclud- ing the data for Malta, run a regression of Growth on TradeShare.
a. Is the estimated regression slope statistically significant? This is, can you reject the null hypothesis H0: b1 = 0 vs. a two-sided alternative hypothesis at the 10%, 5%, or 1% significance level?
Empirical Exercises 175
176 CHAPTER 5
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
b. What is the p-value associated with the coefficient’s t-statistic?
c. Construct a 90% confidence interval for b1.
E5.3 On the text website, http://www.pearsonhighered.com/stock_watson/, you will find the data file Birthweight_Smoking, which contains data for a ran- dom sample of babies born in Pennsylvania in 1989. The data include the baby’s birth weight together with various characteristics of the mother, including whether she smoked during the pregnancy.2 A detailed descrip- tion is given in Birthweight_Smoking_Description, also available on the website. In this exercise you will investigate the relationship between birth weight and smoking during pregnancy.
a. In the sample:
i. ii. iii.
b. i. ii.
iii.
What is the average value of Birthweight for all mothers?
For mothers who smoke?
For mothers who do not smoke?
Use the data in the sample to estimate the difference in average birth weight for smoking and nonsmoking mothers.
What is the standard error for the estimated difference in (i)?
Construct a 95% confidence interval for the difference in the average birth weight for smoking and nonsmoking mothers.
c.
d.
Run a regression of Birthweight on the binary variable Smoker.
i. Explain how the estimated slope and intercept are related to your
answers in parts (a) and (b).
ii. Explain how the SE(bn1) is related to your answer in b(ii).
iii. Construct a 95% confidence interval for the effect of smoking on birth weight.
Do you think smoking is uncorrelated with other factors that cause low birth weight? That is, do you think that the regression error term, say ui, has a conditional mean of zero, given Smoking (Xi)? (You will investigate this further in Birthweight and Smoking exercises in later chapters.)
2These data were provided by Professors Douglas Almond (Columbia University), Ken Chay (Brown University), and David Lee (Princeton University) and were used in their paper “The Costs of Low Birth Weight,” Quarterly Journal of Economics, August 2005, 120(3): 1031–1083.
APPENDIX
5.1
Formulas for OLS Standard Errors
This appendix discusses the formulas for OLS standard errors. These are first presented under the least squares assumptions in Key Concept 4.3, which allow for heteroskedasticity; these are the “heteroskedasticity-robust” standard errors. Formulas for the variance of the OLS estima- tors and the associated standard errors are then given for the special case of homoskedasticity.
Heteroskedasticity-Robust Standard Errors
The estimator sn 2n defined in Equation (5.4) is obtained by replacing the population vari- b1
ances in Equation (4.21) by the corresponding sample variances, with a modification. The variance in the numerator of Equation (4.21) is estimated by 1 g n (X – X )2 un 2, where
the divisor n – 2 (instead of n) incorporates a degrees-of-freedom adjustment to correct
for downward bias, analogously to the degrees-of-freedom adjustment used in the defini-
tion of the SER in Section 4.3. The variance in the denominator is estimated by
(1>n)gni = 1(Xi – X)2. Replacing var3(Xi – mX)ui4 and var(Xi) in Equation (4.21) by these
two estimators yields sn 2n in Equation (5.4). The consistency of heteroskedasticity-robust b1
standard errors is discussed in Section 17.3. The estimator of the variance of bn0 is
1nn22 n – 2 a H i un i
Formulas for OLS Standard Errors 177
sn 2n = 1 * i = 1 ,
bnn2 (5.26)
n-2 i=1 i i
0
a n1 a Hn 2i b i=1
where Hni = 1 – (X>n1gni=1 X2i)Xi. The standard error of bn0 is SE(bn0) = 2sn2n . The rea- b0
soning behind the estimator sn 2n is the same as behind sn 2n and stems from replacing popu-
b0
lation expectations with sample averages.
Homoskedasticity-Only Variances
b1
Under homoskedasticity, the conditional variance of ui given Xi is a constant: var(ui Xi) = s2u. If the errors are homoskedastic, the formulas in Key Concept 4.4 simplify to
s2n = s2u and b1 ns2
X
s2n = E(X2i)s2. b0 nsX2 u
(5.27) (5.28)
178 CHAPTER 5
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
To derive Equation (5.27), write the numerator in Equation (4.21) as var3(Xi – mX)ui4 = E(5(Xi – mX)ui – E3(Xi – mX)ui462) = E53(Xi – mX)ui426 = E3(Xi – mX)2u2i 4 = E3(Xi – mX)2 var(ui Xi)4, where the second equality follows because E3(Xi – mX)ui4 = 0 (by the first least squares assumption) and where the final equality follows from the law of iterated expectations (Section 2.3). If ui is homoskedastic, then var(ui Xi) = s2u, so E3(Xi – mX)2 var(ui Xi)4 = s2uE3(Xi – mX)24 = s2us2X. The result in Equation (5.27) fol- lows by substituting this expression into the numerator of Equation (4.21) and simplifying. A similar calculation yields Equation (5.28).
Homoskedasticity-Only Standard Errors
The homoskedasticity-only standard errors are obtained by substituting sample means and variances for the population means and variances in Equations (5.27) and (5.28) and by estimating the variance of ui by the square of the SER. The homoskedasticity-only estima- tors of these variances are
s2 s∼2n = un
b1 n
a(Xi -X)2
i=1
(homoskedasticity@only) and
(homoskedasticity@only),
(5.29)
(5.30)
n
a1 X2bs2
nai un s∼2n = i = 1
b0 n
a(Xi -X)2
i=1
where s2 is given in Equation (4.19). The homoskedasticity-only standard errors are the un
square roots of s∼2n and s∼n2 . b0 b1
5.2
APPENDIX
The Gauss–Markov Conditions and
a Proof of the Gauss–Markov Theorem
As discussed in Section 5.5, the Gauss–Markov theorem states that if the Gauss–Markov conditions hold, then the OLS estimator is the best (most efficient) conditionally linear unbi- ased estimator (is BLUE). This appendix begins by stating the Gauss–Markov conditions and showing that they are implied by the three least squares condition plus homoskedasticity.
The Gauss–Markov Conditions and a Proof of the Gauss–Markov Theorem 179 We next show that the OLS estimator is a linear conditionally unbiased estimator. Finally,
we turn to the proof of the theorem.
The Gauss–Markov Conditions
The three Gauss–Markov conditions are
(i) E(ui0X1,c,Xn)=0
(ii) var(ui0X1,c,Xn)=s2u, 06s2u6∞
(iii) E(uiuj 0X1, c, Xn) = 0, i ≠ j,
(5.31)
where the conditions hold for i, j = 1, c, n. The three conditions, respectively, state that ui has mean zero, that ui has a constant variance, and that the errors are uncorrelated for different observations, where all these statements hold conditionally on all observed X’s (X1, c, Xn).
The Gauss–Markov conditions are implied by the three least squares assumptions (Key Concept 4.3), plus the additional assumptions that the errors are homoskedastic. Because the observations are i.i.d. (Assumption 2), E(ui X1, c, Xn) = E(ui Xi), and by Assumption 1, E(uiXi) = 0; thus condition (i) holds. Similarly, by Assumption 2, var(uiX1,c,Xn) = var(uiXi),andbecausetheerrorsareassumedtobehomoskedastic, var(uiXi) = s2u, which is constant. Assumption 3 (nonzero finite fourth moments) ensures that 0 6 s2u 6 ∞, so condition (ii) holds. To show that condition (iii) is implied by the least squares assumptions, note that E(uiuj X1, c, Xn) = E(uiuj Xi, Xj) because (Xi, Yi) are i.i.d. by Assumption 2. Assumption 2 also implies that E(uiuj Xi, Xj) = E(ui Xi) E(uj Xj) for i ≠ j; because E(ui Xi) = 0 for all i, it follows that E(uiuj X1, c, Xn) = 0 for all i ≠ j, so condition (iii) holds. Thus the least squares assumptions in Key Concept 4.3, plus homoskedasticity of the errors, imply the Gauss–Markov conditions in Equation (5.31).
The OLS Estimator bn1 Is a Linear Conditionally
Unbiased Estimator
To show that bn1 is linear, first note that, because g ni = 1(Xi – X) = 0 (by the definition of X ), gni=1(Xi – X)(Yi – Y) = gni=1(Xi – X)Yi – Ygni=1(Xi – X) = gni=1(Xi – X)Yi. Sub- stituting this result into the formula for bn1 in Equation (4.7) yields
n
a(Xi – X)Yi n (X – X)
bn = i = 1 = an Y , where an = i (5.32) 1naiiin
a(Xj – X)2 i=1 a (Xj – X)2 j=1 j=1
180 CHAPTER 5
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
Because the weights ani, i = 1, c, n, in Equation (5.32) depend on X1, c, Xn but not on Y1, c, Yn, the OLS estimator bn1 is a linear estimator.
Under the Gauss–Markov conditions, bn1 is conditionally unbiased, and the variance of the conditional distribution of bn1, given X1, c, Xn, is
var(bn X , c, X ) = s2u . 11nn
a(Xi – X)2 i=1
The result that bn1 is conditionally unbiased was previously shown in Appendix 4.3. Proof of the Gauss–Markov Theorem
(5.33)
We start by deriving some facts that hold for all linear conditionally unbiased estimators—
that is, for all estimators b∼ satisfying Equations (5.24) and (5.25). Substituting 1
Y
= b + b X + u into b∼ = g n a Y and collecting terms, we have that i01ii1i=1ii
(5.34) a E(u 0 X , c, X ) = 0;
nnn
b∼ =ba ab+ba aXb+ au.
1 0ai 1aii aii i=1 i=1 i=1
By the first Gauss–Markov condition, E( g n a u X , c, X ) = g n i=1ii1ni=1ii1n
thus taking conditional expectations of both sides of Equation (5.34) yields E( b∼1 0 X1, c, Xn) = b (gn a ) + b (gn a X ). Because b∼ is conditionally unbiased by assumption, it must
0i=1i1i=1ii 1
be that b0(gni = 1ai) + b1(gni = 1aiXi) = b1, but for this equality to hold for all values of b0 and
b , it must be the case that, for b∼ to be conditionally unbiased, 11
nn
aai = 0 and aaiXi = 1. (5.35) i=1 i=1
Under the Gauss–Markov conditions, the variance of b∼ , conditional on X , c, X , has a 11n
simple form. Substituting Equation (5.35) into Equation (5.34) yields b∼ – b = g n a u . Thus 11i=1ii
v a r ( b∼ X , c , X ) = v a r ( g n a u X , c , X ) = g n g n a a c o v ( u , u X , c , X ) ; 11 n i=1ii1 n i=1j=1ij ij1 n
applying the second and third Gauss–Markov conditions, the cross terms in the double summation vanish, and the expression for the conditional variance simplifies to
n
var(b∼1 X1,c,Xn) = s2uaa2i. (5.36)
i=1
Note that Equations (5.35) and (5.36) apply to bn1 with weights ai = ani, given in Equation (5.32).
We now show that the two restrictions in Equation (5.35) and the expression for the
conditional variance in Equation (5.36) imply that the conditional variance of b∼ exceeds 1
theconditionalvarianceofbn unlessb∼ =bn.Leta =an +d,sogn a2 =gn (an +d)2 = 111iiii=1ii=1ii
gni=1 an2i + 2gni=1anidi + gni=1d2i.
The Gauss–Markov Conditions and a Proof of the Gauss–Markov Theorem 181 Using the definition of ani in Equation (5.32), we have that
nnn
n a(Xi – X)di adiXi – Xadi
an d = i = 1 = i = 1 i = 1 aiinn
i=1 a(Xj – X)2 a(Xj – X)2
j=1 j=1
nnnn
a aaiXi – aaniXib – Xa aai – aanib
= i=1 i=1 i=1 i=1 =0, n
a(Xj -X)2 j=1
where the penultimate equality follows from di = ai – ani and the final equality follows from Equation (5.35) (which holds for both ai and ani). Thus s2ugni=1a2i = s2ugni=1an2i + s2ugni=1d2i = var(bn1 X1,c,Xn) + s2ugni=1d2i;substitutingthisresultintoEquation(5.36) yields
n
var(b∼1 X1, c, Xn) – var(bn1 X1, c, Xn) = s2u ad2i . (5.37)
i=1
Thus b∼ has a greater conditional variance than bn if d is nonzero for any i = 1, c, n. But
11i
if d = 0 for all i, then a = an and b∼ = bn , which proves that OLS is BLUE.
i ii11
The Gauss–Markov Theorem When X Is Nonrandom
With a minor change in interpretation, the Gauss–Markov theorem also applies to nonrandom regressors; that is, it applies to regressors that do not change their values over repeated samples. Specifically, if the second least squares assumption is replaced by the assumption that X1, c, Xn are nonrandom (fixed over repeated samples) and u1, c, un are i.i.d., then the foregoing statement and proof of the Gauss–Markov theorem apply directly, except that all of the “conditional on X1, c, Xn” statements are unnecessary because X1, c, Xn take on the same values from one sample to the next.
The Sample Average Is the Efficient
Linear Estimator of E(Y )
An implication of the Gauss–Markov theorem is that the sample average, Y, is the most efficient linear estimator of E(Yi) when Yi , c, Yn are i.i.d. To see this, consider the case of regression without an “X” so that the only regressor is the constant regressor X0i = 1. Then the OLS estimator bn0 = Y. It follows that, under the Gauss–Markov assumptions, Y is BLUE. Note that the Gauss–Markov requirement that the error be homoskedastic is automatically satisfied in this case because there is no regressor, so it follows that Y is BLUE if Y1, c, Yn are i.i.d. This result was stated previously in Key Concept 3.3.
182
Linear Regression
C h6a p t e r
with Multiple Regressors
C hapter 5 ended on a worried note. Although school districts with lower student–teacher ratios tend to have higher test scores in the California data
set, perhaps students from districts with small classes have other advantages that help them perform well on standardized tests. Could this have produced misleading results, and, if so, what can be done?
Omitted factors, such as student characteristics, can, in fact, make the ordinary least squares (OLS) estimator of the effect of class size on test scores misleading or, more precisely, biased. This chapter explains this “omitted variable bias” and intro- duces multiple regression, a method that can eliminate omitted variable bias. The key idea of multiple regression is that if we have data on these omitted variables, then we can include them as additional regressors and thereby estimate the effect of one regressor (the student–teacher ratio) while holding constant the other vari- ables (such as student characteristics).
This chapter explains how to estimate the coefficients of the multiple linear regression model. Many aspects of multiple regression parallel those of regression with a single regressor, studied in Chapters 4 and 5. The coefficients of the multiple regression model can be estimated from data using OLS; the OLS estimators in multiple regression are random variables because they depend on data from a random sample; and in large samples the sampling distributions of the OLS estimators are approximately normal.
6.1
Omitted Variable Bias
By focusing only on the student–teacher ratio, the empirical analysis in Chapters 4 and 5 ignored some potentially important determinants of test scores by collect- ing their influences in the regression error term. These omitted factors include school characteristics, such as teacher quality and computer usage, and student characteristics, such as family background. We begin by considering an omitted student characteristic that is particularly relevant in California because of its large immigrant population: the prevalence in the school district of students who are still learning English.
By ignoring the percentage of English learners in the district, the OLS estima- tor of the slope in the regression of test scores on the student–teacher ratio could be biased; that is, the mean of the sampling distribution of the OLS estimator might not equal the true effect on test scores of a unit change in the student– teacher ratio. Here is the reasoning. Students who are still learning English might perform worse on standardized tests than native English speakers. If districts with large classes also have many students still learning English, then the OLS regres- sion of test scores on the student–teacher ratio could erroneously find a correla- tion and produce a large estimated coefficient, when in fact the true causal effect of cutting class sizes on test scores is small, even zero. Accordingly, based on the analysis of Chapters 4 and 5, the superintendent might hire enough new teachers to reduce the student–teacher ratio by 2, but her hoped-for improvement in test scores will fail to materialize if the true coefficient is small or zero.
A look at the California data lends credence to this concern. The correlation between the student–teacher ratio and the percentage of English learners (stu- dents who are not native English speakers and who have not yet mastered Eng- lish) in the district is 0.19. This small but positive correlation suggests that districts with more English learners tend to have a higher student–teacher ratio (larger classes). If the student–teacher ratio were unrelated to the percentage of English learners, then it would be safe to ignore English proficiency in the regression of test scores against the student–teacher ratio. But because the student–teacher ratio and the percentage of English learners are correlated, it is possible that the OLS coefficient in the regression of test scores on the student–teacher ratio reflects that influence.
Definition of Omitted Variable Bias
If the regressor (the student–teacher ratio) is correlated with a variable that has been omitted from the analysis (the percentage of English learners) and that determines, in part, the dependent variable (test scores), then the OLS estimator will have omitted variable bias.
Omitted variable bias occurs when two conditions are true: (1) when the omitted variable is correlated with the included regressor and (2) when the omit- ted variable is a determinant of the dependent variable. To illustrate these condi- tions, consider three examples of variables that are omitted from the regression of test scores on the student–teacher ratio.
Example #1: Percentage of English learners. Because the percentage of English learners is correlated with the student–teacher ratio, the first condition for omitted
6.1 Omitted Variable Bias 183
184 ChapteR 6 Linear Regression with Multiple Regressors
variable bias holds. It is plausible that students who are still learning English will do worse on standardized tests than native English speakers, in which case the percentage of English learners is a determinant of test scores and the second con- dition for omitted variable bias holds. Thus the OLS estimator in the regression of test scores on the student–teacher ratio could incorrectly reflect the influence of the omitted variable, the percentage of English learners. That is, omitting the percentage of English learners may introduce omitted variable bias.
Example #2: Time of day of the test. Another variable omitted from the analysis is the time of day that the test was administered. For this omitted variable, it is plausible that the first condition for omitted variable bias does not hold but that the second condition does. For example, if the time of day of the test varies from one district to the next in a way that is unrelated to class size, then the time of day and class size would be uncorrelated so the first condition does not hold. Conversely, the time of day of the test could affect scores (alertness varies through the school day), so the second condition holds. However, because in this example the time of day the test is administered is uncorrelated with the student–teacher ratio, the student–teacher ratio could not be incorrectly picking up the “time of day” effect. Thus omitting the time of day of the test does not result in omitted variable bias.
Example #3: Parking lot space per pupil. Another omitted variable is parking lot space per pupil (the area of the teacher parking lot divided by the number of students). This variable satisfies the first but not the second condition for omitted variable bias. Specifically, schools with more teachers per pupil probably have more teacher parking space, so the first condition would be satisfied. However, under the assumption that learning takes place in the classroom, not the parking lot, parking lot space has no direct effect on learning; thus the second condition does not hold. Because parking lot space per pupil is not a determinant of test scores, omitting it from the analysis does not lead to omitted variable bias.
Omitted variable bias is summarized in Key Concept 6.1.
Omitted variable bias and the first least squares assumption. Omitted variable bias means that the first least squares assumption—that E(ui Xi) = 0, as listed in Key Concept 4.3—is incorrect. To see why, recall that the error term ui in the linear regression model with a single regressor represents all factors, other than Xi, that are determinants of Yi. If one of these other factors is correlated with Xi, this means that the error term (which contains this factor) is correlated with Xi. In other words, if an omitted variable is a determinant of Yi, then it is in the error term, and if it is correlated with Xi, then the error term is correlated with Xi.
6.1 Omitted Variable Bias 185
Omitted Variable Bias in Regression with a Single Regressor
Key ConCept
6.1
Omitted variable bias is the bias in the OLS estimator that arises when the regres- sor, X, is correlated with an omitted variable. For omitted variable bias to occur, two conditions must be true:
1. X is correlated with the omitted variable.
2. The omitted variable is a determinant of the dependent variable, Y.
Because ui and Xi are correlated, the conditional mean of ui given Xi is nonzero. This correlation therefore violates the first least squares assumption, and the con- sequence is serious: The OLS estimator is biased. This bias does not vanish even in very large samples, and the OLS estimator is inconsistent.
A Formula for Omitted Variable Bias
The discussion of the previous section about omitted variable bias can be sum- marized mathematically by a formula for this bias. Let the correlation between Xi and ui be corr(Xi, ui) = rXu. Suppose that the second and third least squares assumptions hold, but the first does not because rXu is nonzero. Then the OLS estimator has the limit (derived in Appendix 6.1)
np su
b1 ¡ b1 + rXus . (6.1)
X
The formula in Equation (6.1) summarizes several of the ideas discussed above about omitted variable bias:
1. Omitted variable bias is a problem whether the sample size is large or small.
That is, as the sample size increases, bn1 is close to b1 + rXu(su > sX) with increas- ingly high probability.
Because bn1 does not converge in probability to the true value b1, bn1 is biased
omitted variable bias. The term r (s > s ) in Equation (6.1) is the bias in Xu u X
and inconsistent; that is, bn1 is not a consistent estimator of b1 when there is
bn1 that persists even in large samples.
186 ChapteR 6 Linear Regression with Multiple Regressors the Mozart effect: omitted Variable Bias?
Astudy published in Nature in 1993 (Rauscher, Shaw, and Ky, 1993) suggested that listening to Mozart for 10 to 15 minutes could temporarily raise your IQ by 8 or 9 points. That study made big news—and politicians and parents saw an easy way to make their children smarter. For a while, the state of Georgia even distributed classical music CDs to all infants in the state.
What is the evidence for the “Mozart effect”? A review of dozens of studies found that students who take optional music or arts courses in high school do, in fact, have higher English and math test scores than those who don’t.1 A closer look at these stud- ies, however, suggests that the real reason for the better test performance has little to do with those courses. Instead, the authors of the review suggested that the correlation between testing well and taking art or music could arise from any number of things. For example, the academically better students might have more time to take optional music courses or more interest in doing so, or those schools with a deeper music curriculum might just be better schools across the board.
In the terminology of regression, the estimated relationship between test scores and taking optional
music courses appears to have omitted variable bias. By omitting factors such as the student’s innate abil- ity or the overall quality of the school, studying music appears to have an effect on test scores when in fact it has none.
So is there a Mozart effect? One way to find out is to do a randomized controlled experiment. (As discussed in Chapter 4, randomized controlled experiments eliminate omitted variable bias by randomly assigning participants to “treatment” and “control” groups.) Taken together, the many con- trolled experiments on the Mozart effect fail to show that listening to Mozart improves IQ or general test performance. For reasons not fully understood, however, it seems that listening to classical music does help temporarily in one narrow area: fold- ing paper and visualizing shapes. So the next time you cram for an origami exam, try to fit in a little Mozart, too.
1See the fall/winter 2000 issue of Journal of Aesthetic Education 34, especially the article by Ellen Winner and Monica Cooper (pp. 11–76) and the one by Lois Hetland (pp. 105–148).
r between the regressor and the error term. The larger 0 r Xu Xu
0 is, the larger
2. Whether this bias is large or small in practice depends on the correlation
the bias.
3. The direction of the bias in bn1 depends on whether X and u are positively
or negatively correlated. For example, we speculated that the percentage of students learning English has a negative effect on district test scores (stu- dents still learning English have lower scores), so that the percentage of English learners enters the error term with a negative sign. In our data, the fraction of English learners is positively correlated with the student–teacher
ratio (districts with more English learners have larger classes). Thus the student– teacher ratio (X ) would be negatively correlated with the error term (u), so rXu 6 0 and the coefficient on the student–teacher ratio bn1 would be biased toward a negative number. In other words, having a small percentage of English learners is associated both with high test scores and low student– teacher ratios, so one reason that the OLS estimator suggests that small classes improve test scores may be that the districts with small classes have fewer English learners.
Addressing Omitted Variable Bias by Dividing
the Data into Groups
What can you do about omitted variable bias? Our superintendent is considering increasing the number of teachers in her district, but she has no control over the fraction of immigrants in her community. As a result, she is interested in the effect of the student–teacher ratio on test scores, holding constant other factors, includ- ing the percentage of English learners. This new way of posing her question sug- gests that, instead of using data for all districts, perhaps we should focus on districts with percentages of English learners comparable to hers. Among this subset of districts, do those with smaller classes do better on standardized tests?
Table 6.1 reports evidence on the relationship between class size and test scores within districts with comparable percentages of English learners. Districts are divided into eight groups. First, the districts are broken into four categories
Differences in test Scores for California School Districts with Low and high Student–teacher Ratios, by the percentage of english Learners in the District
6.1 Omitted Variable Bias 187
taBLe 6.1
All districts
Percentage of English learners
6 1.9% 1.9–8.8%
8.8–23.0%
7 23.0%
Student–teacher ratio < 20
Student–teacher ratio ≥ 20
average
test Score n
650.0 182
665.4 27
661.8 44
649.7 50
634.8 61
Difference in test Scores, Low vs. high Str
average test Score
n
Difference
7.4
- 0.9
3.3
5.2
1.9
t-statistic 4.04
- 0.30
1.13
1.72
0.68
657.4 238
664.5 76
665.2 64
654.9 54
636.7 44
188 ChapteR 6 Linear Regression with Multiple Regressors
that correspond to the quartiles of the distribution of the percentage of English learners across districts. Second, within each of these four categories, districts are further broken down into two groups, depending on whether the student–teacher ratio is small (STR 6 20) or large (STR Ú 20).
The first row in Table 6.1 reports the overall difference in average test scores between districts with low and high student–teacher ratios, that is, the difference in test scores between these two groups without breaking them down further into the quartiles of English learners. (Recall that this difference was previously reported in regression form in Equation (5.18) as the OLS estimate of the coeffi- cient on Di in the regression of TestScore on Di, where Di is a binary regressor that equals 1 if STRi 6 20 and equals 0 otherwise.) Over the full sample of 420 districts, the average test score is 7.4 points higher in districts with a low student–teacher ratio than a high one; the t-statistic is 4.04, so the null hypothesis that the mean test score is the same in the two groups is rejected at the 1% significance level.
The final four rows in Table 6.1 report the difference in test scores between districts with low and high student–teacher ratios, broken down by the quartile of the percentage of English learners. This evidence presents a different picture. Of the districts with the fewest English learners (6 1.9%), the average test score for those 76 with low student–teacher ratios is 664.5 and the average for the 27 with high student–teacher ratios is 665.4. Thus, for the districts with the fewest English learners, test scores were on average 0.9 points lower in the districts with low student–teacher ratios! In the second quartile, districts with low student–teacher ratios had test scores that averaged 3.3 points higher than those with high student– teacher ratios; this gap was 5.2 points for the third quartile and only 1.9 points for the quartile of districts with the most English learners. Once we hold the percent- age of English learners constant, the difference in performance between districts with high and low student–teacher ratios is perhaps half (or less) of the overall estimate of 7.4 points.
At first this finding might seem puzzling. How can the overall effect of test scores be twice the effect of test scores within any quartile? The answer is that the districts with the most English learners tend to have both the highest student– teacher ratios and the lowest test scores. The difference in the average test score between districts in the lowest and highest quartile of the percentage of English learners is large, approximately 30 points. The districts with few English learners tend to have lower student–teacher ratios: 74% (76 of 103) of the districts in the first quartile of English learners have small classes (STR 6 20), while only 42% (44 of 105) of the districts in the quartile with the most English learners have small classes. So, the districts with the most English learners have both lower test scores and higher student–teacher ratios than the other districts.
This analysis reinforces the superintendent’s worry that omitted variable bias is present in the regression of test scores against the student–teacher ratio. By looking within quartiles of the percentage of English learners, the test score dif- ferences in the second part of Table 6.1 improve on the simple difference-of- means analysis in the first line of Table 6.1. Still, this analysis does not yet provide the superintendent with a useful estimate of the effect on test scores of changing class size, holding constant the fraction of English learners. Such an estimate can be provided, however, using the method of multiple regression.
6.2
The Multiple Regression Model
The multiple regression model extends the single variable regression model of Chapters 4 and 5 to include additional variables as regressors. This model permits estimating the effect on Yi of changing one variable (X1i) while holding the other regressors (X2i, X3i, and so forth) constant. In the class size problem, the multiple regression model provides a way to isolate the effect on test scores (Yi) of the student–teacher ratio (X1i) while holding constant the percentage of students in the district who are English learners (X2i).
The Population Regression Line
Suppose for the moment that there are only two independent variables, X1i and X2i. In the linear multiple regression model, the average relationship between these two independent variables and the dependent variable, Y, is given by the linear function
E(YiX1i = x1,X2i = x2) = b0 + b1x1 + b2x2, (6.2)
where E(Yi X1i = x1, X2i = x2) is the conditional expectation of Yi given that X1i = x1 and X2i = x2. That is, if the student–teacher ratio in the ith district (X1i) equals some value x1 and the percentage of English learners in the ith district (X2i) equals x2, then the expected value of Yi given the student–teacher ratio and the percentage of English learners is given by Equation (6.2).
Equation (6.2) is the population regression line or population regression func- tion in the multiple regression model. The coefficient b0 is the intercept; the coef- ficient b1 is the slope coefficient of X1i or, more simply, the coefficient on X1i; and the coefficient b2 is the slope coefficient of X2i or, more simply, the coefficient on X2i. One or more of the independent variables in the multiple regression model are sometimes referred to as control variables.
6.2 The Multiple Regression Model 189
190 ChapteR 6 Linear Regression with Multiple Regressors
The interpretation of the coefficient b1 in Equation (6.2) is different than it was when X1i was the only regressor: In Equation (6.2), b1 is the effect on Y of a unit change in X1, holding X2 constant or controlling for X2.
This interpretation of b1 follows from the definition that the expected effect on Y of a change in X1, ∆X1, holding X2 constant, is the difference between the expected value of Y when the independent variables take on the values X1 + ∆X1 and X2 and the expected value of Y when the independent variables take on the values X1 and X2. Accordingly, write the population regression function in Equation (6.2) as Y = b0 + b1X1 + b2X2 and imagine changing X1 by the amount ∆X1 while not changing X2, that is, while holding X2 constant. Because X1 has changed, Y will change by some amount, say ∆Y. After this change, the new value of Y, Y + ∆Y, is
Y+∆Y=b0 +b1(X1 +∆X1)+b2X2. (6.3)
An equation for ∆Y in terms of ∆X1 is obtained by subtracting the equation Y = b0 + b1X1 + b2X2 from Equation (6.3), yielding ∆Y = b1∆X1. Rearranging this equation shows that
b1 = ∆Y holding X2 constant. (6.4) ∆X1
The coefficient b1 is the effect on Y (the expected change in Y ) of a unit change in X1, holding X2 fixed. Another phrase used to describe b1 is the partial effect on Y of X1, holding X2 fixed.
The interpretation of the intercept in the multiple regression model, b0, is similar to the interpretation of the intercept in the single-regressor model: It is the expected value of Yi when X1i and X2i are zero. Simply put, the intercept b0 deter- mines how far up the Y axis the population regression line starts.
The Population Multiple Regression Model
The population regression line in Equation (6.2) is the relationship between Y and X1 and X2 that holds on average in the population. Just as in the case of regression with a single regressor, however, this relationship does not hold exactly because many other factors influence the dependent variable. In addition to the student– teacher ratio and the fraction of students still learning English, for example, test scores are influenced by school characteristics, other student characteristics, and luck. Thus the population regression function in Equation (6.2) needs to be aug- mented to incorporate these additional factors.
Just as in the case of regression with a single regressor, the factors that deter- mine Yi in addition to X1i and X2i are incorporated into Equation (6.2) as an “error” term ui. This error term is the deviation of a particular observation (test scores in the ith district in our example) from the average population relationship. Accordingly, we have
Yi = b0 + b1X1i + b2X2i + ui,i = 1,c,n, (6.5)
where the subscript i indicates the ith of the n observations (districts) in the sample. Equation (6.5) is the population multiple regression model when there are
two regressors, X1i and X2i.
In regression with binary regressors, it can be useful to treat b0 as the coeffi-
cient on a regressor that always equals 1; think of b0 as the coefficient on X0i, where X0i = 1 for i = 1, c, n. Accordingly, the population multiple regression model in Equation (6.5) can alternatively be written as
Yi = b0X0i + b1X1i + b2X2i + ui, where X0i = 1, i = 1, c, n. (6.6)
The variable X0i is sometimes called the constant regressor because it takes on the same value—the value 1—for all observations. Similarly, the intercept, b0, is sometimes called the constant term in the regression.
The two ways of writing the population regression model, Equations (6.5) and (6.6), are equivalent.
The discussion so far has focused on the case of a single additional variable, X2. In practice, however, there might be multiple factors omitted from the single- regressor model. For example, ignoring the students’ economic background might result in omitted variable bias, just as ignoring the fraction of English learners did. This reasoning leads us to consider a model with three regressors or, more gener- ally, a model that includes k regressors. The multiple regression model with k regressors, X1i, X2i, c, Xki, is summarized as Key Concept 6.2.
The definitions of homoskedasticity and heteroskedasticity in the multiple regression model are extensions of their definitions in the single-regressor model. The error term ui in the multiple regression model is homoskedastic if the variance of the conditional distribution of ui given X1i, c, Xki, var(ui X1i, c, Xki), is constant for i = 1, c, n and thus does not depend on the values of X1i, c, Xki. Otherwise, the error term is heteroskedastic.
The multiple regression model holds out the promise of providing just what the superintendent wants to know: the effect of changing the student– teacher ratio, holding constant other factors that are beyond her control.
6.2 The Multiple Regression Model 191
192 ChapteR 6 Linear Regression with Multiple Regressors
the Multiple Regression Model
6.2
Key ConCept
The multiple regression model is
Yi =b0 +b1X1i +b2X2i + g+bkXki +ui,i=1,c,n,
(6.7)
where
• Yi is ith observation on the dependent variable; X1i, X2i, c, Xki are the ith
observations on each of the k regressors; and ui is the error term.
• The population regression line is the relationship that holds between Y and
the X’s on average in the population:
E(YX1i = x1,X2i = x2,c,Xki = xk)= b0 + b1x1 + b2x2 + g+ bkxk.
• b1 is the slope coefficient on X1, b2 is the coefficient on X2, and so on. The coefficient b1 is the expected change in Yi resulting from changing X1i by one unit, holding constant X2i, c, Xki. The coefficients on the other X’s are interpreted similarly.
• The intercept b0 is the expected value of Y when all the X’s equal 0. The intercept can be thought of as the coefficient on a regressor, X0i, that equals 1 for all i.
These factors include not just the percentage of English learners, but other measurable factors that might affect test performance, including the economic background of the students. To be of practical help to the superintendent, however, we need to provide her with estimates of the unknown population coefficients b0, c, bk of the population regression model calculated using a sample of data. Fortunately, these coefficients can be estimated using ordinary least squares.
6.3
The OLS Estimator in Multiple Regression
This section describes how the coefficients of the multiple regression model can be estimated using OLS.
6.3 The OLS Estimator in Multiple Regression 193
The OLS Estimator
Section 4.2 shows how to estimate the intercept and slope coefficients in the single-
regressor model by applying OLS to a sample of observations of Y and X. The
key idea is that these coefficients can be estimated by minimizing the sum of
minimize g (Y - b - b X ) . The estimators that do so are the OLS estima- ni=1i 0 1i2
squared prediction mistakes, that is, by choosing the estimators b0 and b1 so as to
tors, bn0 and bn1.
The method of OLS also can be used to estimate the coefficients b0, b1, c, bk
in the multiple regression model. Let b0, b1, c, bk be estimates of b0, b1, c, bk. The predicted value of Yi, calculated using these estimates, is b0 + b1X1i + g+ bkXki, and the mistake in predicting Yi is Yi - (b0 + b1X1i + g + bkXki) = Yi - b0 - b1X1i - g - bkXki. The sum of these squared prediction mistakes over all n observations is thus
an
i=1(Yi - b0 - b1X1i - g- bkXki)2. (6.8)
The sum of the squared mistakes for the linear regression model in Expression (6.8) is the extension of the sum of the squared mistakes given in Equation (4.6) for the linear regression model with a single regressor.
The estimators of the coefficients b0, b1, c, bk that minimize the sum of
squared mistakes in Expression (6.8) are called the ordinary least squares (OLS)
estimators of B , B , c, B . The OLS estimators are denoted b , b , c, b . 01 k n0n1 nk
The terminology of OLS in the linear multiple regression model is the same
as in the linear regression model with a single regressor. The OLS regression line
is the straight line constructed using the OLS estimators: b + b X + g + b X . n0 n1 1 nk k
The predicted value of Yi given X1i, c, Xki, based on the OLS regression line, is Y = b + bX + g+ bX .TheOLSresidualforthei observationisthe
ni n0 n11i nkki th
difference between Yi and its OLS predicted value; that is, the OLS residual is
n un i = Y i - Y i .
The OLS estimators could be computed by trial and error, repeatedly trying different values of b0, c, bk until you are satisfied that you have minimized the total sum of squares in Expression (6.8). It is far easier, however, to use explicit formulas for the OLS estimators that are derived using calculus. The formulas for the OLS estimators in the multiple regression model are similar to those in Key Concept 4.2 for the single-regressor model. These formulas are incorporated into modern statistical software. In the multiple regression model, the formulas are best expressed and discussed using matrix notation, so their presentation is deferred to Section 18.1.
194 ChapteR 6 Linear Regression with Multiple Regressors
Key ConCept
6.3
the OLS estimators, predicted Values, and Residuals in the Multiple Regression Model
The OLS estimators b , b , c, b are the values of b , b , c, b that minimize n0n1 nk 01 k
thesumofsquaredpredictionmistakesg (Y -b -bX -g-bX ). ni=1i 0 11i kki2
The OLS predicted values Yni and residuals uni are
Y = b + b X + g + b X , i = 1, c, n, and
ni n0 n1 1i nk ki
un = Y - Y,i = 1,c,n.
(6.9) (6.10)
i i ni
The OLS estimators b , b , c, b and residual un are computed from a sample
n0 n1 nk i
of n observations of (X1i, c, Xki, Yi ), i = 1, c, n. These are estimators of the
unknown true population coefficients b0, b1, c, bk and error term, ui.
The definitions and terminology of OLS in multiple regression are summa- rized in Key Concept 6.3.
Application to Test Scores and
the Student–Teacher Ratio
In Section 4.2, we used OLS to estimate the intercept and slope coefficient of the regression relating test scores (TestScore) to the student–teacher ratio (STR), using our 420 observations for California school districts; the estimated OLS regression line, reported in Equation (4.11), is
TestScore = 698.9 - 2.28 * STR. (6.11)
Our concern has been that this relationship is misleading because the student– teacher ratio might be picking up the effect of having many English learners in districts with large classes. That is, it is possible that the OLS estimator is subject to omitted variable bias.
We are now in a position to address this concern by using OLS to estimate a multiple regression in which the dependent variable is the test score (Yi) and there are two regressors: the student–teacher ratio (X1i) and the percentage of
6.3 The OLS Estimator in Multiple Regression 195 English learners in the school district (X2i) for our 420 districts (i = 1, c, 420).
The estimated OLS regression line for this multiple regression is
TestScore = 686.0 - 1.10 * STR - 0.65 * PctEL, (6.12)
where PctEL is the percentage of students in the district who are English learners. The OLS estimate of the intercept (bn0) is 686.0, the OLS estimate of the coeffi- cient on the student–teacher ratio (bn1) is -1.10, and the OLS estimate of the coefficient on the percentage English learners (bn2) is -0.65.
The estimated effect on test scores of a change in the student–teacher ratio in the multiple regression is approximately half as large as when the student–teacher ratio is the only regressor: In the single-regressor equation [Equation (6.11)], a unit decrease in the STR is estimated to increase test scores by 2.28 points, but in the multiple regression equation [Equation (6.12)], it is estimated to increase test scores by only 1.10 points. This difference occurs because the coefficient on STR in the multiple regression is the effect of a change in STR, holding constant (or controlling for) PctEL, whereas in the single-regressor regression, PctEL is not held constant.
These two estimates can be reconciled by concluding that there is omitted variable bias in the estimate in the single-regressor model in Equation (6.11). In Section 6.1, we saw that districts with a high percentage of English learners tend to have not only low test scores but also a high student–teacher ratio. If the frac- tion of English learners is omitted from the regression, reducing the student– teacher ratio is estimated to have a larger effect on test scores, but this estimate reflects both the effect of a change in the student–teacher ratio and the omitted effect of having fewer English learners in the district.
We have reached the same conclusion that there is omitted variable bias in the relationship between test scores and the student–teacher ratio by two differ- ent paths: the tabular approach of dividing the data into groups (Section 6.1) and the multiple regression approach [Equation (6.12)]. Of these two methods, mul- tiple regression has two important advantages. First, it provides a quantitative estimate of the effect of a unit decrease in the student–teacher ratio, which is what the superintendent needs to make her decision. Second, it readily extends to more than two regressors so that multiple regression can be used to control for measur- able factors other than just the percentage of English learners.
The rest of this chapter is devoted to understanding and using OLS in the multiple regression model. Much of what you learned about the OLS estimator with a single regressor carries over to multiple regression with few or no modifica- tions, so we will focus on that which is new with multiple regression. We begin by discussing measures of fit for the multiple regression model.
196 ChapteR 6 Linear Regression with Multiple Regressors
6.4
Measures of Fit in Multiple Regression
Three commonly used summary statistics in multiple regression are the standard error of the regression, the regression R2, and the adjusted R2 (also known as R 2). All three statistics measure how well the OLS estimate of the multiple regression line describes, or “fits,” the data.
The Standard Error of the Regression (SER)
(6.13)
The standard error of the regression (SER) estimates the standard deviation of the error term ui. Thus the SER is a measure of the spread of the distribution of Y around the regression line. In multiple regression, the SER is
222 SER=s =3s wheres = un =
nu un un 1ani SSR n-k-1i=1 n-k-1
and where SSR is the sum of squared residuals, SSR = g ni = 1 un 2i .
The only difference between the definition in Equation (6.13) and the defini- tion of the SER in Section 4.3 for the single-regressor model is that here the divi- sorisn - k - 1ratherthann - 2.InSection4.3,thedivisorn - 2(ratherthann) adjusts for the downward bias introduced by estimating two coefficients (the slope and intercept of the regression line). Here, the divisor n - k - 1 adjusts for the downwardbiasintroducedbyestimatingk + 1coefficients(thekslopecoefficients plus the intercept). As in Section 4.3, using n - k - 1 rather than n is called a degrees-of-freedom adjustment. If there is a single regressor, then k = 1, so the formula in Section 4.3 is the same as in Equation (6.13). When n is large, the effect
of the degrees-of-freedom adjustment is negligible.
The R2
The regression R2 is the fraction of the sample variance of Yi explained by (or predicted by) the regressors. Equivalently, the R2 is 1 minus the fraction of the variance of Yi not explained by the regressors.
The mathematical definition of the R2 is the same as for regression with a single regressor:
R2 =ESS=1-SSR, TSS TSS
(6.14)
2
where the explained sum of squares is ESS = gn (Y - Y) and the total sum
of squares is TSS = g ni = 1(Yi - Y )2.
i=1 ni
6.4 Measures of Fit in Multiple Regression 197
In multiple regression, the R2 increases whenever a regressor is added, unless the estimated coefficient on the added regressor is exactly zero. To see this, think about starting with one regressor and then adding a second. When you use OLS to estimate the model with both regressors, OLS finds the values of the coefficients that minimize the sum of squared residuals. If OLS happens to choose the coefficient on the new regressor to be exactly zero, then the SSR will be the same whether or not the second variable is included in the regression. But if OLS chooses any value other than zero, then it must be that this value reduced the SSR relative to the regression that excludes this regressor. In prac- tice, it is extremely unusual for an estimated coefficient to be exactly zero, so in general the SSR will decrease when a new regressor is added. But this means that the R2 generally increases (and never decreases) when a new regressor is added.
The “Adjusted R2”
Because the R2 increases when a new variable is added, an increase in the R2 does not mean that adding a variable actually improves the fit of the model. In this sense, the R2 gives an inflated estimate of how well the regression fits the data. One way to correct for this is to deflate or reduce the R2 by some factor, and this is what the adjusted R2, or R 2, does.
The adjusted R2, or R2, is a modified version of the R2 that does not neces- sarily increase when a new regressor is added. The R 2 is
n-1 SSR s2 R2=1-n-k-1TSS=1- un. (6.15)
s2 Y
2
The difference between this formula and the second definition of the R2 in Equa- tion (6.14) is that the ratio of the sum of squared residuals to the total sum of squares is multiplied by the factor (n - 1) > (n – k – 1). As the second expres- sion in Equation (6.15) shows, this means that the adjusted R2 is 1 minus the ratio of the sample variance of the OLS residuals [with the degrees-of-freedom correc- tion in Equation (6.13)] to the sample variance of Y.
TherearethreeusefulthingstoknowabouttheR .First,(n – 1)>(n – k – 1) is always greater than 1, so R 2 is always less than R2.
Second, adding a regressor has two opposite effects on the R2. On the one
(n – 1) > (n – k – 1) increases. Whether the R on which of these two effects is stronger.
hand, the SSR falls, which increases the R2. On the other hand, the factor
2
increases or decreases depends
198 ChapteR 6 Linear Regression with Multiple Regressors
Third, the R2 can be negative. This happens when the regressors, taken together, reduce the sum of squared residuals by such a small amount that this reduction fails to offset the factor (n – 1) > (n – k – 1).
Application to Test Scores
Equation (6.12) reports the estimated regression line for the multiple regression relating test scores (TestScore) to the student–teacher ratio (STR) and the per- centage of English learners (PctEL). The R2 for this regression line is R2 = 0.426 , the adjusted R2 is R2 = 0.424, and the standard error of the regression is SER = 14.5.
Comparing these measures of fit with those for the regression in which PctEL is excluded [Equation (5.8)] shows that including PctEL in the regression increased the R2 from 0.051 to 0.426. When the only regressor is STR, only a small fraction of the variation in TestScore is explained; however, when PctEL is added to the regression, more than two-fifths (42.6%) of the variation in test scores is explained. In this sense, including the percentage of English learners substantially improves the fit of the regression. Because n is large and only two regressors appear in Equation (6.12), the difference between R2 and adjusted R2 is very small (R2 = 0.426 versus R 2 = 0.424).
The SER for the regression excluding PctEL is 18.6; this value falls to 14.5 when PctEL is included as a second regressor. The units of the SER are points on the standardized test. The reduction in the SER tells us that predictions about standardized test scores are substantially more precise if they are made using the regression with both STR and PctEL than if they are made using the regression with only STR as a regressor.
Using the R2 and adjusted R2. The R2 is useful because it quantifies the extent to which the regressors account for, or explain, the variation in the dependent variable. Nevertheless, heavy reliance on the R2 (or R2) can be a trap. In appli- cations, “maximize the R2” is rarely the answer to any economically or statisti- cally meaningful question. Instead, the decision about whether to include a variable in a multiple regression should be based on whether including that variable allows you better to estimate the causal effect of interest. We return to the issue of how to decide which variables to include—and which to exclude—in Chapter 7. First, however, we need to develop methods for quan- tifying the sampling uncertainty of the OLS estimator. The starting point for doing so is extending the least squares assumptions of Chapter 4 to the case of multiple regressors.
6.5 The Least Squares Assumptions in Multiple Regression 199 The Least Squares Assumptions
6.5
in Multiple Regression
There are four least squares assumptions in the multiple regression model. The first three are those of Section 4.3 for the single regressor model (Key Concept 4.3), extended to allow for multiple regressors, and these are discussed only briefly. The fourth assumption is new and is discussed in more detail.
Assumption #1: The Conditional Distribution of ui Given X1i, X2i, c, Xki Has a Mean of Zero
The first assumption is that the conditional distribution of ui given X1i, c, Xki has a mean of zero. This assumption extends the first least squares assumption with a single regressor to multiple regressors. This assumption means that some- times Yi is above the population regression line and sometimes Yi is below the population regression line, but on average over the population Yi falls on the population regression line. Therefore, for any value of the regressors, the expected value of ui is zero. As is the case for regression with a single regressor, this is the key assumption that makes the OLS estimators unbiased. We return to omitted variable bias in multiple regression in Section 7.5.
Assumption #2: (X1i, X2i, c, Xki,Yi), i = 1, c, n,
Are i.i.d.
The second assumption is that (X1i, c, Xki, Yi ), i = 1, c, n, are independently and identically distributed (i.i.d.) random variables. This assumption holds automati- cally if the data are collected by simple random sampling. The comments on this assumption appearing in Section 4.3 for a single regressor also apply to multiple regressors.
Assumption #3: Large Outliers Are Unlikely
The third least squares assumption is that large outliers—that is, observations with values far outside the usual range of the data—are unlikely. This assumption serves as a reminder that, as in single-regressor case, the OLS estimator of the coefficients in the multiple regression model can be sensitive to large outliers.
The assumption that large outliers are unlikely is made mathematically pre- cise by assuming that X1i, c, Xki, and Yi have nonzero finite fourth moments: 0 6 E ( X 41 i ) 6 ∞ , c , 0 6 E ( X 4k i ) 6 ∞ a n d 0 6 E ( Y 4i ) 6 ∞ . A n o t h e r w a y t o state this assumption is that the dependent variable and regressors have finite
200 ChapteR 6 Linear Regression with Multiple Regressors
kurtosis. This assumption is used to derive the properties of OLS regression sta-
tistics in large samples.
Assumption #4: No Perfect Multicollinearity
The fourth assumption is new to the multiple regression model. It rules out an inconvenient situation, called perfect multicollinearity, in which it is impos- sible to compute the OLS estimator. The regressors are said to exhibit perfect multicollinearity, (or to be perfectly multicollinear) if one of the regressors is a perfect linear function of the other regressors. The fourth least squares assumption is that the regressors are not perfectly multicollinear.
Why does perfect multicollinearity make it impossible to compute the OLS estimator? Suppose you want to estimate the coefficient on STR in a regression of TestScorei on STRi and PctELi, except that you make a typographical error and accidentally type in STRi a second time instead of PctELi; that is, you regress TestScorei on STRi and STRi. This is a case of perfect multicollinearity because one of the regressors (the first occurrence of STR) is a perfect linear function of another regressor (the second occurrence of STR). Depending on how your soft- ware package handles perfect multicollinearity, if you try to estimate this regres- sion the software will do one of two things: Either it will drop one of the occurrences of STR or it will refuse to calculate the OLS estimates and give an error message. The mathematical reason for this failure is that perfect multicollinearity produces division by zero in the OLS formulas.
At an intuitive level, perfect multicollinearity is a problem because you are asking the regression to answer an illogical question. In multiple regression, the coefficient on one of the regressors is the effect of a change in that regressor, hold- ing the other regressors constant. In the hypothetical regression of TestScore on STR and STR, the coefficient on the first occurrence of STR is the effect on test scores of a change in STR, holding constant STR. This makes no sense, and OLS cannot estimate this nonsensical partial effect.
The solution to perfect multicollinearity in this hypothetical regression is sim- ply to correct the typo and to replace one of the occurrences of STR with the variable you originally wanted to include. This example is typical: When perfect multicollinearity occurs, it often reflects a logical mistake in choosing the regres- sors or some previously unrecognized feature of the data set. In general, the solu- tion to perfect multicollinearity is to modify the regressors to eliminate the problem.
Additional examples of perfect multicollinearity are given in Section 6.7, which also defines and discusses imperfect multicollinearity.
6.6 The Distribution of the OLS Estimators in Multiple Regression 201
the Least Squares assumptions in the Multiple Regression Model
Yi =b0 +b1X1i +b2X2i +g+bkXki +ui,i=1,c,n, where
1. ui has conditional mean zero given X1i, X2i, c, Xki; that is, E(ui X1i, X2i, c, Xki) = 0
2. (X1i, X2i, c, Xki, Yi), i = 1, c, n, are independently and identically distrib- uted (i.i.d.) draws from their joint distribution.
3. Large outliers are unlikely: X1i, c, Xki and Yi have nonzero finite fourth moments.
4. There is no perfect multicollinearity.
Key ConCept
6.4
The least squares assumptions for the multiple regression model are summa- rized in Key Concept 6.4.
6.6
The Distribution of the OLS Estimators in Multiple Regression
Because the data differ from one sample to the next, different samples produce dif- ferent values of the OLS estimators. This variation across possible samples gives rise to the uncertainty associated with the OLS estimators of the population regression coefficients, b0, b1, c, bk. Just as in the case of regression with a single regressor, this variation is summarized in the sampling distribution of the OLS estimators.
Recall from Section 4.4 that, under the least squares assumptions, the OLS estimators (bn0 and bn1) are unbiased and consistent estimators of the unknown coefficients (b0 and b1) in the linear regression model with a single regressor. In addition, in large samples, the sampling distribution of bn0 and bn1 is well approxi- mated by a bivariate normal distribution.
These results carry over to multiple regression analysis. That is, under the least squares assumptions of Key Concept 6.4, the OLS estimators b , b , c, b
n0 n1 nk are unbiased and consistent estimators of b0, b1, c, bk in the linear multiple
202 ChapteR 6 Linear Regression with Multiple Regressors
Key ConCept
6.5
Large-Sample Distribution of b , b , c, b n0 n1 nk
If the least squares assumptions (Key Concept 6.4) hold, then in large samples the OLS estimators b , b , c, b are jointly normally distributed and each b is
distributed N(bj, s2n ), j = 0, c, k. bj
n0 n1 nk nj
regression model. In large samples, the joint sampling distribution of b , b , c, b n0 n1 nk
is well approximated by a multivariate normal distribution, which is the extension of the bivariate normal distribution to the general case of two or more jointly normal random variables (Section 2.4).
Although the algebra is more complicated when there are multiple regressors,
the central limit theorem applies to the OLS estimators in the multiple regression
model for the same reason that it applies to Y and to the OLS estimators when
there is a single regressor: The OLS estimators b , b , c, b are averages of the n0 n1 nk
randomly sampled data, and if the sample size is sufficiently large, the sampling distribution of those averages becomes normal. Because the multivariate normal distribution is best handled mathematically using matrix algebra, the expressions for the joint distribution of the OLS estimators are deferred to Chapter 18.
Key Concept 6.5 summarizes the result that, in large samples, the distribution of the OLS estimators in multiple regression is approximately jointly normal. In general, the OLS estimators are correlated; this correlation arises from the correlation between the regressors. The joint sampling distribution of the OLS estimators is discussed in more detail for the case that there are two regressors and homoskedastic errors in Appendix (6.2), and the general case is discussed in Section 18.2.
6.7
Multicollinearity
As discussed in Section 6.5, perfect multicollinearity arises when one of the regressors is a perfect linear combination of the other regressors. This section provides some examples of perfect multicollinearity and discusses how perfect multicollinearity can arise, and can be avoided, in regressions with multiple binary regressors. Imperfect multicollinearity arises when one of the regressors is very highly correlated—but not perfectly correlated—with the other regressors. Unlike perfect multicollinearity, imperfect multicollinearity does not prevent estimation of the regression, nor does it imply a logical problem with the choice of regressors. However, it does mean that one or more regression coefficients could be estimated imprecisely.
Examples of Perfect Multicollinearity
We continue the discussion of perfect multicollinearity from Section 6.5 by exam- ining three additional hypothetical regressions. In each, a third regressor is added to the regression of TestScorei on STRi and PctELi in Equation (6.12).
Example #1: Fraction of English learners. Let FracELi be the fraction of English learners in the ith district, which varies between 0 and 1. If the variable FracELi were included as a third regressor in addition to STRi and PctELi, the regressors would be perfectly multicollinear. The reason is that PctEL is the percentage of English learn- ers, so that PctELi = 100 * FracELi for every district. Thus one of the regressors (PctELi) can be written as a perfect linear function of another regressor (FracELi).
Because of this perfect multicollinearity, it is impossible to compute the OLS estimates of the regression of TestScorei on STRi, PctELi, and FracELi. At an intuitive level, OLS fails because you are asking, What is the effect of a unit change in the percentage of English learners, holding constant the fraction of English learners? Because the percentage of English learners and the fraction of English learners move together in a perfect linear relationship, this question makes no sense and OLS cannot answer it.
Example #2: “Not very small” classes. Let NVSi be a binary variable that equals 1 if the student–teacher ratio in the ith district is “not very small,” specifically, NVSi equals 1 if STRi Ú 12 and equals 0 otherwise. This regression also exhibits perfect multicollinearity, but for a more subtle reason than the regression in the previous example. There are in fact no districts in our data set with STRi 6 12; as you can see in the scatterplot in Figure 4.2, the smallest value of STR is 14. Thus NVSi = 1 for all observations. Now recall that the linear regression model with an intercept can equivalently be thought of as including a regressor, X0i, that equals 1 for all i, as shown in Equation (6.6). Thus we can write NVSi = 1 * X0i for all the obser- vations in our data set; that is, NVSi can be written as a perfect linear combination of the regressors; specifically, it equals X0i.
This illustrates two important points about perfect multicollinearity. First, when the regression includes an intercept, then one of the regressors that can be implicated in perfect multicollinearity is the constant regressor X0i. Second, perfect multicollinearity is a statement about the data set you have on hand. While it is possible to imagine a school district with fewer than 12 students per teacher, there are no such districts in our data set so we cannot analyze them in our regression.
Example #3: Percentage of English speakers. Let PctESi be the percentage of “English speakers” in the ith district, defined to be the percentage of students who are not English learners. Again the regressors will be perfectly multicollinear.
6.7 Multicollinearity 203
204 ChapteR 6 Linear Regression with Multiple Regressors
Like the previous example, the perfect linear relationship among the regressors involves the constant regressor X0i: For every district, PctESi = 100 – PctELi = 100 * X0i – PctELi, because X0i = 1 for all i.
This example illustrates another point: Perfect multicollinearity is a feature of the entire set of regressors. If either the intercept (that is, the regressor X0i) or PctELi were excluded from this regression, the regressors would not be perfectly multicollinear.
Thedummyvariabletrap. Anotherpossiblesourceofperfectmulticollinearityarises when multiple binary, or dummy, variables are used as regressors. For example, sup- pose you have partitioned the school districts into three categories: rural, suburban, and urban. Each district falls into one (and only one) category. Let these binary vari- ables be Rurali, which equals 1 for a rural district and equals 0 otherwise; Suburbani; and Urbani. If you include all three binary variables in the regression along with a constant, the regressors will be perfect multicollinearity: Because each district belongs to one and only one category, Rurali +Suburbani + Urbani = 1 = X0i, where X0i denotes the constant regressor introduced in Equation (6.6). Thus, to estimate the regression, you must exclude one of these four variables, either one of the binary indicators or the constant term. By convention, the constant term is retained, in which case one of the binary indicators is excluded. For example, if Rurali were excluded, then the coefficient on Suburbani would be the average difference between test scores in suburban and rural districts, holding constant the other variables in the regression.
In general, if there are G binary variables, if each observation falls into one and only one category, if there is an intercept in the regression, and if all G binary variables are included as regressors, then the regression will fail because of perfect multicollinearity. This situation is called the dummy variable trap. The usual way to avoid the dummy variable trap is to exclude one of the binary variables from the multiple regression, so only G – 1 of the G binary variables are included as regressors. In this case, the coefficients on the included binary variables represent the incremental effect of being in that category, relative to the base case of the omitted category, holding constant the other regressors. Alternatively, all G binary regressors can be included if the intercept is omitted from the regression.
Solutions to perfect multicollinearity. Perfect multicollinearity typically arises when a mistake has been made in specifying the regression. Sometimes the mis- take is easy to spot (as in the first example) but sometimes it is not (as in the second example). In one way or another, your software will let you know if you make such a mistake because it cannot compute the OLS estimator if you have.
When your software lets you know that you have perfect multicollinearity, it is important that you modify your regression to eliminate it. Some software is
unreliable when there is perfect multicollinearity, and at a minimum you will be ceding control over your choice of regressors to your computer if your regressors are perfectly multicollinear.
Imperfect Multicollinearity
Despite its similar name, imperfect multicollinearity is conceptually quite differ- ent from perfect multicollinearity. Imperfect multicollinearity means that two or more of the regressors are highly correlated in the sense that there is a linear function of the regressors that is highly correlated with another regressor. Imper- fect multicollinearity does not pose any problems for the theory of the OLS esti- mators; indeed, a purpose of OLS is to sort out the independent influences of the various regressors when these regressors are potentially correlated.
If the regressors are imperfectly multicollinear, then the coefficients on at least one individual regressor will be imprecisely estimated. For example, consider the regression of TestScore on STR and PctEL. Suppose we were to add a third regres- sor, the percentage of the district’s residents who are first-generation immigrants. First-generation immigrants often speak English as a second language, so the vari- ables PctEL and percentage immigrants will be highly correlated: Districts with many recent immigrants will tend to have many students who are still learning English. Because these two variables are highly correlated, it would be difficult to use these data to estimate the partial effect on test scores of an increase in PctEL, holding constant the percentage immigrants. In other words, the data set provides little information about what happens to test scores when the percentage of Eng- lish learners is low but the fraction of immigrants is high, or vice versa. If the least squares assumptions hold, then the OLS estimator of the coefficient on PctEL in this regression will be unbiased; however, it will have a larger variance than if the regressors PctEL and percentage immigrants were uncorrelated.
The effect of imperfect multicollinearity on the variance of the OLS estimators can be seen mathematically by inspecting Equation (6.17) in Appendix (6.2), which is the variance of bn1 in a multiple regression with two regressors (X1 and X2) for the special case of a homoskedastic error. In this case, the variance of bn1 is inversely proportional to 1 – r2X1,X2, where rX1, X2 is the correlation between X1 and X2. The larger the correlation between the two regressors, the closer this term is to zero and the larger is the variance of bn1. More generally, when multiple regressors are imperfectly multicollinear, the coefficients on one or more of these regressors will be imprecisely estimated—that is, they will have a large sampling variance.
Perfect multicollinearity is a problem that often signals the presence of a logical error. In contrast, imperfect multicollinearity is not necessarily an error,
6.7 Multicollinearity 205
206 ChapteR 6 Linear Regression with Multiple Regressors
but rather just a feature of OLS, your data, and the question you are trying to answer. If the variables in your regression are the ones you meant to include—the ones you chose to address the potential for omitted variable bias—then imperfect multicollinearity implies that it will be difficult to estimate precisely one or more of the partial effects using the data at hand.
6.8
Conclusion
Regression with a single regressor is vulnerable to omitted variable bias: If an omitted variable is a determinant of the dependent variable and is correlated with the regressor, then the OLS estimator of the slope coefficient will be biased and will reflect both the effect of the regressor and the effect of the omitted variable. Multiple regression makes it possible to mitigate omitted variable bias by includ- ing the omitted variable in the regression. The coefficient on a regressor, X1, in multiple regression is the partial effect of a change in X1, holding constant the other included regressors. In the test score example, including the percentage of English learners as a regressor made it possible to estimate the effect on test scores of a change in the student–teacher ratio, holding constant the percentage of English learners. Doing so reduced by half the estimated effect on test scores of a change in the student–teacher ratio.
The statistical theory of multiple regression builds on the statistical theory of regression with a single regressor. The least squares assumptions for multiple regres- sion are extensions of the three least squares assumptions for regression with a single regressor, plus a fourth assumption ruling out perfect multicollinearity. Because the regression coefficients are estimated using a single sample, the OLS estimators have a joint sampling distribution and therefore have sampling uncertainty. This sampling uncertainty must be quantified as part of an empirical study, and the ways to do so in the multiple regression model are the topic of the next chapter.
Summary
1. Omitted variable bias occurs when an omitted variable (1) is correlated with an included regressor and (2) is a determinant of Y.
2. The multiple regression model is a linear regression model that includes multiple regressors, X1, X2, c, Xk. Associated with each regressor is a regression coefficient, b1, b2, c, bk. The coefficient b1 is the expected change in Y associated with a one-unit change in X1, holding the other regressors constant. The other regression coefficients have an analogous interpretation.
3. The coefficients in multiple regression can be estimated by OLS. When the four least squares assumptions in Key Concept 6.4 are satisfied, the OLS esti- mators are unbiased, consistent, and normally distributed in large samples.
4. Perfect multicollinearity, which occurs when one regressor is an exact linear function of the other regressors, usually arises from a mistake in choosing which regressors to include in a multiple regression. Solving perfect multi- collinearity requires changing the set of regressors.
5. The standard error of the regression, the R2, and the R 2 are measures of fit for the multiple regression model.
Key Terms
omitted variable bias (183)
multiple regression model (189)
population regression line (189)
population regression function (189)
intercept (189)
slope coefficient of X1i (189)
coefficient on X1i (189)
slope coefficient of X2i (189)
coefficient on X2i (189)
holding X2 constant (190)
controlling for X (190) 2
partial effect (190)
population multiple regression model
(191)
constant regressor (191) constant term (191) homoskedastic (191) heteroskedastic (191) ordinary least squares (OLS)
22
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Review the Concepts
6.1 A researcher is interested in the effect on test scores of computer usage. Using school district data like that used in this chapter, she regresses district
Review the Concepts 207
estimators of b0, b1, c, bk (193) OLS regression line (193)
predicted value (193)
OLS residual (193)
R2 (196)
adjusted R 1R 2 (197)
perfect multicollinearity (200) dummy variable trap (204) imperfect multicollinearity (205)
MyEconLab Can Help You Get a Better Grade
208 ChapteR 6 Linear Regression with Multiple Regressors
average test scores on the number of computers per student. Will bn1 be an unbiased estimator of the effect on test scores of increasing the number of computers per student? Why or why not? If you think bn1 is biased, is it biased up or down? Why?
6.2 A multiple regression includes two regressors: Yi = b0 + b1X1i + b2X2i + ui. What is the expected change in Y if X1 increases by 3 units and X2 is unchanged? What is the expected change in Y if X2 decreases by 5 units and X1 is unchanged? What is the expected change in Y if X1 increases by 3 units and X2 decreases by 5 units?
6.3 How does R 2 differ from R2? Why is R 2 useful in a regression model with multiple regressors?
6.4 Explain why two perfectly multicollinear regressors cannot be included in a linear multiple regression. Give two examples of a pair of perfectly multicollinear regressors.
6.5 Explain why it is difficult to estimate precisely the partial effect of X1, hold- ing X2 constant, if X1 and X2 are highly correlated.
Exercises
The first four exercises refer to the table of estimated regressions on page 209, computed using data for 2012 from the CPS. The data set consists of information on 7440 full-time, full-year workers. The highest educational achievement for each worker was either a high school diploma or a bachelor’s degree. The workers’ ages ranged from 25 to 34 years. The data set also contains information on the region of the country where the person lived, marital status, and number of chil- dren. For the purposes of these exercises, let
AHE = average hourly earnings (in 2012 dollars)
College = binary variable (1 if college, 0 if high school)
Female = binary variable (1 if female, 0 if male)
Age = age (in years)
Ntheast = binary variable (1 if Region = Northeast, 0 otherwise) Midwest = binary variable (1 if Region = Midwest, 0 otherwise) South = binary variable (1 if Region = South, 0 otherwise)
West = binary variable (1 if Region = West, 0 otherwise)
6.1 Compute R 2 for each of the regressions.
6.2 Using the regression results in column (1):
a. Do workers with college degrees earn more, on average, than workers
with only high school degrees? How much more?
b. Do men earn more than women, on average? How much more?
6.3 Using the regression results in column (2):
a. Is age an important determinant of earnings? Explain.
b. Sally is a 29-year-old female college graduate. Betsy is a 34-year-old female college graduate. Predict Sally’s and Betsy’s earnings.
6.4 Using the regression results in column (3):
a. Do there appear to be important regional differences?
b. Why is the regressor West omitted from the regression? What would happen if it were included?
Results of Regressions of average hourly earnings on Gender and education Binary Variables and Other Characteristics, Using 2012 Data from the Current population Survey
College 1X 2 1
Exercises 209
Dependent variable: average hourly earnings (ahe).
(1) (2)
8.31 8.32
(3)
8.34 – 3.80 0.52 0.18 – 1.23 – 0.43 2.05
9.67 0.182
7440
regressor
Female 1X 2 3
-3.85 -3.81
Age1X22 0.51
Northeast 1X 2 4
Midwest 1X 2 South 1X 2 5
6
Summary Statistics
SER R2 R2
n
17.02 1.87
9.79 9.68 0.162 0.180
7440 7440
Intercept
210 ChapteR 6 Linear Regression with Multiple Regressors
c. Juanita is a 28-year-old female college graduate from the South. Jennifer is a 28-year-old female college graduate from the Midwest. Calculate the expected difference in earnings between Juanita and Jennifer.
6.5 Data were collected from a random sample of 220 home sales from a com- munity in 2013. Let Price denote the selling price (in $1000), BDR denote the number of bedrooms, Bath denote the number of bathrooms, Hsize denote the size of the house (in square feet), Lsize denote the lot size (in square feet), Age denote the age of the house (in years), and Poor denote a binary variable that is equal to 1 if the condition of the house is reported as “poor.” An estimated regression yields
Price = 119.2 + 0.485BDR + 23.4Bath + 0.156Hsize + 0.002Lsize + 0.090Age – 48.8Poor, R 2 = 0.72, SER = 41.5.
a. Suppose that a homeowner converts part of an existing family room in her house into a new bathroom. What is the expected increase in the value of the house?
b. Suppose that a homeowner adds a new bathroom to her house, which increases the size of the house by 100 square feet. What is the expected increase in the value of the house?
c. What is the loss in value if a homeowner lets his house run down so that its condition becomes “poor”?
d. Compute the R2 for the regression.
6.6 A researcher plans to study the causal effect of police on crime, using data from a random sample of U.S. counties. He plans to regress the county’s crime rate on the (per capita) size of the county’s police force.
a. Explain why this regression is likely to suffer from omitted variable bias. Which variables would you add to the regression to control for important omitted variables?
b. Use your answer to (a) and the expression for omitted variable bias
given in Equation (6.1) to determine whether the regression will
likely over- or underestimate the effect of police on the crime rate.
6.7 Critique each of the following proposed research plans. Your critique should explain any problems with the proposed research and describe how the research plan might be improved. Include a discussion of any additional
(That is, do you think that b1 7 b1 or b1 6 b1?)
nn
Exercises 211 data that need to be collected and the appropriate statistical techniques for
analyzing those data.
a. A researcher is interested in determining whether a large aerospace firm is guilty of gender bias in setting wages. To determine potential bias, the researcher collects salary and gender information for all of the firm’s engineers. The researcher then plans to conduct a “difference in means” test to determine whether the average salary for women is significantly less than the average salary for men.
b. A researcher is interested in determining whether time spent in prison has a permanent effect on a person’s wage rate. He collects data on
a random sample of people who have been out of prison for at least 15 years. He collects similar data on a random sample of people who have never served time in prison. The data set includes information on each person’s current wage, education, age, ethnicity, gender, tenure (time in current job), occupation, and union status, as well as whether the person has ever been incarcerated. The researcher plans to estimate the effect of incarceration on wages by regressing wages on an indicator variable for incarceration, including in the regression the other potential determinants of wages (education, tenure, union status, and so on).
6.8 A recent study found that the death rate for people who sleep 6 to 7 hours per night is lower than the death rate for people who sleep 8 or more hours. The 1.1 million observations used for this study came from a random sur- vey of Americans aged 30 to 102. Each survey respondent was tracked for 4 years. The death rate for people sleeping 7 hours was calculated as the ratio of the number of deaths over the span of the study among people sleeping 7 hours to the total number of survey respondents who slept 7 hours. This calculation was then repeated for people sleeping 6 hours and so on. Based on this summary, would you recommend that Americans who sleep 9 hours per night consider reducing their sleep to 6 or 7 hours if they want to prolong their lives? Why or why not? Explain.
6.9 (Yi, X1i, X2i) satisfy the assumptions in Key Concept 6.4. You are interested in b1, the causal effect of X1 on Y. Suppose that X1 and X2 are uncorrelated. You estimate b1 by regressing Y onto X1 (so that X2 is not included in the regression). Does this estimator suffer from omitted variable bias? Explain.
6.10 (Yi, X1i, X2i) satisfy the assumptions in Key Concept 6.4; in addition, var(ui X1i, X2i) = 4 and var(X1i) = 6. A random sample of size n = 400 is drawn from the population.
212 ChapteR 6 Linear Regression with Multiple Regressors
a. Assume that X1 and X2 are uncorrelated. Compute the variance of bn1.
[Hint: Look at Equation (6.17) in Appendix 6.2.]
b. Assume that corr(X1, X2) = 0.5. Compute the variance of bn1.
c. Comment on the following statements: “When X1 and X2 are corre- lated, the variance of bn1 is larger than it would be if X1 and X2 were uncorrelated. Thus, if you are interested in b1, it is best to leave X2 out of the regression if it is correlated with X1.”
6.11 (Requires calculus) Consider the regression model Yi = b1X1i + b2X2i + ui
for i = 1, c, n. (Notice that there is no constant term in the regression.) Following analysis like that used in Appendix (4.2):
a. Specify the least squares function that is minimized by OLS.
b. Compute the partial derivatives of the objective function with respect
to b1 and b2.
c. Suppose that g X X = 0. Show that b = g X Y > g X .
nnnn2
i=1 1i 2i 1
d. Suppose that g X X ≠ 0. Derive an expression for b as a func-
ni=1 1i 2i
tion of the data (Yi, X1i, X2i), i = 1, c, n.
n1
e. Suppose that the model includes an intercept: Yi = b0 + b1X1i +
b2X2i + ui. Show that the least squares estimators satisfy bn0 = nn
Y – b1X1 – b2X2.
f. As in (e), suppose that the model contains an intercept. Also
suppose that g (X – X )(X – X ) = 0. Show that b = ni=1 1i 1 2i 2 n1
g (X – X )(Y – Y)>g (X – X ) . How does this compare ni=1 1i 1 i ni=1 1i 12
to the OLS estimator of b1 from the regression that omits X2? Empirical Exercises
(Only two empirical exercises for this chapter are given in the text, but you can find more on the text website, http://www.pearsonhighered.com/stock_watson/.)
E6.1 Use the Birthweight_Smoking data set introduced in Empirical Exercise E5.3 to answer the following questions.
a. Regress Birthweight on Smoker. What is the estimated effect of smoking on birth weight?
i=1 1i i i=1 1i
b.
Empirical Exercises 213 Regress Birthweight on Smoker, Alcohol, and Nprevist.
i. Using the two conditions in Key Concept 6.1, explain why the exclusion of Alcohol and Nprevist could lead to omitted variable bias in the regression estimated in (a).
ii. Is the estimated effect of smoking on birth weight substantially different from the regression that excludes Alcohol and Nprevist? Does the regression in (a) seem to suffer from omitted variable bias?
iii. Jane smoked during her pregnancy, did not drink alcohol, and had 8 prenatal care visits. Use the regression to predict the birth weight of Jane’s child.
iv. Compute R2 and R 2. Why are they so similar?
Estimate the coefficient on Smoking for the multiple regression model in (b), using the three-step process in Appendix (6.3) (the Frisch-Waugh theorem). Verify that the three-step process yields the same estimated coefficient for Smoking as that obtained in (b).
An alternative way to control for prenatal visits is to use the binary variables Tripre0 through Tripre3. Regress Birthweight on Smoker, Alcohol, Tripre0, Tripre2, and Tripre3.
i. Why is Tripre1 excluded from the regression? What would happen if you included it in the regression?
ii. The estimated coefficient on Tripre0 is large and negative. What does this coefficient measure? Interpret its value.
iii. Interpret the value of the estimated coefficients on Tripre2 and Tripre3.
iv. Does the regression in (d) explain a larger fraction of the variance in birth weight than the regression in (b)?
c.
d.
the data set Growth described in Empirical Exercise E4.1, but
E6.2 Using
excluding the data for Malta, carry out the following exercises.
a. Construct a table that shows the sample mean, standard deviation, and minimum and maximum values for the series Growth, Trade- Share, YearsSchool, Oil, Rev_Coups, Assassinations, and RGDP60. Include the appropriate units for all entries.
b. Run a regression of Growth on TradeShare, YearsSchool, Rev_Coups, Assassinations, and RGDP60. What is the value of the coefficient on
214 ChapteR 6 Linear Regression with Multiple Regressors
Rev_Coups? Interpret the value of this coefficient. Is it large or small
in a real-world sense?
c. Use the regression to predict the average annual growth rate for a country that has average values for all regressors.
d. Repeat (c) but now assume that the country’s value for TradeShare is one standard deviation above the mean.
e. Why is Oil omitted from the regression? What would happen if it were included?
6.1
appenDix
Derivation of Equation (6.1)
This appendix presents a derivation of the formula for omitted variable bias in Equation (6.1). Equation (4.30) in Appendix (4.3) states
(X – X)u n1ani i
bn=b+ i=1
1 1 n1an i 2
.
(6.16)
np
i=1ii iiXuuX
tion (6.16) yields Equation (6.1).
(X – X) i=1
n2p2 Under the last two assumptions in Key Concept 4.3, (1 > n) g (X – X ) ¡ s and
i=1i X (1>n)g (X – X)u ¡ cov(u , X ) = r s s . Substitution of these limits into Equa-
appenDix
6.2
Distribution of the OLS Estimators When There Are Two Regressors and Homoskedastic Errors
Although the general formula for the variance of the OLS estimators in multiple regression is complicated, if there are two regressors (k = 2) and the errors are homoskedastic, then the formula simplifies enough to provide some insights into the distribution of the OLS estimators.
The Frisch–Waugh Theorem 215 Because the errors are homoskedastic, the conditional variance of ui can be written as
var(u 0X ,X ) = s . When there are two regressors, X and X , and the error term is i1i2i 2u 1i 2i
homoskedastic, in large samples the sampling distribution of bn is N(b , s2 ) where the
variance of this distribution, s2 , is bn 1
s2n =n1a 1 bs2u, (6.17) b 1 1 – r 2X 1 , X 2 s 2X 1
where rX1, X2 is the population correlation between the two regressors X1 and X2 and s2X1 is the population variance of X1.
The variance s2n of the sampling distribution of bn1 depends on the squared correla- b1
tion between the regressors. If X1 and X2 are highly correlated, either positively or negatively, then r2X1,X2 is close to 1, and thus the term 1 – r2X1,X2 in the denominator of Equation (6.17) is small and the variance of bn1 is larger than it would be if rX1, X2 were close to 0.
Another feature of the joint normal large-sample distribution of the OLS estimators is that bn1 and bn2 are in general correlated. When the errors are homoskedastic, the correla- tion between the OLS estimators bn1 and bn2 is the negative of the correlation between the two regressors:
1 1bn1
appenDix
6.3
corr(b1, b2) = -rX1, X2.
The Frisch–Waugh Theorem
nn
(6.18)
The OLS estimator in multiple regression can be computed by a sequence of shorter regressions. Consider the multiple regression model in Equation (6.7). The OLS estimator of b1 can be computed in three steps:
1. Regress X1 on X2, X3, c , Xk, and let X∼1 denote the residuals from this regression; 2. RegressYonX2,X3,c,Xk,andletY∼denotetheresidualsfromthisregression;and 3. Regress Y∼ on X∼1,
where the regressions include a constant term (intercept). The Frisch-Waugh theorem states that the OLS coefficient in step 3 equals the OLS coefficient on X1 in the multiple regression model [Equation (6.7)].
This result provides a mathematical statement of how the multiple regression coeffi- cient bn1 estimates the effect on Y of X1, controlling for the other X’s: Because the first two
216 ChapteR 6 Linear Regression with Multiple Regressors
regressions (steps 1 and 2) remove from Y and X1 their variation associated with the other X’s, the third regression estimates the effect on Y of X1 using what is left over after remov- ing (controlling for) the effect of the other X’s. The Frisch-Waugh theorem is proven in Exercise 18.17.
This theorem suggests how Equation (6.17) can be derived from Equation (5.27).
Because bn1 is the OLS regression coefficient from the regression of Y∼ onto X∼1, Equation (5.27)
suggests that the homoskedasticity-only variance of bn1 is s 2n = s 2u , where s∼2 is the ∼∼ b1ns∼2X1
X1
variance of X1. Because X1 is the residual from the regression of X1 onto X2 (recall that
Equation (6.17) pertains to the model with k = 2 regressors), Equation (6.15) implies that s2∼ = (1 – R2X ,X )s2X , where R2X ,X is the adjusted R2 from the regression of X1 onto X2.
X1 121 12
2p22p2 2p2
Equation (6.17) follows from s∼ ¡
rX ,X and sX ¡
sX .
s∼ , RX ,X ¡
X1 X112 12 1 1
CHAPTER
7
Hypothesis Tests and Confidence Intervals in Multiple Regression
As discussed in Chapter 6, multiple regression analysis provides a way to mitigate the problem of omitted variable bias by including additional regressors, thereby controlling for the effects of those additional regressors. The coefficients of the multiple regression model can be estimated by OLS. Like all estimators, the OLS estimator has sam- pling uncertainty because its value differs from one sample to the next.
This chapter presents methods for quantifying the sampling uncertainty of the OLS estimator through the use of standard errors, statistical hypothesis tests, and confidence intervals. One new possibility that arises in multiple regression is a hypothesis that simultaneously involves two or more regression coefficients. The general approach to testing such “joint” hypotheses involves a new test statistic, the F-statistic.
Section 7.1 extends the methods for statistical inference in regression with a single regressor to multiple regression. Sections 7.2 and 7.3 show how to test hypotheses that involve two or more regression coefficients. Section 7.4 extends the notion of confidence intervals for a single coefficient to confidence sets for mul- tiple coefficients. Deciding which variables to include in a regression is an impor- tant practical issue, so Section 7.5 discusses ways to approach this problem. In Section 7.6, we apply multiple regression analysis to obtain improved estimates of the effect on test scores of a reduction in the student–teacher ratio using the California test score data set.
7.1
Hypothesis Tests and Confidence Intervals for a Single Coefficient
This section describes how to compute the standard error, how to test hypotheses, and how to construct confidence intervals for a single coefficient in a multiple regression equation.
Standard Errors for the OLS Estimators
Recall that, in the case of a single regressor, it was possible to estimate the variance of the OLS estimator by substituting sample averages for expectations, which
217
218 CHAPTER 7
Hypothesis Tests and Confidence Intervals in Multiple Regression
2
led to the estimator sn N given in Equation (5.4). Under the least squares assump-
b1
tions, the law of large numbers implies that these sample averages converge to
22p
their population counterparts, so, for example, sn N > s N ¡ 1. The square root
b1 b1
of sn N is the standard error of b , SE(b ), an estimator of the standard deviation
2nn b1 11
of the sampling distribution of bn1.
All this extends directly to multiple regression. The OLS estimator bnj of the
jth regression coefficient has a standard deviation, and this standard deviation is estimated by its standard error, SE(bnj). The formula for the standard error is most easily stated using matrices (see Section 18.2). The important point is that, as far as standard errors are concerned, there is nothing conceptually different between the single- or multiple-regressor cases. The key ideas—the large-sample normal- ity of the estimators and the ability to estimate consistently the standard deviation of their sampling distribution—are the same whether one has one, two, or 12 regressors.
Hypothesis Tests for a Single Coefficient
Suppose that you want to test the hypothesis that a change in the student–teacher ratio has no effect on test scores, holding constant the percentage of English learn- ers in the district. This corresponds to hypothesizing that the true coefficient b1 on the student–teacher ratio is zero in the population regression of test scores on STR and PctEL. More generally, we might want to test the hypothesis that the true coef- ficient bj on the jth regressor takes on some specific value, bj,0. The null value bj,0 comes either from economic theory or, as in the student–teacher ratio example, from the decision-making context of the application. If the alternative hypothesis is two-sided, then the two hypotheses can be written mathematically as
H0 : bj = bj,0 vs. H1 : bj ≠ bj,0 (two-sided alternative). (7.1)
For example, if the first regressor is STR, then the null hypothesis that changing the student–teacher ratio has no effect on class size corresponds to the null hypothesis that b1 = 0 (so b1,0 = 0). Our task is to test the null hypothesis H0 against the alternative H1 using a sample of data.
Key Concept 5.2 gives a procedure for testing this null hypothesis when there is a single regressor. The first step in this procedure is to calculate the standard error of the coefficient. The second step is to calculate the t-statistic using the general formula in Key Concept 5.1. The third step is to compute the p-value of the test using the cumulative normal distribution in Appendix Table 1 or, alterna- tively, to compare the t-statistic to the critical value corresponding to the
7.1 Hypothesis Tests and Confidence Intervals for a Single Coefficient 219
Testing the Hypothesis bj = bj,0 Against the Alternative bj Z bj,0
1. Compute the standard error of bnj, SE(bnj).
2. Compute the t-statistic,
KEY CONCEPT
7.1
3. Compute the p-value,
t = bnj – bj,0. SE(bnj)
p-value = 2Φ(-tact),
(7.2)
(7.3)
where tact is the value of the t-statistic actually computed. Reject the hypothesis at the 5% significance level if the p-value is less than 0.05 or, equivalently, if tact 7 1.96. The standard error and (typically) the t-statistic and p-value testing bj = 0 are
computed automatically by regression software.
desired significance level of the test. The theoretical underpinnings of this proce- dure are that the OLS estimator has a large-sample normal distribution that, under the null hypothesis, has as its mean the hypothesized true value and that the variance of this distribution can be estimated consistently.
This underpinning is present in multiple regression as well. As stated in Key Concept 6.5, the sampling distribution of bnj is approximately normal. Under the null hypothesis the mean of this distribution is bj,0. The variance of this distribu- tion can be estimated consistently. Therefore we can simply follow the same pro- cedure as in the single-regressor case to test the null hypothesis in Equation (7.1).
The procedure for testing a hypothesis on a single coefficient in multiple regression is summarized as Key Concept 7.1. The t-statistic actually computed is denoted tact in this box. However, it is customary to denote this simply as t, and we adopt this simplified notation for the rest of the book.
Confidence Intervals for a Single Coefficient
The method for constructing a confidence interval in the multiple regression model is also the same as in the single-regressor model. This method is summa- rized as Key Concept 7.2.
220 CHAPTER 7 Hypothesis Tests and Confidence Intervals in Multiple Regression
KEY CONCEPT
7.2
Confidence Intervals for a Single Coefficient in Multiple Regression
A 95% two-sided confidence interval for the coefficient bj is an interval that con- tains the true value of bj with a 95% probability; that is, it contains the true value of bj in 95% of all possible randomly drawn samples. Equivalently, it is the set of values of bj that cannot be rejected by a 5% two-sided hypothesis test. When the sample size is large, the 95% confidence interval is
95% confidence interval for bj = 3bnj – 1.96SE(bnj), bnj + 1.96SE(bnj)4. (7.4)
A 90% confidence interval is obtained by replacing 1.96 in Equation (7.4) with 1.64.
The method for conducting a hypothesis test in Key Concept 7.1 and the method for constructing a confidence interval in Key Concept 7.2 rely on the large-sample normal approximation to the distribution of the OLS estimator bnj. Accordingly, it should be kept in mind that these methods for quantifying the sampling uncertainty are only guaranteed to work in large samples.
Application to Test Scores and
the Student–Teacher Ratio
Can we reject the null hypothesis that a change in the student–teacher ratio has no effect on test scores, once we control for the percentage of English learners in the district? What is a 95% confidence interval for the effect on test scores of a change in the student–teacher ratio, controlling for the percentage of English learners? We are now able to find out. The regression of test scores against STR and PctEL, estimated by OLS, was given in Equation (6.12) and is restated here with standard errors in parentheses below the coefficients:
TestScore = 686.0 – 1.10 * STR – 0.650 * PctEL. (7.5) (8.7) (0.43) (0.031)
To test the hypothesis that the true coefficient on STR is 0, we first need to com- pute the t-statistic in Equation (7.2). Because the null hypothesis says that the true value of this coefficient is zero, the t-statistic is t = (-1.10 – 0)>0.43 = -2.54.
7.1 Hypothesis Tests and Confidence Intervals for a Single Coefficient 221
Theassociatedp-valueis2Φ(-2.54) = 1.1%;thatis,thesmallestsignificancelevel at which we can reject the null hypothesis is 1.1%. Because the p-value is less than 5%, the null hypothesis can be rejected at the 5% significance level (but not quite at the 1% significance level).
A 95% confidence interval for the population coefficient on STR is – 1.10 { 1.96 * 0.43 = ( – 1.95, – 0.26); that is, we can be 95% confident that the true value of the coefficient is between -1.95 and -0.26. Interpreted in the con- text of the superintendent’s interest in decreasing the student–teacher ratio by 2, the 95% confidence interval for the effect on test scores of this reduction is
(-1.95 * 2, -0.26 * 2) = (-3.90, -0.52).
Adding expenditures per pupil to the equation. Your analysis of the multiple regression in Equation (7.5) has persuaded the superintendent that, based on the evidence so far, reducing class size will improve test scores in her district. Now, however, she moves on to a more nuanced question. If she is to hire more teach- ers, she can pay for those teachers either through cuts elsewhere in the budget (no new computers, reduced maintenance, and so on) or by asking for an increase in her budget, which taxpayers do not favor. What, she asks, is the effect on test scores of reducing the student–teacher ratio, holding expenditures per pupil (and the percentage of English learners) constant?
This question can be addressed by estimating a regression of test scores on the student–teacher ratio, total spending per pupil, and the percentage of English learners. The OLS regression line is
TestScore = 649.6 – 0.29 * STR + 3.87 * Expn – 0.656 * PctEL, (7.6) (15.5) (0.48) (1.59) (0.032)
where Expn is total annual expenditures per pupil in the district in thousands of dollars.
The result is striking. Holding expenditures per pupil and the percentage of English learners constant, changing the student–teacher ratio is estimated to have a very small effect on test scores: The estimated coefficient on STR is -1.10 in Equation (7.5) but, after adding Expn as a regressor in Equation (7.6), it is only
-0.29. Moreover, the t-statistic for testing that the true value of the coefficient is zeroisnowt = (-0.29 – 0)>0.48 = -0.60,sothehypothesisthatthepopulation value of this coefficient is indeed zero cannot be rejected even at the 10% signifi- cance level ( – 0.60 6 1.64). Thus Equation (7.6) provides no evidence that hir- ing more teachers improves test scores if overall expenditures per pupil are held constant.
222 CHAPTER 7
Hypothesis Tests and Confidence Intervals in Multiple Regression
7.2
Tests of Joint Hypotheses
This section describes how to formulate joint hypotheses on multiple regression coefficients and how to test them using an F-statistic.
Testing Hypotheses on Two or More Coefficients
Joint null hypotheses. Consider the regression in Equation (7.6) of the test score against the student–teacher ratio, expenditures per pupil, and the percentage of English learners. Our angry taxpayer hypothesizes that neither the student– teacher ratio nor expenditures per pupil have an effect on test scores, once we control for the percentage of English learners. Because STR is the first regressor in Equation (7.6) and Expn is the second, we can write this hypothesis mathemati- cally as
H0:b1 = 0andb2 = 0vs.H1:b1 ≠ 0and>orb2 ≠ 0. (7.7)
One interpretation of the regression in Equation (7.6) is that, in these Califor- nia data, school administrators allocate their budgets efficiently. Suppose, counter- factually, that the coefficient on STR in Equation (7.6) were negative and large. If so, school districts could raise their test scores simply by decreasing funding for other purposes (textbooks, technology, sports, and so on) and transferring those funds to hire more teachers, thereby reducing class sizes while holding expenditures constant. However, the small and statistically insignificant coefficient on STR in Equation (7.6) indicates that this transfer would have little effect on test scores. Put differently, districts are already allocating their funds efficiently.
Note that the standard error on STR increased when Expn was added, from 0.43 in Equation (7.5) to 0.48 in Equation (7.6). This illustrates the general point, introduced in Section 6.7 in the context of imperfect multicollinearity, that cor- relation between regressors (the correlation between STR and Expn is – 0.62) can make the OLS estimators less precise.
What about our angry taxpayer? He asserts that the population values of both the coefficient on the student–teacher ratio (b1) and the coefficient on spending per pupil (b2) are zero; that is, he hypothesizes that both b1 = 0 and b2 = 0. Although it might seem that we can reject this hypothesis because the t-statistic testing b2 = 0 in Equation (7.6) is t = 3.87>1.59 = 2.43, this reasoning is flawed. The taxpayer’s hypothesis is a joint hypothesis, and to test it we need a new tool, the F-statistic.
The hypothesis that both the coefficient on the student–teacher ratio (b1) and the coefficient on expenditures per pupil (b2) are zero is an example of a joint hypothesis on the coefficients in the multiple regression model. In this case, the null hypothesis restricts the value of two of the coefficients, so as a matter of ter- minology we can say that the null hypothesis in Equation (7.7) imposes two restrictions on the multiple regression model: b1 = 0 and b2 = 0.
In general, a joint hypothesis is a hypothesis that imposes two or more restric- tions on the regression coefficients. We consider joint null and alternative hypoth- eses of the form
H0 : bj = bj,0, bm = bm,0, c, for a total of q restrictions, vs.
H1 : one or more of the q restrictions under H0 does not hold, (7.8)
where bj, bm, c, refer to different regression coefficients and bj,0, bm,0, c, refer to the values of these coefficients under the null hypothesis. The null hypothesis in Equation (7.7) is an example of Equation (7.8). Another example is that, in a regression with k = 6 regressors, the null hypothesis is that the coefficients on the 2nd, 4th, and 5th regressors are zero; that is, b2 = 0, b4 = 0, and b5 = 0 so that there are q = 3 restrictions. In general, under the null hypothesis H0 there are q such restrictions.
If any one (or more than one) of the equalities under the null hypothesis H0 in Equation (7.8) is false, then the joint null hypothesis itself is false. Thus the alternative hypothesis is that at least one of the equalities in the null hypothesis H0 does not hold.
Why can’t I just test the individual coefficients one at a time? Although it seems it should be possible to test a joint hypothesis by using the usual t-statistics to test the restrictions one at a time, the following calculation shows that this approach is unreliable. Specifically, suppose that you are interested in testing the joint null hypothesis in Equation (7.6) that b1 = 0 and b2 = 0. Let t1 be the t-statistic for testing the null hypothesis that b1 = 0 and let t2 be the t-statistic for testing the null hypothesis that b2 = 0. What happens when you use the “one-at-a-time” testing procedure: Reject the joint null hypothesis if either t1 or t2 exceeds 1.96 in absolute value?
Because this question involves the two random variables t1 and t2, answering it requires characterizing the joint sampling distribution of t1 and t2. As mentioned in Section 6.6, in large samples bn1 and bn2 have a joint normal distribution, so under the joint null hypothesis the t-statistics t1 and t2 have a bivariate normal distribu- tion, where each t-statistic has mean equal to 0 and variance equal to 1.
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First consider the special case in which the t-statistics are uncorrelated and thus are independent. What is the size of the one-at-a-time testing procedure; that is, what is the probability that you will reject the null hypothesis when it is true? More than 5%! In this special case we can calculate the rejection probability of this method exactly. The null is not rejected only if both t1 … 1.96 and t2 … 1.96. Because the t-statistics are independent, Pr( t1 … 1.96 and t2 … 1.96) = Pr(t1 … 1.96) * Pr(t2 … 1.96) = 0.952 = 0.9025 = 90.25%. So the proba- bilityofrejectingthenullhypothesiswhenitistrueis1 – 0.952 = 9.75%.This“one at a time” method rejects the null too often because it gives you too many chances: If you fail to reject using the first t-statistic, you get to try again using the second.
If the regressors are correlated, the situation is even more complicated. The size of the “one at a time” procedure depends on the value of the correlation between the regressors. Because the “one at a time” testing approach has the wrong size—that is, its rejection rate under the null hypothesis does not equal the desired significance level—a new approach is needed.
One approach is to modify the “one at a time” method so that it uses different critical values that ensure that its size equals its significance level. This method, called the Bonferroni method, is described in Appendix (7.1). The advantage of the Bonferroni method is that it applies very generally. Its disadvantage is that it can have low power: It frequently fails to reject the null hypothesis when in fact the alternative hypothesis is true.
Fortunately, there is another approach to testing joint hypotheses that is more powerful, especially when the regressors are highly correlated. That approach is based on the F-statistic.
The F-Statistic
The F-statistic is used to test joint hypothesis about regression coefficients. The formulas for the F-statistic are integrated into modern regression software. We first discuss the case of two restrictions, then turn to the general case of q restrictions.
The F-statistic with q = 2 restrictions. When the joint null hypothesis has the two restrictions that b1 = 0 and b2 = 0, the F-statistic combines the two t-statistics t1 and t2 using the formula
F = 1at21 + t2 – 2rnt1,t2t1t2 b, (7.9) 2 1 – rn 2t 1 , t 2
where rnt1,t2 is an estimator of the correlation between the two t-statistics.
To understand the F-statistic in Equation (7.9), first suppose that we know that the t-statistics are uncorrelated so we can drop the terms involving rnt1, t2. If so, Equation (7.9) simplifies and F = 12(t21 + t2); that is, the F-statistic is the average of the squared t-statistics. Under the null hypothesis, t1 and t2 are independent stan- dard normal random variables (because the t-statistics are uncorrelated by assump- tion), so under the null hypothesis F has an F2, ∞ distribution (Section 2.4). Under the alternative hypothesis that either b1 is nonzero or b2 is nonzero (or both), then either t21 or t2 (or both) will be large, leading the test to reject the null hypothesis.
In general the t-statistics are correlated, and the formula for the F-statistic in Equation (7.9) adjusts for this correlation. This adjustment is made so that, under the null hypothesis, the F-statistic has an F2, ∞ distribution in large samples whether or not the t-statistics are correlated.
The F-statistic with q restrictions. The formula for the heteroskedasticity-robust F-statistic testing the q restrictions of the joint null hypothesis in Equation (7.8) is given in Section 18.3. This formula is incorporated into regression software, making the F-statistic easy to compute in practice.
Under the null hypothesis, the F-statistic has a sampling distribution that, in large samples, is given by the Fq, ∞ distribution. That is, in large samples, under the null hypothesis
the F @statistic is distributed Fq, ∞ . (7.10)
Thus the critical values for the F-statistic can be obtained from the tables of the Fq, ∞ distribution in Appendix Table 4 for the appropriate value of q and the desired significance level.
Computing the heteroskedasticity-robust F-statistic in statistical software. If the F-statistic is computed using the general heteroskedasticity-robust formula, its large-n distribution under the null hypothesis is Fq, ∞ regardless of whether the errors are homoskedastic or heteroskedastic. As discussed in Section 5.4, for his- torical reasons most statistical software computes homoskedasticity-only standard errors by default. Consequently, in some software packages you must select a “robust” option so that the F-statistic is computed using heteroskedasticity-robust standard errors (and, more generally, a heteroskedasticity-robust estimate of the “covariance matrix”). The homoskedasticity-only version of the F-statistic is dis- cussed at the end of this section.
Computing the p-value using the F-statistic. The p-value of the F-statistic can be computed using the large-sample Fq, ∞ approximation to its distribution. Let
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F act denote the value of the F-statistic actually computed. Because the F-statistic has a large-sample Fq, ∞ distribution under the null hypothesis, the p-value is
p@value = Pr3Fq, ∞ 7 Fact4. (7.11)
The p-value in Equation (7.11) can be evaluated using a table of the Fq,∞ distribu- tion (or, alternatively, a table of the x2q distribution, because a x2q-distributed ran- dom variable is q times an Fq,∞-distributed random variable). Alternatively, the p-value can be evaluated using a computer, because formulas for the cumulative chi-squared and F distributions have been incorporated into most modern statistical software.
The “overall” regression F-statistic. The “overall” regression F-statistic tests the joint hypothesis that all the slope coefficients are zero. That is, the null and alter- native hypotheses are
H0:b1 = 0,b2 = 0,c,bk = 0vs.H1:bj ≠ 0,atleastonej,j = 1,c,k. (7.12)
Under this null hypothesis, none of the regressors explains any of the variation in Yi, although the intercept (which under the null hypothesis is the mean of Yi) can be nonzero. The null hypothesis in Equation (7.12) is a special case of the general null hypothesis in Equation (7.8), and the overall regression F-statistic is the F-statistic computed for the null hypothesis in Equation (7.12). In large samples, the overall regressionF-statistichasanFk,∞ distributionwhenthenullhypothesisistrue.
The F-statistic when q = 1. When q = 1, the F-statistic tests a single restriction. Then the joint null hypothesis reduces to the null hypothesis on a single regression coefficient, and the F-statistic is the square of the t-statistic.
Application to Test Scores
and the Student–Teacher Ratio
We are now able to test the null hypothesis that the coefficients on both the student–teacher ratio and expenditures per pupil are zero, against the alternative that at least one coefficient is nonzero, controlling for the percentage of English learners in the district.
To test this hypothesis, we need to compute the heteroskedasticity-robust F-statistic of the test that b1 = 0 and b2 = 0 using the regression of TestScore on STR, Expn, and PctEL reported in Equation (7.6). This F-statistic is 5.43. Under
the null hypothesis, in large samples this statistic has an F2,∞ distribution. The 5% critical value of the F2,∞ distribution is 3.00 (Appendix Table 4), and the 1% crit- ical value is 4.61. The value of the F-statistic computed from the data, 5.43, exceeds 4.61, so the null hypothesis is rejected at the 1% level. It is very unlikely that we would have drawn a sample that produced an F-statistic as large as 5.43 if the null hypothesis really were true (the p-value is 0.005). Based on the evidence in Equa- tion (7.6) as summarized in this F-statistic, we can reject the taxpayer’s hypothesis that neither the student–teacher ratio nor expenditures per pupil have an effect on test scores (holding constant the percentage of English learners).
The Homoskedasticity-Only F-Statistic
One way to restate the question addressed by the F-statistic is to ask whether relaxing the q restrictions that constitute the null hypothesis improves the fit of the regression by enough that this improvement is unlikely to be the result merely of random sampling variation if the null hypothesis is true. This restatement sug- gests that there is a link between the F-statistic and the regression R2: A large F-statistic should, it seems, be associated with a substantial increase in the R2. In fact, if the error ui is homoskedastic, this intuition has an exact mathematical expression. Specifically, if the error term is homoskedastic, the F-statistic can be written in terms of the improvement in the fit of the regression as measured either by the decrease in the sum of squared residuals or by the increase in the regression R2. The resulting F-statistic is referred to as the homoskedasticity-only F-statistic, because it is valid only if the error term is homoskedastic. In contrast, the hetero- skedasticity-robust F-statistic computed using the formula in Section 18.3 is valid whether the error term is homoskedastic or heteroskedastic. Despite this signifi- cant limitation of the homoskedasticity-only F-statistic, its simple formula sheds light on what the F-statistic is doing. In addition, the simple formula can be com- puted using standard regression output, such as might be reported in a table that includes regression R2’s but not F-statistics.
The homoskedasticity-only F-statistic is computed using a simple formula based on the sum of squared residuals from two regressions. In the first regression, called the restricted regression, the null hypothesis is forced to be true. When the null hypothesis is of the type in Equation (7.8), where all the hypothesized values are zero, the restricted regression is the regression in which those coefficients are set to zero; that is, the relevant regressors are excluded from the regression. In the second regression, called the unrestricted regression, the alternative hypothesis is allowed to be true. If the sum of squared residuals is sufficiently smaller in the unre- stricted than the restricted regression, then the test rejects the null hypothesis.
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The homoskedasticity-only F-statistic is given by the formula (SSRrestricted – SSRunrestricted)>q
F = SSR >(n – k – 1), (7.13) unrestricted unrestricted
where SSRrestricted is the sum of squared residuals from the restricted regression, SSRunrestricted is the sum of squared residuals from the unrestricted regression, q is the number of restrictions under the null hypothesis, and kunrestricted is the number of regressors in the unrestricted regression. An alternative equivalent formula for the homoskedasticity-only F-statistic is based on the R2 of the two regressions:
F = (R2unrestricted – R2restricted)>q . (7.14) (1 – R2unrestricted)(n – kunrestricted – 1)
If the errors are homoskedastic, then the difference between the homoskedasticity- only F-statistic computed using Equation (7.13) or (7.14) and the heteroskedasticity- robust F-statistic vanishes as the sample size n increases. Thus, if the errors are homoskedastic, the sampling distribution of the homoskedasticity-only F-statistic under the null hypothesis is, in large samples, Fq,∞.
These formulas are easy to compute and have an intuitive interpretation in terms of how well the unrestricted and restricted regressions fit the data. Unfor- tunately, the formulas apply only if the errors are homoskedastic. Because homo- skedasticity is a special case that cannot be counted on in applications with economic data, or more generally with data sets typically found in the social sci- ences, in practice the homoskedasticity-only F-statistic is not a satisfactory substi- tute for the heteroskedasticity-robust F-statistic.
Using the homoskedasticity-only F-statistic when n is small. If the errors are homoskedastic and are i.i.d. normally distributed, then the homoskedasticity-only F-statistic defined in Equations (7.13) and (7.14) has an Fq,n – kunrestricted – 1 distribu- tion under the null hypothesis. Critical values for this distribution, which depend on both q and n – kunrestricted – 1, are given in Appendix Table 5. As discussed in Section 2.4, the Fq,n – kunrestricted – 1 distribution converges to the Fq,∞ distribution as n increases; for large sample sizes, the differences between the two distribu- tions are negligible. For small samples, however, the two sets of critical values differ.
Application to test scores and the student–teacher ratio. To test the null hypothesis that the population coefficients on STR and Expn are 0, controlling for PctEL, we need to compute the R2 (or SSR) for the restricted and unrestricted
7.3 Testing Single Restrictions Involving Multiple Coefficients 229
regression. The unrestricted regression has the regressors STR, Expn, and PctEL, and is given in Equation (7.6); its R2 is 0.4366; that is, R2unrestricted = 0.4366. The restricted regression imposes the joint null hypothesis that the true coefficients on STR and Expn are zero; that is, under the null hypothesis STR and Expn do not enter the population regression, although PctEL does (the null hypothesis does not restrict the coefficient on PctEL). The restricted regression, estimated by OLS, is
TestScore = 664.7 – 0.671 * PctEL, R2 = 0.4149, (7.15) (1.0) (0.032)
so R2restricted = 0.4149. The number of restrictions is q = 2, the number of observa- tions is n = 420, and the number of regressors in the unrestricted regression is k = 3. The homoskedasticity-only F-statistic, computed using Equation (7.14), is
F = (0.4366 – 0.4149)>2 = 8.01. (1 – 0.4366)(420 – 3 – 1)
Because 8.01 exceeds the 1% critical value of 4.61, the hypothesis is rejected at the 1% level using the homoskedasticity-only test.
This example illustrates the advantages and disadvantages of the homoskedasticity- only F-statistic. Its advantage is that it can be computed using a calculator. Its disad- vantage is that the values of the homoskedasticity-only and heteroskedasticity-robust F-statistics can be very different: The heteroskedasticity-robust F-statistic testing this joint hypothesis is 5.43, quite different from the less reliable homoskedasticity- only value of 8.01.
7.3
Testing Single Restrictions Involving Multiple Coefficients
Sometimes economic theory suggests a single restriction that involves two or more regression coefficients. For example, theory might suggest a null hypothesis of the form b1 = b2; that is, the effects of the first and second regressor are the same. In this case, the task is to test this null hypothesis against the alternative that the two coefficients differ:
H0:b1 = b2 vs.H1:b1 ≠ b2. (7.16)
This null hypothesis has a single restriction, so q = 1, but that restriction involves multiple coefficients (b1 and b2). We need to modify the methods presented so far
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to test this hypothesis. There are two approaches; which is easier depends on your software.
Approach #1: Test the restriction directly. Some statistical packages have a spe- cialized command designed to test restrictions like Equation (7.16) and the result is an F-statistic that, because q = 1, has an F1,∞ distribution under the null hypoth- esis. (Recall from Section 2.4 that the square of a standard normal random vari- able has an F1,∞ distribution, so the 95% percentile of the F1,∞ distribution is 1.962 = 3.84.)
Approach #2: Transform the regression. If your statistical package cannot test the restriction directly, the hypothesis in Equation (7.16) can be tested using a trick in which the original regression equation is rewritten to turn the restriction in Equation (7.16) into a restriction on a single regression coefficient. To be con- crete, suppose there are only two regressors, X1i and X2i, in the regression, so the population regression has the form
Yi = b0 + b1X1i + b2X2i + ui. (7.17)
Here is the trick: By subtracting and adding b2X1i, we have that b1X1i + b2X2i = b1X1i – b2X1i + b2X1i + b2X2i = (b1 – b2)X1i + b2(X1i + X2i) = g1X1i + b2Wi, where g1 = b1 – b2 and Wi = X1i + X2i. Thus the population regression in Equation (7.17) can be rewritten as
Yi =b0 +g1X1i +b2Wi +ui. (7.18)
Becausethecoefficientg1inthisequationisg1 =b1 -b2,underthenullhypoth- esis in Equation (7.16), g1 = 0, while under the alternative, g1 ≠ 0. Thus, by turning Equation (7.17) into Equation (7.18), we have turned a restriction on two regression coefficients into a restriction on a single regression coefficient.
Because the restriction now involves the single coefficient g1, the null hypoth- esis in Equation (7.16) can be tested using the t-statistic method of Section 7.1. In practice, this is done by first constructing the new regressor Wi as the sum of the two original regressors, then estimating the regression of Yi on X1i and Wi. A 95% confidence interval for the difference in the coefficients b1 – b2 can be calculated as gn1 { 1.96 SE(gn1).
This method can be extended to other restrictions on regression equations using the same trick (see Exercise 7.9).
The two methods (Approaches #1 and #2) are equivalent, in the sense that the F-statistic from the first method equals the square of the t-statistic from the second method.
7.4 Confidence Sets for Multiple Coefficients 231
Extension to q + 1. In general, it is possible to have q restrictions under the null hypothesis in which some or all of these restrictions involve multiple coefficients. The F-statistic of Section 7.2 extends to this type of joint hypothesis. The F-statistic can be computed by either of the two methods just discussed for q = 1. Precisely how best to do this in practice depends on the specific regression software being used.
7.4
Confidence Sets for Multiple Coefficients
This section explains how to construct a confidence set for two or more regression coefficients. The method is conceptually similar to the method in Section 7.1 for constructing a confidence set for a single coefficient using the t-statistic, except that the confidence set for multiple coefficients is based on the F-statistic.
A 95% confidence set for two or more coefficients is a set that contains the true population values of these coefficients in 95% of randomly drawn samples. Thus a confidence set is the generalization to two or more coefficients of a confi- dence interval for a single coefficient.
Recall that a 95% confidence interval is computed by finding the set of values of the coefficients that are not rejected using a t-statistic at the 5% sig- nificance level. This approach can be extended to the case of multiple coefficients. To make this concrete, suppose you are interested in constructing a confidence set for two coefficients, b1 and b2. Section 7.2 showed how to use the F-statistic to test a joint null hypothesis that b1 = b1,0 and b2 = b2,0. Suppose you were to test every possible value of b1,0 and b2,0 at the 5% level. For each pair of candidates (b1,0, b2,0), you compute the F-statistic and reject it if it exceeds the 5% critical value of 3.00. Because the test has a 5% significance level, the true population values of b1 and b2 will not be rejected in 95% of all samples. Thus the set of val- ues not rejected at the 5% level by this F-statistic constitutes a 95% confidence set for b1 and b2.
Although this method of trying all possible values of b1,0 and b2,0 works in theory, in practice it is much simpler to use an explicit formula for the confidence set. This formula for the confidence set for an arbitrary number of coefficients is based on the formula for the F-statistic. When there are two coefficients, the resulting confidence sets are ellipses.
As an illustration, Figure 7.1 shows a 95% confidence set (confidence ellipse) for the coefficients on the student–teacher ratio and expenditure per pupil, hold- ing constant the percentage of English learners, based on the estimated regression in Equation (7.6). This ellipse does not include the point (0,0). This means that the
232 CHAPTER 7 Hypothesis Tests and Confidence Intervals in Multiple Regression
FIGURE 7.1 95% Confidence Set for Coefficients on STR and Expn from Equation (7.6)
The 95% confidence set for
the coefficients on STR (b1) 9
Coefficient on Expn (β2) 8
and Expn (b2) is an ellipse.
The ellipse contains the pairs
of values of b1 and b2 that 6 cannot be rejected using the
F-statistic at the 5% 5 significance level. The point
(b1, b2) = (0, 0) is not
contained in the confidence
set, so the null hypothesis 1 H0: b1 = 0 and b2 = 0
is rejected at the 5% 0 significance level.
-2.0
7 95% confidence set
4 3 2
-1
1
1
(b1, b2) = (–0.29, 3.87) (b , b ) = (0, 0)
^^
12
-1.5
-1.0 -0.5
0.0 0.5
1.0 1.5
null hypothesis that these two coefficients are both zero is rejected using the F-statistic at the 5% significance level, which we already knew from Section 7.2. The confidence ellipse is a fat sausage with the long part of the sausage oriented in the lower-left/upper-right direction. The reason for this orientation is that the estimated correlation between bn1 and bn2 is positive, which in turn arises because the correlation between the regressors STR and Expn is negative (schools that spend more per pupil tend to have fewer students per teacher).
7.5
Model Specification for Multiple Regression
The job of determining which variables to include in multiple regression—that is, the problem of choosing a regression specification—can be quite challenging, and no single rule applies in all situations. But do not despair, because some useful guidelines are available. The starting point for choosing a regression specification is thinking through the possible sources of omitted variable bias. It is important to rely on your expert knowledge of the empirical problem and to focus on obtain- ing an unbiased estimate of the causal effect of interest; do not rely solely on purely statistical measures of fit such as the R2 or R2.
Coefficient on STR (β1)
7.5 Model Specification for Multiple Regression 233
Omitted Variable Bias in Multiple Regression
KEY CONCEPT
7.3
Omitted variable bias is the bias in the OLS estimator that arises when one or more included regressors are correlated with an omitted variable. For omitted variable bias to arise, two things must be true:
1. At least one of the included regressors must be correlated with the omitted variable.
2. The omitted variable must be a determinant of the dependent variable, Y.
Omitted Variable Bias in Multiple Regression
The OLS estimators of the coefficients in multiple regression will have omitted variable bias if an omitted determinant of Yi is correlated with at least one of the regressors. For example, students from affluent families often have more learning opportunities outside the classroom (reading material at home, travel, museum visits, etc.) than do their less affluent peers, which could lead to better test scores. Moreover, if the district is a wealthy one, then the schools will tend to have larger budgets and lower student–teacher ratios. If so, the availability of outside learning opportunities and the student–teacher ratio would be negatively correlated, and the OLS estimate of the coefficient on the student–teacher ratio would pick up the effect of outside learning opportunities, even after controlling for the percent- age of English learners. In short, omitting outside learning opportunities (and other variables related to the students’ economic background) could lead to omit- ted variable bias in the regression of test scores on the student–teacher ratio and the percentage of English learners.
The general conditions for omitted variable bias in multiple regression are similar to those for a single regressor: If an omitted variable is a determinant of Yi and if it is correlated with at least one of the regressors, then the OLS estimator of at least one of the coefficients will have omitted variable bias. The two condi- tions for omitted variable bias in multiple regression are summarized in Key Concept 7.3.
At a mathematical level, if the two conditions for omitted variable bias are satisfied, then at least one of the regressors is correlated with the error term. This means that the conditional expectation of ui given X1i, c, Xki is nonzero, so the first least squares assumption is violated. As a result, the omitted variable bias
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persists even if the sample size is large; that is, omitted variable bias implies that the OLS estimators are inconsistent.
The Role of Control Variables in Multiple Regression
So far, we have implicitly distinguished between a regressor for which we wish to estimate a causal effect—that is, a variable of interest—and control variables. We now discuss this distinction in more detail.
A control variable is not the object of interest in the study; rather it is a regres- sor included to hold constant factors that, if neglected, could lead the estimated causal effect of interest to suffer from omitted variable bias. The least squares assumptions for multiple regression (Section 6.5) treat the regressors symmetri- cally. In this subsection, we introduce an alternative to the first least squares assumption in which the distinction between a variable of interest and a control variable is explicit. If this alternative assumption holds, the OLS estimator of the effect of interest is unbiased, but the OLS coefficients on control variables are in general biased and do not have a causal interpretation.
For example, consider the potential omitted variable bias arising from omit- ting outside learning opportunities from a test score regression. Although “out- side learning opportunities” is a broad concept that is difficult to measure, those opportunities are correlated with the students’ economic background, which can be measured. Thus a measure of economic background can be included in a test score regression to control for omitted income-related determinants of test scores, like outside learning opportunities. To this end, we augment the regression of test scores on STR and PctEL with the percentage of students receiving a free or sub- sidized school lunch (LchPct). Because students are eligible for this program if their family income is less than a certain threshold (approximately 150% of the poverty line), LchPct measures the fraction of economically disadvantaged chil- dren in the district. The estimated regression is
TestScore = 700.2 – 1.00 * STR – 0.122 * PctEL – 0.547 * LchPct. (7.19) (5.6) (0.27) (0.033) (0.024)
Including the control variable LchPct does not substantially change any conclu- sions about the class size effect: The coefficient on STR changes only slightly from its value of -1.10 in Equation (7.5) to -1.00 in Equation (7.19), and it remains statistically significant at the 1% level.
What does one make of the coefficient on LchPct in Equation (7.19)? That coef- ficientisverylarge:ThedifferenceintestscoresbetweenadistrictwithLchPct = 0%
7.5 Model Specification for Multiple Regression 235
and one with LchPct = 50% is estimated to be 27.4 points 3 = 0.547 * (50 – 0)4, approximately the difference between the 75th and 25th percentiles of test scores in Table 4.1. Does this coefficient have a causal interpretation? Suppose that upon seeing Equation (7.19) the superintendent proposed eliminating the reduced- price lunch program so that, for her district, LchPct would immediately drop to zero. Would eliminating the lunch program boost her district’s test scores? Com- mon sense suggests that the answer is no; in fact, by leaving some students hungry, eliminating the reduced-price lunch program could have the opposite effect. But does it make sense to treat the coefficient on the variable of interest STR as causal, but not the coefficient on the control variable LchPct?
The distinction between variables of interest and control variables can be made mathematically precise by replacing the first least squares assumption of Key Concept 6.4—that is, the conditional mean-zero assumption—with an assumption called conditional mean independence. Consider a regression with two variables, in which X1i is the variable of interest and X2i is the control vari- able. Conditional mean independence requires that the conditional expectation of ui given X1i and X2i does not depend on (is independent of) X1i, although it can depend on X2i. That is
E(ui X1i, X2i) = E(ui X2i) (conditional mean independence). (7.20)
As is shown in Appendix (7.2), under the conditional mean independence assump- tion in Equation (7.20), the coefficient on X1i has a causal interpretation but the coefficient on X2i does not.
The idea of conditional mean independence is that once you control for X2i, X1i can be treated as if it were randomly assigned, in the sense that the con- ditional mean of the error term no longer depends on X1i. Including X2i as a control variable makes X1i uncorrelated with the error term so that OLS can estimate the causal effect on Y1i of a change in X1i. The control variable, however, remains correlated with the error term, so the coefficient on the control variable is subject to omitted variable bias and does not have a causal interpretation.
The terminology of control variables can be confusing. The control variable X2i is included because it controls for omitted factors that affect Yi and are cor- related with X1i and because it might (but need not) have a causal effect itself. Thus the coefficient on X1i is the effect on Yi of X1i, using the control variable X2i both to hold constant the direct effect of X2i and to control for factors correlated with X2i. Because this terminology is awkward, it is conventional simply to say that the coefficient on X1i is the effect on Yi, controlling for X2i. When a control variable is used, it is controlling both for its own direct causal effect (if any) and for the effect
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Hypothesis Tests and Confidence Intervals in Multiple Regression
of correlated omitted factors, with the aim of ensuring that conditional mean inde- pendence holds.
In the class size example, LchPct can be correlated with factors, such as learn- ing opportunities outside school, that enter the error term; indeed, it is because of this correlation that LchPct is a useful control variable. This correlation between LchPct and the error term means that the estimated coefficient on LchPct does not have a causal interpretation. What the conditional mean independence assumption requires is that, given the control variables in the regression (PctEL and LchPct), the mean of the error term does not depend on the student–teacher ratio. Said differently, conditional mean independence says that among schools with the same values of PctEL and LchPct, class size is “as if” randomly assigned: including PctEL and LchPct in the regression controls for omitted factors so that STR is uncorrelated with the error term. If so, the coefficient on the student– teacher ratio has a causal interpretation even though the coefficient on LchPct does not: For the superintendent struggling to increase test scores, there is no free lunch.
Model Specification in Theory and in Practice
In theory, when data are available on the omitted variable, the solution to omit- ted variable bias is to include the omitted variable in the regression. In practice, however, deciding whether to include a particular variable can be difficult and requires judgment.
Our approach to the challenge of potential omitted variable bias is twofold. First, a core or base set of regressors should be chosen using a combination of expert judgment, economic theory, and knowledge of how the data were collected; the regression using this base set of regressors is sometimes referred to as a base specification. This base specification should contain the variables of primary inter- est and the control variables suggested by expert judgment and economic theory. Expert judgment and economic theory are rarely decisive, however, and often the variables suggested by economic theory are not the ones on which you have data. Therefore the next step is to develop a list of candidate alternative specifications, that is, alternative sets of regressors. If the estimates of the coefficients of interest are numerically similar across the alternative specifications, then this provides evi- dence that the estimates from your base specification are reliable. If, on the other hand, the estimates of the coefficients of interest change substantially across speci- fications, this often provides evidence that the original specification had omitted variable bias. We elaborate on this approach to model specification in Section 9.2 after studying some tools for specifying regressions.
7.5 Model Specification for Multiple Regression 237 Interpreting the R2 and the Adjusted R2 in Practice
An R2 or an R2 near 1 means that the regressors are good at predicting the values of the dependent variable in the sample, and an R2 or an R2 near 0 means that they are not. This makes these statistics useful summaries of the predictive ability of the regression. However, it is easy to read more into them than they deserve.
There are four potential pitfalls to guard against when using the R2 or R2:
1. An increase in the R2 or R2 does not necessarily mean that an added variable is statistically significant. The R2 increases whenever you add a regressor, whether or not it is statistically significant. The R2 does not always increase, but if it does, this does not necessarily mean that the coefficient on that added regressor is statistically significant. To ascertain whether an added variable is statistically significant, you need to perform a hypothesis test using the t-statistic.
2. A high R2 or R2 does not mean that the regressors are a true cause of the dependent variable. Imagine regressing test scores against parking lot area per pupil. Parking lot area is correlated with the student–teacher ratio, with whether the school is in a suburb or a city, and possibly with district income—all things that are correlated with test scores. Thus the regression of test scores on parking lot area per pupil could have a high R2 and R2, but the relationship is not causal (try telling the superintendent that the way to increase test scores is to increase parking space!).
3. A high R2 or R2 does not mean that there is no omitted variable bias. Recall the discussion of Section 6.1, which concerned omitted variable bias in the regression of test scores on the student–teacher ratio. The R2 of the regres- sion never came up because it played no logical role in this discussion. Omit- ted variable bias can occur in regressions with a low R2, a moderate R2, or a high R2. Conversely, a low R2 does not imply that there necessarily is omit- ted variable bias.
4. A high R2 or R2 does not necessarily mean that you have the most appro- priate set of regressors, nor does a low R2 or R2 necessarily mean that you have an inappropriate set of regressors. The question of what constitutes the right set of regressors in multiple regression is difficult, and we return to it throughout this textbook. Decisions about the regressors must weigh issues of omitted variable bias, data availability, data quality, and, most importantly, economic theory and the nature of the substantive questions being addressed. None of these questions can be answered simply by having a high (or low) regression R2 or R2.
These points are summarized in Key Concept 7.4.
238 CHAPTER 7 Hypothesis Tests and Confidence Intervals in Multiple Regression
R2 and R 2: What They Tell You—and What They Don’t
7.4
KEY CONCEPT
The R 2 and R 2 tell you whether the regressors are good at predicting, or “explain- ing,” the values of the dependent variable in the sample of data on hand. If the R2 (or R2) is nearly 1, then the regressors produce good predictions of the dependent variable in that sample, in the sense that the variance of the OLS residual is small compared to the variance of the dependent variable. If the R2 (or R2) is nearly 0, the opposite is true.
The R2 and R2 do NOT tell you whether:
1. An included variable is statistically significant,
2. The regressors are a true cause of the movements in the dependent variable,
3. There is omitted variable bias, or
4. You have chosen the most appropriate set of regressors.
7.6
Analysis of the Test Score Data Set
This section presents an analysis of the effect on test scores of the student–teacher ratio using the California data set. Our primary purpose is to provide an example in which multiple regression analysis is used to mitigate omitted variable bias. Our secondary purpose is to demonstrate how to use a table to summarize regression results.
Discussion of the base and alternative specifications. This analysis focuses on estimating the effect on test scores of a change in the student–teacher ratio, hold- ing constant student characteristics that the superintendent cannot control. Many factors potentially affect the average test score in a district. Some of these factors are correlated with the student–teacher ratio, so omitting them from the regres- sion results in omitted variable bias. Because these factors, such as outside learn- ing opportunities, are not directly measured, we include control variables that are correlated with these omitted factors. If the control variables are adequate in the sense that the conditional mean independence assumption holds, then the coef- ficient on the student–teacher ratio is the effect of a change in the student–teacher ratio, holding constant these other factors.
Here we consider three variables that control for background characteristics of the students that could affect test scores: the fraction of students who are still
7.6 Analysis of the Test Score Data Set 239
learning English, the percentage of students who are eligible for receiving a sub- sidized or free lunch at school, and a new variable, the percentage of students in the district whose families qualify for a California income assistance program. Eligibility for this income assistance program depends in part on family income, with a lower (stricter) threshold than the subsidized lunch program. The final two variables thus are different measures of the fraction of economically disadvantaged children in the district (their correlation coefficient is 0.74). Theory and expert judg- ment do not tell us which of these two variables to use to control for determinants of test scores related to economic background. For our base specification, we use the percentage eligible for a subsidized lunch, but we also consider an alternative specification that uses the fraction eligible for the income assistance program.
Scatterplots of tests scores and these variables are presented in Figure 7.2. Each of these variables exhibits a negative correlation with test scores. The correla- tion between test scores and the percentage of English learners is – 0.64; between test scores and the percentage eligible for a subsidized lunch is -0.87; and between test scores and the percentage qualifying for income assistance is -0.63.
What scale should we use for the regressors? A practical question that arises in regression analysis is what scale you should use for the regressors. In Figure 7.2, the units of the variables are percent, so the maximum possible range of the data is 0 to 100. Alternatively, we could have defined these variables to be a decimal fraction rather than a percent; for example, PctEL could be replaced by the fraction of English learners, FracEL( = PctEL > 100), which would range between 0 and 1 instead of between 0 and 100. More generally, in regression analysis some decision usually needs to be made about the scale of both the dependent and independent variables. How, then, should you choose the scale, or units, of the variables?
The general answer to the question of choosing the scale of the variables is to make the regression results easy to read and to interpret. In the test score applica- tion, the natural unit for the dependent variable is the score of the test itself. In the regression of TestScore on STR and PctEL reported in Equation (7.5), the coefficient on PctEL is -0.650. If instead the regressor had been FracEL, the regression would have had an identical R2 and SER; however, the coefficient on FracEL would have been -65.0. In the specification with PctEL, the coefficient is the predicted change in test scores for a 1-percentage-point increase in English learners, holding STR constant; in the specification with FracEL, the coefficient is the predicted change in test scores for an increase by 1 in the fraction of English learners—that is, for a 100-percentage-point-increase—holding STR constant. Although these two specifications are mathematically equivalent, for the pur- poses of interpretation the one with PctEL seems, to us, more natural.
240
CHAPTER 7 Hypothesis Tests and Confidence Intervals in Multiple Regression
FIGURE 7.2
Test score
720 700 680 660 640 620
6000 25 50 75 100 Percent
(a) Percentage of English language learners Test score
720 700 680 660 640 620 600
(c)
Scatterplots of Test Scores vs. Three Student Characteristics
0 25 50 75 100
Percent
Percentage qualifying for income assistance
Test score
720 700 680 660 640 620
6000 25 50 75 100
Percent
(b) Percentage qualifying for reduced price lunch
The scatterplots show a negative relationship between test scores and (a) the percentage of English learners (correla- tion = – 0.64), (b) the percentage of students qualifying for a reduced price lunch (correlation = – 0.87); and (c) the percentage qualifying for income assistance (correlation = – 0.63).
Another consideration when deciding on a scale is to choose the units of the regressors so that the resulting regression coefficients are easy to read. For exam- ple, if a regressor is measured in dollars and has a coefficient of 0.00000356, it is easier to read if the regressor is converted to millions of dollars and the coefficient 3.56 is reported.
Tabular presentation of result. We are now faced with a communication prob- lem. What is the best way to show the results from several multiple regressions that contain different subsets of the possible regressors? So far, we have presented
7.6 Analysis of the Test Score Data Set 241 TABLE 7.1 Results of Regressions of Test Scores on the Student–Teacher Ratio and Student
Characteristic Control Variables Using California Elementary School Districts
Dependent variable: average test score in the district.
Regressor
Student–teacher ratio (X1)
Percent English learners (X2)
Percent eligible for subsidized lunch (X3) Percent on public income assistance (X4) Intercept
Summary Statistics
SER R2
n
(1)
-2.28** (0.52)
698.9** (10.4)
18.58
0.049 420
(2)
-1.10* (0.43)
-0.650** (0.031)
686.0** (8.7)
14.46
0.424 420
(3)
-1.00** (0.27)
-0.122** (0.033)
-0.547* (0.024)
700.2** (5.6)
9.08
0.773 420
(4)
-1.31* (0.34)
-0.488** (0.030)
-0.790** (0.068)
698.0** (6.9)
11.65
0.626 420
(5)
-1.01* (0.27)
-0.130** (0.036)
-0.529* (0.038)
0.048 (0.059)
700.4** (5.5)
9.08
0.773
420
These regressions were estimated using the data on K–8 school districts in California, described in Appendix (4.1). Heteroskedasticity- robust standard errors are given in parentheses under coefficients. The individual coefficient is statistically significant at the *5% level or **1% significance level using a two-sided test.
regression results by writing out the estimated regression equations, as in Equa- tions (7.6) and (7.19). This works well when there are only a few regressors and only a few equations, but with more regressors and equations this method of presentation can be confusing. A better way to communicate the results of several regressions is in a table.
Table 7.1 summarizes the results of regressions of the test score on various sets of regressors. Each column summarizes a separate regression. Each regression has the same dependent variable, test score. The entries in the first five rows are the estimated regression coefficients, with their standard errors below them in paren- theses. The asterisks indicate whether the t-statistics, testing the hypothesis that the relevant coefficient is zero, is significant at the 5% level (one asterisk) or the 1% level (two asterisks). The final three rows contain summary statistics for the regres- sion (the standard error of the regression, SER, and the adjusted R2, R2) and the sample size (which is the same for all of the regressions, 420 observations).
All the information that we have presented so far in equation format appears as a column of this table. For example, consider the regression of the test score
242 CHAPTER 7
Hypothesis Tests and Confidence Intervals in Multiple Regression
against the student–teacher ratio, with no control variables. In equation form, this regression is
TestScore = 698.9 – 2.28 * STR, R 2 = 0.049, SER = 18.58, n = 420. (7.21) (10.4) (0.52)
All this information appears in column (1) of Table 7.1. The estimated coeffi- cient on the student–teacher ratio (-2.28) appears in the first row of numerical entries, and its standard error (0.52) appears in parentheses just below the esti- mated coefficient. The intercept (698.9) and its standard error (10.4) are given in the row labeled “Intercept.” (Sometimes you will see this row labeled “con- stant” because, as discussed in Section 6.2, the intercept can be viewed as the coefficient on a regressor that is always equal to 1.) Similarly, the R2 (0.049), the SER (18.58), and the sample size n (420) appear in the final rows. The blank entries in the rows of the other regressors indicate that those regressors are not included in this regression.
Although the table does not report t-statistics, they can be computed from the information provided; for example, the t-statistic testing the hypothesis that the coef- ficientonthestudent–teacherratioincolumn(1)iszerois-2.28>0.52 = -4.38.This hypothesis is rejected at the 1% level, which is indicated by the double asterisk next to the estimated coefficient in the table.
Regressions that include the control variables measuring student characteris- tics are reported in columns (2) through (5). Column (2), which reports the regres- sion of test scores on the student–teacher ratio and on the percentage of English learners, was previously stated as Equation (7.5).
Column (3) presents the base specification, in which the regressors are the student–teacher ratio and two control variables, the percentage of English learners and the percentage of students eligible for a free lunch.
Columns (4) and (5) present alternative specifications that examine the effect of changes in the way the economic background of the students is measured. In column (4) the percentage of students on income assistance is included as a regres- sor, and in column (5) both of the economic background variables are included.
Discussion of empirical results. These results suggest three conclusions:
1. Controlling for these student characteristics cuts the effect of the student– teacher ratio on test scores approximately in half. This estimated effect is not very sensitive to which specific control variables are included in the regression. In all cases the coefficient on the student–teacher ratio remains statistically significant at the 5% level. In the four specifications with control variables, regressions (2) through (5), reducing the student–teacher ratio
7.7 Conclusion 243 by one student per teacher is estimated to increase average test scores by
approximately 1 point, holding constant student characteristics.
2. The student characteristic variables are potent predictors of test scores. The student–teacher ratio alone explains only a small fraction of the variation in test scores: The R 2 in column (1) is 0.049. The R 2 jumps, however, when the student characteristic variables are added. For example, the R2 in the base specification, regression (3), is 0.773. The signs of the coefficients on the student demographic variables are consistent with the patterns seen in Figure 7.2: Districts with many English learners and districts with many poor children have lower test scores.
3. The control variables are not always individually statistically significant: In specification (5), the hypothesis that the coefficient on the percent- age qualifying for income assistance is zero is not rejected at the 5% level (the t-statistic is -0.82). Because adding this control variable to the base specification (3) has a negligible effect on the estimated coefficient for the student–teacher ratio and its standard error, and because the coefficient on this control variable is not significant in specification (5), this additional control variable is redundant, at least for the purposes of this analysis.
7.7
Conclusion
Chapter 6 began with a concern: In the regression of test scores against the student–teacher ratio, omitted student characteristics that influence test scores might be correlated with the student–teacher ratio in the district, and, if so, the student–teacher ratio in the district would pick up the effect on test scores of these omitted student characteristics. Thus the OLS estimator would have omitted vari- able bias. To mitigate this potential omitted variable bias, we augmented the regression by including variables that control for various student characteristics (the percentage of English learners and two measures of student economic back- ground). Doing so cuts the estimated effect of a unit change in the student–teacher ratio in half, although it remains possible to reject the null hypothesis that the population effect on test scores, holding these control variables constant, is zero at the 5% significance level. Because they eliminate omitted variable bias arising from these student characteristics, these multiple regression estimates, hypothesis tests, and confidence intervals are much more useful for advising the superintendent than the single-regressor estimates of Chapters 4 and 5.
The analysis in this and the preceding chapter has presumed that the popula- tion regression function is linear in the regressors—that is, that the conditional
244 CHAPTER 7
Hypothesis Tests and Confidence Intervals in Multiple Regression
expectation of Yi given the regressors is a straight line. There is, however, no particular reason to think this is so. In fact, the effect of reducing the student– teacher ratio might be quite different in districts with large classes than in districts that already have small classes. If so, the population regression line is not linear in the X’s but rather is a nonlinear function of the X’s. To extend our analysis to regression functions that are nonlinear in the X’s, however, we need the tools developed in the next chapter.
Summary
1. Hypothesis tests and confidence intervals for a single regression coefficient are carried out using essentially the same procedures used in the one-vari- able linear regression model of Chapter 5. For example, a 95% confidence interval for b1 is given by bn1 { 1.96 SE(bn1).
2. Hypotheses involving more than one restriction on the coefficients are called joint hypotheses. Joint hypotheses can be tested using an F-statistic.
3. Regression specification proceeds by first determining a base specification cho- sen to address concern about omitted variable bias. The base specification can be modified by including additional regressors that address other potential sources of omitted variable bias. Simply choosing the specification with the highest R2 can lead to regression models that do not estimate the causal effect of interest.
Key Terms
restrictions (223)
joint hypothesis (223)
F-statistic (224)
restricted regression (227) unrestricted regression (227) homoskedasticity-only F-statistic (228)
95% confidence set (231)
control variable (234)
conditional mean independence (235) base specification (236)
alternative specifications (236) Bonferroni test (251)
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Review the Concepts
7.1 Explain how you would test the null hypothesis that b1 = 0 in the multiple regression model Yi = b0 + b1X1i + b2X2i + ui. Explain how you would test the null hypothesis that b2 = 0. Explain how you would test the joint hypothesis that b1 = 0 and b2 = 0. Why isn’t the result of the joint test implied by the results of the first two tests?
7.2 Provide an example of a regression that arguably would have a high value of R2 but would produce biased and inconsistent estimators of the regres- sion coefficient(s). Explain why the R2 is likely to be high. Explain why the OLS estimators would be biased and inconsistent.
7.3 What is a control variable, and how does it differ from a variable of inter- est? Looking at Table 7.1, which variables are control variables? What is the variable of interest? Do coefficients on control variables measure causal effects? Explain.
Exercises
The first six exercises refer to the table of estimated regressions on page 246, computed using data for 2012 from the CPS. The data set consists of information on 7440 full-time, full-year workers. The highest educational achievement for each worker was either a high school diploma or a bachelor’s degree. The workers’ ages ranged from 25 to 34 years. The data set also contains information on the region of the country where the person lived, marital status, and number of children. For the purposes of these exercises, let
AHE = average hourly earnings (in 2012 dollars)
College = binary variable (1 if college, 0 if high school)
Female = binary variable (1 if female, 0 if male)
Age = age (in years)
Ntheast = binary variable (1 if Region = Northeast, 0 otherwise) Midwest = binary variable (1 if Region = Midwest, 0 otherwise) South = binary variable (1 if Region = South, 0 otherwise)
West = binary variable (1 if Region = West, 0 otherwise)
7.1 Add * (5%) and ** (1%) to the table to indicate the statistical significance of the coefficients.
Exercises 245
246 CHAPTER 7
Hypothesis Tests and Confidence Intervals in Multiple Regression
7.2 Using the regression results in column (1):
a. Is the college–high school earnings difference estimated from this
regression statistically significant at the 5% level? Construct a 95% confidence interval of the difference.
b. Is the male–female earnings difference estimated from this regression statistically significant at the 5% level? Construct a 95% confidence interval for the difference.
7.3 Using the regression results in column (2):
a. Is age an important determinant of earnings? Use an appropriate sta-
tistical test and/or confidence interval to explain your answer.
b. Sally is a 29-year-old female college graduate. Betsy is a 34-year-old female college graduate. Construct a 95% confidence interval for the expected difference between their earnings.
7.4 Using the regression results in column (3):
a. Do there appear to be important regional differences? Use an appro-
priate hypothesis test to explain your answer.
Results of Regressions of Average Hourly Earnings on Gender and Education Binary Variables and Other Characteristics Using 2012 Data from the Current Population Survey
Dependent variable: average hourly earnings (AHE). Regressor
College (X1) Female (X2) Age (X3) Northeast (X4) Midwest (X5) South (X6) Intercept
Summary Statistics and Joint Tests
F-statistic for regional effects = 0 SER
R2
n
(1)
8.31 (0.23)
-3.85 (0.23)
(2) (3)
8.32 8.34 (0.22) (0.22)
-3.81 -3.80 (0.22) (0.22)
0.51 0.52 (0.04) (0.04)
0.18 (0.36)
-1.23 (0.31)
-0.43 (0.30)
1.87 2.05 (1.18) (1.18)
7.38 9.68 9.67
0.180 0.182 7440 7440
17.02 (0.17)
9.79
0.162 7440
b. Juanita is a 28-year-old female college graduate from the South. Molly is a 28-year-old female college graduate from the West. Jennifer is a 28-year-old female college graduate from the Midwest.
i. Construct a 95% confidence interval for the difference in expected earnings between Juanita and Molly.
ii. Explain how you would construct a 95% confidence interval for the difference in expected earnings between Juanita and Jennifer. (Hint: What would happen if you included West and excluded Midwest from the regression?)
7.5 The regression shown in column (2) was estimated again, this time using data from 1992 (4000 observations selected at random from the March 1993 CPS, converted into 2012 dollars using the consumer price index). The results are
AHE = 1.26 + 8.66College – 4.24Female + 0.65Age,SER = 9.57,R2 = 0.21. (1.60) (0.33) (0.29) (0.05)
Comparing this regression to the regression for 2012 shown in column (2), was there a statistically significant change in the coefficient on College?
7.6 Evaluate the following statement: “In all of the regressions, the coeffi- cient on Female is negative, large, and statistically significant. This pro- vides strong statistical evidence of gender discrimination in the U.S. labor market.”
7.7 Question 6.5 reported the following regression (where standard errors have been added):
Price = 119.2 + 0.485BDR + 23.4Bath + 0.156Hsize + 0.002Lsize (23.9) (2.61) (8.94) (0.011) (0.00048)
+ 0.090Age – 48.8Poor,R2 = 0.72,SER = 41.5
(0.311) (10.5)
a. Is the coefficient on BDR statistically significantly different from zero?
b. Typically five-bedroom houses sell for much more than two-bedroom houses. Is this consistent with your answer to (a) and with the regres- sion more generally?
c. A homeowner purchases 2000 square feet from an adjacent lot. Construct a 99% confident interval for the change in the value of her house.
Exercises 247
248 CHAPTER 7
Hypothesis Tests and Confidence Intervals in Multiple Regression
d. Lot size is measured in square feet. Do you think that another scale might be more appropriate? Why or why not?
e. The F-statistic for omitting BDR and Age from the regression is
F = 0.08. Are the coefficients on BDR and Age statistically different from zero at the 10% level?
7.8 Referring to Table 7.1 in the text:
a. Construct the R2 for each of the regressions.
b. Construct the homoskedasticity-only F-statistic for testing
b3 = b4 = 0 in the regression shown in column (5). Is the statistic significant at the 5% level?
c. Test b3 = b4 = 0 in the regression shown in column (5) using the Bonferroni test discussed in Appendix 7.1.
d. Construct a 99% confidence interval for b1 for the regression in column (5).
7.9 Consider the regression model Yi = b0 + b1X1i + b2X2i + ui. Use Ap- proach #2 from Section 7.3 to transform the regression so that you can use a t-statistic to test
a. b1 = b2.
b. b1 +2b2 =0.
c. b1 + b2 = 1. (Hint: You must redefine the dependent variable in the regression.)
7.10 Equations (7.13) and (7.14) show two formulas for the homoskedasticity- only F-statistic. Show that the two formulas are equivalent.
7.11 A school district undertakes an experiment to estimate the effect of class size on test scores in second-grade classes. The district assigns 50% of its previous year’s first graders to small second-grade classes (18 students per classroom) and 50% to regular-size classes (21 students per classroom). Students new to the district are handled differently: 20% are randomly assigned to small classes and 80% to regular-size classes. At the end of the second-grade school year, each student is given a standardized exam. Let Yi denote the exam score for the ith student, X1i denote a binary variable that equals 1 if the student is assigned to a small class, and X2i denote a binary variable that equals 1 if the student is newly enrolled. Let b1 denote the causal effect on test scores of reducing class size from regular to small.
a. Consider the regression Yi = b0 + b1X1i + ui. Do you think that E(ui X1i) = 0? Is the OLS estimator of b1 unbiased and consistent? Explain.
b. Consider the regression Yi = b0 + b1X1i + b2X2i + ui. Do you think that E(ui X1i, X2i) depends on X1? Is the OLS estimator of b1 unbiased and consistent? Explain. Do you think that E(ui X1i, X2i) depends on X2? Will the OLS estimator of b2 provide an unbiased and consistent estimate of the causal effect of transferring to a new school (that is, being a newly enrolled student)? Explain.
Empirical Exercises
(Only two empirical exercises for this chapter are given in the text, but you can find more on the text website, http://www.pearsonhighered.com/stock_watson/.)
E7.1 Use the Birthweight_Smoking data set introduced in Empirical Exercise E5.3 to answer the following questions. To begin, run three regressions:
(1) Birthweight on Smoker
(2) Birthweight on Smoker, Alcohol, and Nprevist
(3) Birthweight on Smoker, Alcohol, Nprevist, and Unmarried
a. What is the value of the estimated effect of smoking on birth weight in each of the regressions?
b. Construct a 95% confidence interval for the effect of smoking on birth weight, using each of the regressions.
c. Does the coefficient on Smoker in regression (1) suffer from omitted variable bias? Explain.
d. Does the coefficient on Smoker in regression (2) suffer from omitted variable bias? Explain.
e. Consider the coefficient on Unmarried in regression (3).
i. Construct a 95% confidence interval for the coefficient.
ii. Is the coefficient statistically significant? Explain.
iii. Is the magnitude of the coefficient large? Explain.
iv. A family advocacy group notes that the large coefficient suggests that public policies that encourage marriage will lead, on average, to healthier babies. Do you agree? (Hint: Review the discussion of control variables in Section 7.5. Discuss some of the various
Empirical Exercises 249
250 CHAPTER 7
Hypothesis Tests and Confidence Intervals in Multiple Regression
factors that Unmarried may be controlling for and how this affects the interpretation of its coefficient.)
f. Consider the various other control variables in the data set. Which do you think should be included in the regression? Using a table like Table 7.1, examine the robustness of the confidence interval you con- structed in (b). What is a reasonable 95% confidence interval for the effect of smoking on birth weight?
E7.2 In the empirical exercises on earning and height in Chapters 4 and 5, you esti- mated a relatively large and statistically significant effect of a worker’s height on his or her earnings. One explanation for this result is omitted variable bias: Height is correlated with an omitted factor that affects earnings. For example, Case and Paxson (2008) suggest that cognitive ability (or intelligence) is the omitted factor. The mechanism they describe is straightforward: Poor nutri- tion and other harmful environmental factors in utero and in early childhood have, on average, deleterious effects on both cognitive and physical develop- ment. Cognitive ability affects earnings later in life and thus is an omitted variable in the regression.
a. Suppose that the mechanism described above is correct. Explain how this leads to omitted variable bias in the OLS regression of Earnings on Height. Does the bias lead the estimated slope to be too large or too small? [Hint: Review Equation (6.1).]
If the mechanism described above is correct, the estimated effect of height on earnings should disappear if a variable measuring cognitive ability is included in the regression. Unfortunately, there isn’t a direct measure of cognitive ability in the data set, but the data set does include “years of education” for each indi- vidual. Because students with higher cognitive ability are more likely to attend school longer, years of education might serve as a control variable for cognitive ability; in this case, including education in the regression will eliminate, or at least attenuate, the omitted variable bias problem.
Use the years of education variable (educ) to construct four indicator variables for whether a worker has less than a high school diploma (LT_ HS = 1ifeduc 6 12,0otherwise),ahighschooldiploma(HS = 1ifeduc = 12, 0 otherwise), some college (Some_Col = 1 if 12 6 educ 6 16, 0 other- wise),orabachelor’sdegreeorhigher(College = 1ifeduc Ú 16,0otherwise).
b. Focusing first on women only, run a regression of (1) Earnings on Height and (2) Earnings on Height, including LT_HS, HS, and Some_ Col as control variables.
APPENDIX
i. ComparetheestimatedcoefficientonHeightinregressions(1) and (2). Is there a large change in the coefficient? Has it changed in a way consistent with the cognitive ability explanation? Explain.
ii. The regression omits the control variable College. Why?
iii. Test the joint null hypothesis that the coefficients on the education
variables are equal to zero.
iv. Discuss the values of the estimated coefficients on LT_HS, HS, and Some_Col. (Each of the estimated coefficients is negative, and the coefficient on LT_HS is more negative than the coefficient on HS, which in turn is more negative than the coefficient on Some_Col. Why? What do the coefficients measure?)
c. Repeat (b), using data for men.
7.1
The Bonferroni Test of a Joint Hypothesis 251
The Bonferroni Test of a Joint Hypothesis
The method of Section 7.2 is the preferred way to test joint hypotheses in multiple regres- sion. However, if the author of a study presents regression results but did not test a joint restriction in which you are interested and if you do not have the original data, then you will not be able to compute the F-statistic as in Section 7.2. This appendix describes a way to test joint hypotheses that can be used when you only have a table of regression results. This method is an application of a very general testing approach based on Bonferroni’s inequality.
The Bonferroni test is a test of a joint hypothesis based on the t-statistics for the individual hypotheses; that is, the Bonferroni test is the one-at-a-time t-statistic test of Section 7.2 done properly. The Bonferroni test of the joint null hypothesis b1 = b1,0 and b2 = b2,0 based on the critical value c 7 0, uses the following rule:
Accept if t1 … c and if t2 … c; otherwise, reject (7.22) (Bonferroni one-at-a-time t-statistic test)
where t1 and t2 are the t-statistics that test the restrictions on b1 and b2, respectfully.
The trick is to choose the critical value c in such a way that the probability that the one-at-a-time test rejects when the null hypothesis is true is no more than the desired significance level, say 5%. This is done by using Bonferroni’s inequality to choose the critical value c to allow both for the fact that two restrictions are being tested and for any
possible correlation between t1 and t2.
252 CHAPTER 7
Hypothesis Tests and Confidence Intervals in Multiple Regression
Bonferroni’s Inequality
Bonferroni’s inequality is a basic result of probability theory. Let A and B be events. Let A x B be the event “both A and B” (the intersection of A and B), and let A h B be the event“AorBorboth”(theunionofAandB).ThenPr(AhB) = Pr(A) + Pr(B)- Pr(A x B). Because Pr(A x B) Ú 0, it follows that Pr(A h B) … Pr(A) + Pr(B).1 Now let A be the event that t1 7 candB be the event that t2 7 c. Then the inequality Pr(A h B) … Pr(A) + Pr(B) yields
Pr(t1 7 cort2 7 corboth) … Pr(t1 7 c) + Pr(t2 7 c). (7.23) Bonferroni Tests
Because the event “ t1 7 c or t2 7 c or both” is the rejection region of the one-at-a- time test, Equation (7.23) leads to a valid critical value for the one-at-a-time test. Under the null hypothesis in large samples, Pr(t1 7 c) = Pr(t2 7 c) = Pr(Z 7 c). Thus Equation (7.23) implies that, in large samples, the probability that the one-at-a-time test rejects under the null is
PrH0 (one-at-a-time test rejects) … 2Pr( Z 7 c). (7.24)
The inequality in Equation (7.24) provides a way to choose a critical value c so that the prob- ability of the rejection under the null hypothesis equals the desired significance level. The Bonferroni approach can be extended to more than two coefficients; if there are q restrictions under the null, the factor of 2 on the right-hand side in Equation (7.24) is replaced by q.
Table 7.2 presents critical values c for the one-at-a-time Bonferroni test for various significance levels and q = 2, 3, and 4. For example, suppose the desired significance level is 5% and q = 2. According to Table 7.2, the critical value c is 2.241. This critical value is
TABLE 7.2 Bonferroni Critical Values c for the One-at-a-Time t-Statistic Test of a Joint Hypothesis
Number of Restrictions (q)
2
3
4
10%
1.960
2.128
2.241
Significance Level
5% 1%
2.241 2.807
2.394 2.935
2.498 3.023
1This inequality can be used to derive other interesting inequalities. For example, it implies that 1 – Pr(AhB) Ú 1 – 3Pr(A) + Pr(B)4.LetAc andBc bethecomplementsofAandB—thatis, the events “not A” and “not B.” Because the complement of A h B is Ac x Bc,1 – Pr(A h B) = Pr(Ac x Bc), which yields Bonferroni’s inequality, Pr(Ac x Bc) Ú 1 – 3Pr(A) + Pr(B)4.
APPENDIX
the 1.25% percentile of the standard normal distribution, so Pr(Z 7 2.241) = 2.5%. Thus Equation (7.24) tells us that, in large samples, the one-at-a-time test in Equa- tion (7.22) will reject at most 5% of the time under the null hypothesis.
The critical values in Table 7.2 are larger than the critical values for testing a single restriction. For example, with q = 2, the one-at-a-time test rejects if at least one t-statistic exceeds 2.241 in absolute value. This critical value is greater than 1.96 because it properly corrects for the fact that, by looking at two t-statistics, you get a second chance to reject the joint null hypothesis, as discussed in Section 7.2.
If the individual t-statistics are based on heteroskedasticity-robust standard errors, then the Bonferroni test is valid whether or not there is heteroskedasticity, but if the t- statistics are based on homoskedasticity-only standard errors, the Bonferroni test is valid only under homoskedasticity.
Application to Test Scores
The t-statistics testing the joint null hypothesis that the true coefficients on test scores and expenditures per pupil in Equation (7.6) are, respectively, t1 = -0.60 and t2 = 2.43. Although t1 6 2.241, because t2 7 2.241, we can reject the joint null hypothesis at the 5% significance level using the Bonferroni test. However, both t1 and t2 are less than 2.807 in absolute value, so we cannot reject the joint null hypothesis at the 1% significance level using the Bonferroni test. In contrast, using the F-statistic in Section 7.2, we were able to reject this hypothesis at the 1% significance level.
7.2
Conditional Mean Independence 253
Conditional Mean Independence
This appendix shows that, under the assumption of conditional mean independence intro- duced in Section 7.5 [Equation (7.20)], the OLS coefficient estimator is unbiased for the variable of interest but not for the control variable.
Consider a regression with two regressors, Yi = b0 + b1X1i + b2X2i + ui. If E(ui X1i, X2i) = 0, as would be true if X1i and X2i are randomly assigned in an experiment, then the OLS estimators bn1 and bn2 are unbiased estimators of the causal effects b1 and b2.
Now suppose that X1i is the variable of interest and X2i is a control variable that is correlated with omitted factors in the error term. Although the conditional mean zero assumption does not hold, suppose that conditional mean independence does so that E(uiX1i,X2i) = E(uiX2i).Forconvenience,furthersupposethatE(uiX2i)islinearinX2i so that E(ui X2i) = g0 + g2X2i, where g0 and g1 are constants. (This linearity assumption is discussed below.) Define vi to be the difference between ui and the conditional expectation of
254 CHAPTER 7
Hypothesis Tests and Confidence Intervals in Multiple Regression
ui given X1i and X2i—that is, vi = ui – E(ui X1i, X2i)—so that vi has conditional mean zero: E(viX1i, X2i) = E3ui – E(uiX1i, X2i)X1i, X2i4 = E(uiX1i, X2i) – E(ui X1i, X2i) = 0. Thus,
Yi =b0 +b1X1i +b2X2i +ui
= b0 + b1X1i + b2X2i + E(ui X1i, X2i) + vi (using the definition of vi)
= b0 + b1X1i + b2X2i + E(ui X2i) + vi (using conditional mean independence)
= b0 + b1X1i + b2X2i + (g0 + g2X2i) + vi = (b0 + g0) + b1X1i + (b2 + g2)X2i + vi = d0 + b1X1i + d2X2i + vi,
3using linearity of E(ui X2i)4 (collecting terms)
whered0 =b0 +g0andd2 =b2 +g2.
The error vi in Equation (7.25) has conditional mean zero; that is, E(vi X1i, X2i) = 0.
Therefore, the first least squares assumption for multiple regression applies to the final line of Equation (7.25), and if the other three least squares assumptions for multiple regression also hold, then the OLS regression of Yi on a constant, X1i, and X2i will yield unbiased and consistent estimators of d0, b1, and d2. Thus the OLS estimator of the coefficient on X1i is unbiased for the causal effect b1. However, the OLS estimator of the coefficient on X2i is not unbiased for b2 and instead estimates the sum of the causal effect b2 and the coefficient g2 arising from the correlation of the control variable X2i with the original error term ui.
The derivation in Equation (7.25) works for any value of b2, including zero. A variable X2i is a useful control variable if conditional mean independence holds; it need not have a direct causal effect on Yi.
The fourth line in Equation (7.25) uses the assumption that E(ui X2i) is linear in X2i. As discussed in Section 2.4, this will be true if ui and X2i are jointly normally distributed. The assumption of linearity can be relaxed using methods discussed in Chapter 8. Exercise 18.9 works through the steps in Equation (7.25) for multiple variables of interest and mul- tiple control variables.
In terms of the example in Section 7.5 [the regression in Equation (7.19)], if X2i is LchPct, then b2 is the causal effect of the subsidized lunch program (b2 is positive if the program’s nutritional benefits improve test scores), g2 is negative because LchPct is nega- tively correlated with (controls for) omitted learning advantages that improve test scores, and d2 = b2 + g2 would be negative if the omitted variable bias contribution through g2 outweights the positive causal effect b2.
To better understand the conditional mean independence assumption, return to the con- cept of an ideal randomized controlled experiment. As discussed in Section 4.4, if X1i is ran- domly assigned, then in a regression of Yi on X1i, the conditional mean zero assumption holds. If, however, X1i is randomly assigned, conditional on another variable X2i, then the conditional mean independence assumption holds, but if X2i is correlated with ui, the conditional mean zero
(7.25)
assumption does not. For example, consider an experiment to study the effect on grades in econometrics of mandatory versus optional homework. Among economics majors (X2i = 1), 75%areassignedtothetreatmentgroup(mandatoryhomework:X1i = 1),whileamongnon- economicsmajors(X2i = 0),only25%areassignedtothetreatmentgroup.Becausetreatment is randomly assigned within majors and within nonmajors, ui is independent of X1i, given X2i, soinparticular,E(uiX1i,X2i) = E(uiX2i).Ifchoiceofmajoriscorrelatedwithothercharac- teristics (like prior math) that determine performance in an econometrics course, then E(ui X2i) ≠ 0, and the regression of the final exam grade (Yi) on X1i alone will be subject to omitted variable bias (X1i is correlated with major and thus with other omitted determinants of grade). Including major (X2i) in the regression eliminates this omitted variable bias (treat- ment is randomly assigned, given major), making the OLS estimator of the coefficient on X1i an unbiased estimator of the causal effect on econometrics grades of requiring homework. However, the OLS estimator of the coefficient on major is not unbiased for the causal effect of switching into economics because major is not randomly assigned and is correlated with other omitted factors that would not change (like prior math) were a student to switch majors.
Conditional Mean Independence 255
256
C h8a p t e r
Nonlinear Regression Functions
In Chapters 4 through 7, the population regression function was assumed to be linear. In other words, the slope of the population regression function was constant, so the effect on Y of a unit change in X does not itself depend on the value of X. But what if the effect on Y of a change in X does depend on the value of one
or more of the independent variables? If so, the population regression function is nonlinear.
This chapter develops two groups of methods for detecting and modeling nonlinear population regression functions. The methods in the first group are useful when the effect on Y of a change in one independent variable, X1, depends on the value of X1 itself. For example, reducing class sizes by one student per teacher might have a greater effect if class sizes are already manageably small than if they are so large that the teacher can do little more than keep the class under control. If so, the test score (Y) is a nonlinear function of the student–teacher ratio (X1), where this function is steeper when X1 is small. An example of a nonlinear regression function with this feature is shown in Figure 8.1. Whereas the linear population regression function in Figure 8.1a has a constant slope, the nonlinear population regression function in Figure 8.1b has a steeper slope when X1 is small than when it is large. This first group of methods is presented in Section 8.2.
The methods in the second group are useful when the effect on Y of a change in X1 depends on the value of another independent variable, say X2. For example, students still learning English might especially benefit from having more one-on-one attention; if so, the effect on test scores of reducing the student–teacher ratio will be greater in districts with many students still learning English than in districts with few English learners. In this example, the effect on test scores (Y) of a reduction in the student–teacher ratio (X1) depends on the percentage of English learners in the district (X2). As shown in Figure 8.1c, the slope of this type of population regression function depends on the value of X2. This second group of methods is presented in Section 8.3.
In the models of Sections 8.2 and 8.3, the population regression function is a nonlinear function of the independent variables; that is, the conditional expectation E(Yi X1i, c, Xki) is a nonlinear function of one or more of the X ’s. Although they are nonlinear in the X’s, these models are linear functions of the unknown coefficients
Nonlinear Regression Functions 257 Figure 8.1 Population Regression Functions with Different Slopes
YY
Rise
Run
Rise
Run
Rise
Run
(a) Constant slope Y
(b)
Slope depends on the value of X1
X1
X1
Rise
Run
Rise
Run
Population regression function when X2 = 1
Population regression function when X2 = 0
X1 (c) Slope depends on the value of X2
In Figure 8.1a, the population regression function has a constant slope. In Figure 8.1b, the slope of the population regression function depends on the value of X1. In Figure 8.1c, the slope of the population regression function depends on the value of X2.
(or parameters) of the population regression model and thus are versions of the multiple regression model of Chapters 6 and 7. Therefore, the unknown parameters of these nonlinear regression functions can be estimated and tested using OLS and the methods of Chapters 6 and 7.
Sections 8.1 and 8.2 introduce nonlinear regression functions in the context of regression with a single independent variable, and Section 8.3 extends this to two independent variables. To keep things simple, additional control variables are omitted in the empirical examples of Sections 8.1 through 8.3. In practice, however, it is important to analyze nonlinear regression functions in models that control for omitted factors by including control variables as well. In Section 8.5, we combine nonlinear regression functions and additional control variables when we take a close
258 ChaPteR 8 Nonlinear Regression Functions
8.1
A General Strategy for Modeling Nonlinear Regression Functions
This section lays out a general strategy for modeling nonlinear population regres- sion functions. In this strategy, the nonlinear models are extensions of the multi- ple regression model and therefore can be estimated and tested using the tools of Chapters 6 and 7. First, however, we return to the California test score data and consider the relationship between test scores and district income.
Test Scores and District Income
In Chapter 7, we found that the economic background of the students is an impor- tant factor in explaining performance on standardized tests. That analysis used two economic background variables (the percentage of students qualifying for a subsidized lunch and the percentage of district families qualifying for income assistance) to measure the fraction of students in the district coming from poor families. A different, broader measure of economic background is the average annual per capita income in the school district (“district income”). The California data set includes district income measured in thousands of 1998 dollars. The sam- ple contains a wide range of income levels: For the 420 districts in our sample, the median district income is 13.7 (that is, $13,700 per person), and it ranges from 5.3 ($5300 per person) to 55.3 ($55,300 per person).
Figure 8.2 shows a scatterplot of fifth-grade test scores against district income for the California data set, along with the OLS regression line relating these two variables. Test scores and average income are strongly positively correlated, with a correlation coefficient of 0.71; students from affluent districts do better on the tests than students from poor districts. But this scatterplot has a peculiarity: Most of the points are below the OLS line when income is very low (under $10,000) or very high (over $40,000), but are above the line when income is between $15,000 and $30,000. There seems to be some curvature in the relationship between test scores and income that is not captured by the linear regression.
look at possible nonlinearities in the relationship between test scores and the student–teacher ratio, holding student characteristics constant. In some applications, the regression function is a nonlinear function of the X ’s and of the parameters. If so, the parameters cannot be estimated by OLS, but they can be estimated using nonlinear least squares. Appendix 8.1 provides examples of such functions and describes the nonlinear least squares estimator.
8.1 A General Strategy for Modeling Nonlinear Regression Functions 259 Figure 8.2 Scatterplot of test Score vs. District Income with a Linear OLS Regression Function
There is a positive correlation between test scores and district income (correlation = 0.71),
but the linear OLS regression line does not adequately describe the relationship between these variables.
680
660
640
620
600
In short, it seems that the relationship between district income and test scores is not a straight line. Rather, it is nonlinear. A nonlinear function is a function with a slope that is not constant: The function ƒ(X) is linear if the slope of ƒ(X) is the same for all values of X, but if the slope depends on the value of X, then ƒ(X) is nonlinear.
If a straight line is not an adequate description of the relationship between district income and test scores, what is? Imagine drawing a curve that fits the points in Figure 8.2. This curve would be steep for low values of district income and then would flatten out as district income gets higher. One way to approximate such a curve mathematically is to model the relationship as a quadratic function. That is, we could model test scores as a function of income and the square of income.
A quadrEatlieccptropnuiclaPtiuobnlisrehginrgesSsieornvimceosdeInlcr.elating test scores and income is written mathSemtoactki/cWalalytsaosn, Econometrics 1e
STOC.ITEM.0022
TestScore = b + b Income + b Income2 + u , (8.1) Fig.06.02i 0 1 i 2 i i
Test score
740 720
700
0 10 20 30 40 50 60
District income (thousands of dollars)
1st Proof 2nd Proof 3rd Proof Final
where b0, b1, and b2 are coefficients, Incomei is the income in the ith district, Income2i is the square of income in the ith district, and ui is an error term that, as usual, represents all the other factors that determine test scores. Equation (8.1) is called the quadratic regression model because the population regression function,
260 ChaPteR 8 Nonlinear Regression Functions
E(TestScoreiIncomei) = b0 + b1Incomei + b2Income2i,isaquadraticfunctionof the independent variable, Income.
If you knew the population coefficients b0, b1, and b2 in Equation (8.1), you could predict the test score of a district based on its average income. But these population coefficients are unknown and therefore must be estimated using a sample of data.
At first, it might seem difficult to find the coefficients of the quadratic func- tion that best fits the data in Figure 8.2. If you compare Equation (8.1) with the multiple regression model in Key Concept 6.2, however, you will see that Equa- tion (8.1) is in fact a version of the multiple regression model with two regressors: The first regressor is Income, and the second regressor is Income2. Mechanically, you can create this second regressor by generating a new variable that equals the square of Income, for example as an additional column in a spreadsheet. Thus, after defining the regressors as Income and Income2, the nonlinear model in Equation (8.1) is simply a multiple regression model with two regressors!
Because the quadratic regression model is a variant of multiple regression, its unknown population coefficients can be estimated and tested using the OLS methods described in Chapters 6 and 7. Estimating the coefficients of Equation (8.1) using OLS for the 420 observations in Figure 8.2 yields
TestScore = 607.3 + 3.85Income – 0.0423Income2,R2 = 0.554, (8.2) (2.9) (0.27) (0.0048)
where (as usual) standard errors of the estimated coefficients are given in parenthe- ses. The estimated regression function of Equation (8.2) is plotted in Figure 8.3, superimposed over the scatterplot of the data. The quadratic function captures the curvature in the scatterplot: It is steep for low values of district income but flat- tens out when district income is high. In short, the quadratic regression function seems to fit the data better than the linear one.
We can go one step beyond this visual comparison and formally test the hypothesis that the relationship between income and test scores is linear against the alternative that it is nonlinear. If the relationship is linear, then the regression function is correctly specified as Equation (8.1), except that the regressor Income2 is absent; that is, if the relationship is linear, then Equation (8.1) holds with b2 = 0. Thus we can test the null hypothesis that the population regression function is linear against the alternative that it is quadratic by testing the null hypothesis that b2 = 0 against the alternative that b2 ≠ 0.
Because Equation (8.1) is just a variant of the multiple regression model, the null hypothesis that b2 = 0 can be tested by constructing the t-statistic for this
8.1 A General Strategy for Modeling Nonlinear Regression Functions 261 Figure 8.3 Scatterplot of test Score vs. District Income with Linear and Quadratic Regression Functions
The quadratic OLS regression function fits the data better than the linear OLS regression function.
700
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660
640
620
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The Effect on Y of a Change in X
in Nonlinear Specifications
Test score
740 720
Linear regression
Quadratic regression
0 10 20 30 40 50 60
District income (thousands of dollars)
hypothesis. This t-statistic is t = (b – 0)>SE(b ), which from Equation (8.2) is n2 n2
t = -0.0423>0.0048 = -8.81. In absolute value, this exceeds the 5% critical value of this test (which is 1.96). Indeed the p-value for the t-statistic is less than 0.01%, so we can reject the hypothesis that b2 = 0 at all conventional significance levels. Thus this formal hypothesis test supports our informal inspection of Fig- ures 8.2 and 8.3: The quadratic model fits the data better than the linear model.
Put aside the test score example for a moment and consider a general problem.
Electronic Publishing Services Inc.
You want to know how the dependent variable Y is expected to change when the Stock/Watson, Econometrics 1e
independent variable X changes by the amount ∆X , holding constant other 11
STOC.ITEM.0023
independent variables X , c, X . When the population regression function is 2k
Fig. 06.03
linear, this effect is easy to calculate: As shown in Equation (6.4), the expected
1st Proof 2nd Proof 3rd Proof Final
change in Y is ∆Y = b1∆X1, where b1 is the population regression coefficient multiplying X1. When the regression function is nonlinear, however, the expected change in Y is more complicated to calculate because it can depend on the values of the independent variables.
262 Chapter 8 Nonlinear Regression Functions Ageneralformulaforanonlinearpopulationregressionfunction.1 Thenonlinear
population regression models considered in this chapter are of the form
Yi = f(X1i,X2i, c,Xki) + ui,i = 1, c,n, (8.3)
where f(X1i, X2i, c, Xki) is the population nonlinear regression function, a pos- sibly nonlinear function of the independent variables X1i, X2i, c, Xki, and ui is the error term. For example, in the quadratic regression model in Equation (8.1), only one independent variable is present, so X1 is Income and the population regression function is f(Incomei) = b0 + b1Incomei + b2Income2i .
Because the population regression function is the conditional expectation of Yi given X1i, X2i, c, Xki, in Equation (8.3) we allow for the possibility that this conditional expectation is a nonlinear function of X1i, X2i, c, Xki; that is, E(Yi X1i, X2i, c, Xki) = f(X1i, X2i, c, Xki), where ƒ can be a nonlinear function. If the population regression function is linear, then f(X1i, X2i, c, Xki) = b0 + b1X1i + b2X2i + g+ bkXki, and Equation (8.3) becomes the linear regression model in Key Concept 6.2. However, Equation (8.3) allows for nonlinear regression functions as well.
The effect on Y of a change in X1. As discussed in Section 6.2, the effect on Y of a change in X1, ∆X1, holding X2, c, Xk constant, is the difference in the expected valueofYwhentheindependentvariablestakeonthevaluesX1 + ∆X1,X2,c,Xk and the expected value of Y when the independent variables take on the values X1, X2, c, Xk. The difference between these two expected values, say ∆Y, is what happens to Y on average in the population when X1 changes by an amount ∆X1, holding constant the other variables X2, c, Xk. In the nonlinear regression model of Equation (8.3), this effect on Y is ∆Y = f(X1 + ∆X1, X2, c, Xk) – f(X1, X2, c, Xk).
Because the regression function f is unknown, the population effect on Y of a change in X1 is also unknown. To estimate the population effect, first estimate the population regression function. At a general level, denote this estimated function
1The term nonlinear regression applies to two conceptually different families of models. In the first family, the population regression function is a nonlinear function of the X’s but is a linear function of the unknown parameters (the b’s). In the second family, the population regression function is a nonlinear function of the unknown parameters and may or may not be a nonlinear function of the X’s. The models in the body of this chapter are all in the first family. Appendix 8.1 takes up models from the second family.
8.1 A General Strategy for Modeling Nonlinear Regression Functions 263
the expected Change on Y of a Change in X1 in the Nonlinear Regression Model (8.3)
Key ConCept
8.1
The expected change in Y, ∆Y, associated with the change in X1, ∆X1, holding X2, c, Xk constant, is the difference between the value of the population regres- sion function before and after changing X1, holding X2, c, Xk constant. That is, the expected change in Y is the difference:
∆Y = f(X1 + ∆X1, X2, c, Xk) – f(X1, X2, c, Xk). (8.4) The estimator of this unknown population difference is the difference between
the predicted values for these two cases. Let f (X , X , c, X ) be the predicted 12k
value of Y based on the estimator fn of the population regression function. Then the predicted change in Y is
∆Y=f(X +∆X,X,c,X)-f(X,X,c,X). (8.5) nn112kn12k
n
by fn ; an example of such an estimated function is the estimated quadratic regres- sion function in Equation (8.2). The estimated effect on Y (denoted ∆Yn) of the change in X1 is the difference between the predicted value of Y when the inde- pendent variables take on the values X1 + ∆X1, X2, c, Xk and the predicted value of Y when they take on the values X1, X2, c, Xk.
The method for calculating the expected effect on Y of a change in X1 is sum- marized in Key Concept 8.1. The method in Key Concept 8.1 always works, whether ∆X1 is large or small and whether the regressors are continuous or dis- crete. Appendix 8.2 shows how to evaluate the slope using calculus for the special case of a single continuous regressor when ∆X1 small.
Application to test scores and income. What is the predicted change in test scores associated with a change in district income of $1000, based on the estimated qua- dratic regression function in Equation (8.2)? Because that regression function is quadratic, this effect depends on the initial district income. We therefore consider two cases: an increase in district income from 10 to 11 (i.e., from $10,000 per capita to $11,000) and an increase in district income from 40 to 41.
264 ChaPteR 8 Nonlinear Regression Functions
To compute ∆Yn associated with the change in income from 10 to 11, we can apply the general formula in Equation (8.5) to the quadratic regression model. Doing so yields
nnnn2nnn2 ∆Y=(b0 +b1 *11+b2 *11)-(b0 +b1 *10+b2 *10), (8.6)
where bn0, bn1, and bn2 are the OLS estimators.
The term in the first set of parentheses in Equation (8.6) is the predicted value
of Y when Income = 11, and the term in the second set of parentheses is the predicted value of Y when Income = 10. These predicted values are calculated using the OLS estimates of the coefficients in Equation (8.2). Accordingly, when Income = 10, the predicted value of test scores is 607.3 + 3.85 * 10 – 0.0423 * 102 = 641.57. When Income = 11, the predicted value is 607.3 + 3.85 * 11 – 0.0423 * 112 = 644.53. The difference in these two predicted values is ∆Yn = 644.53 – 641.57 = 2.96 points; that is, the predicted difference in test scores between a district with average income of $11,000 and one with average income of $10,000 is 2.96 points.
In the second case, when income changes from $40,000 to $41,000, the difference
n
Standard errors of estimated effects. The estimator of the effect on Y of changing X1 depends on the estimator of the population regression function, fn, which varies from one sample to the next. Therefore, the estimated effect contains a sampling error. One way to quantify the sampling uncertainty associated with the estimated effect is to compute a confidence interval for the true population effect. To do so, we need to compute the standard error of ∆Yn in Equation (8.5).
in the predicted values in Equation (8.6) is ∆Y = (607.3 + 3.85 * 41 – 0.0423 * 412) – (607.3 + 3.85 * 40 – 0.0423 * 402) = 694.04 – 693.62 = 0.42 points. Thus a change of income of $1000 is associated with a larger change in predicted test scores if the initial income is $10,000 than if it is $40,000 (the predicted changes are 2.96 points versus 0.42 point). Said differently, the slope of the estimated qua- dratic regression function in Figure 8.3 is steeper at low values of income (like $10,000) than at the higher values of income (like $40,000).
It is easy to compute a standard error for ∆Yn when the regression function is n
linear. The estimated effect of a change in X1 is b1∆X1, so the standard error of nnn
∆Y is SE(∆Y) = SE(b1)∆X1 and a 95% confidence interval for the estimated change is b ∆X { 1.96SE(b )∆X .
n11 n11
In the nonlinear regression models of this chapter, the standard error of ∆Yn can
be computed using the tools introduced in Section 7.3 for testing a single restriction involving multiple coefficients. To illustrate this method, consider the estimated change in test scores associated with a change in income from 10 to 11 in
8.1 A General Strategy for Modeling Nonlinear Regression Functions 265 nnn22nn
Equation(8.6),whichis∆Y = b1 * (11 – 10) + b2 * (11 – 10 ) = b1 + 21b2. The standard error of the predicted change therefore is
nnn
SE(∆Y) = SE(b1 + 21b2). (8.7) nn
Thus, if we can compute the standard error of b1 + 21b2, then we have computed the standard error of ∆Yn. There are two methods for doing this using standard regression software, which correspond to the two approaches in Section 7.3 for testing a single restriction on multiple coefficients.
The first method is to use Approach #1 of Section 7.3, which is to compute the F-statistic testing the hypothesis that b1 + 21b2 = 0. The standard error of ∆Yn is then given by2
SE(∆Yn)= ∆Yn. (8.8) 2F
Equation (8.8) gives SE(∆Y) = 2.96 > 2299.94 = 0.17. Thus a 95% confidence interval for the change in the expected value of Y is 2.96 { 1.96 * 0.17 or (2.63, 3.29).
When applied to the quadratic regression in Equation (8.2), the F-statistic testing the hypothesis that b1 + 21b2 = 0 is F = 299.94. Because ∆Yn = 2.96, applying
n
The second method is to use Approach #2 of Section 7.3, which entails transforming the regressors so that, in the transformed regression, one of the coefficients is b1 + 21b2. Doing this transformation is left as an exercise (Exercise 8.9).
A comment on interpreting coefficients in nonlinear specifications. In the mul- tiple regression model of Chapters 6 and 7, the regression coefficients had a natural interpretation. For example, b1 is the expected change in Y associated with a change in X1, holding the other regressors constant. But, as we have seen, this is not generally the case in a nonlinear model. That is, it is not very helpful to think of b1 in Equation (8.1) as being the effect of changing the dis- trict’s income, holding the square of the district’s income constant. In nonlinear models the regression function is best interpreted by graphing it and by calcu- lating the predicted effect on Y of changing one or more of the independent variables.
2 n1 n2 n1 n12 n n2
hypothesis—that is, F = t = [(b + 21b )>SE(b + 21b )] = [∆Y>SE(∆Y)] —and solving for
2Equation (8.8) is derived by noting that the F-statistic is the square of the t-statistic testing this
S E ( ∆ Yn ) .
266 ChaPteR 8 Nonlinear Regression Functions
8.2
Nonlinear Functions of a Single Independent Variable
This section provides two methods for modeling a nonlinear regression function. To keep things simple, we develop these methods for a nonlinear regression func- tion that involves only one independent variable, X. As we see in Section 8.5, however, these models can be modified to include multiple independent variables.
A General Approach to Modeling Nonlinearities
Using Multiple Regression
The general approach to modeling nonlinear regression functions taken in this chapter has five elements:
1. Identify a possible nonlinear relationship. The best thing to do is to use economic theory and what you know about the application to suggest a pos- sible nonlinear relationship. Before you even look at the data, ask yourself whether the slope of the regression function relating Y and X might reason- ably depend on the value of X or on another independent variable. Why might such nonlinear dependence exist? What nonlinear shapes does this suggest? For example, thinking about classroom dynamics with 11-year-olds suggests that cutting class size from 18 students to 17 could have a greater effect than cutting it from 30 to 29.
2. Specify a nonlinear function and estimate its parameters by OLS. Sections 8.2 and 8.3 contain various nonlinear regression functions that can be esti- mated by OLS. After working through these sections you will understand the characteristics of each of these functions.
3. Determine whether the nonlinear model improves upon a linear model. Just because you think a regression function is nonlinear does not mean it really is! You must determine empirically whether your nonlinear model is appro- priate. Most of the time you can use t-statistics and F-statistics to test the null hypothesis that the population regression function is linear against the alternative that it is nonlinear.
4. Plot the estimated nonlinear regression function. Does the estimated regression function describe the data well? Looking at Figures 8.2 and 8.3 suggested that the quadratic model fit the data better than the linear model.
5. Estimate the effect on Y of a change in X. The final step is to use the esti- mated regression to calculate the effect on Y of a change in one or more regressors X using the method in Key Concept 8.1.
8.2 Nonlinear Functions of a Single Independent Variable 267
The first method discussed in this section is polynomial regression, an exten- sion of the quadratic regression used in the last section to model the relationship between test scores and income. The second method uses logarithms of X, of Y, or of both X and Y. Although these methods are presented separately, they can be used in combination.
Appendix 8.2 provides a calculus-based treatment of the models in this section.
Polynomials
One way to specify a nonlinear regression function is to use a polynomial in X. In general, let r denote the highest power of X that is included in the regression. The polynomial regression model of degree r is
Yi =b0 +b1Xi +b2X2i +g+brXri +ui. (8.9)
When r = 2, Equation (8.9) is the quadratic regression model discussed in Sec- tion 8.1. When r = 3 so that the highest power of X included is X3, Equation (8.9) is called the cubic regression model.
The polynomial regression model is similar to the multiple regression model of Chapter 6 except that in Chapter 6 the regressors were distinct independent variables whereas here the regressors are powers of the same dependent variable, X; that is, the regressors are X, X2, X3, and so on. Thus the techniques for estima- tion and inference developed for multiple regression can be applied here. In par- ticular, the unknown coefficients b0, b1, c, br in Equation (8.9) can be estimated by OLS regression of Yi against Xi, X2i , c, X ri .
Testing the null hypothesis that the population regression function is linear. If the population regression function is linear, then the quadratic and higher-degree terms do not enter the population regression function. Accordingly, the null hypothesis (H0) that the regression is linear and the alternative (H1) that it is a polynomial of degree r correspond to
H0 : b2 = 0, b3 = 0, c, br = 0 vs. H1 : at least one bj ≠ 0, j = 2, c, r. (8.10)
The null hypothesis that the population regression function is linear can be tested against the alternative that it is a polynomial of degree r by testing H0 against H1 in Equation (8.10). Because H0 is a joint null hypothesis with q = r – 1 restric- tions on the coefficients of the population polynomial regression model, it can be tested using the F-statistic as described in Section 7.2.
268 ChaPteR 8 Nonlinear Regression Functions
Which degree polynomial should I use? That is, how many powers of X should be included in a polynomial regression? The answer balances a trade-off between flexibil- ity and statistical precision. Increasing the degree r introduces more flexibility into the regression function and allows it to match more shapes; a polynomial of degree r can have up to r – 1 bends (that is, inflection points) in its graph. But increasing r means adding more regressors, which can reduce the precision of the estimated coefficients.
Thus the answer to the question of how many terms to include is that you should include enough to model the nonlinear regression function adequately, but no more. Unfortunately, this answer is not very useful in practice!
A practical way to determine the degree of the polynomial is to ask whether the coefficients in Equation (8.9) associated with largest values of r are zero. If so, then these terms can be dropped from the regression. This procedure, which is called sequential hypothesis testing because individual hypotheses are tested sequentially, is summarized in the following steps:
2. Use the t-statistic to test the hypothesis that the coefficient on X 3b in Equa- r
tion (8.9)] is zero. If you reject this hypothesis, then Xr belongs in the regres- sion, so use the polynomial of degree r.
3. If you do not reject br = 0 in step 2, eliminate Xr from the regression and estimate a polynomial regression of degree r – 1. Test whether the coef- ficient on X r – 1 is zero. If you reject, use the polynomial of degree r – 1.
4. If you do not reject br – 1 = 0 in step 3, continue this procedure until the coef- ficient on the highest power in your polynomial is statistically significant.
This recipe has one missing ingredient: the initial degree r of the polynomial. In many applications involving economic data, the nonlinear functions are smooth, that is, they do not have sharp jumps, or “spikes.” If so, then it is appropriate to choose a small maximum degree for the polynomial, such as 2, 3, or 4—that is, begin with r = 2 or 3 or 4 in step 1.
Application to district income and test scores. The estimated cubic regression function relating district income to test scores is
1. Pick a maximum value of r and estimate the polynomial regression for that r.
TestScore = 600.1 + 5.02Income – 0.096Income2 + 0.00069Income3, (5.1) (0.71) (0.029) (0.00035)
R2 = 0.555.
(8.11)
The t-statistic on Income3 is 1.97, so the null hypothesis that the regression func- tion is a quadratic is rejected against the alternative that it is a cubic at the 5%
r
8.2 Nonlinear Functions of a Single Independent Variable 269
level. Moreover, the F-statistic testing the joint null hypothesis that the coeffi- cients on Income2 and Income3 are both zero is 37.7, with a p-value less than 0.01%, so the null hypothesis that the regression function is linear is rejected against the alternative that it is either a quadratic or a cubic.
Interpretation of coefficients in polynomial regression models. The coefficients in polynomial regressions do not have a simple interpretation. The best way to interpret polynomial regressions is to plot the estimated regression function and calculate the estimated effect on Y associated with a change in X for one or more values of X.
Logarithms
Another way to specify a nonlinear regression function is to use the natural loga- rithm of Y and/or X. Logarithms convert changes in variables into percentage changes, and many relationships are naturally expressed in terms of percentages. Here are some examples:
• A box in Chapter 3, “The Gender Gap of Earnings of College Graduates in the United States,” examined the wage gap between male and female col- lege graduates. In that discussion, the wage gap was measured in terms of dollars. However, it is easier to compare wage gaps across professions and over time when they are expressed in percentage terms.
• In Section 8.1, we found that district income and test scores were nonlinearly related. Would this relationship be linear using percentage changes? That is, might it be that a change in district income of 1%—rather than $1000—is associated with a change in test scores that is approximately constant for different values of income?
• In the economic analysis of consumer demand, it is often assumed that a 1% increase in price leads to a certain percentage decrease in the quan- tity demanded. The percentage decrease in demand resulting from a 1% increase in price is called the price elasticity.
Regression specifications that use natural logarithms allow regression models to estimate percentage relationships such as these. Before introducing those specifi- cations, we review the exponential and natural logarithm functions.
The exponential function and the natural logarithm. The exponential function and its inverse, the natural logarithm, play an important role in modeling nonlinear regression functions. The exponential function of x is ex (that is, e raised
270 ChaPteR 8 Nonlinear Regression Functions Figure 8.4 the Logarithm Function, Y = ln(X)
The logarithmic function Y = ln(X) is steeper for
small than for large values of X, is only defined for 5
Y
X 7 0, and has slope 1>X.
4
Y = ln(X)
3
2 1
00 20 40 60 80 100 120 X
to the power x), where e is the constant 2.71828 . . . ; the exponential function is also written as exp(x). The natural logarithm is the inverse of the exponential function; that is, the natural logarithm is the function for which x = ln(ex) or, equivalently, x = ln3exp(x)4. The base of the natural logarithm is e. Although there are logarithms in other bases, such as base 10, in this book we consider only logarithms in base e—that is, the natural logarithm—so when we use the term logarithm we always mean “natural logarithm.”
The logarithm function, y = ln(x), is graphed in Figure 8.4. Note that the logarithm function is defined only for positive values of x. The logarithm function has a slope that is steep at first and then flattens out (although the function con- tinues to increase). The slope of the logarithm function ln(x) is 1>x.
The logarithm function has the following useful properties:
ln(1>x) = -ln(x); Stock/Watson, Econometrics 1e
Electronic Publishing Services Inc.
(8.12)
(8.13)
(8.14)
Final
(8.15)
STOlCn.(IaTxE)M=.0l0n2(a4) + ln(x);
ln(x>a) = ln(x) – ln(a); and 1st Proof 2nd Proof
ln(xa) = a ln(x).
Fig. 06.04
Logarithms and percentages. The link between the logarithm and percentages relies on a key fact: When ∆x is small, the difference between the logarithm of
3rd Proof
8.2 Nonlinear Functions of a Single Independent Variable 271 x + ∆x and the logarithm of x is approximately ∆x>x, the percentage change in x
divided by 100. That is,
ln(x + ∆x) – ln(x) ≅ ¢when is smallb, (8.16)
∆x ∆x xx
where “_” means “approximately equal to.” The derivation of this approximation relies on calculus, but it is readily demonstrated by trying out some values of x and ∆x. For example, when x = 100 and ∆x = 1, then ∆x>x = 1>100 = 0.01 (or 1%),whileln(x + ∆x) – ln(x) = ln(101) – ln(100) = 0.00995(or0.995%).Thus ∆x>x (which is 0.01) is very close to ln(x + ∆x) – ln(x) (which is 0.00995). When ∆x = 5, ∆x>x = 5>100 = 0.05, while ln(x + ∆x) – ln(x) = ln(105) – ln(100) = 0.04879.
The three logarithmic regression models. There are three different cases in which logarithms might be used: when X is transformed by taking its logarithm but Y is not; when Y is transformed to its logarithm but X is not; and when both Y and X are transformed to their logarithms. The interpretation of the regression coefficients is different in each case. We discuss these three cases in turn.
Case I: X is in logarithms, Y is not. In this case, the regression model is
Yi = b0 + b1ln(Xi) + ui,i = 1,c,n. (8.17)
Because Y is not in logarithms but X is, this is sometimes referred to as a linear- log model.
In the linear-log model, a 1% change in X is associated with a change in Y of 0.01b1. To see this, consider the difference between the population regression
function at values of X that differ by ∆X: This is 3b + b ln(X + ∆X)4 – 010
3b + b ln(X)4 = b 3ln(X + ∆X) – ln(X)4 ≅ b (∆X>X), where the final step uses
111
the approximation in Equation (8.16). If X changes by 1%, then ∆X>X = 0.01;
thus in this model a 1% change in X is associated with a change of Y of 0.01b1. The only difference between the regression model in Equation (8.17) and the regression model of Chapter 4 with a single regressor is that the right-hand vari- able is now the logarithm of X rather than X itself. To estimate the coefficients b0 and b1 in Equation (8.17), first compute a new variable, ln(X), which is readily done using a spreadsheet or statistical software. Then b0 and b1 can be estimated by the OLS regression of Yi on ln(Xi), hypotheses about b1 can be tested using the t-statistic, and a 95% confidence interval for b1 can be constructed as
b { 1.96SE(b ). n1 n1
272 Chapter 8 Nonlinear Regression Functions
As an example, return to the relationship between district income and test scores. Instead of the quadratic specification, we could use the linear-log specifi- cation in Equation (8.17). Estimating this regression by OLS yields
TestScore = 557.8 + 36.42ln(Income),R2 = 0.561. (3.8) (1.40)
(8.18)
According to Equation (8.18), a 1% increase in income is associated with an increase in test scores of 0.01 * 36.42 = 0.36 point.
To estimate the effect on Y of a change in X in its original units of thousands
of dollars (not in logarithms), we can use the method in Key Concept 8.1. For
example, what is the predicted difference in test scores for districts with aver-
ference between the predicted values: ∆Y = 3557.8 + 36.42ln(11)4 – 3557.8 + 36.42ln(10)4 = 36.42 * 3ln(11) – ln(10)4 = 3.47.Similarly,thepredicteddiffer- ence between a district with average income of $40,000 and a district with average income of $41,000 is 36.42 * 3ln(41) – ln(40)4 = 0.90. Thus, like the quadratic specification, this regression predicts that a $1000 increase in income has a larger effect on test scores in poor districts than it does in affluent districts.
The estimated linear-log regression function in Equation (8.18) is plotted in Figure 8.5. Because the regressor in Equation (8.18) is the natural logarithm of income rather than income, the estimated regression function is not a straight line. Like the quadratic regression function in Figure 8.3, it is initially steep but then flattens out for higher levels of income.
Case II: Y is in logarithms, X is not. In this case, the regression model is
ln(Yi) = b0 + b1Xi + ui. (8.19)
Because Y is in logarithms but X is not, this is referred to as a log-linear model. In the log-linear model, a one-unit change in X (∆X = 1) is associated with a (100 * b1) % change in Y. To see this, compare the expected values of ln(Y) for
valuesofXthatdifferby∆X.Theexpectedvalueofln(Y)givenXisln(Y) = b0 +
age incomes of $10,000 versus $11,000? The estimated value of ∆Y is the dif- n
b1X. When X is X + ∆X, the expected value is given by ln(Y + ∆Y) = b0 +
ln(Y) = 3b + b (X + ∆X)4 – 3b + b X4 = b ∆X.Fromtheapproximationin 01011
b1(X + ∆X). Thus the difference between these expected values is ln(Y + ∆Y) –
Equation (8.16), however, if b ∆X is small, then ln(Y + ∆Y) – ln(Y) ≅ ∆Y>Y. 1
Thus∆Y>Y ≅ b ∆X.If∆X = 1sothatXchangesbyoneunit,then∆Y>Ychanges 1
8.2 Nonlinear Functions of a Single Independent Variable 273 Figure 8.5 the Linear-Log Regression Function
The estimated linear-log regression function
Yn = bn0 + bn1 ln(X) captures much of the nonlinear
relation between test scores and district income.
700
680
660
640
620
6000 10 20 30 40 50 60 District income (thousands of dollars)
by b1. Translated into percentages, a one-unit change in X is associated with a (100 * b1)% change in Y.
As an illustration, we return to the empirical example of Section 3.7, the rela-
tionship between age and earnings of college graduates. Many employment con-
tracts specify that, for each additional year of service, a worker gets a certain
percentage increase in his or her wage. This percentage relationship suggests esti-
mating the log-linear specification in Equation (8.19) so that each additional year
of age (X) is, on average in the population, associated with some constant percent-
age increase in earnings (Y). By first computing the new dependent variable,
ln(Earningsi), the unknown coefficients b0 and b1 can be estimated by the OLS
regression of ln(Earningsi) against Agei. When estimated using the 14,752 obser- Electronic Publishing Services Inc.
(8.20)
Final
Test score
740 720
Linear-log regression
vations on college graduates in the March 2013 Current Population Survey (the
Stock/Watson, Econometrics 1e
data are described in Appendix 3.1), this relationship is
STOC.ITEM.0025
Fig. 06.05 2
ln(Earnings) = 2.811 + 0.0096 Age, R = 0.034.
1st (P0r.0o1o8f ) (0.00024n)d Proof 3rd Proof
According to this regression, earnings are predicted to increase by 0.96% 3(100 * 0.0096)%4 for each additional year of age.
274 Chapter 8 Nonlinear Regression Functions
Case III: Both X and Y are in logarithms. In this case, the regression model is
ln(Yi) = b0 + b1ln(Xi) + ui. (8.21)
Because both Y and X are specified in logarithms, this is referred to as a log-log model.
In the log-log model, a 1% change in X is associated with a b1% change in Y.
apply Key Concept 8.1; thus ln(Y + ∆Y) – ln(Y) = 3b + b ln(X + ∆X)4 – 01
Thus in this specification b1 is the elasticity of Y with respect to X. To see this, again
3b + b ln(X)4 = b 3ln(X + ∆X) – ln(X)4. Application of the approximation 011
in Equation (8.16) to both sides of this equation yields
Thus in the log-log specification b1 is the ratio of the percentage change in Y associated with the percentage change in X. If the percentage change in X is 1% (that is, if ∆X = 0.01X), then b1 is the percentage change in Y associated with a 1% change in X. That is, b1 is the elasticity of Y with respect to X.
As an illustration, return to the relationship between income and test scores. When this relationship is specified in this form, the unknown coefficients are esti- mated by a regression of the logarithm of test scores against the logarithm of income. The resulting estimated equation is
ln(TestScore) = 6.336 + 0.0554ln(Income),R2 = 0.557. (8.23) (0.006) (0.0021)
According to this estimated regression function, a 1% increase in income is esti- mated to correspond to a 0.0554% increase in test scores.
The estimated log-log regression function in Equation (8.23) is plotted in Figure 8.6. Because Y is in logarithms, the vertical axis in Figure 8.6 is the loga- rithm of the test score and the scatterplot is the logarithm of test scores versus district income. For comparison purposes, Figure 8.6 also shows the estimated regression function for a log-linear specification, which is
ln(TestScore) = 6.439 + 0.00284Income,R2 = 0.497. (8.24) (0.003) (0.00018)
∆Y ≅ b ∆X or Y1X
∆Y>Y 100 * (∆Y>Y) percentage change in Y
b1 = ∆X>X = 100 * (∆X>X) = percentage change in X. (8.22)
8.2 Nonlinear Functions of a Single Independent Variable 275 Figure 8.6 the Log-Linear and Log-Log Regression Functions
In the log-linear regression function, ln(Y) is a linear function of X. In the log-log regression function, ln(Y) is a linear function of ln(X).
ln(Test score)
6.60
6.55
6.50
6.45
6.40
0 10 20 30 40 50 60
District income (thousands of dollars)
Log-linear regression
Log-log regression
Because the vertical axis is in logarithms, the regression function in Equation (8.24) is the straight line in Figure 8.6.
As you can see in Figure 8.6, the log-log specification fits better than the log- linear specification. This is consistent with the higher R 2 for the log-log regression (0.557) than for the log-linear regression (0.497). Even so, the log-log specification does not fit the data especially well: At the lower values of income most of the observations fall below the log-log curve, while in the middle income range most of the observations fall above the estimated regression function.
The three logarithmic regression models are summarized in Key Concept 8.2.
A difficulty with comparing logarithmic specifications. Which of the log regres-
sion models best fits the data? As we saw in the discussion of Equations (8.23) and
(8.24),theR2 canbeusEeldecttorocnoimcpPaurbelitshheinloggS-leinrevaicreasnIdnclo.g-logmodels;asithap- 22
pened, the log-log moSdteolckh/aWdatthseonh,igEhceornRom. eStirmicisla1rely, the R can be used to
compare the linear-logSrTeOgrCe.sIsTioEnMi.n00E2q6uation (8.18) and the linear regression of
Y against X. In the testFsicgo. r0e6a.0n6d income regression, the linear-log regression has
an R2 of 0.561 while the linear regression has an R2 of 0.508, so the linear-log model fits the data better. 1st Proof 2nd Proof 3rd Proof Final
How can we compare the linear-log model and the log-log model? Unfortu- nately, the R2 cannot be used to compare these two regressions because their dependent variables are different [one is Y, the other is ln(Y)]. Recall that the R2
276 Chapter 8 Nonlinear Regression Functions
Logarithms in regression: three Cases
8.2
Key ConCept
Logarithms can be used to transform the dependent variable Y, an independent variable X, or both (but the variable being transformed must be positive). The fol- lowing table summarizes these three cases and the interpretation of the regression coefficient b1. In each case, b1 can be estimated by applying OLS after taking the logarithm of the dependent and/or independent variable.
Case Regression Specification
I Yi = b0 + b1ln(Xi) + ui
II ln(Yi) = b0 + b1Xi + ui
III ln(Yi) = b0 + b1ln(Xi) + ui
Interpretation of B1
A 1% change in X is associated with a change in Y of 0.01b1.
A change in X by one unit (∆X = 1)
is associated with a 100b1% change in Y.
A 1% change in X is associated with a b1% change in Y, so b1 is the elasticity of Y with respect to X.
measures the fraction of the variance of the dependent variable explained by the regressors. Because the dependent variables in the log-log and linear-log models are different, it does not make sense to compare their R 2’s.
Because of this problem, the best thing to do in a particular application is to decide, using economic theory and either your or other experts’ knowledge of the problem, whether it makes sense to specify Y in logarithms. For example, labor economists typically model earnings using logarithms because wage comparisons, contract wage increases, and so forth are often most naturally discussed in per- centage terms. In modeling test scores, it seems (to us, anyway) natural to discuss test results in terms of points on the test rather than percentage increases in the test scores, so we focus on models in which the dependent variable is the test score rather than its logarithm.
Computing predicted values of Y when Y is in logarithms.3 If the dependent vari- able Y has been transformed by taking logarithms, the estimated regression can be used to compute directly the predicted value of ln(Y). However, it is a bit trickier to compute the predicted value of Y itself.
3 This material is more advanced and can be skipped without loss of continuity.
8.2 Nonlinear Functions of a Single Independent Variable 277
To see this, consider the log-linear regression model in Equation (8.19) and rewrite it so that it is specified in terms of Y rather than ln(Y). To do so, take the exponential function of both sides of the Equation (8.19); the result is
Yi = exp(b0 + b1Xi + ui) = eb0 + b1Xieui. (8.25)
The expected value of Yi given Xi is E(YiXi) = E(eb0+b1XieuiXi) = eb0+b1Xi
E(euiXi). The problem is that even if E(uiXi) = 0,E(euiXi) ≠ 1. Thus the
appropriate predicted value of Yi is not simply obtained by taking the exponential nn
function of bn0 + bn1Xi, that is, by setting Yn i = eb0 + b1Xi: This predicted value is biased because of the missing factor E(eui Xi).
One solution to this problem is to estimate the factor E(eui Xi) and use this estimate when computing the predicted value of Y. Exercise 17.12 works through several ways to estimate E(eui Xi), but this gets complicated, particularly if ui is heteroskedastic, and we do not pursue it further.
Another solution, which is the approach used in this book, is to compute predicted values of the logarithm of Y but not transform them to their original units. In practice, this is often acceptable because when the dependent variable is specified as a logarithm, it is often most natural just to use the logarithmic speci- fication (and the associated percentage interpretations) throughout the analysis.
Polynomial and Logarithmic Models
of Test Scores and District Income
In practice, economic theory or expert judgment might suggest a functional form to use, but in the end the true form of the population regression function is unknown. In practice, fitting a nonlinear function therefore entails deciding which method or combination of methods works best. As an illustration, we compare logarithmic and polynomial models of the relationship between district income and test scores.
Polynomial specifications. We considered two polynomial specifications specified using powers of Income, quadratic [Equation (8.2)] and cubic [Equation (8.11)]. Because the coefficient on Income3 in Equation (8.11) was significant at the 5% level, the cubic specification provided an improvement over the quadratic, so we select the cubic model as the preferred polynomial specification.
Logarithmic specifications. The logarithmic specification in Equation (8.18) seemed to provide a good fit to these data, but we did not test this formally. One way to do so is to augment it with higher powers of the logarithm of income. If
278 ChaPteR 8 Nonlinear Regression Functions
8.3
Interactions Between Independent Variables
In the introduction to this chapter we wondered whether reducing the student– teacher ratio might have a bigger effect on test scores in districts where many students are still learning English than in those with few still learning English. This could arise, for example, if students who are still learning English benefit differentially from one-on-one or small-group instruction. If so, the presence of many English learners in a district would interact with the student–teacher ratio in such a way that the effect on test scores of a change in the student–teacher ratio would depend on the fraction of English learners.
these additional terms are not statistically different from zero, then we can con- clude that the specification in Equation (8.18) is adequate in the sense that it cannot be rejected against a polynomial function of the logarithm. Accordingly, the estimated cubic regression (specified in powers of the logarithm of income) is
TestScore = 486.1 + 113.4ln(Income) – 26.93ln(Income)24 (79.4) (87.9) (31.7)
32
+ 3.063ln(Income)4 , R = 0.560. (8.26)
(3.74)
The t-statistic on the coefficient on the cubic term is 0.818, so the null hypothesis that the true coefficient is zero is not rejected at the 10% level. The F-statistic testing the joint hypothesis that the true coefficients on the quadratic and cubic term are both zero is 0.44, with a p-value of 0.64, so this joint null hypothesis is not rejected at the 10% level. Thus the cubic logarithmic model in Equation (8.26) does not pro- vide a statistically significant improvement over the model in Equation (8.18), which is linear in the logarithm of income.
Comparing the cubic and linear-log specifications. Figure 8.7 plots the estimated regression functions from the cubic specification in Equation (8.11) and the linear- log specification in Equation (8.18). The two estimated regression functions are quite similar. One statistical tool for comparing these specifications is the R 2. The R2 of the logarithmic regression is 0.561, and for the cubic regression it is 0.555. Because the logarithmic specification has a slight edge in terms of the R2 and because this specification does not need higher-degree polynomials in the logarithm of income to fit these data, we adopt the logarithmic specification in Equation (8.18).
8.3 Interactions Between Independent Variables 279 Figure 8.7 the Linear-Log and Cubic Regression Functions
The estimated cubic regression function
[Equation (8.11)] and the estimated linear-log 740
regression function [Equation (8.18)] are nearly identical in this sample.
720
700
680
660
640
620
6000 10 20 30 40 50 60 District income (thousands of dollars)
Test score
Linear-log regression
Cubic regression
This section explains how to incorporate such interactions between two inde- pendent variables into the multiple regression model. The possible interaction between the student–teacher ratio and the fraction of English learners is an exam- ple of the more general situation in which the effect on Y of a change in one independent variable depends on the value of another independent variable. We consider three cases: when both independent variables are binary, when one is binary and the other is continuous, and when both are continuous.
Interactions Between Two Binary Variables
Consider the population regression of log earnings [Yi, where Yi = ln(Earningsi)]
against two binary variables: whether a worker has a college degree (D1i, where
D1i = 1 if the ith persoEnlegcrtarodnuiactePdubfrloismhincgolSleegrev)icaensdInthce. worker’s gender (D2i, th
where D2i = 1 if the i Stpoecrks/oWnaistsfoenm,aElec)o. Tnohme eptorpicusla1teion linear regression of Yi on these two binary vaSriTaOblCes.ITisEM.0027
Fig. 06.07
Y=b+bD +bD +u. (8.27) i 0 11i 22i i
1st Proof 2nd Proof 3rd Proof Final
In this regression model, b1 is the effect on log earnings of having a college degree, holding gender constant, and b2 is the effect of being female, holding schooling constant.
280 ChaPteR 8 Nonlinear Regression Functions
The specification in Equation (8.27) has an important limitation: The effect of having a college degree in this specification, holding constant gender, is the same for men and women. There is, however, no reason that this must be so. Phrased mathematically, the effect on Yi of D1i, holding D2i constant, could depend on the value of D2i. In other words, there could be an interaction between having a college degree and gender so that the value in the job market of a degree is different for men and women.
Although the specification in Equation (8.27) does not allow for this interac- tion between having a college degree and gender, it is easy to modify the specifica- tion so that it does by introducing another regressor, the product of the two binary variables, D1i * D2i. The resulting regression is
Yi = b0 + b1D1i + b2D2i + b3(D1i * D2i) + ui. (8.28)
The new regressor, the product D1i * D2i, is called an interaction term or an interacted regressor, and the population regression model in Equation (8.28) is called a binary variable interaction regression model.
The interaction term in Equation (8.28) allows the population effect on log earnings (Yi) of having a college degree (changing D1i from D1i = 0 to D1i = 1) to depend on gender (D2i). To show this mathematically, calculate the population effect of a change in D1i using the general method laid out in Key Concept 8.1. The first step is to compute the conditional expectation of Yi for D1i = 0, given a valueofD2i;thisisE(YiD1i =0,D2i =d2)=b0 +b1 *0+b2 *d2 +b3* (0 * d2) = b0 + b2d2, where we use the conditional mean zero assumption, E(uiD1i,D2i) = 0.ThenextstepistocomputetheconditionalexpectationofYiafter the change—that is, for D1i = 1—given the same value of D2i; this is E(Yi D1i = 1, D2i =d2)=b0 +b1 *1+b2 *d2 +b3 *(1*d2)=b0 +b1 +b2d2 +b3d2. The effect of this change is the difference of expected values [that is, the difference in Equation (8.4)], which is
E(YiD1i = 1, D2i = d2) – E(YiD1i = 0, D2i = d2) = b1 + b3d2. (8.29)
Thus, in the binary variable interaction specification in Equation (8.28), the effect of acquiring a college degree (a unit change in D1i) depends on the person’s gender [the value of D2i, which is d2 in Equation (8.29)]. If the person is male (d2 = 0), the effect of acquiring a college degree is b1, but if the person is female (d2 = 1), the effect is b1 + b3. The coefficient b3 on the interaction term is the dif- ference in the effect of acquiring a college degree for women versus men.
8.3 Interactions Between Independent Variables 281
a Method for Interpreting Coefficients in Regressions with Binary Variables
Key ConCept
8.3
First compute the expected values of Y for each possible case described by the set of binary variables. Next compare these expected values. Each coefficient can then be expressed either as an expected value or as the difference between two or more expected values.
Although this example was phrased using log earnings, having a college degree, and gender, the point is a general one. The binary variable interaction regression allows the effect of changing one of the binary independent variables to depend on the value of the other binary variable.
The method we used here to interpret the coefficients was, in effect, to work through each possible combination of the binary variables. This method, which applies to all regressions with binary variables, is summarized in Key Concept 8.3.
Application to the student–teacher ratio and the percentage of English learners. Let HiSTRi be a binary variable that equals 1 if the student–teacher ratio is 20 or more and equals 0 otherwise, and let HiELi be a binary variable that equals 1 if the percentage of English learners is 10% or more and equals 0 otherwise. The inter- acted regression of test scores against HiSTRi and HiELi is
TestScore = 664.1 – 1.9 HiSTR – 18.2 HiEL – 3.5(HiSTR * HiEL), (1.4) (1.9) (2.3) (3.1)
R2 = 0.290.
(8.30)
The predicted effect of moving from a district with a low student–teacher ratio to one with a high student–teacher ratio, holding constant whether the percentage of English learners is high or low, is given by Equation (8.29), with estimated coef- ficients replacing the population coefficients. According to the estimates in Equation (8.30), this effect thus is – 1.9 – 3.5HiEL. That is, if the fraction of English learners is low (HiEL = 0), then the effect on test scores of moving from HiSTR = 0 to HiSTR = 1isfortestscorestodeclineby1.9points.IfthefractionofEnglishlearn- ers is high, then test scores are estimated to decline by 1.9 + 3.5 = 5.4 points.
The estimated regression in Equation (8.30) also can be used to estimate the mean test scores for each of the four possible combinations of the binary variables.
282 ChaPteR 8 Nonlinear Regression Functions
This is done using the procedure in Key Concept 8.3. Accordingly, the sample aver- age test score for districts with low student–teacher ratios (HiSTRi = 0) and low fractions of English learners (HiELi = 0) is 664.1. For districts with HiSTRi = 1 (high student–teacher ratios) and HiELi = 0 (low fractions of English learners), the sample average is 662.2 ( = 664.1 – 1.9). When HiSTRi = 0 and HiELi = 1, the sample average is 645.9 (= 664.1 – 18.2), and when HiSTRi = 1 and HiELi = 1, the sample average is 640.5 (= 664.1 – 1.9 – 18.2 – 3.5).
Interactions Between a Continuous and
a Binary Variable
Next consider the population regression of log earnings [Yi = ln(Earningsi)] against one continuous variable, the individual’s years of work experience (Xi), and one binary variable, whether the worker has a college degree (Di, where Di = 1 if the ith person is a college graduate). As shown in Figure 8.8, the popula- tion regression line relating Y and the continuous variable X can depend on the binary variable D in three different ways.
In Figure 8.8a, the two regression lines differ only in their intercept. The corresponding population regression model is
Yi =b0 +b1Xi +b2Di +ui. (8.31)
This is the familiar multiple regression model with a population regression func- tion that is linear in Xi and Di. When Di = 0, the population regression function is b0 + b1Xi, so the intercept is b0 and the slope is b1. When Di = 1, the population regression function is b0 + b1Xi + b2, so the slope remains b1 but the intercept is b0 + b2. Thus b2 is the difference between the intercepts of the two regression lines, as shown in Figure 8.8a. Stated in terms of the earnings example, b1 is the effect on log earnings of an additional year of work experience, holding college degree status constant, and b2 is the effect of a college degree on log earnings, holding years of experience constant. In this specification, the effect of an addi- tional year of work experience is the same for college graduates and nongraduates; that is, the two lines in Figure 8.8a have the same slope.
In Figure 8.8b, the two lines have different slopes and intercepts. The differ- ent slopes permit the effect of an additional year of work to differ for college graduates and nongraduates. To allow for different slopes, add an interaction term to Equation (8.31):
Yi =b0 +b1Xi +b2Di +b3(Xi *Di)+ui, (8.32)
8.3 Interactions Between Independent Variables 283 Figure 8.8 Regression Functions Using Binary and Continuous Variables
b0 +b2 b0
b0 +b2 b0
(a)
Different intercepts, same slope
Y b0 + (b1 +b2)X
slope = b1+b2
XX (b) Different intercepts, different slopes
YY
(b0 +b2)+(b1 +b3)X
slope = b1+b3
slope = b1
b0 +b1X
(b0 +b2)+b1X
b0 +b1X slope = b1
b0
b0 +b1X
slope = b1
X
(c) Same intercept, different slopes
Interactions of binary variables and continuous variables can produce three different population regression functions: (a) b0 + b1X + b2D allows for different intercepts but has the same slope, (b) b0 + b1X + b2D + b3(X * D) allows for different intercepts and different slopes, and (c) b0 + b1X + b2(X * D) has the same intercept but allows for different slopes.
where Xi * Di is a new variable, the product of Xi and Di. To interpret the coef- ficients of this regression, apply the procedure in Key Concept 8.3. Doing so shows that, if Di = 0, the population regression function is b0 + b1Xi, whereas if Di = 1, the population regression function is (b0 + b2) + (b1 + b3)Xi. Thus this specification allows for two different population regression functions relating Yi and Xi, depending on the value of Di, as is shown in Figure 8.8b. The difference between the two intercepts is b2, and the difference between the two slopes is b3. In the earnings example, b1 is the effect of an additional year of work experience for nongraduates (Di = 0) and b1 + b3 is this effect for graduates, so b3 is the dif- ference in the effect of an additional year of work experience for college graduates versus nongraduates.
284 ChaPteR 8 Nonlinear Regression Functions
Interactions Between Binary and Continuous Variables
8.4
Key ConCept
Through the use of the interaction term Xi * Di, the population regression line relating Yi and the continuous variable Xi can have a slope that depends on the binary variable Di. There are three possibilities:
1. Different intercept, same slope (Figure 8.8a):
Yi =b0 +b1Xi +b2Di +ui;
2. Different intercept and slope (Figure 8.8b):
Yi =b0 +b1Xi +b2Di +b3(Xi *Di)+ui;
3. Same intercept, different slope (Figure 8.8c):
Yi =b0 +b1Xi +b2(Xi *Di)+ui.
A third possibility, shown in Figure 8.8c, is that the two lines have different slopes but the same intercept. The interacted regression model for this case is
Yi =b0 +b1Xi +b2(Xi *Di)+ui. (8.33)
The coefficients of this specification also can be interpreted using Key Concept 8.3. In terms of the earnings example, this specification allows for different effects of experience on log earnings between college graduates and nongraduates, but requires that expected log earnings be the same for both groups when they have no prior experience. Said differently, this specification corresponds to the popula- tion mean entry-level wage being the same for college graduates and nongraduates. This does not make much sense in this application, and in practice this specification is used less frequently than Equation (8.32), which allows for different intercepts and slopes.
All three specifications—Equations (8.31), (8.32), and (8.33)—are versions of the multiple regression model of Chapter 6, and once the new variable Xi * Di is created, the coefficients of all three can be estimated by OLS.
The three regression models with a binary and a continuous independent variable are summarized in Key Concept 8.4.
Application to the student–teacher ratio and the percentage of English learners. Does the effect on test scores of cutting the student–teacher ratio depend on whether the percentage of students still learning English is high or low? One way to answer this question is to use a specification that allows for two
8.3 Interactions Between Independent Variables 285
different regression lines, depending on whether there is a high or a low percent- age of English learners. This is achieved using the different intercept/different slope specification:
TestScore = 682.2 – 0.97STR + 5.6HiEL – 1.28(STR * HiEL), (11.9) (0.59) (19.5) (0.97)
R2 = 0.305,
(8.34)
where the binary variable HiELi equals 1 if the percentage of students still learn- ing English in the district is greater than 10% and equals 0 otherwise.
For districts with a low fraction of English learners (HiELi = 0), the esti- mated regression line is 682.2 – 0.97STRi. For districts with a high fraction of English learners (HiELi = 1), the estimated regression line is 682.2 + 5.6 – 0.97STRi – 1.28STRi = 687.8 – 2.25STRi. According to these estimates, reducing the student–teacher ratio by 1 is predicted to increase test scores by 0.97 point in districts with low fractions of English learners but by 2.25 points in districts with high fractions of English learners. The difference between these two effects, 1.28 points, is the coefficient on the interaction term in Equation (8.34).
The interaction regression model in Equation (8.34) allows us to estimate the effect of more nuanced policy interventions than the across-the-board class size reduc- tion considered so far. For example, suppose that the state considered a policy to reduce the student–teacher ratio by 2 in districts with a high fraction of English learn- ers (HiELi = 1) but to leave class size unchanged in other districts. Applying the method of Key Concept 8.1 to Equations (8.32) and (8.34) shows that the estimated effect of this reduction for the districts for which HiEL = 1 is -2(bn1 + bn3) = 4.50. The standard error of this estimated effect is SE( – 2bn1 – 2bn3) = 1.53, which can be computed using Equation (8.8) and the methods of Section 7.3.
The OLS regression in Equation (8.34) can be used to test several hypotheses about the population regression line. First, the hypothesis that the two lines are in fact the same can be tested by computing the F-statistic testing the joint hypothesis that the coefficient on HiELi and the coefficient on the interaction term STRi * HiELi are both zero. This F-statistic is 89.9, which is significant at the 1% level.
Second, the hypothesis that two lines have the same slope can be tested by testing whether the coefficient on the interaction term is zero. The t-statistic, -1.28>0.97 = -1.32, is less than 1.64 in absolute value, so the null hypothesis that the two lines have the same slope cannot be rejected using a two-sided test at the
10% significance level.
Third, the hypothesis that the lines have the same intercept corresponds to the
restriction that the population coefficient on HiEL is zero. The t-statistic testing
286 ChaPteR 8 Nonlinear Regression Functions
this restriction is t = 5.6>19.5 = 0.29, so the hypothesis that the lines have the same intercept cannot be rejected at the 5% level.
These three tests produce seemingly contradictory results: The joint test using the F-statistic rejects the joint hypothesis that the slope and the intercept are the same, but the tests of the individual hypotheses using the t-statistic fail to reject. The reason is that the regressors, HiEL and STR * HiEL, are highly correlated. This results in large standard errors on the individual coefficients. Even though it is impossible to tell which of the coefficients is nonzero, there is strong evidence against the hypothesis that both are zero.
Finally, the hypothesis that the student–teacher ratio does not enter this spec- ification can be tested by computing the F-statistic for the joint hypothesis that the coefficients on STR and on the interaction term are both zero. This F-statistic is 5.64, which has a p-value of 0.004. Thus the coefficients on the student–teacher ratio are statistically significant at the 1% significance level.
Interactions Between Two Continuous Variables
Now suppose that both independent variables (X1i and X2i) are continuous. An example is when Yi is log earnings of the ith worker, X1i is his or her years of work experience, and X2i is the number of years he or she went to school. If the population regression function is linear, the effect on wages of an additional year of experience does not depend on the number of years of education, or, equivalently, the effect of an additional year of education does not depend on the number of years of work experience. In reality, however, there might be an interaction between these two variables so that the effect on wages of an additional year of experience depends on the number of years of education. This interaction can be modeled by augmenting the linear regression model with an interaction term that is the product of X1i and X2i:
Yi = b0 + b1X1i + b2X2i + b3(X1i * X2i) + ui. (8.35)
The interaction term allows the effect of a unit change in X1 to depend on X2. To see this, apply the general method for computing effects in nonlinear regression models in Key Concept 8.1. The difference in Equation (8.4), computed for the interacted regression function in Equation (8.35), is ∆Y = (b1 + b3X2)∆X1 [Exercise 8.10(a)]. Thus the effect on Y of a change in X1, holding X2 constant, is
∆Y = b1 + b3X2, (8.36) ∆X1
which depends on X2. For example, in the earnings example, if b3 is positive, then the effect on log earnings of an additional year of experience is greater, by the amount b3, for each additional year of education the worker has.
8.3 Interactions Between Independent Variables 287 the return to education and the gender gap
In addition to its intellectual pleasures, education has economic rewards. As the boxes in Chapters 3 and 5 show, workers with more education tend to earn more than their counterparts with less educa- tion. The analysis in those boxes was incomplete, however, for at least three reasons. First, it failed to control for other determinants of earnings that might be correlated with educational achievement, so the OLS estimator of the coefficient on educa- tion could have omitted variable bias. Second, the functional form used in Chapter 5—a simple linear relation—implies that earnings change by a constant dollar amount for each additional year of education, whereas one might suspect that the dollar change in earnings is actually larger at higher levels of educa- tion. Third, the box in Chapter 5 ignores the gen- der differences in earnings highlighted in the box in Chapter 3.
All these limitations can be addressed by a multiple regression analysis that controls for determinants of earnings that, if omitted, could cause omitted variable bias and that uses a nonlin- ear functional form relating education and earn- ings. Table 8.1 summarizes regressions estimated using data on full-time workers, ages 30 through 64, from the Current Population Survey (the CPS data are described in Appendix 3.1). The depen- dent variable is the logarithm of hourly earnings, so another year of education is associated with a constant percentage increase (not dollar increase) in earnings.
Table 8.1 has four salient results. First, the omis- sion of gender in regression (1) does not result in sub- stantial omitted variable bias: Even though gender enters regression (2) significantly and with a large
coefficient, gender and years of education are uncor- related; that is, on average men and women have nearly the same levels of education. Second, the returns to education are economically and statisti- cally significantly different for men and women: In regression (3), the t-statistic testing the hypothesis that they are the same is 4.55 ( = 0.008 > 0.0018). Third, regression (4) controls for the region of the country in which the individual lives, thereby addressing potential omitted variable bias that might arise if years of education differ systematically by region. Controlling for region makes a small dif- ference to the estimated coefficients on the educa- tion terms, relative to those reported in regression (3). Fourth, regression (4) controls for the potential experience of the worker, as measured by years since completion of schooling. The estimated coeffi- cients imply a declining marginal value for each year of potential experience.
The estimated economic return to education in regression (4) is 11.26% for each year of educa- tion for men and 12.25% (= 0.1126 + 0.0099, in percent) for women. Because the regression func- tions for men and women have different slopes, the gender gap depends on the years of education. For 12 years of education, the gender gap is estimated to be 27.3% (= 0.0099 * 12 – 0.392, in percent); for 16 years of education, the gender gap is less in percentage terms, 23.4%.
These estimates of the return to education and the gender gap still have limitations, including the possibility of other omitted variables, notably the native ability of the worker, and potential problems associated with the way variables are measured in the CPS. Nevertheless, the estimates in Table 8.1
continued on next page
288
ChaPteR 8 Nonlinear Regression Functions
taBLe 8.1 the Return to education and the Gender Gap: Regression Results
for the United States in 2012
Dependent variable: logarithm of Hourly Earnings.
regressor
Years of education
Female
Female * Years of education
Potential experience
Potential experience2
Midwest
South
West
Intercept
R2
(1)
0.1082** (0.0009)
(2) (3)
0.1111** 0.1078** (0.0009) (0.0012)
– 0.251** – 0.367** (0.005) (0.026)
0.0081** (0.0018)
(4)
0.1126** (0.0012)
– 0.392** (0.025)
0.0099** (0.0018)
0.0186** (0.0012)
-0.000263** (0.000024)
– 0.080** (0.007)
– 0.083** (0.007)
– 0.018** (0.007)
1.335** (0.024)
0.276
1.515** (0.013)
0.221
1.585** 1.632** (0.013) (0.016)
0.263 0.264
The data are from the March 2013 Current Population Survey (see Appendix 3.1). The sample size is n = 50,174 observa- tions for each regression. Female is an indicator variable that equals 1 for women and 0 for men. Midwest, South, and West are indicator variables denoting the region of the United States in which the worker lives: For example, Midwest equals 1 if the worker lives in the Midwest and equals 0 otherwise (the omitted region is Northeast). Standard errors are reported in parentheses below the estimated coefficients. Individual coefficients are statistically significant at the *5% or **1% sig- nificance level.
are consistent with those obtained by economists who carefully address these limitations. A survey by the econometrician David Card (1999) of dozens of empirical studies concludes that labor economists’ best estimates of the return to education generally fall
between 8% and 11%, and that the return depends on the quality of the education. If you are interested in learning more about the economic return to edu- cation, see Card (1999).
8.3 Interactions Between Independent Variables 289
Interactions in Multiple Regression
Key ConCept
8.5
The interaction term between the two independent variables X1 and X2 is their product X1 * X2. Including this interaction term allows the effect on Y of a change in X1 to depend on the value of X2 and, conversely, allows the effect of a change in X2 to depend on the value of X1.
The coefficient on X1 * X2 is the effect of a one-unit increase in X1 and X2, above and beyond the sum of the individual effects of a unit increase in X1 alone and a unit increase in X2 alone. This is true whether X1 and/or X2 are continuous or binary.
A similar calculation shows that the effect on Y of a change ∆X2 in X2, hold- ing X constant, is ∆Y>∆X = (b + b X ).
12231
Putting these two effects together shows that the coefficient b3 on the
interaction term is the effect of a unit increase in X1 and X2, above and beyond the sum of the effects of a unit increase in X1 alone and a unit increase in X2 alone. That is, if X1 changes by ∆X1 and X2 changes by ∆X2, then the expected change in Y is ∆Y = (b1 + b3X2)∆X1 + (b2 + b3X1)∆X2 + b3∆X1∆X2 [Exercise 8.10(c)]. The first term is the effect from changing X1 holding X2 constant; the second term is the effect from changing X2 holding X1 constant; and the final term, b3∆X1∆X2, is the extra effect from changing both X1 and X2.
Interactions between two variables are summarized as Key Concept 8.5.
When interactions are combined with logarithmic transformations, they can be used to estimate price elasticities when the price elasticity depends on the characteristics of the good (see the box “The Demand for Economics Journals” on page 290 for an example).
Application to the student–teacher ratio and the percentage of English learners. The previous examples considered interactions between the student– teacher ratio and a binary variable indicating whether the percentage of English learners is large or small. A different way to study this interaction is to examine the interaction between the student–teacher ratio and the continuous variable,
290 ChaPteR 8 Nonlinear Regression Functions the Demand for economics Journals
Professional economists follow the most recent research in their areas of specialization. Most research in economics first appears in economics journals, so economists—or their libraries—sub- scribe to economics journals.
How elastic is the demand by libraries for eco- nomics journals? To find out, we analyzed the rela- tionship between the number of subscriptions to a journal at U.S. libraries (Yi) and the journal’s library
subscription price using data for the year 2000 for 180 economics journals. Because the product of a journal is not the paper on which it is printed but rather the ideas it contains, its price is logically measured not in dollars per year or dollars per page but instead in dol- lars per idea. Although we cannot measure “ideas” directly, a good indirect measure is the number of times that articles in a journal are subsequently cited by other researchers. Accordingly, we measure price
Figure 8.9
Subscriptions
1200 1000 800 600 400 200
00 5
Library Subscriptions and Prices of economics Journals
10
15
20
25
ln(Subscriptions)
8 7 6 5 4 3 2 1
0-6 -5 -4 -3 -2 -1 0 1 2 3 4
ln(Price per citation)
(b) ln(Subscriptions) and ln(Price per citation)
There is a nonlinear inverse relation between the number of U.S. library subscriptions (quantity) and the library price per citation (price), as shown in Fig- ure 8.9a for 180 economics journals in 2000. But as seen in Figure 8.9b, the relation between log quan- tity and log price appears to be approximately lin- ear. Figure 8.9c shows that demand is more elastic for young journals (Age = 5) than for old journals (Age = 80).
continued on next page
(a) Subscriptions and Price per citation
ln(Subscriptions)
8 7 6 5 4 3 2 1
0-6-5-4-3-2-1 0 1 2 3 4
ln(Price per citation)
(c) ln(Subscriptions) and ln(Price per citation)
Price per citation
Demand when Age = 5
Demand when Age = 80
as the “price per citation” in the journal. The price range is enormous, from 12¢ per citation (the Ameri- can Economic Review) to 20¢ per citation or more. Some journals are expensive per citation because they have few citations, others because their library sub- scription price per year is very high. In 2014, a library print subscription to the Journal of Econometrics cost $4089, compared to only $455 for a bundled subscrip- tion to all seven journals published by the American
Economics Association, including the American Economic Review!
Because we are interested in estimating elastici- ties, we use a log-log specification (Key Concept 8.2). The scatterplots in Figures 8.9a and 8.9b provide empirical support for this transformation. Because some of the oldest and most prestigious journals are the cheapest per citation, a regression of log quantity against log price could have omitted variable bias.
8.3 Interactions Between Independent Variables 291
taBLe 8.2 estimates of the Demand for economics Journals
Dependent variable: logarithm of subscriptions at u.S. libraries in the year 2000; 180 observations.
regressor
ln(Price per citation)
[ln(Price per citation)]2
[ln(Price per citation)]3
ln(Age)
ln(Age) * ln(Price per citation)
ln(Characters , 1,000,000)
Intercept
F Statistics and Summary Statistics
F-statistic testing coefficients on quadratic and cubic terms (p-value)
SER
R2
(1)
-0.533** (0.034)
(2)
-0.408** (0.044)
0.424** (0.119)
0.206* (0.098)
3.21** (0.38)
0.705
0.607
(4)
-0.899** (0.145)
0.374** (0.118)
0.141** (0.040)
0.229* (0.096)
3.43** (0.38)
0.688
0.626
4.77** (0.055)
0.750
0.555
(3)
-0.961** (0.160)
0.017 (0.025)
0.0037 (0.0055)
0.373** (0.118)
0.156** (0.052)
0.235* (0.098)
3.41** (0.38)
0.25 (0.779)
0.691
0.622
continued on next page
The F-statistic tests the hypothesis that the coefficients on 3ln(Price per citation)42 and 3ln(Price per citation)43 are both zero. Standard errors are given in parentheses under coefficients, and p-values are given in parentheses under F-statistics. Individual coefficients are statistically significant at the *5% level or **1% level.
292 ChaPteR 8 Nonlinear Regression Functions Our regressions therefore include two control vari-
ables: the logarithm of age and the logarithm of the number of characters per year in the journal.
The regression results are summarized in Table 8.2. Those results yield the following conclusions (see if you can find the basis for these conclusions in the table!):
1. Demand is less elastic for older than for newer journals.
2. The evidence supports a linear, rather than a cubic, function of log price.
3. Demandisgreaterforjournalswithmorecharac- ters, holding price and age constant.
So what is the elasticity of demand for econom-
ics journals? It depends on the age of the journal. Demand curves for an 80-year-old journal and a 5-year-old upstart are superimposed on the scatterplot
in Figure 8.9c; the older journal’s demand elasticity is -0.28 (SE = 0.06), while the younger journal’s is
-0.67(SE = 0.08).
This demand is very inelastic: Demand is very
insensitive to price, especially for older journals. For libraries, having the most recent research on hand is a necessity, not a luxury. By way of comparison, experts estimate the demand elasticity for cigarettes to be in the range of -0.3 to -0.5. Economics jour- nals are, it seems, as addictive as cigarettes, but a lot better for your health!1
1These data were graciously provided by Professor Theo- dore Bergstrom of the Department of Economics at the University of California, Santa Barbara. If you are inter- ested in learning more about the economics of economics journals, see Bergstrom (2001).
the percentage of English learners (PctEL). The estimated interaction regres- sion is
TestScore = 686.3 – 1.12STR – 0.67PctEL + 0.0012(STR * PctEL), (11.8) (0.59) (0.37) (0.019)
R2 = 0.422.
(8.37)
When the percentage of English learners is at the median (PctEL = 8.85), the slope of the line relating test scores and the student–teacher ratio is estimated to be -1.11 (= -1.12 + 0.0012 * 8.85). When the percentage of English learners is at the 75th percentile (PctEL = 23.0), this line is estimated to be flatter, with a slope of -1.09 (= -1.12 + 0.0012 * 23.0). That is, for a district with 8.85% Eng- lish learners, the estimated effect of a one-unit reduction in the student–teacher ratio is to increase test scores by 1.11 points, but for a district with 23.0% English learners, reducing the student–teacher ratio by one unit is predicted to increase test scores by only 1.09 points. The difference between these estimated effects is not statistically significant, however: The t-statistic testing whether the coefficient
8.4 Nonlinear Effects on Test Scores of the Student–Teacher Ratio 293
on the interaction term is zero is t = 0.0012>0.019 = 0.06, which is not significant at the 10% level.
To keep the discussion focused on nonlinear models, the specifications in Sections 8.1 through 8.3 exclude additional control variables such as the students’ economic background. Consequently, these results arguably are subject to omit- ted variable bias. To draw substantive conclusions about the effect on test scores of reducing the student–teacher ratio, these nonlinear specifications must be aug- mented with control variables, and it is to such an exercise that we now turn.
8.4
Nonlinear Effects on Test Scores of the Student–Teacher Ratio
This section addresses three specific questions about test scores and the student– teacher ratio. First, after controlling for differences in economic characteristics of different districts, does the effect on test scores of reducing the student–teacher ratio depend on the fraction of English learners? Second, does this effect depend on the value of the student–teacher ratio? Third, and most important, after taking economic factors and nonlinearities into account, what is the estimated effect on test scores of reducing the student–teacher ratio by two students per teacher, as our superintendent from Chapter 4 proposes to do?
We answer these questions by considering nonlinear regression specifications of the type discussed in Sections 8.2 and 8.3, extended to include two measures of the economic background of the students: the percentage of students eligible for a subsi- dized lunch and the logarithm of average district income. The logarithm of income is used because the empirical analysis of Section 8.2 suggests that this specification captures the nonlinear relationship between test scores and income. As in Section 7.6, we do not include expenditures per pupil as a regressor and in so doing we are considering the effect of decreasing the student–teacher ratio, allowing expenditures per pupil to increase (that is, we are not holding expenditures per pupil constant).
Discussion of Regression Results
The OLS regression results are summarized in Table 8.3. The columns labeled (1) through (7) each report separate regressions. The entries in the table are the coefficients, standard errors, certain F-statistics and their p-values, and summary statistics, as indicated by the description in each row.
The first column of regression results, labeled regression (1) in the table, is regression (3) in Table 7.1 repeated here for convenience. This regression does not
294 ChaPteR 8 Nonlinear Regression Functions
taBLe 8.3 Nonlinear Regression Models of test Scores Dependent variable: average test score in district; 420 observations.
regressor
Student–teacher ratio (STR)
STR2
STR3
% English learners
% English learners
Ú 10%? (Binary, HiEL)
HiEL * STR
HiEL * STR2
HiEL * STR3
% Eligible for subsidized lunch
Average district income (logarithm)
Intercept
(1) (2)
-1.00** -0.73** (0.27) (0.26)
-0.122** -0.176** (0.033) (0.034)
(3)
-0.97 (0.59)
5.64 (19.51)
-1.28 (0.97)
(4)
– 0.53 (0.34)
5.50 (9.80)
– 0.58 (0.50)
– 0.411** (0.029)
12.12** (1.80)
653.6** (9.9)
5.92 (0.003)
8.63
0.795
(5)
64.33** (24.86)
– 3.42** (1.25)
0.059** (0.021)
-5.47** (1.03)
– 0.420** (0.029)
11.75** (1.78)
252.0 (163.6)
(6)
83.70** (28.50)
– 4.38** (1.44)
0.075** (0.024)
816.1* (327.7)
– 123.3* (50.2)
6.12* (2.54)
– 0.101* (0.043)
– 0.418** (0.029)
11.80** (1.78)
122.3 (185.5)
4.96 (6 0.001)
5.81 (0.003)
2.69 (0.046)
8.55
0.799
(7)
65.29** (25.26)
– 3.47** (1.27)
0.060** (0.021)
– 0.166** (0.034)
-0.547** -0.398** (0.024) (0.033)
11.57** (1.81)
– 0.402** (0.033)
11.51** (1.81)
244.8 (165.7)
5.91 (0.001)
5.96 (0.003)
8.57
0.798
700.2** 658.6** 682.2**
(5.6)
(8.6)
(11.9)
5.64 (0.004)
15.88
0.305
F-Statistics and p-Values on Joint hypotheses
(a) All STR variables and interactions = 0
(b) STR2, STR3 = 0
(c) HiEL * STR, HiEL * STR2, HiEL * STR3 = 0
SER
R2
(6
(6
6.31 0.001)
6.17 0.001)
8.56
0.798
9.08
0.773
8.64
0.794
These regressions were estimated using the data on K–8 school districts in California, described in Appendix 4.1. Standard errors are given in parentheses under coefficients, and p-values are given in parentheses under F-statistics. Individual coefficients are statistically significant at the *5% or **1% significance level.
8.4 Nonlinear Effects on Test Scores of the Student–Teacher Ratio 295
control for income, so the first thing we do is check whether the results change substantially when log income is included as an additional economic control vari- able. The results are given in regression (2) in Table 8.3. The log of income is statistically significant at the 1% level and the coefficient on the student–teacher ratio becomes somewhat closer to zero, falling from -1.00 to -0.73, although it remains statistically significant at the 1% level. The change in the coefficient on STR is large enough between regressions (1) and (2) to warrant including the loga- rithm of income in the remaining regressions as a deterrent to omitted variable bias.
Regression (3) in Table 8.3 is the interacted regression in Equation (8.34) with the binary variable for a high or low percentage of English learners, but with no eco- nomic control variables. When the economic control variables (percentage eligible for subsidized lunch and log income) are added [regression (4) in the table], the coeffi- cients change, but in neither case is the coefficient on the interaction term significant at the 5% level. Based on the evidence in regression (4), the hypothesis that the effect of STR is the same for districts with low and high percentages of English learners cannot be rejected at the 5% level (the t-statistic is t = -0.58>0.50 = -1.16).
Regression (5) examines whether the effect of changing the student–teacher ratio depends on the value of the student–teacher ratio by including a cubic spec- ification in STR in addition to the other control variables in regression (4) [the interaction term, HiEL * STR, was dropped because it was not significant in regression (4) at the 10% level]. The estimates in regression (5) are consistent with the student–teacher ratio having a nonlinear effect. The null hypothesis that the relationship is linear is rejected at the 1% significance level against the alter- native that it is cubic (the F-statistic testing the hypothesis that the true coeffi- cients on STR2 and STR3 are zero is 6.17, with a p-value of 6 0.001).
Regression (6) further examines whether the effect of the student–teacher ratio depends not just on the value of the student–teacher ratio but also on the fraction of English learners. By including interactions between HiEL and STR, STR2, and STR3, we can check whether the (possibly cubic) population regressions functions relating test scores and STR are different for low and high percentages of English learners. To do so, we test the restriction that the coefficients on the three interac- tion terms are zero. The resulting F-statistic is 2.69, which has a p-value of 0.046 and thus is significant at the 5% but not the 1% significance level. This provides some evidence that the regression functions are different for districts with high and low percentages of English learners; however, comparing regressions (6) and (4) makes it clear that these differences are associated with the quadratic and cubic terms.
Regression (7) is a modification of regression (5), in which the continuous variable PctEL is used instead of the binary variable HiEL to control for the per- centage of English learners in the district. The coefficients on the other regressors
296 ChaPteR 8 Nonlinear Regression Functions
Figure 8.10 three Regression Functions Relating test Scores and Student–teacher Ratio
The cubic regressions from columns (5) and (7) of Table 8.3 are nearly identical. They indicate a small amount of nonlinearity in the relation between test scores and student–teacher ratio.
Test score
720
700
680
660
640
620
600
Cubic regression (5) Cubic regression (7) Linear regression (2)
12 14 16 18 20 22 24 26 28
Student–teacher ratio
do not change substantially when this modification is made, indicating that the results in regression (5) are not sensitive to what measure of the percentage of English learners is actually used in the regression.
In all the specifications, the hypothesis that the student–teacher ratio does not enter the regressions is rejected at the 1% level.
The nonlinear specifications in Table 8.3 are most easily interpreted graphi- cally. Figure 8.10 graphs the estimated regression functions relating test scores and the student–teacher ratio for the linear specification (2) and the cubic specifications (5) and (7), along with a scatterplot of the data.4 These estimated regression func- tions show the predicted value of test scores as a function of the student–teacher ratio, holding fixed other values of the independent variables in the regression. The estimated regression functions are all close to one another, although the cubic regressions flatten out for large values of the student–teacher ratio.
Regression (6) indicates a statistically significant difference in the cubic regres- sion functions relatingEtlesctrsocnoircesPaunbdlisShTinRg, dSeprevnicdeisngInocn. whether the percentage of English learners in Sthteocdkis/Wtriactsisonla,rgEecornsomaellt.rFicisgu1re 8.11 graphs these two esti- mated regression funcStiToOnsCs.oITtEhMat.0w0e3c0an see whether this difference, in addition
Fig. 06.10
4For each curve, the predicted value was computed by setting each independent variable, other than
1st Proof 2nd Proof 3rd Proof Final STR, to its sample average value and computing the predicted value by multiplying these fixed values of
the independent variables by the respective estimated coefficients from Table 8.3. This was done for vari- ous values of STR, and the graph of the resulting adjusted predicted values is the estimated regression function relating test scores and the STR, holding the other variables constant at their sample averages.
8.4 Nonlinear Effects on Test Scores of the Student–Teacher Ratio 297 Figure 8.11 Regression Functions for Districts with high and Low Percentages of english Learners
Districts with low percentages of English learners (HiEL = 0) are shown by gray dots, and districts with HiEL = 1 are shown by colored dots. The cubic regression function for HiEL = 1 from regression (6) in Table 8.3 is approximately 10 points below the
cubic regression function for HiEL = 0 for 17 … STR … 23, but otherwise the two functions have similar shapes and slopes in this range. The slopes of the regression functions differ most for very large and small values of STR, for which there are few observations.
Test score
720
700
680
660
640
Regression function (HiEL = 0)
Regression function (HiEL = 1)
620 600
600
12 14 16 18 20 22 24 26 28
Student–teacher ratio
to being statistically significant, is of practical importance. As Figure 8.11 shows, for student–teacher ratios between 17 and 23—a range that includes 88% of the observations—the two functions are separated by approximately 10 points but otherwise are very similar; that is, for STR between 17 and 23, districts with a lower percentage of English learners do better, holding constant the student– teacher ratio, but the effect of a change in the student–teacher ratio is essentially the same for the two groups. The two regression functions are different for student– teacher ratios below 16.5, but we must be careful not to read more into this than is justified. The districts with STR 6 16.5 constitute only 6% of the observations, so the differences between the nonlinear regression functions are reflecting dif- ferences in these very few districts with very low student–teacher ratios. Thus, based on Figure 8.11, we conclude that the effect on test scores of a change in the student–teacher ratio does not depend on the percentage of English learners for the range of student–teacher ratios for which we have the most data.
Summary of Findings
Electronic Publishing Services Inc.
Stock/Watson, Econometrics 1e
These results let us answer the three questions raised at the start of this section.
STOC.ITEM.0031
First, after controlling for economic background, whether there are many or few
Fig. 06.11
English learners in the district does not have a substantial influence on the effect on
1st Proof 2nd Proof 3rd Proof Final test scores of a change in the student–teacher ratio. In the linear specifications, there
is no statistically significant evidence of such a difference. The cubic specification in regression (6) provides statistically significant evidence (at the 5% level) that the
298 ChaPteR 8 Nonlinear Regression Functions
regression functions are different for districts with high and low percentages of Eng- lish learners; as shown in Figure 8.11, however, the estimated regression functions have similar slopes in the range of student–teacher ratios containing most of our data.
Second, after controlling for economic background, there is evidence of a nonlinear effect on test scores of the student–teacher ratio. This effect is statisti- cally significant at the 1% level (the coefficients on STR2 and STR3 are always significant at the 1% level).
Third, we now can return to the superintendent’s problem that opened Chap- ter 4. She wants to know the effect on test scores of reducing the student–teacher ratio by two students per teacher. In the linear specification (2), this effect does not depend on the student–teacher ratio itself, and the estimated effect of this reductionistoimprovetestscoresby1.46(= -0.73 * -2)points.Inthenonlinear specifications, this effect depends on the value of the student–teacher ratio. If her district currently has a student–teacher ratio of 20 and she is considering cutting it to 18, then based on regression (5) the estimated effect of this reduction is to improve test scores by 3.00 points, while based on regression (7) this estimate is 2.93. If her district currently has a student–teacher ratio of 22 and she is consid- ering cutting it to 20, then based on regression (5) the estimated effect of this reduction is to improve test scores by 1.93 points, while based on regression (7) this estimate is 1.90. The estimates from the nonlinear specifications suggest that cutting the student–teacher ratio has a greater effect if this ratio is already small.
8.5
Conclusion
This chapter presented several ways to model nonlinear regression functions. Because these models are variants of the multiple regression model, the unknown coefficients can be estimated by OLS, and hypotheses about their values can be tested using t- and F-statistics as described in Chapter 7. In these models, the expected effect on Y of a change in one of the independent variables, X1, holding the other independent vari- ables X2, c, Xk constant in general depends on the values of X1, X2, c, Xk.
There are many different models in this chapter, and you could not be blamed for being a bit bewildered about which to use in a given application. How should you analyze possible nonlinearities in practice? Section 8.1 laid out a general approach for such an analysis, but this approach requires you to make decisions and exercise judgment along the way. It would be convenient if there were a single recipe you could follow that would always work in every application, but in prac- tice data analysis is rarely that simple.
The single most important step in specifying nonlinear regression functions is to “use your head.” Before you look at the data, can you think of a reason, based on
economic theory or expert judgment, why the slope of the population regression function might depend on the value of that, or another, independent variable? If so, what sort of dependence might you expect? And, most important, which nonlinearities (if any) could have major implications for the substantive issues addressed by your study? Answering these questions carefully will focus your analysis. In the test score application, for example, such reasoning led us to investigate whether hiring more teachers might have a greater effect in districts with a large percentage of students still learning English, perhaps because those students would differentially benefit from more personal attention. By making the question precise, we were able to find a precise answer: After controlling for the economic background of the students, we found no statistically significant evidence of such an interaction.
Summary
1. In a nonlinear regression, the slope of the population regression function depends on the value of one or more of the independent variables.
2. The effect on Y of a change in the independent variable(s) can be com- puted by evaluating the regression function at two values of the independent variable(s). The procedure is summarized in Key Concept 8.1.
3. A polynomial regression includes powers of X as regressors. A quadratic regression includes X and X2, and a cubic regression includes X, X2, and X3.
4. Small changes in logarithms can be interpreted as proportional or percent- age changes in a variable. Regressions involving logarithms are used to esti-
mate proportional changes and elasticities.
5. The product of two variables is called an interaction term. When interaction
terms are included as regressors, they allow the regression slope of one vari- able to depend on the value of another variable.
Key Terms
quadratic regression model (259) nonlinear regression function (262) polynomial regression model (267) cubic regression model (267) elasticity (269)
exponential function (269) natural logarithm (270) linear-log model (271)
log-linear model (272)
log-log model (274)
interaction term (280)
interacted regressor (280) interaction regression model (280) nonlinear least squares (311) nonlinear least squares
estimators (311)
Key Terms 299
300 Chapter 8 Nonlinear Regression Functions
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Review the Concepts
8.1 Sketch a regression function that is increasing (has a positive slope) and is steep for small values of X but less steep for large values of X. Explain how you would specify a nonlinear regression to model this shape. Can you think of an economic relationship with a shape like this?
8.2 A “Cobb–Douglas” production function relates production (Q) to factors of production, capital (K), labor (L), and raw materials (M), and an error term u using the equation Q = lKb1Lb2Mb3eu, where l, b1, b2, and b3 are production parameters. Suppose that you have data on production and the factors of production from a random sample of firms with the same Cobb– Douglas production function. How would you use regression analysis to estimate the production parameters?
8.3 Can you use R 2 to compare the fit of a log-log and log-linear regression? Why? Can you use R 2to compare the fit of a log-log and linear-log regression? Why?
8.4 Suppose the regression in Equation (8.30) is estimated using LoSTR and LoEL in place of HiSTR and HiEL, where LoSTR = 1 – HiSTR is an indicator for a low-class-size district and LoEL = 1 – HiEL is an indica- tor for a district with a low percentage of English learners. What are the values of the estimated regression coefficients?
8.5 Suppose that in Exercise 8.2 you thought that the value of b2 was not constant but rather increased when K increased. How could you use an interaction term to capture this effect?
8.6 You have estimated a linear regression model relating Y to X. Your professor says, “I think that the relationship between Y and X is nonlinear.” Explain how you would test the adequacy of your linear regression.
Exercises
8.1 Sales in a company are $196 million in 2013 and increase to $198 million in 2014.
a. Compute the percentage increase in sales, using the usual formula 100 * (Sales2014 – Sales2013). Compare this value to the approximation
Sales2013
100 * 3ln(Sales ) – ln(Sales )4.
Exercises 301
2014 2013
b. Repeat (a), assuming that Sales2014 = 205, Sales2014 = 250, and
Sales2014 = 500.
c. How good is the approximation when the change is small? Does the quality of the approximation deteriorate as the percentage change increases?
8.2 Suppose that a researcher collects data on houses that have sold in a particular neighborhood over the past year and obtains the regression results in the table shown below.
a. Using the results in column (1), what is the expected change in price of building a 500-square-foot addition to a house? Construct a 95% confidence interval for the percentage change in price.
b. Comparing columns (1) and (2), is it better to use Size or ln(Size) to explain house prices?
c. Using column (2), what is the estimated effect of pool on price? (Make sure you get the units right.) Construct a 95% confidence interval for this effect.
d. The regression in column (3) adds the number of bedrooms to the regression. How large is the estimated effect of an additional bed- room? Is the effect statistically significant? Why do you think the estimated effect is so small? (Hint: Which other variables are being held constant?)
e. Is the quadratic term ln(Size)2 important?
f. Use the regression in column (5) to compute the expected change in price when a pool is added to a house that doesn’t have a view. Repeat the exercise for a house that has a view. Is there a large difference? Is the difference statistically significant?
302 ChaPteR 8 Nonlinear Regression Functions
Regression Results for exercise 8.2
Dependent variable: ln(Price) regressor
Size
ln(Size)
ln(Size)2
Bedrooms
Pool
View
Pool * View
Condition
Intercept
Summary Statistics
SER
R2
(1)
0.00042 (0.000038)
0.082 (0.032)
0.037 (0.029)
0.13 (0.045)
10.97 (0.069)
0.102
0.72
(2)
0.69 (0.054)
0.071 (0.034)
0.027 (0.028)
0.12 (0.035)
6.60 (0.39)
0.098
0.74
(3)
0.68 (0.087)
0.0036 (0.037)
0.071 (0.034)
0.026 (0.026)
0.12 (0.035)
6.63 (0.53)
0.099
0.73
(4)
0.57 (2.03)
0.0078 (0.14)
0.071 (0.036)
0.027 (0.029)
0.12 (0.036)
7.02 (7.50)
0.099
0.73
(5)
0.69 (0.055)
0.071 (0.035)
0.027 (0.030)
0.0022 (0.10)
0.12 (0.035)
6.60 (0.40)
0.099
0.73
Variable definitions: Price = sale price ($); Size = house size (in square feet); Bedrooms = number of bedrooms; Pool = binary variable (1 if house has a swimming pool, 0 otherwise); View = binary variable (1 if house has a nice view, 0 otherwise); Condition = binary variable (1 if real estate agent reports house is in excellent condition, 0 otherwise).
8.3 After reading this chapter’s analysis of test scores and class size, an educator comments, “In my experience, student performance depends on class size, but not in the way your regressions say. Rather, students do well when class size is less than 20 students and do very poorly when class size is greater than 25. There are no gains from reducing class size below 20 students, the rela- tionship is constant in the intermediate region between 20 and 25 students, and there is no loss to increasing class size when it is already greater than 25.” The educator is describing a “threshold effect” in which performance is constant for class sizes less than 20, then jumps and is constant for class
Exercises 303 sizes between 20 and 25, and then jumps again for class sizes greater than 25.
To model these threshold effects, define the binary variables STRsmall = 1 if STR 6 20, and STRsmall = 0 otherwise;
STRmoderate = 1 if 20 … STR … 25, and STRmoderate = 0 otherwise; and STRlarge = 1 if STR 7 25, and STRlarge = 0 otherwise.
a. Consider the regression TestScorei = b0 + b1STRsmalli + b2STRlargei + ui. Sketch the regression function relating TestScore to STR for hypothetical values of the regression coefficients that are consistent
with the educator’s statement.
b. A researcher tries to estimate the regression TestScorei = b0 + b1STRsmalli + b2STRmoderatei + b3STRlargei + ui and finds that the software gives an error message. Why?
8.4 Read the box “The Return to Education and the Gender Gap” in Section 8.3.
a. Consider a man with 16 years of education and 2 years of experience who is from a western state. Use the results from column (4)
of Table 8.1 and the method in Key Concept 8.1 to estimate the expected change in the logarithm of average hourly earnings (AHE) associated with an additional year of experience.
b. Repeat (a), assuming 10 years of experience.
c. Explain why the answers to (a) and (b) are different.
d. Is the difference in the answers to (a) and (b) statistically significant at the 5% level? Explain.
e. Would your answers to (a) through (d) change if the person were a woman? If the person were from the South? Explain.
f. How would you change the regression if you suspected that the effect of experience on earnings was different for men than for women?
8.5 Read the box “The Demand for Economics Journals” in Section 8.3.
a. The box reaches three conclusions. Looking at the results in the table,
what is the basis for each of these conclusions?
b. Using the results in regression (4), the box reports that the elasticity
of demand for an 80-year-old journal is -0.28.
i. How was this value determined from the estimated regression?
ii. The box reports that the standard error for the estimated elasticity is 0.06. How would you calculate this standard error?
304 ChaPteR 8 Nonlinear Regression Functions
(Hint: See the discussion “Standard errors of estimated effects” on
page 264.)
c. Suppose that the variable Characters had been divided by 1000 instead of 1,000,000. How would the results in column (4) change?
8.6 Refer to Table 8.3.
a. A researcher suspects that the effect of %Eligible for subsidized lunch has a nonlinear effect on test scores. In particular, he conjectures that increases in this variable from 10% to 20% have little effect on test scores but that changes from 50% to 60% have a much larger effect.
i. Describe a nonlinear specification that can be used to model this form of nonlinearity.
ii. How would you test whether the researcher’s conjecture was better than the linear specification in column (7) of Table 8.3?
b. A researcher suspects that the effect of income on test scores is different in districts with small classes than in districts with large classes.
i. Describe a nonlinear specification that can be used to model this form of nonlinearity.
ii. How would you test whether the researcher’s conjecture was better than the linear specification in column (7) of Table 8.3?
8.7 This problem is inspired by a study of the “gender gap” in earnings in top corporate jobs [Bertrand and Hallock (2001)]. The study compares total compensation among top executives in a large set of U.S. public corpo- rations in the 1990s. (Each year these publicly traded corporations must report total compensation levels for their top five executives.)
a.
Let Female be an indicator variable that is equal to 1 for females and 0 for males. A regression of the logarithm of earnings onto Female yields
ln (Earnings) = 6.48 – 0.44 Female, SER = 2.65. (0.01) (0.05)
i. The estimated coefficient on Female is -0.44. Explain what this value means.
ii. The SER is 2.65. Explain what this value means.
iii. Does this regression suggest that female top executives earn less
than top male executives? Explain.
iv. Does this regression suggest that there is gender discrimination? Explain.
b. Two new variables, the market value of the firm (a measure of firm size, in millions of dollars) and stock return (a measure of firm performance, in percentage points), are added to the regression:
ln(Earnings) = 3.86 – 0.28Female + 0.37ln(MarketValue) + 0.004Return, (0.03) (0.04) (0.004) (0.003)
n = 46,670,R2 = 0.345.
i. The coefficient on ln(MarketValue) is 0.37. Explain what this
value means.
ii. The coefficient on Female is now -0.28. Explain why it has changed from the regression in (a).
c. Are large firms more likely than small firms to have female top exec- utives? Explain.
8.8 X is a continuous variable that takes on values between 5 and 100. Z is a binary variable. Sketch the following regression functions (with values of X between 5 and 100 on the horizontal axis and values of Yn on the vertical axis):
e.
Yn = 1.0 + 125.0X – 0.01X2.
Yn=2.0+3.0*ln(X).
a. b.
d. i. Yn = 2.0 + 3.0 * ln(X) + 4.0Z – 1.0 * Z * ln(X), with Z = 1. ii. Same as (i), but with Z = 0.
Yn=2.0-3.0*ln(X).
c. i. Yn = 2.0 + 3.0 * ln(X) + 4.0Z, with Z = 1.
ii. Same as (i), but with Z = 0.
8.9 Explain how you would use Approach #2 from Section 7.3 to calculate the confidence interval discussed below Equation (8.8). [Hint: This requires estimating a new regression using a different definition of the regressors and the dependent variable. See Exercise (7.9).]
8.10 ConsidertheregressionmodelYi = b0 + b1X1i + b2X2i + b3(X1i * X2i) + ui. Use Key Concept 8.1 to show:
a. ∆Y>∆X = b + b X (effect of change in X , holding X constant).
b. ∆Y>∆X1 = b1 + b3X2 (effect of change in X1, holding X2 constant).
223121 c. If X1 changes by ∆X1 and X2 changes by ∆X2, then ∆Y =
(b1 + b3X2)∆X1 + (b2 + b3X1)∆X2 + b3∆X1∆X2.
8.11 Derive the expressions for the elasticities given in Appendix 8.2 for the linear and log-log models. (Hint: For the log-log model, assume that u
Exercises 305
306 ChaPteR 8 Nonlinear Regression Functions
and X are independent, as is done in Appendix 8.2 for the log-linear
model.)
8.12 The discussion following Equation (8.28) interprets the coefficient on
interacted binary variables using the conditional mean zero assump- tion. This exercise shows that interpretation also applies under con- ditional mean independence. Consider the hypothetical experiment in Exercise 7.11.
a. Suppose that you estimate the regression Yi = g0 + g1X1i + ui using only the data on returning students. Show that g1 is the class size effect for returning students—that is, that g1 = E(Yi X1i = 1, X2i = 0) – E(Yi X1i = 0, X2i = 0). Explain why gn 1 is an unbiased estimator of g1.
b. Suppose that you estimate the regression Yi = d0 + d1X1i + ui using only the data on new students. Show that d1 is the class size effect for new students—that is, that d1 = E(YiX1i = 1, X2i = 1) – E(YiX1i = 0, X2i = 1). Explain why dn1 is an unbiased estimator of d1.
c. Consider the regression for both returning and new students,
Yi = b0 + b1X1i + b2X2i + b3(X1i * X2i) + ui. Use the conditional mean independence assumption E(ui X1i, X2i) = E(ui X2i) to show
that b1 = g1, b1 + b3 = d1, and b3 = d1 – g1 (the difference in the class size effects).
d. Suppose that you estimate the interaction regression in (c) using the combined data and that E(ui X1i, X2i) = E(ui X2i). Show that bn1 and bn3 are unbiased but that bn2 is in general biased.
Empirical Exercises
(Only two empirical exercises for this chapter are given in the text, but you can find more on the text website http://www.pearsonhighered.com/stock_watson/.)
E8.1 Lead is toxic, particularly for young children, and for this reason govern- ment regulations severely restrict the amount of lead in our environment. But this was not always the case. In the early part of the 20th century, the underground water pipes in many U.S. cities contained lead, and lead from these pipes leached into drinking water. In this exercise you will investigate the effect of these lead water pipes on infant mortality. On the text website http://www.pearsonhighered.com/stock_watson/, you will find the data file Lead_Mortality, which contains data on infant mortality, type of water pipes (lead or non-lead), water acidity (pH), and several demographic variables
for 172 U.S. cities in 1900.5 A detailed description is given in Lead_Mortality_ Description, also available on the website.
a. b.
Compute the average infant mortality rate (Inf ) for cities with lead pipes and for cities with non-lead pipes. Is there a statistically signifi- cant difference in the averages?
The amount of lead leached from lead pipes depends on the chemis- try of the water running through the pipes. The more acidic the water (that is, the lower its pH), the more lead is leached. Run a regression of Inf on Lead, pH, and the interaction term Lead * pH.
i. The regression includes four coefficients (the intercept and the three coefficients multiplying the regressors). Explain what each coefficient measures.
ii. Plot the estimated regression function relating Inf to pH for
Lead = 0 and for Lead = 1. Describe the differences in the regression functions and relate these differences to the coefficients you discussed in (i).
iii. Does Lead have a statistically significant effect on infant mortality? Explain.
iv. Does the effect of Lead on infant mortality depend on pH? Is this dependence statistically significant?
v. What is the average value of pH in the sample? At this pH level, what is the estimated effect of Lead on infant mortality? What
is the standard deviation of pH? Suppose that the pH level is one standard deviation lower than the average level of pH in the sample; what is the estimated effect of Lead on infant mortality? What if pH is one standard deviation higher than the average value?
vi. Construct a 95% confidence interval for the effect of Lead on infant mortality when pH = 6.5.
The analysis in (b) may suffer from omitted variable bias because it neglects factors that affect infant mortality and that might potentially be correlated with Lead and pH. Investigate this concern, using the other variables in the data set.
c.
Empirical Exercises 307
5These data were provided by Professor Karen Clay of Carnegie Mellon University and were used in her paper with Werner Troesken and Michael Haines, “Lead and Mortality,” The Review of Economics and Statistics, 2014, 96(3).
308 ChaPteR 8 Nonlinear Regression Functions
E8.2 On the text website http://www.pearsonhighered.com/stock_watson/ you will find a data file CPS12, which contains data for full-time, full-year workers, ages 25–34, with a high school diploma or B.A./B.S. as their high- est degree. A detailed description is given in CPS12_Description, also available on the website. (These are the same data as in CPS92_12, used in Empirical Exercise 3.1, but are limited to the year 2012.) In this exercise, you will investigate the relationship between a worker’s age and earnings. (Generally, older workers have more job experience, leading to higher productivity and higher earnings.)
a. Run a regression of average hourly earnings (AHE) on age (Age), gender (Female), and education (Bachelor). If Age increases from
25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?
b. Run a regression of the logarithm of average hourly earnings, ln(AHE), on Age, Female, and Bachelor. If Age increases from 25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?
c. Run a regression of the logarithm of average hourly earnings, ln(AHE), on ln(Age), Female, and Bachelor. If Age increases from 25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?
d. Run a regression of the logarithm of average hourly earnings, ln(AHE), on Age, Age2, Female, and Bachelor. If Age increases from 25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?
e. Do you prefer the regression in (c) to the regression in (b)? Explain.
f. Do you prefer the regression in (d) to the regression in (b)? Explain.
g. Do you prefer the regression in (d) to the regression in (c)? Explain.
h. Plot the regression relation between Age and ln(AHE) from (b), (c), and (d) for males with a high school diploma. Describe the similari- ties and differences between the estimated regression functions. Would your answer change if you plotted the regression function for females with college degrees?
i. Run a regression of ln(AHE) on Age, Age2, Female, Bachelor, and the interaction term Female * Bachelor. What does the coef- ficient on the interaction term measure? Alexis is a 30-year-old female with a bachelor’s degree. What does the regression predict
appenDix
Regression Functions That Are Nonlinear in the Parameters 309
for her value of ln(AHE)? Jane is a 30-year-old female with a high school degree. What does the regression predict for her value of ln(AHE)? What is the predicted difference between Alexis’s and Jane’s earnings? Bob is a 30-year-old male with a bachelor’s degree. What does the regression predict for his value of ln(AHE)? Jim is a 30-year-old male with a high school degree. What does the regres- sion predict for his value of ln(AHE)? What is the predicted differ- ence between Bob’s and Jim’s earnings?
j. Is the effect of Age on earnings different for men than for women? Specify and estimate a regression that you can use to answer this question.
k. Is the effect of Age on earnings different for high school graduates than for college graduates? Specify and estimate a regression that you can use to answer this question.
l. After running all these regressions (and any others that you want to run), summarize the effect of age on earnings for young workers.
8.1
Regression Functions That Are Nonlinear in the Parameters
The nonlinear regression functions considered in Sections 8.2 and 8.3 are nonlinear func- tions of the X’s but are linear functions of the unknown parameters. Because they are linear in the unknown parameters, those parameters can be estimated by OLS after defin- ing new regressors that are nonlinear transformations of the original X’s. This family of nonlinear regression functions is both rich and convenient to use. In some applications, however, economic reasoning leads to regression functions that are not linear in the param- eters. Although such regression functions cannot be estimated by OLS, they can be esti- mated using an extension of OLS called nonlinear least squares.
Functions That Are Nonlinear in the Parameters
We begin with two examples of functions that are nonlinear in the parameters. We then provide a general formulation.
Logisticcurve. Supposethatyouarestudyingthemarketpenetrationofatechnology,such as the adoption of database management software in different industries. The dependent variable is the fraction of firms in the industry that have adopted the software, a single
310 ChaPteR 8 Nonlinear Regression Functions
Figure 8.12
two Functions that are Nonlinear in their Parameters
YY
b0
0
1
(a)
0
A logistic curve (b) A negative exponential growth curve
XX
Part (a) plots the logistic function of Equation (8.38), which has predicted values that lie between 0 and 1. Part (b) plots the negative exponential growth function of Equation (8.39), which has a slope that is always positive and decreases as X increases, and an asymptote at b0 as X tends to infinity.
independent variable X describes an industry characteristic, and you have data on n indus- tries. The dependent variable is between 0 (no adopters) and 1 (100% adoption). Because a linear regression model could produce predicted values less than 0 or greater than 1, it makes sense to use instead a function that produces predicted values between 0 and 1.
The logistic function smoothly increases from a minimum of 0 to a maximum of 1. The logistic regression model with a single X is
Yi = 1 + ui. (8.38) 1 + e-(b0 + b1Xi)
The logistic function with a single X is graphed in Figure 8.12a. As can be seen in the graph, the logistic function has an elongated “S” shape. For small values of X, the value of the function is nearly 0 and the slope is flat; the curve is steeper for moderate values of X; and for large values of X, the function approaches 1 and the slope is flat again.
Negative exponential growth. The functions used in Section 8.2 to model the relation between test scores and income have some deficiencies. For example, the polynomial mod- els can produce a negative slope for some values of income, which is implausible. The logarithmic specification has a positive slope for all values of income; however, as income gets very large, the predicted values increase without bound, so for some incomes the pre- dicted value for a district will exceed the maximum possible score on the test.
The negative exponential growth model provides a nonlinear specification that has a positive slope for all values of income, has a slope that is greatest at low values of income
Regression Functions That Are Nonlinear in the Parameters 311 and decreases as income rises, and has an upper bound (that is, an asymptote as income
increases to infinity). The negative exponential growth regression model is
Yi = b031 – e-b1(Xi – b2)4 + ui. (8.39)
The negative exponential growth function is graphed in Figure 8.12b. The slope is steep for low values of X, but as X increases, it reaches an asymptote of b0.
Generalfunctionsthatarenonlinearintheparameters. Thelogisticandnegativeexponen- tial growth regression models are special cases of the general nonlinear regression model
Yi = f(X1i, c, Xki; b0, c, bm) + ui, (8.40)
in which there are k independent variables and m + 1 parameters, b0, c, bm. In the mod- els of Sections 8.2 and 8.3, the X’s entered this function nonlinearly, but the parameters entered linearly. In the examples of this appendix, the parameters enter nonlinearly as well. If the parameters are known, then predicted effects can be computed using the method described in Section 8.1. In applications, however, the parameters are unknown and must be estimated from the data. Parameters that enter nonlinearly cannot be estimated by OLS, but they can be estimated by nonlinear least squares.
Nonlinear Least Squares Estimation
Nonlinear least squares is a general method for estimating the unknown parameters of a regression function when those parameters enter the population regression function nonlinearly.
Recall the discussion in Section 5.3 of the OLS estimator of the coefficients of the linear multiple regression model. The OLS estimator minimizes the sum of squared predic-
n
tionmistakesinEquation(5.8), g 3Y – (b + b X + g+ b X )4 .Inprinciple,the
i011i kki2
OLS estimator can be computed by checking many trial values of b0, c, bk and settling
i=1
on the values that minimize the sum of squared mistakes.
This same approach can be used to estimate the parameters of the general nonlinear
regression model in Equation (8.40). Because the regression function is nonlinear in the coefficients, this method is called nonlinear least squares. For a set of trial parameter values b0, b1, c, bm construct the sum of squared prediction mistakes:
an 3Yi – f(X1i, c, Xki, b1, c, bm)42. (8.41) i=1
The nonlinear least squares estimators of b0, b1, c, bm are the values of b0, b1, c, bm that minimize the sum of squared prediction mistakes in Equation (8.41).
312 ChaPteR 8 Nonlinear Regression Functions
Figure 8.13 the Negative exponential Growth and Linear-Log Regression Functions
The negative exponential growth regression function [Equation (8.42)] and the linear-log regression function [Equation (8.18)] both capture the nonlinear relation between test scores and district income. One difference between the two functions is that the negative exponential growth model has
an asymptote as Income increases to infinity, but the linear-log regression function does not.
600 0
In linear regression, a relatively simple formula expresses the OLS estimator as a function of the data. Unfortunately, no such general formula exists for nonlinear least squares, so the nonlinear least squares estimator must be found numerically using a computer. Regression software incorporates algorithms for solving the nonlinear least squares minimization problem, which simplifies the task of computing the nonlinear least squares estimator in practice.
Under general conditions on the function f and the X’s, the nonlinear least squares estima- tor shares two key properties with the OLS estimator in the linear regression model: It is con- sistent, and it is normally distributed in large samples. In regression software that supports nonlinear least squares estimation, the output typically reports standard errors for the esti- mated parameters. As a consequence, inference concerning the parameters can proceed as usual; in particular, t-statistics can be constructed using the general approach in Key Concept 5.1, and a 95% confidence interval can be constructed as the estimated coefficient, plus or minus 1.96 standard errors. Just as in linear regression, the error term in the nonlinear regres- sion model can be heteroskedastic, so heteroskedasticity-robust standard errors should be used.
Application to the Test Score–Income Relation
A negative exponential growth model, fit to district income (X) and test scores (Y), has the desirable features of a slope that is always positive [if b1 in Equation (8.39) is positive] and an asymptote of b0 as income increases to infinity. The result of estimating b0, b1, and b2 in
Test score
700
650
Linear-log regression
Negative exponential growth regression
20 40 60
District income
Slopes and Elasticities for Nonlinear Regression Functions 313 n
Equation (8.39) using the California test score data yields b0 = 703.2 (heteroskedasticity- nn
robust standard error = 4.44), b1 = 0.0552 (SE = 0.0068), and b2 = -34.0 (SE = 4.48). Thus the estimated nonlinear regression function (with standard errors reported below the parameter estimates) is
TestScore = 703.231 – e-0.0552(Income+34.0)4. (4.44) (0.0068) (4.48)
(8.42)
This estimated regression function is plotted in Figure 8.13, along with the logarithmic regression function and a scatterplot of the data. The two specifications are, in this case, quite similar. One difference is that the negative exponential growth curve flattens out at the highest levels of income, consistent with having an asymptote.
8.2
appenDix
Slopes and Elasticities for Nonlinear Regression Functions
This appendix uses calculus to evaluate slopes and elasticities of nonlinear regression func- tions with continuous regressors. We focus on the case of Section 8.2, in which there is a single X. This approach extends to multiple X’s, using partial derivatives.
Consider the nonlinear regression model, Yi = f(Xi) + ui, with E(uiXi) = 0. The
derivative of f, that is, df(X)>dX . For the polynomial regression function in Equation X=x
slope of the population regression function, f(X), evaluated at the point X = x, is the
0122rraa-1
(8.9), f(X) = b + b X + b X + g+ bX anddX >dX = aX foranyconstanta,so
df(X)>dX = b + 2b x + g+ rb x .Theestimatedslopeatxisdf(X)>dX = X=x 1 2 rr-1 n X=x
n1 n2 nr r-1 n1 n2
b +2bx+g+rbx .ThestandarderroroftheestimatedslopeisSE(b +2bx+
n r-1
g + rbr x ); for a given value of x, this is the standard error of a weighted sum of regression
coefficients, which can be computed using the methods of Section 7.3 and Equation (8.8). The elasticity of Y with respect to X is the percentage change in Y for a given percent- age change in X. Formally, this definition applies in the limit that the percentage change in X goes to zero, so the slope appearing in the definition in Equation (8.22) is replaced by
the derivative and the elasticity is
elasticityofYwithrespecttoX = dY * X = dlnY. dX Y d ln X
314 Chapter 8 Nonlinear Regression Functions
In a regression model, Y depends both on X and on the error term u. Because u is random, it is conventional to evaluate the elasticity as the percentage change not of Y but of the predicted component of Y—that is, the percentage change in E(Y X). Accordingly, the elasticity of E(Y X) with respect to X is
dE(YX) * X = d lnE(YX). dX E(YX) dlnX
The elasticities for the linear model and for the three logarithmic models summarized in Key Concept 8.2 are given in the table below.
Case
linear
linear-log
log-linear
log-log
Population Regression Model
Y=b0 +b1X+u
Y = b0 + b1ln(X) + u
ln(Y) = b0 + b1X + u
ln(Y) = b0 + b1ln(X) + u
Elasticity of E(Y|X ) with Respect to X
b1X
b0 + b1X
b1
b0 + b1ln(X)
b1X b1
The log-log specification has a constant elasticity, but in the other three specifications, the elasticity depends on X.
log model, E(YX) = b + b ln(X). Because dln(X)>dX = 1>X, applying the chain rule 01
We now derive the expressions for the linear-log and log-linear models. For the linear-
yields dE(YX)>dX = b >X. Thus the elasticity is dE(YX)>dX * X>E(YX) = 1
(b >X) * X>[b + b ln(X)] = b >[b + b ln(X)], as is given in the table. For the log-linear 101101
model, it is conventional to make the additional assumption that u and X are independently
distributed, so the expression for E(YX) given following Equation (8.25) becomes
the additional assumption that u and X are independent. Thus dE(YX)>dX = ce b b0+b1X 1 b0+b1X 1
E(YX) = ceb0+b1X, where c = E(eu) is a constant that does not depend on X because of b0+b1X 1
and the elasticity is dE(YX)>dX * X>E(YX) = ce b * X>(ce ) = b X. The derivations for the linear and log-log models are left as Exercise 8.11.
CHAPTER
9
Assessing Studies Based on Multiple Regression
The preceding five chapters explain how to use multiple regression to analyze the relationship among variables in a data set. In this chapter, we step back and ask, What makes a study that uses multiple regression reliable or unreliable? We focus on statistical studies that have the objective of estimating the causal effect of a change in some independent variable, such as class size, on a dependent variable, such as test scores. For such studies, when will multiple regression provide a useful estimate of the causal effect, and, just as importantly, when will it fail to do so?
To answer these questions, this chapter presents a framework for assessing statistical studies in general, whether or not they use regression analysis. This framework relies on the concepts of internal and external validity. A study is internally valid if its statistical inferences about causal effects are valid for the population and setting studied; it is externally valid if its inferences can be generalized to other populations and settings. In Sections 9.1 and 9.2, we discuss internal and external validity, list a variety of possible threats to internal and external validity, and discuss how to identify those threats in practice. The discussion in Sections 9.1 and 9.2 focuses on the estimation of causal effects from observational data. Section 9.3 discusses a different use of regression models—forecasting—and provides an introduction to the threats to the validity of forecasts made using regression models.
As an illustration of the framework of internal and external validity, in Section 9.4 we assess the internal and external validity of the study of the effect on test scores of cutting the student–teacher ratio presented in Chapters 4 through 8.
9.1
Internal and External Validity
The concepts of internal and external validity, defined in Key Concept 9.1, pro- vide a framework for evaluating whether a statistical or econometric study is use- ful for answering a specific question of interest.
Internal and external validity distinguish between the population and setting studied and the population and setting to which the results are generalized. The population studied is the population of entities—people, companies, school dis- tricts, and so forth—from which the sample was drawn. The population to which
315
316 CHAPTER 9 Assessing Studies Based on Multiple Regression
Internal and External Validity
9.1
KEY CONCEPT
A statistical analysis is said to have internal validity if the statistical inferences about causal effects are valid for the population being studied. The analysis is said to have external validity if its inferences and conclusions can be generalized from the population and setting studied to other populations and settings.
the results are generalized, or the population of interest, is the population of enti- ties to which the causal inferences from the study are to be applied. For example, a high school (grades 9 through 12) principal might want to generalize our findings on class sizes and test scores in California elementary school districts (the popula- tion studied) to the population of high schools (the population of interest).
By “setting,” we mean the institutional, legal, social, and economic environ- ment. For example, it would be important to know whether the findings of a laboratory experiment assessing methods for growing organic tomatoes could be generalized to the field, that is, whether the organic methods that work in the set- ting of a laboratory also work in the setting of the real world. We provide other examples of differences in populations and settings later in this section.
Threats to Internal Validity
Internal validity has two components. First, the estimator of the causal effect should be unbiased and consistent. For example, if bnSTR is the OLS estimator of the effect on test scores of a unit change in the student–teacher ratio in a certain regression, then bnSTR should be an unbiased and consistent estimator of the true population causal effect of a change in the student–teacher ratio, bSTR.
Second, hypothesis tests should have the desired significance level (the actual rejection rate of the test under the null hypothesis should equal its desired sig- nificance level), and confidence intervals should have the desired confidence level. For example, if a confidence interval is constructed as bnSTR { 1.96 SE(bnSTR), this confidence interval should contain the true population causal effect, bSTR, with probability 95% over repeated samples.
In regression analysis, causal effects are estimated using the estimated regres- sion function, and hypothesis tests are performed using the estimated regression coefficients and their standard errors. Accordingly, in a study based on OLS regression, the requirements for internal validity are that the OLS estimator is unbiased and consistent, and that standard errors are computed in a way that
makes confidence intervals have the desired confidence level. For various reasons these requirements might not be met, and these reasons constitute threats to internal validity. These threats lead to failures of one or more of the least squares assumptions in Key Concept 6.4. For example, one threat that we have discussed at length is omitted variable bias; it leads to correlation between one or more regressors and the error term, which violates the first least squares assumption. If data are available on the omitted variable or on an adequate control variable, then this threat can be avoided by including that variable as an additional regressor.
Section 9.2 provides a detailed discussion of the various threats to internal validity in multiple regression analysis and suggests how to mitigate them.
Threats to External Validity
Potential threats to external validity arise from differences between the popula- tion and setting studied and the population and setting of interest.
Differences in populations. Differences between the population studied and the population of interest can pose a threat to external validity. For example, laboratory studies of the toxic effects of chemicals typically use animal populations like mice (the population studied), but the results are used to write health and safety regulations for human populations (the population of interest). Whether mice and men differ suffi- ciently to threaten the external validity of such studies is a matter of debate.
More generally, the true causal effect might not be the same in the population studied and the population of interest. This could be because the population was chosen in a way that makes it different from the population of interest, because of differences in characteristics of the populations, because of geographical differ- ences, or because the study is out of date.
Differences in settings. Even if the population being studied and the population of interest are identical, it might not be possible to generalize the study results if the settings differ. For example, a study of the effect on college binge drinking of an antidrinking advertising campaign might not generalize to another identical group of college students if the legal penalties for drinking at the two colleges differ. In this case, the legal setting in which the study was conducted differs from the legal setting to which its results are applied.
More generally, examples of differences in settings include differences in the institutional environment (public universities versus religious universities), differ- ences in laws (differences in legal penalties), or differences in the physical environ- ment (tailgate-party binge drinking in southern California versus Fairbanks, Alaska).
9.1 Internal and External Validity 317
318 CHAPTER 9
Assessing Studies Based on Multiple Regression
Application to test scores and the student–teacher ratio. Chapters 7 and 8 reported statistically significant, but substantively small, estimated improvements in test scores resulting from reducing the student–teacher ratio. This analysis was based on test results for California school districts. Suppose for the moment that these results are internally valid. To what other populations and settings of interest could this finding be generalized?
The closer the population and setting of the study are to those of interest, the stronger the case for external validity. For example, college students and college instruction are very different from elementary school students and instruction, so it is implausible that the effect of reducing class sizes estimated using the California elementary school district data would generalize to colleges. On the other hand, ele- mentary school students, curriculum, and organization are broadly similar through- out the United States, so it is plausible that the California results might generalize to performance on standardized tests in other U.S. elementary school districts.
How to assess the external validity of a study. External validity must be judged using specific knowledge of the populations and settings studied and those of interest. Important differences between the two will cast doubt on the external validity of the study.
Sometimes there are two or more studies on different but related populations. If so, the external validity of both studies can be checked by comparing their results. For example, in Section 9.4 we analyze test score and class size data for elementary school districts in Massachusetts and compare the Massachusetts and California results. In general, similar findings in two or more studies bolster claims to external validity, while differences in their findings that are not readily explained cast doubt on their external validity.1
Howtodesignanexternallyvalidstudy. Becausethreatstoexternalvaliditystem from a lack of comparability of populations and settings, these threats are best minimized at the early stages of a study, before the data are collected. Study design is beyond the scope of this textbook, and the interested reader is referred to Shadish, Cook, and Campbell (2002).
1A comparison of many related studies on the same topic is called a meta-analysis. The discussion in the box “The Mozart Effect: Omitted Variable Bias?” in Chapter 6 is based on a meta-analysis, for example. Performing a meta-analysis of many studies has its own challenges. How do you sort the good studies from the bad? How do you compare studies when the dependent variables differ? Should you put more weight on studies with larger samples? A discussion of meta-analysis and its challenges goes beyond the scope of this textbook. The interested reader is referred to Hedges and Olkin (1985) and Cooper and Hedges (1994).
9.2 Threats to Internal Validity of Multiple Regression Analysis 319 Threats to Internal Validity
9.2
of Multiple Regression Analysis
Studies based on regression analysis are internally valid if the estimated regression coefficients are unbiased and consistent, and if their standard errors yield confidence intervals with the desired confidence level. This section surveys five reasons why the OLS estimator of the multiple regression coefficients might be biased, even in large samples: omitted variables, misspecification of the functional form of the regression function, imprecise measurement of the independent variables (“errors in variables”), sample selection, and simultaneous causality. All five sources of bias arise because the regressor is correlated with the error term in the population regression, violating the first least squares assumption in Key Concept 6.4. For each, we discuss what can be done to reduce this bias. The section concludes with a discussion of circumstances that lead to inconsistent standard errors and what can be done about it.
Omitted Variable Bias
Recall that omitted variable bias arises when a variable that both determines Y and is correlated with one or more of the included regressors is omitted from the regression. This bias persists even in large samples, so the OLS estimator is inconsistent. How best to minimize omitted variable bias depends on whether or not variables that adequately control for the potential omitted variable are available.
Solutions to omitted variable bias when the variable is observed or there are ade- quate control variables. If you have data on the omitted variable, then you can include that variable in a multiple regression, thereby addressing the problem. Alternatively, if you have data on one or more control variables and if these con- trol variables are adequate in the sense that they lead to conditional mean inde- pendence [Equation (7.20)], then including those control variables eliminates the potential bias in the coefficient on the variable of interest.
Adding a variable to a regression has both costs and benefits. On the one hand, omitting the variable could result in omitted variable bias. On the other hand, including the variable when it does not belong (that is, when its population regression coefficient is zero) reduces the precision of the estimators of the other regression coefficients. In other words, the decision whether to include a variable involves a trade-off between bias and variance of the coefficient of interest. In practice, there are four steps that can help you decide whether to include a vari- able or set of variables in a regression.
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The first step is to identify the key coefficient or coefficients of interest in your regression. In the test score regressions, this is the coefficient on the student– teacher ratio, because the question originally posed concerns the effect on test scores of reducing the student–teacher ratio.
The second step is to ask yourself: What are the most likely sources of important omitted variable bias in this regression? Answering this question requires applying economic theory and expert knowledge, and should occur before you actually run any regressions; because this step is done before analyzing the data, it is referred to as a priori (“before the fact”) reasoning. In the test score example, this step entails identifying those determinants of test scores that, if ignored, could bias our estimator of the class size effect. The results of this step are a base regression specification, the starting point for your empirical regression analysis, and a list of additional “ques- tionable” variables that might help to mitigate possible omitted variable bias.
The third step is to augment your base specification with the additional ques- tionable control variables identified in the second step. If the coefficients on the additional control variables are statistically significant or if the estimated coeffi- cients of interest change appreciably when the additional variables are included, then they should remain in the specification and you should modify your base specification. If not, then these variables can be excluded from the regression.
The fourth step is to present an accurate summary of your results in tabular form. This provides “full disclosure” to a potential skeptic, who can then draw his or her own conclusions. Table 7.1 and 8.3 are examples of this strategy. For exam- ple, in Table 8.3, we could have presented only the regression in column (7), because that regression summarizes the relevant effects and nonlinearities in the other regressions in that table. Presenting the other regressions, however, permits the skeptical reader to draw his or her own conclusions.
These steps are summarized in Key Concept 9.2.
Solutions to omitted variable bias when adequate control variables are not available. Adding an omitted variable to a regression is not an option if you do not have data on that variable and if there are no adequate control variables. Still, there are three other ways to solve omitted variable bias. Each of these three solu- tions circumvents omitted variable bias through the use of different types of data.
The first solution is to use data in which the same observational unit is observed at different points in time. For example, test score and related data might be collected for the same districts in 1995 and again in 2000. Data in this form are called panel data. As explained in Chapter 10, panel data make it possible to control for unobserved omitted variables as long as those omitted variables do not change over time.
9.2 Threats to Internal Validity of Multiple Regression Analysis 321
Omitted Variable Bias: Should I Include More Variables in My Regression?
KEY CONCEPT
9.2
If you include another variable in your multiple regression, you will eliminate the possibility of omitted variable bias from excluding that variable, but the variance of the estimator of the coefficients of interest can increase. Here are some guide- lines to help you decide whether to include an additional variable:
1. Be specific about the coefficient or coefficients of interest.
2. Use a priori reasoning to identify the most important potential sources of omitted variable bias, leading to a base specification and some “questionable” variables.
3. Test whether additional “questionable” control variables have nonzero coef- ficients.
4. Provide “full disclosure” representative tabulations of your results so that others can see the effect of including the questionable variables on the coefficient(s) of interest. Do your results change if you include a question- able control variable?
The second solution is to use instrumental variables regression. This method relies on a new variable, called an instrumental variable. Instrumental variables regression is discussed in Chapter 12.
The third solution is to use a study design in which the effect of interest (for example, the effect of reducing class size on student achievement) is studied using a randomized controlled experiment. Randomized controlled experiments are discussed in Chapter 13.
Misspecification of the Functional Form
of the Regression Function
If the true population regression function is nonlinear but the estimated regression is linear, then this functional form misspecification makes the OLS estimator biased. This bias is a type of omitted variable bias, in which the omitted variables are the terms that reflect the missing nonlinear aspects of the regression function. For example, if the population regression function is a quadratic polynomial, then a regression that omits the square of the independent variable would suffer from omitted variable bias. Bias arising from functional form misspecification is summarized in Key Concept 9.3.
322 CHAPTER 9 Assessing Studies Based on Multiple Regression
Functional Form Misspecification
9.3
KEY CONCEPT
Functional form misspecification arises when the functional form of the estimated regression function differs from the functional form of the population regression function. If the functional form is misspecified, then the estimator of the partial effect of a change in one of the variables will, in general, be biased. Functional form misspecification often can be detected by plotting the data and the estimated regression function, and it can be corrected by using a different functional form.
Solutionstofunctionalformmisspecification. Whenthedependentvariableiscon- tinuous (like test scores), this problem of potential nonlinearity can be solved using the methods of Chapter 8. If, however, the dependent variable is discrete or binary (for example, Yi equals 1 if the ith person attended college and equals 0 otherwise), things are more complicated. Regression with a discrete dependent variable is discussed in Chapter 11.
Measurement Error and Errors-in-Variables Bias
Suppose that in our regression of test scores against the student–teacher ratio we had inadvertently mixed up our data so that we ended up regressing test scores for fifth graders on the student–teacher ratio for tenth graders in that district. Although the student–teacher ratio for elementary school students and tenth graders might be correlated, they are not the same, so this mix-up would lead to bias in the estimated coefficient. This is an example of errors-in-variables bias because its source is an error in the measurement of the independent variable. This bias persists even in very large samples, so the OLS estimator is inconsistent if there is measurement error.
There are many possible sources of measurement error. If the data are collected through a survey, a respondent might give the wrong answer. For example, one ques- tion in the Current Population Survey involves last year’s earnings. A respondent might not know his or her exact earnings or might misstate the amount for some other reason. If instead the data are obtained from computerized administrative records, there might have been typographical errors when the data were first entered.
To see that errors in variables can result in correlation between the regressor and the error term, suppose that there is a single regressor Xi (say, actual earn- ings) but that Xi is measured imprecisely by X∼i (the respondent’s stated earnings). Because X∼i, not Xi, is observed, the regression equation actually estimated is the
9.2 Threats to Internal Validity of Multiple Regression Analysis 323 one based on X∼i. Written in terms of the imprecisely measured variable X∼i, the
population regression equation Yi = b0 + b1Xi + ui is
Y = b + b X∼ + 3 b ( X – X∼ ) + u 4
i01i1iii
= b0 + b1X∼i + vi, (9.1)
where vi = b1(Xi – X∼i) + ui. Thus the population regression equation written in terms of X∼i has an error term that contains the measurement error, the difference between X∼i and Xi. If this difference is correlated with the measured value X∼i, then the regressor X∼i will be correlated with the error term and bn1 will be biased and inconsistent.
The precise size and direction of the bias in bn1 depend on the correlation ∼∼
between Xi and the measurement error, Xi – Xi. This correlation depends, in turn, on the specific nature of the measurement error.
For example, suppose that the measured value, X∼i , equals the actual, unmea- sured value, Xi, plus a purely random component, wi, which has mean zero and variance s2w. Because the error is purely random, we might suppose that wi is uncorrelated with Xi and with the regression error ui. This assumption consti- tutes the classical measurement error model in which X∼i = Xi + wi, where corr(wi, Xi) = 0 and corr(wi, ui) = 0. Under the classical measurement error model, a bit of algebra2 shows that bn1 has the probability limit
np sX2
b1¡s2 +s2b1. (9.2)
Xw
That is, if the measurement error has the effect of simply adding a random element
to the actual value of the independent variable, then bn1 is inconsistent. Because the
ratio s2X islessthan1,bn willbebiasedtoward0,eveninlargesamples.Inthe s2X + s2w 1
extreme case that the measurement error is so large that essentially no information about Xi remains, the ratio of the variances in the final expression in Equation (9.2) is 0 and bn1 converges in probability to 0. In the other extreme, when there is no measurement error, s2w = 0, so bn1 ¡p b1.
A different model of measurement error supposes that the respondent makes his or her best estimate of the true value. In this “best guess” model the response
2 U n d e r t h i s m e a s u r e m e n t e r r o r a s s u m p t i o n , v = b ( X – X∼ ) + u = – b w + u , c o v ( X , u ) = 0 , a n d i1iii1iiii
cov(X∼,w)=cov(X +w,w)=s2,socov(X∼,v)=-bcov(X∼,w)+cov(X∼,u)=-bs2.Thus, 1iiiiwii11iii1w
from Equation (6.1), b ¡p
b – b s >(s + s ) =
[s2>(s2 +s2)]b. XXw1
n22222n222
b – b s >s∼. Now s∼ = s + s , so b ¡p 111wXXXw111wXw
324 CHAPTER 9 Assessing Studies Based on Multiple Regression
Errors-in-Variables Bias
9.4
KEY CONCEPT
Errors-in-variables bias in the OLS estimator arises when an independent vari- able is measured imprecisely. This bias depends on the nature of the measurement error and persists even if the sample size is large. If the measured variable equals the actual value plus a mean-zero, independently distributed measurement error, then the OLS estimator in a regression with a single right-hand variable is biased toward zero, and its probability limit is given in Equation (9.2).
X∼i is modeled as the conditional mean of Xi, given the information available to the r e s p o n d e n t . B e c a u s e X∼ i i s t h e b e s t g u e s s , t h e m e a s u r e m e n t e r r o r X∼ i – X i i s u n c o r – related with the response X∼i (if the measurement error were correlated with X∼i , then that would be useful information for predicting Xi, in which case X∼i would not have been the best guess of Xi). That is, E3(X∼i – Xi)X∼i4 = 0, and if the respondent’s information is uncorrelated with ui, then X∼i is uncorrelated with the error term vi. Thus, in this “best guess” measurement error model, bn1 is consistent, but because var(vi) 7 var(ui), the variance of bn1 is larger than it would be absent measurement error. The “best guess” measurement error model is examined further in Exercise 9.12.
Problems created by measurement error can be even more complicated if there is intentional misreporting. For example, suppose that survey respondents provide the income reported on their income taxes but intentionally underreport their true tax- able income by not including cash payments. If, for example, all respondents report only 90% of income, then X∼i = 0.90Xi and bn1 will be biased up by 10%.
Although the result in Equation (9.2) is specific to classical measurement error, it illustrates the more general proposition that if the independent variable is measured imprecisely, then the OLS estimator is biased, even in large samples. Errors-in-variables bias is summarized in Key Concept 9.4.
Measurement error in Y. The effect of measurement error in Y is different from
measurement error in X. If Y has classical measurement error, then this measure-
ment error increases the variance of the regression and of bn1 but does not induce
bias in bn . To see this, suppose that measured Y is ∼Y, which equals true Y plus 1iii
random measurement error wi. Then the regression model estimated is ∼Yi = b0 + b1Xi + vi,wherevi = wi + ui.Ifwi istrulyrandom,thenwi andXi are independently distributed so that E(wi 0 Xi) = 0, in which case E(vi 0 Xi) = 0, so bn1 is unbiased. However, because var(vi) 7 var(ui), the variance of bn1 is larger
9.2 Threats to Internal Validity of Multiple Regression Analysis 325
than it would be without measurement error. In the test score/class size example, suppose that test scores have purely random grading errors that are independent of the regressors; then the classical measurement error model of this paragraph applies to ∼Yi, and bn1 is unbiased. More generally, measurement error in Y that has condi- tional mean zero given the regressors will not induce bias in the OLS coefficients.
Solutionstoerrors-in-variablesbias. Thebestwaytosolvetheerrors-in-variables problem is to get an accurate measure of X. If this is impossible, however, econo- metric methods can be used to mitigate errors-in-variables bias.
One such method is instrumental variables regression. It relies on having another variable (the “instrumental” variable) that is correlated with the actual value Xi but is uncorrelated with the measurement error. This method is studied in Chapter 12.
A second method is to develop a mathematical model of the measurement error and, if possible, to use the resulting formulas to adjust the estimates. For example, if a researcher believes that the classical measurement error model applies and if she knows or can estimate the ratio s2w>s2X, then she can use Equation (9.2) to compute an estimator of b1 that corrects for the downward bias. Because this approach requires specialized knowledge about the nature of the measurement error, the details typically are specific to a given data set and its measurement prob- lems and we shall not pursue this approach further in this textbook.
Missing Data and Sample Selection
Missing data are a common feature of economic data sets. Whether missing data pose a threat to internal validity depends on why the data are missing. We con- sider three cases: when the data are missing completely at random, when the data are missing based on X, and when the data are missing because of a selection process that is related to Y beyond depending on X.
When the data are missing completely at random—that is, for random rea- sons unrelated to the values of X or Y—the effect is to reduce the sample size but not introduce bias. For example, suppose that you conduct a simple random sam- ple of 100 classmates, then randomly lose half the records. It would be as if you had never surveyed those individuals. You would be left with a simple random sample of 50 classmates, so randomly losing the records does not introduce bias.
When the data are missing based on the value of a regressor, the effect also is to reduce the sample size but not introduce bias. For example, in the class size/ student–teacher ratio example, suppose that we used only the districts in which the student–teacher ratio exceeds 20. Although we would not be able to draw conclusions about what happens when STR … 20, this would not introduce bias into our analysis of the class size effect for districts with STR 7 20.
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Sample Selection Bias
9.5
KEY CONCEPT
Sample selection bias arises when a selection process influences the availability of data and that process is related to the dependent variable, beyond depending on the regressors. Sample selection induces correlation between one or more regres- sors and the error term, leading to bias and inconsistency of the OLS estimator.
In contrast to the first two cases, if the data are missing because of a selection process that is related to the value of the dependent variable (Y), beyond depend- ing on the regressors (X), then this selection process can introduce correlation between the error term and the regressors. The resulting bias in the OLS estimator is called sample selection bias. An example of sample selection bias in polling was given in the box “Landon Wins!” in Section 3.1. In that example, the sample selec- tion method (randomly selecting phone numbers of automobile owners) was related to the dependent variable (who the individual supported for president in 1936), because in 1936 car owners with phones were more likely to be Republicans. The sample selection problem can be cast either as a consequence of nonrandom sampling or as a missing data problem. In the 1936 polling example, the sample was a random sample of car owners with phones, not a random sample of voters. Alter- natively, this example can be cast as a missing data problem by imagining a random sample of voters, but with missing data for those without cars and phones. The mechanism by which the data are missing is related to the dependent variable, leading to sample selection bias.
The box “Do Stock Mutual Funds Outperform the Market?” provides an example of sample selection bias in financial economics. Sample selection bias is summarized in Key Concept 9.5.3
Solutions to selection bias. The methods we have discussed so far cannot elimi- nate sample selection bias. The methods for estimating models with sample selec- tion are beyond the scope of this book. Those methods build on the techniques introduced in Chapter 11, where further references are provided.
Simultaneous Causality
So far, we have assumed that causality runs from the regressors to the dependent variable (X causes Y). But what if causality also runs from the dependent variable to
3Exercise 18.16 provides a mathematical treatment of the three missing data cases discussed here.
9.2 Threats to Internal Validity of Multiple Regression Analysis 327 Do Stock Mutual Funds Outperform the Market?
Stock mutual funds are investment vehicles that hold a portfolio of stocks. By purchasing shares in a mutual fund, a small investor can hold a broadly diversified portfolio without the hassle and expense (transaction cost) of buying and selling shares in indi- vidual companies. Some mutual funds simply track the market (for example, by holding the stocks in the S&P 500), whereas others are actively managed by full-time professionals whose job is to make the fund earn a better return than the overall market—and competitors’ funds. But do these actively managed funds achieve this goal? Do some mutual funds con- sistently beat other funds and the market?
One way to answer these questions is to compare future returns on mutual funds that had high returns over the past year to future returns on other funds and on the market as a whole. In making such compari- sons, financial economists know that it is important to select the sample of mutual funds carefully. This task is not as straightforward as it seems, however. Some databases include historical data on funds currently available for purchase, but this approach means that the dogs—the most poorly performing funds—are omitted from the data set because they went out of business or were merged into other funds. For this
reason, a study using data on historical performance of currently available funds is subject to sample selec- tion bias: The sample is selected based on the value of the dependent variable, returns, because funds with the lowest returns are eliminated. The mean return of all funds (including the defunct) over a ten-year period will be less than the mean return of those funds still in existence at the end of those ten years, so a study of only the latter funds will overstate per- formance. Financial economists refer to this selection bias as “survivorship bias” because only the better funds survive to be in the data set.
When financial econometricians correct for survivorship bias by incorporating data on defunct funds, the results do not paint a flattering portrait of mutual fund managers. Corrected for survivor- ship bias, the econometric evidence indicates that actively managed stock mutual funds do not out- perform the market on average and that past good performance does not predict future good perfor- mance. For further reading on mutual funds and sur- vivorship bias, see Malkiel (2012), Chapter 11 and Carhart (1997). The problem of survivorship bias also arises in evaluating hedge fund performance; for further reading, see Aggarwal and Jorion (2010).
one or more regressors (Y causes X)? If so, causality runs “backward” as well as for- ward; that is, there is simultaneous causality. If there is simultaneous causality, an OLS regression picks up both effects, so the OLS estimator is biased and inconsistent.
For example, our study of test scores focused on the effect on test scores of reducing the student–teacher ratio, so causality is presumed to run from the student–teacher ratio to test scores. Suppose, however, that a government initiative subsidized hiring teachers in school districts with poor test scores. If so, causality would run in both directions: For the usual educational reasons low student–teacher ratios would arguably lead to high test scores, but because of the government program low test scores would lead to low student–teacher ratios.
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Simultaneous causality leads to correlation between the regressor and the error term. In the test score example, suppose that there is an omitted factor that leads to poor test scores; because of the government program, this factor that produces low scores in turn results in a low student–teacher ratio. Thus a negative error term in the population regression of test scores on the student–teacher ratio reduces test scores, but because of the government program it also leads to a decrease in the student–teacher ratio. In other words, the student–teacher ratio is positively correlated with the error term in the population regression. This in turn leads to simultaneous causality bias and inconsistency of the OLS estimator.
This correlation between the error term and the regressor can be made math- ematically precise by introducing an additional equation that describes the reverse causal link. For convenience, consider just the two variables X and Y and ignore other possible regressors. Accordingly, there are two equations, one in which X causes Y and one in which Y causes X:
Yi =b0 +b1Xi +ui and (9.3) Xi =g0 +g1Yi +vi. (9.4)
Equation (9.3) is the familiar one in which b1 is the effect on Y of a change in X, where u represents other factors. Equation (9.4) represents the reverse causal effect of Y on X. In the test score problem, Equation (9.3) represents the educa- tional effect of class size on test scores, while Equation (9.4) represents the reverse causal effect of test scores on class size induced by the government program.
Simultaneous causality leads to correlation between Xi and the error term ui in Equation (9.3). To see this, imagine that ui is positive, which increases Yi. How- ever, this higher value of Yi affects the value of Xi through the second of these equations, and if g1 is positive, a high value of Yi will lead to a high value of Xi. Thus, if g1 is positive, Xi and ui will be positively correlated.4
Because this can be expressed mathematically using two simultaneous equa- tions, the simultaneous causality bias is sometimes called simultaneous equations bias. Simultaneous causality bias is summarized in Key Concept 9.6.
Solutions to simultaneous causality bias. There are two ways to mitigate simul- taneous causality bias. One is to use instrumental variables regression, the topic
4To show this mathematically, note that Equation (9.4) implies that cov(Xi, ui) = cov(g0 + g1Yi + vi, ui) = g1cov(Yi, ui) + cov(vi, ui). Assuming that cov(vi, ui) = 0 by Equation (9.3) this in turn implies that cov(Xi, ui) = g1cov(Yi, ui) = g1cov(b0 + b1Xi + ui, ui) = g1b1cov(Xi, ui) + g1s2u. Solving for cov(Xi, ui) then yields the result cov(Xi, ui) = g1s2u>(1 – g1b1).
9.2 Threats to Internal Validity of Multiple Regression Analysis 329
Simultaneous Causality Bias
KEY CONCEPT
9.6
Simultaneous causality bias, also called simultaneous equations bias, arises in a regression of Y on X when, in addition to the causal link of interest from X to Y, there is a causal link from Y to X. This reverse causality makes X correlated with the error term in the population regression of interest.
of Chapter 12. The second is to design and implement a randomized controlled experiment in which the reverse causality channel is nullified, and such experi- ments are discussed in Chapter 13.
Sources of Inconsistency of OLS Standard Errors
Inconsistent standard errors pose a different threat to internal validity. Even if the OLS estimator is consistent and the sample is large, inconsistent standard errors will produce hypothesis tests with size that differs from the desired significance level and “95%” confidence intervals that fail to include the true value in 95% of repeated samples.
There are two main reasons for inconsistent standard errors: improperly han- dled heteroskedasticity and correlation of the error term across observations.
Heteroskedasticity. As discussed in Section 5.4, for historical reasons some regression software report homoskedasticity-only standard errors. If, however, the regression error is heteroskedastic, those standard errors are not a reliable basis for hypothesis tests and confidence intervals. The solution to this problem is to use heteroskedasticity-robust standard errors and to construct F-statistics using a heteroskedasticity-robust variance estimator. Heteroskedasticity-robust standard errors are provided as an option in modern software packages.
Correlation of the error term across observations. In some settings, the population regression error can be correlated across observations. This will not happen if the data are obtained by sampling at random from the population because the random- ness of the sampling process ensures that the errors are independently distributed from one observation to the next. Sometimes, however, sampling is only partially random. The most common circumstance is when the data are repeated observations on the same entity over time, such as the same school district for different years. If the omitted variables that constitute the regression error are persistent (like district demographics), “serial” correlation is induced in the regression error over time.
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Threats to the Internal Validity of a Multiple Regression Study
9.7
KEY CONCEPT
There are five primary threats to the internal validity of a multiple regression study:
1. Omitted variables
2. Functional form misspecification
3. Errors in variables (measurement error in the regressors)
4. Sample selection
5. Simultaneous causality
Each of these, if present, results in failure of the first least squares assumption, E(ui 0 X1i, c, Xki) ≠ 0, which in turn means that the OLS estimator is biased and inconsistent.
Incorrect calculation of the standard errors also poses a threat to internal validity. Homoskedasticity-only standard errors are invalid if heteroskedasticity is present. If the variables are not independent across observations, as can arise in panel and time series data, then a further adjustment to the standard error formula is needed to obtain valid standard errors.
Applying this list of threats to a multiple regression study provides a system- atic way to assess the internal validity of that study.
Serial correlation in the error term can arise in panel data (data on multiple districts for multiple years) and in time series data (data on a single district for multiple years). Another situation in which the error term can be correlated across observa- tions is when sampling is based on a geographical unit. If there are omitted vari- ables that reflect geographic influences, these omitted variables could result in
correlation of the regression errors for adjacent observations.
Correlation of the regression error across observations does not make the OLS
estimator biased or inconsistent, but it does violate the second least squares assump- tion in Key Concept 6.4. The consequence is that the OLS standard errors—both homoskedasticity-only and heteroskedasticity-robust—are incorrect in the sense that they do not produce confidence intervals with the desired confidence level.
In many cases, this problem can be fixed by using an alternative formula for standard errors. We provide formulas for computing standard errors that are robust to both heteroskedasticity and serial correlation in Chapter 10 (regression with panel data) and in Chapter 15 (regression with time series data).
Key Concept 9.7 summarizes the threats to internal validity of a multiple regression study.
9.3 Internal and External Validity When the Regression Is Used for Forecasting 331 Internal and External Validity When
9.3
the Regression Is Used for Forecasting
Up to now, the discussion of multiple regression analysis has focused on the esti- mation of causal effects. Regression models can be used for other purposes, how- ever, including forecasting. When regression models are used for forecasting, concerns about external validity are very important, but concerns about unbiased estimation of causal effects are not.
Using Regression Models for Forecasting
Chapter 4 began by considering the problem of a school superintendent who wants to know how much test scores would increase if she reduced class sizes in her school district; that is, the superintendent wants to know the causal effect on test scores of a change in class size. Accordingly, Chapters 4 through 8 focused on using regression analysis to estimate causal effects using observational data.
Now consider a different problem. A parent moving to a metropolitan area plans to choose where to live based in part on the quality of the local schools. The parent would like to know how different school districts perform on standardized tests. Suppose, however, that test score data are not available (perhaps they are confidential) but data on class sizes are. In this situation, the parent must guess at how well the different districts perform on standardized tests based on a limited amount of information. That is, the parent’s problem is to forecast average test scores in a given district based on information related to test scores—in particular, class size.
How can the parent make this forecast? Recall the regression of test scores on the student–teacher ratio (STR) from Chapter 4:
TestScore = 698.9 – 2.28 * STR. (9.5)
We concluded that this regression is not useful for the superintendent: The OLS estimator of the slope is biased because of omitted variables such as the compo- sition of the student body and students’ other learning opportunities outside school.
Nevertheless, Equation (9.5) could be useful to the parent trying to choose a home. To be sure, class size is not the only determinant of test performance, but from the parent’s perspective what matters is whether it is a reliable predictor of test performance. The parent interested in forecasting test scores does not care whether the coefficient in Equation (9.5) estimates the causal effect on test scores of class size. Rather, the parent simply wants the regression to explain much of
332 CHAPTER 9
Assessing Studies Based on Multiple Regression
9.4
Example: Test Scores and Class Size
The framework of internal and external validity helps us to take a critical look at what we have learned—and what we have not—from our analysis of the Califor- nia test score data.
External Validity
Whether the California analysis can be generalized—that is, whether it is exter- nally valid—depends on the population and setting to which the generalization is made. Here, we consider whether the results can be generalized to performance on other standardized tests in other elementary public school districts in the United States.
Section 9.1 noted that having more than one study on the same topic provides an opportunity to assess the external validity of both studies by comparing their results. In the case of test scores and class size, other comparable data sets are, in
the variation in test scores across districts and to be stable—that is, to apply to the districts to which the parent is considering moving. Although omitted variable bias renders Equation (9.5) useless for answering the causal question, it still can be useful for forecasting purposes.
More generally, regression models can produce reliable forecasts, even if their coefficients have no causal interpretation. This recognition underlies much of the use of regression models for forecasting.
Assessing the Validity of Regression
Models for Forecasting
Because the superintendent’s problem and the parent’s problem are conceptually very different, the requirements for the validity of the regression are different for their respective problems. To obtain credible estimates of causal effects, we must address the threats to internal validity summarized in Key Concept 9.7.
In contrast, if we are to obtain reliable forecasts, the estimated regression must have good explanatory power, its coefficients must be estimated precisely, and it must be stable in the sense that the regression estimated on one set of data can be reliably used to make forecasts using other data. When a regression model is used for forecasting, a paramount concern is that the model is externally valid in the sense that it is stable and quantitatively applicable to the circumstance in which the forecast is made. In Part IV, we return to the problem of assessing the validity of a regression model for forecasting future values of time series data.
TABLE 9.1
fact, available. In this section, we examine a different data set, based on standard- ized test results for fourth graders in 220 public school districts in Massachusetts in 1998. Both the Massachusetts and California tests are broad measures of student knowledge and academic skills, although the details differ. Similarly, the organization of classroom instruction is broadly similar at the elementary school level in the two states (as it is in most U.S. elementary school districts), although aspects of elementary school funding and curriculum differ. Thus finding similar results about the effect of the student–teacher ratio on test performance in the California and Massachusetts data would be evidence of external validity of the findings in California. Conversely, finding different results in the two states would raise questions about the internal or external validity of at least one of the studies.
Comparison of the California and Massachusetts data. Like the California data, the Massachusetts data are at the school district level. The definitions of the vari- ables in the Massachusetts data set are the same as those in the California data set, or nearly so. More information on the Massachusetts data set, including defi- nitions of the variables, is given in Appendix 9.1.
Table 9.1 presents summary statistics for the California and Massachusetts samples. The average test score is higher in Massachusetts, but the test is differ- ent, so a direct comparison of scores is not appropriate. The average student– teacher ratio is higher in California (19.6 versus 17.3). Average district income is 20% higher in Massachusetts, but the standard deviation of income is greater in
Summary Statistics for California and Massachusetts Test Score Data Sets
9.4 Example: Test Scores and Class Size 333
Test scores Student–teacher ratio
% English learners
% Receiving lunch subsidy Average district income ($) Number of observations Year
Average
654.1 19.6
15.8%
44.7% $15,317
California
Standard Deviation
19.1 1.9
18.3% 27.1% $7226
Average
709.8 17.3
1.1% 15.3%
$18,747
Massachusetts
Standard Deviation
15.1 2.3
2.9% 15.1% $5808
420 1999
220 1998
334
CHAPTER 9
Assessing Studies Based on Multiple Regression
FIGURE 9.1
California; that is, there is a greater spread in average district incomes in Califor- nia than in Massachusetts. The average percentage of students still learning Eng- lish and the average percentage of students receiving subsidized lunches are both much higher in the California than in the Massachusetts districts.
Testscoresandaveragedistrictincome. Tosavespace,wedonotpresentscatterplots of all the Massachusetts data. Because it was a focus in Chapter 8, however, it is interesting to examine the relationship between test scores and average district income in Massachusetts. This scatterplot is presented in Figure 9.1. The general pattern of this scatterplot is similar to that in Figure 8.2 for the California data: The relationship between income and test scores appears to be steep for low values of income and flatter for high values. Evidently, the linear regression plot- ted in the figure misses this apparent nonlinearity. Cubic and logarithmic regres- sion functions are also plotted in Figure 9.1. The cubic regression function has a slightly higher R 2 than the logarithmic specification (0.486 versus 0.455). Compar- ing Figures 8.7 and 9.1 shows that the general pattern of nonlinearity found in the California income and test score data is also present in the Massachusetts data.
Test Scores vs. Income for Massachusetts Data
The estimated linear regression function does not capture the nonlinear relation between income and test scores in the Massachusetts data.
The estimated linear-log and cubic regression functions are similar
for district incomes between $13,000 and $30,000, the region containing most of the observations.
Test score
780
760
740
720
700
680
660
640
620
0 10 20 30 40 50
District income (thousands of dollars)
Linear regression Linear-log regression
Cubic regression
TABLE 9.2
The precise functional forms that best describe this nonlinearity differ, however, with the cubic specification fitting best in Massachusetts but the linear-log speci- fication fitting best in California.
Multiple regression results. Regression results for the Massachusetts data are presented in Table 9.2. The first regression, reported in column (1) in the table, has only the student–teacher ratio as a regressor. The slope is negative (-1.72),
Multiple Regression Estimates of the Student–Teacher Ratio and Test Scores: Data from Massachusetts
Dependent variable: average combined English, math, and science test score in the school district, fourth grade; 220 observations.
Regressor
Student–teacher ratio (STR)
STR2
STR3
% English learners
% English learners 7 median? (Binary, HiEL)
HiEL * STR
% Eligible for free lunch
District income (logarithm)
District income
District income2
District income3
Intercept
(1) (2)
-1.72** -0.69* (0.50) (0.27)
(3)
– 0.64* (0.27)
– 0.437 (0.303)
– 0.582** (0.097)
– 3.07 (2.35)
0.164 (0.085)
– 0.0022* (0.0010)
744.0** (21.3)
(4)
12.4 (14.0)
– 0.680 (0.737)
0.011 (0.013)
– 0.434 (0.300)
– 0.587** (0.104)
– 3.38 (2.49)
0.174 (0.089)
– 0.0023* (0.0010)
665.5** (81.3)
(5)
– 1.02** (0.37)
-12.6 (9.8)
0.80 (0.56)
– 0.709** (0.091)
– 3.87* (2.49)
0.184* (0.090)
– 0.0023* (0.0010)
759.9** (23.2)
(6)
– 0.67* (0.27)
9.4 Example: Test Scores and Class Size 335
-0.411 (0.306)
-0.521** (0.077)
16.53** (3.15)
– 0.653** (0.72)
– 3.22 (2.31)
0.165 (0.085)
– 0.0022* (0.0010)
747.4** (20.3)
739.6** (8.6)
682.4** (11.5)
(Table 9.2 continued)
336 CHAPTER 9 Assessing Studies Based on Multiple Regression
(Table 9.2 continued)
F-Statistics and p-Values Testing Exclusion of Groups of Variables
(1)
(2)
(3)
7.74 (6 0.001)
8.61
0.676
(4)
(5) (6)
All STR variables and interac- tions = 0
STR2, STR3 = 0 Income2, Income3 HiEL, HiEL * STR SER
R2
2.86 4.01
14.64
0.063
8.69
0.670
(0.038)
0.45 (0.641)
7.75 (6 0.001)
8.63
0.675
(0.020)
5.85 6.55 (0.003) (0.002)
1.58 (0.208)
8.62 8.64
0.675 0.674
These regressions were estimated using the data on Massachusetts elementary school districts described in Appendix 9.1. Stan- dard errors are given in parentheses under the coefficients, and p-values are given in parentheses under the F-statistics. Individual coefficients are statistically significant at the *5% level or **1% level.
and the hypothesis that the coefficient is zero can be rejected at the 1% signifi- cance level (t = -1.72>0.50 = -3.44).
The remaining columns report the results of including additional variables that control for student characteristics and of introducing nonlinearities into the esti- mated regression function. Controlling for the percentage of English learners, the percentage of students eligible for a free lunch, and the average district income reduces the estimated coefficient on the student–teacher ratio by 60%, from – 1.72 in regression (1) to -0.69 in regression (2) and -0.64 in regression (3).
Comparing the R2’s of regressions (2) and (3) indicates that the cubic speci- fication (3) provides a better model of the relationship between test scores and income than does the logarithmic specification (2), even holding constant the student–teacher ratio. There is no statistically significant evidence of a nonlinear relationship between test scores and the student–teacher ratio: The F-statistic in regression (4) testing whether the population coefficients on STR2 and STR3 are zero has a p-value of 0.641. Similarly, there is no evidence that a reduction in the student–teacher ratio has a different effect in districts with many English learners than with few [the t-statistic on HiEL * STR in regression (5) is 0.80>0.56 = 1.43]. Finally, regression (6) shows that the estimated coefficient on the student–teacher
9.4 Example: Test Scores and Class Size 337
ratio does not change substantially when the percentage of English learners [which is insignificant in regression (3)] is excluded. In short, the results in regres- sion (3) are not sensitive to the changes in functional form and specification consid- ered in regressions (4) through (6) in Table 9.2. Therefore, we adopt regression (3) as our base estimate of the effect in test scores of a change in the student–teacher ratio based on the Massachusetts data.
Comparison of Massachusetts and California results. For the California data, we found the following:
1. Adding variables that control for student background characteristics reduced the coefficient on the student–teacher ratio from -2.28 [Table 7.1, regres- sion (1)] to – 0.73 [Table 8.3, regression (2)], a reduction of 68%.
2. The hypothesis that the true coefficient on the student–teacher ratio is zero was rejected at the 1% significance level, even after adding variables that control for student background and district economic characteristics.
3. The effect of cutting the student–teacher ratio did not depend in an impor- tant way on the percentage of English learners in the district.
4. There is some evidence that the relationship between test scores and the student–teacher ratio is nonlinear.
Do we find the same things in Massachusetts? For findings (1), (2), and (3), the
answer is yes. Including the additional control variables reduces the coefficient on the student–teacher ratio from -1.72 [Table 9.2, regression (1)] to -0.69 [Table 9.2, regression (2)], a reduction of 60%. The coefficients on the student–teacher ratio remain significant after adding the control variables. Those coefficients are only significant at the 5% level in the Massachusetts data, whereas they are sig- nificant at the 1% level in the California data. However, there are nearly twice as many observations in the California data, so it is not surprising that the California estimates are more precise. As in the California data, there is no statistically sig- nificant evidence in the Massachusetts data of an interaction between the student– teacher ratio and the binary variable indicating a large percentage of English learners in the district.
Finding (4), however, does not hold up in the Massachusetts data: The hypothesis that the relationship between the student–teacher ratio and test scores is linear cannot be rejected at the 5% significance level when tested against a cubic specification.
Because the two standardized tests are different, the coefficients themselves cannot be compared directly: One point on the Massachusetts test is not the same as one point on the California test. If, however, the test scores are put into the
338 CHAPTER 9 Assessing Studies Based on Multiple Regression
TABLE 9.3
Student–Teacher Ratios and Test Scores: Comparing the Estimates from California and Massachusetts
Estimated Effect of Two Fewer Students per Teacher, In Units of:
California
Linear: Table 9.3(2) Cubic: Table 9.3(7)
Reduce STR from 20 to 18
Cubic: Table 9.3(7)
Reduce STR from 22 to 20
Massachusetts
Linear: Table 9.2(3)
Standard errors are given in parentheses.
OLS Estimate
βnSTR -0.73
(0.26) —
—
-0.64 (0.27)
Standard Deviation of Test Scores Across Districts
19.1 19.1 19.1
15.1
Points on the Test
1.46 (0.52)
2.93 (0.70)
1.90 (0.69)
1.28 (0.54)
Standard Deviations
0.076 (0.027)
0.153 (0.037)
0.099 (0.036)
0.085 (0.036)
same units, then the estimated class size effects can be compared. One way to do this is to transform the test scores by standardizing them: Subtract the sample average and divide by the standard deviation so that they have a mean of 0 and a variance of 1. The slope coefficients in the regression with the standardized test score equal the slope coefficients in the original regression divided by the standard deviation of the test. Thus the coefficient on the student–teacher ratio divided by the standard deviation of test scores can be compared across the two data sets.
This comparison is undertaken in Table 9.3. The first column reports the OLS estimates of the coefficient on the student–teacher ratio in a regression with the percentage of English learners, the percentage of students eligible for a free lunch, and the average district income included as control variables. The second column reports the standard deviation of the test scores across districts. The final two columns report the estimated effect on test scores of reducing the student–teacher ratio by two students per teacher (our superintendent’s proposal), first in the units of the test and second in standard deviation units. For the linear specification, the OLS coefficient estimate using California data is -0.73, so cutting the student– teacherratiobytwoisestimatedtoincreasedistricttestscoresby-0.73 * (-2) = 1.46
9.4 Example: Test Scores and Class Size 339
points. Because the standard deviation of test scores is 19.1 points, this corresponds to 1.46>19.1 = 0.076 standard deviation of the distribution of test scores across districts. The standard error of this estimate is 0.26 * 2>19.1 = 0.027. The esti- mated effects for the nonlinear models and their standard errors were computed using the method described in Section 8.1.
Based on the linear model using California data, a reduction of two students per teacher is estimated to increase test scores by 0.076 standard deviation unit, with a standard error of 0.027. The nonlinear models for California data suggest a somewhat larger effect, with the specific effect depending on the initial student– teacher ratio. Based on the Massachusetts data, this estimated effect is 0.085 stan- dard deviation unit, with a standard error of 0.036.
These estimates are essentially the same. Cutting the student–teacher ratio is predicted to raise test scores, but the predicted improvement is small. In the Cal- ifornia data, for example, the difference in test scores between the median district and a district at the 75th percentile is 12.2 test score points (Table 4.1), or 0.64 (= 12.2>19.1) standard deviations. The estimated effect from the linear model is just over one-tenth this size; in other words, according to this estimate, cutting the student teacher–ratio by two would move a district only one-tenth of the way from the median to the 75th percentile of the distribution of test scores across districts. Reducing the student–teacher ratio by two is a large change for a district, but the estimated benefits shown in Table 9.3, while nonzero, are small.
This analysis of Massachusetts data suggests that the California results are externally valid, at least when generalized to elementary school districts else- where in the United States.
Internal Validity
The similarity of the results for California and Massachusetts does not ensure their internal validity. Section 9.2 listed five possible threats to internal validity that could induce bias in the estimated effect on test scores on class size. We con- sider these threats in turn.
Omitted variables. The multiple regressions reported in this and previous chapters control for a student characteristic (the percentage of English learners), a family eco- nomic characteristic (the percentage of students receiving a subsidized lunch), and a broader measure of the affluence of the district (average district income).
If these control variables are adequate, then for the purpose of regression analysis it is as if the student–teacher ratio is randomly assigned among districts with the same values of these control variables, in which case the conditional
340 CHAPTER 9
Assessing Studies Based on Multiple Regression
mean independence assumption holds. There still could be, however, some omitted factors for which these three variables might not be adequate controls. For example, if the student–teacher ratio is correlated with teacher quality even among districts with the same fraction of immigrants and the same socioeconomic characteristics (perhaps because better teachers are attracted to schools with smaller student– teacher ratios) and if teacher quality affects test scores, then omission of teacher quality could bias the coefficient on the student–teacher ratio. Similarly, among dis- tricts with the same socioeconomic characteristics, districts with a low student– teacher ratio might have families that are more committed to enhancing their children’s learning at home. Such omitted factors could lead to omitted variable bias.
One way to eliminate omitted variable bias, at least in theory, is to conduct an experiment. For example, students could be randomly assigned to different size classes, and their subsequent performance on standardized tests could be compared. Such a study was in fact conducted in Tennessee, and we examine it in Chapter 13.
Functionalform. TheanalysishereandinChapter8exploredavarietyoffunctional forms. We found that some of the possible nonlinearities investigated were not statis- tically significant, while those that were did not substantially alter the estimated effect of reducing the student–teacher ratio. Although further functional form analysis could be carried out, this suggests that the main findings of these studies are unlikely to be sensitive to using different nonlinear regression specifications.
Errorsinvariables. Theaveragestudent–teacherratiointhedistrictisabroadand potentially inaccurate measure of class size. For example, because students move in and out of districts, the student–teacher ratio might not accurately represent the actual class sizes experienced by the students taking the test, which in turn could lead to the estimated class size effect being biased toward zero. Another variable with potential measurement error is average district income. Those data were taken from the 1990 census, while the other data pertain to 1998 (Massachusetts) or 1999 (California). If the economic composition of the district changed substantially over the 1990s, this would be an imprecise measure of the actual average district income.
Selection. TheCaliforniaandtheMassachusettsdatacoverallthepublicelemen- tary school districts in the state that satisfy minimum size restrictions, so there is no reason to believe that sample selection is a problem here.
Simultaneous causality. Simultaneous causality would arise if the performance on standardized tests affected the student–teacher ratio. This could happen, for example, if there is a bureaucratic or political mechanism for increasing the funding
9.4 Example: Test Scores and Class Size 341
of poorly performing schools or districts that in turn resulted in hiring more teach- ers. In Massachusetts, no such mechanism for equalization of school financing was in place during the time of these tests. In California, a series of court cases led to some equalization of funding, but this redistribution of funds was not based on student achievement. Thus in neither Massachusetts nor California does simulta- neous causality appear to be a problem.
Heteroskedasticityandcorrelationoftheerrortermacrossobservations. Allthe results reported here and in earlier chapters use heteroskedastic-robust standard errors, so heteroskedasticity does not threaten internal validity. Correlation of the error term across observations, however, could threaten the consistency of the stan- dard errors because simple random sampling was not used (the sample consists of all elementary school districts in the state). Although there are alternative standard error formulas that could be applied to this situation, the details are complicated and specialized and we leave them to more advanced texts.
Discussion and Implications
The similarity between the Massachusetts and California results suggest that these studies are externally valid, in the sense that the main findings can be generalized to performance on standardized tests at other elementary school districts in the United States.
Some of the most important potential threats to internal validity have been addressed by controlling for student background, family economic background, and district affluence, and by checking for nonlinearities in the regression function. Still, some potential threats to internal validity remain. A leading candidate is omitted variable bias, perhaps arising because the control variables do not capture other characteristics of the school districts or extracurricular learning opportunities.
Based on both the California and the Massachusetts data, we are able to answer the superintendent’s question from Section 4.1: After controlling for fam- ily economic background, student characteristics, and district affluence, and after modeling nonlinearities in the regression function, cutting the student–teacher ratio by two students per teacher is predicted to increase test scores by approxi- mately 0.08 standard deviation of the distribution of test scores across districts. This effect is statistically significant, but it is quite small. This small estimated effect is in line with the results of the many studies that have investigated the effects on test scores of class size reductions.5
5If you are interested in learning more about the relationship between class size and test scores, see the reviews by Ehrenberg et al. (2001a, 2001b).
342 CHAPTER 9
Assessing Studies Based on Multiple Regression
9.5
Conclusion
The concepts of internal and external validity provide a framework for assessing what has been learned from an econometric study.
A study based on multiple regression is internally valid if the estimated coeffi- cients are unbiased and consistent, and if standard errors are consistent. Threats to the internal validity of such a study include omitted variables, misspecification of functional form (nonlinearities), imprecise measurement of the independent vari- ables (errors in variables), sample selection, and simultaneous causality. Each of these introduces correlation between the regressor and the error term, which in turn makes OLS estimators biased and inconsistent. If the errors are correlated across observations, as they can be with time series data, or if they are heteroskedastic but the standard errors are computed using the homoskedasticity-only formula, then internal validity is compromised because the standard errors will be inconsistent. These latter problems can be addressed by computing the standard errors properly.
A study using regression analysis, like any statistical study, is externally valid if its findings can be generalized beyond the population and setting studied. Some- times it can help to compare two or more studies on the same topic. Whether or not there are two or more such studies, however, assessing external validity requires making judgments about the similarities of the population and setting studied and the population and setting to which the results are being generalized.
The next two parts of this textbook develop ways to address threats to internal validity that cannot be mitigated by multiple regression analysis alone. Part III extends the multiple regression model in ways designed to mitigate all five sources of potential bias in the OLS estimator; Part III also discusses a different approach to obtaining internal validity, randomized controlled experiments. Part IV devel- ops methods for analyzing time series data and for using time series data to esti- mate so-called dynamic causal effects, which are causal effects that vary over time.
The superintendent can now use this estimate to help her decide whether to reduce class sizes. In making this decision, she will need to weigh the costs of the proposed reduction against the benefits. The costs include teacher salaries and expenses for additional classrooms. The benefits include improved academic per- formance, which we have measured by performance on standardized tests, but there are other potential benefits that we have not studied, including lower drop- out rates and enhanced future earnings. The estimated effect of the proposal on standardized test performance is one important input into her calculation of costs and benefits.
Summary
1. Statistical studies are evaluated by asking whether the analysis is internally and externally valid. A study is internally valid if the statistical inferences about causal effects are valid for the population being studied. A study is externally valid if its inferences and conclusions can be generalized from the population and setting studied to other populations and settings.
2. In regression estimation of causal effects, there are two types of threats to internal validity. First, OLS estimators are biased and inconsistent if the regressors and error terms are correlated. Second, confidence intervals and hypothesis tests are not valid when the standard errors are incorrect.
3. Regressors and error terms may be correlated when there are omitted variables, an incorrect functional form is used, one or more of the regressors are measured with error, the sample is chosen nonrandomly from the population, or there is simultaneous causality between the regressors and dependent variables.
4. Standard errors are incorrect when the errors are heteroskedastic and the computer software uses the homoskedasticity-only standard errors, or when the error term is correlated across different observations.
5. When regression models are used solely for forecasting, it is not necessary for the regression coefficients to be unbiased estimates of causal effects. It is critical, however, that the regression model be externally valid for the forecasting application at hand.
Key Terms
population studied (315)
internal validity (316)
external validity (316)
population of interest (316) functional form misspecification (321)
errors-in-variables bias (322)
classical measurement error model (323) sample selection bias (326) simultaneous causality (327) simultaneous equations bias (328)
Key Terms 343
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344 CHAPTER 9
Assessing Studies Based on Multiple Regression
Review the Concepts
9.1 What is the difference between internal and external validity? Between the population studied and the population of interest?
9.2 Key Concept 9.2 describes the problem of variable selection in terms of a trade-off between bias and variance. What is this trade-off? Why could including an additional regressor decrease bias? Increase variance?
9.3 Economic variables are often measured with error. Does this mean that regression analysis is unreliable? Explain.
9.4 Suppose that a state offered voluntary standardized tests to all its third graders and that these data were used in a study of class size on student per- formance. Explain how sample selection bias might invalidate the results.
9.5 A researcher estimates the effect on crime rates of spending on police by using city-level data. Explain how simultaneous causality might invalidate the results.
9.6 A researcher estimates a regression using two different software packages. The first uses the homoskedasticity-only formula for standard errors. The second uses the heteroskedasticity-robust formula. The standard errors are very different. Which should the researcher use? Why?
Exercises
9.1 Suppose that you have just read a careful statistical study of the effect of advertising on the demand for cigarettes. Using data from New York during the 1970s, the study concluded that advertising on buses and sub- ways was more effective than print advertising. Use the concept of external validity to determine if these results are likely to apply to Boston in the 1970s, Los Angeles in the 1970s, and New York in 2014.
9.2 Consider the one-variable regression model Yi = b0 + b1Xi + ui and sup- pose that it satisfies the least squares assumptions in Key Concept 4.3. Suppose that Yi is measured with error, so the data are ∼Yi = Yi + wi, where wi is the measurement error, which is i.i.d. and independent of Yi and Xi. Consider the population regression ∼Yi = b0 + b1Xi + vi, where vi is the regression error, using the mismeasured dependent variable, ∼Yi.
a. Show that vi = ui + wi.
b. Show that the regression ∼Yi = b0 + b1Xi + vi satisfies the least squares assumptions in Key Concept 4.3. (Assume that wi is inde- pendent of Yj and Xj for all values of i and j and has a finite fourth moment.)
c. Are the OLS estimators consistent?
d. Can confidence intervals be constructed in the usual way?
e. Evaluate these statements: “Measurement error in the X’s is a serious problem. Measurement error in Y is not.”
9.3 Labor economists studying the determinants of women’s earnings dis- covered a puzzling empirical result. Using randomly selected employed women, they regressed earnings on the women’s number of children and a set of control variables (age, education, occupation, and so forth). They found that women with more children had higher wages, controlling for these other factors. Explain how sample selection might be the cause of this result. (Hint: Notice that women who do not work outside the home are missing from the sample.) [This empirical puzzle motivated James Heckman’s research on sample selection that led to his 2000 Nobel Prize in Economics. See Heckman (1974).]
9.4 Using the regressions shown in column (2) of Table 9.3 and column (2) of Table 9.2, construct a table like Table 9.3 to compare the estimated effects of a 10% increase in district income on test scores in California and Mas- sachusetts.
9.5 The demand for a commodity is given by Q = b0 + b1P + u, where Q denotes quantity, P denotes price, and u denotes factors other than price that determine demand. Supply for the commodity is given by Q = g0 + g1P + v, where v denotes factors other than price that deter- mine supply. Suppose that u and v both have a mean of zero, have variances s2u and s2v, and are mutually uncorrelated.
a. Solve the two simultaneous equations to show how Q and P depend on u and v.
b. Derive the means of P and Q.
c. Derive the variance of P, the variance of Q, and the covariance
between Q and P.
d. A random sample of observations of (Qi, Pi) is collected, and Qi is regressed on Pi. (That is, Qi is the regressand, and Pi is the regressor.) Suppose that the sample is very large.
Exercises 345
346 CHAPTER 9
Assessing Studies Based on Multiple Regression
i. Use your answers to (b) and (c) to derive values of the regression coefficients. [Hint: Use Equations (4.7) and (4.8).]
ii. A researcher uses the slope of this regression as an estimate of the slope of the demand function (b1). Is the estimated slope too large or too small? (Hint: Remember that demand curves slope down and supply curves slope up.)
9.6 Suppose that n = 100 i.i.d. observations for (Yi, Xi) yield the following regression results:
Yn = 32.1 + 66.8X, SER = 15.1, R2 = 0.81. (15.1) (12.2)
Another researcher is interested in the same regression, but he makes an error when he enters the data into his regression program: He enters each observation twice, so he has 200 observations (with observation 1 entered twice, observation 2 entered twice, and so forth).
a. Using these 200 observations, what results will be produced by his regression program? (Hint: Write the “incorrect” values of the sam- ple means, variances, and covariances of Y and X as functions of the “correct” values. Use these to determine the regression statistics.)
Yn = ____ + ____X, SER = ____, R2 = ____. (____) (____)
b. Which (if any) of the internal validity conditions are violated?
9.7 Are the following statements true or false? Explain your answer.
a. “An ordinary least squares regression of Y onto X will not be inter- nally valid if X is correlated with the error term.”
b. “Each of the five primary threats to internal validity implies that X is correlated with the error term.”
9.8 Would the regression in Equation (9.5) be useful for predicting test scores in a school district in Massachusetts? Why or why not?
9.9 Consider the linear regression of TestScore on Income shown in Figure 8.2 and the nonlinear regression in Equation (8.18). Would either of these regressions provide a reliable estimate of the effect of income on test scores? Would either of these regressions provide a reliable method for forecasting test scores? Explain.
9.10 Read the box “The Return to Education and the Gender Gap” in Section 8.3. Discuss the internal and external validity of the estimated effect of education on earnings.
9.11 Read the box “The Demand for Economics Journals” in Section 8.3. Dis- cuss the internal and external validity of the estimated effect of price per citation on subscriptions.
9.12 Consider the one-variable regression model Yi = b0 + b1Xi + ui and sup- pose that it satisfies the least squares assumptions in Key Concept 4.3. The regressor Xi is missing, but data on a related variable, Zi, are available, and the value of X is estimated usingX∼ = E(X 0Z). Let w = X∼ – X.
i iiiiii
a. Show that X∼i is the minimum mean square error estimator of Xi using Zi. That is, let Xn i = g(Zi) be some other guess of Xi based on Zi, and show that E3(Xni – Xi)24 Ú E3(X∼i – Xi)24. (Hint: Review Exercise 2.27.)
b. Show that E(w 0 X∼ ) = 0. ii
c. Suppose that E(ui 0 Zi) = 0 and that X∼i is used as the regressor in place of Xi. Show that bn1 is consistent. Is bn0 consistent?
9.13 Assume that the regression model Yi = b0 + b1Xi + ui satisfies the least squares assumptions in Key Concept 4.3 in Section 4.4. You and a friend collect a random sample of 300 observations on Y and X.
a. Your friend reports the he inadvertently scrambled the X observa- tions for 20% of the sample. For these scrambled observations, the value of X does not correspond to Xi for the ith observation; rather,
it corresponds to the value of X for some other observation. In the notation of Section 9.2, the measured value of the regressor, X∼i , is equal to Xi for 80% of the observations, but it is equal to a randomly selected Xj for the remaining 20% of the observations. You regress Yi on X∼i . Show that E(bn1) = 0.8b1.
b. Explain how you could construct an unbiased estimate of b1 using the OLS estimator in (a).
c. Suppose now that your friend tells you that the X’s were scrambled for the first 60 observations but that the remaining 240 observations are correct. You estimate b1 by regressing Y on X, using only the cor- rectly measured 240 observations. Is this estimator of b1 better than the estimator you proposed in (b)? Explain.
Exercises 347
348 CHAPTER 9
Assessing Studies Based on Multiple Regression
Empirical Exercises
(Only two empirical exercises for this chapter are given in the text, but you can find more on the text website, http://www.pearsonhighered.com/stock_ watson/.)
E9.1 Use the data set CPS12, described in Empirical Exercise 8.2, to answer the following questions.
a. Discuss the internal validity of the regressions that you used to answer Empirical Exercise 8.2(l). Include a discussion of possible omitted variable bias, misspecification of the functional form of the regression, errors in variables, sample selection, simultaneous causal- ity, and inconsistency of the OLS standard errors.
b. The data set CPS92_12 described in Empirical Exercise 3.1 includes data from 2012 and 1992. Use these data to investigate the (temporal) external validity of the conclusions that you reached in Empirical Exercise 8.2(l). [Note: Remember to adjust for inflation, as explained in Empirical Exercise 3.1(b).]
E9.2 Use the data set Birthweight_Smoking introduced in Empirical Exercise
5.1 to answer the following questions.
a. In Empirical Exercise 7.1(f), you estimated several regressions and were asked: “What is a reasonable 95% confidence interval for the effect of smoking on birth weight?”
i. In Chapter 8 you learned about nonlinear regressions. Can you think of any nonlinear regressions that can potentially improve your answer to Empirical Exercise E7.1(f)? After estimating these additional regressions, what is a reasonable 95% confidence interval for the effect of smoking on birth weight?
ii. Discuss the internal validity of the regressions you used to con- struct the confidence interval. Include a discussion of possible omitted variable bias, misspecification of the functional form of the regression, errors in variables, sample selection, simultaneous causality, and inconsistency of the OLS standard errors.
b. The data set Birthweight_Smoking includes babies born in Pennsylvania in 1989. Discuss the external validity of your analysis for (i) California in 1989, (ii) Illinois in 2015, and (iii) South Korea in 2014.
APPENDIX
9.1
The Massachusetts Elementary School Testing Data
The Massachusetts data are districtwide averages for public elementary school districts in 1998. The test score is taken from the Massachusetts Comprehensive Assessment System (MCAS) test administered to all fourth graders in Massachusetts public schools in the spring of 1998. The test is sponsored by the Massachusetts Department of Education and is mandatory for all public schools. The data analyzed here are the overall total score, which is the sum of the scores on the English, math, and science portions of the test.
Data on the student–teacher ratio, the percentage of students receiving a subsidized lunch, and the percentage of students still learning English are averages for each elemen- tary school district for the 1997–1998 school year and were obtained from the Massachu- setts Department of Education. Data on average district income were obtained from the 1990 U.S. Census.
The Massachusetts Elementary School Testing Data 349
CHAPTER
10
Regression with Panel Data
Multiple regression is a powerful tool for controlling for the effect of variables on which we have data. If data are not available for some of the variables, however, they cannot be included in the regression and the OLS estimators of the regression coefficients could have omitted variable bias.
This chapter describes a method for controlling for some types of omitted variables without actually observing them. This method requires a specific type of data, called panel data, in which each observational unit, or entity, is observed at two or more time periods. By studying changes in the dependent variable over time, it is possible to eliminate the effect of omitted variables that differ across entities but are constant over time.
The empirical application in this chapter concerns drunk driving: What are the effects of alcohol taxes and drunk driving laws on traffic fatalities? We address this question using data on traffic fatalities, alcohol taxes, drunk driving laws, and related variables for the 48 contiguous U.S. states for each of the seven years from 1982 to 1988. This panel data set lets us control for unobserved variables that differ from one state to the next, such as prevailing cultural attitudes toward drinking and driving, but do not change over time. It also allows us to control for variables that vary through time, like improvements in the safety of new cars, but do not vary across states.
Section 10.1 describes the structure of panel data and introduces the drunk driving data set. Fixed effects regression, the main tool for regression analysis of panel data, is an extension of multiple regression that exploits panel data to control for variables that differ across entities but are constant over time. Fixed effects regression is introduced in Sections 10.2 and 10.3, first for the case of only two time periods and then for multiple time periods. In Section 10.4, these methods are extended to incorporate so-called time fixed effects, which control for unobserved variables that are constant across entities but change over time. Section 10.5 dis- cusses the panel data regression assumptions and standard errors for panel data regression. In Section 10.6, we use these methods to study the effect of alcohol taxes and drunk driving laws on traffic deaths.
350
10.1 Panel Data 351
Notation for Panel Data
KEY CONCEPT
10.1
Panel data consist of observations on the same n entities at two or more time periods T, as is illustrated in Table 1.3. If the data set contains observations on the variables X and Y, then the data are denoted
(Xit,Yit),i = 1,c,nandt = 1,c,T, (10.1)
where the first subscript, i, refers to the entity being observed and the second subscript, t, refers to the date at which it is observed.
10.1
Panel Data
Recall from Section 1.3 that panel data (also called longitudinal data) refers to data for n different entities observed at T different time periods. The state traffic fatality data studied in this chapter are panel data. Those data are for n = 48 enti- ties (states), where each entity is observed in T = 7 time periods (each of the years 1982, . . . , 1988), for a total of 7 * 48 = 336 observations.
When describing cross-sectional data it was useful to use a subscript to denote the entity; for example, Yi referred to the variable Y for the ith entity. When describing panel data, we need some additional notation to keep track of both the entity and the time period. We do so by using two subscripts rather than one: The first, i, refers to the entity, and the second, t, refers to the time period of the obser- vation. Thus Yit denotes the variable Y observed for the ith of n entities in the tth of T periods. This notation is summarized in Key Concept 10.1.
Some additional terminology associated with panel data describes whether some observations are missing. A balanced panel has all its observations; that is, the variables are observed for each entity and each time period. A panel that has some missing data for at least one time period for at least one entity is called an unbalanced panel. The traffic fatality data set has data for all 48 contiguous U.S. states for all seven years, so it is balanced. If, however, some data were missing (for example, if we did not have data on fatalities for some states in 1983), then the data set would be unbalanced. The methods presented in this chapter are described for a balanced panel; however, all these methods can be used with an unbalanced panel, although precisely how to do so in practice depends on the regression software being used.
352 CHAPTER 10
Regression with Panel Data
Example: Traffic Deaths and Alcohol Taxes
There are approximately 40,000 highway traffic fatalities each year in the United States. Approximately one-fourth of fatal crashes involve a driver who was drink- ing, and this fraction rises during peak drinking periods. One study (Levitt and Porter, 2001) estimates that as many as 25% of drivers on the road between 1 a.m. and 3 a.m. have been drinking and that a driver who is legally drunk is at least 13 times as likely to cause a fatal crash as a driver who has not been drinking.
In this chapter, we study how effective various government policies designed to discourage drunk driving actually are in reducing traffic deaths. The panel data set contains variables related to traffic fatalities and alcohol, including the number of traffic fatalities in each state in each year, the type of drunk driving laws in each state in each year, and the tax on beer in each state. The measure of traffic deaths we use is the fatality rate, which is the number of annual traffic deaths per 10,000 people in the population in the state. The measure of alcohol taxes we use is the “real” tax on a case of beer, which is the beer tax, put into 1988 dollars by adjusting for inflation.1 The data are described in more detail in Appendix 10.1.
Figure 10.1a is a scatterplot of the data for 1982 on two of these variables, the fatality rate and the real tax on a case of beer. A point in this scatterplot repre- sents the fatality rate in 1982 and the real beer tax in 1982 for a given state. The OLS regression line obtained by regressing the fatality rate on the real beer tax is also plotted in the figure; the estimated regression line is
FatalityRate = 2.01 + 0.15BeerTax (1982 data). (10.2) (0.15) (0.13)
The coefficient on the real beer tax is positive, but not statistically significant at the 10% level.
Because we have data for more than one year, we can reexamine this relation- ship for another year. This is done in Figure 10.1b, which is the same scatterplot as before except that it uses the data for 1988. The OLS regression line through these data is
FatalityRate = 1.86 + 0.44BeerTax (1988 data). (10.3) (0.11) (0.13)
1To make the taxes comparable over time, they are put into “1988 dollars” using the Consumer Price Index (CPI). For example, because of inflation a tax of $1 in 1982 corresponds to a tax of $1.23 in 1988 dollars.
FIGURE 10.1 The Traffic Fatality Rate and the Tax on Beer
10.1
Panel Data 353
Panel (a) is a scatterplot of traffic fatality rates and the real tax
on a case of beer (in 1988 dollars) for 48 states in 1982. Panel (b) shows the data for 1988. Both plots show a positive relationship between the fatality rate and the real beer tax.
Fatality rate (fatalities per 10,000)
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0 0.5 1.0
(a) 1982 data
Fatality rate (fatalities per 10,000)
4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
0.0 0.5 1.0
(b) 1988 data
FatalityRate = 2.01 + 0.15BeerTax
1.5 2.0
2.5
3.0
Beer tax (dollars per case $1988)
FatalityRate = 1.86 + 0.44BeerTax
1.5 2.0
2.5 3.0
Beer tax (dollars per case $1988)
354
CHAPTER 10
Regression with Panel Data
10.2
Panel Data with Two Time Periods: “Before and After” Comparisons
When data for each state are obtained for T = 2 time periods, it is possible to compare values of the dependent variable in the second period to values in the first period. By focusing on changes in the dependent variable, this “before and after” or “differences” comparison in effect holds constant the unobserved factors that differ from one state to the next but do not change over time within the state.
Let Zi be a variable that determines the fatality rate in the ith state but does not change over time (so the t subscript is omitted). For example, Zi might be the local cultural attitude toward drinking and driving, which changes slowly and thus could be considered to be constant between 1982 and 1988. Accordingly, the pop- ulation linear regression relating Zi and the real beer tax to the fatality rate is
FatalityRateit = b0 + b1BeerTaxit + b2Zi + uit, (10.4) where uit is the error term and i = 1, c, n and t = 1, c, T.
In contrast to the regression using the 1982 data, the coefficient on the real beer tax is statistically significant at the 1% level (the t-statistic is 3.43). Curiously, the estimated coefficients for the 1982 and the 1988 data are positive: Taken liter- ally, higher real beer taxes are associated with more, not fewer, traffic fatalities.
Should we conclude that an increase in the tax on beer leads to more traffic deaths? Not necessarily, because these regressions could have substantial omitted variable bias. Many factors affect the fatality rate, including the quality of the automobiles driven in the state, whether the state highways are in good repair, whether most driving is rural or urban, the density of cars on the road, and whether it is socially acceptable to drink and drive. Any of these factors may be correlated with alcohol taxes, and if they are, they will lead to omitted variable bias. One approach to these potential sources of omitted variable bias would be to collect data on all these variables and add them to the annual cross-sectional regressions in Equations (10.2) and (10.3). Unfortunately, some of these variables, such as the cultural acceptance of drinking and driving, might be very hard or even impossible to measure.
If these factors remain constant over time in a given state, however, then another route is available. Because we have panel data, we can in effect hold these factors constant even though we cannot measure them. To do so, we use OLS regression with fixed effects.
10.2 Panel Data with Two Time Periods: “Before and After” Comparisons 355
Because Zi does not change over time, in the regression model in Equation (10.4) it will not produce any change in the fatality rate between 1982 and 1988. Thus, in this regression model, the influence of Zi can be eliminated by analyzing the change in the fatality rate between the two periods. To see this mathemati- cally, consider Equation (10.4) for each of the two years 1982 and 1988:
FatalityRatei1982 = b0 + b1BeerTaxi1982 + b2Zi + ui1982, (10.5) FatalityRatei1988 = b0 + b1BeerTaxi1988 + b2Zi + ui1988. (10.6)
Subtracting Equation (10.5) from Equation (10.6) eliminates the effect of Zi: FatalityRatei1988 – FatalityRatei1982
= b1(BeerTaxi1988 – BeerTaxi1982) + ui1988 – ui1982. (10.7)
This specification has an intuitive interpretation. Cultural attitudes toward drink- ing and driving affect the level of drunk driving and thus the traffic fatality rate in a state. If, however, they did not change between 1982 and 1988, then they did not produce any change in fatalities in the state. Rather, any changes in traffic fatali- ties over time must have arisen from other sources. In Equation (10.7), these other sources are changes in the tax on beer and changes in the error term (which captures changes in other factors that determine traffic deaths).
Specifying the regression in changes in Equation (10.7) eliminates the effect of the unobserved variables Zi that are constant over time. In other words, analyzing changes in Y and X has the effect of controlling for variables that are constant over time, thereby eliminating this source of omitted variable bias.
Figure 10.2 presents a scatterplot of the change in the fatality rate between 1982 and 1988 against the change in the real beer tax between 1982 and 1988 for the 48 states in our data set. A point in Figure 10.2 represents the change in the fatality rate and the change in the real beer tax between 1982 and 1988 for a given state. The OLS regression line, estimated using these data and plotted in the figure, is
FatalityRate1988 – FatalityRate1982 = -0.072 – 1.04(BeerTax1988 – BeerTax1982). (0.065) (0.36) (10.8)
Including an intercept in Equation (10.8) allows for the possibility that the mean change in the fatality rate, in the absence of a change in the real beer tax, is non- zero. For example, the negative intercept (-0.072) could reflect improvements in auto safety from 1982 to 1988 that reduced the average fatality rate.
356 CHAPTER 10 Regression with Panel Data
FIGURE 10.2 Changes in Fatality Rates and Beer Taxes, 1982–1988
This is a scatterplot
of the change in
the traffic fatality 1.0 rate and the change
in real beer taxes
between 1982 and
1988 for 48 states.
There is a nega-
tive relationship
between changes –0.5 in the fatality rate
and changes in the
beer tax.
FatalityRate1988 – FatalityRate1982 = –0.072 – 1.04(BeerTax1988 – BeerTax1982)
Change in fatality rate (fatalities per 10,000)
0.5 0.0
–1.0 –1.5
–0.4 –0.2
0.0 0.2
0.4 0.6
–0.6
In contrast to the cross-sectional regression results, the estimated effect of a change in the real beer tax is negative, as predicted by economic theory. The hypoth- esis that the population slope coefficient is zero is rejected at the 5% significance level. According to this estimated coefficient, an increase in the real beer tax by $1 per case reduces the traffic fatality rate by 1.04 deaths per 10,000 people. This estimated effect is very large: The average fatality rate is approximately 2 in these data (that is, 2 fatal- ities per year per 10,000 members of the population), so the estimate suggests that traf- fic fatalities can be cut in half merely by increasing the real tax on beer by $1 per case.
By examining changes in the fatality rate over time, the regression in Equa- tion (10.8) controls for fixed factors such as cultural attitudes toward drinking and driving. But there are many factors that influence traffic safety, and if they change over time and are correlated with the real beer tax, then their omission will pro- duce omitted variable bias. In Section 10.5, we undertake a more careful analysis that controls for several such factors, so for now it is best to refrain from drawing any substantive conclusions about the effect of real beer taxes on traffic fatalities.
This “before and after” analysis works when the data are observed in two dif- ferent years. Our data set, however, contains observations for seven different years, and it seems foolish to discard those potentially useful additional data. But the “before and after” method does not apply directly when T 7 2. To analyze all the observations in our panel data set, we use the method of fixed effects regression.
Change in beer tax (dollars per case $1988)
10.3
Fixed Effects Regression
Fixed effects regression is a method for controlling for omitted variables in panel data when the omitted variables vary across entities (states) but do not change over time. Unlike the “before and after” comparisons of Section 10.2, fixed effects regres- sion can be used when there are two or more time observations for each entity.
The fixed effects regression model has n different intercepts, one for each entity. These intercepts can be represented by a set of binary (or indicator) vari- ables. These binary variables absorb the influences of all omitted variables that differ from one entity to the next but are constant over time.
The Fixed Effects Regression Model
Consider the regression model in Equation (10.4) with the dependent variable (FatalityRate) and observed regressor (BeerTax) denoted as Yit and Xit, respectively:
Yit = b0 + b1Xit + b2Zi + uit, (10.9)
where Zi is an unobserved variable that varies from one state to the next but does not change over time (for example, Zi represents cultural attitudes toward drink- ing and driving). We want to estimate b1, the effect on Y of X holding constant the unobserved state characteristics Z.
Because Zi varies from one state to the next but is constant over time, the popu- lation regression model in Equation (10.9) can be interpreted as having n intercepts, one for each state. Specifically, let ai = b0 + b2Zi. Then Equation (10.9) becomes
Yit = b1Xit + ai + uit. (10.10)
Equation (10.10) is the fixed effects regression model, in which a1, c, an are treated as unknown intercepts to be estimated, one for each state. The interpretation of ai as a state-specific intercept in Equation (10.10) comes from considering the popu- lation regression line for the ith state; this population regression line is ai + b1Xit. The slope coefficient of the population regression line, b1, is the same for all states, but the intercept of the population regression line varies from one state to the next.
Because the intercept ai in Equation (10.10) can be thought of as the “effect” of being in entity i (in the current application, entities are states), the terms a1, c, an are known as entity fixed effects. The variation in the entity fixed effects comes from omitted variables that, like Zi in Equation (10.9), vary across entities but not over time.
10.3 Fixed Effects Regression 357
358 CHAPTER 10
Regression with Panel Data
The state-specific intercepts in the fixed effects regression model also can be expressed using binary variables to denote the individual states. Section 8.3 con- sidered the case in which the observations belong to one of two groups and the population regression line has the same slope for both groups but different inter- cepts (see Figure 8.8a). That population regression line was expressed mathemat- ically using a single binary variable indicating one of the groups (case #1 in Key Concept 8.4). If we had only two states in our data set, that binary variable regres- sion model would apply here. Because we have more than two states, however, we need additional binary variables to capture all the state-specific intercepts in Equation (10.10).
To develop the fixed effects regression model using binary variables, let D1i be a binary variable that equals 1 when i = 1 and equals 0 otherwise, let D2i equal 1 when i = 2 and equal 0 otherwise, and so on. We cannot include all n binary variables plus a common intercept, for if we do the regressors will be perfectly multicollinear (this is the “dummy variable trap” of Section 6.7), so we arbitrarily omit the binary variable D1i for the first group. Accordingly, the fixed effects regression model in Equation (10.10) can be written equivalently as
Yit = b0 + b1Xit + g2D2i + g3D3i + g+ gnDni + uit, (10.11)
where b0, b1, g2, c, gn are unknown coefficients to be estimated. To derive the relationship between the coefficients in Equation (10.11) and the intercepts in Equation (10.10), compare the population regression lines for each state in the two equations. In Equation (10.11), the population regression equation for the first state is b0 + b1Xit, so a1 = b0. For the second and remaining states, it is b0 + b1Xit + gi, so ai = b0 + gi for i Ú 2.
Thus there are two equivalent ways to write the fixed effects regression model, Equations (10.10) and (10.11). In Equation (10.10), it is written in terms of n state- specific intercepts. In Equation (10.11), the fixed effects regression model has a common intercept and n – 1 binary regressors. In both formulations, the slope coefficient on X is the same from one state to the next. The state-specific intercepts in Equation (10.10) and the binary regressors in Equation (10.11) have the same source: the unobserved variable Zi that varies across states but not over time.
Extension to multiple X’s. If there are other observed determinants of Y that are correlated with X and that change over time, then these should also be included in the regression to avoid omitted variable bias. Doing so results in the fixed effects regression model with multiple regressors, summarized in Key Concept 10.2.
10.3 Fixed Effects Regression 359
The Fixed Effects Regression Model
KEY CONCEPT
10.2
The fixed effects regression model is
Yit = b1X1,it + g + bkXk,it + ai + uit, (10.12)
where i = 1, c, n; t = 1, c, T; X1,it is the value of the first regressor for entity i in time period t, X2,it is the value of the second regressor, and so forth; and a1, c, an are entity-specific intercepts.
Equivalently, the fixed effects regression model can be written in terms of a commonintercept,theX’s,andn – 1binaryvariablesrepresentingallbutoneentity:
Yit = b0 + b1X1,it + g+ bkXk,it + g2D2i
+ g3D3i + g + gnDni + uit, (10.13)
where D2i = 1 if i = 2 and D2i = 0 otherwise, and so forth.
Estimation and Inference
In principle the binary variable specification of the fixed effects regression model [Equation (10.13)] can be estimated by OLS. This regression, however, has k + n regressors (the k X’s, the n – 1 binary variables, and the intercept), so in practice this OLS regression is tedious or, in some software packages, impossible to imple- ment if the number of entities is large. Econometric software therefore has special routines for OLS estimation of fixed effects regression models. These special rou- tines are equivalent to using OLS on the full binary variable regression, but are faster because they employ some mathematical simplifications that arise in the algebra of fixed effects regression.
The “entity-demeaned” OLS algorithm. Regression software typically computes the OLS fixed effects estimator in two steps. In the first step, the entity-specific average is subtracted from each variable. In the second step, the regression is esti- mated using “entity-demeaned” variables. Specifically, consider the case of a single regressor in the version of the fixed effects model in Equation (10.10) and take the average of both sides of Equation (10.10); then Yi = b1Xi + ai + ui, where
Y = (1>T)gT Y , andX and u are defined similarly. Thus Equation (10.10) i t=1it i i
360 CHAPTER 10
Regression with Panel Data
impliesthatYit – Yi = b1(Xit – Xi) + (uit – ui).LetY∼it = Yit – Yi,X∼it = Xit – Xi and ∼uit = uit – ui; accordingly,
∼Yit = b1X∼it + ∼uit. (10.14)
Thus b1 can be estimated by the OLS regression of the “entity-demeaned” vari- ables ∼Yit on X∼it . In fact, this estimator is identical to the OLS estimator of b1 obtained by estimation of the fixed effects model in Equation (10.11) using n – 1 binary variables (Exercise 18.6).
The “before and after” (differences) regression versus the binary variables specification. Although Equation (10.11) with its binary variables looks quite dif- ferent from the “before and after” regression model in Equation (10.7), in the special case that T = 2 the OLS estimator of b1 from the binary variable specification and that from the “before and after” specification are identical if the intercept is excluded from the “before and after” specifications. Thus, when T = 2, there are three ways to estimate b1 by OLS: the “before and after” specification in Equation (10.7) (with- out an intercept), the binary variable specification in Equation (10.11), and the “entity-demeaned” specification in Equation (10.14). These three methods are equivalent; that is, they produce identical OLS estimates of b1 (Exercise 10.11).
The sampling distribution, standard errors, and statistical inference. In multiple regression with cross-sectional data, if the four least squares assumptions in Key Concept 6.4 hold, then the sampling distribution of the OLS estimator is normal in large samples. The variance of this sampling distribution can be estimated from the data, and the square root of this estimator of the variance—that is, the stan- dard error—can be used to test hypotheses using a t-statistic and to construct confidence intervals.
Similarly, in multiple regression with panel data, if a set of assumptions— called the fixed effects regression assumptions—hold, then the sampling distribu- tion of the fixed effects OLS estimator is normal in large samples, the variance of that distribution can be estimated from the data, the square root of that estimator is the standard error, and the standard error can be used to construct t-statistics and confidence intervals. Given the standard error, statistical inference—testing hypotheses (including joint hypotheses using F-statistics) and constructing confi- dence intervals—proceeds in exactly the same way as in multiple regression with cross-sectional data.
The fixed effects regression assumptions and standard errors for fixed effects regression are discussed further in Section 10.5.
Application to Traffic Deaths
The OLS estimate of the fixed effects regression line relating the real beer tax to the fatality rate, based on all 7 years of data (336 observations), is
FatalityRate = -0.66BeerTax + StateFixedEffects, (10.15) (0.29)
where, as is conventional, the estimated state fixed intercepts are not listed to save space and because they are not of primary interest in this application.
Like the “differences” specification in Equation (10.8), the estimated coeffi- cient in the fixed effects regression in Equation (10.15) is negative, so, as pre- dicted by economic theory, higher real beer taxes are associated with fewer traffic deaths, which is the opposite of what we found in the initial cross-sectional regres- sions of Equations (10.2) and (10.3). The two regressions are not identical because the “differences” regression in Equation (10.8) uses only the data for 1982 and 1988 (specifically, the difference between those two years), whereas the fixed effects regression in Equation (10.15) uses the data for all 7 years. Because of the additional observations, the standard error is smaller in Equation (10.15) than in Equation (10.8).
Including state fixed effects in the fatality rate regression lets us avoid omitted variables bias arising from omitted factors, such as cultural attitudes toward drink- ing and driving, that vary across states but are constant over time within a state. Still, a skeptic might suspect that other factors could lead to omitted variables bias. For example, over this period cars were getting safer and occupants were increasingly wearing seat belts; if the real tax on beer rose on average during the mid-1980s, then BeerTax could be picking up the effect of overall automobile safety improvements. If, however, safety improvements evolved over time but were the same for all states, then we can eliminate their influence by including time fixed effects.
10.4
10.4 Regression with Time Fixed Effects 361
Regression with Time Fixed Effects
Just as fixed effects for each entity can control for variables that are constant over time but differ across entities, so can time fixed effects control for variables that are constant across entities but evolve over time.
Because safety improvements in new cars are introduced nationally, they serve to reduce traffic fatalities in all states. So, it is plausible to think of automo- bile safety as an omitted variable that changes over time but has the same value
362 CHAPTER 10
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for all states. The population regression in Equation (10.9) can be modified to make explicit the effect of automobile safety, which we will denote St:
Yit = b0 + b1Xit + b2Zi + b3St + uit, (10.16)
where St is unobserved and where the single t subscript emphasizes that safety changes over time but is constant across states. Because b3St represents variables that determine Yit, if St is correlated with Xit, then omitting St from the regression leads to omitted variable bias.
Time Effects Only
For the moment, suppose that the variables Zi are not present so that the term b2Zi can be dropped from Equation (10.16), although the term b3St remains. Our objective is to estimate b1, controlling for St.
Although St is unobserved, its influence can be eliminated because it varies over time but not across states, just as it is possible to eliminate the effect of Zi, which varies across states but not over time. In the entity fixed effects model, the presence of Zi leads to the fixed effects regression model in Equation (10.10), in which each state has its own intercept (or fixed effect). Similarly, because St varies over time but not over states, the presence of St leads to a regression model in which each time period has its own intercept.
The time fixed effects regression model with a single X regressor is
Yit = b1Xit + lt + uit. (10.17)
This model has a different intercept, lt, for each time period. The intercept lt in Equation (10.17) can be thought of as the “effect” on Y of year t (or, more gener- ally, time period t), so the terms l1, c, lT are known as time fixed effects. The variation in the time fixed effects comes from omitted variables that, like St in Equation (10.16), vary over time but not across entities.
Just as the entity fixed effects regression model can be represented using n – 1 binary indicators, so, too, can the time fixed effects regression model be represented using T – 1 binary indicators:
Yit = b0 + b1Xit + d2B2t + g +dTBTt + uit, (10.18)
where d2, c, dT are unknown coefficients and where B2t = 1 if t = 2 and B2t = 0 otherwise, and so forth. As in the fixed effects regression model in Equation (10.11),
10.4 Regression with Time Fixed Effects 363
in this version of the time effects model the intercept is included, and the first binary variable (B1t) is omitted to prevent perfect multicollinearity.
When there are additional observed “X” regressors, then these regressors appear in Equations (10.17) and (10.18) as well.
In the traffic fatalities regression, the time fixed effects specification allows us to eliminate bias arising from omitted variables like nationally intro- duced safety standards that change over time but are the same across states in a given year.
Both Entity and Time Fixed Effects
If some omitted variables are constant over time but vary across states (such as cultural norms) while others are constant across states but vary over time (such as national safety standards), then it is appropriate to include both entity (state) and time effects.
The combined entity and time fixed effects regression model is
Yit = b1Xit + ai + lt + uit, (10.19)
where ai is the entity fixed effect and lt is the time fixed effect. This model can equivalently be represented using n – 1 entity binary indicators and T – 1 time binary indicators, along with an intercept:
Yit =b0 +b1Xit +g2D2i +g+gnDni
+ d2B2t + g + dTBTt + uit, (10.20)
where b0, b1, g2, c, gn, and d2, c, dT are unknown coefficients.
When there are additional observed “X” regressors, then these appear in
Equations (10.19) and (10.20) as well.
The combined state and time fixed effects regression model eliminates omit-
ted variables bias arising both from unobserved variables that are constant over time and from unobserved variables that are constant across states.
Estimation. The time fixed effects model and the entity and time fixed effects model are both variants of the multiple regression model. Thus their coefficients can be estimated by OLS by including the additional time binary variables. Alter- natively, in a balanced panel the coefficients on the X’s can be computed by first deviating Y and the X’s from their entity and time-period means and then by
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estimating the multiple regression equation of deviated Y on the deviated X’s. This algorithm, which is commonly implemented in regression software, eliminates the need to construct the full set of binary indicators that appear in Equation (10.20). An equivalent approach is to deviate Y, the X’s, and the time indicators from their entity (but not time) means and to estimate k + T coefficients by mul- tiple regression of the deviated Y on the deviated X’s and the deviated time indi- cators. Finally, if T = 2, the entity and time fixed effects regression can be estimated using the “before and after” approach of Section 10.2, including the intercept in the regression. Thus the “before and after” regression reported in Equation (10.8), in which the change in FatalityRate from 1982 to 1988 is regressed on the change in BeerTax from 1982 to 1988 including an intercept, provides the same estimate of the slope coefficient as the OLS regression of FatalityRate on BeerTax, including entity and time fixed effects, estimated using data for the two years 1982 and 1988.
Applicationtotrafficdeaths. Addingtimeeffectstothestatefixedeffectsregres- sion results in the OLS estimate of the regression line:
FatalityRate = -0.64BeerTax + StateFixedEffects + TimeFixedEffects. (10.21) (0.36)
This specification includes the beer tax, 47 state binary variables (state fixed effects), 6 single-year binary variables (time fixed effects), and an intercept, so this regression actually has 1 + 47 + 6 + 1 = 55 right-hand variables! The coeffi- cients on the time and state binary variables and the intercept are not reported because they are not of primary interest.
Including time effects has little impact on the coefficient on the real beer tax [compare Equations (10.15) and (10.21)]. Although this coefficient is less pre- cisely estimated when time effects are included, it is still significant at the 10%, but not 5%, significance level (t = -0.64>0.36 = -1.78).
This estimated relationship between the real beer tax and traffic fatalities is immune to omitted variable bias from variables that are constant either over time or across states. However, many important determinants of traffic deaths do not fall into this category, so this specification could still be subject to omitted variable bias. Section 10.6 therefore undertakes a more complete empirical examination of the effect of the beer tax and of laws aimed directly at eliminating drunk driv- ing, controlling for a variety of factors. Before turning to that study, we first dis- cuss the assumptions underlying panel data regression and the construction of standard errors for fixed effects estimators.
10.5 The Fixed Effects Regression Assumptions and Standard Errors for Fixed Effects Regression 365
10.5
The Fixed Effects Regression Assumptions and Standard Errors for Fixed Effects Regression
In panel data, the regression error can be correlated over time within an entity. Like heteroskedasticity, this correlation does not introduce bias into the fixed effects estimator, but it affects the variance of the fixed effects estimator and therefore it affects how one computes standard errors. The standard errors for fixed effects regressions reported in this chapter are so-called clustered standard errors, which are robust both to heteroskedasticity and to correlation over time within an entity. When there are many entities (when n is large), hypothesis tests and confidence intervals can be computed using the usual large-sample normal and F critical values.
This section describes clustered standard errors. We begin with the fixed effects regression assumptions, which extend the least squares regression assump- tions to panel data; under these assumptions, the fixed effects estimator is asymp- totically normally distributed when n is large. To keep the notation as simple as possible, this section focuses on the entity fixed effects regression model of Section 10.3, in which there are no time effects.
The Fixed Effects Regression Assumptions
The four fixed effects regression assumptions are summarized in Key Concept 10.3. These assumptions extend the four least squares assumptions, stated for cross- sectional data in Key Concept 6.4, to panel data.
The first assumption is that the error term has conditional mean zero, given all T values of X for that entity. This assumption plays the same role as the first least squares assumption for cross-sectional data in Key Concept 6.4 and implies that there is no omitted variable bias. The requirement that the conditional mean of uit not depend on any of the values of X for that entity—past, present, or future—adds an important subtlety beyond the first least squares assumption for cross-sectional data. This assumption is violated if current uit is correlated with past, present, or future values of X.
The second assumption is that the variables for one entity are distributed iden- tically to, but independently of, the variables for another entity; that is, the variables are i.i.d. across entities for i = 1, c, n. Like the second least squares assumption in Key Concept 6.4, the second assumption for fixed effects regression holds if enti- ties are selected by simple random sampling from the population.
366 CHAPTER 10 Regression with Panel Data
KEY CONCEPT
10.3
The Fixed Effects Regression Assumptions
Yit = b1Xit + ai + uit,i = 1,c,n,t = 1,c,T,
where
1. uit has conditional mean zero: E(uit Xi1, Xi2, c, XiT, ai) = 0.
2. (Xi1, Xi2, c, XiT, ui1, ui2, c, uiT), i = 1, c, n are i.i.d. draws from their joint distribution.
3. Large outliers are unlikely: (Xit, uit) have nonzero finite fourth moments.
4. There is no perfect multicollinearity.
For multiple regressors, Xit should be replaced by the full list X1,it, X2,it, c, Xk,it.
The third and fourth assumptions for fixed effects regression are analogous to the third and fourth least squares assumptions for cross-sectional data in Key Concept 6.4. Under the least squares assumptions for panel data in Key Concept 10.3, the fixed effects estimator is consistent and is normally distributed when n is large.
The details are discussed in Appendix 10.2.
An important difference between the panel data assumptions in Key Concept
10.3 and the assumptions for cross-sectional data in Key Concept 6.4 is Assump- tion 2. The cross-sectional counterpart of Assumption 2 holds that each observa- tion is independent, which arises under simple random sampling. In contrast, Assumption 2 for panel data holds that the variables are independent across enti- ties but makes no such restriction within an entity. For example, Assumption 2 allows Xit to be correlated over time within an entity.
If Xit is correlated with Xis for different values of s and t—that is, if Xit is cor- related over time for a given entity—then Xit is said to be autocorrelated (correlated with itself, at different dates) or serially correlated. Autocorrelation is a pervasive feature of time series data: What happens one year tends to be correlated with what happens the next year. In the traffic fatality example, Xit, the beer tax in state i in year t, is autocorrelated: Most of the time, the legislature does not change the beer tax, so if it is high one year relative to its mean value for state i, it will tend to be high the next year, too. Similarly, it is possible to think of reasons why uit would be auto- correlated. Recall that uit consists of time-varying factors that are determinants of Yit but are not included as regressors, and some of these omitted factors might be autocorrelated. For example, a downturn in the local economy might produce
10.5 The Fixed Effects Regression Assumptions and Standard Errors for Fixed Effects Regression 367
layoffs and diminish commuting traffic, thus reducing traffic fatalities for 2 or more years. Similarly, a major road improvement project might reduce traffic accidents not only in the year of completion but also in future years. Such omitted factors, which persist over multiple years, produce autocorrelated regression errors. Not all omitted factors will produce autocorrelation in uit; for example, severe winter driving conditions plausibly affect fatalities, but if winter weather conditions for a given state are independently distributed from one year to the next, then this com- ponent of the error term would be serially uncorrelated. In general, though, as long as some omitted factors are autocorrelated, then uit will be autocorrelated.
Standard Errors for Fixed Effects Regression
If the regression errors are autocorrelated, then the usual heteroskedasticity-robust standard error formula for cross-section regression [Equations (5.3) and (5.4)] is not valid. One way to see this is to draw an analogy to heteroskedasticity. In a regres- sion with cross-sectional data, if the errors are heteroskedastic, then (as discussed in Section 5.4) the homoskedasticity-only standard errors are not valid because they were derived under the false assumption of homoskedasticity. Similarly, if the errors are autocorrelated, then the usual standard errors will not be valid because they were derived under the false assumption of no serial correlation.
Standard errors that are valid if uit is potentially heteroskedastic and poten- tially correlated over time within an entity are referred to as heteroskedasticity- and autocorrelation-consistent (HAC) standard errors. The standard errors used in this chapter are one type of HAC standard errors, clustered standard errors. The term clustered arises because these standard errors allow the regression errors to have an arbitrary correlation within a cluster, or grouping, but assume that the regression errors are uncorrelated across clusters. In the context of panel data, each cluster consists of an entity. Thus clustered standard errors allow for hetero- skedasticity and for arbitrary autocorrelation within an entity, but treat the errors as uncorrelated across entities. That is, clustered standard errors allow for hetero- skedasticity and autocorrelation in a way that is consistent with the second fixed effects regression assumption in Key Concept 10.3.
Like heteroskedasticity-robust standard errors in regression with cross-sectional data, clustered standard errors are valid whether or not there is heteroskedasticity, autocorrelation, or both. If the number of entities n is large, inference using clustered standard errors can proceed using the usual large-sample normal critical values for t-statisticsandFq,∞ criticalvaluesforF-statisticstestingqrestrictions.
In practice, there can be a large difference between clustered standard errors and standard errors that do not allow for autocorrelation of uit. For example, the usual (cross-sectional data) heteroskedasticity-robust standard error for the BeerTax
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10.6
Drunk Driving Laws and Traffic Deaths
Alcohol taxes are only one way to discourage drinking and driving. States differ in their punishments for drunk driving, and a state that cracks down on drunk driving could do so by toughening driving laws as well as raising taxes. If so, omit- ting these laws could produce omitted variable bias in the OLS estimator of the effect of real beer taxes on traffic fatalities, even in regressions with state and time fixed effects. In addition, because vehicle use depends in part on whether drivers have jobs and because tax changes can reflect economic conditions (a state budget deficit can lead to tax hikes), omitting state economic conditions also could result in omitted variable bias. In this section, we therefore extend the preceding analy- sis of traffic fatalities to include other driving laws and economic conditions.
The results are summarized in Table 10.1. The format of the table is the same as that of the tables of regression results in Chapters 7 through 9: Each column reports a different regression, and each row reports a coefficient estimate and standard error, F-statistic and p-value, or other information about the regression.
Column (1) in Table 10.1 presents results for the OLS regression of the fatal- ity rate on the real beer tax without state and time fixed effects. As in the cross- sectional regressions for 1982 and 1988 [Equations (10.2) and (10.3)], the coefficient on the real beer tax is positive (0.36): According to this estimate, increasing beer taxes increases traffic fatalities! However, the regression in col- umn (2) [reported previously as Equation (10.15)], which includes state fixed effects, suggests that the positive coefficient in regression (1) is the result of omit- ted variable bias (the coefficient on the real beer tax is – 0.66). The regression R 2 jumps from 0.091 to 0.889 when fixed effects are included; evidently, the state fixed effects account for a large amount of the variation in the data.
Little changes when time effects are added, as reported in column (3) [reported previously as Equation (10.21)], except that the beer tax coefficient is now estimated less precisely. The results in columns (1) through (3) are consistent with the omitted fixed factors—historical and cultural factors, general road condi- tions, population density, attitudes toward drinking and driving, and so forth— being important determinants of the variation in traffic fatalities across states.
coefficient in Equation (10.21) is 0.25, substantially smaller than the clustered stan- dard error, 0.36, and the respective t-statistics testing b1 = 0 are -2.51 and -1.78. The reason we report the clustered standard error is that it allows for serial correla- tion of uit within an entity, whereas the usual heteroskedasticity-robust standard error does not. The formula for clustered standard errors is given in Appendix 10.2.
TABLE 10.1 Regression Analysis of the Effect of Drunk Driving Laws on Traffic Deaths Dependent variable: Traffic fatality rate (deaths per 10,000).
10.6 Drunk Driving Laws and Traffic Deaths 369
Regressor
Beer tax
Drinking age 18
Drinking age 19
Drinking age 20
Drinking age
Mandatory jail
or community service?
Average vehicle miles per driver
Unemployment rate
Real income per capita (logarithm)
Years
State effects?
Time effects?
Clustered standard errors?
(1)
0.36** (0.05)
(2)
– 0.66* (0.29)
(3)
– 0.64+ (0.36)
(4)
– 0.45 (0.30)
0.028 (0.070)
-0.018 (0.050)
0.032 (0.051)
0.038 (0.103)
0.008 (0.007)
-0.063** (0.013)
1.82** (0.64)
(5)
– 0.69* (0.35)
-0.010 (0.083)
– 0.076 (0.068)
– 0.100+ (0.056)
0.085 (0.112)
0.017 (0.011)
1982–88 yes yes yes
3.48 (0.006)
(6) (7)
– 0.46 – 0.93** (0.31) (0.34)
1982–88 1982–88 1982–88 1982–88
– 0.002 (0.021)
0.039 (0.103)
0.009 (0.007)
-0.063** (0.013)
1.79** (0.64)
1982–88 yes yes yes
10.28
( 6 0.001)
0.037 (0.102)
– 0.065 (0.099)
– 0.113 (0.125)
0.089 (0.164)
0.124 (0.049)
-0.091** (0.021)
1.00 (0.68)
1982 & 1988 only yes
yes
yes
37.49
( 6 0.001)
no no no
yes yes yes no yes yes yes yes yes
F-Statistics and p-Values Testing Exclusion of Groups of Variables
Time effects = 0
Drinking age coefficients = 0
4.22 (0.002)
0.891
10.12
( 6 0.001)
0.35 (0.786)
29.62 (6 0.001)
0.926
0.42 25.20
(6 0.001) 0.899
Unemployment rate, income per capita = 0
R2
0.091
0.889
0.893
31.96 (6 0.001)
0.926
1.41
(0.253) (0.738)
These regressions were estimated using panel data for 48 U.S. states. Regressions (1) through (6) use data for all years 1982 to 1988, and regression (7) uses data from 1982 and 1988 only. The data set is described in Appendix 10.1. Standard errors are given in parentheses under the coefficients, and p-values are given in parentheses under the F-statistics. The individual coefficient is statistically significant at the +10%, *5%, or **1% significance level.
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The next four regressions in Table 10.1 include additional potential determi- nants of fatality rates along with state and time effects. The base specification, reported in column (4), includes variables related to drunk driving laws plus vari- ables that control for the amount of driving and overall state economic conditions. The first legal variables are the minimum legal drinking age, represented by three binary variables for a minimum legal drinking age of 18, 19, and 20 (so the omitted group is a minimum legal drinking age of 21 or older). The other legal variable is the punishment associated with the first conviction for driving under the influence of alcohol, either mandatory jail time or mandatory community service (the omitted group is less severe punishment). The three measures of driving and economic condi- tions are average vehicle miles per driver, the unemployment rate, and the logarithm of real (1988 dollars) personal income per capita (using the logarithm of income permits the coefficient to be interpreted in terms of percentage changes of income; see Section 8.2). The final regression in Table 10.1 follows the “before and after” approach of Section 10.2 and uses only data from 1982 and 1988; thus regression (7) extends the regression in Equation (10.8) to include the additional regressors.
The regression in column (4) has four interesting results.
1. Including the additional variables reduces the estimated effect of the beer tax from -0.64 in column (3) to -0.45 in column (4). One way to evaluate the magnitude of this coefficient is to imagine a state with an average real beer tax doubling its tax; because the average real beer tax in these data is approximately $0.50 per case (in 1988 dollars), this entails increasing the tax by $0.50 per case. The estimated effect of a $0.50 increase in the beer tax is to decrease the expected fatality rate by 0.45 * 0.50 = 0.23 death per 10,000. This estimated effect is large: Because the average fatality rate is 2 per 10,000, a reduction of 0.23 corresponds to reducing traffic deaths by nearly one-eighth. This said, the estimate is quite imprecise: Because the standard error on this coefficient is 0.30, the 95% confidence interval for this effect is -0.45 * 0.50 { 1.96 * 0.30 * 0.50 = (-0.52, 0.07). This wide 95% confidence interval includes zero, so the hypothesis that the beer tax has no effect cannot be rejected at the 5% significance level.
2. The minimum legal drinking age is precisely estimated to have a small effect on traffic fatalities. According to the regression in column (4), the 95% confi- dence interval for the increase in the fatality rate in a state with a minimum legal drinking age of 18, relative to age 21, is ( – 0.11, 0.17). The joint hypothesis that the coefficients on the minimum legal drinking age variables are zero can- not be rejected at the 10% significance level: The F-statistic testing the joint hypothesis that the three coefficients are zero is 0.35, with a p-value of 0.786.
10.6 Drunk Driving Laws and Traffic Deaths 371
3. The coefficient on the first offense punishment variable is also estimated to be small and is not significantly different from zero at the 10% significance level.
4. The economic variables have considerable explanatory power for traffic fa- talities. High unemployment rates are associated with fewer fatalities: An increase in the unemployment rate by one percentage point is estimated to reduce traffic fatalities by 0.063 death per 10,000. Similarly, high values of real per capita income are associated with high fatalities: The coefficient is 1.82, so a 1% increase in real per capita income is associated with an increase in traffic fatalities of 0.0182 death per 10,000 (see Case I in Key Concept 8.2 for interpretation of this coefficient). According to these estimates, good economic conditions are associated with higher fatalities, perhaps because of increased traffic density when the unemployment rate is low or greater alcohol consumption when income is high. The two economic variables are jointly significant at the 0.1% significance level (the F-statistic is 29.62).
Columns (5) through (7) of Table 10.1 report regressions that check the sen-
sitivity of these conclusions to changes in the base specification. The regression in column (5) drops the variables that control for economic conditions. The result is an increase in the estimated effect of the real beer tax, which becomes significant at the 5% level, but no appreciable change in the other coefficients. The sensitivity of the estimated beer tax coefficient to including the economic variables, com- bined with the statistical significance of the coefficients on those variables in col- umn (4), indicates that the economic variables should remain in the base specification. The regression in column (6) shows that the results in column (4) are not sensitive to changing the functional form when the three drinking age indicator variables are replaced by the drinking age itself. When the coefficients are estimated using the changes of the variables from 1982 to 1988 [column (7)], as in Section 10.2, the findings from column (4) are largely unchanged except that the coefficient on the beer tax is larger and is significant at the 1% level.
The strength of this analysis is that including state and time fixed effects mit- igates the threat of omitted variable bias arising from unobserved variables that either do not change over time (like cultural attitudes toward drinking and driv- ing) or do not vary across states (like safety innovations). As always, however, it is important to think about possible threats to validity. One potential source of omitted variable bias is that the measure of alcohol taxes used here, the real tax on beer, could move with other alcohol taxes, which suggests interpreting the results as pertaining more broadly than just to beer. A subtler possibility is that hikes in the real beer tax could be associated with public education campaigns. If
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10.7
Conclusion
This chapter showed how multiple observations over time on the same entity can be used to control for unobserved omitted variables that differ across entities but are constant over time. The key insight is that if the unobserved variable does not change over time, then any changes in the dependent variable must be due to influences other than these fixed characteristics. If cultural attitudes toward drink- ing and driving do not change appreciably over 7 years within a state, then expla- nations for changes in the traffic fatality rate over those 7 years must lie elsewhere.
To exploit this insight, you need data in which the same entity is observed at two or more time periods; that is, you need panel data. With panel data, the mul- tiple regression model of Part II can be extended to include a full set of entity binary variables; this is the fixed effects regression model, which can be estimated by OLS. A twist on the fixed effects regression model is to include time fixed effects, which control for unobserved variables that change over time but are constant across entities. Both entity and time fixed effects can be included in the regression to control for variables that vary across entities but are constant over time and for variables that vary over time but are constant across entities.
Despite these virtues, entity and time fixed effects regression cannot control for omitted variables that vary both across entities and over time. And, obviously, panel data methods require panel data, which often are not available. Thus there
2For further analysis of these data, see Ruhm (1996). A recent meta-analysis of 112 studies of the effect of alcohol prices and taxes on consumption found elasticities of -0.46 for beer, -0.69 for wine, and -0.80 for spirits, and concluded that alcohol taxes have large effects on reducing consumption, relative to other programs [Wagenaar, Salois, and Komro (2009)]. To learn more about drunk driving and alcohol, and about the economics of alcohol more generally, also see Cook and Moore (2000), Chaloupka, Grossman, and Saffer (2002), Young and Bielinska-Kwapisz (2006), and Dang (2008).
so, changes in the real beer tax could pick up the effect of a broader campaign to reduce drunk driving.
Taken together, these results present a provocative picture of measures to control drunk driving and traffic fatalities. According to these estimates, neither stiff punishments nor increases in the minimum legal drinking age have important effects on fatalities. In contrast, there is some evidence that increasing alcohol taxes, as measured by the real tax on beer, does reduce traffic deaths, presumably through reduced alcohol consumption. The imprecision of the estimated beer tax coefficient means, however, that we should be cautious about drawing policy con- clusions from this analysis and that additional research is warranted.2
remains a need for a method that can eliminate the influence of unobserved omit- ted variables when panel data methods cannot do the job. A powerful and general method for doing so is instrumental variables regression, the topic of Chapter 12.
Summary
1. Panel data consist of observations on multiple (n) entities—states, firms, people, and so forth—where each entity is observed at two or more time periods (T).
2. Regression with entity fixed effects controls for unobserved variables that differ from one entity to the next but remain constant over time.
3. When there are two time periods, fixed effects regression can be estimated by a “before and after” regression of the change in Y from the first period to the second on the corresponding change in X.
4. Entity fixed effects regression can be estimated by including binary variables for n – 1 entities plus the observable independent variables (the X’s) and an intercept.
5. Time fixed effects control for unobserved variables that are the same across entities but vary over time.
6. A regression with time and entity fixed effects can be estimated by includ- ing binary variables for n – 1 entities and binary variables for T – 1 time periods plus the X’s and an intercept.
7. In panel data, variables are typically autocorrelated—that is, correlated over time within an entity. Standard errors need to allow both for this autocor- relation and for potential heteroskedasticity, and one way to do so is to use clustered standard errors.
Key Terms
panel data (351)
balanced panel (351)
unbalanced panel (351)
fixed effects regression model (357) entity fixed effects (357)
time fixed effects regression model
(362)
time fixed effects (362)
entity and time fixed effects regression model (363)
autocorrelated (366) serially correlated (366) heteroskedasticity- and
autocorrelation-consistent
(HAC) standard errors (367) clustered standard errors (367)
Key Terms 373
374 CHAPTER 10
Regression with Panel Data
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Review the Concepts
10.1 Why is it necessary to use two subscripts, i and t, to describe panel data? What does i refer to? What does t refer to?
10.2 A researcher is using a panel data set on n = 1000 workers over T = 10 years (from 2001 through 2010) that contains the workers’ earnings, gender, education, and age. The researcher is interested in the effect of education on earnings. Give some examples of unobserved person-specific variables that are correlated with both education and earnings. Can you think of examples of time-specific variables that might be correlated with education and earnings? How would you control for these person-specific and time- specific effects in a panel data regression?
10.3 Can the regression that you suggested in response to Question 10.2 be used to estimate the effect of gender on an individual’s earnings? Can that regression be used to estimate the effect of the national unemployment rate on an individual’s earnings? Explain.
10.4 In the context of the regression you suggested for Question 10.2, explain why the regression error for a given individual might be serially correlated.
Exercises
10.1 This exercise refers to the drunk driving panel data regression summarized in Table 10.1.
a. New Jersey has a population of 8.1 million people. Suppose that New Jersey increased the tax on a case of beer by $1 (in 1988 dollars). Use the results in column (4) to predict the number of lives that would be saved over the next year. Construct a 95% confidence interval for your answer.
b. The drinking age in New Jersey is 21. Suppose that New Jersey lowered its drinking age to 18. Use the results in column (4) to predict the change in the number of traffic fatalities in the next year. Construct a 95% confidence interval for your answer.
c. Suppose that real income per capita in New Jersey increases by 1% in the next year. Use the results in column (4) to predict the change in the number of traffic fatalities in the next year. Construct a 90% confidence interval for your answer.
d. Should time effects be included in the regression? Why or why not?
e. A researcher conjectures that the unemployment rate has a different effect on traffic fatalities in the western states than in the other states. How would you test this hypothesis? (Be specific about the specification of the regression and the statistical test you
would use.)
10.2 Consider the binary variable version of the fixed effects model in Equation (10.11), except with an additional regressor, D1i; that is, let
Yit =b0 +b1Xit +g1D1i +g2D2i +g+gnDni +uit.
a. Suppose that n = 3. Show that the binary regressors and the “con- stant” regressor are perfectly multicollinear; that is, express one of the variables D1i, D2i, D3i, and X0,it as a perfect linear function of the others, where X0,it = 1 for all i, t.
b. Show the result in (a) for general n.
c. What will happen if you try to estimate the coefficients of the
regression by OLS?
10.3 Section 9.2 gave a list of five potential threats to the internal validity of a regression study. Apply that list to the empirical analysis in Section 10.6 and thereby draw conclusions about its internal validity.
10.4 Using the regression in Equation (10.11), what is the slope and intercept for
a. Entity 1 in time period 1?
b. Entity 1 in time period 3?
c. Entity 3 in time period 1?
d. Entity 3 in time period 3?
Exercises 375
376 CHAPTER 10
Regression with Panel Data
10.5 Consider the model with a single regressor Yit = b1X1,it + ai + lt + uit. This model also can be written as
Yit =b0 +b1X1,it +d2B2t +g+dTBTt +g2D2i +g+gnDni +uit,
where B2t = 1 if t = 2 and 0 otherwise, D2i = 1 if i = 2 and 0 otherwise, and so forth. How are the coefficients (b0, d2, c, dT, g2, c, gn) related to the coefficients (a1, c, an, l1, c, lT)?
10.6 Do the fixed effects regression assumptions in Key Concept 10.3 imply that cov(∼v it,∼v is) = 0 for t ≠ s in Equation (10.28)? Explain.
10.7 A researcher believes that traffic fatalities increase when roads are icy and thinks that therefore states with more snow will have more fatalities than other states. Comment on the following methods designed to estimate the effect of snow on fatalities:
a. The researcher collects data on the average snowfall for each state and adds this regressor (AverageSnowi) to the regressions given in Table 10.1.
b. The researcher collects data on the snowfall in each state for each year in the sample (Snowit) and adds this regressor to the regressions.
10.8 Consider observations (Yit, Xit) from the linear panel data model Yit =Xitb1 +ai +lit+uit,
where t = 1,c,T;i = 1,c,n; and ai + lit is an unobserved entity- specific time trend. How would you estimate b1?
10.9 a.
In the fixed effects regression model, are the fixed entity effects, ai, consistently estimated as n ¡ ∞ with T fixed? (Hint: Analyze the model with no X’s: Yit = ai + uit.)
b. If n is large (say, n = 2000) but T is small (say, T = 4), do you think that the estimated values of ai are approximately normally distrib- uted? Why or why not? (Hint: Analyze the model Yit = ai + uit.)
10.10 Inastudyoftheeffectonearningsofeducationusingpaneldataonannual earnings for a large number of workers, a researcher regresses earnings in a given year on age, education, union status, and the worker’s earnings in the previous year, using fixed effects regression. Will this regression give
reliable estimates of the effects of the regressors (age, education, union status, and previous year’s earnings) on earnings? Explain. (Hint: Check the fixed effects regression assumptions in Section 10.5.)
10.11 Let bnDM denote the entity-demeaned estimator given in Equation (10.22), 1
and let bnBA denote the “before and after” estimator without an intercept, 1
sothatbnBA=3Σn (X -X)(Y -Y)4>3Σn (X -X)24.Show 1 i=1 i2 i1 i2 i1 i=1 i2 i1
that, if T = 2, bnDM = bnBA. [Hint: Use the definition of X∼ before Equa- 11 it
∼1 ∼1
tion (10.22) to show that Xi1 = -2(Xi2 – Xi1) and Xi2 = 2(Xi2 – Xi1).]
Empirical Exercises
(Only two empirical exercises for this chapter are given in the text, but you can find more on the text website, http://www.pearsonhighered.com/stock_watson/.)
E10.1 Some U.S. states have enacted laws that allow citizens to carry concealed weapons. These laws are known as “shall-issue” laws because they instruct local authorities to issue a concealed weapons permit to all applicants who are citizens, are mentally competent, and have not been convicted of a felony. (Some states have some additional restrictions.) Proponents argue that if more people carry concealed weapons, crime will decline because crimi- nals will be deterred from attacking other people. Opponents argue that crime will increase because of accidental or spontaneous use of the weap- ons. In this exercise, you will analyze the effect of concealed weapons laws on violent crimes. On the textbook website, http://www.pearsonhighered. com/stock_watson, you will find the data file Guns, which contains a balanced panel of data from the 50 U.S. states plus the District of Columbia for the years 1977 through 1999.3 A detailed description is given in Guns_ Description, available on the website.
a. Estimate (1) a regression of ln(vio) against shall and (2) a regres- sion of ln(vio) against shall, incarc_rate, density, avginc, pop, pb1064, pw1064, and pm1029.
i. Interpret the coefficient on shall in regression (2). Is this estimate large or small in a “real-world” sense?
3These data were provided by Professor John Donohue of Stanford University and were used in his paper with Ian Ayres, “Shooting Down the ‘More Guns Less Crime’ Hypothesis,” Stanford Law Review, 2003, 55: 1193–1312.
Empirical Exercises 377
378 CHAPTER 10
Regression with Panel Data
ii. Does adding the control variables in regression (2) change the estimated effect of a shall-carry law in regression (1) as measured by statistical significance? As measured by the “real-world” signifi- cance of the estimated coefficient?
iii. Suggest a variable that varies across states but plausibly varies little—or not at all—over time and that could cause omitted vari- able bias in regression (2).
b. Do the results change when you add fixed state effects? If so, which set of regression results is more credible, and why?
c. Do the results change when you add fixed time effects? If so, which set of regression results is more credible, and why?
d. Repeat the analysis using ln(rob) and ln(mur) in place of ln(vio).
e. In your view, what are the most important remaining threats to the
internal validity of this regression analysis?
f. Based on your analysis, what conclusions would you draw about the effects of concealed weapons laws on these crime rates?
E10.2 Do citizens demand more democracy and political freedom as their incomes grow? That is, is democracy a normal good? On the textbook website, http://www.pearsonhighered.com/stock_watson, you will find the data file Income_Democracy, which contains a panel data set from 195 countries for the years 1960, 1965, . . . , 2000. A detailed description is given in Income_Democracy_Description, available on the website.4 The data- set contains an index of political freedom/democracy for each country in each year, together with data on the country’s income and various demo- graphic controls. (The income and demographic controls are lagged five years relative to the democracy index to allow time for democracy to adjust to changes in these variables.)
a. Is the data set a balanced panel? Explain.
b. The index of political freedom/democracy is labeled Dem_ind.
i. What are the minimum and maximum values of Dem_ind in the data set? What are the mean and standard deviation of Dem_ind
4These data were provided by Daron Acemoglu of M.I.T. and were used in his paper with Simon Johnson, James Robinson, and Pierre Yared, “Income and Democracy,” American Economic Review, 2008, 98:3, 808–842.
c.
Empirical Exercises 379 in the data set? What are the 10th, 25th, 50th, 75th, and 90th
percentiles of its distribution?
ii. What is the value of Dem_ind for the United States in 2000? Averaged over all years in the data set?
iii. What is the value of Dem_ind for Libya in 2000? Averaged over all years in the data set?
iv. List five countries with an average value of Dem_ind greater than 0.95; less than 0.10; and between 0.3 and 0.7.
The logarithm of per capita income is labeled Log_GDPPC. Regress Dem_ind on Log_GDPPC. Use standard errors that are clustered by country.
i. How large is the estimated coefficient on Log_GDPPC? Is the coefficient statistically significant?
ii. If per capita income in a country increases by 20%, by how much is Dem_ind predicted to increase? What is a 95% confidence interval for the prediction? Is the predicted increase in Dem_ind large or small? (Explain what you mean by large or small.)
iii. Why is it important to use clustered standard errors for the regression? Do the results change if you do not use clustered standard errors?
i. Suggest a variable that varies across countries but plausibly varies little—or not at all—over time and that could cause omitted vari- able bias in the regression in (c).
ii. Estimate the regression in (c), allowing for country fixed effects. How do your answers to (c)(i) and (c)(ii) change?
iii. Exclude the data for Azerbaijan and rerun the regression. Do the results change? Why or why not?
iv. Suggest a variable that varies over time but plausibly varies little—or not at all—across countries and that could cause omitted variable bias in the regression in (c).
v. Estimate the regression in (c), allowing for time and country fixed effects. How do your answers to (c)(i) and (c)(ii) change?
vi. There are addition demographic controls in the data set. Should these variables be included in the regression? If so, how do the results change when they are included?
Based on your analysis, what conclusions do you draw about the effects of income on democracy?
d.
e.
380 CHAPTER 10
Regression with Panel Data
APPENDIX
10.1
The State Traffic Fatality Data Set
The data are for the contiguous 48 U.S. states (excluding Alaska and Hawaii), annually for 1982 through 1988. The traffic fatality rate is the number of traffic deaths in a given state in a given year, per 10,000 people living in that state in that year. Traffic fatality data were obtained from the U.S. Department of Transportation Fatal Accident Reporting System. The beer tax (the tax on a case of beer) was obtained from Beer Institute’s Brewers Almanac. The drinking age variables in Table 10.1 are binary vari- ables indicating whether the legal drinking age is 18, 19, or 20. The binary punishment variable in Table 10.1 describes the state’s minimum sentencing requirements for an initial drunk driving conviction: This variable equals 1 if the state requires jail time or community service and equals 0 otherwise (a lesser punishment). Data on the total vehicle miles traveled annually by state were obtained from the Department of Trans- portation. Personal income data were obtained from the U.S. Bureau of Economic Analysis, and the unemployment rate was obtained from the U.S. Bureau of Labor Statistics.
These data were graciously provided by Professor Christopher J. Ruhm of the Depart- ment of Economics at the University of North Carolina.
APPENDIX
10.2
Standard Errors for Fixed Effects Regression
This appendix provides formulas for standard errors for fixed effects regression with a single regressor. These formulas are extended to multiple regressors in Exercise 18.15.
The Asymptotic Distribution of the Fixed Effects
Estimator with Large n
Thefixedeffectsestimator. Thefixedeffectsestimatorofb1istheOLSestimatorobtained using the entity-demeaned regression of Equation (10.14), in which Y∼it is regressed on X∼it, whereY∼ =Y -Y,∼X =X -X,Y =T-1gT Y,andX =T-1gT X.Thefor-
i i t=1 it i t=1 it
it it i it it
mula for the OLS estimator is obtained by replacing Xi – X by X∼it and Yi – Y by ∼Yit in
Standard Errors for Fixed Effects Regression 381 Equation (4.7) and by replacing the single summation in Equation (4.7) by two summa-
tions, one over entities (i = 1, c, n) and one over time periods (t = 1, c, T),5 so
nT∼∼ a aXitYit
bn1 = i=1t=1 .
n T ∼2 a a X it i=1t=1
(10.22)
The derivation of the sampling distribution of bn1 parallels the derivation in Appendix 4.3 of the sampling distribution of the OLS estimator with cross-sectional data. First, substitute ∼Yit = b1X∼it + ∼uit [Equation (10.14)] into the numerator of Equation (10.22) to obtain the panel data counterpart of Equation (4.30):
bn1 = b1 +
Next, rearrange this expression and multiply both sides by 2nT to obtain
1n
Anahi 1T 1nT
(10.23)
1nT∼∼ nT a a Xit u it
i=1t=1 . 1 n T ∼2
nT a a X it i=1t=1
2nT(bn1 – b1) = i = 1 , where hi = a X∼it∼u it and Qn ∼ = a a X∼2it. (10.24) QnX∼ ATt=1 X nTi=1t=1
The scaling factor in Equation (10.24), nT, is the total number of observations.
Distribution and standard errors when n is large. In most panel data applications, n is much larger than T, which motivates approximating sampling distributions by letting n S ∞ while keeping T fixed. Under the fixed effects regression assumptions of Key Con- cept10.3,QnX∼¡p QX∼ =ET-1gTt=1X∼2itasnS∞.Also,hiisi.i.d.overi=1,c,n(by Assumption 2) with mean zero (by Assumption 1) and variance s2h (which is finite by Assumption 3), so by the central limit theorem, 11>ngni = 1hi ¡d N(0, s2h). It follows from Equation (10.24) that
1nT(bn1 – b1) ¡d Na0, s2h b, 2
5The double summation is the extension to double subscripts of a single summation:
n
= a(Xi1 + Xi2 + g+ XiT)
(10.25)
nTnT
aaXit = aaaXitb i=1t=1 i=1 t=1
i=1
=(X11 +X12 +g+X1T)+(X21 +X22 +g+X2T)+g+(Xn1 +Xn2 +g+XnT).
QX∼
382 CHAPTER 10
Regression with Panel Data
From Equation (10.25), the variance of the large-sample distribution of bn1 is
var(bn ) = 1 s2h (10.26) 1 nT2.
QX∼
The clustered standard error formula replaces the population moments in Equation (10.26)
by their sample counterparts:
SE(bn1)= 1s2h, CnT n2 QX∼
1n 1n wheres2hn = n – 1a(hni – hn)2 = n – 1ahn2i
(10.27) wherehn = 21>TgT X∼un isthesamplecounterpartofh 3hn ish inEquation(10.24),
with ∼u it
i
i=1 i=1 t=1itit ii i
replaced by the fixed effects regression residual un 4 and hn = (1>n)gn hn . The it i=1 i
final equality in Equation (10.27) arises because hn = 0, which in turn follows from the residuals and regressors being uncorrelated [Equation (4.34)]. Note that s2hn is just the sam- ple variance of hni [see Equation (3.7)].
The estimator s2hn is a consistent estimator of s2h as n S ∞ , even if there is heteroskedasticity or autocorrelation (Exercise 18.15); thus the clustered standard error in Equation (10.27) is heteroskedasticity- and autocorrelation-consistent. Because the clustered standard error is consistent, the t-statistic testing b1 = b1,0 has a standard normal distribution under the null hypothesis as n S ∞ .
All the foregoing results apply if there are multiple regressors. In addition, if n is large, then the F-statistic testing q restrictions (computed using the clustered variance formula) has its usual asymptotic Fq, ∞ distribution.
Why isn’t the usual heteroskedasticity-robust estimator of Chapter 5 valid for panel data? There are two reasons. The most important reason is that the heteroskedasticity- robust estimator of Chapter 5 does not allow for serial correlation within a cluster. Recall that, for two random variables U and V, var(U + V) = var(U) + var(V) + 2cov(U, V). The variance hi in Equation (10.24) therefore can be written as the sum of variances plus covariances. Let ∼v it = X∼it∼u it; then
(10.28)
T
var(h)=vara 1 ∼vb=1var(∼v +∼v +g+∼v )
i ATaitTi1i2 iT t=1
= T13var(∼vi1) + var(∼vi2) + g + var(∼viT)
+ 2cov(∼vi1,∼vi2) + g+ 2cov(∼viT-1,∼viT)4.
Standard Errors for Fixed Effects Regression 383
The heteroskedasticity-robust variance formula of Chapter 5 misses all the covariances in the final part of Equation (10.28), so if there is serial correlation, the usual heteroskedasticity- robust variance estimator is inconsistent.
The second reason is that if T is small, the estimation of the fixed effects introduces bias into the Chapter 5 heteroskedasticity-robust variance estimator. This problem does not arise in cross-sectional regression.
The one case in which the usual heteroskedasticity-robust standard errors can be used with panel data is with fixed effects regression with T = 2 observations. In this case, fixed effects regression is equivalent to the “before and after” differences regression in Section 10.2, and heteroskedasticity-robust and clustered standard errors are equivalent.
For empirical examples showing the importance of using clustered standard errors in economic panel data, see Bertrand, Duflo, and Mullainathan (2004).
Standard Errors When uit Is Correlated Across Entities. In some cases, uit might be corre- lated across entities. For example, in a study of earnings, suppose that the sampling scheme selects families by simple random sampling, then tracks all siblings within a family. Because the omitted factors that enter the error term could have common elements for siblings, it is not reasonable to assume that the errors are independent for siblings (even though they are independent across families).
In the siblings example, families are natural clusters, or groupings, of observations, where uit is correlated within the cluster but not across clusters. The derivation leading to Equation (10.27) can be modified to allow for clusters across entities (for example, fami- lies) or across both entities and time, as long as there are many clusters.
Distribution and Standard Errors When n Is Small
If n is small and T is large, then it remains possible to use clustered standard errors; how- ever, t-statistics need to be compared with critical values from the tn – 1 tables, and the F-statistic testing q restrictions needs to be compared to the Fq, n – q critical value multiplied by (n – 1)>(n – q). These distributions are valid under the assumptions in Key Concept 10.3, plus some additional assumptions on the joint distribution of Xit and uit over time within an entity. Although the validity of the t-distribution in cross-sectional regression requires normality and homoskedasticity of the regression errors (Section 5.6), neither requirement is needed to justify using the t-distribution with clustered standard errors in panel data when T is large.
To see why the cluster t-statistic has a tn – 1 distribution when n is small and T is large, even if uit is neither normally distributed nor homoskedastic, first note that if T is large, then under additional assumptions,hi in Equation (10.24) will obey a central limit theorem, so hi ¡d N(0, s2h ). (The additional assumptions required for this result are substantial and technical, and we defer further discussion of them to our treatment of time series data
384 CHAPTER 10
Regression with Panel Data
in Chapter 14.) Thus, if T is large, then 2nT(bn1 – b1) in Equation (10.24) is a scaled aver-
age of the n normal random variables hi. Moreover, the clustered formula s2hn in Equation
(10.27) is the usual formula for the sample variance, and if it could be computed using hi,
then (n – 1)s2h >s2h would have a x2n – 1 distribution, so the t-statistic would have a tn – 1 dis-
tribution [see Section 3.6]. Using the residuals to compute hni and s2hn does not change this
conclusion. In the case of multiple regressors, analogous reasoning leads to the conclusion
that the F-statistic testing q restrictions, computed using the cluster variance estimator, is n-1
distributed as (n – q)Fq, n – q. [For example, the 5% critical value for this F-statistic when n = 10andq = 4is(10 – 1) * 4.53 = 6.80,where4.53isthe5%criticalvaluefromthe
10 – 4
F4,6 distribution given in Appendix Table 5B.] Note that, as n increases, the tn – 1 and
n-1 6 (n – q)Fq, n – q distributions approach the usual standard normal and Fq, ∞ distributions.
If both n and T are small, then in general bn1 will not be normally distributed, and clustered standard errors will not provide reliable inference.
6 n-1
Not all software implements clustered standard errors using the tn – 1 and (n – q)Fq, n – q distributions
that apply if n is small, so you should check how your software implements and treats clustered stan- dard errors.
CHAPTER
11
Regression with a Binary Dependent Variable
Two people, identical but for their race, walk into a bank and apply for a mortgage, a large loan so that each can buy an identical house. Does the bank treat them the same way? Are they both equally likely to have their mortgage application accepted? By law they must receive identical treatment. But whether they actually do is a matter of great concern among bank regulators.
Loans are made and denied for many legitimate reasons. For example, if the proposed loan payments take up most or all of the applicant’s monthly income, a loan officer might justifiably deny the loan. Also, even loan officers are human and they can make honest mistakes, so the denial of a single minority applicant does not prove anything about discrimination. Many studies of discrimination thus look for statistical evidence of discrimination, that is, evidence contained in large data sets showing that whites and minorities are treated differently.
But how, precisely, should one check for statistical evidence of discrimination in the mortgage market? A start is to compare the fraction of minority and white applicants who were denied a mortgage. In the data examined in this chapter, gathered from mortgage applications in 1990 in the Boston, Massachusetts, area, 28% of black applicants were denied mortgages but only 9% of white applicants were denied. But this comparison does not really answer the question that opened this chapter, because the black applicants and the white applicants were not necessarily “identical but for their race.” Instead, we need a method for comparing rates of denial, holding other applicant characteristics constant.
This sounds like a job for multiple regression analysis—and it is, but with a twist. The twist is that the dependent variable—whether the applicant is denied—is binary. In Part II, we regularly used binary variables as regressors, and they caused no particular problems. But when the dependent variable is binary, things are more difficult: What does it mean to fit a line to a dependent variable that can take on only two values, 0 and 1?
The answer to this question is to interpret the regression function as a condi- tional probability. This interpretation is discussed in Section 11.1, and it allows us to apply the multiple regression models from Part II to binary dependent variables. Section 11.1 goes over this “linear probability model.” But the predicted probability interpretation also suggests that alternative, nonlinear regression models can do a
385
386
CHAPTER 11
Regression with a Binary Dependent Variable
11.1
Binary Dependent Variables
and the Linear Probability Model
Whether a mortgage application is accepted or denied is one example of a binary variable. Many other important questions also concern binary outcomes. What is the effect of a tuition subsidy on an individual’s decision to go to college? What determines whether a teenager takes up smoking? What determines whether a country receives foreign aid? What determines whether a job applicant is success- ful? In all these examples, the outcome of interest is binary: The student does or does not go to college, the teenager does or does not take up smoking, a country does or does not receive foreign aid, the applicant does or does not get a job.
This section discusses what distinguishes regression with a binary dependent variable from regression with a continuous dependent variable and then turns to the simplest model to use with binary dependent variables, the linear probability model.
Binary Dependent Variables
The application examined in this chapter is whether race is a factor in denying a mortgage application; the binary dependent variable is whether a mortgage appli- cation is denied. The data are a subset of a larger data set compiled by researchers at the Federal Reserve Bank of Boston under the Home Mortgage Disclosure Act (HMDA) and relate to mortgage applications filed in the Boston, Massachusetts, area in 1990. The Boston HMDA data are described in Appendix 11.1.
Mortgage applications are complicated and so is the process by which the bank loan officer makes a decision. The loan officer must forecast whether the
better job modeling these probabilities. These methods, called “probit” and “logit” regression, are discussed in Section 11.2. Section 11.3, which is optional, discusses the method used to estimate the coefficients of the probit and logit regressions, the method of maximum likelihood estimation. In Section 11.4, we apply these meth- ods to the Boston mortgage application data set to see whether there is evidence of racial bias in mortgage lending.
The binary dependent variable considered in this chapter is an example of a dependent variable with a limited range; in other words, it is a limited dependent variable. Models for other types of limited dependent variables, for example, dependent variables that take on multiple discrete values, are surveyed in Appendix 11.3.
FIGURE 11.1
applicant will make his or her loan payments. One important piece of information is the size of the required loan payments relative to the applicant’s income. As anyone who has borrowed money knows, it is much easier to make payments that are 10% of your income than 50%! We therefore begin by looking at the relation- ship between two variables: the binary dependent variable deny, which equals 1 if the mortgage application was denied and equals 0 if it was accepted, and the continuous variable P/I ratio, which is the ratio of the applicant’s anticipated total monthly loan payments to his or her monthly income.
Figure 11.1 presents a scatterplot of deny versus P/I ratio for 127 of the 2380 observations in the data set. (The scatterplot is easier to read using this subset of the data.) This scatterplot looks different from the scatterplots of Part II because the variable deny is binary. Still, it seems to show a relationship between deny and P/I ratio: Few applicants with a payment-to-income ratio less than 0.3 have their application denied, but most applicants with a payment-to-income ratio exceeding 0.4 are denied.
This positive relationship between P/I ratio and deny (the higher the P/I ratio, the greater the fraction of denials) is summarized in Figure 11.1 by the OLS regres- sion line estimated using these 127 observations. As usual, this line plots the pre- dicted value of deny as a function of the regressor, the payment-to-income ratio. For example, when P/I ratio = 0.3, the predicted value of deny is 0.20. But what, pre- cisely, does it mean for the predicted value of the binary variable deny to be 0.20?
Scatterplot of Mortgage Application Denial and the Payment-to-Income Ratio
11.1 Binary Dependent Variables and the Linear Probability Model 387
Mortgage applicants with
a high ratio of debt payments to income (P/I ratio) are more likely to have their application denied (deny = 1 if denied, deny = 0 if approved). The linear probability model
uses a straight line to model the probability of denial, conditional on the P/I ratio.
Deny
1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
– 0.2
– 0.4
0.0 0.1
e denied
e approved
0.6 0.7 0.8
P/I ratio
Mortgag Linear probability model
Mortgag
0.2 0.3
0.4 0.5
388 CHAPTER 11
Regression with a Binary Dependent Variable
The key to answering this question—and more generally to understanding regression with a binary dependent variable—is to interpret the regression as modeling the probability that the dependent variable equals 1. Thus the predicted value of 0.20 is interpreted as meaning that, when P/I ratio is 0.3, the probability of denial is estimated to be 20%. Said differently, if there were many applications with P/I ratio = 0.3, then 20% of them would be denied.
This interpretation follows from two facts. First, from Part II, the population regression function is the expected value of Y given the regressors, E(Y X1, c, Xk). Second, from Section 2.2, if Y is a 0–1 binary variable, its expected value (or mean) is the probability that Y = 1; that is, E(Y) = 0 * Pr(Y = 0) + 1 * Pr(Y = 1) = Pr(Y = 1). In the regression context the expected value is conditional on the value of the regressors, so the probability is conditional on X. Thus for a binary variable, E(YX1, c, Xk) = Pr(Y = 1 X1, c, Xk). In short, for a binary dependent variable, the predicted value from the population regression is the probability that Y = 1, given X.
The linear multiple regression model applied to a binary dependent variable is called the linear probability model: “linear” because it is a straight line and “probability model” because it models the probability that the dependent variable equals 1 (in our example, the probability of loan denial).
The Linear Probability Model
The linear probability model is the name for the multiple regression model of Part II when the dependent variable is binary rather than continuous. Because the dependent variable Y is binary, the population regression function corresponds to the probability that the dependent variable equals 1, given X. The population coefficient b1 on a regressor X is the change in the probability that Y = 1 associ- ated with a unit change in X. Similarly, the OLS predicted value, Yni, computed using the estimated regression function, is the predicted probability that the dependent variable equals 1, and the OLS estimator bn1 estimates the change in the probability that Y = 1 associated with a unit change in X.
Almost all of the tools of Part II carry over to the linear probability model. The coefficients can be estimated by OLS. Ninety-five percent confidence inter- vals can be formed as {1.96 standard errors, hypotheses concerning several coefficients can be tested using the F-statistic discussed in Chapter 7, and inter- actions between variables can be modeled using the methods of Section 8.3. Because the errors of the linear probability model are always heteroskedastic (Exercise 11.8), it is essential that heteroskedasticity-robust standard errors be used for inference.
11.1 Binary Dependent Variables and the Linear Probability Model 389
One tool that does not carry over is the R2. When the dependent variable is continuous, it is possible to imagine a situation in which the R2 equals 1: All the data lie exactly on the regression line. This is impossible when the dependent vari- able is binary, unless the regressors are also binary. Accordingly, the R2 is not a particularly useful statistic here. We return to measures of fit in the next section.
The linear probability model is summarized in Key Concept 11.1.
Application to the Boston HMDA data. The OLS regression of the binary depen- dent variable, deny, against the payment-to-income ratio, P/I ratio, estimated using all 2380 observations in our data set is
deny = -0.080 + 0.604 P>I ratio. (11.1) (0.032) (0.098)
The estimated coefficient on P/I ratio is positive, and the population coefficient is statistically significantly different from zero at the 1% level (the t-statistic is 6.13). Thus applicants with higher debt payments as a fraction of income are more likely to have their application denied. This coefficient can be used to compute the predicted change in the probability of denial, given a change in the regressor.
The Linear Probability Model
The linear probability model is the linear multiple regression model,
KEY CONCEPT
11.1
Yi = b0 + b1X1i + b2X2i + g+bkXki + ui, (11.2)
applied to a binary dependent variable Yi. Because Y is binary, E(Y X1, X2, c, Xk) = Pr(Y = 1 X1, X2, c, Xk), so for the linear probability model,
Pr(Y = 1X1,X2,c,Xk) = b0 + b1X1 + b2X2 + g+bkXk.
The regression coefficient b1 is the change in the probability that Y = 1 associ- ated with a unit change in X1, holding constant the other regressors, and so forth for b2, c , bk. The regression coefficients can be estimated by OLS, and the usual (heteroskedasticity-robust) OLS standard errors can be used for confidence intervals and hypothesis tests.
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For example, according to Equation (11.1), if P/I ratio increases by 0.1, the probability of denial increases by 0.604 * 0.1 ≅ 0.060, that is, by 6.0 percentage points.
The estimated linear probability model in Equation (11.1) can be used to com- pute predicted denial probabilities as a function of P/I ratio. For example, if projected debt payments are 30% of an applicant’s income, P/I ratio is 0.3 and the predicted value from Equation (11.1) is -0.080 + 0.604 * 0.3 = 0.101. That is, according to this linear probability model, an applicant whose projected debt payments are 30% of income has a probability of 10.1% that his or her application will be denied. [This is different from the probability of 20% based on the regression line in Figure 11.1, because that line was estimated using only 127 of the 2380 observations used to estimate Equation (11.1).]
What is the effect of race on the probability of denial, holding constant the P/I ratio? To keep things simple, we focus on differences between black appli- cants and white applicants. To estimate the effect of race, holding constant P/I ratio, we augment Equation (11.1) with a binary regressor that equals 1 if the applicant is black and equals 0 if the applicant is white. The estimated linear probability model is
deny = -0.091 + 0.559P>Iratio + 0.177black. (11.3) (0.029) (0.089) (0.025)
The coefficient on black, 0.177, indicates that an African American applicant has a 17.7% higher probability of having a mortgage application denied than a white applicant, holding constant their payment-to-income ratio. This coefficient is significant at the 1% level (the t-statistic is 7.11).
Taken literally, this estimate suggests that there might be racial bias in mort- gage decisions, but such a conclusion would be premature. Although the payment- to-income ratio plays a role in the loan officer’s decision, so do many other factors, such as the applicant’s earning potential and the individual’s credit history. If any of these variables are correlated with the regressors black or P/I ratio, their omis- sion from Equation (11.3) will cause omitted variable bias. Thus we must defer any conclusions about discrimination in mortgage lending until we complete the more thorough analysis in Section 11.3.
Shortcomingsofthelinearprobabilitymodel. Thelinearitythatmakesthelinear probability model easy to use is also its major flaw. Because probabilities cannot exceed 1, the effect on the probability that Y = 1 of a given change in X must be nonlinear: Although a change in P/I ratio from 0.3 to 0.4 might have a large effect on the probability of denial, once P/I ratio is so large that the loan
is very likely to be denied, increasing P/I ratio further will have little effect. In contrast, in the linear probability model, the effect of a given change in P/I ratio is constant, which leads to predicted probabilities in Figure 11.1 that drop below 0 for very low values of P/I ratio and exceeds 1 for high values! But this is non- sense: A probability cannot be less than 0 or greater than 1. This nonsensical feature is an inevitable consequence of the linear regression. To address this prob- lem, we introduce new nonlinear models specifically designed for binary depen- dent variables, the probit and logit regression models.
11.2
Probit and Logit Regression
Probit and logit1 regression are nonlinear regression models specifically designed for binary dependent variables. Because a regression with a binary dependent variable Y models the probability that Y = 1, it makes sense to adopt a nonlinear formulation that forces the predicted values to be between 0 and 1. Because cumulative probability distribution functions (c.d.f.’s) produce probabilities between 0 and 1 (Section 2.1), they are used in logit and probit regressions. Probit regression uses the standard normal c.d.f. Logit regression, also called logistic regression, uses the “logistic” c.d.f.
Probit Regression
Probit regression with a single regressor. The probit regression model with a single regressor X is
Pr(Y = 1X) = Φ(b0 + b1X), (11.4)
where Φ is the cumulative standard normal distribution function (tabulated in Appendix Table 1).
For example, suppose that Y is the binary mortgage denial variable (deny), X is the payment-to-income ratio (P/I ratio), b0 = – 2, and b1 = 3. What then is the probability of denial if P/I ratio = 0.4? According to Equation (11.4), this probabil- ity is Φ(b0 + b1P/I ratio) = Φ(-2 + 3P/I ratio) = Φ(-2 + 3 * 0.4) =Φ(-0.8). According to the cumulative normal distribution table (Appendix Table 1), Φ(-0.8) = Pr(Z … -0.8) = 21.2%. That is, when P/I ratio is 0.4, the predicted
1Pronounced pro- -bit and lo- -jit.
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Regression with a Binary Dependent Variable
probability that the application will be denied is 21.2%, computed using the probit model with the coefficients b0 = -2 and b1 = 3.
Intheprobitmodel,thetermb0 +b1Xplaystheroleof“z”inthecumulative standard normal distribution table in Appendix Table 1. Thus the calculation in the previous paragraph can, equivalently, be done by first computing the “z-value,”z = b0 + b1X = -2 + 3 * 0.4 = -0.8,andthenlookinguptheprob- ability in the tail of the normal distribution to the left of z = – 0.8, which is 21.2%.
The probit coefficient b1 in Equation (11.4) is the change in the z-value asso- ciated with a unit change in X. If b1 is positive, an increase in X increases the z-value and thus increases the probability that Y = 1; if b1 is negative, an increase in X decreases the probability that Y = 1. Although the effect of X on the z-value is linear, its effect on the probability is nonlinear. Thus in practice the easiest way to interpret the coefficients of a probit model is to compute the predicted prob- ability, or the change in the predicted probability, for one or more values of the regressors. When there is just one regressor, the predicted probability can be plotted as a function of X.
Figure 11.2 plots the estimated regression function produced by the probit regression of deny on P/I ratio for the 127 observations in the scatterplot. The esti- mated probit regression function has a stretched “S” shape: It is nearly 0 and flat for small values of P/I ratio, it turns and increases for intermediate values, and it
FIGURE 11.2
The probit model uses the
cumulative normal distribution 1.4
Probit Model of the Probability of Denial, Given P/I Ratio
Deny
Probit model
Mortgag
Mortgag
function to model the probability
of denial given the payment-to-
income ratio or, more generally, 1.0 to model Pr(Y = 1 X). Unlike 0.8 the linear probability model,
the probit conditional
probabilities are always
between 0 and 1. 0.2
0.0
– 0.2
– 0.4
0.0 0.1
1.2
0.6 0.4
e denied
e approved
0.6 0.7 0.8
P/I ratio
0.2 0.3
0.4 0.5
flattens out again and is nearly 1 for large values. For small values of the payment- to-income ratio, the probability of denial is small. For example, for P/I ratio = 0.2, the estimated probability of denial based on the estimated probit function in Fig- ure 11.2 is Pr(deny = 1 P/I ratio = 0.2) = 2.1% When P/I ratio = 0.3, the esti- mated probability of denial is 16.1%. When P/I ratio = 0.4, the probability of denial increases sharply to 51.9%, and when P/I ratio = 0.6, the denial probability is 98.3%. According to this estimated probit model, for applicants with high payment- to-income ratios, the probability of denial is nearly 1.
Probitregressionwithmultipleregressors. Inalltheregressionproblemswehave studied so far, leaving out a determinant of Y that is correlated with the included regressors results in omitted variable bias. Probit regression is no exception. In linear regression, the solution is to include the additional variable as a regressor. This is also the solution to omitted variable bias in probit regression.
The probit model with multiple regressors extends the single-regressor probit model by adding regressors to compute the z-value. Accordingly, the probit pop- ulation regression model with two regressors, X1 and X2, is
Pr(Y = 1X1,X2) = Φ(b0 + b1X1 + b2X2). (11.5)
For example, suppose that b0 = -1.6, b1 = 2, and b2 = 0.5. If X1 = 0.4 and X2 = 1,thez-valueisz = -1.6 + 2 * 0.4 + 0.5 * 1 = -0.3.So,theprobability that Y=1 given X1 =0.4 and X2 =1 is Pr(Y=1X1 =0.4,X2 =1)= Φ(-0.3) = 38%.
Effect of a change in X. In general, the regression model can be used to determine the expected change in Y arising from a change in X. When Y is binary, its condi- tional expectation is the conditional probability that it equals 1, so the expected change in Y arising from a change in X is the change in the probability that Y = 1.
Recall from Section 8.1 that, when the population regression function is a nonlinear function of X, this expected change is estimated in three steps: First, compute the predicted value at the original value of X using the estimated regression function; next, computethepredictedvalueatthechangedvalueofX,X + ∆X;finally,computethe difference between the two predicted values. This procedure is summarized in Key Concept 8.1. As emphasized in Section 8.1, this method always works for computing predicted effects of a change in X, no matter how complicated the nonlinear model. When applied to the probit model, the method of Key Concept 8.1 yields the estimated effect on the probability that Y = 1 of a change in X.
The probit regression model, predicted probabilities, and estimated effects are summarized in Key Concept 11.2.
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394 CHAPTER 11 Regression with a Binary Dependent Variable
KEY CONCEPT
11.2
The Probit Model, Predicted Probabilities, and Estimated Effects
The population probit model with multiple regressors is
Pr(Y = 1X1, X2, c, Xk) = Φ(b0 + b1X1 + b2X2 + g+ bkXk)
where the dependent variable Y is binary, Φ is the cumulative standard normal distri- bution function, and X1, X2, and so on are regressors. The model is best interpreted by computing predicted probabilities and the effect of a change in a regressor.
The predicted probability that Y = 1, given values of X1, X2, c, Xk, is cal- culated by computing the z-value, z = b0 + b1X1 + b2X2 + g + bkXk, and then looking up this z-value in the normal distribution table (Appendix Table 1).
The coefficient b1 is the change in the z-value arising from a unit change in X1, holding constant X2, c, Xk.
The effect on the predicted probability of a change in a regressor is computed by (1) computing the predicted probability for the initial value of the regres- sors, (2) computing the predicted probability for the new or changed value of the regressors, and (3) taking their difference.
(11.6)
Application to the mortgage data. As an illustration, we fit a probit model to the 2380 observations in our data set on mortgage denial (deny) and the payment-to- income ratio (P/I ratio):
Pr(deny = 1 0 P>I ratio) = Φ(-2.19 + 2.97P>I ratio). (11.7) (0.16) (0.47)
The estimated coefficients of – 2.19 and 2.97 are difficult to interpret because they affect the probability of denial via the z-value. Indeed, the only things that can be readily concluded from the estimated probit regression in Equation (11.7) are that the payment-to-income ratio is positively related to probability of denial (the coefficient on P/I ratio is positive) and that this relationship is statistically signifi- cant (t = 2.97>0.47 = 6.32).
What is the change in the predicted probability that an application will be denied when the payment-to-income ratio increases from 0.3 to 0.4? To answer this question, we follow the procedure in Key Concept 8.1: Compute the probability
ofdenialforP/Iratio = 0.3andthenforP/Iratio = 0.4,andthencomputethediffer- ence. The probability of denial when P/I ratio = 0.3 is Φ( – 2.19 + 2.97 * 0.3) = Φ( – 1.30) = 0.097. The probability of denial when P/I ratio = 0.4 is Φ( – 2.19 + 2.97 * 0.4) = Φ(-1.00) = 0.159. The estimated change in the probability of denial is 0.159 – 0.097 = 0.062. That is, an increase in the payment-to-income ratio from 0.3 to 0.4 is associated with an increase in the probability of denial of 6.2 percentage points, from 9.7% to 15.9%.
Because the probit regression function is nonlinear, the effect of a change in X depends on the starting value of X. For example, if P/I ratio = 0.5, the esti- mated denial probability based on Equation (11.7) is Φ(-2.19 + 2.97 * 0.5) = Φ(-0.71) = 0.239. Thus the change in the predicted probability when P/I ratio increases from 0.4 to 0.5 is 0.239 – 0.159, or 8.0 percentage points, larger than the increase of 6.2 percentage points when P/I ratio increases from 0.3 to 0.4.
What is the effect of race on the probability of mortgage denial, holding con- stant the payment-to-income ratio? To estimate this effect, we estimate a probit regression with both P/I ratio and black as regressors:
Pr(deny = 1P>I ratio, black) = Φ(-2.26 + 2.74 P>I ratio + 0.71black). (11.8) (0.16) (0.44) (0.083)
Again, the values of the coefficients are difficult to interpret but the sign and statis- tical significance are not. The coefficient on black is positive, indicating that an African American applicant has a higher probability of denial than a white appli- cant, holding constant their payment-to-income ratio. This coefficient is statistically significant at the 1% level (the t-statistic on the coefficient multiplying black is 8.55). For a white applicant with P/I ratio = 0.3, the predicted denial probability is 7.5%, while for a black applicant with P/I ratio = 0.3, it is 23.3%; the difference in denial probabilities between these two hypothetical applicants is 15.8 percentage points.
Estimation of the probit coefficients. The probit coefficients reported here were estimated using the method of maximum likelihood, which produces efficient (minimum variance) estimators in a wide variety of applications, including regres- sion with a binary dependent variable. The maximum likelihood estimator is con- sistent and normally distributed in large samples, so t-statistics and confidence intervals for the coefficients can be constructed in the usual way.
Regression software for estimating probit models typically uses maximum likelihood estimation, so this is a simple method to apply in practice. Standard errors produced by such software can be used in the same way as the standard errors of regression coefficients; for example, a 95% confidence interval for the
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396 CHAPTER 11 Regression with a Binary Dependent Variable
Logit Regression
11.3
KEY CONCEPT
The population logit model of the binary dependent variable Y with multiple regressors is
Pr(Y = 1X1,X2,c,Xk) = F(b0 + b1X1 + b2X2 + g+bkXk) =1. (11.9)
Logit regression is similar to probit regression except that the cumulative distri- bution function is different.
1 + e-(b0 +b1X1 +b2X2 +g+ bkXk)
true probit coefficient can be constructed as the estimated coefficient {1.96 standard errors. Similarly, F-statistics computed using maximum likelihood estimators can be used to test joint hypotheses. Maximum likelihood estima- tion is discussed further in Section 11.3, with additional details given in Appendix 11.2.
Logit Regression
The logit regression model. The logit regression model is similar to the probit regression model except that the cumulative standard normal distribution func- tion Φ in Equation (11.6) is replaced by the cumulative standard logistic distribu- tion function, which we denote by F. Logit regression is summarized in Key Concept 11.3. The logistic cumulative distribution function has a specific func- tional form, defined in terms of the exponential function, which is given as the final expression in Equation (11.9).
As with probit, the logit coefficients are best interpreted by computing pre- dicted probabilities and differences in predicted probabilities.
The coefficients of the logit model can be estimated by maximum likelihood. The maximum likelihood estimator is consistent and normally distributed in large samples, so t-statistics and confidence intervals for the coefficients can be con- structed in the usual way.
The logit and probit regression functions are similar. This is illustrated in Figure 11.3, which graphs the probit and logit regression functions for the dependent variable deny and the single regressor P/I ratio, estimated by maximum likelihood
11.2 Probit and Logit Regression 397 FIGURE 11.3 Probit and Logit Models of the Probability of Denial, Given P/I Ratio
These logit and probit
models produce nearly 1.4
identical estimates of the
probability that a mortgage application will be denied, 1.0 given the payment-to-income 0.8 ratio.
– 0.2 – 0.4
0.0 0.1
Deny
1.2
0.6 0.4 0.2 0.0
Probit model
0.2 0.3
Logit model
0.4 0.5 0.6
Mortgage denied
Mortgage approved
0.7 0.8
P/I ratio
using the same 127 observations as in Figures 11.1 and 11.2. The differences between the two functions are small.
Historically, the main motivation for logit regression was that the logistic cumulative distribution function could be computed faster than the normal cumu- lative distribution function. With the advent of more efficient computers, this distinction is no longer important.
Application to the Boston HMDA data. A logit regression of deny against P/I ratio and black, using the 2380 observations in the data set, yields the estimated regression function
Pr(deny = 1P>I ratio, black) = F(-4.13 + 5.37P>I ratio + 1.27black). (11.10) (0.35) (0.96) (0.15)
The coefficient on black is positive and statistically significant at the 1% level (the t-statistic is 8.47). The predicted denial probability of a white applicant with P/Iratio = 0.3 is 1>31 + e-(-4.13+5.37*0.3+1.27*0)4 = 1>31 + e2.524 = 0.074, or 7.4%. The predicted denial probability of an African American applicant with P/I ratio = 0.3 is 1>31 + e1.254 = 0.222, or 22.2 %, so the difference between the two probabilities is 14.8 percentage points.
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11.3
Estimation and Inference in the Logit and Probit Models2
The nonlinear models studied in Sections 8.2 and 8.3 are nonlinear functions of the independent variables but are linear functions of the unknown coefficients (“parameters”). Consequently, the unknown coefficients of those nonlinear regression functions can be estimated by OLS. In contrast, the probit and logit regression functions are a nonlinear function of the coefficients. That is, the probit coefficients b0, b1, c, bk, in Equation (11.6) appear inside the cumulative stan- dard normal distribution function Φ, and the logit coefficients in Equation (11.9)
2This section contains more advanced material that can be skipped without loss of continuity.
Comparing the Linear Probability, Probit,
and Logit Models
All three models—linear probability, probit, and logit—are just approximations to the unknown population regression function E(YX) = Pr(Y = 1X). The linear probability model is easiest to use and to interpret, but it cannot capture the nonlinear nature of the true population regression function. Probit and logit regressions model this nonlinearity in the probabilities, but their regression coef- ficients are more difficult to interpret. So which should you use in practice?
There is no one right answer, and different researchers use different models. Probit and logit regressions frequently produce similar results. For example, according to the estimated probit model in Equation (11.8), the difference in denial probabilities between a black applicant and a white applicant with P/I ratio = 0.3 was estimated to be 15.8 percentage points, whereas the logit esti- mate of this gap, based on Equation (11.10), was 14.9 percentage points. For prac- tical purposes the two estimates are very similar. One way to choose between logit and probit is to pick the method that is easiest to use in your statistical software.
The linear probability model provides the least sensible approximation to the nonlinear population regression function. Even so, in some data sets there may be few extreme values of the regressors, in which case the linear probability model still can provide an adequate approximation. In the denial probability regression in Equation (11.3), the estimated black/white gap from the linear probability model is 17.7 percentage points, larger than the probit and logit estimates but still qualitatively similar. The only way to know this, however, is to estimate both a linear and nonlinear model and to compare their predicted probabilities.
11.3 Estimation and Inference in the Logit and Probit Models 399
appear inside the cumulative standard logistic distribution function F. Because the population regression function is a nonlinear function of the coefficients b0, b1, c, bk, those coefficients cannot be estimated by OLS.
This section provides an introduction to the standard method for estimation of probit and logit coefficients, maximum likelihood; additional mathematical details are given in Appendix 11.2. Because it is built into modern statistical software, maximum likelihood estimation of the probit and logit coefficients is easy in practice. The theory of maximum likelihood estimation, however, is more complicated than the theory of least squares. We therefore first discuss another estimation method, nonlinear least squares, before turning to maximum likelihood.
Nonlinear Least Squares Estimation
Nonlinear least squares is a general method for estimating the unknown param- eters of a regression function when, like the probit coefficients, those parameters enter the population regression function nonlinearly. The nonlinear least squares estimator, which was introduced in Appendix 8.1, extends the OLS estimator to regression functions that are nonlinear functions of the parameters. Like OLS, nonlinear least squares finds the values of the parameters that minimize the sum of squared prediction mistakes produced by the model.
To be concrete, consider the nonlinear least squares estimator of the param- eters of the probit model. The conditional expectation of Y given the X’s is E(YX1, c, Xk) = Pr(Y = 1X1, c, Xk) = Φ(b0 + b1X1 + g+ bkXk). Estimation by nonlinear least squares fits this conditional expectation function, which is a nonlinear function of the parameters, to the dependent variable. That is, the nonlinear least squares estimator of the probit coefficients are those values of b0, c, bk that minimize the sum of squared prediction mistakes:
an 3Yi – Φ(b0 + b1X1i + g+ bkXki)42. (11.11) i=1
The nonlinear least squares estimator shares two key properties with the OLS estimator in linear regression: It is consistent (the probability that it is close to the true value approaches 1 as the sample size gets large), and it is normally distrib- uted in large samples. There are, however, estimators that have a smaller variance than the nonlinear least squares estimator; that is, the nonlinear least squares estimator is inefficient. For this reason, the nonlinear least squares estimator of the probit coefficients is rarely used in practice, and instead the parameters are estimated by maximum likelihood.
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Maximum Likelihood Estimation
The likelihood function is the joint probability distribution of the data, treated as a function of the unknown coefficients. The maximum likelihood estimator (MLE) of the unknown coefficients consists of the values of the coefficients that maximize the likelihood function. Because the MLE chooses the unknown coef- ficients to maximize the likelihood function, which is in turn the joint probability distribution, in effect the MLE chooses the values of the parameters to maximize the probability of drawing the data that are actually observed. In this sense, the MLEs are the parameter values “most likely” to have produced the data.
To illustrate maximum likelihood estimation, consider two i.i.d. observations, Y1 and Y2, on a binary dependent variable with no regressors. Thus Y is a Ber- noulli random variable, and the only unknown parameter to estimate is the prob- ability p that Y = 1, which is also the mean of Y.
To obtain the maximum likelihood estimator, we need an expression for the likelihood function, which in turn requires an expression for the joint probability distribution of the data. The joint probability distribution of the two observations Y1 and Y2 is Pr (Y1 = y1, Y2 = y2). Because Y1 and Y2 are independently distrib- uted, the joint distribution is the product of the individual distributions [Equation (2.23)], so Pr(Y1 = y1, Y2 = y2) = Pr(Y1 = y1)Pr(Y2 = y2). The Bernoulli distri- bution can be summarized in the formula Pr(Y = y) = py(1 – p)1-y: When y=1,Pr(Y=1)=p1(1-p)0 =p,andwheny=0,Pr(Y=0)=p0(1-p)1= 1 – p. Thus the joint probability distribution of Y1 and Y2 is Pr(Y1 = y1, Y2 = y2) = 3py1(1 – p)1-y14 * 3py2(1 – p)1-y24 = p(y1+y2)(1 – p)2-(y1+y2).
The likelihood function is the joint probability distribution, treated as a function of the unknown coefficients. For n = 2 i.i.d. observations on Bernoulli random variables, the likelihood function is
f(p; Y1, Y2) = p(Y1 + Y2)(1 – p)2 – (Y1 + Y2). (11.12)
The maximum likelihood estimator of p is the value of p that maximizes the like- lihood function in Equation (11.12). As with all maximization or minimization problems, this can be done by trial and error; that is, you can try different values of p and compute the likelihood f(p; Y1, Y2) until you are satisfied that you have maximized this function. In this example, however, maximizing the likelihood function using calculus produces a simple formula for the MLE: The MLE is pn = 12 (Y1 + Y2). In other words, the MLE of p is just the sample average! In fact, for general n, the MLE pn of the Bernoulli probability p is the sample average; that is, pn = Y (this is shown in Appendix 11.2). In this example, the MLE is the usual estimator of p, the fraction of times Yi = 1 in the sample.
11.3 Estimation and Inference in the Logit and Probit Models 401
This example is similar to the problem of estimating the unknown coefficients of the probit and logit regression models. In those models, the success probability p is not constant, but rather depends on X; that is, it is the success probability con- ditional on X, which is given in Equation (11.6) for the probit model and Equation (11.9) for the logit model. Thus the probit and logit likelihood functions are similar to the likelihood function in Equation (11.12) except that the success probability varies from one observation to the next (because it depends on Xi). Expressions for the probit and logit likelihood functions are given in Appendix 11.2.
Like the nonlinear least squares estimator, the MLE is consistent and nor- mally distributed in large samples. Because regression software commonly com- putes the MLE of the probit coefficients, this estimator is easy to use in practice. All the estimated probit and logit coefficients reported in this chapter are MLEs.
Statistical inference based on the MLE. Because the MLE is normally distributed in large samples, statistical inference about the probit and logit coefficients based on the MLE proceeds in the same way as inference about the linear regression function coefficients based on the OLS estimator. That is, hypothesis tests are performed using the t-statistic and 95% confidence intervals are formed as { 1.96 standard errors. Tests of joint hypotheses on multiple coefficients use the F-statistic in a way similar to that discussed in Chapter 7 for the linear regression model. All of this is completely analogous to statistical inference in the linear regression model.
An important practical point is that some statistical software reports tests of joint hypotheses using the F-statistic, while other software uses the chi-squared statistic. The chi-squared statistic is q * F, where q is the number of restrictions being tested. Because the F-statistic is, under the null hypothesis, distributed as x2q > q in large samples, q * F is distributed as x2q in large samples. Because the two approaches differ only in whether they divide by q, they produce identical infer- ences, but you need to know which approach is implemented in your software so that you use the correct critical values.
Measures of Fit
In Section 11.1, it was mentioned that the R2 is a poor measure of fit for the linear probability model. This is also true for probit and logit regression. Two measures of fit for models with binary dependent variables are the “fraction correctly pre- dicted” and the “pseudo-R2.” The fraction correctly predicted uses the following rule: If Yi = 1 and the predicted probability exceeds 50% or if Yi = 0 and the predicted probability is less than 50%, then Yi is said to be correctly predicted.
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11.4
Application to the Boston HMDA Data
The regressions of the previous two sections indicated that denial rates were higher for black than white applicants, holding constant their payment-to-income ratio. Loan officers, however, legitimately weigh many factors when deciding on a mortgage application, and if any of those other factors differ systematically by race, the estimators considered so far have omitted variable bias.
In this section, we take a closer look at whether there is statistical evidence of discrimination in the Boston HMDA data. Specifically, our objective is to esti- mate the effect of race on the probability of denial, holding constant those appli- cant characteristics that a loan officer might legally consider when deciding on a mortgage application.
The most important variables available to loan officers through the mortgage applications in the Boston HMDA data set are listed in Table 11.1; these are the variables we will focus on in our empirical models of loan decisions. The first two variables are direct measures of the financial burden the proposed loan would place on the applicant, measured in terms of his or her income. The first of these is the P/I ratio; the second is the ratio of housing-related expenses to income. The next variable is the size of the loan, relative to the assessed value of the home; if the loan-to-value ratio is nearly 1, the bank might have trouble recouping the full amount of the loan if the applicant defaults on the loan and the bank forecloses. The final three financial variables summarize the applicant’s credit history. If an appli- cant has been unreliable paying off debts in the past, the loan officer legitimately
Otherwise, Yi is said to be incorrectly predicted. The “fraction correctly predicted” is the fraction of the n observations Y1, c,Yn that are correctly predicted.
An advantage of this measure of fit is that it is easy to understand. A disadvan- tage is that it does not reflect the quality of the prediction: If Yi = 1, the observation is treated as correctly predicted whether the predicted probability is 51% or 90%.
The pseudo-R2 measures the fit of the model using the likelihood function. Because the MLE maximizes the likelihood function, adding another regressor to a probit or logit model increases the value of the maximized likelihood, just like adding a regressor necessarily reduces the sum of squared residuals in linear regression by OLS. This suggests measuring the quality of fit of a probit model by comparing values of the maximized likelihood function with all the regressors to the value of the likelihood with none. This is, in fact, what the pseudo-R2 does. A formula for the pseudo-R2 is given in Appendix 11.2.
11.4 Application to the Boston HMDA Data 403 TABLE 11.1 Variables Included in Regression Models of Mortgage Decisions
Variable
Financial Variables
P/I ratio
housing expense-to- income ratio
loan-to-value ratio consumer credit score
mortgage credit score
public bad credit record
Definition
Ratio of total monthly debt payments to total monthly income Ratio of monthly housing expenses to total monthly income
Ratio of size of loan to assessed value of property
1 if no “slow” payments or delinquencies
2 if one or two slow payments or delinquencies
3 if more than two slow payments
4 if insufficient credit history for determination
5 if delinquent credit history with payments 60 days overdue 6 if delinquent credit history with payments 90 days overdue
1 if no late mortgage payments
2 if no mortgage payment history
3 if one or two late mortgage payments
4 if more than two late mortgage payments
1 if any public record of credit problems (bankruptcy, charge-offs, collection actions)
0 otherwise
Sample Average
0.331 0.255
0.738 2.1
1.7
0.074
0.020
0.116 0.393 0.984 3.8 0.288 0.142 0.120
Additional Applicant Characteristics
denied mortgage insurance
1 if applicant applied for mortgage insurance and was denied, 0 otherwise
1 if self-employed, 0 otherwise
1 if applicant reported being single, 0 otherwise
1 if applicant graduated from high school, 0 otherwise
1989 Massachusetts unemployment rate in the applicant’s industry 1 if unit is a condominium, 0 otherwise
1 if applicant is black, 0 if white
1 if mortgage application denied, 0 otherwise
self-employed
single
high school diploma unemployment rate condominium
black
deny
might worry about the applicant’s ability or desire to make mortgage payments in the future. The three variables measure different types of credit histories, which the loan officer might weigh differently. The first concerns consumer credit, such as credit card debt; the second is previous mortgage payment history; and the third
404 CHAPTER 11
Regression with a Binary Dependent Variable
measures credit problems so severe that they appeared in a public legal record, such as filing for bankruptcy.
Table 11.1 also lists some other variables relevant to the loan officer’s deci- sion. Sometimes the applicant must apply for private mortgage insurance.3 The loan officer knows whether that application was denied, and that denial would weigh negatively with the loan officer. The next three variables, which concern the employment status, marital status, and educational attainment of the applicant, relate to the prospective ability of the applicant to repay. In the event of foreclo- sure, characteristics of the property are relevant as well, and the next variable indi- cates whether the property is a condominium. The final two variables in Table 11.1 are whether the applicant is black or white and whether the application was denied or accepted. In these data, 14.2% of applicants are black and 12.0% of applications are denied.
Table 11.2 presents regression results based on these variables. The base specifications, reported in columns (1) through (3), include the financial variables in Table 11.1 plus the variables indicating whether private mortgage insurance was denied and whether the applicant is self-employed. In the 1990s, loan officers commonly used thresholds, or cutoff values, for the loan-to-value ratio, so the base specification for that variable uses binary variables for whether the loan- to-value ratio is high (Ú 0.95), medium (between 0.8 and 0.95), or low (60.8; this case is omitted to avoid perfect multicollinearity). The regressors in the first three columns are similar to those in the base specification considered by the Federal Reserve Bank of Boston researchers in their original analysis of these data.4 The regressions in columns (1) through (3) differ only in how the denial probability is modeled, using a linear probability model, a logit model, and a pro- bit model, respectively.
Because the regression in column (1) is a linear probability model, its coeffi- cients are estimated changes in predicted probabilities arising from a unit change in the independent variable. Accordingly, an increase in P/I ratio of 0.1 is estimated to
3Mortgage insurance is an insurance policy under which the insurance company makes the monthly payment to the bank if the borrower defaults. During the period of this study, if the loan-to-value ratio exceeds 80%, the applicant typically was required to buy mortgage insurance.
4The difference between the regressors in columns (1) through (3) and those in Munnell et al. (1996), table 2(1), is that Munnell et al. include additional indicators for the location of the home and the identity of the lender, data that are not publicly available; an indicator for a multifamily home, which is irrelevant here because our subset focuses on single-family homes; and net wealth, which we omit because this variable has a few very large positive and negative values and thus risks making the results sensitive to a few specific outlier observations.
TABLE 11.2 Mortgage Denial Regressions Using the Boston HMDA Data
Dependent variable: deny = 1 if mortgage application is denied, = 0 if accepted; 2380 observations.
Regression Model LPM
Regressor (1)
black 0.084** (0.023)
Logit
(2)
0.688** (0.182)
4.76** (1.33)
-0.11 (1.29)
0.46** (0.16)
1.49** (0.32)
0.29** (0.04)
0.28* (0.14)
1.23** (0.20)
4.55** (0.57)
0.67** (0.21)
Probit
(3)
0.389** (0.098)
2.44** (0.61)
-0.18 (0.68)
0.21** (0.08)
0.79** (0.18)
0.15** (0.02)
0.15* (0.07)
0.70** (0.12)
2.56** (0.30)
0.36** (0.11)
Probit
(4)
0.371** (0.099)
2.46** (0.60)
-0.30 (0.68)
0.22** (0.08)
0.79** (0.18)
0.16** (0.02)
0.11 (0.08)
0.70** (0.12)
2.59** (0.29)
0.35** (0.11)
0.23** (0.08)
-0.61** (0.23)
0.03 (0.02)
no
-2.57** (0.34)
Probit
(5)
0.363** (0.100)
2.62** (0.61)
-0.50 (0.70)
0.22** (0.08)
0.84** (0.18)
0.34** (0.11)
0.16 (0.10)
0.72** (0.12)
2.59** (0.30)
0.34** (0.11)
0.23** (0.08)
-0.60* (0.24)
0.03 (0.02)
−0.05 (0.09)
yes
-2.90** (0.39)
Probit
(6)
0.246 (0.448)
2.57** (0.66)
-0.54 (0.74)
0.22** (0.08)
0.79** (0.18)
0.16** (0.02)
0.11 (0.08)
0.70** (0.12)
2.59** (0.29)
0.35** (0.11)
0.23** (0.08)
-0.62** (0.23)
0.03 (0.02)
-0.58 (1.47)
1.23 (1.69)
no
-2.54** (0.35)
(continued)
11.4 Application to the Boston HMDA Data 405
P/I ratio
housing expense-to- income ratio
medium loan-to-value ratio
(0.80 … loan-value ratio … 0.95)
high loan-to-value ratio (loan-value ratio 7 0.95)
consumer credit score
mortgage credit score
public bad credit record
denied mortgage insurance
self-employed
single
high school diploma
unemployment rate
condominium
black * P/I ratio
black * housing expense- to-income ratio
additional credit rating indicator variables
0.449** (0.114)
-0.048 (0.110)
0.031* (0.013)
0.189** (0.050)
0.031** (0.005)
0.021 (0.011)
0.197** (0.035)
0.702** (0.045)
0.060** (0.021)
constant -0.183** (0.028)
no
no
-5.71** (0.48)
no
-3.04** (0.23)
406 CHAPTER 11 Regression with a Binary Dependent Variable
(Table 11.2 continued)
F-Statistics and p-Values Testing Exclusion of Groups of Variables
applicant single;
high school diploma; industry unemployment rate
additional credit rating indicator variables
race interactions and black
race interactions only
difference in predicted probability of denial, white vs. black (percentage points)
1.22 (0.291)
(1)
(2)
(3)
(4)
5.85
( 6 0.001)
(5) (6)
5.22 5.79 (0.001) ( 6 0.001)
4.96 (0.002)
0.27 (0.766)
8.4%
6.0%
7.1%
6.6%
6.3% 6.5%
These regressions were estimated using the n = 2380 observations in the Boston HMDA data set described in Appendix 11.1. The linear probability model was estimated by OLS, and probit and logit regressions were estimated by maximum likelihood. Standard errors are given in parentheses under the coefficients, and p-values are given in parentheses under the F-statistics. The change in pre- dicted probability in the final row was computed for a hypothetical applicant whose values of the regressors, other than race, equal the sample mean. Individual coefficients are statistically significant at the *5% or **1% level.
increase the probability of denial by 4.5 percentage points (the coefficient on P/I ratio in column (1) is 0.449, and 0.449 * 0.1 ≅ 0.045). Similarly, having a high loan-to-value ratio increases the probability of denial: A loan-to-value ratio exceeding 95% is associated with an 18.9 percentage point increase (the coeffi- cient is 0.189) in the denial probability, relative to the omitted case of a loan-to- value ratio less than 80%, holding the other variables in column (1) constant. Applicants with a poor credit rating also have a more difficult time getting a loan, all else being constant, although interestingly the coefficient on consumer credit is statistically significant but the coefficient on mortgage credit is not. Applicants with a public record of credit problems, such as filing for bankruptcy, have much greater difficulty obtaining a loan: All else equal, a public bad credit record is estimated to increase the probability of denial by 0.197, or 19.7 percentage points. Being denied private mortgage insurance is estimated to be virtually decisive: The estimated coefficient of 0.702 means that being denied mortgage insurance increases your chance of being denied a mortgage by 70.2 percentage points, all else equal. Of the nine variables (other than race) in the regression, the coefficients on all but two are statistically significant at the 5% level, which is consistent with loan officers’ considering many factors when they make their decisions.
11.4 Application to the Boston HMDA Data 407
The coefficient on black in regression (1) is 0.084, indicating that the differ- ence in denial probabilities for black and white applicants is 8.4 percentage points, holding constant the other variables in the regression. This is statistically signifi- cant at the 1% significance level (t = 3.65).
The logit and probit estimates reported in columns (2) and (3) yield similar conclusions. In the logit and probit regressions, eight of the nine coefficients on variables other than race are individually statistically significantly different from zero at the 5% level, and the coefficient on black is statistically significant at the 1% level. As discussed in Section 11.2, because these models are nonlinear, specific values of all the regressors must be chosen to compute the difference in predicted probabilities for white applicants and black applicants. A conven- tional way to make this choice is to consider an “average” applicant who has the sample average values of all the regressors other than race. The final row in Table 11.2 reports this estimated difference in probabilities, evaluated for this average applicant. The estimated racial differentials are similar to each other: 8.4 percentage points for the linear probability model [column (1)], 6.0 percent- age points for the logit model [column (2)], and 7.1 percentage points for the probit model [column (3)]. These estimated race effects and the coefficients on black are less than in the regressions of the previous sections, in which the only regressors were P/I ratio and black, indicating that those earlier estimates had omitted variable bias.
The regressions in columns (4) through (6) investigate the sensitivity of the results in column (3) to changes in the regression specification. Column (4) modifies column (3) by including additional applicant characteristics. These characteristics help to predict whether the loan is denied; for example, having at least a high school diploma reduces the probability of denial (the estimate is negative and the coefficient is statistically significant at the 1% level). However, controlling for these personal characteristics does not change the estimated coefficient on black or the estimated difference in denial probabilities (6.6%) in an important way.
Column (5) breaks out the six consumer credit categories and four mortgage credit categories to test the null hypothesis that these two variables enter linearly; this regression also adds a variable indicating whether the property is a condo- minium. The null hypothesis that the credit rating variables enter the expression for the z-value linearly is not rejected, nor is the condominium indicator signifi- cant, at the 5% level. Most importantly, the estimated racial difference in denial probabilities (6.3%) is essentially the same as in columns (3) and (4).
Column (6) examines whether there are interactions. Are different standards applied to evaluating the payment-to-income and housing expense-to-income ratios
408 CHAPTER 11
Regression with a Binary Dependent Variable
for black applicants versus white applicants? The answer appears to be no: The interaction terms are not jointly statistically significant at the 5% level. However, race continues to have a significant effect, because the race indicator and the interaction terms are jointly statistically significant at the 1% level. Again, the estimated racial difference in denial probabilities (6.5%) is essentially the same as in the other probit regressions.
In all six specifications, the effect of race on the denial probability, hold- ing other applicant characteristics constant, is statistically significant at the 1% level. The estimated difference in denial probabilities between black appli- cants and white applicants ranges from 6.0 percentage points to 8.4 percentage points.
One way to assess whether this differential is large or small is to return to a variation on the question posed at the beginning of this chapter. Suppose two individuals apply for a mortgage, one white and one black, but otherwise having the same values of the other independent variables in regression (3); specifically, aside from race, the values of the other variables in regression (3) are the sample average values in the HMDA data set. The white applicant faces a 7.4% chance of denial, but the black applicant faces a 14.5% chance of denial. The estimated racial difference in denial probabilities, 7.1 percentage points, means that the black applicant is nearly twice as likely to be denied as the white applicant.
The results in Table 11.2 (and in the original Boston Fed study) provide sta- tistical evidence of racial patterns in mortgage denial that, by law, ought not be there. This evidence played an important role in spurring policy changes by bank regulators.5 But economists love a good argument, and not surprisingly these results have also stimulated a vigorous debate.
Because the suggestion that there is (or was) racial discrimination in lending is charged, we briefly review some points of this debate. In so doing, it is useful to adopt the framework of Chapter 9, that is, to consider the internal and external validity of the results in Table 11.2, which are representative of previous analyses of the Boston HMDA data. A number of the criticisms made of the original Fed- eral Reserve Bank of Boston study concern internal validity: possible errors in the data, alternative nonlinear functional forms, additional interactions, and so forth. The original data were subjected to a careful audit, some errors were found, and the results reported here (and in the final published Boston Fed study) are based
5These policy shifts include changes in the way that fair lending examinations were done by federal bank regulators, changes in inquiries made by the U.S. Department of Justice, and enhanced education programs for banks and other home loan origination companies.
on the “cleaned” data set. Estimation of other specifications—different functional forms and/or additional regressors—also produces estimates of racial differentials comparable to those in Table 11.2. A potentially more difficult issue of internal validity is whether there is relevant nonracial financial information obtained during in-person loan interviews, not recorded on the loan application itself, that is corre- lated with race; if so, there still might be omitted variable bias in the Table 11.2 regressions. Finally, some have questioned external validity: Even if there was racial discrimination in Boston in 1990, it is wrong to implicate lenders elsewhere today. Moreover, racial discrimination might be less likely using modern online applications, because the mortgage can be approved or denied without a face- to-face meeting. The only way to resolve the question of external validity is to consider data from other locations and years.6
11.5
Conclusion
When the dependent variable Y is binary, the population regression function is the probability that Y = 1, conditional on the regressors. Estimation of this pop- ulation regression function entails finding a functional form that does justice to its probability interpretation, estimating the unknown parameters of that function, and interpreting the results. The resulting predicted values are predicted proba- bilities, and the estimated effect of a change in a regressor X is the estimated change in the probability that Y = 1 arising from the change in X.
A natural way to model the probability that Y = 1 given the regressors is to use a cumulative distribution function, where the argument of the c.d.f. depends on the regressors. Probit regression uses a normal c.d.f. as the regression function, and logit regression uses a logistic c.d.f. Because these models are nonlinear func- tions of the unknown parameters, those parameters are more complicated to esti- mate than linear regression coefficients. The standard estimation method is maximum likelihood. In practice, statistical inference using the maximum likeli- hood estimates proceeds the same way as it does in linear multiple regression; for
6If you are interested in further reading on this topic, a good place to start is the symposium on racial discrimination and economics in the Spring 1998 issue of the Journal of Economic Perspectives. The article in that symposium by Helen Ladd (1998) surveys the evidence and debate on racial discrimina- tion in mortgage lending. A more detailed treatment is given in Goering and Wienk (1996). The U.S. mortgage market has changed dramatically since the Boston Fed study, including a relaxation of lend- ing standards, a bubble in housing prices, the financial crisis of 2008–2009, and a return to tighter lend- ing standards. For an introduction to changes in mortgage markets, see Green and Wachter (2008).
11.5 Conclusion 409
410 CHAPTER 11 Regression with a Binary Dependent Variable
James Heckman and Daniel McFadden, Nobel Laureates
The 2000 Nobel Prize in economics was awarded jointly to two econometricians, James J. Heckman of the University of Chicago and Daniel L. McFad- den of the University of California at Berkeley, for fundamental contributions to the analysis of data on individuals and firms. Much of their work addressed difficulties that arise with limited dependent variables.
Heckman was awarded the prize for develop- ing tools for handling sample selection. As discussed in Section 9.2, sample selection bias occurs when the availability of data is influenced by a selection process related to the value of dependent variable. For example, suppose you want to estimate the relationship between earnings and some regressor, X, using a random sample from the population. If you estimate the regression using the subsample of employed workers—that is, those reporting positive earnings—the OLS estimate could be subject to selection bias. Heckman’s solution was to specify a preliminary equation with a binary dependent variable indicating whether the worker is in or out of the labor force (in or out of the subsample) and to treat this equation and the earnings equation as a system of simultaneous equations. This general strategy has been extended to selection problems that arise in many fields, ranging from labor economics to industrial organization to finance.
McFadden was awarded the prize for develop- ing models for analyzing discrete choice data (does a high school graduate join the military, go to college, or get a job?). He started by considering the problem of an individual maximizing the expected utility of each possible choice, which could depend on observ- able variables (such as wages, job characteristics, and family background). He then derived models for the individual choice probabilities with unknown coeffi- cients, which in turn could be estimated by maximum likelihood. These models and their extensions have proven widely useful in analyzing discrete choice data in many fields, including labor economics, health eco- nomics, and transportation economics.
For more information on these and other Nobel laureates in economics, visit the Nobel Foundation website, http://www.nobel.se/economics.
James J. Heckman
Daniel L. McFadden
example, 95% confidence intervals for a coefficient are constructed as the esti- mated coefficient {1.96 standard errors.
Despite its intrinsic nonlinearity, sometimes the population regression func- tion can be adequately approximated by a linear probability model, that is, by the straight line produced by linear multiple regression. The linear probability model, probit regression, and logit regression all give similar “bottom line” answers when they are applied to the Boston HMDA data: All three methods estimate substantial
differences in mortgage denial rates for otherwise similar black applicants and white applicants.
Binary dependent variables are the most common example of limited dependent variables, which are dependent variables with a limited range. The final quarter of the twentieth century saw important advances in econometric methods for analyzing other limited dependent variables (see the Nobel Laureates box). Some of these methods are reviewed in Appendix 11.3.
Summary
1. When Y is a binary variable, the population regression function shows the probability that Y = 1 given the value of the regressors, X1, X2, c, Xk.
2. The linear multiple regression model is called the linear probability model when Y is a binary variable because the probability that Y = 1 is a linear function of the regressors.
3. Probit and logit regression models are nonlinear regression models used when Y is a binary variable. Unlike the linear probability model, probit and logit regressions ensure that the predicted probability that Y = 1 is between 0 and 1 for all values of X.
4. Probit regression uses the standard normal cumulative distribution function. Logit regression uses the logistic cumulative distribution function. Logit and probit coefficients are estimated by maximum likelihood.
5. The values of coefficients in probit and logit regressions are not easy to interpret. Changes in the probability that Y = 1 associated with changes in one or more of the X’s can be calculated using the general procedure for nonlinear models outlined in Key Concept 8.1.
6. Hypothesis tests on coefficients in the linear probability, logit, and probit models are performed using the usual t- and F-statistics.
Key Terms
limited dependent variable (386) linear probability model (388) probit (391)
logit (391)
logistic regression (391)
likelihood function (400) maximum likelihood estimator
(MLE) (400)
fraction correctly predicted (401) pseudo-R2 (402)
Key Terms 411
412 CHAPTER 11
Regression with a Binary Dependent Variable
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www.pearsonhighered.com/stock_watson.
Review the Concepts
11.1 Suppose that a linear probability model yields a predicted value of Y that is equal to 1.3. Explain why this is nonsensical.
11.2 In Table 11.2 the estimated coefficient on black is 0.084 in column (1), 0.688 in column (2), and 0.389 in column (3). In spite of these large differ- ences, all three models yield similar estimates of the marginal effect of race on the probability of mortgage denial. How can this be?
11.3 One of your friends is using data on individuals to study the determi- nants of smoking at your university. She asks you whether she should use a probit, logit, or linear probability model. What advice do you give her? Why?
11.4 Why are the coefficients of probit and logit models estimated by maximum likelihood instead of OLS?
Exercises
Exercises 11.1 through 11.5 are based on the following scenario: Four hundred driver’s license applicants were randomly selected and asked whether they passed their driving test (Passi = 1) or failed their test (Passi = 0); data were also col- lected on their gender (Malei = 1 if male and = 0 if female) and their years of driving experience (Experiencei, in years). The following table summarizes several estimated models.
11.1 Using the results in column (1):
a. Does the probability of passing the test depend on Experience? Explain.
Dependent Variable: Pass
Experience
Male
Male * Experience Constant
Probit
(1)
0.031 (0.009)
0.712 (0.126)
b. c. d.
11.2 a. b.
11.3 a. b.
Linear Logit Probability
(2) (3)
0.040 0.006 (0.016) (0.002)
1.059 0.774 (0.221) (0.034)
Probit (4)
– 0.333 (0.161)
1.282 (0.124)
Logit (5)
– 0.622 (0.303)
2.197 (0.242)
Linear Probability
(6)
– 0.071 (0.034)
0.900 (0.022)
Probit (7)
0.041 (0.156)
– 0.074 (0.259)
-0.015 (0.019)
0.806 (0.200)
Exercises 413
Matthew has 10 years of driving experience. What is the probability that he will pass the test?
Christopher is a new driver (zero years of experience). What is the probability that he will pass the test?
The sample included values of Experience between 0 and 40 years, and only four people in the sample had more than 30 years of driv- ing experience. Jed is 95 years old and has been driving since he was 15. What is the model’s prediction for the probability that Jed will pass the test? Do you think that this prediction is reliable? Why or why not?
Answer (a) through (c) from Exercise 11.1 using the results in column (2).
Sketch the predicted probabilities from the probit and logit in col- umns (1) and (2) for values of Experience between 0 and 60. Are the probit and logit models similar?
Answer (a) through (c) from Exercise 11.1 using the results in column (3).
Sketch the predicted probabilities from the probit and linear prob- ability in columns (1) and (3) as a function of Experience for values of Experience between 0 and 60. Do you think that the linear probability is appropriate here? Why or why not?
11.4 Using the results in columns (4) through (6):
a. Compute the estimated probability of passing the test for men and for
women.
b. Are the models in (4) through (6) different? Why or why not?
414 CHAPTER 11
Regression with a Binary Dependent Variable
11.5 Using the results in column (7):
a. Akira is a man with 10 years of driving experience. What is the prob-
ability that he will pass the test?
b. Jane is a woman with 2 years of driving experience. What is the prob- ability that she will pass the test?
c. Does the effect of experience on test performance depend on gender? Explain.
11.6 Use the estimated probit model in Equation (11.8) to answer the following questions:
a. A black mortgage applicant has a P/I ratio of 0.35. What is the prob- ability that his application will be denied?
b. Suppose that the applicant reduced this ratio to 0.30. What effect would this have on his probability of being denied a mortgage?
c. Repeat (a) and (b) for a white applicant.
d. Does the marginal effect of the P/I ratio on the probability of mortgage
denial depend on race? Explain.
11.7 Repeat Exercise 11.6 using the logit model in Equation (11.10). Are the logit and probit results similar? Explain.
11.8 Consider the linear probability model Yi = b0 + b1Xi + ui, where Pr(Yi = 1Xi) = b0 + b1Xi.
a. Show that E(ui Xi) = 0.
b. Show that var(ui Xi) = (b0 + b1Xi)[1 – (b0 + b1Xi)]. [Hint: Review
Equation (2.7).]
c. Is ui heteroskedastic? Explain.
d. (Requires Section 11.3) Derive the likelihood function.
11.9 Use the estimated linear probability model shown in column (1) of
Table 11.2 to answer the following:
a. Two applicants, one white and one black, apply for a mortgage. They have the same values for all the regressors other than race. How much more likely is the black applicant to be denied a mortgage?
b. Construct a 95% confidence interval for your answer to (a).
c. Think of an important omitted variable that might bias the answer in (a). What is it, and how would it bias the results?
11.10 (RequiresSection11.3andcalculus)SupposethatarandomvariableYhas the following probability distribution: Pr(Y = 1) = p, Pr(Y = 2) = q, and Pr(Y = 3) = 1 – p – q.Arandomsampleofsizenisdrawnfrom this distribution, and the random variables are denoted Y1, Y2, c, Yn.
a. Derive the likelihood function for the parameters p and q.
b. Derive formulas for the MLE of p and q.
11.11 (Requires Appendix 11.3) Which model would you use for:
a. A study explaining the number of minutes that a person spends
talking on a cell phone during the month?
b. A study explaining grades (A through F) in a large Principles of Economics class?
c. A study of consumers’ choices for Coke, Pepsi, or generic cola?
d. A study of the number of cell phones owned by a family?
Empirical Exercises
(Only two empirical exercises for this chapter are given in the text, but you can find more on the text website, http://www.pearsonhighered.com/stock_watson/.)
E11.1 In April 2008 the unemployment rate in the United States stood at 5.0%. By April 2009 it had increased to 9.0%, and it had increased further, to 10.0%, by October 2009. Were some groups of workers more likely to lose their jobs than others during the Great Recession? For example, were young workers more likely to lose their jobs than middle-aged workers? What about workers with a college degree versus those without a degree, or women versus men? On the textbook website, http://www.pearsonhighered .com/stock_watson, you will find the data file Employment_08_09, which contains a random sample of 5440 workers who were surveyed in April 2008 and reported that they were employed full time. A detailed descrip- tion is given in Employment_08_09_Description, available on the website. These workers were surveyed one year later, in April 2009, and asked about their employment status (employed, unemployed, or out of the labor force). The data set also includes various demographic measures for each individual. Use these data to answer the following questions.
a. What fraction of workers in the sample were employed in April 2009? Use your answer to compute a 95% confidence interval for
Empirical Exercises 415
416 CHAPTER 11
Regression with a Binary Dependent Variable
the probability that a worker was employed in April 2009, conditional on being employed in April 2008.
b. Regress Employed on Age and Age2, using a linear probability model.
i. Based on this regression, was age a statistically significant determi-
nant of employment in April 2009?
ii. Is there evidence of a nonlinear effect of age on the probability of being employed?
iii. Compute the predicted probability of employment for a 20-year- old worker, a 40-year-old worker, and a 60-year-old worker.
c. Repeat (b) using a probit regression.
d. Repeat (b) using a logit regression.
e. Are there important differences in your answers to (b)–(d)? Explain.
f. The data set includes variables measuring the workers’ educational attainment, sex, race, marital status, region of the country, and weekly earnings in April 2008.
i. Construct a table like Table 11.2 to investigate whether the con- clusions on the effect of age on employment from (b)–(d) are affected by omitted variable bias.
ii. Use the regressions in your table to discuss the characteristics of workers who were hurt most by the Great Recession.
g. The results in (a)–(f) were based on the probability of employment. Workers who are not employed can either be (i) unemployed or
(ii) out the labor force. Do the conclusions you reached in (a)–(f) also hold for workers who became unemployed? (Hint: Use the binary variable Unemployed instead of Employed.)
h. These results have covered employment transitions during the Great Recession, but what about transitions during normal times? On the textbook website, you will find the data file Employment_06_07, which measures the same variables but for the years 2006–2007. Ana- lyze these data and comment on the differences in employment tran- sitions during recessions and normal times.
E11.2 Believe it or not, workers used to be able to smoke inside office buildings. Smoking bans were introduced in several areas during the 1990s. In addi- tion to eliminating the externality of secondhand smoke, supporters of these bans argued that they would encourage smokers to quit by reducing their opportunities to smoke. In this assignment you will estimate the effect
of workplace smoking bans on smoking, using data on a sample of 10,000 U.S. indoor workers from 1991 to 1993, available on the textbook website, http://www.pearsonhighered.com/stock_watson, in the file Smoking. The data set contains information on whether individuals were or were not subject to a workplace smoking ban, whether the individuals smoked, and other individual characteristics.7 A detailed description is given in Smoking_ Description, available on the website.
a. Estimate the probability of smoking for (i) all workers, (ii) workers affected by workplace smoking bans, and (iii) workers not affected by workplace smoking bans.
b. What is the difference in the probability of smoking between workers affected by a workplace smoking ban and workers not affected by a workplace smoking ban? Use a linear probability model to determine whether this difference is statistically significant.
c. Estimate a linear probability model with smoker as the dependent variable and the following regressors: smkban, female, age, age2, hsdrop, hsgrad, colsome, colgrad, black, and hispanic. Compare the estimated effect of a smoking ban from this regression with your answer from (b). Suggest a reason, based on the substance of this regression, explaining the change in the estimated effect of a smoking ban between (b) and (c).
d. Test the hypothesis that the coefficient on smkban is zero in the pop- ulation version of the regression in (c) against the alternative that it is nonzero, at the 5% significance level.
e. Test the hypothesis that the probability of smoking does not depend on the level of education in the regression in (c). Does the probability of smoking increase or decrease with the level of education?
f. Repeat (c)–(e) using a probit model.
g. Repeat (c)–(e) using a logit model.
h. i. Mr. A is white, non-Hispanic, 20 years old, and a high school dropout. Using the probit regression and assuming that Mr. A is not subject to a workplace smoking ban, calculate the probability that Mr. A smokes. Carry out the calculation again, assuming that
7These data were provided by Professor William Evans of the University of Maryland and were used in his paper with Matthew Farrelly and Edward Montgomery, “Do Workplace Smoking Bans Reduce Smoking?” American Economic Review, 1999, 89(4): 728–747.
Empirical Exercises 417
418 CHAPTER 11
Regression with a Binary Dependent Variable
he is subject to a workplace smoking ban. What is the effect of the smoking ban on the probability of smoking?
ii. Repeat (i) for Ms. B, a female, black, 40-year-old college graduate.
iii. Repeat (i)–(ii) using the linear probability model.
iv. Repeat (i)–(ii) using the logit model.
v. Based on your answers to (i)–(iv), do the logit, probit, and linear probability models differ? If they do, which results make most sense? Are the estimated effects large in a real work sense?
The Boston HMDA Data Set
The Boston HMDA data set was collected by researchers at the Federal Reserve Bank of Boston. The data set combines information from mortgage applications and a follow-up survey of the banks and other lending institutions that received these mortgage applications. The data pertain to mortgage applications made in 1990 in the greater Boston metropolitan area. The full data set has 2925 observations, consisting of all mortgage applications by blacks and Hispanics plus a random sample of mortgage applications by whites.
To narrow the scope of the analysis in this chapter, we use a subset of the data for single-family residences only (thereby excluding data on multifamily homes) and for black applicants and white applicants only (thereby excluding data on applicants from other minority groups). This leaves 2380 observations. Definitions of the variables used in this chapter are given in Table 11.1.
These data were graciously provided to us by Geoffrey Tootell of the Research Department of the Federal Reserve Bank of Boston. More information about this data set, along with the conclusions reached by the Federal Reserve Bank of Boston researchers, is available in the article by Alicia H. Munnell, Geoffrey M. B. Tootell, Lynne E. Browne, and James McEneaney, “Mortgage Lending in Boston: Interpreting HMDA Data,” Amer- ican Economic Review, 1996, pp. 25–53.
11.2
APPENDIX
11.1
APPENDIX
Maximum Likelihood Estimation
This appendix provides a brief introduction to maximum likelihood estimation in the con- text of the binary response models discussed in this chapter. We start by deriving the MLE of the success probability p for n i.i.d. observations of a Bernoulli random variable. We then
Maximum Likelihood Estimation 419 turn to the probit and logit models and discuss the pseudo-R2. We conclude with a discussion
of standard errors for predicted probabilities. This appendix uses calculus at two points.
MLE for n i.i.d. Bernoulli Random Variables
The first step in computing the MLE is to derive the joint probability distribution. For n i.i.d. observations on a Bernoulli random variable, this joint probability distribution is the extension of the n = 2 case in Section 11.3 to general n:
Pr(Y1 = y1,Y2 = y2,c,Yn = yn)
= 3py1(1 – p)(1-y1)4 * 3py2(1 – p)(1-y2)4 * g* 3pyn(1 – p)(1-yn)4
= p(y1 +g+ yn)(1 – p)n-(y1 +g+ yn). (11.13) The likelihood function is the joint probability distribution, treated as a function of the
unknown coefficients. Let S = g ni = 1Yi; then the likelihood function is
fBernoulli(p; Y1, c, Yn) = pS(1 – p)n – S. (11.14)
The MLE of p is the value of p that maximizes the likelihood in Equation (11.14). The likelihood function can be maximized using calculus. It is convenient to maximize not the likelihood but rather its logarithm (because the logarithm is a strictly increasing function, maximizing the likelihood or its logarithm gives the same estimator). The log likelihood is Sln(p) + (n – S)ln(1 – p), and the derivative of the log likelihood with respect to p is
d ln 3f (p; Y , c, Y )4 = S – n – S. (11.15) dp Bernoulli 1 n p 1-p
Setting the derivative in Equation (11.15) to zero and solving for p yields the MLE pn = S > n = Y .
MLE for the Probit Model
For the probit model, the probability that Yi = 1, conditional on X1i, c, Xki, is
pi = Φ(b0 + b1X1i + g+ bkXki). The conditional probability distribution for the ith
observationisPr[Y = yX ,c,X ] = pyi(1 – p)1-yi.Assumingthat(X ,c,X ,Y) i i 1i ki i i 1i ki i
are i.i.d., i = 1, c, n, the joint probability distribution of Y1, c, Yn, conditional on the X’s, is
Pr(Y1 = y1,c,Yn = ynX1i,c,Xki,i = 1,c,n)
= Pr(Y1 = y1X11, c, Xk1) * g* Pr(Yn = ynX1n, c, Xkn)
= py1(1 – p )1-y1 * g* pyn(1 – p )1-yn. (11.16) 11 nn
420 CHAPTER 11
Regression with a Binary Dependent Variable
The likelihood function is the joint probability distribution, treated as a function of the unknown coefficients. It is conventional to consider the logarithm of the likelihood. Accordingly, the log likelihood function is
(11.17)
ln3fprobit(b0,c,bk;Y1,c,YnX1i,c,Xki,i = 1,c,n)4 n
= aYiln3Φ(b0 +b1X1i +g+bkXki)4 i=1
n
+ a(1 – Yi) ln31 – Φ(b0 + b1X1i + g+ bkXki)4,
i=1
where this expression incorporates the probit formula for the conditional probability, pi = Φ(b0 + b1X1i + g+ bkXki).
The MLE for the probit model maximizes the likelihood function or, equivalently, the logarithm of the likelihood function given in Equation (11.17). Because there is no simple formula for the MLE, the probit likelihood function must be maximized using a numerical algorithm on the computer.
Under general conditions, maximum likelihood estimators are consistent and have a normal sampling distribution in large samples.
MLE for the Logit Model
The likelihood for the logit model is derived in the same way as the likelihood for the probit model. The only difference is that the conditional success probability pi for the logit model is given by Equation (11.9). Accordingly, the log likelihood of the logit model is given by Equation (11.17), with Φ(b0 + b1X1i + g+ bkXki) replaced by 31 + e-(b0 + b1X1i + b2X2i + g+ bkXki)4-1. As with the probit model, there is no simple formula for the MLE of the logit coefficients, so the log likelihood must be maximized numerically.
Pseudo-R2
The pseudo-R2 compares the value of the likelihood of the estimated model to the value of the likelihood when none of the X’s are included as regressors. Specifically, the pseudo-R2 for the probit model is
ln(fmax )
pseudo@R2 = 1 – probit , (11.18)
wherefmax probit
is the value of the maximized probit likelihood (which includes the X’s) and f max is the value of the maximized Bernoulli likelihood (the probit model excluding all
Bernoulli the X’s).
ln(f max ) Bernoulli
Other Limited Dependent Variable Models 421 Standard Errors for Predicted Probabilities
For simplicity, consider the case of a single regressor in the probit model. Then the pre-
nMLE nMLE
dicted probability at a fixed value of that regressor, x, is pn(x) = Φ(b0 + b1 x), where
bnMLE and bnMLE are the MLEs of the two probit coefficients. Because this predicted prob- 01
ability depends on the estimators bnMLE and bnMLE, and because those estimators have a 01
sampling distribution, the predicted probability will also have a sampling distribution.
The variance of the sampling distribution of pn(x) is calculated by approximating the
function Φ(bnMLE + bnMLEx), a nonlinear function of bnMLE and bnMLE, by a linear function of 01 01
bnMLE and bnMLE. Specifically, let 01
pn(x) = Φ(bnMLE + bnMLEx) ≅ c + a(bnMLE – b) + a(bnMLE – b) (11.19) 01000111
where the constant c and factors a0 and a1 depend on x and are obtained from calculus. [Equation (11.19) is a first-order Taylor series expansion; c = Φ(b0 + b1x); and
N MLE N MLE
a0 and a1 are the partial derivatives, a0 = 0Φ(b0 + b1x)>0b0b0 , b1 1
and a = .] The variance of pn(x) now can be calculated using the approx- imation in Equation (11.19) and the expression for the variance of the sum of two random
N MLE N MLE 0Φ(b0 + b1x)>0b1b0 ,b1
variables in Equation (2.31):
var3pn(x)4 ≅ var3c + a(bnMLE – b) + a(bnMLE – b)4 000111
= a2var(bnMLE) + a2var(bnMLE) + 2a a cov(bnMLE, bnMLE). (11.20) 00 11 0101
Using Equation (11.20), the standard error of pn(x) can be calculated using estimates of the variances and covariance of the MLEs.
11.3
APPENDIX
Other Limited Dependent Variable Models
This appendix surveys some models for limited dependent variables, other than binary vari- ables, found in econometric applications. In most cases the OLS estimators of the parameters of limited dependent variable models are inconsistent, and estimation is routinely done using maximum likelihood. There are several advanced references available to the reader inter- ested in further details; see, for example, Ruud (2000) and Wooldridge (2010).
Censored and Truncated Regression Models
Suppose that you have cross-sectional data on car purchases by individuals in a given year. Car buyers have positive expenditures, which can reasonably be treated as continuous
422 CHAPTER 11
Regression with a Binary Dependent Variable
random variables, but nonbuyers spent $0. Thus the distribution of car expenditures is a combination of a discrete distribution (at zero) and a continuous distribution.
Nobel laureate James Tobin developed a useful model for a dependent variable with a partly continuous and partly discrete distribution (Tobin, 1958). Tobin suggested model- ing the ith individual in the sample as having a desired level of spending, Y*i , that is related to the regressors (for example, family size) according to a linear regression model. That is, when there is a single regressor, the desired level of spending is
Y*i = b0 + b1Xi + ui,i = 1,c, n. (11.21)
If Y*i (what the consumer wants to spend) exceeds some cutoff, such as the minimum price of a car, the consumer buys the car and spends Yi = Y*i , which is observed. However, if Y*i is less than the cutoff, spending of Yi = 0 is observed instead of Y*i .
When Equation (11.21) is estimated using observed expenditures Yi in place of Y*i , the OLS estimator is inconsistent. Tobin solved this problem by deriving the likelihood func- tion using the additional assumption that ui has a normal distribution, and the resulting MLE has been used by applied econometricians to analyze many problems in economics. In Tobin’s honor, Equation (11.21), combined with the assumption of normal errors, is called the tobit regression model. The tobit model is an example of a censored regression model, so called because the dependent variable has been “censored” above or below a certain cutoff.
Sample Selection Models
In the censored regression model, there are data on buyers and nonbuyers, as there would be if the data were obtained via simple random sampling of the adult population. If, how- ever, the data are collected from sales tax records, then the data would include only buyers: There would be no data at all for nonbuyers. Data in which observations are unavailable above or below a threshold (data for buyers only) are called truncated data. The truncated regression model is a regression model applied to data in which observations are simply unavailable when the dependent variable is above or below a certain cutoff.
The truncated regression model is an example of a sample selection model, in which the selection mechanism (an individual is in the sample by virtue of buying a car) is related to the value of the dependent variable (expenditure on a car). As discussed in the box in Section 11.4, one approach to estimation of sample selection models is to develop two equations, one for Y*i andoneforwhetherY*i isobserved.Theparametersofthemodelcanthenbeestimatedby maximum likelihood, or in a stepwise procedure, estimating the selection equation first and then estimating the equation for Y*i . For additional discussion, see Ruud (2000, Chapter 28), Greene (2012, Chapter 19), or Wooldridge (2010, Chapter 17).
Other Limited Dependent Variable Models 423
Count Data
Count data arise when the dependent variable is a counting number—for example, the number of restaurant meals eaten by a consumer in a week. When these numbers are large, the variable can be treated as approximately continuous, but when they are small, the continuous approxi- mation is a poor one. The linear regression model, estimated by OLS, can be used for count data, even if the number of counts is small. Predicted values from the regression are interpreted as the expected value of the dependent variable, conditional on the regressors. So, when the dependent variable is the number of restaurant meals eaten, a predicted value of 1.7 means, on average, 1.7 restaurant meals per week. As in the binary regression model, however, OLS does not take advantage of the special structure of count data and can yield nonsense predictions, for example, – 0.2 restaurant meal per week. Just as probit and logit eliminate nonsense predic- tions when the dependent variable is binary, special models do so for count data. The two most widely used models are the Poisson and negative binomial regression models.
Ordered Responses
Ordered response data arise when mutually exclusive qualitative categories have a natural ordering, such as obtaining a high school degree, obtaining some college education (but not graduating), or graduating from college. Like count data, ordered response data have a natural ordering, but unlike count data, they do not have natural numerical values.
Because there are no natural numerical values for ordered response data, OLS is inap- propriate. Instead, ordered data are often analyzed using a generalization of probit called the ordered probit model, in which the probabilities of each outcome (e.g., a college educa- tion), conditional on the independent variables (such as parents’ income), are modeled using the cumulative normal distribution.
Discrete Choice Data
A discrete choice or multiple choice variable can take on multiple unordered qualitative values. One example in economics is the mode of transport chosen by a commuter: She might take the subway, ride the bus, drive, or make her way under her own power (walk, bicycle). If we were to analyze these choices, the dependent variable would have four pos- sible outcomes (subway, bus, car, human-powered). These outcomes are not ordered in any natural way. Instead, the outcomes are a choice among distinct qualitative alternatives.
The econometric task is to model the probability of choosing the various options, given various regressors such as individual characteristics (how far the commuter’s house is from the subway station) and the characteristics of each option (the price of the subway). As discussed in the box in Section 11.3, models for analysis of discrete choice data can be developed from prin- ciples of utility maximization. Individual choice probabilities can be expressed in probit or logit form, and those models are called multinomial probit and multinomial logit regression models.
Chapter
Instrumental Variables Regression
Chapter 9 discussed several problems, including omitted variables, errors in variables, and simultaneous causality, that make the error term correlated with the regressor. Omitted variable bias can be addressed directly by including the omitted variable in a multiple regression, but this is only feasible if you have data on the omitted variable. And sometimes, such as when causality runs both from X to Y and from Y to X so that there is simultaneous causality bias, multiple regression simply cannot eliminate the bias. If a direct solution to these problems is either infeasible or unavailable, a new method is required.
Instrumental variables (IV) regression is a general way to obtain a consistent estimator of the unknown coefficients of the population regression function when the regressor, X, is correlated with the error term, u. To understand how IV regres- sion works, think of the variation in X as having two parts: one part that, for what- ever reason, is correlated with u (this is the part that causes the problems) and a second part that is uncorrelated with u. If you had information that allowed you to isolate the second part, you could focus on those variations in X that are uncorre- lated with u and disregard the variations in X that bias the OLS estimates. This is, in fact, what IV regression does. The information about the movements in X that are uncorrelated with u is gleaned from one or more additional variables, called instru- mental variables or simply instruments. Instrumental variables regression uses these additional variables as tools or “instruments” to isolate the movements in X that are uncorrelated with u, which in turn permit consistent estimation of the regression coefficients.
The first two sections of this chapter describe the mechanics and assumptions of IV regression: why IV regression works, what is a valid instrument, and how to implement and to interpret the most common IV regression method, two stage least squares. The key to successful empirical analysis using instrumental variables is finding valid instruments, and Section 12.3 takes up the question of how to assess whether a set of instruments is valid. As an illustration, Section 12.4 uses IV regression to estimate the elasticity of demand for cigarettes. Finally, Section 12.5 turns to the difficult question of where valid instruments come from in the first place.
12
424
12.1 The IV Estimator with a Single Regressor and a Single Instrument 425 The IV Estimator with a Single Regressor
12.1
and a Single Instrument
We start with the case of a single regressor, X, which might be correlated with the regression error, u. If X and u are correlated, the OLS estimator is inconsistent; that is, it may not be close to the true value of the regression coefficient even when the sample is very large [see Equation (6.1)]. As discussed in Section 9.2, this cor- relation between X and u can stem from various sources, including omitted vari- ables, errors in variables (measurement errors in the regressors), and simultaneous causality (when causality runs “backward” from Y to X as well as “forward” from X to Y). Whatever the source of the correlation between X and u, if there is a valid instrumental variable, Z, the effect on Y of a unit change in X can be esti- mated using the instrumental variables estimator.
The IV Model and Assumptions
The population regression model relating the dependent variable Yi and regressor Xi is
Yi = b0 + b1Xi + ui,i = 1,c,n, (12.1)
where as usual ui is the error term representing omitted factors that determine Yi. If Xi and ui are correlated, the OLS estimator is inconsistent. Instrumental vari- ables estimation uses an additional, “instrumental” variable Z to isolate that part of X that is uncorrelated with ui.
Endogeneityandexogeneity. Instrumentalvariablesregressionhassomespecial- ized terminology to distinguish variables that are correlated with the population error term u from ones that are not. Variables correlated with the error term are called endogenous variables, while variables uncorrelated with the error term are called exogenous variables. The historical source of these terms traces to models with multiple equations, in which an “endogenous” variable is determined within the model while an “exogenous” variable is determined outside the model. For example, Section 9.2 considered the possibility that if low test scores produced decreases in the student–teacher ratio because of political intervention and increased funding, causality would run both from the student–teacher ratio to test scores and from test scores to the student–teacher ratio. This was represented math- ematically as a system of two simultaneous equations [Equations (9.3) and (9.4)],
426 ChapteR 12 Instrumental Variables Regression
one for each causal connection. As discussed in Section 9.2, because both test scores and the student–teacher ratio are determined within the model, both are correlated with the population error term u; that is, in this example, both variables are endogenous. In contrast, an exogenous variable, which is determined outside the model, is uncorrelated with u.
Thetwoconditionsforavalidinstrument. Avalidinstrumentalvariable(“instru- ment”) must satisfy two conditions, known as the instrument relevance condition and the instrument exogeneity condition:
1. Instrument relevance: corr (Zi, Xi) ≠ 0. 2. Instrument exogeneity: corr (Zi, ui) = 0.
If an instrument is relevant, then variation in the instrument is related to varia- tion in Xi. If in addition the instrument is exogenous, then that part of the variation of Xi captured by the instrumental variable is exogenous. Thus an instrument that is relevant and exogenous can capture movements in Xi that are exogenous. This exogenous variation can in turn be used to estimate the population coefficient b1.
The two conditions for a valid instrument are vital for instrumental variables regression, and we return to them (and their extension to a multiple regressors and multiple instruments) repeatedly throughout this chapter.
The Two Stage Least Squares Estimator
If the instrument Z satisfies the conditions of instrument relevance and exogene- ity, the coefficient b1 can be estimated using an IV estimator called two stage least squares (TSLS). As the name suggests, the two stage least squares estimator is calculated in two stages. The first stage decomposes X into two components: a problematic component that may be correlated with the regression error and another problem-free component that is uncorrelated with the error. The second stage uses the problem-free component to estimate b1.
The first stage begins with a population regression linking X and Z: X=p+pZ+v, (12.2)
i01ii
where p0 is the intercept, p1 is the slope, and vi is the error term. This regression
provides the needed decomposition of X . One component is p + p Z , the part i01i
of Xi that can be predicted by Zi. Because Zi is exogenous, this component of Xi is uncorrelated with ui, the error term in Equation (12.1). The other component of Xi is vi, which is the problematic component of Xi that is correlated with ui.
12.1 The IV Estimator with a Single Regressor and a Single Instrument 427 The idea behind TSLS is to use the problem-free component of X , p + p Z ,
i01i and to disregard vi. The only complication is that the values of p0 and p1 are
unknown, so p + p Z cannot be calculated. Accordingly, the first stage of TSLS 01i
applies OLS to Equation (12.2) and uses the predicted value from the OLS regres- sion, Xni = pn0 + pn1Zi, where pn0 and pn1 are the OLS estimates.
The second stage of TSLS is easy: Regress Yi on Xn i using OLS. The resulting estimators from the second-stage regression are the TSLS estimators, bnTSLS and
bnTSLS. 1
Why Does IV Regression Work?
Two examples provide some intuition for why IV regression solves the problem of correlation between Xi and ui.
Example#1:PhilipWright’sproblem. Themethodofinstrumentalvariablesesti- mation was first published in 1928 in an appendix to a book written by Philip G. Wright (Wright, 1928), although the key ideas of IV regression appear to have been developed collaboratively with his son, Sewall Wright (see the box). Philip Wright was concerned with an important economic problem of his day: how to set an import tariff (a tax on imported goods) on animal and vegetable oils and fats, such as butter and soy oil. In the 1920s, import tariffs were a major source of tax revenue for the United States. The key to understanding the economic effect of a tariff was having quantitative estimates of the demand and supply curves of the goods. Recall that the supply elasticity is the percentage change in the quantity supplied arising from a 1% increase in the price and that the demand elasticity is the percentage change in the quantity demanded arising from a 1% increase in the price. Philip Wright needed estimates of these elasticities of supply and demand.
To be concrete, consider the problem of estimating the elasticity of demand for butter. Recall from Key Concept 8.2 that the coefficient in a linear equation relating ln(Yi) to ln(Xi) has the interpretation of the elasticity of Y with respect to X. In Wright’s problem, this suggests the demand equation
ln(Qbutter) = b + b ln(Pbutter) + u , (12.3) i01ii
where Qbutter is the ith observation on the quantity of butter consumed, Pbutter is its ii
price, and ui represents other factors that affect demand, such as income and consumer tastes. In Equation (12.3), a 1% increase in the price of butter yields a b1 percent change in demand, so b1 is the demand elasticity.
0
428 ChapteR 12 Instrumental Variables Regression
Who Invented Instrumental Variables regression?
I nstrumental variables regression was first pro- posed as a solution to the simultaneous causation problem in econometrics in the appendix to Philip G. Wright’s 1928 book, The Tariff on Animal and Vegetable Oils. If you want to know how animal and vegetable oils were produced, transported, and sold in the early twentieth century, the first 285 pages of the book are for you. Econometricians, how- ever, will be more interested in Appendix B. The appendix provides two derivations of “the method of introducing external factors”—what we now call the instrumental variables estimator—and uses IV regression to estimate the supply and demand elas- ticities for butter and flaxseed oil. Philip was an obscure economist with a scant intellectual legacy other than this appendix, but his son Sewall went on to become a preeminent population geneticist and statistician. Because the mathematical material in the appendix is so different than the rest of the book, many econometricians assumed that Philip’s son Sewall Wright wrote the appendix anonymously.
So who wrote Appendix B?
In fact, either father or son could have been the
author. Philip Wright (1861–1934) received a mas- ter’s degree in economics from Harvard University in 1887, and he taught mathematics and economics (as well as literature and physical education) at a small college in Illinois. In a book review [Wright (1915)], he used a figure like Figures 12.1a and 12.1b to show how a regression of quantity on price will not, in general, estimate a demand curve, but instead estimates a combination of the supply and demand curves. In the early 1920s, Sewall Wright (1889–1988) was researching the statistical analysis of multiple equations with multiple causal variables
in the context of genetics, research that in part led to his assuming a professorship in 1930 at the Uni- versity of Chicago.
Although it is too late to ask Philip or Sewall who wrote Appendix B, it is never too late to do some statistical detective work. Stylometrics is the subfield of statistics, invented by Frederick Mosteller and David Wallace (1963), that uses subtle, subconscious differences in writing styles to identify authorship of disputed texts using statisti- cal analysis of grammatical constructions and word choice. The field has had verified successes, such as Donald Foster’s (1996) uncovering of Joseph Klein as the author of the political novel Primary Colors. When Appendix B is compared statistically to texts known to have been written independently by Philip and by Sewall, the results are clear: Philip was the author.
Does this mean that Philip G. Wright invented IV regression? Not quite. Recently, correspondence between Philip and Sewall in the mid-1920s has come to light, and this correspondence shows that the development of IV regression was a joint intel- lectual collaboration between father and son. To learn more, see Stock and Trebbi (2003).
Philip G. Wright
Sewall Wright
12.1 The IV Estimator with a Single Regressor and a Single Instrument 429
Philip Wright had data on total annual butter consumption and its average
annual price in the United States for 1912 to 1922. It would have been easy to use
these data to estimate the demand elasticity by applying OLS to Equation (12.3),
but he had a key insight: Because of the interactions between supply and demand,
the regressor, ln (Pbutter), was likely to be correlated with the error term. i
To see this, look at Figure 12.1a, which shows the market demand and supply curves for butter for three different years. The demand and supply curves for the first period are denoted D1 and S1, and the first period’s equilibrium price and quantity are determined by their intersection. In year 2, demand increases from D1 to D2 (say, because of an increase in income) and supply decreases from S1 to S2 (because of an increase in the cost of producing butter); the equilibrium price and quantity are determined by the intersection of the new supply and demand curves. In year 3, the factors affecting demand and supply change again; demand increases again to D3, supply increases to S3, and a new equilibrium quantity and price are determined. Figure 12.1b shows the equilibrium quantity and price pairs for these three periods and for eight subsequent years, where in each year the supply and demand curves are subject to shifts associated with factors other than price that affect market supply and demand. This scatterplot is like the one that Wright would have seen when he plotted his data. As he reasoned, fitting a line to these points by OLS will estimate neither a demand curve nor a supply curve, because the points have been determined by changes in both demand and supply.
Wright realized that a way to get around this problem was to find some third variable that shifted supply but did not shift demand. Figure 12.1c shows what hap- pens when such a variable shifts the supply curve, but demand remains stable. Now all of the equilibrium price and quantity pairs lie on a stable demand curve, and the slope of the demand curve is easily estimated. In the instrumental variable formula- tion of Wright’s problem, this third variable—the instrumental variable—is corre- lated with price (it shifts the supply curve, which leads to a change in price) but is uncorrelated with u (the demand curve remains stable). Wright considered several potential instrumental variables; one was the weather. For example, below-average rainfall in a dairy region could impair grazing and thus reduce butter production at a given price (it would shift the supply curve to the left and increase the equilibrium price), so dairy-region rainfall satisfies the condition for instrument relevance. But dairy-region rainfall should not have a direct influence on the demand for butter, so the correlation between dairy-region rainfall and ui would be zero; that is, dairy- region rainfall satisfies the condition for instrument exogeneity.
Example #2: Estimating the effect on test scores of class size. Despite controlling for student and district characteristics, the estimates of the effect on test scores of
430 ChapteR 12 Instrumental Variables Regression FIgure 12.1 equilibrium price and Quantity Data
(a) Price and quantity are determined by the intersection of the supply and demand curves. The equilibrium in the first period is determined by the intersection of the demand curve D1 and the supply curve S1. Equilibrium in the
second period is the intersection of D2 and S2, and equilib- rium in the third period is the intersection of D3 and S3.
(b) This scatterplot shows equilibrium price and quantity in 11 different time periods. The demand and supply curves are hidden. Can you determine the demand and supply curves from the points on the scatterplot?
Price
Period 2 equilibrium
Period 1 equilibrium
(c) When the supply curve shifts from S1 to S2 to S3 but the demand curve remains at D1, the equilibrium prices and quantities trace out the demand curve.
time periods
Price
(a) Demand and supply in three time periods Price
S2
S1
D3 D2
D1
Quantity
S3
Period 3 equilibrium
Quantity (b) Equilibrium price and quantity for 11
(c) Equilibrium price and quantity when only the supply curve shifts
S
2
S1
S3
D1
Quantity
12.1 The IV Estimator with a Single Regressor and a Single Instrument 431
class size reported in Part II still might have omitted variables bias resulting from unmeasured variables such as learning opportunities outside school or the quality of the teachers. If data on these variables are unavailable, this omitted variables bias cannot be addressed by including the variables in the multiple regressions.
Instrumental variables regression provides an alternative approach to this problem. Consider the following hypothetical example: Some California schools are forced to close for repairs because of a summer earthquake. Districts closest to the epicenter are most severely affected. A district with some closed schools needs to “double up” its students, temporarily increasing class size. This means that dis- tance from the epicenter satisfies the condition for instrument relevance because it is correlated with class size. But if distance to the epicenter is unrelated to any of the other factors affecting student performance (such as whether the students are still learning English, or other disruptive effects of the earthquake on student per- formance), then it will be exogenous because it is uncorrelated with the error term. Thus the instrumental variable, distance to the epicenter, could be used to circum- vent omitted variables bias and to estimate the effect of class size on test scores.
The Sampling Distribution of the TSLS Estimator
The exact distribution of the TSLS estimator in small samples is complicated. However, like the OLS estimator, its distribution in large samples is simple: The TSLS estimator is consistent and is normally distributed.
Formula for the TSLS estimator. Although the two stages of TSLS make the esti- mator seem complicated, when there is a single X and a single instrument Z, as we assume in this section, there is a simple formula for the TSLS estimator. Let sZY be the sample covariance between Z and Y and let sZX be the sample covari- ance between Z and X. As shown in Appendix 12.2, the TSLS estimator with a single instrument is
bnTSLS = sZY. (12.4) 1 sZX
That is, the TSLS estimator of b1 is the ratio of the sample covariance between Z and Y to the sample covariance between Z and X.
Sampling distribution of bnTSLS when the sample size is large. The formula in 1
Equation (12.4) can be used to show that bnTSLS is consistent and, in large samples, 1
normally distributed. The argument is summarized here, with mathematical details given in Appendix 12.3.
432 ChapteR 12 Instrumental Variables Regression
The argument that bnTSLS is consistent combines the assumptions that Z is
1i relevant and exogenous with the consistency of sample covariances for population
covariances. To begin, note that because Yi = b0 + b1Xi + ui in Equation (12.1),
cov(Zi,Yi) = cov3Zi, (b0 + bXi + ui)4 = b1cov(Zi, Xi) + cov(Zi, ui), (12.5)
where the second equality follows from the properties of covariances [Equation (2.33)]. By the instrument exogeneity assumption, cov(Zi, ui) = 0, and by the instrument relevance assumption, cov (Zi, Xi) ≠ 0. Thus, if the instrument is valid, Equation (12.5) implies that
b1 = cov (Zi, Yi ). (12.6) cov (Zi, Xi)
That is, the population coefficient b1 is the ratio of the population covariance between Z and Y to the population covariance between Z and X.
As discussed in Section 3.7, the sample covariance is a consistent estimator of the populationcovariance;thatis,sZY ¡p cov(Zi,Yi)andsZX ¡p cov(Zi,Xi).It follows from Equations (12.4) and (12.6) that the TSLS estimator is consistent:
bnTSLS = sZY ¡p cov(Zi,Yi) = b. (12.7) 1 sZX cov (Zi, Xi) 1
The formula in Equation (12.4) also can be used to show that the sampling distri-
bution of bnTSLS is normal in large samples. The reason is the same as for every 1
other least squares estimator we have considered: The TSLS estimator is an aver-
age of random variables, and when the sample size is large, the central limit theo-
rem tells us that averages of random variables are normally distributed.
1
s = g (Z -Z)(Y -Y),anaverageof(Z -Z)(Y -Y).Abitof
Specifically, the numerator of the expression for bnTSLS in Equation (12.4) is
1n
ZY n-1 i=1 i i i i
algebra, sketched out in Appendix 12.3, shows that because of this averaging the central limit theorem implies that, in large samples, bnTSLS has a sampling distribu-
tion that is approximately N(b , s2 ), where 1 nTSLS
sbn1TSLS n 3cov(Zi, Xi)42
can be estimated by estimating the variance and covariance terms appearing in
2 1 var 3(Zi – mZ)ui4 =.
b1
(12.8) Statistical inference using the large-sample distribution. The variance s2
1
bn 1T S L S
12.1 The IV Estimator with a Single Regressor and a Single Instrument 433 Equation (12.8), and the square root of the estimate of s2 is the standard error
bn 1T S L S
of the IV estimator. This is done automatically in TSLS regression commands in
econometric software packages. Because bnTSLS is normally distributed in large 1
samples, hypothesis tests about b1 can be performed by computing the t-statistic, nTSLS nTSLS
and a 95% large-sample confidence interval is given by b { 1.96 SE(b ). 11
Application to the Demand for Cigarettes
Philip Wright was interested in the demand elasticity of butter, but today other commodities, such as cigarettes, figure more prominently in public policy debates. One tool in the quest for reducing illnesses and deaths from smoking—and the costs, or externalities, imposed by those illnesses on the rest of society—is to tax cigarettes so heavily that current smokers cut back and potential new smokers are discouraged from taking up the habit. But precisely how big a tax hike is needed to make a dent in cigarette consumption? For example, what would the after-tax sales price of cigarettes need to be to achieve a 20% reduction in cigarette consumption?
The answer to this question depends on the elasticity of demand for ciga- rettes. If the elasticity is -1, then the 20% target in consumption can be achieved by a 20% increase in price. If the elasticity is -0.5, then the price must rise 40% to decrease consumption by 20%. Of course, we do not know the demand elastic- ity of cigarettes: We must estimate it from data on prices and sales. But, as with butter, because of the interactions between supply and demand, the elasticity of demand for cigarettes cannot be estimated consistently by an OLS regression of log quantity on log price.
We therefore use TSLS to estimate the elasticity of demand for cigarettes using annual data for the 48 contiguous U.S. states for 1985 through 1995 (the data are described in Appendix 12.1). For now, all the results are for the cross section of states in 1995; results using data for earlier years (panel data) are presented in Section 12.4.
The instrumental variable, SalesTaxi, is the portion of the tax on cigarettes
arising from the general sales tax, measured in dollars per pack (in real dollars,
deflated by the Consumer Price Index). Cigarette consumption, Qcigarettes, is the i
number of packs of cigarettes sold per capita in the state, and the price, P cigarettes, i
is the average real price per pack of cigarettes including all taxes.
Before using TSLS it is essential to ask whether the two conditions for instru- ment validity hold. We return to this topic in detail in Section 12.3, where we provide some statistical tools that help in this assessment. Even with those statis- tical tools, judgment plays an important role, so it is useful to think about whether
the sales tax on cigarettes plausibly satisfies the two conditions.
434 ChapteR 12 Instrumental Variables Regression
First consider instrument relevance. Because a high sales tax increases the
after-tax sales price Pcigarettes, the sales tax per pack plausibly satisfies the condi- i
tion for instrument relevance.
Next consider instrument exogeneity. For the sales tax to be exogenous, it
must be uncorrelated with the error in the demand equation; that is, the sales tax must affect the demand for cigarettes only indirectly through the price. This seems plausible: General sales tax rates vary from state to state, but they do so mainly because different states choose different mixes of sales, income, property, and other taxes to finance public undertakings. Those choices about public finance are driven by political considerations, not by factors related to the demand for ciga- rettes. We discuss the credibility of this assumption more in Section 12.4, but for now we keep it as a working hypothesis.
In modern statistical software, the first stage of TSLS is estimated automati- cally, so you do not need to run this regression yourself to compute the TSLS estimator. Even so, it is a good idea to look at the first-stage regression. Using data for the 48 states in 1995, it is
ln(Pcigarettes) = 4.62 + 0.031SalesTax. (12.9) ii
(0.03) (0.005)
As expected, higher sales taxes mean higher after-tax prices. The R2 of this regres- sion is 47%, so the variation in sales tax on cigarettes explains 47% of the variance of cigarette prices across states.
In the second stage of TSLS, ln(Qcigarettes) is regressed on ln(Pcigarettes) using ii
OLS. The resulting estimated regression function is
ln(Qcigarettes) = 9.72 – 1.08ln(Pcigarettes). (12.10)
This estimated regression function is written using the regressor in the second stage, the predicted value ln(P cigarettes). It is, however, conventional and less cum-
bersome simply to report the estimated regression function with ln(Pcigarettes) i
rather than ln(Pcigarettes). Reported in this notation, the TSLS estimates and i
ii
i
heteroskedasticity-robust standard errors are
ln(Qcigarettes) = 9.72 – 1.08ln(Pcigarettes). (12.11)
(1.53) (0.32)
The TSLS estimate suggests that the demand for cigarettes is surprisingly elastic, in light of their addictive nature: An increase in the price of 1% reduces consumption
ii
by 1.08%. But, recalling our discussion of instrument exogeneity, perhaps this esti- mate should not yet be taken too seriously. Even though the elasticity was estimated using an instrumental variable, there might still be omitted variables that are corre- lated with the sales tax per pack. A leading candidate is income: States with higher incomes might depend relatively less on a sales tax and more on an income tax to finance state government. Moreover, the demand for cigarettes presumably depends on income. Thus we would like to reestimate our demand equation including income as an additional regressor. To do so, however, we must first extend the IV regression model to include additional regressors.
12.2
12.2 The General IV Regression Model 435
The General IV Regression Model
The general IV regression model has four types of variables: the dependent vari- able, Y; problematic endogenous regressors, like the price of cigarettes, which are correlated with the error term and which we will label X; additional regressors, called included exogenous variables, which we will label W; and instrumental vari- ables, Z. In general, there can be multiple endogenous regressors (X’s), multiple included exogenous regressors (W’s), and multiple instrumental variables (Z’s).
For IV regression to be possible, there must be at least as many instrumental variables (Z’s) as endogenous regressors (X’s). In Section 12.1, there was a single endogenous regressor and a single instrument. Having (at least) one instrument for this single endogenous regressor was essential. Without the instrument we could not have computed the instrumental variables estimator: there would be no first-stage regression in TSLS.
The relationship between the number of instruments and the number of endogenous regressors has its own terminology. The regression coefficients are said to be exactly identified if the number of instruments (m) equals the number of endogenous regressors (k); that is, m = k. The coefficients are overidentified if the number of instruments exceeds the number of endogenous regressors; that is, m 7 k. They are underidentified if the number of instruments is less than the number of endogenous regressors; that is, m 6 k. The coefficients must be either exactly identified or overidentified if they are to be estimated by IV regression.
The general IV regression model and its terminology are summarized in Key Concept 12.1.
IncludedexogenousvariablesandcontrolvariablesinIVregression. TheWvari-
E(u 0W) = 0, or they can be control variables that need not have a causal ii
ables in Equation (12.12) can be either exogenous variables, in which case
436 ChapteR 12 Instrumental Variables Regression
Key ConCept
12.1
the General Instrumental Variables Regression Model and terminology
The general IV regression model is
Yi = b0 + b1X1i + g+ bkXki + bk+1W1i + g+ bk+rWri + ui,
i= 1, c, n, where
• Yi is the dependent variable;
• b0, b1, c, bk + r are unknown regression coefficients;
• X1i, c, Xki are k endogenous regressors, which are potentially correlated with ui;
• W1i, c, Wri are r included exogenous regressors, which are uncorrelated with ui or are control variables;
• ui is the error term, which represents measurement error and/or omitted factors; and
• Z1i, c, Zmi are m instrumental variables.
The coefficients are overidentified if there are more instruments than endogenous regressors (m 7 k), they are underidentified if m 6 k, and they are exactly identi- fied if m = k. Estimation of the IV regression model requires exact identification or overidentification.
(12.12)
interpretation but are included to ensure that the instrument is uncorrelated with the error term. For example, Section 12.1 raised the possibility that the sales tax might be correlated with income, which economic theory tells us is a determinant of cigarette demand. If so, the sales tax would be correlated with the error term in the cigarette demand equation, ln(Qci igarettes) = b0 + b1ln(Picigarettes) + ui, and thus would not be an exogenous instrument. Including income in the regression, or including variables that control for income, would remove this source of poten- tial correlation between the instrument and the error term. In general, if W is an effective control variable in IV regression, then including W makes the instrument uncorrelated with u, so the TSLS estimator of the coefficient on X is consistent; if W is correlated with u, however, then the TSLS coefficient on W is subject to omitted variable bias and does not have a causal interpretation. The logic of con- trol variables in IV regression therefore parallels the logic of control variables in OLS, discussed in Section 7.5.
12.2 The General IV Regression Model 437
The mathematical condition for W to be an effective control variable in IV
regression is similar to the condition on control variables in OLS discussed in
Section 7.5. Specifically, including W must ensure that the conditional mean of u
E(u 0Z, W) = E(u 0W). For clarity, in the body of this chapter we focus on the iii ii
does not depend on Z, so conditional mean independence holds; that is,
case that W variables are exogenous so that E(u 0 W ) = 0. Appendix 12.6 explains ii
how the results of this chapter extend to the case that W is a control variable, in which case the conditional mean zero condition, E(u 0 W ) = 0, is replaced by the
conditional mean independence condition, E(u 0 Z , W ) = E(u 0 W ). iii ii
ii
TSLS with a single endogenous regressor. When there is a single endogenous regressor X and some additional included exogenous variables, the equation of interest is
Yi = b0 + b1Xi + b2W1i + g+ b1+rWri + ui, (12.13)
where, as before, Xi might be correlated with the error term, but W1i, c, Wri are not. The population first-stage regression of TSLS relates X to the exogenous
TSLS in the General IV Model
variables, that is, the W’s and the instruments (Z’s):
X =p +pZ +g+pZ +p W +g+p W +v, (12.14)
where p0, p1, c, pm + r are unknown regression coefficients and vi is an error term. Equation (12.14) is sometimes called the reduced form equation for X. It relates the endogenous variable X to all the available exogenous variables, both
those included in the regression of interest (W) and the instruments (Z).
In the first stage of TSLS, the unknown coefficients in Equation (12.14) are estimated by OLS, and the predicted values from this regression are Xn 1, c, Xn n. In the second stage of TSLS, Equation (12.13) is estimated by OLS, except that Xi is replaced by its predicted value from the first stage. That is, Yi is regressed on Xn i, W1i, c, Wri using OLS. The resulting estimator of b0, b1, c, b1 + r is the
TSLS estimator.
Extension to multiple endogenous regressors. When there are multiple endoge- nous regressors X1i, c, Xki, the TSLS algorithm is similar, except that each endogenous regressor requires its own first-stage regression. Each of these first- stage regressions has the same form as Equation (12.14); that is, the dependent
i 0 1 1i m mi m+1 1i m+r ri i
438 ChapteR 12 Instrumental Variables Regression
two Stage Least Squares
12.2
Key ConCept
The TSLS estimator in the general IV regression model in Equation (12.12) with multiple instrumental variables is computed in two stages:
1. First-stage regression(s): Regress X1i on the instrumental variables (Z1i, c, Zmi) and the included exogenous variables (W1i, c, Wri) using OLS, including an intercept. Compute the predicted values from this regres- sion; call these Xn 1i. Repeat this for all the endogenous regressorsX2i, c, Xki, thereby computing the predicted values X , c, X .
n1i nki
2. Second-stageregression:RegressYionthepredictedvaluesoftheendogenous
variables (X , c, X ) and the included exogenous variables (W , c, W ) n1i nki 1i ri
nTSLS nTSLS using OLS, including an intercept. The TSLS estimators b , c, b are
the estimators from the second-stage regression.
In practice, the two stages are done automatically within TSLS estimation com- mands in modern econometric software.
0 k+r
variable is one of the X’s, and the regressors are all the instruments (Z’s) and all the included exogenous variables (W’s). Together, these first-stage regressions produce predicted values of each of the endogenous regressors.
In the second stage of TSLS, Equation (12.12) is estimated by OLS, except that the endogenous regressors (X’s) are replaced by their respective predicted values (Xn ’s). The resulting estimator of b0, b1, c, bk + r is the TSLS estimator.
In practice, the two stages of TSLS are done automatically within TSLS esti- mation commands in modern econometric software. The general TSLS estimator is summarized in Key Concept 12.2.
Instrument Relevance and Exogeneity
in the General IV Model
The conditions of instrument relevance and exogeneity need to be modified for the general IV regression model.
When there is one included endogenous variable but multiple instruments, the condition for instrument relevance is that at least one Z is useful for predicting X, given W. When there are multiple included endogenous variables, this condi- tion is more complicated because we must rule out perfect multicollinearity in the
12.2 The General IV Regression Model 439
the two Conditions for Valid Instruments
Key ConCept
12.3
A set of m instruments Z1i, c, Zmi must satisfy the following two conditions to be valid:
1. Instrument Relevance
• In general, let Xn *1i be the predicted value of X1i from the population regres-
sion of X1i on the instruments (Z’s) and the included exogenous regressors
(W’s), and let “1” denote the constant regressor that takes on the value 1 for
all observations. Then (X , c, X , W , c, W , 1) are not perfectly multi- n *1 i n *k i 1 i r i
collinear.
• If there is only one X, then for the previous condition to hold, at least one Z must have a non-zero coefficient in the population regression of X on the Z’s and the W’s.
2. Instrument Exogeneity
The instruments are uncorrelated with the error term; that is, corr(Z1i, ui) = 0, c, corr(Zmi, ui) = 0.
second-stage population regression. Intuitively, when there are multiple included endogenous variables, the instruments must provide enough information about the exogenous movements in these variables to sort out their separate effects on Y.
The general statement of the instrument exogeneity condition is that each instrument must be uncorrelated with the error term ui. The general conditions for valid instruments are given in Key Concept 12.3.
The IV Regression Assumptions and Sampling
Distribution of the TSLS Estimator
Under the IV regression assumptions, the TSLS estimator is consistent and has a sampling distribution that, in large samples, is approximately normal.
TheIVregressionassumptions. TheIVregressionassumptionsaremodificationsof the least squares assumptions for the multiple regression model in Key Concept 6.4. The first IV regression assumption modifies the conditional mean assumption in Key Concept 6.4 to apply only to the included exogenous variables. Just like the second least squares assumption for the multiple regression model, the second
440 ChapteR 12 Instrumental Variables Regression
the IV Regression assumptions
12.4
Key ConCept
The variables and errors in the IV regression model in Key Concept 12.1 satisfy the following:
1. E(ui W1i, c, Wri) = 0;
2. (X1i, c, Xki, W1i, c, Wri, Z1i, c, Zmi,Yi) are i.i.d. draws from their joint
distribution;
3. Large outliers are unlikely: The X’s, W’s, Z’s, and Y have nonzero finite fourth moments; and
4. The two conditions for a valid instrument in Key Concept 12.3 hold.
IV regression assumption is that the draws are i.i.d., as they are if the data are collected by simple random sampling. Similarly, the third IV assumption is that large outliers are unlikely.
The fourth IV regression assumption is that the two conditions for instrument validity in Key Concept 12.3 hold. The instrument relevance condition in Key Concept 12.3 subsumes the fourth least squares assumption in Key Concept 4.6 (no perfect multicollinearity) by assuming that the regressors in the second-stage regression are not perfectly multicollinear. The IV regression assumptions are summarized in Key Concept 12.4.
Sampling distribution of the TSLS estimator. Under the IV regression assump- tions, the TSLS estimator is consistent and normally distributed in large samples. This is shown in Section 12.1 (and Appendix 12.3) for the special case of a single endogenous regressor, a single instrument, and no included exogenous variables. Conceptually, the reasoning in Section 12.1 carries over to the general case of multiple instruments and multiple included endogenous variables. The expressions in the general case are complicated, however, and are deferred to Chapter 18.
Inference Using the TSLS Estimator
Because the sampling distribution of the TSLS estimator is normal in large sam- ples, the general procedures for statistical inference (hypothesis tests and confi- dence intervals) in regression models extend to TSLS regression. For example, 95% confidence intervals are constructed as the TSLS estimator {1.96 standard errors. Similarly, joint hypotheses about the population values of the coefficients can be tested using the F-statistic, as described in Section 7.2.
12.2 The General IV Regression Model 441
Calculation of TSLS standard errors. There are two points to bear in mind about TSLS standard errors. First, the standard errors reported by OLS estimation of the second-stage regression are incorrect because they do not recognize that it is the second stage of a two-stage process. Specifically, the second-stage OLS stan- dard errors fail to adjust for the second-stage regression using the predicted val- ues of the included endogenous variables. Formulas for standard errors that make the necessary adjustment are incorporated into (and automatically used by) TSLS regression commands in econometric software. Therefore, this issue is not a con- cern in practice if you use a specialized TSLS regression command.
Second, as always the error u might be heteroskedastic. It is therefore impor- tant to use heteroskedasticity-robust versions of the standard errors for precisely the same reason as it is important to use heteroskedasticity-robust standard errors for the OLS estimators of the multiple regression model.
Application to the Demand for Cigarettes
In Section 12.1, we estimated the elasticity of demand for cigarettes using data on annual consumption in 48 U.S. states in 1995 using TSLS with a single regressor (the logarithm of the real price per pack) and a single instrument (the real sales tax per pack). Income also affects demand, however, so it is part of the error term of the population regression. As discussed in Section 12.1, if the state sales tax is related to state income, it is correlated with a variable in the error term of the cigarette demand equation, which violates the instrument exogeneity condition. If so, the IV estimator in Section 12.1 is inconsistent. That is, the IV regression suffers from a version of omitted variable bias. To solve this problem, we need to include income in the regression.
We therefore consider an alternative specification in which the logarithm of
income is included in the demand equation. In the terminology of Key Concept
12.1, the dependent variable Y is the logarithm of consumption, ln (Q cigarettes); the i
endogenous regressorX is the logarithm of the real after-tax price, ln (P cigarettes); i
the included exogenous variable W is the logarithm of the real per capita state income, ln(Inci); and the instrument Z is the real sales tax per pack, SalesTaxi. The TSLS estimates and (heteroskedasticity-robust) standard errors are
ln(Qcigarettes) = 9.43 – 1.14ln(Pcigarettes) + 0.21ln(Inc). (12.15) iii
(1.26) (0.37) (0.31)
This regression uses a single instrument, SalesTaxi, but in fact another candidate instrument is available. In addition to general sales taxes, states levy special taxes
442
ChapteR 12 Instrumental Variables Regression
that apply only to cigarettes and other tobacco products. These cigarette-specific taxes (CigTaxi) constitute a possible second instrumental variable. The cigarette- specific tax increases the price of cigarettes paid by the consumer, so it arguably meets the condition for instrument relevance. If it is uncorrelated with the error term in the state cigarette demand equation, it is an exogenous instrument.
With this additional instrument in hand, we now have two instrumental vari-
ables, the real sales tax per pack and the real state cigarette-specific tax per pack.
With two instruments and a single endogenous regressor, the demand elasticity is
overidentified; that is, the number of instruments (SalesTaxi and CigTaxi, so m = 2)
exceedsthenumberofincludedendogenousvariables(Pcigarettes,sok = 1).Wecan i
estimate the demand elasticity using TSLS, where the regressors in the first-stage regression are the included exogenous variable, ln(Inci), and both instruments.
The resulting TSLS estimate of the regression function using the two instru- ments SalesTaxi and CigTaxi is
ln(Qcigarettes) = 9.89 – 1.28 ln(P cigarettes) + 0.28 ln(Inc ) (12.16) iii.
(0.96) (0.25)
(0.25)
Compare Equations (12.15) and (12.16): The standard error of the estimated price elasticity is smaller by one-third in Equation (12.16) [0.25 in Equation (12.16) versus 0.37 in Equation (12.15)]. The reason the standard error is smaller in Equa- tion (12.16) is that this estimate uses more information than Equation (12.15): In Equation (12.15), only one instrument is used (the sales tax), but in Equation (12.16), two instruments are used (the sales tax and the cigarette-specific tax). Using two instruments explains more of the variation in cigarette prices than using just one, and this is reflected in smaller standard errors on the estimated demand elasticity.
Are these estimates credible? Ultimately, credibility depends on whether the set of instrumental variables—here, the two taxes—plausibly satisfies the two con- ditions for valid instruments. It is therefore vital that we assess whether these instruments are valid, and it is to this topic that we now turn.
12.3
Checking Instrument Validity
Whether instrumental variables regression is useful in a given application hinges on whether the instruments are valid: Invalid instruments produce meaningless results. It therefore is essential to assess whether a given set of instruments is valid in a particular application.
12.3 Checking Instrument Validity 443 Assumption #1: Instrument Relevance
The role of the instrument relevance condition in IV regression is subtle. One way to think of instrument relevance is that it plays a role akin to the sample size: The more relevant the instruments—that is, the more the variation in X is explained by the instruments—the more information is available for use in IV regression. A more relevant instrument produces a more accurate estimator, just as a larger sample size produces a more accurate estimator. Moreover, statistical inference using TSLS is predicated on the TSLS estimator having a normal sampling distri- bution, but according to the central limit theorem the normal distribution is a good approximation in large—but not necessarily small—samples. If having a more relevant instrument is like having a larger sample size, this suggests, cor- rectly, that the more relevant is the instrument, the better is the normal approxi- mation to the sampling distribution of the TSLS estimator and its t-statistic.
Instruments that explain little of the variation in X are called weak instru- ments. In the cigarette example, the distance of the state from cigarette manufac- turing plants arguably would be a weak instrument: Although a greater distance increases shipping costs (thus shifting the supply curve in and raising the equilib- rium price), cigarettes are lightweight, so shipping costs are a small component of the price of cigarettes. Thus the amount of price variation explained by shipping costs, and thus distance to manufacturing plants, probably is quite small.
This section discusses why weak instruments are a problem, how to check for weak instruments, and what to do if you have weak instruments. It is assumed throughout that the instruments are exogenous.
Why weak instruments are a problem. If the instruments are weak, then the nor- mal distribution provides a poor approximation to the sampling distribution of the TSLS estimator, even if the sample size is large. Thus there is no theoretical jus- tification for the usual methods for performing statistical inference, even in large samples. In fact, if instruments are weak, then the TSLS estimator can be badly biased in the direction of the OLS estimator. In addition, 95% confidence inter- vals constructed as the TSLS estimator {1.96 standard errors can contain the true value of the coefficient far less than 95% of the time. In short, if instruments are weak, TSLS is no longer reliable.
To see that there is a problem with the large-sample normal approximation to the sampling distribution of the TSLS estimator, consider the special case, introduced in Section 12.1, of a single included endogenous variable, a single instrument, and no included exogenous regressor. If the instrument is valid, then
bnTSLS is consistent because the sample covariances s and s are consistent; that 1 ZY ZX
444 ChapteR 12 Instrumental Variables Regression
a Rule of thumb for Checking for Weak Instruments
12.5
Key ConCept
The first-stage F-statistic is the F-statistic testing the hypothesis that the coef- ficients on the instruments Z1i, c, Zmi equal zero in the first stage of two stage least squares. When there is a single endogenous regressor, a first-stage F-statistic less than 10 indicates that the instruments are weak, in which case the TSLS estima- tor is biased (even in large samples) and TSLS t-statistics and confidence intervals are unreliable.
nTSLS
is, b = s /s ¡p cov(Z, Y )>cov(Z, X) = b [Equation (12.7)]. But
1ZYZX iiii1
now suppose that the instrument is not just weak but irrelevant so that
cov(Zi, Xi) = 0. Then sZX ¡p cov(Zi, Xi) = 0, so, taken literally, the denomi-
nator on the right-hand side of the limit cov(Zi, Yi)/cov(Zi, Xi) is zero! Clearly,
the argument that bnTSLS is consistent breaks down when the instrument relevance 1
condition fails. As shown in Appendix 12.4, this breakdown results in the TSLS
estimator having a nonnormal sampling distribution, even if the sample size is
very large. In fact, when the instrument is irrelevant, the large-sample distribution
of bnTSLS is not that of a normal random variable, but rather the distribution of a 1
ratio of two normal random variables!
While this circumstance of totally irrelevant instruments might not be encoun-
tered in practice, it raises a question: How relevant must the instruments be for the normal distribution to provide a good approximation in practice? The answer to this question in the general IV model is complicated. Fortunately, however, there is a simple rule of thumb available for the most common situation in practice, the case of a single endogenous regressor.
Checkingforweakinstrumentswhenthereisasingleendogenousregressor. One way to check for weak instruments when there is a single endogenous regressor is to compute the F-statistic testing the hypothesis that the coefficients on the instru- ments are all zero in the first-stage regression of TSLS. This first-stage F-statistic provides a measure of the information content contained in the instruments: The more information content, the larger is the expected value of the F-statistic. One simple rule of thumb is that you do not need to worry about weak instruments if the first-stage F-statistic exceeds 10. (Why 10? See Appendix 12.5.) This is sum- marized in Key Concept 12.5.
What do I do if I have weak instruments? If you have many instruments, some of those instruments are probably weaker than others. If you have a small number of strong instruments and many weak ones, you will be better off discarding the weakest instruments and using the most relevant subset for your TSLS analysis. Your TSLS standard errors might increase when you drop weak instruments, but keep in mind that your original standard errors were not meaningful anyway!
If, however, the coefficients are exactly identified, you cannot discard the weak instruments. Even if the coefficients are overidentified, you might not have enough strong instruments to achieve identification, so discarding some weak instruments will not help. In this case, you have two options. The first option is to find additional, stronger instruments. This is easier said than done: It requires an intimate knowledge of the problem at hand and can entail redesigning the data set and the nature of the empirical study. The second option is to proceed with your empirical analysis using the weak instruments, but employing methods other than TSLS. Although this chapter has focused on TSLS some other methods for instrumental variable analysis are less sensitive to weak instruments than TSLS, and some of these methods are discussed in Appendix 12.5.
Assumption #2: Instrument Exogeneity
If the instruments are not exogenous, then TSLS is inconsistent: The TSLS estimator converges in probability to something other than the population coefficient in the regression. After all, the idea of instrumental variables regression is that the instru- ment contains information about variation in Xi that is unrelated to the error term ui. If, in fact, the instrument is not exogenous, it cannot pinpoint this exogenous variation in Xi, and it stands to reason that IV regression fails to provide a consistent estimator. The math behind this argument is summarized in Appendix 12.4.
Canyoustatisticallytesttheassumptionthattheinstrumentsareexogenous? Yes and no. On the one hand, it is not possible to test the hypothesis that the instru- ments are exogenous when the coefficients are exactly identified. On the other hand, if the coefficients are overidentified, it is possible to test the overidentifying restrictions, that is, to test the hypothesis that the “extra” instruments are exog- enous under the maintained assumption that there are enough valid instruments to identify the coefficients of interest.
First consider the case that the coefficients are exactly identified, so you have as many instruments as endogenous regressors. Then it is impossible to develop a statistical test of the hypothesis that the instruments are in fact exogenous. That is, empirical evidence cannot be brought to bear on the question of whether these
12.3 Checking Instrument Validity 445
446 ChapteR 12 Instrumental Variables Regression a Scary regression
One way to estimate the percentage increase in earnings from going to school for another year (the “return to education”) is to regress the logarithm of earnings against years of school using data on indi- viduals. But if more able individuals are both more successful in the labor market and attend school lon- ger (perhaps because they find it easier), then years of schooling will be correlated with the omitted variable, innate ability, and the OLS estimator of the return to education will be biased. Because innate ability is extremely difficult to measure and thus cannot be used as a regressor, some labor economists have turned to IV regression to estimate the return to education. But what variable is correlated with years of education but not the error term in the earnings regression? That is, what is a valid instrumental variable?
Your birthday, suggested labor economists Joshua Angrist and Alan Krueger. Because of mandatory schooling laws, they reasoned, your birthday is corre- lated with your years of education: If the law requires you to attend school until your 16th birthday and you turn 16 in January while you are in tenth grade, you might drop out—but if you turn 16 in July you already will have completed tenth grade. If so, your birthday satisfies the instrument relevance condition. But being born in January or July should have no direct effect on your earnings (other than through years of education), so your birthday satisfies the instrument exogeneity condition. They implemented this idea by using the individual’s quarter (three-month period) of birth as an instrumental variable. They used a very large sample of data from the U.S. Census (their regres- sions had at least 329,000 observations!), and they controlled for other variables such as the worker’s age.
But John Bound, another labor economist, was skeptical. He knew that weak instruments cause TSLS to be unreliable and worried that, despite the
extremely large sample size, the quarter of birth might be a weak instrument in some of their specifications. So when Bound and Krueger next met over lunch, the conversation inevitably turned to whether the Angrist–Krueger instruments were weak. Krueger thought not and suggested a creative way to find out: Why not rerun the regressions using a truly irrelevant instrument—replace each individual’s real quarter of birth by a fake quarter of birth, randomly generated by the computer—and compare the results using the real and fake instruments? What they found was amazing: It didn’t matter whether you used the real quarter of birth or the fake one as the instrument— TSLS gave basically the same answer!
This was a scary regression for labor econome- tricians. The TSLS standard error computed using the real data suggests that the return to education is precisely estimated—but so does the standard error computed using the fake data. Of course, the fake data cannot estimate the return to education pre- cisely, because the fake instrument is totally irrel- evant. The worry, then, is that the TSLS estimates based on the real data are just as unreliable as those based on the fake data.
The problem is that the instruments are in fact very weak in some of Angrist and Krueger’s regressions. In some of their specifications, the first-stage F-statistic is less than 2, far less than the rule-of-thumb cutoff of 10. In other specifications, Angrist and Krueger have larger first-stage F-statistics, and in those cases the TSLS inferences are not subject to the problem of weak instruments. By the way, in those specifications the return to education is estimated to be approxi- mately 8%, somewhat greater than estimated by OLS.1
1The original IV regressions are reported in Angrist and Krueger (1991), and the re-analysis using the fake instru- ments is published in Bound, Jaeger, and Baker (1995).
instruments satisfy the exogeneity restriction. In this case, the only way to assess whether the instruments are exogenous is to draw on expert opinion and your personal knowledge of the empirical problem at hand. For example, Philip Wright’s knowledge of agricultural supply and demand led him to suggest that below-average rainfall would plausibly shift the supply curve for butter but would not directly shift the demand curve.
Assessing whether the instruments are exogenous necessarily requires making an expert judgment based on personal knowledge of the application. If, however, there are more instruments than endogenous regressors, then there is a statistical tool that can be helpful in this process: the so-called test of overidentifying restrictions.
The overidentifying restrictions test. Suppose that you have a single endogenous regressor and two instruments. Then you could compute two different TSLS esti- mators: one using the first instrument, the other using the second. These two estimators will not be the same because of sampling variation, but if both instru- ments are exogenous, then they will tend to be close to each other. But what if these two instruments produce very different estimates? You might sensibly con- clude that there is something wrong with one or the other of the instruments, or both. That is, it would be reasonable to conclude that one or the other, or both, of the instruments are not exogenous.
The test of overidentifying restrictions implicitly makes this comparison. We
say implicitly, because the test is carried out without actually computing all of the
different possible IV estimates. Here is the idea. Exogeneity of the instruments
means that they are uncorrelated with ui. This suggests that the instruments
should be approximately uncorrelated with unTSLS, where unTSLS = Y – (bnTSLS + iii0
bnTSLSX + g + bnTSLSW ) is the residual from the estimated TSLS regression 11i k+rri
using all the instruments (approximately rather than exactly because of sampling
variation). (Note that these residuals are constructed using the true X’s rather
than their first-stage predicted values.) Accordingly, if the instruments are in fact
exogenous, then the coefficients on the instruments in a regression of unTSLS on the i
instruments and the included exogenous variables should all be zero, and this hypothesis can be tested.
This method for computing the overidentifying restriction test is summarized in Key Concept 12.6. This statistic is computed using the homoskedasticity-only F-statistic. The test statistic is commonly called the J-statistic and is computed as J = mF.
In large samples, if the instruments are not weak and the errors are homoskedastic, then, under the null hypothesis that the instruments are exogenous, the J-statistic has a chi-squared distribution with m − k degrees of freedom (x2m – k). It is important to remember that even though the number of restrictions being tested
12.3 Checking Instrument Validity 447
448 ChapteR 12 Instrumental Variables Regression
Key ConCept
12.6
the Overidentifying Restrictions test (the J-Statistic)
Let unTSLS be the residuals from TSLS estimation of Equation (12.12). Use OLS to
i
estimate the regression coefficients in unTSLS=d+dZ+g+dZ+d W+g+d W+e,
where ei is the regression error term. Let F denote the homoskedasticity-only F-statistic testing the hypothesis that d1 = g = dm = 0. The overidentifying restrictions test statistic is J = mF. Under the null hypothesis that all the instru- ments are exogenous, if ei is homoskedastic, in large samples J is distributed x2m – k, where m – k is the “degree of overidentification,” that is, the number of instru- ments minus the number of endogenous regressors.
i 0 1 1i m mi m+1 1i m+r ri i
is m, the degrees of freedom of the asymptotic distribution of the J-statistic is m – k. The reason is that it is only possible to test the overidentifying restrictions, of which there are m – k. The modification of the J-statistic for heteroskedastic errors is given in Section 18.7.
The easiest way to see that you cannot test the exogeneity of the regressors when the coefficients are exactly identified (m = k) is to consider the case of a single included endogenous variable (k = 1). If there are two instruments, then you can compute two TSLS estimators, one for each instrument, and you can compare them to see if they are close. But if you have only one instrument, then you can compute only one TSLS estimator and you have nothing to compare it to. In fact,ifthecoefficientsareexactlyidentified,sothatm = k,thentheoveridentifying test statistic J is exactly zero.
12.4
Application to the Demand for Cigarettes1
Our attempt to estimate the elasticity of demand for cigarettes left off with the TSLS estimates summarized in Equation (12.16), in which income was an included exoge- nous variable and there were two instruments, the general sales tax and the cigarette- specific tax. We can now undertake a more thorough evaluation of these instruments.
1This section assumes knowledge of the material in Sections 10.1 and 10.2 on panel data with T = 2 time periods.
12.4 Application to the Demand for Cigarettes 449
As in Section 12.1, it makes sense that the two instruments are relevant because taxes are a big part of the after-tax price of cigarettes, and shortly we will look at this empirically. First, however, we focus on the difficult question of whether the two tax variables are plausibly exogenous.
The first step in assessing whether an instrument is exogenous is to think through the arguments for why it may or may not be. This requires thinking about which factors account for the error term in the cigarette demand equation and whether these factors are plausibly related to the instruments.
Why do some states have higher per capita cigarette consumption than others? One reason might be variation in incomes across states, but state income is included in Equation (12.16), so this is not part of the error term. Another reason is that there are historical factors influencing demand. For example, states that grow tobacco have higher rates of smoking than most other states. Could this factor be related to taxes? Quite possibly: If tobacco farming and cigarette production are important industries in a state, then these industries could exert influence to keep cigarette-specific taxes low. This suggests that an omitted factor in cigarette demand—whether the state grows tobacco and produces cigarettes—could be correlated with cigarette-specific taxes.
One solution to this possible correlation between the error term and the instru- ment would be to include information on the size of the tobacco and cigarette indus- try in the state; this is the approach we took when we included income as a regressor in the demand equation. But because we have panel data on cigarette consumption, a different approach is available that does not require this information. As discussed in Chapter 10, panel data make it possible to eliminate the influence of variables that vary across entities (states) but do not change over time, such as the climate and historical circumstances that lead to a large tobacco and cigarette industry in a state. Two methods for doing this were given in Chapter 10: constructing data on changes in the variables between two different time periods and using fixed effects regression. To keep the analysis here as simple as possible, we adopt the former approach and perform regressions of the type described in Section 10.2, based on the changes in the variables between two different years.
The time span between the two different years influences how the estimated elasticities are to be interpreted. Because cigarettes are addictive, changes in price will take some time to alter behavior. At first, an increase in the price of cigarettes might have little effect on demand. Over time, however, the price increase might contribute to some smokers’ desire to quit, and, importantly, it could discourage nonsmokers from taking up the habit. Thus the response of demand to a price increase could be small in the short run but large in the long run. Said differently, for an addictive product like cigarettes, demand might be inelastic in the short
450 ChapteR 12 Instrumental Variables Regression the externalities of Smoking
S moking imposes costs that are not fully borne by the smoker; that is, it generates externalities. One economic justification for taxing cigarettes therefore is to “internalize” these externalities. In theory, the tax on a pack of cigarettes should equal the dollar value of the externalities created by smoking that pack. But what, precisely, are the externalities of smoking, mea-
sured in dollars per pack?
Several studies have used econometric methods to
estimate the externalities of smoking. The negative externalities—costs—borne by others include medi- cal costs paid by the government to care for ill smok- ers, health care costs of nonsmokers associated with secondhand smoke, and fires caused by cigarettes.
But, from a purely economic point of view, smok- ing also has positive externalities, or benefits. The biggest economic benefit of smoking is that smok- ers tend to pay much more in Social Security (public pension) taxes than they ever get back. There are also large savings in nursing home expenditures on the very old—smokers tend not to live that long. Because the negative externalities of smok- ing occur while the smoker is alive but the positive
ones accrue after death, the net present value of the per-pack externalities (the value of the net costs per pack, discounted to the present) depends on the dis- count rate.
The studies do not agree on a specific dollar value of the net externalities. Some suggest that the net externalities, properly discounted, are quite small, less than current taxes. In fact, the most extreme estimates suggest that the net externalities are posi- tive, so smoking should be subsidized! Other studies, which incorporate costs that are probably important but difficult to quantify (such as caring for babies who are unhealthy because their mothers smoke), suggest that externalities might be $1 per pack, possibly even more. But all the studies agree that, by tending to die in late middle age, smokers pay far more in taxes than they ever get back in their brief retirement.1
1An early calculation of the externalities of smoking was reported by Willard G. Manning et al. (1989). A calcula- tion suggesting that health care costs would go up if every- one stopped smoking is presented in Barendregt et al. (1997). Other studies of the externalities of smoking are reviewed by Chaloupka and Warner (2000).
run—that is, it might have a short-run elasticity near zero—but it might be more elastic in the long run.
In this analysis, we focus on estimating the long-run price elasticity. We do
this by considering quantity and price changes that occur over 10-year periods.
Specifically, in the regressions considered here, the 10-year change in log quantity,
ln(Qcigarettes) – ln(Qcigarettes), is regressed against the 10-year change in log price, i,1995 i,1985
ln(P cigarettes) – ln(P cigarettes), and the 10-year change in log income, ln(Inc ) – i,1995 i,1985 i,1995
ln(Inci,1985). Two instruments are used: the change in the sales tax over 10 years, SalesTaxi,1995 – SalesTaxi,1985, and the change in the cigarette-specific tax over 10 years, CigTaxi,1995 – CigTaxi,1985.
12.4 Application to the Demand for Cigarettes 451 taBLe 12.1 two Stage Least Squares estimates of the Demand for Cigarettes Using
panel Data for 48 U.S. States
Dependent variable: ln(Qcigarettes) – ln(Qcigarettes) i,1995 i,1985
regressor
ln(Pcigarettes) – ln(Pcigarettes) i,1995 i,1985
ln(Inci,1995) – ln(Inci,1985)
Intercept
Instrumental variable(s)
First-stage F-statistic
Overidentifying restrictions J-test and p-value
(1)
– 0.94** (0.21)
0.53 (0.34)
– 0.12 (0.07)
Sales tax
33.70
—
(2)
– 1.34** (0.23)
0.43 (0.30)
– 0.02 (0.07)
Cigarette-specific tax
107.20
—
(3)
– 1.20** (0.20)
0.46 (0.31)
– 0.05 (0.06)
Both sales tax and cigarette-specific tax
88.60
4.93 (0.026)
These regressions were estimated using data for 48 U.S. states (48 observations on the 10-year differences). The data are described in Appendix 12.1. The J-test of overidentifying restrictions is described in Key Concept 12.6 (its p-value is given in parentheses), and the first-stage F-statistic is described in Key Concept 12.5. Individual coefficients are statistically significant at the *5% significance level or **1% significance level.
The results are presented in Table 12.1. As usual, each column in the table presents the results of a different regression. All regressions have the same regres- sors, and all coefficients are estimated using TSLS; the only difference between the three regressions is the set of instruments used. In column (1), the only instru- ment is the sales tax; in column (2), the only instrument is the cigarette-specific tax; and in column (3), both taxes are used as instruments.
In IV regression, the reliability of the coefficient estimates hinges on the validity of the instruments, so the first things to look at in Table 12.1 are the diag- nostic statistics assessing the validity of the instruments.
First, are the instruments relevant? We need to look at the first-stage F-statistics. The first-stage regression in column (1) is
ln(Pcigarettes) – ln(Pcigarettes) = 0.53 – 0.223ln(Inc ) – ln(Inc )4 i,1995 i,1985 i,1995 i,1985
(0.03) (0.22)
+ 0.0255(SalesTaxi,1995 – SalesTaxi,1985). (0.0044)
(12.18)
452 ChapteR 12 Instrumental Variables Regression
Because there is only one instrument in this regression, the first-stage F-statistic
22 SalesTax – SalesTax , is zero; this is F = t = (0.0255>0.0044) = 33.7.
is the square of the t-statistic testing that the coefficient on the instrumental variable,
i,1995 i,1985
For the regressions in columns (2) and (3), the first-stage F-statistics are 107.2 and
88.6, so in all three cases the first-stage F-statistics exceed 10. We conclude that the instruments are not weak, so we can rely on the standard methods for statistical inference (hypothesis tests, confidence intervals) using the TSLS coefficients and standard errors.
Second, are the instruments exogenous? Because the regressions in columns (1) and (2) each have a single instrument and a single included endogenous regres- sor, the coefficients in those regressions are exactly identified. Thus we cannot deploy the J-test in either of those regressions. The regression in column (3), however, is overidentified because there are two instruments and a single included endogenous regressor, so there is one (m – k = 2 – 1 = 1) overidentifying restriction. The J-statistic is 4.93; this has a x21 distribution, so the 5% critical value is 3.84 (Appendix Table 3) and the null hypothesis that both the instruments are exogenous is rejected at the 5% significance level (this deduction also can be made directly from the p-value of 0.026, reported in the table).
The reason the J-statistic rejects the null hypothesis that both instruments are exogenous is that the two instruments produce rather different estimated coefficients. When the only instrument is the sales tax [column (1)], the estimated price elasticity is – 0.94, but when the only instrument is the cigarette-specific tax, the estimated price elasticity is – 1.34. Recall the basic idea of the J-statistic: If both instruments are exog- enous, then the two TSLS estimators using the individual instruments are consistent and differ from each other only because of random sampling variation. If, however, one of the instruments is exogenous and one is not, then the estimator based on the endogenous instrument is inconsistent, which is detected by the J-statistic. In this application, the difference between the two estimated price elasticities is sufficiently large that it is unlikely to be the result of pure sampling variation, so the J-statistic rejects the null hypothesis that both the instruments are exogenous.
The J-statistic rejection means that the regression in column (3) is based on invalid instruments (the instrument exogeneity condition fails). What does this imply about the estimates in columns (1) and (2)? The J-statistic rejection says that at least one of the instruments is endogenous, so there are three logical pos- sibilities: The sales tax is exogenous but the cigarette-specific tax is not, in which case the column (1) regression is reliable; the cigarette-specific tax is exogenous but the sales tax is not, so the column (2) regression is reliable; or neither tax is exogenous, so neither regression is reliable. The statistical evidence cannot tell us which possibility is correct, so we must use our judgment.
12.5 Where Do Valid Instruments Come From? 453
We think that the case for the exogeneity of the general sales tax is stronger than that for the cigarette-specific tax, because the political process can link changes in the cigarette-specific tax to changes in the cigarette market and smok- ing policy. For example, if smoking decreases in a state because it falls out of fashion, there will be fewer smokers and a weakened lobby against cigarette- specific tax increases, which in turn could lead to higher cigarette-specific taxes. Thus changes in tastes (which are part of u) could be correlated with changes in cigarette-specific taxes (the instrument). This suggests discounting the IV esti- mates that use the cigarette-only tax as an instrument and adopting the price elasticity estimated using the general sales tax as an instrument, -0.94.
The estimate of – 0.94 indicates that cigarette consumption is somewhat elas- tic: An increase in price of 1% leads to a decrease in consumption of 0.94%. This may seem surprising for an addictive product like cigarettes. But remember that this elasticity is computed using changes over a 10-year period, so it is a long-run elasticity. This estimate suggests that increased taxes can make a substantial dent in cigarette consumption, at least in the long run.
When the elasticity is estimated using 5-year changes from 1985 to 1990 rather than the 10-year changes reported in Table 12.1, the elasticity (estimated with the general sales tax as the instrument) is -0.79; for changes from 1990 to 1995, the elasticity is -0.68. These estimates suggest that demand is less elastic over hori- zons of 5 years than over 10 years. This finding of greater price elasticity at longer horizons is consistent with the large body of research on cigarette demand. Demand elasticity estimates in that literature typically fall in the range -0.3 to
-0.5, but these are mainly short-run elasticities; some studies suggest that the long-run elasticity could be perhaps twice the short-run elasticity.2
12.5
Where Do Valid Instruments Come From?
In practice the most difficult aspect of IV estimation is finding instruments that are both relevant and exogenous. There are two main approaches, which reflect two different perspectives on econometric and statistical modeling.
The first approach is to use economic theory to suggest instruments. For exam- ple, Philip Wright’s understanding of the economics of agricultural markets led him to look for an instrument that shifted the supply curve but not the demand
2A sobering economic study by Adda and Cornaglia (2006) suggests that smokers compensate for higher taxes by smoking more intensively, thus extracting more nicotine per cigarette. If you are inter- ested in learning more about the economics of smoking, see Chaloupka and Warner (2000), Gruber (2001), and Carpenter and Cook (2008).
454 ChapteR 12 Instrumental Variables Regression
curve; this in turn led him to consider weather conditions in agricultural regions. One area where this approach has been particularly successful is the field of finan- cial economics. Some economic models of investor behavior involve statements about how investors forecast, which then imply sets of variables that are uncorre- lated with the error term. Those models sometimes are nonlinear in the data and in the parameters, in which case the IV estimators discussed in this chapter cannot be used. An extension of IV methods to nonlinear models, called generalized method of moments estimation, is used instead. Economic theories are, however, abstractions that often do not take into account the nuances and details necessary for analyzing a particular data set. Thus this approach does not always work.
The second approach to constructing instruments is to look for some exoge- nous source of variation in X arising from what is, in effect, a random phenome- non that induces shifts in the endogenous regressor. For example, in our hypothetical example in Section 12.1, earthquake damage increased average class size in some school districts, and this variation in class size was unrelated to poten- tial omitted variables that affect student achievement. This approach typically requires knowledge of the problem being studied and careful attention to the details of the data, and it is best explained through examples.
Three Examples
We now turn to three empirical applications of IV regression that provide exam- ples of how different researchers used their expert knowledge of their empirical problem to find instrumental variables.
Does putting criminals in jail reduce crime? This is a question only an economist would ask. After all, a criminal cannot commit a crime outside jail while in prison, and that some criminals are caught and jailed serves to deter others. But the mag- nitude of the combined effect—the change in the crime rate associated with a 1% increase in the prison population—is an empirical question.
One strategy for estimating this effect is to regress crime rates (crimes per 100,000 members of the general population) against incarceration rates (prisoners per 100,000), using annual data at a suitable level of jurisdiction (for example, U.S. states). This regression could include some control variables measuring economic conditions (crime increases when general economic conditions worsen), demo- graphics (youths commit more crimes than the elderly), and so forth. There is, how- ever, a serious potential for simultaneous causality bias that undermines such an analysis: If the crime rate goes up and the police do their job, there will be more prisoners. On the one hand, increased incarceration reduces the crime rate; on the other hand, an increased crime rate increases incarceration. As in the butter example
12.5 Where Do Valid Instruments Come From? 455
in Figure 12.1, because of this simultaneous causality an OLS regression of the crime rate on the incarceration rate will estimate some complicated combination of these two effects. This problem cannot be solved by finding better control variables.
This simultaneous causality bias, however, can be eliminated by finding a suitable instrumental variable and using TSLS. The instrument must be correlated with the incarceration rate (it must be relevant), but it must also be uncorrelated with the error term in the crime rate equation of interest (it must be exogenous). That is, it must affect the incarceration rate but be unrelated to any of the unob- served factors that determine the crime rate.
Where does one find something that affects incarceration but has no direct effect on the crime rate? One place is exogenous variation in the capacity of exist- ing prisons. Because it takes time to build a prison, short-term capacity restrictions can force states to release prisoners prematurely or otherwise reduce incarceration rates. Using this reasoning, Levitt (1996) suggested that lawsuits aimed at reducing prison overcrowding could serve as an instrumental variable, and he implemented this idea using panel data for the U.S. states from 1972 to 1993.
Are variables measuring overcrowding litigation valid instruments? Although Levitt did not report first-stage F-statistics, the prison overcrowding litigation slowed the growth of prisoner incarcerations in his data, suggesting that this instrument is relevant. To the extent that overcrowding litigation is induced by prison conditions but not by the crime rate or its determinants, this instrument is exogenous. Because Levitt breaks down overcrowding legislation into several types and thus has several instruments, he is able to test the overidentifying restrictions and fails to reject them using the J-statistic, which bolsters the case that his instruments are valid.
Using these instruments and TSLS, Levitt estimated the effect on the crime rate of incarceration to be substantial. This estimated effect was three times larger than the effect estimated using OLS, suggesting that OLS suffered from large simultaneous causality bias.
Doescuttingclasssizesincreasetestscores? Aswesawintheempiricalanalysisof Part II, schools with small classes tend to be wealthier, and their students have access to enhanced learning opportunities both in and out of the classroom. In Part II, we used multiple regression to tackle the threat of omitted variables bias by controlling for various measures of student affluence, ability to speak English, and so forth. Still, a skeptic could wonder whether we did enough: If we left out something important, our estimates of the class size effect would still be biased.
This potential omitted variables bias could be addressed by including the right control variables, but if these data are unavailable (some, like outside learn- ing opportunities, are hard to measure), then an alternative approach is to use
456 ChapteR 12 Instrumental Variables Regression
IV regression. This regression requires an instrumental variable correlated with class size (relevance) but uncorrelated with the omitted determinants of test per- formance that make up the error term, such as parental interest in learning, learn- ing opportunities outside the classroom, quality of the teachers and school facilities, and so forth (exogeneity).
Where does one look for an instrument that induces random, exogenous variation in class size, but is unrelated to the other determinants of test perfor- mance? Hoxby (2000) suggested biology. Because of random fluctuations in tim- ings of births, the size of the incoming kindergarten class varies from one year to the next. Although the actual number of children entering kindergarten might be endogenous (recent news about the school might influence whether parents send a child to a private school), she argued that the potential number of children enter- ing kindergarten—the number of 4-year-olds in the district—is mainly a matter of random fluctuations in the birth dates of children.
Is potential enrollment a valid instrument? Whether it is exogenous depends on whether it is correlated with unobserved determinants of test performance. Surely biological fluctuations in potential enrollment are exogenous, but potential enrollment also fluctuates because parents with young children choose to move into an improving school district and out of one in trouble. If so, an increase in potential enrollment could be correlated with unobserved factors such as the quality of school management, rendering this instrument invalid. Hoxby addressed this problem by reasoning that growth or decline in the potential student pool for this reason would occur smoothly over several years, whereas random fluctuations in birth dates would produce short-term “spikes” in potential enrollment. Thus, she used as her instrument not potential enrollment, but the deviation of potential enrollment from its long-term trend. These deviations satisfy the criterion for instrument relevance (the first-stage F-statistics all exceed 100). She makes a good case that this instrument is exogenous, but, as in all IV analysis, the credibility of this assumption is ultimately a matter of judgment.
Hoxby implemented this strategy using detailed panel data on elementary schools in Connecticut in the 1980s and 1990s. The panel data set permitted her to include school fixed effects, which, in addition to the instrumental variables strategy, attack the problem of omitted variables bias at the school level. Her TSLS estimates suggested that the effect on test scores of class size is small; most of her estimates were statistically insignificantly different from zero.
Does aggressive treatment of heart attacks prolong lives? Aggressive treat- ments for victims of heart attacks (technically, acute myocardial infarctions, or AMI) hold the potential for saving lives. Before a new medical procedure—in this
12.5 Where Do Valid Instruments Come From? 457
example, cardiac catheterization3—is approved for general use, it goes through clinical trials, a series of randomized controlled experiments designed to measure its effects and side effects. But strong performance in a clinical trial is one thing; actual performance in the real world is another.
A natural starting point for estimating the real-world effect of cardiac catheter- ization is to compare patients who received the treatment to those who did not. This leads to regressing the length of survival of the patient against the binary treatment variable (whether the patient received cardiac catheterization) and other control variables that affect mortality (age, weight, other measured health conditions, and so forth). The population coefficient on the indicator variable is the increment to the patient’s life expectancy provided by the treatment. Unfortunately, the OLS estimator is subject to bias: Cardiac catheterization does not “just happen” to a patient randomly; rather, it is performed because the doctor and patient decide that it might be effective. If their decision is based in part on unobserved factors relevant to health outcomes not in the data set, the treatment decision will be correlated with the regression error term. If the healthiest patients are the ones who receive the treatment, the OLS estimator will be biased (treatment is correlated with an omitted variable), and the treatment will appear more effective than it really is.
This potential bias can be eliminated by IV regression using a valid instru- mental variable. The instrument must be correlated with treatment (must be rel- evant) but must be uncorrelated with the omitted health factors that affect survival (must be exogenous).
Where does one look for something that affects treatment but not the health outcome, other than through its effect on treatment? McClellan, McNeil, and Newhouse (1994) suggested geography. Most hospitals in their data set did not specialize in cardiac catheterization, so many patients were closer to “regular” hospitals that did not offer this treatment than to cardiac catheterization hospitals. McClellan, McNeil, and Newhouse therefore used as an instrumental variable the difference between the distance from the AMI patient’s home to the nearest cardiac catheterization hospital and the distance to the nearest hospital of any sort; this distance is zero if the nearest hospital is a cardiac catheterization hospital, and otherwise it is positive. If this relative distance affects the probability of receiving this treatment, then it is relevant. If it is distributed randomly across AMI victims, then it is exogenous.
Is relative distance to the nearest cardiac catheterization hospital a valid instru- ment? McClellan, McNeil, and Newhouse do not report first-stage F-statistics, but they do provide other empirical evidence that it is not weak. Is this distance
3Cardiac catheterization is a procedure in which a catheter, or tube, is inserted into a blood vessel and guided all the way to the heart to obtain information about the heart and coronary arteries.
458
ChapteR 12 Instrumental Variables Regression
measure exogenous? They make two arguments. First, they draw on their medical expertise and knowledge of the health care system to argue that distance to a hos- pital is plausibly uncorrelated with any of the unobservable variables that determine AMI outcomes. Second, they have data on some of the additional variables that affect AMI outcomes, such as the weight of the patient, and in their sample distance is uncorrelated with these observable determinants of survival; this, they argue, makes it more credible that distance is uncorrelated with the unobservable determi- nants in the error term as well.
Using 205,021 observations on Americans aged at least 64 who had an AMI in 1987, McClellan, McNeil, and Newhouse reached a striking conclusion: Their TSLS estimates suggest that cardiac catheterization has a small, possibly zero, effect on health outcomes; that is, cardiac catheterization does not substantially prolong life. In contrast, the OLS estimates suggest a large positive effect. They interpret this difference as evidence of bias in the OLS estimates.
McClellan, McNeil, and Newhouse’s IV method has an interesting interpreta- tion. The OLS analysis used actual treatment as the regressor, but because actual treatment is itself the outcome of a decision by patient and doctor, they argue that the actual treatment is correlated with the error term. Instead, TSLS uses pre- dicted treatment, where the variation in predicted treatment arises because of variation in the instrumental variable: Patients closer to a cardiac catheterization hospital are more likely to receive this treatment.
This interpretation has two implications. First, the IV regression actually esti- mates the effect of the treatment not on a “typical” randomly selected patient, but rather on patients for whom distance is an important consideration in the treat- ment decision. The effect on those patients might differ from the effect on a typi- cal patient, which provides one explanation of the greater estimated effectiveness of the treatment in clinical trials than in McClellan, McNeil, and Newhouse’s IV study. Second, it suggests a general strategy for finding instruments in this type of setting: Find an instrument that affects the probability of treatment, but does so for reasons that are unrelated to the outcome except through their effect on the likelihood of treatment. Both these implications have applicability to experimen- tal and “quasi-experimental” studies, the topic of Chapter 13.
12.6
Conclusion
From the humble start of estimating how much less butter people will buy if its price rises, IV methods have evolved into a general approach for estimating regres- sions when one or more variables are correlated with the error term. Instrumental variables regression uses the instruments to isolate variation in the endogenous
regressors that is uncorrelated with the error in the regression of interest; this is the first stage of two stage least squares. This in turn permits estimation of the effect of interest in the second stage of two stage least squares.
Successful IV regression requires valid instruments, that is, instruments that are both relevant (not weak) and exogenous. If the instruments are weak, then the TSLS estimator can be biased, even in large samples, and statistical inferences based on TSLS t-statistics and confidence intervals can be misleading. Fortu- nately, when there is a single endogenous regressor, it is possible to check for weak instruments simply by checking the first-stage F-statistic.
If the instruments are not exogenous—that is, if one or more instruments is correlated with the error term—the TSLS estimator is inconsistent. If there are more instruments than endogenous regressors, instrument exogeneity can be examined by using the J-statistic to test the overidentifying restrictions. However, the core assumption—that there are at least as many exogenous instruments as there are endogenous regressors—cannot be tested. It is there- fore incumbent on both the empirical analyst and the critical reader to use their own understanding of the empirical application to evaluate whether this assumption is reasonable.
The interpretation of IV regression as a way to exploit known exogenous vari- ation in the endogenous regressor can be used to guide the search for potential instrumental variables in a particular application. This interpretation underlies much of the empirical analysis in the area that goes under the broad heading of program evaluation, in which experiments or quasi-experiments are used to esti- mate the effect of programs, policies, or other interventions on some outcome measure. A variety of additional issues arises in those applications—for example, the interpretation of IV results when, as in the cardiac catheterization example, different “patients” might have different responses to the same “treatment.” These and other aspects of empirical program evaluation are taken up in Chapter 13.
Summary
1. Instrumental variables regression is a way to estimate regression coefficients when one or more regressors are correlated with the error term.
2. Endogenous variables are correlated with the error term in the equation of interest; exogenous variables are uncorrelated with this error term.
3. For an instrument to be valid, it must be (1) correlated with the included endogenous variable and (2) exogenous.
4. IV regression requires at least as many instruments as included endogenous variables.
12.6 Summary 459
460 Chapter 12 Instrumental Variables Regression
5. The TSLS estimator has two stages. First, the included endogenous variables are regressed against the included exogenous variables and the instruments. Second, the dependent variable is regressed against the included exogenous variables and the predicted values of the included endogenous variables from the first-stage regression(s).
6. Weak instruments (instruments that are nearly uncorrelated with the included endogenous variables) make the TSLS estimator biased and TSLS confidence intervals and hypothesis tests unreliable.
7. If an instrument is not exogenous, the TSLS estimator is inconsistent.
Key Terms
instrumental variables (IV) regression (424)
instrumental variable (instrument) (424)
endogenous variable (425)
exogenous variable (425)
instrument relevance condition (426) instrument exogeneity condition (426) two stage least squares (426)
included exogenous variables (435)
exactly identified (435) overidentified (435) underidentified (435)
reduced form (437)
first-stage regression (438) second-stage regression (438) weak instruments (443) first-stage F-statistic (444)
test of overidentifying restrictions
(445)
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Review the Concepts
12.1 In the demand curve regression model of Equation (12.3), is ln(Pbutter) i
positively or negatively correlated with the error, ui? If b1 is estimated by OLS, would you expect the estimated value to be larger or smaller than the true value of b1? Explain.
12.2 In the study of cigarette demand in this chapter, suppose that we used as an instrument the number of trees per capita in the state. Is this instrument relevant? Is it exogenous? Is it a valid instrument?
12.3 In his study of the effect of incarceration on crime rates, suppose that Levitt had used the number of lawyers per capita as an instrument. Would this instrument be relevant? Would it be exogenous? Would it be a valid instrument?
12.4 In their study of the effectiveness of cardiac catheterization, McClellan, McNeil, and Newhouse (1994) used as an instrument the difference in distance to cardiac catheterization and regular hospitals. How could you determine whether this instrument is relevant? How could you determine whether this instrument is exogenous?
Exercises
12.1 This question refers to the panel data regressions summarized in Table 12.1.
a. Suppose that the federal government is considering a new tax on ciga- rettes that is estimated to increase the retail price by $0.50 per pack. If the current price per pack is $7.50, use the regression in column (1) to predict the change in demand. Construct a 95% confidence interval for the change in demand.
b. Suppose that the United States enters a recession, and income falls by 2%. Use the regression in column (1) to predict the change in demand.
c. Suppose that the recession lasts less than 1 year. Do you think that the regression in column (1) will provide a reliable answer to the question in (b)? Why or why not?
d. Suppose that the F-statistic in column (1) were 3.7 instead of 33.7. Would the regression provide a reliable answer to the question posed in (a)? Why or why not?
12.2 Consider the regression model with a single regressor: Yi = b0 + b1Xi + ui. Suppose that the least squares assumptions in Key Concept 4.3 are satisfied.
a. Show that Xi is a valid instrument. That is, show that Key Concept 12.3 is satisfied with Zi = Xi.
12.6 Exercises 461
462 ChapteR 12 Instrumental Variables Regression
b. Show that the IV regression assumptions in Key Concept 12.4 are sat-
isfied with this choice of Zi.
c. Show that the IV estimator constructed using Zi = Xi is identical to
the OLS estimator.
12.3 A classmate is interested in estimating the variance of the error term in Equation (12.1).
g
a. Suppose that she uses the estimator from the second-stage regression
n nTSLS nTSLS n 2 n
(Y – b – b X ) , where X is the fit-
2 1 of TSLS: sn =
an-2i=1i01ii
ted value from the first-stage regression. Is this estimator consistent?
(For the purposes of this question, suppose that the sample is very large and the TSLS estimators are essentially identical to b0 and b1.)
b. Is sn2 = 1 gn (Y – bnTSLS – bnTSLSX)2 consistent? bn-2i=1i01i
12.4 Consider TSLS estimation with a single included endogenous variable and
a single instrument. Then the predicted value from the first-stage regres-
sion is Xn i = pn0 + pn1Zi. Use the definition of the sample variance and cova-
riancetoshowthats = pn s ands2 = pn2s2.Usethisresulttofillinthe XnY 1 ZY Xn 1 Z
steps of the derivation in Appendix 12.2 of Equation (12.4).
12.5 Consider the instrumental variable regression model
Yi =b0 +b1Xi +b2Wi +ui,
where Xi is correlated with ui and Zi is an instrument. Suppose that the first three assumptions in Key Concept 12.4 are satisfied. Which IV assumption is not satisfied when:
a. Zi is independent of (Yi, Xi, Wi)?
b. Zi = Wi?
c. Wi = 1 for all i?
d. Zi = Xi?
12.6 In an instrumental variable regression model with one regressor, Xi, and one instrument, Zi, the regression of Xi onto Zi has R2 = 0.05 and n = 100. Is Zi a strong instrument? [Hint: See Equation (7.14).] Would your answer change if R2 = 0.05 and n = 500?
12.7 In an instrumental variable regression model with one regressor, Xi, and two instruments, Z1i and Z2i, the value of the J-statistic is J = 18.2.
12.6 Exercises 463 a. Does this suggest that E(u 0 Z , Z ) ≠ 0? Explain.
i 1i 2i
b. Does this suggest that E(u 0 Z ) ≠ 0? Explain.
i 1i
12.8 Consider a product market with a supply function Qsi = b0 + b1Pi + usi, a demand function Qdi = g0 + udi , and a market equilibrium condition Qsi = Qdi , where usi and udi are mutually independent i.i.d. random vari- ables, both with a mean of zero.
a. Show that Pi and usi are correlated.
b. Show that the OLS estimator of b1 is inconsistent.
c. How would you estimate b0, b1, and g0?
12.9 A researcher is interested in the effect of military service on human capital. He collects data from a random sample of 4000 workers aged 40 and runs the OLS regression Yi = b0 + b1Xi + ui, where Yi is a worker’s annual earnings and Xi is a binary variable that is equal to 1 if the person served in the military and is equal to 0 otherwise.
a. Explain why the OLS estimates are likely to be unreliable. (Hint: Which variables are omitted from the regression? Are they correlated with military service?)
b. During the Vietnam War there was a draft in which priority for the draft was determined by a national lottery. (The days of the year were randomly reordered 1 through 365. Those with birthdates ordered first were drafted before those with birthdates ordered second, and so forth.) Explain how the lottery might be used as an instrument to estimate the effect of military service on earnings.
(For more about this issue, see Joshua D. Angrist, “Lifetime Earnings and the Vietnam Era Draft Lottery: Evidence from Social Security Administration Records,” American Economic Review, June 1990: 313–336.)
12.10 Consider the instrumental variable regression model Yi = b0 + b1Xi + b2Wi + ui, where Zi is an instrument. Suppose that data on Wi are not available and the model is estimated omitting Wi from the regression.
a. Suppose that Zi and Wi are uncorrelated. Is the IV estimator consistent?
b. Suppose that Zi and Wi are correlated. Is the IV estimator consistent?
464 ChapteR 12 Instrumental Variables Regression Empirical Exercises
(Only three empirical exercises for this chapter are given in the text, but you can find more on the text website, http://www.pearsonhighered.com/stock_ watson/.)
E12.1 Howdoesfertilityaffectlaborsupply?Thatis,howmuchdoesawoman’s labor supply fall when she has an additional child? In this exercise you will estimate this effect using data for married women from the 1980 U.S. Census.4 The data are available on the textbook website, http://www .pearsonhighered.com/stock_watson, in the file Fertility and described in the file Fertility_Description. The data set contains information on married women aged 21–35 with two or more children.
a. Regress weeksworked on the indicator variable morekids, using OLS. On average, do women with more than two children work less than women with two children? How much less?
b. Explain why the OLS regression estimated in (a) is inappropriate for estimating the causal effect of fertility (morekids) on labor supply (weeksworked).
c. The data set contains the variable samesex, which is equal to 1 if the first two children are of the same sex (boy–boy or girl–girl) and equal to 0 otherwise. Are couples whose first two children are of the same sex more likely to have a third child? Is the effect large? Is it statisti- cally significant?
d. Explain why samesex is a valid instrument for the instrumental vari- able regression of weeksworked on morekids.
e. Is samesex a weak instrument?
f. Estimate the regression of weeksworked on morekids, using samesex
as an instrument. How large is the fertility effect on labor supply?
g. Do the results change when you include the variables agem1, black, hispan, and othrace in the labor supply regression (treating these vari- able as exogenous)? Explain why or why not.
4These data were provided by Professor William Evans of the University of Maryland and were used in his paper with Joshua Angrist, “Children and Their Parents’ Labor Supply: Evidence from Exog- enous Variation in Family Size,” American Economic Review, 1998, 88(3): 450–477.
E12.2 Does viewing a violent movie lead to violent behavior? If so, the inci- dence of violent crimes, such as assaults, should rise following the release of a violent movie that attracts many viewers. Alternatively, movie view- ing may substitute for other activities (such as alcohol consumption) that lead to violent behavior, so that assaults should fall when more viewers are attracted to the cinema. On the textbook website, http://www .pearsonhighered.com/stock_watson, you will find the data file Movies, which contains data on the number of assaults and movie attendance for 516 weekends from 1995 through 2004.5 A detailed description is given in Movies_Description, available on the website. The dataset includes week- end U.S. attendance for strongly violent movies (such as Hannibal), mildly violent movies (such as Spider-Man), and nonviolent movies (such as Find- ing Nemo). The dataset also includes a count of the number of assaults for the same weekend in a subset of counties in the United States. Finally, the dataset includes indicators for year, month, whether the weekend is a holiday, and various measures of the weather.
a.
b.
i. Regress the logarithm of the number of assaults [ln_assaults = ln(assaults)] on the year and month indicators. Is there evidence of seasonality in assaults? That is, do there tend to be more assaults in some months than others? Explain.
ii. Regress total movie attendance (attend = attend_v + attend_m + attend_n) on the year and month indicators. Is there evidence of seasonality in movie attendance? Explain.
Regress ln_assaults on attend_v, attend_m, attend_n, the year and month indicators, and the weather and holiday control variables available in the data set.
i. Based on the regression, does viewing a strongly violent movie increase or decrease assaults? By how much? Is the estimated effect statistically significant?
ii. Does attendance at strongly violent movies affect assaults differ- ently than attendance at moderately violent movies? Differently than attendance at nonviolent movies?
iii. A strongly violent blockbuster movie is released, and the week- end’s attendance at strongly violent movies increases by 6 million;
12.6 Empirical Exercises 465
5These are aggregated versions of data provided by Gordon Dahl of University of California–San Diego and Stefano DellaVigna of University of California–Berkeley and were used in their paper “Does Movie Violence Increase Violent Crime?” Quarterly Journal of Economics, 2009, 124(2): 677–734.
466 ChapteR 12 Instrumental Variables Regression
meanwhile, attendance falls by 2 million for moderately violent movies and by 1 million for nonviolent movies. What is the pre- dicted effect on assaults? Construct a 95% confidence interval for the change in assaults. [Hint: Review Section 7.3 and material sur- rounding Equations (8.7) and (8.8).)]
c. It is difficult to control for all the variables that affect assaults and that might be correlated with movie attendance. For example, the effect of the weather on assaults and movie attendance is only
crudely approximated by the weather variables in the data set. How- ever, the data set does include a set of instruments, pr_attend_v, pr_attend_m, and pr_attend_n, that are correlated with attendance but are (arguably) uncorrelated with weekend-specific factors (such as the weather) that affect both assaults and movie attendance. These instruments use historical attendance patterns, not information on a particular weekend, to predict a film’s attendance in a given weekend. For example, if a film’s attendance is high in the second week of its release, then this can be used to predict that its attendance was also high in the first week of its release. (The details of the construction
of these instruments are available in the Dahl and DellaVigna paper referenced in footnote 5.) Run the regression from part (b) (includ- ing year, month, holiday, and weather controls) but now using pr_ attend_v, pr_attend_m, and pr_attend_n as instruments for attend_v, attend_m, and attend_n. Use this regression to answer (b)(i)–(b)(iii).
d. The intuition underlying the instruments in (c) is that attendance in a given week is correlated with attendance in surrounding weeks. For each move category, the data set includes attendance in surround- ing weeks. Run the regression using the instruments attend_v_f, attend_m_f, attend_n_f, attend_v_b, attend_m_b, and attend_n_b instead of the instruments used in part (c). Use this regression to answer (b)(i)–(b)(iii).
e. There are nine instruments listed in (c) and (d), but only three are needed for identification. Carry out the test for overidentification summarized in Key Concept 12.6. What do you conclude about the validity of the instruments?
f. Based on your analysis, what do you conclude about the effect of vio- lent movies on (short-run) violent behavior?
E12.3 (This requires Appendix 12.5) On the textbook website, http://www .pearsonhighered.com/stock_watson, you will find the data set WeakInstrument,
Derivation of the Formula for the TSLS Estimator in Equation (12.4) 467 which contains 200 observations on (Yi, Xi, Zi) for the instrumental regres-
sion Yi = b0 + b1Xi + ui.
a. Construct bnTSLS, its standard error, and the usual 95% confidence
1 interval for b1.
b. Compute the F-statistic for the regression of Xi on Zi. Is there evidence of a “weak instrument” problem?
c. Compute a 95% confidence interval for b1, using the Anderson–Rubin procedure. (To implement the procedure, assume that – 5 … b1 … 5.)
d. Comment on the differences in the confidence intervals in (a) and (c). Which is more reliable?
12.1
appenDIx
The Cigarette Consumption Panel Data Set
The data set consists of annual data for the 48 contiguous U.S. states from 1985 to 1995. Quantity consumed is measured by annual per capita cigarette sales in packs per fiscal year, as derived from state tax collection data. The price is the real (that is, inflation-adjusted) average retail cigarette price per pack during the fiscal year, including taxes. Income is real per capita income. The general sales tax is the average tax, in cents per pack, due to the broad-based state sales tax applied to all consumption goods. The cigarette-specific tax is the tax applied to cigarettes only. All prices, income, and taxes used in the regressions in this chapter are deflated by the Consumer Price Index and thus are in constant (real) dollars. We are grateful to Professor Jonathan Gruber of MIT for providing us with these data.
appenDIx
12.2
Derivation of the Formula
for the TSLS Estimator in Equation (12.4)
The first stage of TSLS is to regress Xi on the instrument Zi by OLS and then compute the OLS predicted value Xn i; the second stage is to regress Yi on Xn i by OLS. Accordingly, the formula for the TSLS estimator, expressed in terms of the predicted value Xni, is the formula for the OLS estimator in Key Concept 4.2, with Xni replacing Xi. That is,
468 Chapter 12 Instrumental Variables Regression
Appendix
12.3
Large-Sample Distribution of the TSLS Estimator
This appendix studies the large-sample distribution of the TSLS estimator in the case con- sidered in Section 12.1—that is, with a single instrument, a single included endogenous variable, and no included exogenous variables.
To start, we derive a formula for the TSLS estimator in terms of the errors; this formula forms the basis for the remaining discussion, similar to the expression for the OLS estimator inEquation(4.30)inAppendix4.3.FromEquation(12.1),Yi – Y = b1(Xi – X) + (ui – u). Accordingly, the sample covariance between Z and Y can be expressed as
s
nTSLS 2 2 b = s >s , where s
n
nn
Because Xi is the predicted value of Xi from the first-stage regression, Xi = pn0 + pn1Zi,
1 XnYXn Xn between Yi and Xn i.
is the sample variance of X and s is the sample covariance i XnY
nTSLS 2 2 (Exercise 12.4). Thus, the TSLS estimator can be written as b = s >s = s >(pn s ).
the definitions of sample variances and covariances imply that s = pn s and s2 = pn 2s2 XnY 1ZY Xn 1Z
1 XnYXnZY1Z Finally, pn is the OLS slope coefficient from the first stage of TSLS, so pn = s >s . Sub-
nTSLS 2
stitution of this formula for pn into the formula b = s >(pn s ) yields the formula for
1 1 ZX2Z
1 1ZY1Z the TSLS estimator in Equation (12.4).
= n – 1 (Z – Z)(Y – Y) ZY1ani i
1
= i=1(Z -Z)3b(X -X)+(u -u)4
ani 1i i n – 1 i=1
=bs +n-1i=1(Z -Z)(u -u) 1ZX1ani i
=bs +n-1 (Z -Z)u, 1ZX 1ani i
(12.19) where s = [1>(n – 1)]g (Z – Z)(X – X) and where the final equality follows
i=1 ZX i=1ii
n
because g (Z – Z) = 0. Substituting the definition of s and the final expression in
by (n – 1)>n yields
n
i=1 i ZX
nTSLS Equation (12.19) into the definition of b1
and multiplying the numerator and denominator
ni=1(Z -Z)u 1ani i
1 11ani i ni=1(Z -Z)(X -X)
nTSLS
b =b+
. (12.20)
Large-Sample Distribution of the TSLS Estimator When the Instrument Is Not Valid 469 Large-Sample Distribution of bnTSLS When the IV
1
Regression Assumptions in Key Concept 12.4 Hold
Equation (12.20) for the TSLS estimator is similar to Equation (4.30) in Appendix 4.3 for the OLS estimator, with the exceptions that Z rather than X appears in the numerator and the denominator is the covariance between Z and X rather than the variance of X. Because of these similarities, and because Z is exogenous, the argument in Appendix 4.3 that the OLS estimator is normally distributed in large samples extends to bnTSLS.
1
Specifically, when the sample is large, Z ≅ mZ, so the numerator is approximately
1n
q = (n)g q , where q = (Z – m )u . Because the instrument is exogenous, E(q ) = 0.
i=1iiiZi i
var3(Z – m )u 4. It follows that var (q) = s = s >n, and, by the central limit theorem, iZi qq
By the IV regression assumptions in Key Concept 12.4, qi is i.i.d. with variance s2q =
q>s is, in large samples, distributed N(0, 1). 2 2 q
Because the sample covariance is consistent for the population covariance, sZX ¡p cov(Zi, Xi), which, because the instrument is relevant, is nonzero. Thus, by Equation
nTSLS nTSLS
(12.20) b ≅ b + q>cov(Z , X ), so in large samples b is approximately distributed
22222 N(b,s )wheres = s >3cov(Z,X)4 = (1>n)var3(Z – m )u4>3cov(Z,X)4 ,
1 bn1TSLS
11ii1
bn1TSLS q i i i Z i i i which is the expression given in Equation (12.8).
Appendix
12.4
Large-Sample Distribution of the TSLS Estimator When the Instrument Is Not Valid
This appendix considers the large-sample distribution of the TSLS estimator in the setup of Section 12.1 (one X, one Z) when one or the other of the conditions for instrument validity fails. If the instrument relevance condition fails, the large-sample distribution of the TSLS estimator is not normal; in fact, its distribution is that of a ratio of two normal random vari- ables. If the instrument exogeneity condition fails, the TSLS estimator is inconsistent.
Large-Sample Distribution of bnTSLS When
the Instrument Is Weak
Firstconsiderthecasethattheinstrumentisirrelevantsothatcov(Zi,Xi) = 0.Thentheargu- ment in Appendix 12.3 entails division by zero. To avoid this problem, we need to take a closer look at the behavior of the term in the denominator of Equation (12.20) when the population covariance is zero.
1
470 ChapteR 12 Instrumental Variables Regression
We start by rewriting Equation (12.20). Because of the consistency of the sample average,
of Equation (12.20) is approximately (n)g (Z – m )(X – m ) = (n)g r = r, where i=1iZiX i=1i
in large samples, Z is close to mZ, and X is close to mX. Thus the term in the denominator 1n 1n
r=(Z-m)(X-m).Lets =var3(Z-m)(X-m)4,lets =s>n,andlet i i Z i X 2r i Z i X 2r 2r
q, s2q, and sq2 be as defined in Appendix 12.3. Then Equation (12.20) implies that, in large samples,
qqq qq
b ≅ b + = b + as baq>s b = b + as baq>s b. (12.21)
r r>s sr>s
nTSLS
1 1 1 sr r 1 r r
If the instrument is irrelevant, then E(ri) = cov(Zi, Xi) = 0. Thus r is the sample average
assumption), have variance s = var3(Z – m )(X – m )4 (which is finite by the third IV 2r iZiX
of the random variables ri, i = 1, c, n, which are i.i.d. (by the second least squares
regression assumption), and have a mean of zero (because the instruments are irrelevant). It follows that the central limit theorem applies to r, specifically, r>s is approximately
r
distributed N(0, 1). Therefore, the final expression of Equation (12.21) implies that, in
nTSLS
large samples, the distribution of b – b is the distribution of aS, where a = s >s and
11qr
S is the ratio of two random variables, each of which has a standard normal distribution
(these two standard normal random variables are correlated).
In other words, when the instrument is irrelevant, the central limit theorem applies to
the denominator as well as the numerator of the TSLS estimator, so in large samples the distribution of the TSLS estimator is the distribution of the ratio of two normal random variables. Because Xi and ui are correlated, these normal random variables are correlated, and the large-sample distribution of the TSLS estimator when the instrument is irrelevant is complicated. In fact, the large-sample distribution of the TSLS estimator with irrelevant instruments is centered on the probability limit of the OLS estimator. Thus, when the instrument is irrelevant, TSLS does not eliminate the bias in OLS and, moreover, has a nonnormal distribution, even in large samples.
A weak instrument represents an intermediate case between an irrelevant instrument and the normal distribution derived in Appendix 12.3. When the instrument is weak but not irrelevant, the distribution of the TSLS estimator continues to be nonnormal, so the general lesson here about the extreme case of an irrelevant instrument carries over to weak instruments.
Large-Sample Distribution of bnTSLS When
1 the Instrument Is Endogenous
The numerator in the final expression in Equation (12.20) converges in probability to cov(Zi, ui). If the instrument is exogenous, this is zero, and the TSLS estimator is consistent
Instrumental Variables Analysis with Weak Instruments 471 (assuming that the instrument is not weak). If, however, the instrument is not exogenous,
p
then, if the instrument is not weak, b ¡ b + cov(Z , u )>cov(Z , X ) ≠ b . That is,
if the instrument is not exogenous, the TSLS estimator is inconsistent.
12.5
nTSLS 11iiii1
appenDIx
Instrumental Variables Analysis with Weak Instruments
This appendix discusses some methods for instrumental variables analysis in the presence of potentially weak instruments. The appendix focuses on the case of a single included endogenous regressor [Equations (12.13) and (12.14)].
Testing for Weak Instruments
The rule of thumb in Key Concept 12.5 is that a first-stage F-statistic less than 10 indicates
that the instruments are weak. One motivation for this rule of thumb arises from an
approximate expression for the bias of the TSLS estimator. Let bOLS denote the probabil- 1
ity limit of the OLS estimator b , and let bOLS – b denote the asymptotic bias of the OLS 111
n p OLS
estimator (if the regressor is endogenous, then b1 ¡ b1 ≠ b1). It is possible to show
nTSLS OLS
E(b ) – b ≈ (b – b )>3E(F) – 14, where E(F) is the expectation of the first-
that, when there are many instruments, the bias of the TSLS is approximately
1111
stage F-statistic. If E(F) = 10, then the bias of TSLS, relative to the bias of OLS, is approx- imately 1/9, or just over 10%, which is small enough to be acceptable in many applications. Replacing E(F) 7 10 with F 7 10 yields the rule of thumb in Key Concept 12.5.
The motivation in the previous paragraph involved an approximate formula for the bias of the TSLS estimator when there are many instruments. In most applications, however, the number of instruments, m, is small. Stock and Yogo (2005) provide a formal test for weak instruments that avoids the approximation that m is large. In the Stock– Yogo test, the null hypothesis is that the instruments are weak, and the alternative hypothesis is that the instruments are strong, where strong instruments are defined to be instruments for which the bias of the TSLS estimator is at most 10% of the bias of the OLS estimator. The test entails comparing the first-stage F-statistic (for technical reasons, the homoskedasticity-only version) to a critical value that depends on the number of instruments. As it happens, for a test with a 5% significance level, this critical value ranges between 9.08 and 11.52, so the rule of thumb of comparing F to 10 is a good approximation to the Stock–Yogo test.
472 ChapteR 12 Instrumental Variables Regression
Hypothesis Tests and Confidence Sets for b
If the instruments are weak, the TSLS estimator is biased and has a nonnormal distribu- tion. Thus the TSLS t-test of b1 = b1,0 is unreliable, as is the TSLS confidence interval for b1. There are, however, other tests of b1 = b1,0, along with confidence intervals based on those tests, that are valid whether instruments are strong, weak, or even irrelevant. When there is a single endogenous regressor, the preferred test is Moreira’s (2003) conditional likelihood ratio (CLR) test. An older test, which works for any number of endogenous regressors, is based on the Anderson–Rubin (1949) statistic. Because the Anderson–Rubin (1949) statistic is conceptually less complicated, we describe it first.
The Anderson–Rubin test of b1 = b1,0 proceeds in two steps. In the first step, compute a new variable, Y*i = Yi – b1,0 Xi. In the second step, regress Y*i against the included exog- enous regressors (W’s) and the instruments (Z’s). The Anderson–Rubin statistic is the F-statistic testing the hypothesis that the coefficient on the Z’s are all zero. Under the null hypothesis that b1 = b1,0, if the instruments satisfy the exogeneity condition (condition 2 in Key Concept 12.3), they will be uncorrelated with the error term in this regression, and the null hypothesis will be rejected in 5% of all samples.
As discussed in Sections (3.3) and (7.4), a confidence set can be constructed as the set of values of the parameters that are not rejected by a hypothesis test. Accordingly, the set of values of b1 that are not rejected by a 5% Anderson–Rubin test constitutes a 95% confidence set for b1. When the Anderson–Rubin F-statistic is computed using the homoskedasticity- only formula, the Anderson–Rubin confidence set can be constructed by solving a quadratic equation (see Empirical Exercise 12.3). The logic behind the Anderson–Rubin statistic never assumes instrument relevance, and the Anderson–Rubin confidence set will have a coverage probability of 95% in large samples, whether the instruments are strong, weak, or even irrelevant.
The CLR statistic also tests the hypothesis that b1 = b1,0. Likelihood ratio statistics compare the value of the likelihood (see Appendix 11.2) under the null hypothesis to its value under the alternative and reject it if the likelihood under the alternative is sufficiently greater than under the null. Familiar tests in this book, such as the homoskedasticity-only F-test in multiple regression, can be derived as likelihood ratio tests under the assumption of homoskedastic normally distributed errors. Unlike any of the other tests discussed in this book, however, the critical value of the CLR test depends on the data, specifically on a statistic that measures the strength of the instruments. By using the right critical value, the CLR test is valid whether instruments are strong, weak, or irrelevant. CLR confidence intervals can be computed as the set of b1 that are not rejected by the CLR test.
The CLR test is equivalent to the TSLS t-test when instruments are strong and has very good power when instruments are weak. With suitable software, the CLR test is easy to use. The disadvantage of the CLR test is that it does not generalize readily to more than one endogenous regressor. In that case, the Anderson–Rubin test (and confidence set) is
recommended; however, when instruments are strong (so TSLS is valid) and the coeffi- cients are overidentified, the Anderson–Rubin test is inefficient in the sense that it is less powerful than the TSLS t-test.
Estimation of b
If the instruments are irrelevant, it is not possible to obtain an unbiased estimator of b1, even in large samples. Nevertheless, when instruments are weak, some IV estimators tend to be more centered on the true value of b1 than is TSLS. One such estimator is the limited infor- mation maximum likelihood (LIML) estimator. As its name implies, the LIML estimator is the maximum likelihood estimator of b1 in the system of Equations (12.13) and (12.14). (For a discussion of maximum likelihood estimation, see Appendix 11.2.) The LIML estimator also is the value of b1,0 that minimizes the homoskedasticity-only Anderson–Rubin test statistic. Thus, if the Anderson–Rubin confidence set is not empty, it will contain the LIML estimator. In addition, the CLR confidence interval contains the LIML estimator.
If the instruments are weak, the LIML estimator is more nearly centered on the true value of b1 than is TSLS. If instruments are strong, the LIML and TSLS estimators coincide in large samples. A drawback of the LIML estimator is that it can produce extreme outliers. Confidence intervals constructed around the LIML estimator using the LIML standard error are more reliable than intervals constructed around the TSLS estimator using the TSLS standard error, but are less reliable than Anderson–Rubin or CLR intervals when the instruments are weak.
The problems of estimation, testing, and confidence intervals in IV regression with weak instruments constitute an area of ongoing research. To learn more about this topic, visit the website for this book.
12.6
appenDIx
TSLS with Control Variables 473
TSLS with Control Variables
In Key Concept 12.4, the W variables are assumed to be exogenous. This appendix considers the case in which W is not exogenous, but instead is a control variable included to make Z exogenous. The logic of control variables in TSLS parallels the logic in OLS: If a control variable effectively controls for an omitted factor, then the instrument is uncorrelated with the error term. Because the control variable is correlated with the error term, the coefficient on a control variable does not have a causal interpretation. The mathematics of control variables in TSLS also parallels the mathematics of control variables in OLS and entails relaxing the assumption that the error has conditional mean zero, given Z and W, to be that the conditional mean of the error does not depend on Z. This appendix draws on Appendix 7.2 (Conditional Mean Independence), which should be reviewed first.
474 ChapteR 12 Instrumental Variables Regression
Consider the IV regression model in Equation (12.12) with a single X and a single W:
Yi =b0 +b1Xi +b2Wi +ui. (12.22) WereplaceIVRegressionAssumption#1inKeyConcept12.4[whichstatesthatE(uiWi) = 04
Following Appendix 7.2, we further assume that E(uiWi) is linear in Wi, so E(ui Wi) = g0 + g2Wi, where g0 and g2 are coefficients. Letting ei = ui – E(ui Wi, Zi) and applying the algebra of Equation (7.25) to Equation (12.22), we obtain
with the assumption that, conditional on Wi, the mean of ui does not depend on Zi:
E(ui0Wi,Zi) = E(ui0Wi). (12.23)
Yi =d0 +b1Xi +d2Wi +ei, (12.24) whered = b + g andd = b + g.NowE(e0W,Z) = E3u – E(u0W,Z)0W,Z4=
000222iiiiiiiii E(u 0W,Z) – E(u 0W,Z) = 0,whichinturnimpliescorr(Z,e) = 0.ThusIVRegression
iii iii ii
Assumption #1 and the instrument exogeneity requirement (condition #2 in Key Concept 12.3) both hold for Equation (12.24) with error term ei, Thus, if IV Regression Assumption #1 is replaced by conditional mean independence in Equation (12.23), the original IV regression assumptions in Key Concept 12.4 apply to the modified regression in Equation (12.24).
Because the IV regression assumptions of Key Concept 12.4 hold for Equation (12.24), all the methods of inference (both for weak and strong instruments) discussed in this chap- ter apply to Equation (12.24). In particular, if the instruments are strong, the coefficients in Equation (12.24) will be estimated consistently by TSLS and TSLS tests, and confidence intervals will be valid.
Just as in OLS with control variables, in general the TSLS coefficient on the control variable W does not have a causal interpretation. TSLS consistently estimates d2 in Equa- tion (12.24), but d2 is the sum of b2, the direct causal effect of W, and g2, which reflects the correlation between W and the omitted factors in ui for which W controls.
In the cigarette consumption regressions in Table 12.1, it is tempting to interpret the coefficient on the 10-year change in log income as the income elasticity of demand. If, how- ever, income growth is correlated with increases in education and if more education reduces smoking, income growth would have its own causal effect (b2, the income elasticity) plus an effectarisingfromitscorrelationwitheducation(g2).Ifthelattereffectisnegative(g2 6 0), the income coefficients in Table 12.1 (which estimate d2 = b2 + g2) would underestimate the income elasticity, but if the conditional mean independence assumption in Equation (12.23) holds, the TSLS estimator of the price elasticity is consistent.
CHAPTER
13
Experiments and Quasi-Experiments
In many fields, such as psychology and medicine, causal effects are commonly estimated using experiments. Before being approved for widespread medical use, for example, a new drug must be subjected to experimental trials in which some patients are randomly selected to receive the drug while others are given a harmless ineffective substitute (a “placebo”); the drug is approved only if this randomized controlled experiment provides convincing statistical evidence that the drug is safe and effective.
There are three reasons to study randomized controlled experiments in an econometrics course. First, an ideal randomized controlled experiment provides a conceptual benchmark to judge estimates of causal effects made with observational data. Second, the results of randomized controlled experiments, when conducted, can be very influential, so it is important to understand the limitations and threats to validity of actual experiments as well as their strengths. Third, external circumstances sometimes produce what appears to be randomization; that is, because of external events, the treatment of some individual occurs “as if” it is random, possibly conditional on some control variables. This “as if” randomness produces a “quasi-experiment” or “natural experiment,” and many of the methods developed for analyzing randomized experiments can be applied (with some modifications) to quasi-experiments.
This chapter examines experiments and quasi-experiments in economics. The statistical tools used in this chapter are multiple regression analysis, regression analysis of panel data, and instrumental variables (IV) regression. What distinguishes the discussion in this chapter is not the tools used, but rather the type of data analyzed and the special opportunities and challenges posed when analyzing experiments and quasi-experiments.
The methods developed in this chapter are often used for evaluating social or economic programs. Program evaluation is the field of study that concerns estimating the effect of a program, policy, or some other intervention or “treatment.” What is the effect on earnings of going through a job training program? What is the effect on employment of low-skilled workers of an increase in the minimum wage? What is the effect on college attendance of making low-cost student aid loans available
475
476
CHAPTER 13
Experiments and Quasi-Experiments
13.1
Potential Outcomes, Causal Effects, and Idealized Experiments
This section explains how the population mean of individual-level causal effects can be estimated using a randomized controlled experiment and how data from such an experiment can be analyzed using multiple regression analysis.
Potential Outcomes and the Average Causal Effect
Suppose that you are considering taking a drug for a medical condition, enrolling in a job training program, or doing an optional econometrics problem set. It is reasonable to ask, what are the benefits of doing so—receiving the treatment—for me? You can imagine two hypothetical situations, one in which you receive the treatment and one in which you do not. Under each hypothetical situation, there would be a measurable outcome (the progress of the medical condition, getting a job, your econometrics grade). The difference in these two potential outcomes would be the causal effect, for you, of the treatment.
More generally, a potential outcome is the outcome for an individual under a potential treatment. The causal effect for that individual is the difference in the potential outcome if the treatment is received and the potential outcome if it is not. In general, the causal effect can differ from one individual to the next. For
to middle-class students? This chapter discusses how such programs or policies can be evaluated using experiments or quasi-experiments.
We begin in Section 13.1 by elaborating on the discussions in Chapters 1, 3, and 4 of the estimation of causal effects using randomized controlled experiments. In reality, actual experiments with human subjects encounter practical problems that constitute threats to their internal and external validity; these threats and some econometric tools for addressing them are discussed in Section 13.2. Section 13.3 analyzes an important randomized controlled experiment in which elementary stu- dents were randomly assigned to different-sized classes in the state of Tennessee in the late 1980s.
Section 13.4 turns to the estimation of causal effects using quasi-experiments. Threats to the validity of quasi-experiments are discussed in Section 13.5. One issue that arises in both experiments and quasi-experiments is that treatment effects can differ from one member of the population to the next, and the matter of interpret- ing the resulting estimates of causal effects when the population is heterogeneous is taken up in Section 13.6.
13.1 Potential Outcomes, Causal Effects, and Idealized Experiments 477
example, the effect of a drug could depend on your age, whether you smoke, or other health conditions. The problem is that there is no way to measure the causal effect for a single individual. Because the individual either receives the treatment or does not, one of the potential outcomes can be observed, but not both.
Although the causal effect cannot be measured for a single individual, in many applications it suffices to know the mean causal effect in a population. For example, a job training program evaluation might trade off the average expendi- ture per trainee against average trainee success in finding a job. The mean of the individual causal effects in the population under study is called the average causal effect or the average treatment effect.
The average causal effect for a given population can be estimated, at least in theory, using an ideal randomized controlled experiment. To see how, first sup- pose that the subjects are selected at random from the population of interest. Because the subjects are selected by simple random sampling, their potential outcomes, and thus their causal effects, are drawn from the same distribution, so the expected value of the causal effect in the sample is the average causal effect in the population. Next suppose that subjects are randomly assigned to the treat- ment or the control group. Because an individual’s treatment status is randomly assigned, it is distributed independently of his or her potential outcomes. Thus the expected value of the outcome for those treated minus the expected value of the outcome for those not treated equals the expected value of the causal effect. Thus, when the concept of potential outcomes is combined with (1) random selection of individuals from a population and (2) random experimental assign- ment of treatment to those individuals, the expected value of the difference in outcomes between the treatment and control groups is the average causal effect in the population. That is, as was stated in Section 3.5, the average causal effect on Yi of treatment (Xi = 1) versus no treatment (Xi = 0) is the difference in the conditional expectations, E(Yi Xi = 1) – E(Yi Xi = 0), where E(Yi Xi = 1) and E(Yi Xi = 0) are respectively the expected values of Y for the treatment and control groups in an ideal randomized controlled experiment. Appendix 13.3 provides a mathematical treatment of the foregoing reasoning.
In general, an individual causal effect can be thought of as depending both on observable variables and on unobservable variables. We have already encountered the idea that a causal effect can depend on observable variables; for example, Chapter 8 examined the possibility that the effect of a class size reduction might depend on whether a student is an English learner. For most of this chapter, we focus on the case that variation in causal effects depends only on observable variables. Section 13.6 takes up unobserved heterogeneity in causal effects.
478 CHAPTER 13
Experiments and Quasi-Experiments
Econometric Methods for Analyzing Experimental Data
Data from a randomized controlled experiment can be analyzed by comparing dif- ferences in means or by a regression that includes the treatment indicator and additional control variables. This latter specification, the differences estimator with additional regressors, can also be used in more complicated randomization schemes, in which the randomization probabilities depend on observable covariates.
Thedifferencesestimator. Thedifferencesestimatoristhedifferenceinthesam- ple averages for the treatment and control groups (Section 3.5), which can be com- puted by regressing the outcome variable Y on a binary treatment indicator X:
Yi = b0 + b1Xi + ui,i = 1,c,n. (13.1)
As discussed in Section 4.4, if X is randomly assigned, then E(ui Xi) = 0 and the OLS estimator of b1 in Equation (13.1) is an unbiased and consistent estimator of the causal effect.
Thedifferencesestimatorwithadditionalregressors. Theefficiencyofthediffer- ence estimator often can be improved by including some control variables W in the regression; doing so leads to the differences estimator with additional regressors:
Yi = b0 + b1Xi + b2W1i + g+ b1+rWri + ui,i = 1,c,n. (13.2)
If W helps to explain the variation in Y, then including W reduces the standard error of the regression and, typically, the standard error of bn1. As discussed in Section 7.5 and Appendix 7.2, for the estimator bn1 of the causal effect b1 in Equa- tion (13.2) to be unbiased, the control variables W must be such that ui satisfies conditional mean independence, that is, E(ui Xi,Wi) = E(ui Wi). This condition is satisfied if Wi are pretreatment individual characteristics, such as gender: If Wi is a pretreatment characteristic and Xi is randomly assigned, then Xi is indepen- dent of ui and Wi which implies that E(ui Xi,Wi) = E(ui Wi). The W regressors in Equation (13.2) should not include experimental outcomes (Xi is not randomly assigned, given an experimental outcome). As always with control variables under conditional mean independence, the coefficient on the control variable does not have a causal interpretation.
Estimatingcausaleffectsthatdependonobservables. AsdiscussedinChapter8, variation in causal effects that depends on observables can be estimated by including suitable nonlinear functions of, or interactions with, Xi. For example, if W1i is a
binary indicator denoting gender, then distinct causal effects for men and women canbeestimatedbyincludingtheinteractionvariableW1i *Xiintheregressionin Equation (13.2).
Randomization based on covariates. Randomization in which the probability of assignment to the treatment group depends on one or more observable variables W is called randomization based on covariates. If randomization is based on covariates, then in general the differences estimator based on Equation (13.1) suffers from omitted variable bias. For example, Appendix 7.2 describes a hypo- thetical experiment to estimate the causal effect of mandatory versus optional homework in an econometrics course. In that experiment, economics majors (Wi = 1) were assigned to the treatment group (mandatory homework, Xi = 1 with higher probability than nonmajors (Wi = 0). But if majors tend to do better in the course than nonmajors anyway, then there is omitted variable bias because being in the treatment group is correlated with the omitted variable, being a major.
Because Xi is randomly assigned given Wi, this omitted variable bias can be eliminated by using the differences estimator with the additional control variable Wi. The random assignment of Xi given Wi (combined with the assumption of a linear regression function) implies that, given Wi, Xi is independent of ui in Equation (13.2). This conditional independence in turn implies conditional mean independence, that is, E(ui Xi,Wi) = E(ui Wi) Thus the OLS estimator bn1 in Equation (13.2) is an unbiased estimator of the causal effect when Xi is assigned randomly based on Wi .
13.2
13.2 Threats to Validity of Experiments 479
Threats to Validity of Experiments
Recall from Key Concept 9.1 that a statistical study is internally valid if the statis- tical inferences about causal effects are valid for the population being studied; it is externally valid if its inferences and conclusions can be generalized from the population and setting studied to other populations and settings. Various real- world problems pose threats to the internal and external validity of the statistical analysis of actual experiments with human subjects.
Threats to Internal Validity
Threats to the internal validity of randomized controlled experiments include failure to randomize, failure to follow the treatment protocol, attrition, experimen- tal effects, and small sample sizes.
480 CHAPTER 13
Experiments and Quasi-Experiments
Failure to randomize. If the treatment is not assigned randomly, but instead is based in part on the characteristics or preferences of the subject, then experimen- tal outcomes will reflect both the effect of the treatment and the effect of the nonrandom assignment. For example, suppose that participants in a job training program experiment are assigned to the treatment group depending on whether their last name falls in the first or second half of the alphabet. Because of ethnic differences in last names, ethnicity could differ systematically between the treat- ment and control groups. To the extent that work experience, education, and other labor market characteristics differ by ethnicity, there could be systematic differences between the treatment and control groups in these omitted factors that affect outcomes. In general, nonrandom assignment can lead to correlation between Xi and ui in Equations (13.1) and (13.2), which in turn leads to bias in the estimator of the treatment effect.
It is possible to test for randomization. If treatment is randomly received, then Xi will be uncorrelated with observable pretreatment individual characteris- tics W. Thus, a test for random receipt of treatment entails testing the hypothesis that the coefficients on W1i, c, Wri are zero in a regression of Xi on W1i, c, Wri. In the job training program example, regressing receipt of job training (Xi) on gender, race, and prior education (W’s), and then computing the F-statistic testing whether the coefficients on the W’s are zero, provides a test of the null hypothesis that treatment was randomly received, against the alternative hypothesis that receipt of treatment depends on gender, race, or prior education. If the experi- mental design performs randomization conditional on covariates, then those covariates would be included in the regression and the F-test would test the coef- ficients on the remaining W’s.1
Failure to follow the treatment protocol. In an actual experiment, people do not always do what they are told. In a job training program experiment, for example, some of the subjects assigned to the treatment group might not show up for the training sessions and thus not receive the treatment. Similarly, subjects assigned to the control group might somehow receive the training anyway, perhaps by making a special request to an instructor or administrator.
The failure of individuals to follow completely the randomized treatment pro- tocol is called partial compliance with the treatment protocol. In some cases, the
1In this example, Xi is binary, so, as discussed in Chapter 11, the regression of Xi on W1i, c, Wri is a linear probability model and heteroskedasticity-robust standard errors are essential. Another way to test the hypothesis that E(Xi W1i, c, Wri) does not depend on W1i, c, Wr i when Xi is binary is to use a probit or logit model (see Section 11.2).
13.2 Threats to Validity of Experiments 481
experimenter knows whether the treatment was actually received (for example, whether the trainee attended class), and the treatment actually received is recorded as Xi. With partial compliance, there is an element of choice in whether the subject receives the treatment, so Xi will be correlated with ui even if initially there is random assignment. Thus, failure to follow the treatment protocol leads to bias in the OLS estimator.
If there are data on both treatment actually received (Xi) and on the initial random assignment, then the treatment effect can be estimated by instrumental variables regression. Instrumental variables estimation of the treatment effect entails the estimation of Equation (13.1)—or Equation (13.2) if there are control variables—using the initial random assignment (Zi) as an instrument for the treat- ment actually received (Xi). Recall that a variable must satisfy the two conditions of instrument relevance and instrument exogeneity (Key Concept 12.3) to be a valid instrumental variable. As long as the protocol is partially followed, then the actual treatment level is partially determined by the assigned treatment level, so the instrumental variable Zi is relevant. If initial assignment is random, then Zi is distributed independently of ui (conditional on Wi, if randomization is conditional on covariates), so the instrument is exogenous. Thus, in an experiment with ran- domly assigned treatment, partial compliance, and data on actual treatment, the original random assignment is a valid instrumental variable.
This instrumental variables strategy requires having data on both assigned and received treatment. In some cases, data might not be available on the treat- ment actually received. For example, if a subject in a medical experiment is pro- vided with the drug but, unbeknownst to the researchers, simply does not take it, then the recorded treatment (“received drug”) is incorrect. Incorrect measure- ment of the treatment actually received leads to bias in the differences estimator.
Attrition. Attrition refers to subjects dropping out of the study after being ran- domly assigned to the treatment or control group. Sometimes attrition occurs for reasons unrelated to the treatment program; for example, a participant in a job training study might need to leave town to care for a sick relative. But if the rea- son for attrition is related to the treatment itself, then the attrition results in bias in the OLS estimator of the causal effect. For example, suppose that the most able trainees drop out of the job training program experiment because they get out-of- town jobs acquired using the job training skills, so at the end of the experiment only the least able members of the treatment group remain. Then the distribution of unmeasured characteristics (ability) will differ between the control and treat- ment groups (the treatment enabled the ablest trainees to leave town). In other words, the treatment Xi will be correlated with ui (which includes ability) for those
482 CHAPTER 13 Experiments and Quasi-Experiments
The Hawthorne Effect
During the 1920s and 1930s, the General Elec- tric Company conducted a series of studies of worker productivity at its Hawthorne plant. In one
set of experiments, the researchers varied lightbulb wattage to see how lighting affected the productiv- ity of women assembling electrical parts. In other experiments they increased or decreased rest periods, changed the workroom layout, and shortened work- days. Influential early reports on these studies con- cluded that productivity continued to rise whether the lights were dimmer or brighter, whether work- days were longer or shorter, or whether conditions improved or worsened. Researchers concluded that the productivity improvements were not the conse- quence of changes in the workplace, but instead came
about because their special role in the experiment made the workers feel noticed and valued, so they worked harder and harder. Over the years, the idea that being in an experiment influences subject behav- ior has come to be known as the Hawthorne effect.
But there is a glitch to this story: Careful exami- nation of the actual Hawthorne data reveals no Hawthorne effect (Gillespie, 1991; Jones, 1992)! Still, in some experiments, especially ones in which the subjects have a stake in the outcome, merely being in an experiment could affect behavior. The Hawthorne effect and experimental effects more generally can pose threats to internal validity—even though the Hawthorne effect is not evident in the original Hawthorne data.
who remain in the sample at the end of the experiment and the differences estima- tor will be biased. Because attrition results in a nonrandomly selected sample, attrition that is related to the treatment leads to selection bias (Key Concept 9.4).
Experimental effects. In experiments with human subjects, merely because the subjects are in an experiment can change their behavior, a phenomenon some- times called the Hawthorne effect (see the box on this page).
In some experiments, a “double-blind” protocol can mitigate the effect of being in an experiment: Although subjects and experimenters both know that they are in an experiment, neither knows whether a subject is in the treatment group or the control group. In a medical drug experiment, for example, sometimes the drug and the placebo can be made to look the same so that neither the medical professional dispensing the drug nor the patient knows whether the administered drug is the real thing or the placebo. If the experiment is double blind, then both the treatment and control groups should experience the same experimental effects, and so different outcomes between the two groups can be attributed to the drug.
Double-blind experiments are clearly infeasible in real-world experiments in economics: Both the experimental subject and the instructor know whether the
13.2 Threats to Validity of Experiments 483
subject is attending the job training program. In a poorly designed experiment, this experimental effect could be substantial. For example, teachers in an experimental program might try especially hard to make the program a success if they think their future employment depends on the outcome of the experiment. Deciding whether experimental results are biased because of the experimental effects requires making judgments based on details of how the experiment was conducted.
Small samples. Because experiments with human subjects can be expensive, sometimes the sample size is small. A small sample size does not bias estimators of the causal effect, but it does mean that the causal effect is estimated impre- cisely. A small sample also raises threats to the validity of confidence intervals and hypothesis tests. Because inference based on normal critical values and heteroskedasticity-robust standard errors are justified using large-sample approx- imations, experimental data with small samples are sometimes analyzed under the assumption that the errors are normally distributed (Sections 3.6 and 5.6); however, the assumption of normality is typically as dubious for experimental data as it is for observational data.
Threats to External Validity
Threats to external validity compromise the ability to generalize the results of the study to other populations and settings. Two such threats are when the experi- mental sample is not representative of the population of interest and when the treatment being studied is not representative of the treatment that would be implemented more broadly.
Nonrepresentative sample. The population studied and the population of inter- est must be sufficiently similar to justify generalizing the experimental results. If a job training program is evaluated in an experiment with former prison inmates, then it might be possible to generalize the study results to other former prison inmates. Because a criminal record weighs heavily on the minds of potential employers, however, the results might not generalize to workers who have never committed a crime.
Another example of a nonrepresentative sample can arise when the experi- mental participants are volunteers. Even if the volunteers are randomly assigned to treatment and control groups, these volunteers might be more motivated than the overall population and, for them, the treatment could have a greater effect. More generally, selecting the sample nonrandomly from the population of interest can compromise the ability to generalize the results from the population studied (such as volunteers) to the population of interest.
484
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13.3
Experimental Estimates of the Effect of Class Size Reductions
In this section we return to a question addressed in Part II: What is the effect on test scores of reducing class size in the early grades? In the late 1980s, Tennessee conducted a large, multimillion-dollar randomized controlled experiment to ascertain whether class size reduction was an effective way to improve elementary education. The results of this experiment have strongly influenced our under- standing of the effect of class size reductions.
Nonrepresentative program or policy. The policy or program of interest also must be sufficiently similar to the program studied to permit generalizing the results. One important feature is that the program in a small-scale, tightly moni- tored experiment could be quite different from the program actually imple- mented. If the program actually implemented is widely available, then the scaled-up program might not provide the same quality control as the experimen- tal version or might be funded at a lower level; either possibility could result in the full-scale program being less effective than the smaller experimental program. Another difference between an experimental program and an actual program is its duration: The experimental program only lasts for the length of the experi- ment, whereas the actual program under consideration might be available for longer periods of time.
Generalequilibriumeffects. Anissuerelatedtoscaleanddurationconcernswhat economists call “general equilibrium” effects. Turning a small, temporary exper- imental program into a widespread, permanent program might change the eco- nomic environment sufficiently that the results from the experiment cannot be generalized. A small, experimental job training program, for example, might supplement training by employers, but if the program were made widely avail- able, it could displace employer-provided training, thereby reducing the net ben- efits of the program. Similarly, a widespread educational reform, such as offering school vouchers or sharply reducing class sizes, could increase the demand for teachers and change the type of person who is attracted to teaching, so the even- tual net effect of the widespread reform would reflect these induced changes in school personnel. Phrased in econometric terms, an internally valid small experi- ment might correctly measure a causal effect, holding constant the market or policy environment, but general equilibrium effects mean that these other factors are not, in fact, held constant when the program is implemented broadly.
13.3 Experimental Estimates of the Effect of Class Size Reductions 485 Experimental Design
The Tennessee class size reduction experiment, known as Project STAR (Student– Teacher Achievement Ratio), was a 4-year experiment designed to evaluate the effect on learning of small class sizes. Funded by the Tennessee state legislature, the experiment cost approximately $12 million. The study compared three different class arrangements for kindergarten through third grade: a regular class size, with 22 to 25 students per class, a single teacher, and no aides; a small class size, with 13 to 17 students per class and no aide; and a regular-sized class plus a teacher’s aide.
Each school participating in the experiment had at least one class of each type, and students entering kindergarten in a participating school were randomly assigned to one of these three groups at the beginning of the 1985–1986 academic year. Teachers were also assigned randomly to one of the three types of classes.
According to the original experimental protocol, students would stay in their initially assigned class arrangement for the 4 years of the experiment (kindergar- ten through third grade). However, because of parent complaints, students ini- tially assigned to a regular class (with or without an aide) were randomly reassigned at the beginning of first grade to regular classes with an aide or to regular classes without an aide; students initially assigned to a small class remained in a small class. Students entering school in first grade (kindergarten was optional), in the second year of the experiment, were randomly assigned to one of the three groups. Each year, students in the experiment were given standardized tests (the Stanford Achievement Test) in reading and math.
The project paid for the additional teachers and aides necessary to achieve the target class sizes. During the first year of the study, approximately 6400 stu- dents participated in 108 small classes, 101 regular classes, and 99 regular classes with aides. Over all 4 years of the study, a total of approximately 11,600 students at 80 schools participated in the study.
Deviations from the experimental design. The experimental protocol specified that the students should not switch between class groups, other than through the re-randomization at the beginning of first grade. However, approximately 10% of the students switched in subsequent years for reasons including incompatible chil- dren and behavioral problems. These switches represent a departure from the randomization scheme and, depending on the true nature of the switches, have the potential to introduce bias into the results. Switches made purely to avoid personality conflicts might be sufficiently unrelated to the experiment that they would not introduce bias. If, however, the switches arose because the parents most concerned with their children’s education pressured the school into switch- ing a child into a small class, then this failure to follow the experimental protocol
486 CHAPTER 13
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could bias the results toward overstating the effectiveness of small classes. Another deviation from the experimental protocol was that the class sizes changed over time because students switched between classes and moved in and out of the school district.
Analysis of the STAR Data
Because there are two treatment groups—small class and regular class with aide— the regression version of the differences estimator needs to be modified to handle the two treatment groups and the control group. This modification is done by introducing two binary variables, one indicating whether the student is in a small class and another indicating whether the student is in a regular-sized class with an aide, which leads to the population regression model
Yi = b0 + b1SmallClassi + b2RegAidei + ui, (13.3)
whereYi is a test score, SmallClassi = 1 if the ith student is in a small class and = 0 otherwise, and RegAidei = 1 if the ith student is in a regular class with an aide and = 0 otherwise. The effect on the test score of a small class, relative to a regu- lar class, is b1, and the effect of a regular class with an aide, relative to a regular class, is b2. The differences estimator for the experiment can be computed by estimating b1 and b2 in Equation (13.3) by OLS.
Table 13.1 presents the differences estimates of the effect on test scores of being in a small class or in a regular-sized class with an aide. The dependent
Project STAR: Differences Estimates of Effect on Standardized Test Scores of Class Size Treatment Group
TABLE 13.1
Regressor
Small class
Regular size with aide Intercept
Number of observations
Grade
K 1
13.90** 29.78** (2.45) (2.83)
0.31 11.96** (2.27) (2.65)
918.04** 1039.39** (1.63) (1.78)
5786 6379
2
19.39** (2.71)
3.48 (2.54)
1157.81** (1.82)
6049
3
15.59** (2.40)
–0.29 (2.27)
1228.51** (1.68)
5967
Note: The regressions were estimated using the Project STAR Public Access Data Set described in Appendix 13.1. The dependent variable is the student’s combined score on the math and reading portions of the Stanford Achievement Test. Standard errors are given in parentheses under the coefficients. **The individual coefficient is statistically significant at the 1% significance level using a two-sided test.
13.3 Experimental Estimates of the Effect of Class Size Reductions 487
variable Yi in the regressions in Table 13.1 is the student’s total score on the com- bined math and reading portions of the Stanford Achievement Test. According to the estimates in Table 13.1, for students in kindergarten, the effect of being in a small class is an increase of 13.9 points on the test, relative to being in a regular class; the estimated effect of being in a regular class with an aide is 0.31 point on the test. For each grade, the null hypothesis that small classes provide no improve- ment is rejected at the 1% (two-sided) significance level. However, it is not pos- sible to reject the null hypothesis that having an aide in a regular class provides no improvement, relative to not having an aide, except in first grade. The esti- mated magnitudes of the improvements in small classes are broadly similar in grades K, 2, and 3, although the estimate is larger for first grade.
The differences estimates in Table 13.1 suggest that reducing class size has an effect on test performance, but adding an aide to a regular-sized class has a much smaller effect, possibly zero. As discussed in Section 13.1, augmenting the regressions in Table 13.1 with additional regressors—the W regressors in Equation (13.2)—can provide more efficient estimates of the causal effects. Moreover, if the treatment received is not random because of failures to follow the treatment protocol, then the estimates of the experimental effects based on regressions with additional regressors could differ from the difference estimates reported in Table 13.1. For these two rea- sons, estimates of the experimental effects in which additional regressors are included in Equation (13.3) are reported for kindergarten in Table 13.2; the first column of Table 13.2 repeats the results of the first column (for kindergarten) from Table 13.1, and the remaining three columns include additional regressors that measure teacher, school, and student characteristics.
The main conclusion from Table 13.2 is that the multiple regression estimates of the causal effects of the two treatments (small class and regular-sized class with aide) in the final three columns of Table 13.2 are similar to the differences esti- mate reported in the first column. That adding these observable regressors does not change the estimated causal effects of the different treatments makes it more plausible that the random assignment to the smaller classes also does not depend on unobserved variables. As expected, these additional regressors increase the R 2 of the regression, and the standard error of the estimated class size effect decreases from 2.45 in column (1) to 2.16 in column (4).
Because teachers were randomly assigned to class types within a school, the experiment also provides an opportunity to estimate the effect on test scores of teacher experience. In the terminology of Section 13.1, randomization is condi- tional on the covariates W, where W denotes a full set of binary variables indicat- ing each school; that is, W denotes a full set of school fixed effects. Thus, conditional on W, years of experience are randomly assigned, which in turn
488 CHAPTER 13 Experiments and Quasi-Experiments
TABLE 13.2
Regressor
Small class
Regular size with aide
Teacher’s years of experience
Boy
Free lunch eligible
Black
Race other than black or white
Intercept
School indicator variables?
R2
Number of observations
Project STAR: Differences Estimates with Additional Regressors for Kindergarten
(1)
13.90** (2.45)
0.31 (2.27)
(2)
14.00** (2.45)
–0.60 (2.25)
1.47** (0.17)
(3)
15.93** (2.24)
1.22 (2.04)
0.74** (0.17)
(4)
15.89** (2.16)
1.79 (1.96)
0.66** (0.17)
– 12.09** (1.67)
-34.70** (1.99)
– 25.43** (3.50)
-8.50 (12.52)
yes
0.28 5748
918.04** (1.63)
no
0.01 5786
904.72** (2.22)
no
0.02 5766
yes
0.22 5766
Note: The regressions were estimated using the Project STAR Public Access Data Set described in Appendix 13.1. The dependent variable is the combined test score on the math and reading portions of the Stanford Achievement Test. The number of observa- tions differ in the different regressions because of some missing data. Standard errors are given in parentheses under coefficients. The individual coefficient is statistically significant at the *5% level or **1% significance level using a two-sided test.
implies that ui in Equation (13.2) satisfies conditional mean independence, where the X variables are the class size treatments and the teacher’s years of experience and W is the full set of school fixed effects. Because teachers were not reassigned randomly across schools, without school fixed effects in the regression [Table 13.2, column (2)] years of experience will in general be correlated with the error term, for example wealthier districts might have teachers with more years of experience. When school effects are included, the estimated coefficient on experience is cut in half, from 1.47 in column (2) of Table 13.2 to 0.74 in column (3). Because teach- ers were randomly assigned within a school, column (3) produces an unbiased estimator of the effect on test scores of an additional year of experience. The estimate, 0.74, is statistically significant and moderately large: Ten years of expe- rience corresponds to a predicted increase in test scores of 7.4 points.
13.3 Experimental Estimates of the Effect of Class Size Reductions 489
It is tempting to interpret some of the other coefficients in Table 13.2 but, like coefficients on control variables generally, those coefficients do not have a causal interpretation. For example, kindergarten boys perform worse than girls on these standardized tests. But these individual student characteristics are not randomly assigned (the gender of the student taking the test is not randomly assigned!), so these additional regressors could be correlated with omitted variables. Similarly, if race or eligibility for a free lunch is correlated with reduced learning opportuni- ties outside school (which is omitted from the Table 13.2 regressions), then their estimated coefficients would reflect these omitted influences.
Interpreting the estimated effects of class size. Are the estimated effects of class size reported in Table 13.1 and 13.2 large or small in a practical sense? There are two ways to answer this: first, by translating the estimated changes in raw test scores into units of standard deviations of test scores, so that the estimates in Table 13.1 are comparable across grades; and, second, by comparing the esti- mated class size effect to the other coefficients in Table 13.2.
Because the distribution of test scores is not the same for each grade, the esti- mated effects in Table 13.1 are not directly comparable across grades. We faced this problem in Section 9.4, when we wanted to compare the effect on test scores of a reduction in the student–teacher ratio estimated using data from California to the estimate based on data from Massachusetts. Because the two tests differed, the coefficients could not be compared directly. The solution in Section 9.4 was to translate the estimated effects into units of standard deviations of the test so that a unit decrease in the student–teacher ratio corresponds to a change of an esti- mated fraction of a standard deviation of test scores. We adopt this approach here so that the estimated effects in Table 13.1 can be compared across grades. For example, the standard deviation of test scores for children in kindergarten is 73.7, so the effect of being in a small class in kindergarten, based on the estimate in Table 13.1, is 13.9>73.7 = 0.19, with a standard error of 2.45>73.7 = 0.03. The estimated effects of class size from Table 13.1, converted into units of the stan- dard deviation of test scores across students, are summarized in Table 13.3. Expressed in standard deviation units, the estimated effect of being in a small class is similar for grades K, 2, and 3 and is approximately one-fifth of a standard deviation of test scores. Similarly, the result of being in a regular-sized class with an aide is approximately zero for grades K, 2, and 3. The estimated treatment effects are larger for first grade; however, the estimated difference between the small class and the regular-sized class with an aide is 0.20 for first grade, the same as the other grades. Thus one interpretation of the first-grade results is that the students in the control group—the regular-sized class without an aide—happened
490 CHAPTER 13
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TABLE 13.3
Treatment Group
Small class
Regular size with aide
Sample standard deviation of test scores (sY)
Estimated Class Size Effects in Units of Standard Deviations of the Test Score Across Students
K 1
0.19** 0.33** (0.03) (0.03)
0.00 0.13** (0.03) (0.03)
73.70 91.30
Grade
0.23** (0.03)
0.04 (0.03)
84.10
2
3
0.21** (0.03)
0.00 (0.03)
73.30
Note: The estimates and standard errors in the first two rows are the estimated effects in Table 13.1, divided by the sample standard deviation of the Stanford Achievement Test for that grade (the final row in this table), computed using data on the students in the experiment. Standard errors are given in parentheses under coefficients. **The individual coefficient is statistically significant at the 1% significance level using a two-sided test.
to do poorly on the test that year for some unusual reason, perhaps simply ran- dom sampling variation.
Another way to gauge the magnitude of the estimated effect of being in a small class is to compare the estimated treatment effects with the other coeffi- cients in Table 13.2. In kindergarten, the estimated effect of being in a small class is 13.9 points on the test (first row of Table 13.2). Holding constant race, teacher’s years of experience, eligibility for free lunch, and the treatment group, boys score lower on the standardized test than girls by approximately 12 points according to the estimates in column (4) of Table 13.2. Thus the estimated effect of being in a small class is somewhat larger than the performance gap between girls and boys. As another comparison, the estimated coefficient on the teacher’s years of experi- ence in column (4) is 0.66, so having a teacher with 20 years of experience is esti- mated to improve test performance by 13 points. Thus the estimated effect of being in a small class is approximately the same as the effect of having a 20-year veteran as a teacher, relative to having a new teacher. These comparisons suggest that the estimated effect of being in a small class is substantial.
Additional results. Econometricians, statisticians, and specialists in elementary education have studied this experiment extensively, and we briefly summarize some of those findings here. One is that the effect of a small class is concentrated in the earliest grades as can be seen in Table 13.3; except for the anomalous first- grade results, the test score gap between regular and small classes reported in Table 13.3 is essentially constant across grades (0.19 standard deviation unit in kindergarten, 0.23 in second grade, and 0.21 in third grade). Because the children
13.3 Experimental Estimates of the Effect of Class Size Reductions 491
initially assigned to a small class stayed in that small class, staying in a small class did not result in additional gains; rather, the gains made upon initial assignment were retained in the higher grades, but the gap between the treatment and control groups did not increase. Another finding is that, as indicated in the second row of Table 13.3, this experiment shows little benefit of having an aide in a regular-sized classroom. One potential concern about interpreting the results of the experiment is the failure to follow the treatment protocol for some students (some students switched from the small classes). If initial placement in a kindergarten classroom is random and has no direct effect on test scores, then initial placement can be used as an instrumental variable that partially, but not entirely, influences place- ment. This strategy was pursued by Krueger (1999), who used two stage least squares (TSLS) to estimate the effect on test scores of class size using initial class- room placement as the instrumental variable; he found that the TSLS and OLS estimates were similar, leading him to conclude that deviations from the experi- mental protocol did not introduce substantial bias into the OLS estimates.2
Comparison of the Observational and
Experimental Estimates of Class Size Effects
Part II presented multiple regression estimates of the class size effect based on observational data for California and Massachusetts school districts. In those data, class size was not randomly assigned, but instead was determined by local school officials trying to balance educational objectives against budgetary reali- ties. How do those observational estimates compare with the experimental esti- mates from Project STAR?
To compare the California and Massachusetts estimates with those in Table 13.3, it is necessary to evaluate the same class size reduction and to express the predicted effect in units of standard deviations of test scores. Over the 4 years of the STAR experiment, the small classes had, on average, approximately 7.5 fewer stu- dents than the large classes, so we use the observational estimates to predict the effect on test scores of a reduction of 7.5 students per class. Based on the OLS estimates for the linear specifications summarized in the first column of Table 9.3, the California estimates predict an increase of 5.5 points on the test for a 7.5 student reduction in
2For further reading about Project STAR, see Mosteller (1995), Mosteller, Light, and Sachs (1996), and Krueger (1999). Ehrenberg, Brewer, Gamoran, and Willms (2001a, 2001b) discuss Project STAR and place it in the context of the policy debate on class size and related research on the topic. For some criticisms of Project STAR, see Hanushek (1999a), and for a critical view of the relationship between class size and performance more generally, see Hanushek (1999b). To learn about how the Project STAR subjects performed later in life, see Chetty, Friedman, Hilger, Saez, Schanzenbach, and Yagan (2011).
492 CHAPTER 13
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the student–teacher ratio (0.73 * 7.5 ≅ 5.5 points). The standard deviation of the test across students in California is approximately 38 points, so the estimated effect of the reduction of 7.5 students, expressed in units of standard deviations across students, is 5.5>38 _ 0.14 standard deviations.3 The standard error of the estimated slope coefficient for California is 0.26 (Table 9.3), so the standard error of the estimated effect ofa7.5studentreductioninstandarddeviationunitsis0.26 * 7.5>38 _ 0.05.Thus, based on the California data, the estimated effect of reducing classes by 7.5 students, expressed in units of standard deviation of test scores across students, is 0.14 standard deviation, with a standard error of 0.05. These calculations and similar calculations for Massachusetts are summarized in Table 13.4, along with the STAR estimates for kindergarten taken from column (1) of Table 13.2.
The estimated effects from the California and Massachusetts observational studies are somewhat smaller than the STAR estimates. One reason that estimates from different studies differ, however, is random sampling variability, so it makes sense to compare confidence intervals for the estimated effects from the three studies. Based on the STAR data for kindergarten, the 95% confidence interval for the effect of being in a small class (reported in the final column of Table 13.4) is 0.13 to 0.25. The comparable 95% confidence interval based on the California observational data is 0.04 to 0.24, and for Massachusetts it is 0.02 to 0.22. Thus the 95% confidence intervals from the California and Massachusetts studies contain most of the 95% confidence interval from the STAR kindergarten data. Viewed in this way, the three studies give strikingly similar ranges of estimates.
There are many reasons the experimental and observational estimates might differ. One reason is that, as discussed in Section 9.4, there are remaining threats to the internal validity of the observational studies. For example, because children move into and out of districts, the district student–teacher ratio might not reflect the student–teacher ratio actually experienced by the students, so the coefficient on the student–teacher ratio in the Massachusetts and California studies could be biased toward zero because of errors-in-variables bias. Other reasons concern external validity. The district average student–teacher ratio used in the observa- tional studies is not the same thing as the actual number of children in the class, the STAR experimental variable. Project STAR was in a southern state in the 1980s, potentially different from California and Massachusetts in 1998, and the grades
3In Table 9.3, the estimated effects are presented in terms of the standard deviation of test scores across districts; in Table 13.3, the estimated effects are in terms of the standard deviation of test scores across students. The standard deviation across students is greater than the standard deviation across districts. For California, the standard deviation across students is 38, but the standard deviation across districts is 19.1.
TABLE 13.4
Study
STAR (grade K)
California
Massachusetts
Estimated Effects of Reducing the Student–Teacher Ratio by 7.5 Based on the STAR Data and the California and Massachusetts Observational Data
13.4 Quasi-Experiments 493
Bn1
–13.90** (2.45)
– 0.73** (0.26)
– 0.64* (0.27)
Change in Student–Teacher Ratio
Small class vs. regular class
–7.5
–7.5
Standard Deviation of Test Scores Across Students
73.8 38.0 39.0
Estimated Effect
0.19** (0.03)
0.14** (0.05)
0.12* (0.05)
95% Confidence Interval
(0.13, 0.25) (0.04, 0.24) (0.02, 0.22)
Note: The estimated coefficient bn1 for the STAR study is taken from column (1) of Table 13.2. The estimated coefficients for the California and Massachusetts studies are taken from the first column of Table 9.3. The estimated effect is the effect of being in
a small class versus a regular class (for STAR) or the effect of reducing the student–teacher ratio by 7.5 (for the California and Massachusetts studies). The 95% confidence interval for the reduction in the student–teacher ratio is this estimated effect ± 1.96 standard errors. Standard errors are given in parentheses under estimated effects. The estimated effects are statistically signifi- cantly different from zero at the *5% significance level or **1% significance level using a two-sided test.
being compared differ (K through 3 in STAR, fourth grade in Massachusetts, fifth grade in California). In light of all these reasons to expect different estimates, the findings of the three studies are remarkably similar. That the observational studies are similar to the Project STAR estimates suggests that the remaining threats to the internal validity of the observational estimates are minor.
13.4
Quasi-Experiments
The statistical insights and methods of randomized controlled experiments can carry over to nonexperimental settings. In a quasi-experiment, also called a natu- ral experiment, randomness is introduced by variations in individual circum- stances that make it appear as if the treatment is randomly assigned. These variations in individual circumstances might arise because of vagaries in legal institutions, location, timing of policy or program implementation, natural ran- domness such as birth dates, rainfall, or other factors that are unrelated to the causal effect under study.
There are two types of quasi-experiments. In the first, whether an individual (more generally, an entity) receives treatment is viewed as if it is randomly deter- mined. In this case, the causal effect can be estimated by OLS using the treatment, Xi, as a regressor. In the second type of quasi-experiment, the “as if” random variation only partially determines the treatment. In this case, the causal effect is
494 CHAPTER 13
Experiments and Quasi-Experiments
estimated by instrumental variables regression, where the “as if” random source of variation provides the instrumental variable.
After providing some examples, this section presents some extensions of the econometric methods in Sections 13.1 and 13.2 that can be useful for analyzing data from quasi-experiments.
Examples
We illustrate the two types of quasi-experiments by examples. The first example is a quasi-experiment in which the treatment is “as if” randomly determined. The second and third examples illustrate quasi-experiments in which the “as if” random variation influences, but does not entirely determine, the level of the treatment.
Example #1: Labor market effects of immigration. Does immigration reduce wages? Economic theory suggests that if the supply of labor increases because of an influx of immigrants, the “price” of labor—the wage—should fall. However, all else being equal, immigrants are attracted to cities with high labor demand, so the OLS estimator of the effect on wages of immigration will be biased. An ideal randomized controlled experiment for estimating the effect on wages of immigra- tion would randomly assign different numbers of immigrants (different “treat- ments”) to different labor markets (“subjects”) and measure the effect on wages (the “outcome”). Such an experiment, however, faces severe practical, financial, and ethical problems.
The labor economist David Card (1990) therefore used a quasi-experiment in which a large number of Cuban immigrants entered the Miami, Florida, labor market in the “Mariel boatlift,” which resulted from a temporary lifting of restric- tions on emigration from Cuba in 1980. Half of the immigrants settled in Miami, in part because it had a large preexisting Cuban community. Card estimated the causal effect on wages of an increase in immigration by comparing the change in wages of low-skilled workers in Miami to the change in wages of similar workers in other comparable U.S. cities over the same period. He concluded that this influx of immigrants had a negligible effect on wages of less-skilled workers.
Example #2: Effects on civilian earnings of military service. Does serving in the military improve your prospects on the labor market? The military provides train- ing that future employers might find attractive. However, an OLS regression of individual civilian earnings against prior military service could produce a biased estimator of the effect on civilian earnings of military service because military service is determined, at least in part, by individual choices and characteristics.
For example, the military accepts only applicants who meet minimum physical requirements, and a lack of success in the private sector labor market might make an individual more likely to sign up for the military.
To circumvent this selection bias, Joshua Angrist (1990) used a quasi-experimental design in which he examined labor market histories of those who served in the U.S. military during the Vietnam War. During this period, whether a young man was drafted into the military was determined in part by a national lottery system based on birthdays: Men randomly assigned low lottery numbers were eligible to be drafted, whereas those with high numbers were not. Actual entry into the military was determined by complicated rules, including physical screening and certain exemptions, and some young men volunteered for service, so serving in the military was only partially influenced by whether a man was draft-eligible. Thus being draft- eligible serves as an instrumental variable that partially determines military service but is randomly assigned. In this case, there was true random assignment of draft eligibility via the lottery, but because this randomization was not done as part of an experiment to evaluate the effect of military service, it is a quasi-experiment. Angrist concluded that the long-term effect of military service was to reduce earnings of white, but not nonwhite, veterans.
Example #3: The effect of cardiac catheterization. Section 12.5 described the study by McClellan, McNeil, and Newhouse (1994) in which they used the distance from a heart attack patient’s home to a cardiac catheterization hospital, relative to the distance to a hospital lacking catheterization facilities, as an instrumental variable for actual treatment by cardiac catheterization. This study is a quasi- experiment with a variable that partially determines the treatment. The treatment itself, cardiac catheterization, is determined by personal characteristics of the patient and by the decision of the patient and doctor; however, it is also influenced by whether a nearby hospital is capable of performing this procedure. If the loca- tion of the patient is “as if” randomly assigned and has no direct effect on health outcomes, other than through its effect on the probability of catheterization, then the relative distance to a catheterization hospital is a valid instrumental variable.
Other examples. The quasi-experiment research strategy has been applied in other areas as well. Garvey and Hanka (1999) used variation in U.S. state laws to examine the effect on corporate financial structure (for example, the use of debt by corporations) of anti-takeover laws. Meyer, Viscusi, and Durbin (1995) used large discrete changes in the generosity of unemployment insurance benefits in Kentucky and Michigan, which differentially affected workers with high but not low earnings, to estimate the effect on time out of work of a change in
13.4 Quasi-Experiments 495
496 CHAPTER 13
Experiments and Quasi-Experiments
unemployment benefits. The surveys of Meyer (1995), Rosenzweig and Wolpin (2000), and Angrist and Krueger (2001) give other examples of quasi-experiments in the fields of economics and social policy.
The Differences-in-Differences Estimator
If the treatment in a quasi-experiment is “as if” randomly assigned, conditional on some observed variables W, then the treatment effect can be estimated using the differences regression (13.2). Because the researcher does not have control over the randomization, however, some differences might remain between the treatment and control groups even after controlling for W. One way to adjust for those remaining differences between the two groups is to compare not the out- comes Y, but the change in the outcomes pre- and post-treatment, thereby adjust- ing for differences in pre-treatment values of Y in the two groups. Because this estimator is the difference across groups in the change, or difference over time, this estimator is called the differences-in-differences estimator. For example, in Card’s (1990) study of the effect of immigration on low-skilled workers’ wages, he used a differences-in-differences estimator to compare the change in wages in Miami with the change in wages in other U.S. cities. Another example of the use of the differences-in-differences estimator is given in the box “What Is the Effect on Employment of the Minimum Wage?”
The differences-in-differences estimator. Let Y treatment, before be the sample aver- age of Y for those in the treatment group before the experiment, and let Y treatment, after be the sample average for the treatment group after the experiment. Let Ycontrol, before and Ycontrol, after be the corresponding pretreatment and post treatment sample averages for the control group. The average change in Y over the course of the experiment for those in the treatment group is Ytreatment, after – Ytreatment, before, and the average change in Y over this period for those in the control group is Ycontrol, after – Ycontrol, before. The differences-in- differences estimator is the average change in Y for those in the treatment group, minus the average change in Y for those in the control group:
bndiffs-in-diffs = (Ytreatment,after – Ytreatment,before) – (Ycontrol,after – Ycontrol,before) 1
= ∆Ytreatment – ∆Ycontrol,
(13.4)
where∆Ytreatment is the average change in Y in the treatment group and ∆Ycontrol
is the average change in Y in the control group. If the treatment is randomly assigned,
then bndiffs – in – diffs is an unbiased and consistent estimator of the causal effect. 1
13.4 Quasi-Experiments 497 What Is the Effect on Employment of the Minimum Wage?
How much does an increase in the minimum wage reduce demand for low-skilled workers? Economic theory says that demand falls when the
price rises, but precisely how much is an empirical question. Because prices and quantities are deter- mined by supply and demand, the OLS estimator in a regression of employment against wages has simul- taneous causality bias (Key Concept 9.6). Hypo- thetically, a randomized controlled experiment might randomly assign different minimum wages to different employers and then compare changes in employment (outcomes) in the treatment and con- trol groups, but how could this hypothetical experi- ment be done in practice?
The labor economists David Card and Alan Krueger (1994) decided to conduct such an experi- ment, but to let “nature”—or, more precisely, geog- raphy—perform the randomization for them. In 1992, the minimum wage in New Jersey rose from $4.25 to $5.05 per hour, but the minimum wage in neighboring Pennsylvania stayed constant. In this experiment, the “treatment” of the minimum wage increase—being located in New Jersey instead of Pennsylvania—is viewed “as if” randomly assigned
in the sense that being subject to the wage hike is assumed to be uncorrelated with the other determinants of employment changes over this period. Card and Krueger collected data on employment at fast-food restaurants before and after the wage increase in the two states. When they computed the differences- in-differences estimator, they found a surprising result: There was no evidence that employment fell at New Jersey fast-food restaurants, relative to those in Pennsylvania. In fact, some of their estimates actually suggest that employment increased in New Jersey restaurants after its minimum wage went up, relative to Pennsylvania!
This finding conflicts with basic microeconomic theory and has been quite controversial. Subsequent analysis, using a different source of employment data, suggests that there might have been a small drop in employment in New Jersey after the wage hike, but even so the estimated labor demand curve is very inelastic (Neumark and Wascher, 2000). Although the exact wage elasticity in this quasi-experiment is a matter of debate, the effect on employment of a hike in the minimum wage appears to be smaller than many economists had previously thought.
The differences-in-differences estimator can be written in regression nota- tion. Let ∆Yi be the post experimental value of Y for the ith individual minus the pre-experimental value. The differences-in-differences estimator is the OLS esti- mator of b1 in the regression,
∆Yi = b0 + b1Xi + ui. (13.5)
The differences-in-differences estimator is illustrated in Figure 13.1. In that figure, the sample average of Y for the treatment group is 40 before the experiment, whereas the pretreatment sample average of Y for the control group is 20. Over the course of the experiment, the sample average of Y increases in the control
498 CHAPTER 13 Experiments and Quasi-Experiments FIGURE 13.1 The Differences-in-Differences Estimator
The post-treatment difference between
the treatment and control groups is 90
Outcome
Y treatment, before Ycontrol, before
Y control, after
Y treatment, after ^diffs-in-diffs
b1
80 – 30 = 50, but this overstates the
treatment effect because before the treat-
ment Y was higher for the treatment than
the control group by 40 – 20 = 20. 60 The differences-in-differences estimator is
the difference between the final and ini-
40
(40 – 20) = 50 – 20 = 30. Equivalently, 30
the differences-in-differences estimator is
the average change for the treatment
group minus the average change
tial gaps, so bndiffs – in – diffs = (80 – 30) – 1
for the control group, that is, 0
bn diffs – in – diffs = ∆Y treatment – ∆Y control = 1
(80 – 40) – (30 – 20) = 30.
t=1 t=2
Time period
80 70
50
20 10
group to 30, whereas it increases to 80 for the treatment group. Thus the mean difference of the post-treatment sample averages is 80 – 30 = 50. However, some of this difference arises because the treatment and control groups had dif- ferent pretreatment means: The treatment group started out ahead of the control group. The differences-in-differences estimator measures the gains of the treat- ment group, relative to the control group, which in this example is (80 – 40) – (30 – 20) = 30. By focusing on the change in Y over the course of the experiment, the differences-in-differences estimator removes the influence of initial values of Y that vary between the treatment and control groups.
The differences-in-differences estimator with additional regressors. The differences-in-differences estimator can be extended to include additional regres- sors W1i, c,Wri, which measure individual characteristics prior to the experi- ment. These additional regressors can be incorporated using the multiple regression model
∆Yi = b0 + b1Xi + b2W1i + g+ b1+rWri + ui,i = 1,c,n. (13.6)
The OLS estimator of b1 in Equation (13.6) is the differences-in-differences estimator with additional regressors. If Xi is “as if” randomly assigned, conditional on W1i, c, Wri, then ui satisfies conditional mean independence and the OLS estimator of bn1 in Equation (13.6) is unbiased.
The differences-in-differences estimator described here considers two time periods, before and after the experiment. In some settings there are panel data with multiple time periods. The differences-in-differences estimator can be extended to multiple time periods using the panel data regression methods of Chapter 10.
Differences-in-differencesusingrepeatedcross-sectionaldata. Arepeatedcross- sectional data set is a collection of cross-sectional data sets, where each cross- sectional data set corresponds to a different time period. For example, the data set might contain observations on 400 individuals in the year 2004 and on 500 different individuals in 2005, for a total of 900 different individuals. One example of repeated cross-sectional data is political polling data, in which political preferences are measured by a series of surveys of randomly selected potential voters, where the surveys are taken at different dates and each survey has different respondents.
The premise of using repeated cross-sectional data is that if the individuals (more generally, entities) are randomly drawn from the same population, then the individuals in the earlier cross section can be used as surrogates for the individuals in the treatment and control groups in the later cross section.
When there are two time periods, the regression model for repeated cross- sectional data is
Yit = b0 + b1Xit + b2Gi + b3Dt + b4W1it + g+ b3+rWrit + uit, (13.7)
where Xit is the actual treatment of the ith individual (entity) in the cross section in period t (t = 1, 2), Gi is a binary variable indicating whether the individual is in the treatment group (or in the surrogate treatment group, if the observation is in the pretreatment period), and Dt is the binary indicator that equals 0 in the first period and equals 1 in the second period. The ith individual receives treatment if he or she is in the treatment group in the second period, so in Equation (13.7), Xit = Gi * Dt that is, Xit is the interaction between Gi and Dt.
If the quasi-experiment makes Xit “as if” randomly received, conditional on the W’s, then the causal effect can be estimated by the OLS estimator of b1 in Equation (13.7). If there are more than two time periods, then Equation (13.7) is modified to contain T – 1 binary variables indicating the different time periods (see Section 10.4).
Instrumental Variables Estimators
If the quasi-experiment yields a variable Zi that influences receipt of treatment, if data are available both on Zi and on the treatment actually received (Xi), and if Zi is “as if” randomly assigned (perhaps after controlling for some additional
13.4 Quasi-Experiments 499
500 CHAPTER 13
Experiments and Quasi-Experiments
variables Wi), then Zi is a valid instrument for Xi and the coefficients of Equation (13.2) can be estimated using two-stage least squares. Any control variables appearing in (13.2) also appear as control variables in the first stage of the two- stage least squares estimator of b1.
Regression Discontinuity Estimators
One situation that gives rise to a quasi-experiment is when receipt of the treat- ment depends in whole or in part on whether an observable variable W crosses a threshold value. For example, suppose that students are required to attend sum- mer school if their end-of-year grade point average (GPA) falls below a thresh- old.4 Then one way to estimate the effect of mandatory summer school is to compare outcomes for students whose GPA was just below the threshold (and thus were required to attend) to outcomes for students whose GPA was just above the threshold (so they escaped summer school). The outcome Y could be next year’s GPA, whether the student drops out, or future earnings. As long as there is nothing special about the threshold value other than its use in mandating sum- mer school, it is reasonable to attribute any jump in outcomes at that threshold to summer school. Figure 13.2 illustrates a hypothetical scatterplot of a data set in which the treatment (summer school, X) is required if GPA (W) is less than a threshold value (w0 = 2.0). The scatterplot shows next year’s GPA (Y) for a hypothetical sample of students as a function of this year’s GPA, along with the population regression function. If the only role of the threshold w0 is to mandate summer school, then the jump in next year’s GPA at w0 is an estimate of the effect of summer school on next year’s GPA.
Because of the jump, or discontinuity, in treatment at the threshold, studies that exploit a discontinuity in the probability of receiving treatment at a threshold value are called regression discontinuity designs. There are two types of regres- sion discontinuity designs, sharp and fuzzy.
Sharp regression discontinuity designs. In a sharp regression discontinuity design, receipt of treatment is entirely determined by whether W exceeds the threshold: All students with W 6 w0 attend summer school, and no students with W Ú w0 attend; that is, Xi = 1 if W 6 w0 and Xi = 0 if W Ú w0. In this case, the jump in Y at the threshold equals the average treatment effect for the subpopulation with W = w0, which might be a useful approximation to the average treatment
4This example is a simplified version of the regression discontinuity study of the effect of summer school for elementary and middle school students by Jordan Matsudaira (2008), in which summer school attendance was based in part on end-of-year tests.
13.4 Quasi-Experiments 501 FIGURE 13.2 A Hypothetical Regression Discontinuity Design Scatterplot
Suppose that the binary treatment X is required if W is less than the thresh- old value w0 = 2. As long as the only role of the threshold w0 is to mandate treatment, the treatment effect is given by the magnitude of the jump, or discontinuity,
in the regression function at W = 2.
Y
4.0
3.5
3.0
2.5
2.0
1.5
1.0
Population regression line
Population regression line
1.0 1.5
2.0 2.5 3.0
3.5 4.0
w0 W
effect in the larger population of interest. If the regression function is linear in W, other than for the treatment-induced discontinuity, the treatment effect can be estimated by b1 in the regression:
Yi =b0 +b1Xi +b2Wi +ui. (13.8) If the regression function is nonlinear, then a suitable nonlinear function of W can
be used (Section 8.2).
Fuzzyregressiondiscontinuitydesign. Inafuzzyregressiondiscontinuitydesign, crossing the threshold influences receipt of the treatment but is not the sole deter- minant. For example, suppose that some students whose GPA falls below the threshold are exempted from summer school while some whose GPA exceeds the threshold nevertheless attend. This situation could arise if the threshold rule is part of a more complicated process for determining treatment. In a fuzzy design, Xi will in general be correlated with ui in Equation (13.8). If, however, any special effect of crossing the threshold operates solely by increasing the probability of treat- ment—that is, the direct effect of crossing the threshold is captured by the linear term in W—then an instrumental variables approach is available. Specifically, the binary variable Zi which indicates crossing the threshold (so Zi = 1 if Wi 6 w0 and Zi = 0 if Wi Ú w0) influences receipt of treatment but is uncorrelated with ui,
502
CHAPTER 13
Experiments and Quasi-Experiments
13.5
Potential Problems with Quasi-Experiments
Like all empirical studies, quasi-experiments face threats to internal and external validity. A particularly important potential threat to internal validity is whether the “as if” randomization in fact can be treated reliably as true randomization.
Threats to Internal Validity
The threats to the internal validity of true randomized controlled experiments listed in Section 13.2 also apply to quasi-experiments, but with some modifications.
Failure of randomization. Quasi-experiments rely on differences in individual circumstances—legal changes, sudden unrelated events, and so forth—to provide the “as if” randomization in the treatment level. If this “as if” randomization fails to produce a treatment level X (or an instrumental variable Z) that is random, then in general the OLS estimator is biased (or the instrumental variable estima- tor is not consistent).
As in a true experiment, one way to test for failure of randomization is to check for systematic differences between the treatment and control groups, for example by regressing X (or Z) on the individual characteristics (the W’s) and testing the hypothesis that the coefficients on the W’s are zero. If differences exist that are not readily explained by the nature of the quasi-experiment, then that is evidence that the quasi-experiment did not produce true randomization. Even if there is no relationship between X (or Z) and the W’s, the possibility remains that X (or Z) could be related to some of the unobserved factors in the error term u. Because these factors are unobserved, this possibility cannot be tested, and the validity of the assumption of “as if” randomization must be evaluated using expert knowledge and judgment applied to the application at hand.
Failure to follow the treatment protocol. In a true experiment, failure to follow the treatment protocol arises when members of the treatment group fail to receive treatment, members of the control group actually receive treatment, or both; in consequence, the OLS estimator of the causal effect has selection bias. The coun- terpart to failing to follow the treatment protocol in a quasi-experiment is when the “as if” randomization influences, but does not determine, the treatment level.
so it is a valid instrument for Xi. Thus, in a fuzzy regression discontinuity design, b1 can be estimated by instrumental variables estimation of Equation (13.8), using as an instrument the binary variable indicating that Wi 6 w0.
13.5 Potential Problems with Quasi-Experiments 503 In this case, the instrumental variables estimator based on the quasi-experimental
influence Z can be consistent even though the OLS estimator is not.
Attrition. Attrition in a quasi-experiment is similar to attrition in a true experi- ment in the sense that if it arises because of personal choices or characteristics, then attrition can induce correlation between the treatment level and the error term. The result is sample selection bias, so the OLS estimator of the causal effect is biased and inconsistent.
Experimental effects. An advantage of quasi-experiments is that, because they are not true experiments, there typically is no reason for individuals to think that they are experimental subjects. Thus experimental effects such as the Hawthorne effect generally are not germane in quasi-experiments.
Instrument validity in quasi-experiments. An important step in evaluating a study that uses instrumental variables regression is careful consideration of whether the instrument is in fact valid. This general statement remains true in quasi-experimental studies in which the instrument is “as if” randomly deter- mined. As discussed in Chapter 12, instrument validity requires both instrument relevance and instrument exogeneity. Because instrument relevance can be checked using the statistical methods summarized in Key Concept 12.5, here we focus on the second, more judgmental requirement of instrument exogeneity.
Although it might seem that a randomly assigned instrumental variable is necessarily exogenous, that is not so. Consider the examples of Section 13.4. In Angrist’s (1990) use of draft lottery numbers as an instrumental variable in study- ing the effect on civilian earnings of military service, the lottery number was in fact randomly assigned. But, as Angrist (1990) points out and discusses, if a low draft number results in behavior aimed at avoiding the draft and that avoidance behavior subsequently affects civilian earnings, then a low lottery number (Zi) could be related to unobserved factors that determine civilian earnings (ui); that is, Zi and ui are correlated even though Zi is randomly assigned. As a second example, McClellan, McNeil, and Newhouse’s (1994) study of the effect on heart attack patients of cardiac catheterization treated the relative distance to a cathe- terization hospital as if it were randomly assigned. But, as the authors highlight and examine, if patients who live close to a catheterization hospital are healthier than those who live far away (perhaps because of better access to medical care generally), then the relative distance to a catheterization hospital would be cor- related with omitted variables in the error term of the health outcome equation. In short, just because an instrument is randomly determined or “as if” randomly
504
CHAPTER 13
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13.6
Experimental and Quasi-Experimental Estimates in Heterogeneous Populations
As discussed in Section 13.1, the causal effect can vary from one member of the population to the next. Section 13.1 discusses estimating causal effects that vary depending on observable variables, such as gender. In this section, we consider the consequences of unobserved variation in the causal effect. We refer to unob- served variation in the causal effect as having a heterogeneous population. To keep things simple and to focus on the role of unobserved heterogeneity, in this section we omit control variables W; the conclusions of this section carry over to regressions including control variables.
If the population is heterogeneous, then the ith individual now has his or her own causal effect, b1i, which (in the terminology of Section 13.1) is the difference in the ith individual’s potential outcomes if the treatment is or is not received. For example, b1i might be zero for a resume-writing training program if the ith
determined does not necessarily mean it is exogenous in the sense that corr(Zi, ui) = 0. Thus the case for exogeneity must be scrutinized closely even if the instrument arises from a quasi-experiment.
Threats to External Validity
Quasi-experimental studies use observational data, and the threats to the external validity of a study based on a quasi-experiment are generally similar to the threats discussed in Section 9.1 for conventional regression studies using observational data.
One important consideration is that the special events that create the “as if” randomness at the core of a quasi-experimental study can result in other special features that threaten external validity. For example, Card’s (1990) study of labor market effects of immigration discussed in Section 13.4 used the “as if” random- ness induced by the influx of Cuban immigrants in the Mariel boatlift. There were, however, special features of the Cuban immigrants, Miami, and its Cuban com- munity that might make it difficult to generalize these findings to immigrants from other countries or to other destinations. Similarly, Angrist’s (1990) study of the labor market effects of serving in the U.S. military during the Vietnam War pre- sumably would not generalize to peacetime military service. As usual, whether a study generalizes to a specific population and setting of interest depends on the details of the study and must be assessed on a case-by-case basis.
13.6 Experimental and Quasi-Experimental Estimates in Heterogeneous Populations 505 individual already knows how to write a resume. With this notation, the popula-
tion regression equation can be written
Yi = b0i + b1iXi + ui. (13.9)
Because b1i varies from one individual to the next in the population and the indi- viduals are selected from the population at random, b1i is a random variable that, just like ui, reflects unobserved variation across individuals (for example, varia- tion in preexisting resume-writing skills). The average causal effect is the popula- tion mean value of the causal effect, E(b1i); that is, it is the expected causal effect of a randomly selected member of the population under study.
What do the estimators of Sections 13.1, 13.2, and 13.4 estimate if there is population heterogeneity of the form in Equation (13.9)? We first consider the OLS estimator when Xi is “as if” randomly determined; in this case, the OLS estimator is a consistent estimator of the average causal effect. That is generally not true for the IV estimator, however. Instead, if Xi is partially influenced by Zi, then the IV estimator using the instrument Z estimates a weighted average of the causal effects, where those for whom the instrument is most influential receive the most weight.
OLS with Heterogeneous Causal Effects
If there is heterogeneity in the causal effect and if Xi is randomly assigned, then the differences estimator is a consistent estimator of the average causal effect. This result follows from the discussion in Section 13.1 and Appendix 13.3, which make use of the potential outcome framework; here it is shown without reference to potential outcomes by applying concepts from Chapters 3 and 4 directly to the random coefficients regression model in Equation (13.9).
The OLS estimator of b1 in Equation (13.1) is bn1 = sXY>s2X [Equation (4.7)]. If the observations are i.i.d., then the sample covariance and variance are consis- tent estimators of the population covariance and variance, so bn1 ¡p sXY>s2X. If Xi is randomly assigned, then Xi is distributed independently of other individual characteristics, both observed and unobserved, and in particular is distributed independently of b0i and b1i. Accordingly, the OLS estimator bn1 has the limit
bn1 = sXY ¡p sXY = cov(b0i + b1i Xi + ui,Xi) s2X s2X s2X
= cov(b0i + b1i Xi, Xi) = E(b1i), (13.10) s2X
506 CHAPTER 13
Experiments and Quasi-Experiments
where the third equality uses the facts about covariances in Key Concept 2.3 and cov(ui, Xi) = 0, which is implied by E(ui Xi) = 0 [Equation (2.27)], and where the final equality follows from b0i and b1i being distributed independently of Xi, which they are if Xi is randomly assigned (Exercise 13.9). Thus, if Xi is randomly assigned, bn1 is a consistent estimator of the average causal effect E(b1i).
IV Regression with Heterogeneous Causal Effects
Suppose that the causal effect is estimated by instrumental variables regression of Yi on Xi (treatment actually received) using Zi (initial randomly or “as if” ran- domly assigned treatment) as an instrument. Suppose that Zi is a valid instrument (relevant and exogenous) and that there is heterogeneity in the effect on Xi of Zi. Specifically, suppose that Xi is related to Zi by the linear model
Xi = p0i + p1iZi + vi, (13.11)
where the coefficients p0i and p1i vary from one individual to the next. Equation (13.11) is the first-stage equation of TSLS with the modification that the effect on Xi of a change in Zi is allowed to vary from one individual to the next.
The TSLS estimator is bnTSLS = s >s [Equation (12.4)], the ratio of the 1 ZYZX
sample covariance between Z and Y to the sample covariance between Z and X.
If the observations are i.i.d., then these sample covariances are consistent estima-
tors of the population covariances, so bnTSLS ¡p s >s . Suppose that 1 ZYZX
p0i,p1i,b0i, and b1i are distributed independently of ui, vi, and Zi; that E(u Z) = E(v Z) = 0;andthatE(p ) ≠ 0(instrumentrelevance).Itisshown
iiii 1i
in Appendix 13.2 that, under these assumptions,
bnTSLS = sZY ¡p sZY = E(b1ip1i). (13.12) 1 sZX sZX E(p1i)
That is, the TSLS estimator converges in probability to the ratio of the expected value of the product of b1i and p1i to the expected value of p1i.
The final ratio in Equation (13.12) is a weighted average of the individual causal effects b1i. The weights are p1i>Ep1i, which measure the relative degree to which the instrument influences whether the ith individual receives treatment. Thus the TSLS estimator is a consistent estimator of a weighted average of the individual causal effects, where the individuals who receive the most weight are those for whom the instrument is most influential. The weighted average causal effect that is estimated by TSLS is called the local average treatment effect (LATE). The term “local” emphasizes that it is the weighted average that places the most
13.6 Experimental and Quasi-Experimental Estimates in Heterogeneous Populations 507
weight on those individuals (more generally, entities) whose treatment probabil- ity is most influenced by the instrumental variable.
There are three special cases in which the local average treatment effect equals the average treatment effect:
1. The treatment effect is the same for all individuals. This case corresponds to b1i = b1 for all i. Then the final expression in Equation (13.12) simplifies to E(b1i p1i)>E(p1i) = b1E(p1i) >E(p1i) = b1.
2. The instrument affects each individual equally. This case corresponds to p1i = p1 for all i. In this case, the final expression in Equation (13.12) sim- plifies to E(b1i p1i)>E(p1i) = E(b1i)p1>p1 = E(b1i).
3. The heterogeneity in the treatment effect and heterogeneity in the effect of the instrument are uncorrelated.This case corresponds to b1i and p1i being ran- dom but cov(b1i, p1i) = 0. Because E(b1i p1i) = cov(b1i, p1i) + E(b1i)E(p1i) [Equation (2.34)], if cov(b1i, p1i) = 0, then E(b1i p1i) = E(b1i)E(p1i) and the final expression in Equation (13.12) simplifies to E(b1ip1i)>E(p1i) = E(b1i)E(p1i)>E(p1i) = E(b1i).
In each of these three cases, there is population heterogeneity in the effect of the instrument, in the effect of the treatment, or both, but the local average treatment effect equals the average treatment effect. That is, in all three cases, TSLS is a consistent estimator of the average treatment effect.
Aside from these three special cases, in general the local average treatment
effect differs from the average treatment effect. For example, suppose that Zi has
no influence on the treatment decision for half the population (for them, p1i = 0)
and that Zi has the same, nonzero influence on the treatment decision for the
other half (for them, p1i is a nonzero constant). Then TSLS is a consistent estima-
tor of the average treatment effect in the half of the population for which the
instrument influences the treatment decision. To be concrete, suppose that work-
ers are eligible for a job training program and are randomly assigned a priority
number Z, which influences how likely they are to be admitted to the program.
Half the workers know they will benefit from the program and thus may decide
to enroll in the program; for them, b1i = b1+ 7 0 and p1i = p1+ 7 0. The other
half know that, for them, the program is ineffective so they would not enroll even
if admitted, that is, for them b = 0 and p = 0. The average treatment effect is 1i 1i
E(b1i) = 12(b1+ + 0) = 12b1+.ThelocalaveragetreatmenteffectisE(b1ip1i)>E(p1i). NowE(p ) = 1p+ andE(b p ) = E[b E(p b )] = 1(0 + b+p+) = 1b+p+,so
1i21 1i1i 1i1i1i2 11211 E(b1i p1i)>E(p1i) = b1+. Thus, in this example, the local average treatment effect
is the causal effect for those workers who are likely to enroll in the program, and it gives no weight to those who will not enroll under any circumstances. In contrast,
508 CHAPTER 13
Experiments and Quasi-Experiments
the average treatment effect places equal weight on all individuals, regardless of whether they would enroll. Because individuals decide to enroll based in part on their knowledge of how effective the program will be for them, in this example the local average treatment effect exceeds the average treatment effect.
Implications. If an individual’s decision to receive treatment depends on the effectiveness of the treatment for that individual, then the TSLS estimator in general is not a consistent estimator of the average causal effect. Instead, TSLS estimates a local average treatment effect, where the causal effects of the indi- viduals who are most influenced by the instrument receive the greatest weight. This conclusion leads to a disconcerting situation in which two researchers, armed with different instrumental variables that are both valid in the sense that both are relevant and exogenous, would obtain different estimates of “the” causal effect, even in large samples. The difference arises because each researcher is implicitly estimating a different weighted average of the individual causal effects in the population. In fact, a J-test of overidentifying restrictions can reject if the two instruments estimate different local average treatment effects, even if both instru- ments are valid. Although both estimators provide some insight into the distribu- tion of the causal effects via their respective weighted averages of the form in Equation (13.12), in general neither estimator is a consistent estimator of the average causal effect.5
Example: The cardiac catheterization study. Sections 12.5 and 13.4 discuss McClellan, McNeil, and Newhouse’s (1991) study of the effect on mortality of cardiac catheterization of heart attack patients. The authors used instrumental variables regression, with the relative distance to a cardiac catheterization hospi- tal as the instrumental variable. Based on their TSLS estimates, they found that cardiac catheterization had little or no effect on health outcomes. This result is surprising: Medical procedures such as cardiac catheterization are subjected to rigorous clinical trials prior to approval for widespread use. Moreover, cardiac catheterization allows surgeons to perform medical interventions that would have
5There are several good (but advanced) discussions of the effect of population heterogeneity on program evaluation estimators. They include the survey by Heckman, LaLonde, and Smith (1999, Section 7) and James Heckman’s lecture delivered when he received the Nobel Prize in economics (Heckman, 2001, Section 7). The latter reference and Angrist, Graddy, and Imbens (2000) provide detailed discussion of the random effects model (which treats b1i as varying across individuals) and provide more gen- eral versions of the result in Equation (13.12). The concept of the local average treatment effect was introduced by Angrist and Imbens (1994), who showed that in general it does not equal the average treatment effect.
required major surgery a decade earlier, making these interventions safer and, presumably, better for long-term patient health. How could this econometric study fail to find beneficial effects of cardiac catheterization?
One possible answer is that there is heterogeneity in the treatment effect of cardiac catheterization. For some patients, this procedure is an effective inter- vention, but for others, perhaps those who are healthier, it is less effective or, given the risks involved with any surgery, perhaps on the whole ineffective. Thus the average causal effect in the population of heart attack patients could be, and presumably is, positive. The IV estimator, however, measures a marginal effect, not an average effect, where the marginal effect is the effect of the procedure on those patients for whom distance to the hospital is an important factor in whether they receive treatment. But those patients could be just the relatively healthy patients for whom, on the margin, cardiac catheterization is a relatively ineffective procedure. If so, McClellan, McNeil, and Newhouse’s (1991) TSLS estimator measures the effect of the procedure for the marginal patient (for whom it is relatively ineffective), not for the average patient (for whom it might be effective).
13.7
Conclusion
In Chapter 1, we defined the causal effect in terms of the expected outcome of an ideal randomized controlled experiment. If a randomized controlled experiment is available or can be performed, it can provide compelling evidence on the causal effect under study, although even randomized controlled experiments are subject to potentially important threats to internal and external validity.
Despite their advantages, randomized controlled experiments in economics face considerable hurdles, including ethical concerns and cost. The insights of experimental methods can, however, be applied to quasi-experiments, in which special circumstances make it seem “as if” randomization has occurred. In quasi- experiments, the causal effect can be estimated using a differences-in-differences estimator, possibly augmented with additional regressors; if the “as if” randomiza- tion only partly influences the treatment, then instrumental variables regression can be used instead. An important advantage of quasi-experiments is that the source of the “as if” randomness in the data is usually transparent and thus can be evaluated in a concrete way. An important threat confronting quasi-experiments is that sometimes the “as if” randomization is not really random, so the treatment (or the instrumental variable) is correlated with omitted variables and the result- ing estimator of the causal effect is biased.
13.7 Conclusion 509
510 CHAPTER 13
Experiments and Quasi-Experiments
Quasi-experiments provide a bridge between observational data sets and true randomized controlled experiments. The econometric methods used in this chap- ter for analyzing quasi-experiments are those developed in different contexts, in earlier chapters: OLS, panel data estimation methods, and instrumental variables regression. What differentiates quasi-experiments from the applications exam- ined in Part II and the earlier chapters in Part III are the way in which these methods are interpreted and the data sets to which they are applied. Quasi- experiments provide econometricians with a way to think about how to acquire new data sets, how to think of instrumental variables, and how to evaluate the plausibility of the exogeneity assumptions that underlie OLS and instrumental variables estimation.6
Summary
1. The average causal effect in the population under study is the expected dif- ference in the average outcomes for the treatment and control groups in an ideal randomized controlled experiment. Actual experiments with human subjects deviate from an ideal experiment for various practical reasons, including the failure of people to comply with the experimental protocol.
2. If the actual treatment level Xi is random, then the treatment effect can be estimated by regressing the outcome on the treatment. If the assigned treatment Zi is random but the actual treatment Xi is partly determined by individual choice, then the causal effect can be estimated by instrumental variables regression, using Zi as an instrument. If the treatment (or assigned treatment) is random conditional on some variables W, those control vari- ables need to be included in the regressions.
3. In a quasi-experiment, variations in laws or circumstances or accidents of nature are treated “as if” they induce random assignment to treatment and control groups. If the actual treatment is “as if” random, then the causal effect can be estimated by regression (possibly with additional pretreatment characteristics as regressors); if the assigned treatment is “as if” random, then the causal effect can be estimated by instrumental variables regression.
6Shadish, Cook, and Campbell (2002) provide a comprehensive treatment of experiments and quasi- experiments in the social sciences and in psychology. An important line of research in development economics focuses on experimental evaluations of health and education programs in developing coun- tries. For examples, see Kremer, Miguel, and Thornton (2009) and the website of MIT’s Poverty Action Laboratory (http://www.povertyactionlab.org). Deaton (2010) provides a thoughtful critique of this research.
4. Regression discontinuity estimators are based on quasi-experiments in which treatment depends on whether an observable variable crosses a threshold value.
5. A key threat to the internal validity of a quasi-experimental study is whether the “as if” randomization actually results in exogeneity. Because of behav- ioral responses, the regression error may change in response to the treat- ment induced by the quasi-experiment, so the treatment is not exogenous.
6. When the treatment effect varies from one individual to the next, the OLS estimator is a consistent estimator of the average causal effect if the actual treatment is randomly assigned or “as if” randomly assigned. However, the instrumental variables estimator is a weighted average of the individual treatment effects, where the individuals for whom the instrument is most influential receive the greatest weight.
Key Terms
program evaluation (475)
potential outcome (476)
average causal effect (477)
average treatment effect (477) differences estimator (478) differences estimator with additional
regressors (478) randomization based on covariates
(479)
test for random receipt of treatment
(480)
partial compliance (480)
instrumental variables estimation of the treatment effect (481)
attrition (481) Hawthorne effect (482) quasi-experiment (493) natural experiment (493) differences-in-differences
estimator (496) differences-in-differences estimator
with additional regressors (498) repeated cross-sectional data (499) regression discontinuity (500)
local average treatment effect (507)
Key Terms 511
MyEconLab Can Help You Get a Better Grade
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512 CHAPTER 13
Experiments and Quasi-Experiments
Review the Concepts
13.1 A researcher studying the effects of a new fertilizer on crop yields plans to carry out an experiment in which different amounts of the fertilizer are applied to 100 different 1-acre parcels of land. There will be four treatment levels. Treatment level 1 is no fertilizer, treatment level 2 is 50% of the manufacturer’s recommended amount of fertilizer, treatment level 3 is 100%, and treatment level 4 is 150%. The researcher plans to apply treatment level 1 to the first 25 parcels of land, treatment level 2 to the second 25 parcels, and so forth. Can you suggest a better way to assign treatment levels? Why is your proposal better than the researcher’s method?
13.2 A clinical trial is carried out for a new cholesterol-lowering drug. The drug is given to 500 patients, and a placebo is given to another 500 patients, using random assignment of the patients. How would you estimate the treatment effect of the drug? Suppose that you had data on the weight, age, and gender of each patient. Could you use these data to improve your estimate? Explain. Suppose that you had data on the cholesterol levels of each patient before he or she entered the experiment. Could you use these data to improve your estimate? Explain.
13.3 Researchers studying the STAR data report anecdotal evidence that school principals were pressured by some parents to place their children in the small classes. Suppose that some principals succumbed to this pressure and transferred some children into the small classes. How would such transfers compromise the internal validity of the study? Suppose that you had data on the original random assignment of each student before the principal’s intervention. How could you use this information to restore the internal validity of the study?
13.4 Explain whether experimental effects (like the Hawthorne effect) might be important in each of the experiments in the previous three questions.
13.5 Consider the quasi-experiment described in Section 13.4 involving the draft lottery, military service, and civilian earnings. Explain why there might be heterogeneous effects of military service on civilian earnings; that is, explain why b1i in Equation (13.9) depends on i. Explain why there might be heterogeneous effects of the lottery outcome on the probability of mili- tary service; that is, explain why p1i in Equation (13.11) depends on i. If there are heterogeneous responses of the sort you described, what behav- ioral parameter is being estimated by the TSLS estimator?
Exercises
13.1 Using the results in Table 13.1, calculate the following for each grade: an estimate of the small class treatment effect, relative to the regular class; its standard error; and its 95% confidence interval. (For this exercise, ignore the results for regular classes with aides.)
13.2 For the following calculations, use the results in column (4) of Table 13.2. Consider two classrooms, A and B, with identical values of the regressors in column (4) of Table 13.2, except that:
a. Classroom A is a “small class,” and classroom B is a “regular class.” Construct a 95% confidence interval for the expected difference in average test scores.
b. Classroom A has a teacher with 5 years of experience, and classroom B has a teacher with 10 years of experience. Construct a 95% confi- dence interval for the expected difference in average test scores.
c. Classroom A is a small class with a teacher with 5 years of experi- ence, and classroom B is a regular class with a teacher with 10 years of experience. Construct a 95% confidence interval for the expected difference in average test scores. (Hint: In STAR, the teachers were randomly assigned to the different types of classrooms.)
d. Why is the intercept missing from column (4)?
13.3 Suppose that, in a randomized controlled experiment of the effect of an SAT preparatory course on SAT scores, the following results are reported:
Exercises 513
Average SAT score (X)
Standard deviation of SAT score (SX) Number of men
Number of women
Treatment Group
1241 93.2 55 45
Control Group
1201 97.1 45 55
a. b.
Estimate the average treatment effect on test scores. Is there evidence of nonrandom assignment? Explain.
13.4 Read the box “What Is the Effect on Employment of the Minimum Wage?” in Section 13.4. Suppose, for concreteness, that Card and Krueger collected their data in 1991 (before the change in the New Jersey minimum
514 CHAPTER 13
Experiments and Quasi-Experiments
wage) and in 1993 (after the change in the New Jersey minimum wage). Consider Equation (13.7) with the W regressors excluded.
a. What are the values of Xit, Gi, and Dt for: i. A New Jersey restaurant in 1991?
ii. A New Jersey restaurant in 1993?
iii. A Pennsylvania restaurant in 1991?
iv. A Pennsylvania restaurant in 1993?
b. In terms of the coefficients b0, b1, b2, and b3, what is the expected number of employees in:
i. A New Jersey restaurant in 1991? ii. A New Jersey restaurant in 1993?
iii. A Pennsylvania restaurant in 1991?
iv. A Pennsylvania restaurant in 1993?
c. In terms of the coefficients b0, b1, b2, and b3, what is the average causal effect of the minimum wage on employment?
d. Explain why Card and Krueger used a differences-in-differences estimator of the causal effect instead of the “New Jersey after—New Jersey before” differences estimator or the “1993 New Jersey—1993 Pennsylvania” differences estimator.
13.5 Consider a study to evaluate the effect on college student grades of dorm room Internet connections. In a large dorm, half the rooms are randomly wired for high-speed Internet connections (the treatment group), and final course grades are collected for all residents. Which of the following pose threats to internal validity, and why?
a. Midway through the year, all the male athletes move into a fraternity and drop out of the study. (Their final grades are not observed.)
b. Engineering students assigned to the control group put together a local area network so that they can share a private wireless Internet connection that they pay for jointly.
c. The art majors in the treatment group never learn how to access their Internet accounts.
d. The economics majors in the treatment group provide access to their Internet connection to those in the control group, for a fee.
13.6 Suppose that there are panel data for T = 2 time periods for a random- ized controlled experiment, where the first observation (t = 1) is taken
before the experiment and the second observation (t = 2) is for the post- treatment period. Suppose that the treatment is binary; that is, suppose that Xit = 1 if the ith individual is in the treatment group and t = 2, and Xit = 0 otherwise. Further suppose that the treatment effect can be mod- eled using the specification
Yit = ai + b1Xit + uit,
where ai are individual-specific effects [see Equation (13.11)] with a mean
of zero and a variance of s2a and uit is an error term, where uit is homo-
skedastic, cov(u , u ) = 0, and cov (u , a ) = 0 for all i. Let bndifferences i1 i2 it i 1
denote the differences estimator—that is, the OLS estimator in a regres- sion of Y on X with an intercept—and let bn diffs – in – diffs denote the
i2 i2 1 differences-in-differences estimator—that is, the estimator of b1 based
on the OLS regression of ∆Yi = Yi2 – Yi1 against ∆Xi = Xi2 – Xi1 and an intercept.
a. Show that nvar(bndifferences) ¡ (s2 + s2)>var(X ). (Hint: Use the 1 uai2
homoskedasticity-only formulas for the variance of the OLS estima- tor in Appendix 5.1.)
b. Show that nvar (bndiffs – in – diffs) ¡ 2s2 > var(X ). (Hint: Note that 1 ui2
Xi2 – Xi1 = Xi2. Why?)
c. Based on your answers to (a) and (b), when would you prefer the differences-in-differences estimator over the differences estimator, based purely on efficiency considerations?
13.7 Suppose that you have panel data from an experiment with T = 2 periods (so t = 1, 2). Consider the panel data regression model with fixed indi- vidual and time effects and individual characteristics Wi that do not change over time, such as gender. Let the treatment be binary, so that Xit = 1 for t = 2 for the individuals in the treatment group and let Xit = 0 otherwise. Consider the population regression model
Yit =ai +b1Xit +b2(Dt *Wi)+b0Dt +vit,
where ai are individual fixed effects, Dt is the binary variable that equals 1 if t = 2 and equals 0 if t = 1, Dt * Wi is the product of Dt and Wi , and the a’s and b’s are unknown coefficients. Let ∆Yi = Yi2 – Yi1. Derive Equation (13.6) (in the case of a single W regressor, so r = 1) from this population regression model.
Exercises 515
516 CHAPTER 13
Experiments and Quasi-Experiments
13.8 Suppose that you have the same data as in Exercise 13.7 (panel data with two periods, n observations), but ignore the W regressor. Consider the alternative regression model
Yit =b0 +b1Xit +b2Gi +b3Dt +uit,
where Gi = 1 if the individual is in the treatment group and Gi = 0 if the individual is in the control group. Show that the OLS estimator of b1 is the differences-in-differences estimator in Equation (13.4). (Hint: See Section 8.3.)
13.9 Derive the final equality in Equation (13.10). (Hint: Use the definition of the covariance and that, because the actual treatment Xi is random, b1i and Xi are independently distributed.)
13.10 Consider the regression model with heterogeneous regression coefficients Yi =b0i +b1iXi +vi,
where (vi, Xi, b0i, b1i) are i.i.d. random variables with b0 = E(b0i) and b1 = E(b1i).
a. Show that the model can be written as Yi = b0 + b1Xi + ui, where ui =(b0i -b0)+(b1i -b1)Xi +vi.
b. Suppose that E3b0i Xi4 = b0, E3b1i Xi4 = b1, and E3vi Xi4 = 0 Show that E3ui Xi4 = 0.
c. Show that Assumption #1 and Assumption #2 of Key Concept 4.3 are satisfied.
d. Suppose that outliers are rare so that (ui, Xi) have finite fourth moments. Is it appropriate to use OLS and the methods of Chapters 4 and 5 to estimate and carry out inference about the average values of b0i and b1i?
e. Suppose that b1i and Xi are positively correlated so that observations with larger-than-average values of Xi tend to have larger-than-average values of b1i. Are the assumptions in Key Concept 4.3 satisfied? If not, which assumption(s) is (are) violated? Is it appropriate to use OLS and the methods of Chapters 4 and 5 to estimate and carry out inference about the average value of b0i and b1i?
13.11 In Chapter 12, state-level panel data were used to estimate the price elas- ticity of demand for cigarettes, using the state sales tax as an instrumental variable. Consider in particular regression (1) in Table 12.1. In this case, in your judgment does the local average treatment effect differ from the average treatment effect? Explain.
13.12 ConsiderthepotentialoutcomesframeworkfromAppendix13.3.Suppose that Xi is a binary treatment that is independent of the potential outcomes Yi(1) and Yi(0). Let TEi = Yi (1) – Yi (0) denote the treatment effect for individual i.
a. Can you consistently estimate E [Yi(1)] and E[Yi(0)]? If yes, explain how; if not, explain why not.
b. Can you consistently estimate E (TEi)? If yes, explain how; if not, explain why not.
c. Can you consistently estimate var[Yi(1)] and var[Yi (0)]? If yes, explain how; if not, explain why not.
d. Can you consistently estimate var(TEi)? If yes, explain how; if not, explain why not.
e. Do you think you can consistently estimate the median treatment effect in the population? Explain.
Empirical Exercises
(Only one empirical exercise for this chapter is given in the text, but you can find more on the text Web site http://www.pearsonhighered.com/stock_watson/.)
E13.1 A prospective employer receives two resumes: a resume from a white job applicant and a similar resume from an African American applicant. Is the employer more likely to call back the white applicant to arrange an interview? Marianne Bertrand and Sendhil Mullainathan carried out a randomized controlled experiment to answer this question. Because race is not typically included on a resume, they differentiated resumes on the basis of “white-sounding names” (such as Emily Walsh or Gregory Baker) and “African American–sounding names” (such as Lakisha Washington or Jamal Jones). A large collection of fictitious resumes was created, and the presupposed “race” (based on the “sound” of the name) was ran- domly assigned to each resume. These resumes were sent to prospective employers to see which resumes generated a phone call (a “call back”) from the prospective employer. Data from the experiment and a detailed data description are on the textbook website, http://www.pearsonhighered .com/stock_watson/, in the files Names and Names_Description.7
7These data were provided by Professor Marianne Bertrand of the University of Chicago and were used in her paper with Sendhil Mullainathan, “Are Emily and Greg More Employable Than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination,” American Economic Review, 2004, 94(4): 991–1013.
Empirical Exercises 517
518 CHAPTER 13
Experiments and Quasi-Experiments
APPENDIX
13.1
a. Define the “call-back rate” as the fraction of resumes that generate
a phone call from the prospective employer. What was the call-back rate for whites? For African Americans? Construct a 95% confidence interval for the difference in the call-back rates. Is the difference sta- tistically significant? Is it large in a real-world sense?
b. Is the African American/white call-back rate differential different for men than for women?
c. What is the difference in call-back rates for high-quality versus low- quality resumes? What is the high-quality/low-quality difference for white applicants? For African American applicants? Is there a signifi- cant difference in this high-quality/low-quality difference for whites versus African Americans?
d. The authors of the study claim that race was assigned randomly to the resumes. Is there any evidence of nonrandom assignment?
The Project STAR Data Set
The Project STAR public access data set contains data on test scores, treatment groups, and student and teacher characteristics for the 4 years of the experiment, from academic year 1985–1986 to academic year 1988–1989. The test score data analyzed in this chapter are the sum of the scores on the math and reading portions of the Stanford Achievement Test. The binary variable “Boy” in Table 13.2 indicates whether the student is a boy ( = 1) or girl (= 0); the binary variables “Black” and “Race other than black or white” indicate the student’s race. The binary variable “Free lunch eligible” indicates whether the student is eligible for a free lunch during that school year. The teacher’s years of experience are the total years of experience of the teacher whom the student had in the grade for which the test data apply. The data set also indicates which school the student attended in a given year, making it possible to construct binary school-specific indicator variables.
IV Estimation When the Causal Effect Varies Across Individuals
This appendix derives the probability limit of the TSLS estimator in Equation (13.12) when there is population heterogeneity in the treatment effect and in the influence of the instrument on the receipt of treatment. Specifically, it is assumed that the IV regression
APPENDIX
13.2
IV Estimation When the Causal Effect Varies Across Individuals 519
assumptions in Key Concept 12.4 hold, except that Equations (13.9) and (13.11) hold with heterogeneous effects. Further assume that p0i, p1i, b0i, and b1i are distributed indepen- dently of ui, vi, and Zi; that E(ui Zi) = E(vi Zi) = 0; and that E(p1i) ≠ 0.
Because (Xi, Yi, Zi), i = 1, . . . , n are i.i.d. with four moments, the law of large num- bers in Key Concept 2.6 applies and
bnTSLS = sZY ¡p sZY. (13.13) 1 sZX sZX
(See Appendix 3.3 and Exercise 17.2.) The task thus is to obtain expressions for sZY and sZX in terms of the moments of p1i and b1i. Now sZX = E3(Zi – mZ)(Xi – mX)4 = E3(Zi – mZ)Xi4. Substituting Equation (13.11) into this expression for sZX yields
sZX = E3(Zi – mZ)(p0i + p1iZi + vi)4
= E(p0i) * 0 + E3p1iZi(Zi – mZ)4 + cov(Zi, vi)
= s2ZE(p1i), (13.14)
wherethesecondequalityfollowsbecausecov(Zi,vi) = 0[whichfollowsfromtheassumption E(vi Zi) = 0; see Equation (2.27)], because E[(Zi – mZ)p0i] = E{E3(Zi – mZ)p0i Zi4} = E3(Zi – mZ)E(p0iZi)4 = E(Zi – mz) * E(p0i)(whichusesthelawofiteratedexpectations and the assumption that p0i is independent of Zi), and because E3p1iZi(Zi – mZ)4= E{E[p1iZi(Zi – mZ)Zi)]} = E(p1i)E[Zi(Zi – mZ)] = s2ZE(p1i) (which uses the law of iterated expectations and the assumption that p1i is independent of Zi).
Next consider sZY. Substituting Equation (13.11) into Equation (13.9) yields Yi = b0i + b1i(p0i + p1iZi + vi) + ui, so
sZY = E3(Zi – mZ)Yi4
= E3(Zi – mZ)(b0i + b1i p0i + b1i p1iZi + b1ivi + ui)4 = E(b0i) * 0 + cov(Zi,b1i p0i) + E3b1i p1iZi(Zi – mZ)4
+ E3b1ivi(Zi – mZ)4 + cov(Zi, ui). (13.15)
Because (b1ip0i) and Zi are independently distributed, cov(Zi, b1i p0i) = 0; because b1i is distributed independently of vi and Zi and E(vi Zi) = 0, E[b1ivi(Zi – mZ)] = E(b1i) * E[vi(Zi – mZ)] = 0; because E(ui Zi) = 0, cov(Zi, ui) = 0 ; and because b1i and p1i are distributed independently of Zi, E3b1i p1iZi(Zi – mZ)4 = s2ZE(b1i p1i). Thus the final expression in Equation (13.15) yields
sZY = s2ZE(b1i p1i). (13.16) Substituting Equations (13.14) and (13.16) into Equation (13.13) yields
bnTSLS ¡p s2E(b p )>s2E(p ) = E(b p )>E(p ), which is the result stated in 1 Z1i1iZ1i1i1i1i
Equation (13.12).
520 CHAPTER 13
Experiments and Quasi-Experiments
APPENDIX
13.3
The Potential Outcomes Framework for Analyzing Data from Experiments
This appendix provides a mathematical treatment of the potential outcomes framework discussed in Section 13.1. The potential outcomes framework, combined with a constant treatment effect, implies the regression model in Equation (13.1). If assignment is random, conditional on covariates, the potential outcomes framework leads to Equation (13.2) and conditional mean independence. We consider a binary treatment with Xi = 1 indicating receipt of treatment.
Let Yi(1) denote individual i’s potential outcome if treatment is received and let Yi(0) denote the potential outcome if treatment is not received, so individual i’s treatment effect is Yi(1) – Yi(0). The average treatment effect in the population is E3Yi(1) – Yi(0)4. Because the individual is either treated or not, only one of the two potential outcomes is observed. The observed outcome, Yi, is related to the potential outcomes by
Yi = Yi(1)Xi + Yi(0)(1 – Xi). (13.17)
If some individuals receive the treatment and some do not, the expected difference in observed outcomes between the two groups is E(YiXi = 1) – E(YiXi = 0) = E[Yi(1) Xi = 1] – E[Yi(0) Xi = 0]. This is true no matter how treatment is determined and simply says that the expected difference is the mean treatment outcome for the treated minus the mean no-treatment outcome for the untreated. If in addition the individuals are randomly assigned to the treatment and control groups, then Xi is distributed indepen- dently of all personal attributes and in particular is independent of 3Yi(1), Yi(0)4. With random assignment, the mean difference between the treatment and control groups is
E(YiXi = 1) – E(Yi Xi = 0) = E[Yi(1)Xi = 1] – E[Yi(0)Xi = 0] = E[Yi(1) – Yi(0)], (13.18)
where the second equality uses the fact that 3Yi(1), Yi(0)4 are independent of Xi by ran- dom assignment and the linearity of expectations [Equation (2.28)]. Thus if Xi is ran- domly assigned, the mean difference in the experimental outcomes between the two groups is the average treatment effect in the population from which the subjects were drawn.
The potential outcome framework translates directly into the regression notation used throughout this book. Let ui = Yi(0) – E3Yi(0)4 and denote E3Yi(0)4 = b0. Also denote
The Potential Outcomes Framework for Analyzing Data from Experiments 521 Yi(1) – Yi(0) = b1i, so that b1i is the treatment effect for individual i. Starting with Equa-
tion (13.17), we have
Yi = Yi(1)Xi + Yi(0)(1 – Xi)
= Yi(0) + 3Yi(1) – Yi(0)4Xi
= E3Yi(0)4 + 3Yi(1) – Yi(0)4Xi + 5Yi(0) – E3Yi(0)46
= b0 + b1iXi + ui. (13.19)
Thus, starting with the relationship between observed and potential outcomes and simply changing notation, we obtain the random coefficients regression model in Equation (13.9). [Equation (13.9) has b0 varying across individuals, but that is equivalent to Equation (13.19) because ui also varies across individuals.] If Xi is randomly assigned, then Xi is independent of 3Yi(1), Yi(0)4 and thus is independent of b1i and ui. If the treatment effect is constant, then b1i = b1 and Equation (13.9) becomes Equation (13.1).
As discussed in Appendix 7.2 and Sections 13.1 and 13.3, in some designs Xi is ran- domly assigned based on the value of a third variable, Wi. If Wi and the potential outcomes are not independent, then in general the mean difference between groups does not equal the average treatment effect—that is, Equation (13.18) does not hold. However, random assignment of Xi given Wi implies that, conditional on Wi, Xi and 3Yi(1), Yi(0)4 are inde- pendent. This condition—that Xi and 3Yi(1),Yi(0)4 are independent, conditional on Wi—is often called unconfoundedness.
If the treatment effect does not vary across individuals and if E(Y Xi,Wi) is linear, then unconfoundedness implies conditional mean independence of the regression error in Equation (13.2). To see this, let Yi(0) = b0 + gWi + ui, where g is the causal effect (if any) on Yi(0) of Wi, and let Yi (1) – Yi (0) = b1 (constant treatment effect). Then the logic leading to Equation (13.19) yields Yi = b0 + b1Xi + gWi + ui, which is Equation (13.2). Now E(uiXi, Wi) = E3Yi(0) – b0 – gWiXi, Wi4 = E3Yi(0) – b0 – gWi Wi4 = E(uiWi),where the second equality follows from unconfoundedness (if 3Yi(1),Yi(0)4 is independent of Xi given Wi, then E3Yi(0)Xi,Wi4 = E3Yi(0)Wi4). Thus unconfoundedness implies that E(ui Xi, Wi) = E(ui Wi) in Equation (13.2). The reasoning of Appendix 7.2 implies that, if E(ui Wi) is linear in Wi, then the OLS estimator of b1 in Equation (13.2) is unbiased, although in general the OLS estimator of g is biased because E(ui Wi) ≠ 0.
Chapter
Introduction to Time Series Regression and Forecasting
Time series data—data collected for a single entity at multiple points in time— can be used to answer quantitative questions for which cross-sectional data are inadequate. One such question is, what is the causal effect on a variable of interest, Y, of a change in another variable, X, over time? In other words, what is the dynamic causal effect on Y of a change in X ? For example, what is the effect on traffic fatalities of a law requiring passengers to wear seatbelts, both initially and subsequently, as drivers adjust to the law? Another such question is, what is your best forecast of the value of some variable at a future date? For example, what is your best forecast of next month’s unemployment rate, interest rates, or stock prices? Both of these questions—one about dynamic causal effects, the other about economic forecasting— can be answered using time series data. But time series data pose special challenges, and overcoming those challenges requires some new techniques.
This chapter and Chapters 15 and 16 introduce techniques for econometric analysis of time series data and apply these techniques to the problems of forecast- ing and estimating dynamic causal effects. This chapter introduces the basic con- cepts and tools of regression with time series data and applies them to economic forecasting. Chapter 15 applies the concepts and tools developed in this chapter to the problem of estimating dynamic causal effects using time series data. Chapter 16 takes up some more advanced topics in time series analysis, including forecasting multiple time series and modeling changes in volatility over time.
The empirical problem studied in this chapter is forecasting the growth rate of U.S. Gross Domestic Product (GDP)—that is, the percentage increase in the value of goods and services produced in the U.S. economy. While in a sense forecasting is just an application of regression analysis, forecasting is quite different from the esti- mation of causal effects, the focus of this book until now. As discussed in Section 14.1, models that are useful for forecasting need not have a causal interpretation: If you see pedestrians carrying umbrellas, you might forecast rain, even though carry- ing an umbrella does not cause rain. Section 14.2 introduces some basic concepts of time series analysis and presents some examples of economic time series data. Section 14.3 presents time series regression models in which the regressors are past values of the dependent variable; these “autoregressive” models use the history of GDP to forecast its future. Often, forecasts based on autoregressions can be
14
522
14.1 Using Regression Models for Forecasting 523
improved by adding additional predictor variables and their past values, or “lags,” as regressors, and these so-called autoregressive distributed lag models are introduced in Section 14.4. For example, we find that GDP forecasts made using lagged values of the term spread, the difference between the interest rate on long-term and short- term bonds, improve upon the autoregressive GDP forecasts. A practical issue is deciding how many past values to include in autoregressions and autoregressive distributed lag models, and Section 14.5 describes methods for making this decision.
The assumption that the future will be like the past is an important one in time series regression, sufficiently so that it is given its own name: “stationarity.” Time series variables can fail to be stationary in various ways, but two are especially rele- vant for regression analysis of economic time series data: (1) The series can have persistent, long-run movements—that is, the series can have trends; and (2) the population regression can be unstable over time—that is, the population regres- sion can have breaks. These departures from stationarity jeopardize forecasts and inferences based on time series regression. Fortunately, there are statistical proce- dures for detecting trends and breaks and, once detected, for adjusting the model specification. These procedures are presented in Sections 14.6 and 14.7.
14.1 Using Regression Models for Forecasting
The empirical application of Chapters 4 through 9 focused on estimating the causal effect on test scores of the student–teacher ratio. The simplest regression model relates in Chapter 4 related test scores to the student–teacher ratio (STR):
TestScore = 989.9 – 2.28 * STR. (14.1)
As was discussed in Chapter 6, a school superintendent, contemplating hiring more teachers to reduce class sizes, would not consider this equation to be very helpful. The estimated slope coefficient in Equation (14.1) fails to provide a useful estimate of the causal effect on test scores of the student–teacher ratio because of probable omitted variable bias arising from the omission of school and student characteristics that are determinants of test scores and that are correlated with the student–teacher ratio.
In contrast, as discussed in Chapter 9, a parent who is considering moving to a school district might find Equation (14.1) more helpful. Even though the coef- ficient does not have a causal interpretation, the regression could help the parent forecast test scores in a district for which they are not publicly available. More generally, a regression model can be useful for forecasting even if none of its
524 ChapTeR 14 Introduction to Time Series Regression and Forecasting
coefficients has a causal interpretation. From the perspective of forecasting, what is important is that the model provides as accurate a forecast as possible. Although there is no such thing as a perfect forecast, regression models can nevertheless provide forecasts that are accurate and reliable.
The applications in this chapter differ from the test score/class size prediction problem because this chapter focuses on using time series data to forecast future events. For example, the parent actually would be interested in test scores next year, after his or her child had enrolled in a school. Of course, those tests have not yet been given, so the parent must forecast the scores using currently available information. If test scores are available for past years, then a good starting point is to use data on current and past test scores to forecast future test scores. This reasoning leads directly to the autoregressive models presented in Section 14.3, in which past values of a variable are used in a linear regression to forecast future values of the series. The next step, which is taken in Section 14.4, is to extend these models to include additional predictor variables such as data on class size. Like Equation (14.1), such a regression model can produce accurate and reliable forecasts even if its coefficients have no causal interpretation. In Chapter 15, we return to problems like that faced by the school superintendent and discuss the estimation of causal effects using time series variables.
14.2 Introduction to Time Series Data and Serial Correlation
This section introduces some basic concepts and terminology that arise in time series econometrics. A good place to start any analysis of time series data is by plotting the data, so that is where we begin.
Real GDP in the United States
Gross Domestic Product (GDP) measures the value of goods and services pro- duced in an economy over a given time period. Figure 14.1 a plots values of “real” GDP per year in the United States from 1960 through 2012, where “real” indicates that the values have been adjusted for inflation. The values of GDP are expressed in $1996, which means that the price level is held fixed at its 1996 value. Because U.S. GDP grows at approximately an exponential rate, Figure 14.1 a plots GDP on a logarithmic scale. GDP increased dramatically over a recent 52-year period, from approximately $3 trillion in 1960 to over $15 trillion in 2012. Measured on a logarithmic scale, this five-fold increase corresponds to an increase of 1.6 log points.
14.2 Introduction to Time Series Data and Serial Correlation 525 Figure 14.1 The Logarithm and the Growth Rate of Real GDp in the United States, 1960–2012
Logarithm
9.75
9.50
9.25
9.00
8.75
8.50
8.25
8.00
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
(a) US GDP ($1996, Billions) Percent at an annual rate
20 15 10
5
0
–5
–10
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
(b) Growth Rate in US GDP
GDP increased from $3 trillion per year in 1960 to over $15 trillion per year in 2012, when measured in inflation- adjusted $1996. This five-fold increase corresponds to an increase of 1.6 log points. The growth rate of GDP was not constant, and it varied considerably from quarter to quarter.
The rate of growth was not constant, however, and the figure shows declines in GDP during the recessions of 1960–1961, 1970, 1974–1975, 1980, 1981–1982, 1990–1991, 2001, and 2007–2009, episodes denoted by shading in Figure 14.1.
Lags, First Differences, Logarithms, and Growth Rates
The observation on the time series variable Y made at date t is denoted Yt, and the total number of observations is denoted T. The interval between observations— that is, the period of time between observation t and observation t + 1—is some
526 ChapTeR 14 Introduction to Time Series Regression and Forecasting
unit of time such as weeks, months, quarters (3-month units), or years. For example, the GDP data studied in this chapter are quarterly, so the unit of time (a “period”) is a quarter of a year.
Special terminology and notation are used to indicate future and past values of Y. The value of Y in the previous period is called its first lagged value or, more simply, its first lag, and is denoted Yt – 1. Its jth lagged value (or simply its jth lag) is its value j periods ago, which is Yt – j. Similarly, Yt + 1 denotes the value of Y one period into the future.
The change in the value of Y between period t – 1 and period t is Yt – Yt-1; this change is called the first difference in the variable Yt. In time series data, “∆” is used to represent the first difference, so ∆Yt = Yt – Yt – 1.
Economic time series are often analyzed after computing their logarithms or the changes in their logarithms. One reason for this is that many economic series exhibit growth that is approximately exponential; that is, over the long run, the series tends to grow by a certain percentage per year on average. This implies that the logarithm of the series grows approximately linearly, and is why Figure 14.1a plots the logarithm of U.S. GDP. Another reason is that the standard deviation of many economic time series is approximately proportional to its level; that is, the standard deviation is well expressed as a percentage of the level of the series. This implies that the standard deviation of the logarithm of the series is approxi- mately constant. In either case, it is useful to transform the series so that changes in the transformed series are proportional (or percentage) changes in the original series, and this is achieved by taking the logarithm of the series.1
Lags, first differences, and growth rates are summarized in Key Concept 14.1.
Lags, changes, and percentage changes are illustrated using the U.S. GDP data in Table 14.1. The first column shows the date, or period, where the first quarter of 2012 is denoted 2012:Q1, the second quarter of 2012 is denoted 2012:Q2, and so forth. The second column shows the value of the GDP in that quarter, the third column shows the logarithm of GDP, and the fourth column shows the growth rate of GDP (in percentage points at an annual rate). For example, from the first quar- ter to the second quarter of 2012, GDP increased from $15,382 to $15,428 billion,
variable—that is, ln(X + a) – ln(X) ≅ a>X, where the approximation works best when a/X is small
1The change of the logarithm of a variable is approximately equal to the proportional change of that
[see Equation (8.16) and the surrounding discussion]. Now, replace X with Y and a with∆Y and t-1 t
ods t – 1 and t is approximately ln(Y ) – ln(Y ) = ln(Y + ∆Y ) – ln(Y ) ≅ ∆Y >Y . The tt-1t-1tt-1tt-1
note that Y = Y + ∆Y . This means that the proportional change in the series Y between peri- tt-1t t
expression ln(Y ) – ln(Y ) is the first difference of ln(Y ), that is ∆ln(Y ). Thus ∆ln(Y ) ≅ ∆Y >Y . tt-1ttttt-1
The percentage change is 100 times the fractional change, so the percentage change in the series Yt is approximately 100∆ln(Yt).
14.2 Introduction to Time Series Data and Serial Correlation 527
Lags, First Differences, Logarithms, and Growth Rates
Key ConCept
14.1
• The first lag of a time series Yt is Yt-1; its jth lag is Yt-j.
• The first difference of a series, ∆Yt, is its change between periods t – 1 and t,
that is ∆Yt = Yt – Yt – 1.
• The first difference of the logarithm of Y is ∆ln1Y 2 = ln1Y 2 – ln1Y 2.
approximately 100∆ln1Y 2, where the approximation is most accurate when t
the percentage change is small.
• The percentage change of a time series Yt between periods t – 1 and t is
tttt-1
which is a percentage increase of 100 * (15428 – 15382)>15382 = 0.30%. This is the percentage increase from one quarter to the next. It is conventional to report rates of growth in macroeconomic time series on an annual basis, which is the percentage increase in GDP that would occur over a year if the series were to continue to increase at the same rate. Because there are four quarters in a year, the annualized rate of GDP growth in 2012:Q2 is 0.30 * 4 = 1.20, or 1.20%.
This percentage change can also be computed using the differences-of- logarithms approximation in Key Concept 14.1. The difference in the logarithm ofGDPfrom2012:Q1to2012:Q2isln(15428) – ln(15382) = 0.0030,yieldingthe approximate quarterly percentage difference 100 * 0.0030 = 0.30%. On an annualized basis, this is 0.30 * 4 = 1.20, or 1.20%, the same (to two decimal
tABLe 14.1
Quarter
2012:Q1
2012:Q2
2012:Q3
2012:Q4
2013:Q1
GDP in the United States in 2012 and the First Quarter of 2013
U.S. GDp (billions of $1996), GDPt
15382
15428
15534
15540
15584
Logarithm of GDp, ln(GDPt)
9.641
9.644
9.651
9.651
9.654
Growth Rate of GDp at an Annual
First Lag, GDPGRt−1
4.75
3.64
1.20
2.75
0.15
Rate, GDPGRt
= 400
3.64
1.20
2.75
0.15
1.14
: ∆ln (GDPt)
Note: The quarterly rate of GDP growth is the first difference of the logarithm. This is converted into percentage points at an annual rate by multiplying by 400. The first lag is its value in the previous quarter. All entries are rounded to the nearest decimal.
528 ChapTeR 14 Introduction to Time Series Regression and Forecasting
places), as obtained by directly computing the percentage growth. These calcula-
tions can be summarized as
Annualized rate of GDP Growth = GDPGRt ≅ 4003ln(GDPt) – ln(GDPt – 1)4
= 400∆ln(GDPt), (14.2)
where GDPt is the value of GDP at date t. The factor of 400 arises from converting fractional change to percentages (multiplying by 100) and converting quarterly percentage change to an equivalent annual rate (multiplying by 4).
The final column of Table 14.1 illustrates lags. The first lag GDPGR in 2012:Q2 is 3.64%, the value of GDPGR in 2012:Q1.
Figure 14.1b plots GDPGRt from 1960:Q1 through 2012:Q4. It shows sub- stantial variability in the growth rate of GDP. For example, GDP grew at an annual rate of over 15% in 1978:Q2 and fell at annual rate of over 8% in 2008:Q4. Over the entire period, the growth rate averaged 3.1% (which is responsible for the increase of GDP from $3.1 trillion in 1960 to $15.5 trillion in 2012), and the sample standard deviation was 3.4%.
Autocorrelation
In time series data, the value of Y in one period typically is correlated with its value in the next period. The correlation of a series with its own lagged values is called autocorrelation or serial correlation. The first autocorrelation (or
autocorrelation (Serial Correlation) and autocovariance
14.2
Key ConCept
The j th autocovariance of a series Yt is the covariance between Yt and its j th lag, Yt – j, and the j th autocorrelation coefficient is the correlation between Yt and Yt – j. That is,
th 2var(Y )var(Y ) t t-j
j autocovariance = cov(Yt, Yt – j) (14.3) jthautocorrelation = rj = corr(Yt,Yt – j) = cov(Yt,Yt – j) . (14.4)
The jth autocorrelation coefficient is sometimes called the jth serial correlation coefficient.
14.2 Introduction to Time Series Data and Serial Correlation 529
autocorrelation coefficient) is the correlation between Yt and Yt – 1—that is, the correlation between values of Y at two adjacent dates. The second autocorrelation is the correlation between Yt and Yt – 2, and the jth autocorrelation is the correla- tion between Yt and Yt – j. Similarly, the jth autocovariance is the covariance between Yt and Yt – j. Autocorrelation and autocovariance are summarized in Key Concept 14.2.
The jth population autocovariances and autocorrelations in Key Concept 14.2 can be estimated by the jth sample autocovariances and autocorrelations, cov(Yt, Yt – j) and rnj:
cov(Yt,Yt-j) = T1 aT (Yt – Yj+1:T)(Yt-j – Y1:T-j) (14.5) t=j+1
rnj = cov(Yt, Yt – j), (14.6) var(Yt)
where Yj + 1: T denotes the sample average of Yt computed using the observations t = j + 1, c, T and where var(Yt) is the sample variance of Y.2
The first four sample autocorrelations of GDPGR, the growth rate of GDP, are rn1 = 0.34, rn2 = 0.27, rn3 = 0.13, and rn4 = 0.14. These values suggest that GDP growth rates are mildly positively autocorrelated; if GDP grows faster than average in one period, it tends to also grow faster than average in the following period.
Other Examples of Economic Time Series
Economic time series differ greatly. Four examples of economic time series are plotted in Figure 14.2: the U.S. unemployment rate; the rate of exchange between the dollar and the British pound; the logarithm of an index of industrial produc- tion in Japan; and the daily return on the Wilshire 5000 stock price index.
The U.S. unemployment rate (Figure 14.2a) is the fraction of the labor force out of work, as measured in the Current Population Survey (see Appendix 3.1). Figure 14.2a shows that the unemployment rate increases by large amounts during recessions (the shaded areas in Figure 14.1) and falls during recoveries and expansions.
2The summation in Equation (14.5) is divided by T, whereas in the usual formula for the sample covariance [see Equation (3.24)], the summation is divided by the number of observations in the sum- mation, minus a degrees-of-freedom adjustment. The formula in Equation (14.5) is conventional for the purpose of computing the autocovariance. Equation (14.6) uses the assumption that var(Yt) and var(Yt – j) are the same—an implication of the assumption that Y is stationary, which is discussed in Section 14.4.
530 Chapter 14 Introduction to Time Series Regression and Forecasting
Figure 14.2
Percent
11
10
9
8
7
6
5
4
3
1960 1970 1980 1990 2000 2010 2012
Four economic time Series
(a) U.S. Unemployment Rate Logarithm
5.0 4.5 4.0 3.5 3.0 2.5
1960
(c) Logarithm of Index of Industrial Production
in Japan
Dollars per pound
3.0
2.5
2.0
1.5
1.0
0.5
0.0
1960 1970 1980 1990 2000 2010 2012 (b) U.S. Dollar/British Pound Exchange Rate
Percent per day
12.5 10.0 7.5 5.0 2.5 0.0 –2.5 –5.0 –7.5 –10.0
1970
1980
1990
2000
2010
2012
2006 2010
The four time series have markedly different patterns. The unemployment rate (Figure 14.2a) increases during recessions and declines during recoveries and expansions. The exchange rate between the U.S. dollar and the British pound (Figure 14.2b) shows a discrete change after the 1972 collapse of the Bretton Woods system of fixed exchange rates. The logarithm of the index of industrial production in Japan (Figure 14.2c) shows a pat- tern of decreasing growth. The daily percentage changes in the Wilshire 5000 stock price index (Figure 14.2d) are essentially unpredictable, but the variance changes: This series shows “volatility clustering.”
The dollar/pound exchange rate (Figure 14.2b) is the price of a British pound (£) in U.S. dollars. Before 1972, the developed economies ran a system of fixed exchange rates—called the “Bretton Woods” system—under which governments worked to keep exchange rates from fluctuating. In 1972, inflationary pressures led to the breakdown of this system; thereafter, the major currencies were allowed to “float”; that is, their values were determined by the supply and demand for currencies in the market for foreign exchange. Prior to 1972, the exchange rate
1998 2002 5000 Stock Price Index
1990 1994
(d) Percentage Change in DailyValues of theWilshire
was approximately constant, with the exception of a single devaluation in 1968, in which the official value of the pound, relative to the dollar, was decreased to $2.40. Since 1972 the exchange rate has fluctuated over a very wide range.
The index of industrial production for Japan (Figure 14.2c) measures Japan’s output of industrial commodities. The logarithm of the series is plotted in Figure 14.2c, and changes in this series can be interpreted as (fractional) growth rates. During the 1960s and early 1970s, Japanese industrial production grew quickly, but this growth slowed in the late 1970s and 1980s, and industrial pro- duction has grown little since the early 1990s.
The Wilshire 5000 stock price index is an index of the share prices of all firms traded on exchanges in the United States. Figure 14.2d plots the daily percentage changes in this index for trading days from January 2, 1990, to December 31, 2013 (a total of 4003 observations). Unlike the other series in Figure 14.2, there is very little serial correlation in these daily percentage changes; if there were, then you could predict them using past daily changes and make money by buying when you expect the market to rise and selling when you expect it to fall. Although the changes are essentially unpredictable, inspection of Figure 14.2d reveals patterns in their volatil- ity. For example, the standard deviation of daily percentage changes was relatively large in 1998–2003 and 2007–2008, and it was relatively small in 1994 and 2004. This “volatility clustering” is found in many financial time series, and econometric models for modeling this special type of heteroskedasticity are taken up in Section 16.5.
14.3 Autoregressions
How fast will GDP grow over the next year? Will growth be strong, so it will be a good year for the U.S. economy, or weak—perhaps even negative—signaling that the economy will be in a recession? Firms use growth forecasts when they forecast sales of their products, and local governments use growth forecasts when they develop their budgets for the upcoming year. Economists at central banks, like the U.S. Federal Reserve Bank, use growth forecasts when they set monetary policy. Wall Street investors rely on growth forecasts when deciding how much to pay for stocks and bonds. In this section, we consider forecasts made using an autoregression, a regression model that relates a time series variable to its past values.
The First-Order Autoregressive Model
If you want to predict the future of a time series, a good place to start is in the immediate past. For example, if you want to forecast the rate of GDP growth in the next quarter, you might see how fast GDP grew in the last quarter.
14.3 Autoregressions 531
532 ChapTeR 14 Introduction to Time Series Regression and Forecasting
A systematic way to forecast GDP growth, GDPGRt, using the previous quarter’s value, GDPGRt−1, is to estimate an OLS regression of GDPGRt on GDPGRt−1. Estimated using data from 1962 to 2012, this regression is
GDPGRt = 1.991 + 0.344GDPGRt-1, (14.7) (0.349) (0.075)
where, as usual, standard errors are given in parentheses under the estimated coefficients, and GDPGR is the predicted value of GDPGR based on the estimated regression line. The model in Equation (14.7) is called a first-order autoregression: an autoregression because it is a regression of the series onto its own lag, GDPGRt−1, and first-order because only one lag is used as a regressor. The coefficient in Equation (14.7) is positive, so positive growth of GDP in one quarter is associated with positive growth in the next quarter.
A first-order autoregression is abbreviated AR(1), where the 1 indicates that it is first order. The population AR(1) model for the series Yt is
Yt = b0 + b1Yt-1 + ut, (14.8)
where ut is an error term.
Forecastsandforecasterrors. SupposethatyouhavehistoricaldataonY,andyou want to forecast its future value. If Yt follows the AR(1) model in Equation (14.8) andifb0 andb1 areknown,thentheforecastofYT+1 basedonYT isb0 + b1YT.
In practice, b0 and b1 are unknown, so forecasts must be based on estimates
of b and b . We will use the OLS estimators bn and bn , which are constructed
using historical data. In general, Y will denote the forecast of Y based on 01 nT+10T 01 T+1
information through period T, using a model estimated with data through period T. Accordingly, the forecast based on the AR(1) model in Equation (14.8) is
YnT+10T = bn0 + bn1YT, (14.9)
where bn0 and bn1 are estimated using historical data through time T.
The forecast error is the mistake made by the forecast; this is the difference between the value of YT + 1 that actually occurred and its forecasted value based
on YT:
Forecast error = YT + 1 – YnT + 10T. (14.10)
Forecasts versus predicted values. The forecast is not an OLS predicted value, and the forecast error is not an OLS residual. OLS predicted values are calculated for the observations in the sample used to estimate the regression. In contrast, the forecast is made for some date beyond the data set used to estimate the regression, so the data on the actual value of the forecasted dependent variable are not in the sample used to estimate the regression. Similarly, the OLS residual is the difference between the actual value of Y and its predicted value for observations in the sample, whereas the forecast error is the difference between the future value of Y, which is not contained in the estimation sample, and the forecast of that future value. Said differently, forecasts and forecast errors pertain to “out-of-sample” observations, whereas predicted values and residuals pertain to “in-sample” observations.
Root mean squared forecast error. The root mean squared forecast error (RMSFE) is a measure of the size of the forecast error—that is, of the magnitude of a typical mistake made using a forecasting model. The RMSFE is the square root of the mean squared forecast error:
RMSFE = 3E3(Y
– Y ) 4. (14.11) T+1 nT+10T 2
large, then the RMSFE is approximately 2var(u ), the standard deviation of the t
error ut in the population autoregression [Equation (14.8)]. The standard devia- tion of ut is in turn estimated by the standard error of the regression (SER; see Section 4.3). Thus, if uncertainty arising from estimating the regression coeffi- cients is small enough to be ignored, the RMSFE can be estimated by the standard error of the regression. Estimation of the RMSFE including both sources of fore- cast error is taken up in Section 14.4.
ApplicationtoGDPgrowth. WhatistheforecastofthegrowthrateofGDPinthe first quarter of 2013 (2013:Q1) that a forecaster would have made in 2012:Q4, based on the estimated AR(1) model in Equation (14.7) (which was estimated using data through 2012:Q4)? According to Table 14.1, the growth rate of GDP in 2012:Q4 was 0.15% (so GDPGR 2012:Q4 = 0.15). Plugging this value into Equation (14.7), the forecast of the growth rate of GDP in 2013:Q1 is GDPGR2013:Q12012:Q4 = 1.991 + 0.344 * GDPGR2012:Q4 = 1.991 + 0.344 * 0.15 = 2.0 (rounded to the nearest tenth).Thus, the AR(1) model forecasts that the growth rate of GDP will be 2.0% in 2013:Q1.
The RMSFE has two sources of error: the error arising because future values of
ut are unknown and the error in estimating the coefficients b0 and b1. If the first
source of error is much larger than the second, as it can be if the sample size is
14.3 Autoregressions 533
534 Chapter 14 Introduction to Time Series Regression and Forecasting
How accurate is this AR(1) forecast? Table 14.1 shows that the actual growth rate of GDP in 2013:Q1 was 1.1%, so the AR(1) forecast is high by 0.9 percentage point; that is, the forecast error is −0.9. The R2 of the AR(1) model in Equation (14.7) is only 0.11, so the lagged value of GDP growth explains a small fraction of the variation in GDP growth in the sample used to fit the autoregression. This low R2 is consistent with the poor forecast of GDP growth in 2013:Q1 produced using Equation (14.7). More generally, the low R2 suggests that this AR(1) model will forecast only a small amount of the variation in the growth rate of GDP.
The standard error of the regression in Equation (14.7) is 3.16; ignoring uncertainty arising from estimation of the coefficients, our estimate of the RMSFE for forecasts based on Equation (14.7) is therefore 3.16 percentage points.
The pth-Order Autoregressive Model
The AR(1) model uses Yt – 1 to forecast Yt, but doing so ignores potentially useful information in the more distant past. One way to incorporate this information is to include additional lags in the AR(1) model; this yields the pth-order autoregressive, or AR(p), model.
The pth-order autoregressive model [the AR(p) model] represents Yt as a linear function of p of its lagged values; that is, in the AR(p) model, the regressors are Yt – 1, Yt – 2, c, Yt – p, plus an intercept. The number of lags, p, included in an AR(p) model is called the order, or lag length, of the autoregression.
For example, an AR(2) model of GDP growth uses two lags of GDP growth as regressors. Estimated by OLS over the period 1962–2012, the AR(2) model is
GDPGRt = 1.63 + 0.28GDPGRt-1 + 0.17GDPGRt-2. (14.12) (0 .40) (0 .08) (0 .08)
The coefficient on the additional lag in Equation (14.1312) is significantly differ- ent from zero at the 5% significance level: The t-statistic is 2.27 (p-value = 0.02). This is reflected in an improvement in the R2 from 0.11 for the AR(1) model in Equation (14.7) to 0.14 for the AR(2) model. Similarly, the SER of the AR(2) model in Equation (14.12) is 3.11, an improvement over the SER of the AR(1) model, which is 3.16.
The AR(p) model is summarized in Key Concept 14.3.
PropertiesoftheforecastanderrortermintheAR(p)model. Theassumptionthat
E(u 0 Y , Y , c) = 0], has two important implications. t t-1 t-2
the conditional expectation of ut is zero, given past values of Yt [that is,
14.3 Autoregressions 535
autoregressions
Key ConCept
14.3
The pth-order autoregressive model [the AR(p) model] represents Yt as a linear function of p of its lagged values:
Yt = b0 + b1Yt-1 + b2Yt-2 + g+bpYt-p + ut, (14.13)
where E(ut 0 Yt – 1, Yt – 2, c) = 0. The number of lags p is called the order, or the lag length, of the autoregression.
The first implication is that the best forecast of YT + 1 based on its entire his- tory depends on only the most recent p past values. Specifically, let YT + 1T = E(YT + 1 YT,YT – 1, c) denote the conditional mean of YT + 1, given its entire his- tory. Then YT + 1T has the smallest RMSFE of any forecast, based on the history of Y (Exercise 14.5). If Yt follows an AR(p), then the best forecast of YT + 1 based on YT,YT-1, c is
YT+10T = b0 + b1YT + b2YT-1 + g+ bpYT-p+1, (14.14) which follows from the AR(p) model in Equation (14.13) and the assumption that
E(u0Y ,Y ,c) = 0.Inpractice,thecoefficientsb,b,c,b areunknown, tt-1t-2 01 p
so actual forecasts from an AR(p) use Equation (14.14) with estimated coeffi- cients.
The second implication is that the errors ut are serially uncorrelated, a result that follows from Equation (2.27) (Exercise 14.5).
Application to GDP growth. What is the forecast of the growth rate of GDP in 2013:Q1, using data through 2012:Q4, based on the AR(2) model of GDP growth in Equation (14.12)? To compute this forecast, substitute the values of the GDP growth in 2012:Q3 and 2012:Q4 into Equation (14.12): GDPGR2013:Q12012:Q4 = 1.63 + 0.28GDPGR2012:Q4 + 0.17GDPGR2012:Q3 = 1.63 + 0.28 * 0.15 + 0.17 * 2.75 ≅ 2.1%, where the 2012 values for GDPGR are taken from the fourth col- umn of Table 14.1. The forecast error is the actual value, 1.1%, minus the forecast, or 1.1% − 2.1% = −1.0%, slightly greater in absolute value than the AR(1) fore- cast error of −0.9 percentage point.
536 ChapTeR 14 Introduction to Time Series Regression and Forecasting Can you Beat the Market? part i
Have you ever dreamed of getting rich quickly by beating the stock market? If you think that the market will be going up, you should buy stocks today and sell them later, before the market turns down. If you are good at forecasting swings in stock prices, then this active trading strategy will produce better returns than a passive “buy and hold” strat- egy in which you purchase stocks and just hang onto them. The trick, of course, is having a reliable fore- cast of future stock returns.
Forecasts based on past values of stock returns are sometimes called “momentum” forecasts: If the value of a stock rose this month, perhaps it has momentum and will also rise next month. If so, then returns will be autocorrelated, and the autoregressive model will provide useful forecasts. You can implement a momentum-based strategy for a specific stock or for a stock index that measures the overall value of the market.
taBLe 14.2 autoregressive Models of Monthly excess Stock Returns, 1960:M1–2002:M12
Dependent variable: excess returns on the CrSp value-weighted index
(1)
AR(1)
0.050 (0.051)
0.312 (0.197)
0.968 (0.325)
0.0006
Note: Excess returns are measured in percentage
are estimated over 1960:M1–2002:M12 (T = 516
variables. Entries in the regressor rows are coefficients, with standard errors in parentheses. The final two rows report the F-statistic testing the hypothesis that the coefficients on lags of excess return in the regression are zero, with its p-value in parentheses, and the adjusted R2.
Specification
Regressors
excess returnt -1
excess returnt -2
excess returnt -3
excess returnt -4
Intercept
F-statistic for coefficients on- lags of excess return (p-value)
R2
(2)
AR(2)
0.053 (0.051)
–0.053 (0.048)
0.328 (0.199)
1.342 (0.261)
0.0014
(3)
AR(4)
0.054 (0.051)
–0.054 (0.048)
0.009 (0.050)
−0.016 (0.047)
0.331 (0.202)
0.707 (0.587)
–0.0022
points per month. The data are described in Appendix 14.1. All regressions observations), with earlier observations used for initial values of lagged
14.4 Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model 537
Table 14.2 presents autoregressive models of the excess return on a broad-based index of stock prices, called the CRSP value-weighted index, using monthly data from 1960:M1 to 2002:M12, where “M1” denotes the first month of the year (January), “M2” denotes the second month, and so forth. The monthly excess return is what you earn, in percentage terms, by purchasing a stock at the end of the previ- ous month and selling it at the end of this month, minus what you would have earned had you pur- chased a safe asset (a U.S. Treasury bill). The return on the stock includes the capital gain (or loss) from the change in price plus any dividends you receive during the month. The data are described further in Appendix 14.1.
Sadly, the results in Table 14.2 are negative. The coefficient on lagged returns in the AR(1) model is not statistically significant, and we cannot reject the null hypothesis that the coefficients on lagged
returns are all zero in the AR(2) or AR(4) model. In fact, the adjusted R2 of one of the models is neg- ative and the other two are only slightly positive, suggesting that none of these models is useful for forecasting.
These negative results are consistent with the theory of efficient capital markets, which holds that excess returns should be unpredictable because stock prices already embody all currently available information. The reasoning is simple: If market participants think that a stock will have a positive excess return next month, then they will buy that stock now, but doing so will drive up the price of the stock to exactly the point at which there is no expected excess return. As a result, we should not be able to forecast future excess returns by using past publicly available information, and we cannot do it, at least using the regressions in Table 14.2.
14.4 Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model
Economic theory often suggests other variables that could help forecast a variable of interest. These other variables, or predictors, can be added to an autoregression to produce a time series regression model with multiple predictors. When other variables and their lags are added to an autoregression, the result is an autoregressive distributed lag model.
Forecasting GDP Growth Using the Term Spread
Interest rates on long-term and short-term bonds move together, but not one for one. Figure 14.3a plots interest rates on 10-year U.S. Treasury bonds and 3-Month Treasury bills from 1960 to 2012. Both interest rates show the same long-run tendencies: both were low in the 1960s, both rose through the 1970s and peaked
538 Chapter 14 Introduction to Time Series Regression and Forecasting Figure 14.3 Interest rates and the term Spread, 1960–2012
Percent per annum
16 14 12 10
8
6
4
2 0
10-Year Interest Rate
3-Month Interest Rate
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 (a) 10-Year Interest Rate and Three-Month Interest Rate
Percent per annum
4 3 2 1 0
–1
–2
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
(b) Term Spread
Long-term and short-term interest move together, but not one-for-one. The difference between long-term rates and short-term rates is called the term spread. The term spread has fallen sharply before U.S. recessions, which are shown as shaded regions in the figures.
in the early 1980s, and both fell subsequently. But the gap, or difference, between the two interest rates has not been constant: While short-term rates are generally below long-term rates, the gap between them narrows and even disappears shortly before the start of a recession; recessions are shown as the shaded bars in the figure. This difference between long-term and short-term interest rates is called the term spread and is plotted in Figure 14.3b. The term spread is generally posi- tive, but it falls toward zero or below before recessions.
Figure 14.3 suggests that the term spread might contain information about the future growth of GDP that is not already contained in past values of GDP growth.
14.4 Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model 539 This conjecture is readily checked by augmenting the AR(2) model in Equation
(14.12) to include the first lag of the term spread:
GDPGRt = 0.95 + 0.27GDPGRt-1 + 0.19GDPGRt-2 + 0.44TSpreadt-1. (0.49) (0.08) (0.08) (0.18) (14.15)
The t-statistic on TSpreadt -1 is −2.43, so this term is significant at the 1% level. The R2 of this regression is 0.16, an improvement over the AR(2) R2 of 0.14.
The forecast of the rate of change of GDP in 2013:Q1 is obtained by substitut- ing the 2012:Q3 and 2012:Q4 values of the GDP growth into Equation (14.15), along with the value of the term spread in 2012:Q4 (which is 1.62); the resulting forecast is GDPGR2013:Q12012:Q4 = 2.2%, and the forecast error is −1.1%.
If one lag of the term spread is helpful for forecasting GDP growth, more lags might be even more helpful; adding an additional lag of the term spread yields
GDPGRt = 0.97 + 0.24GDPGRt – 1 + 0.18GDPGRt – 2 (0.47) (0.08) (0.08)
– 0 .14 TSpreadt – 1 + 0 .66 TSpreadt – 2. (14.16) (0.43) (0.43)
The t-statistic testing the significance of the second lag of the term spread is 1.53 (p-value = 0.13), so it falls just short of statistical significance at the 10% level. The R2 of the regression in Equation (14.16) is 0.17, a slight improvement over 0.16 for Equation (14.15). The F-statistic on all the term spread coefficients is 4.43 (p-value = 0.01), indicating that this model represents a statistically significant improvement over the AR(2) model of Section 14.3 [Equation (14.12)]. The stan- dard error of the regression in Equation (14.16) is 3.06, a modest improvement over the SER of 3.11 for the AR(2).
The forecasted rate of growth for GDP in 2013:Q1 using Equation (14.16) is computed by substituting the values of the variables into the equation. The term spread was 1.54 in 2012:Q3 and 1.62 in 2012:Q4. The forecast value of the rate of growth in GDP in 2013:Q1, based on Equation (14.16), is
GDPGR2013:Q12012:Q4 = 0.99 + 0.24 * 0.15 + 0.18 * 2.75
– 0.14 * 1.62 + 0.66 * 1.54 = 2.3. (14.17)
The forecast error is −1.2%.
The autoregressive distributed lag model. Each model in Equations (14.15) and (14.16) is an autoregressive distributed lag (ADL) model: autoregressive
540 ChapTeR 14 Introduction to Time Series Regression and Forecasting
The autoregressive Distributed Lag Model
14.4
Key ConCept
The autoregressive distributed lag model with p lags of Yt and q lags of Xt, denoted ADL(p, q), is
Yt = b0 + b1Yt-1 + b2Yt-2 + g+bpYt-p
+ d1Xt-1 + d2Xt-2 + g + dqXt-q + ut, (14.18)
withE(u0Y ,Y ,c,X ,X ,c) = 0. t t-1 t-2 t-1 t-2
where b0, b1, c, bp, d1, c, dq are unknown coefficients and ut is the error term
because lagged values of the dependent variable are included as regressors, as in an autoregression, and distributed lag because the regression also includes multiple lags (a “distributed lag”) of an additional predictor. In general, an autoregressive distributed lag model with p lags of the dependent variable Yt and q lags of an additional predictor Xt is called an ADL(p, q) model. In this notation, the model in Equation (14.15) is an ADL(2,1) model and the model in Equation (14.16) is an ADL(2,2) model.
The autoregressive distributed lag model is summarized in Key Concept 14.4. With all these regressors, the notation in Equation (14.18) is somewhat cumber- some, and alternative optional notation, based on the so-called lag operator, is presented in Appendix 14.3.
The assumption that the errors in the ADL model have a conditional mean ofzerogivenallpastvaluesofYandX,thatis,thatE(u0Y ,Y ,c,X ,
t t-1 t-2 t-1 Xt – 2, c) = 0 implies that no additional lags of either Y or X belong in the ADL
model. In other words, the lag lengths p and q are the true lag lengths, and the coefficients on additional lags are zero.
The ADL model contains lags of the dependent variable (the autoregressive component) and a distributed lag of a single additional predictor, X. In general, however, forecasts can be improved by using multiple predictors. But before turn- ing to the general time series regression model with multiple predictors, we first introduce the concept of stationarity, which will be used in that discussion.
Stationarity
Regression analysis of time series data necessarily uses data from the past to quantify historical relationships. If the future is like the past, then these historical relationships can be used to forecast the future. But if the future differs
14.4 Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model 541
Stationarity
Key ConCept
14.5
A time series Yt is stationary if its probability distribution does not change over time, that is, if the joint distribution of (Ys + 1, Ys + 2, c, Ys + T) does not depend on s regardless of the value of T; otherwise, Yt is said to be nonstationary. A pair of time series, Xt and Yt, are said to be jointly stationary if the joint distribution of (Xs + 1, Ys + 1, Xs + 2, Ys + 2, c , Xs + T, Ys + T) does not depend on s, regardless of the value of T. Stationarity requires the future to be like the past, at least in a probabilistic sense.
fundamentally from the past, then those historical relationships might not be reli- able guides to the future.
In the context of time series regression, the idea that historical relationships can be generalized to the future is formalized by the concept of stationarity. The precise definition of stationarity, given in Key Concept 14.5, is that the probability distribution of the time series variable does not change over time.
Time Series Regression with Multiple Predictors
The general time series regression model with multiple predictors extends the ADL model to include multiple predictors and their lags. The model is summa- rized in Key Concept 14.6. The presence of multiple predictors and their lags leads to double subscripting of the regression coefficients and regressors.
The time series regression model assumptions. The assumptions in Key Concept 14.6 modify the four least squares assumptions of the multiple regression model for cross-sectional data (Key Concept 6.4) for time series data.
The first assumption is that ut has conditional mean zero, given all the regres- sors and the additional lags of the regressors beyond the lags included in the regression. This assumption extends the assumption used in the AR and ADL models and implies that the best forecast of Yt using all past values of Y and the X’s is given by the regression in Equation (14.19).
The second least squares assumption for cross-sectional data (Key Concept 6.4) is that (X1i, c, Xki, Yi), i = 1, c, n, are independently and identically dis- tributed (i.i.d.). The second assumption for time series regression replaces the i.i.d. assumption by a more appropriate one with two parts. Part (a) is that the data are drawn from a stationary distribution so that the distribution of the data
542 ChapTeR 14 Introduction to Time Series Regression and Forecasting
Time Series Regression with Multiple predictors
14.6
Key ConCept
The general time series regression model allows for k additional predictors, where q1 lags of the first predictor are included, q2 lags of the second predictor are included, and so forth:
Yt = b0 + b1Yt-1 + b2Yt-2 + g+ bpYt-p
+ d11X1t-1 + d12X1t-2 + g + d1q1X1t-q1
+ g+ dk1Xkt-1 + dk2Xkt-2 + g + dkqkXkt-qk + ut,
(14.19)
where
1. E(u0Y ,Y ,c,X ,X ,c,X ,X ,c) = 0;
(b) (Yt, X1t, c, Xkt) and (Yt – j, X1t – j, c, Xkt – j) become independent as j gets large;
3. Large outliers are unlikely: X1t, c, Xkt and Yt have nonzero, finite fourth moments; and
4. There is no perfect multicollinearity.
t t-1 t-2 1t-1 1t-2 kt-1 kt-2
2. (a)Therandomvariables(Yt,X1t,c,Xkt)haveastationarydistribution,and
today is the same as its distribution in the past. This assumption is a time series version of the “identically distributed” part of the i.i.d. assumption: The cross- sectional requirement of each draw being identically distributed is replaced by the time series requirement that the joint distribution of the variables, including lags, does not change over time. In practice, many economic time series appear to be nonstationary, which means that this assumption can fail to hold in applications. If the time series variables are nonstationary, then one or more problems can arise in time series regression: The forecast can be biased, the forecast can be inefficient (there can be alternative forecasts based on the same data with lower MSFE), or conventional OLS-based statistical inferences (for example, performing a hypoth- esis test by comparing the OLS t-statistic to {1.96) can be misleading. Precisely which of these problems occurs, and its remedy, depends on the source of the non- stationarity. In Sections 14.6 and 14.7, we study the problems posed by, tests for, and solutions to two empirically important types of nonstationarity in economic time series: trends and breaks. For now, however, we simply assume that the series are jointly stationary and accordingly focus on regression with stationary variables.
14.4 Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model 543
Granger Causality Tests (Tests of predictive Content)
Key ConCept
14.7
The Granger causality statistic is the F-statistic that tests the hypothesis that the coefficients on all the values of one of the variables in Equation (14.19) (for example, the coefficients on X1t – 1, X1t – 2, c, X1t – q1) are zero. This null hypoth- esis implies that these regressors have no predictive content for Yt beyond that contained in the other regressors, and the test of this null hypothesis is called the Granger causality test.
Part (b) of the second assumption requires that the random variables become independently distributed when the amount of time separating them becomes large. This replaces the cross-sectional requirement that the variables be indepen- dently distributed from one observation to the next with the time series require- ment that they be independently distributed when they are separated by long periods of time. This assumption is sometimes referred to as weak dependence, and it ensures that in large samples there is sufficient randomness in the data for the law of large numbers and the central limit theorem to hold. We do not provide a precise mathematical statement of the weak dependence condition; rather, the reader is referred to Hayashi (2000, Chapter 2).
The third assumption, which is the same as the third least squares assumption for cross-sectional data, that large outliers are unlikely, is made mathematically precise by the assumption that all the variables have nonzero finite fourth moments.
Finally, the fourth assumption, which is also the same as for cross-sectional data, is that the regressors are not perfectly multicollinear.
Statistical inference and the Granger causality test. Under the assumptions of Key Concept 14.6, inference on the regression coefficients using OLS proceeds in the same way as it usually does using cross-sectional data.
One useful application of the F-statistic in time series forecasting is to test whether the lags of one of the included regressors has useful predictive content, above and beyond the other regressors in the model. The claim that a variable has no predictive content corresponds to the null hypothesis that the coeffi- cients on all lags of that variable are zero. The F-statistic testing this null hypothesis is called the Granger causality statistic, and the associated test is called a Granger causality test (Granger, 1969). This test is summarized in Key Concept 14.7.
544 Chapter 14 Introduction to Time Series Regression and Forecasting
Granger causality has little to do with causality in the sense that it is used elsewhere in this book. In Chapter 1, causality was defined in terms of an ideal randomized controlled experiment, in which different values of X are applied experimentally and we observe the subsequent effect on Y. In contrast, Granger causality means that if X Granger-causes Y, then X is a useful predictor of Y, given the other variables in the regression. While “Granger predictability” is a more accurate term than “Granger causality,” the latter has become part of the jargon of econometrics.
As an example, consider the relationship between the growth rate of GDP and its past values and past values of the term spread. Based on the OLS estimates in Equation (14.16), the F-statistic testing the null hypothesis that the coefficients on both lags of the term spread are zero is 4.43 (p-value = 0.01): In the jargon of Key Concept 14.7, we can conclude (at the 1% significance level) that the term spread Granger-causes growth in GDP. This does not necessarily mean that a change in the term spread will cause—in the sense of Chapter 1—a subsequent change in GDP. It does mean that the past values of the term spread appear to contain information that is useful for forecasting changes in GDP, beyond that contained in past values of changes in GDP.
Forecast Uncertainty and Forecast Intervals
In any estimation problem, it is good practice to report a measure of the uncer- tainty of that estimate, and forecasting is no exception. One measure of the uncer- tainty of a forecast is its root mean squared forecast error (RMSFE). Under the additional assumption that the errors ut are normally distributed, the RMSFE can be used to construct a forecast interval—that is, an interval that contains the future value of the variable with a certain probability.
Forecastuncertainty. Theforecasterrorconsistsoftwocomponents:uncertainty arising from estimation of the regression coefficients and uncertainty associated with the future unknown value of ut. For regression with few coefficients and many observations, the uncertainty arising from future ut can be much larger than the uncertainty associated with estimation of the parameters. In general, however, both sources of uncertainty are important, so we now develop an expression for the RMSFE that incorporates these two sources of uncertainty.
To keep the notation simple, consider forecasts of YT + 1 based on an ADL(1,1) model with a single predictor—that is, Yt = b0 + b1Yt – 1 + d1Xt – 1 + ut —and
suppose that ut is homoskedastic. The forecast is YT + 1T = b0 + b1YT + d1XT and the forecast error is
nnnn
14.4 Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model 545 Y -Y =u -3(b -b)+(b -b)Y +(d -d)X4. (14.20)
Because uT + 1 has conditional mean zero and is homoskedastic, uT + 1 has variance s2u and is uncorrelated with the final expression in brackets in Equation (14.20). Thus the mean squared forecast error (MSFE) is
(14.21)
and the RMSFE is the square root of the MSFE.
Estimation of the MSFE entails estimation of the two parts in Equation
(14.21). The first term, s2u, can be estimated by the square of the standard error of the regression, as discussed in Section 14.3. The second term requires estimat- ing the variance of a weighted average of the regression coefficients, and meth- ods for doing so were discussed in Section 8.1 [see the discussion following Equation (8.7)].
An alternative method for estimating the MSFE is to use the variance of pseudo out-of-sample forecasts, a procedure discussed in Section 14.7.
Forecast intervals. A forecast interval is like a confidence interval except that it pertains to a forecast. For example, a 95% forecast interval is an interval that contains the future value of the series in 95% of repeated applications.
One important difference between a forecast interval and a confidence inter- val is that the usual formula for a 95% confidence interval (the estimator {1.96 standard errors) is justified by the central limit theorem and therefore holds for a wide range of distributions of the error term. In contrast, because the forecast errorinEquation(14.20)includesthefuturevalueoftheerroruT + 1,computing a forecast interval requires either estimating the distribution of the error term or making some assumption about that distribution.
In practice, it is convenient to assume that uT + 1 is normally distributed.
If so, Equation (14.20) and the central limit theorem applied to bn0, bn1, and dn1
imply that the forecast error is the sum of two independent, normally distrib-
uted terms, so the forecast error is itself normally distributed, with variance
equaling the MSFE. It follows that a 95% confidence interval is given by
Y { 1.96SE(Y – Y ),whereSE(Y – Y )isanestimatorof nT+1T T+1 nT+1T T+1 nT+1T
the RMSFE.
T+1 nT+1T T+1 n0 0 n1 1 T n1 1 T
n2 MSFE = E3(Y – Y ) 4
T+1 T+1T
=s +var3(b -b)+(b -b)Y +(d -d)X4,
2u n00n11Tn11T
546 Chapter 14 Introduction to Time Series Regression and Forecasting The River of Blood
As part of its efforts to inform the public about monetary policy decisions, the Bank of England regularly publishes forecasts of inflation. These fore- casts combine output from econometric models maintained by professional econometricians at the bank with the expert judgment of the members of the bank’s senior staff and Monetary Policy Committee. The forecasts are presented as a set of forecast inter- vals designed to reflect what these economists con- sider to be the range of probable paths that inflation might take. In its Inflation Report, the bank prints these ranges in red, with the darkest red reserved for the central band. Although the bank prosaically refers to this as the “fan chart,” the press has called these spreading shades of red the “river of blood.”
The river of blood for August 2013 is shown in Figure 14.4. (In this figure the blood is blue, not red, so you will need to use your imagination.) This chart shows that, as of August 2013, the bank’s economists expected the rate of inflation to fall gradually from nearly 3% in early 2013 to 2%, the Bank’s target rate of inflation, in 2015. The economists expressed
considerable uncertainty about the forecast, how- ever. They cited uncertainty about domestic spend- ing, labor productivity, and economic growth in Europe and the emerging economies as important sources of inflation uncertainty. As it turns out, their near-term forecast was overly pessimistic: Inflation fell to their target of 2% by the end of 2013.
The Bank of England has been a pioneer in the movement toward greater openness by central banks, and other central banks now also publish inflation forecasts. The decisions made by monetary policy- makers are difficult ones and affect the lives—and wallets—of many of their fellow citizens. In a democ- racy in the information age, reasoned the economists at the Bank of England, it is particularly important for citizens to understand the bank’s economic out- look and the reasoning behind its difficult decisions.
To see the river of blood in its original red hue, visit the Bank of England’s website, at http://www .bankofengland.co.uk. To learn more about the performance of the Bank of England inflation fore- casts, see Clements (2004).
Figure 14.4 the river of Blood
The Bank of England’s fan chart for August 2013 shows forecast ranges for inflation. The dashed line indicates the third quarter of 2015, 2 years after the release of the report.
Source: Reprinted with permission from the Bank of England.
Percentage increase in prices on a year earlier
7 6 5 4 3 2
1
+
0
2009 10
11
12 13
14 15
1 16 2
–
14.5 Lag Length Selection Using Information Criteria 547
This discussion has focused on the case that the error term, uT + 1, is homoske- dastic. If instead uT + 1 is heteroskedastic, then one needs to develop a model of the heteroskedasticity so that the term s2u in Equation (14.21) can be estimated, given the most recent values of Y and X; methods for modeling this conditional heteroskedasticity are presented in Section 16.5.
Because of uncertainty about future events—that is, uncertainty about uT + 1 —95% forecast intervals can be so wide that they have limited use in decision making. Professional forecasters therefore often report forecast intervals that are tighter than 95%—for example, one standard error forecast intervals (which are 68% forecast intervals if the errors are normally distributed). Alternatively, some forecasters report multiple forecast intervals, as is done by the economists at the Bank of England when they publish their inflation forecasts (see “The River of Blood”on the previous page).
14.5 Lag Length Selection Using Information Criteria
The estimated GDP growth regressions in Sections 14.3 and 14.4 have either one or two lags of the predictors. One lag makes some sense, but why two and not three or four? More generally, how many lags should be included in a time series regression? This section discusses statistical methods for choosing the number of lags, first in an autoregression and then in a time series regression model with multiple predictors.
Determining the Order of an Autoregression
In practice, choosing the order p of an autoregression requires balancing the mar- ginal benefit of including more lags against the marginal cost of additional estima- tion uncertainty. On the one hand, if the order of an estimated autoregression is too low, you will omit potentially valuable information contained in the more distant lagged values. On the other hand, if it is too high, you will be estimating more coefficients than necessary, which in turn introduces additional estimation error into your forecasts.
The F-statistic approach. One approach to choosing p is to start with a model with many lags and to perform hypothesis tests on the final lag. For example, you might start by estimating an AR(6) and test whether the coefficient on the sixth lag is significant at the 5% level; if not, drop it and estimate an AR(5), test the coefficient on the fifth lag, and so forth. The drawback to this method is that
548 ChapTeR 14 Introduction to Time Series Regression and Forecasting
it will produce a model that is too large, at least some of the time: Even if the true AR order is five, so the sixth coefficient is zero, a 5% test using the t-statis- tic will incorrectly reject this null hypothesis 5% of the time just by chance. Thus, when the true value of p is five, this method will estimate p to be six 5% of the time.
The BIC. A way around this problem is to estimate p by minimizing an “informa- tion criterion.” One such information criterion is the Bayes information criterion (BIC), also called the Schwarz information criterion (SIC), which is
BIC(p) = lncSSR(p)d + (p + 1)ln(T), (14.22) TT
where SSR(p) is the sum of squared residuals of the estimated AR(p). The BIC estimator of p, pn, is the value that minimizes BIC(p) among the possible choices p = 0, 1, c, pmax,, where pmax is the largest value of p considered and p = 0 corresponds to the model that contains only an intercept.
The formula for the BIC might look a bit mysterious at first, but it has an intuitive appeal. Consider the first term in Equation (14.22). Because the regression coefficients are estimated by OLS, the sum of squared residuals necessarily decreases (or at least does not increase) when you add a lag. In contrast, the second term is the number of estimated regression coefficients (the number of lags, p, plus one for the intercept) times the factor ln(T)>T. This second term increases when you add a lag. The BIC trades off these two forces so that the number of lags that minimizes the BIC is a consistent esti- mator of the true lag length. Appendix 14.5 provides the mathematics of this argument.
As an example, consider estimating the AR order for an autoregression of the growth rate of GDP. The various steps in the calculation of the BIC are carried outinTable14.3forautoregressionsofmaximumordersix(pmax = 6).Forexample, for the AR(1) model in Equation (14.7), SSR(1)>T = 9.866, so ln3SSR(1)>T4 = 2.289. Because T = 204 (51 years, 4 quarters per year), ln(T)>T= 0.026 and (p + 1)ln(T)>T = 2 * 0.026 = 0.052 Thus BIC(1) = 2.289 + 0.052 = 2.341.
The BIC is smallest when p = 2 in Table 14.3. Thus the BIC estimate of the lag length is 2. As can be seen in Table 14.3, as the number of lags increases, the R2 increases and the SSR decreases. The increase in the R2 is large from zero to one lag, smaller for one to two lags, and smaller yet for other lags. The BIC helps decide precisely how large the increase in the R2 must be to justify including the additional lag.
p
0
1
2
3
4
5
6
SSR(p)/T
ln[SSR( p)/T ]
(p + 1) ln(T )/T
0.026
0.052
0.078
0.104
0.130
0.156
0.183
BIC(p) R2
2.442 0.000
2.341 0.119
2.334 0.148
2.360 0.148
2.382 0.151
2.394 0.164
2.419 0.164
TABLE 14.3
The Bayes Information Criterion (BIC) and the R 2 for Autoregressive Models of U.S. GDP Growth Rates, 1962–2012
14.5 Lag Length Selection Using Information Criteria 549
11.198 2.416
9.866 2.289
9.546 2.256
9.546 2.256
9.508 2.252
9.366 2.237
9.359 2.236
The AIC. The BIC is not the only information criterion; another is the Akaike information criterion (AIC):
AIC(p) = lnc SSR(p) d + (p + 1)2 . (14.23) TT
The difference between the AIC and the BIC is that the term “ln(T)” in the BIC is replaced by “2” in the AIC, so the second term in the AIC is smaller. For example, for the 204 observations used to estimate the GDP autoregressions, ln(T) = ln(204) = 5.32, so the second term for the BIC is more than twice as large as the term in AIC. Thus a smaller decrease in the SSR is needed in the AIC to justify including another lag. As a matter of theory, the second term in the AIC is not large enough to ensure that the correct lag length is chosen, even in large samples, so the AIC estimator of p is not consistent. As is discussed in Appendix 14.5, in large samples the AIC will overestimate p with nonzero probability.
Despite this theoretical blemish, the AIC is widely used in practice. If you are concerned that the BIC might yield a model with too few lags, the AIC provides a reasonable alternative.
A note on calculating information criteria. How well two estimated regressions fit the data is best assessed when they are estimated using the same data sets. Because the BIC and AIC are formal methods for making this comparison, the autoregressions under consideration should be estimated using the same observa- tions. For example, in Table 14.3 all the regressions were estimated using data from 1962:Q1 to 2012:Q4, for a total of 204 observations. Because the autoregressions involve lags of the growth rate of GDP, this means that earlier values of GDP
550 ChapTeR 14 Introduction to Time Series Regression and Forecasting
growth (values before 1962:Q1) were used as regressors for the preliminary obser- vations. Said differently, the regressions examined in Table 14.3 each include observations on GDPGRt, GDPGRt−1, c, GDPGRt−p for t = 1962:Q1, c, 2012:Q4 corresponding to 204 observations on the dependent variable and regres- sors, so T = 204 in Equations (14.22) and (14.24).
Lag Length Selection in Time Series Regression
with Multiple Predictors
The trade-off involved with lag length choice in the general time series regression model with multiple predictors [Equation (14.19)] is similar to that in an autoregression: Using too few lags can decrease forecast accuracy because valuable information is lost, but adding lags increases estimation uncertainty. The choice of lags must balance the benefit of using additional information against the cost of estimating the additional coefficients.
The F-statistic approach. As in the univariate autoregression, one way to deter- mine the number of lags to include is to use F-statistics to test joint hypotheses that sets of coefficients are equal to zero. For example, in the discussion of Equa- tion (14.16), we tested the hypothesis that the coefficient on the second lag of the term spread was equal to zero against the alternative that it was nonzero; this hypothesis was not rejected at the 10% significance level, suggesting that the sec- ond lag of the term spread could be dropped from the regression. If the number of models being compared is small, then this F-statistic method is easy to use. In general, however, the F-statistic method can produce models that are too large, in the sense that the true lag order is overestimated.
Information criteria. As in an autoregression, the BIC and AIC can be used to estimate the number of lags and variables in the time series regression model with multiple predictors. If the regression model has K coefficients (including the inter- cept), the BIC is
BIC(K) = lncSSR(K)d + K ln(T). (14.24) TT
The AIC is defined in the same way, but with 2 replacing ln(T) in Equation (14.24). For each candidate model, the BIC (or AIC) can be evaluated, and the model with the lowest value of the BIC (or AIC) is the preferred model, based on the informa- tion criterion.
There are two important practical considerations when using an information criterion to estimate the lag lengths. First, as is the case for the autoregression, all
the candidate models must be estimated over the same sample; in the notation of
Equation (14.24), the number of observations used to estimate the model, T, must
be the same for all models. Second, when there are multiple predictors, this
approach is computationally demanding because it requires computing many dif-
ferent models (many combinations of the lag parameters). In practice, a conve-
nient shortcut is to require all the regressors to have the same number of lags, that
compared (corresponding to p = 0, 1, c, p 2. Applying this lag-length selec- max
tion method to the ADL for GDP growth and the term spread results in the ADL(2,2) model in Equation (14.16).
14.6 Nonstationarity I: Trends
In Key Concept 14.6, it was assumed that the dependent variable and the regressors are stationary. If this is not the case—that is, if the dependent variable and/or regressors are nonstationary—then conventional hypothesis tests, confidence intervals, and forecasts can be unreliable. The precise problem created by nonstationarity, and the solution to that problem, depends on the nature of that nonstationarity.
In this and the next section, we examine two of the most important types of nonstationarity in economic time series data: trends and breaks. In each section, we first describe the nature of the nonstationarity and then discuss the conse- quences for time series regression if this type of nonstationarity is present but ignored. We next present tests for nonstationarity and discuss remedies for, or solutions to, the problems caused by that particular type of nonstationarity. We begin by discussing trends.
What Is a Trend?
A trend is a persistent long-term movement of a variable over time. A time series variable fluctuates around its trend.
Inspection of Figure 14.1a suggests that the logarithm of U.S. GDP has a clear upwardly increasing trend. The series in Figures 14.2a, b, and c also have trends, but their trends are quite different. The trend in the unemployment rate is increas- ing from the late 1960s through the early 1980s, then decreasing until the early 2000s, and then increasing again through 2012. The $>£ exchange rate clearly had a prolonged downward trend after the collapse of the fixed exchange rate system in 1972. The logarithm of the industrial production index for Japan has a compli- cated trend: fast growth at first, then moderate growth, and finally no growth.
is,torequirethat p = q1 = g= qk,sothatonly pmax + 1modelsneedtobe
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Deterministicandstochastictrends. Therearetwotypesoftrendsintimeseries data: deterministic and stochastic. A deterministic trend is a nonrandom function of time. For example, a deterministic trend might be linear in time; if the loga- rithm of U.S. GDP had a deterministic linear trend so that it increased by 0.75 percentage point per quarter, this trend could be written as 0.75t, where t is mea- sured in quarters. In contrast, a stochastic trend is random and varies over time. For example, a stochastic trend might exhibit a prolonged period of increase followed by a prolonged period of decrease, like the unemployment rate trend in Figure 14.2a. But stochastic trends can be more subtle. For example, if you look carefully at Figure 14.1a, you will notice that the trend growth rate of GDP is not constant; for example, GDP grew faster in the 1960s than in the 1970s (the plot is steeper in the 1960s than in the 1970s), and it grew faster in the 1990s than in the 2000s.
Like many other econometricians, we think it is more appropriate to model economic time series as having stochastic rather than deterministic trends. Eco- nomics is complicated stuff. It is hard to reconcile the predictability implied by a deterministic trend with the complications and surprises faced year after year by workers, businesses, and governments. For example, although the U.S. unemploy- ment rate rose through the 1970s, it was neither destined to rise forever nor des- tined to fall again. Rather, the slow rise of unemployment rates is now understood to have occurred because of a combination of demographic changes (such as an increase in female labor force participation), bad luck (such as oil price shocks and a productivity slowdown), and monetary policy mistakes. Similarly, the $>£ exchange rate trended down from 1972 to 1985 and subsequently drifted up, but these movements too were the consequences of complex economic forces; because these forces change unpredictably, these trends are usefully thought of as having a large unpredictable, or random, component.
For these reasons, our treatment of trends in economic time series focuses on stochastic rather than deterministic trends, and when we refer to “trends” in time series data, we mean stochastic trends unless we explicitly say otherwise. This section presents the simplest model of a stochastic trend, the random walk model; other models of trends are discussed in Section 16.3.
The random walk model of a trend. The simplest model of a variable with a sto- chastic trend is the random walk. A time series Yt is said to follow a random walk if the change in Yt is i.i.d., that is, if
Yt = Yt-1 + ut, (14.25)
where ut is i.i.d. We will, however, use the term random walk more generally to refer to a time series that follows Equation (14.25), where ut has conditional mean zero; that is, E(utYt-1,Yt-2, c) = 0.
The basic idea of a random walk is that the value of the series tomorrow is its
value today, plus an unpredictable change: Because the path followed by Yt con-
Y based on data through time t – 1 is Y ; that is, because E(u0Y , t t-1 tt-1
dom walk, then the best forecast of tomorrow’s value is its value today.
Some series, such as the logarithm of U.S. GDP in Figure 14.1a, have an obvious upward tendency, in which case the best forecast of the series must include an adjustment for the tendency of the series to increase. This adjustment leads to an extension of the random walk model to include a tendency to move, or “drift,” in one direction or the other. This extension is referred to as a random
walk with drift,
Yt = b0 + Yt-1 + ut, (14.26)
where E(utYt-1,Yt-2, c) = 0, and b0 is the “drift” in the random walk. If b0 is positive, then Yt increases on average. In the random walk with drift model, the best forecast of the series tomorrow is the value of the series today, plus the drift b0.
The random walk model (with drift, as appropriate) is simple yet versatile, and it is the primary model for trends used in this book.
A random walk is nonstationary. If Yt follows a random walk, then it is not sta- tionary: The variance of a random walk increases over time, so the distribution of Yt changes over time. One way to see this is to recognize that, because ut is uncor- related with Yt – 1 in Equation (14.25), var(Yt) = var(Yt – 1) + var(ut); for Yt to be stationary, var(Yt) cannot depend on time, so in particular var(Yt) = var(Yt – 1) must hold, but this can happen only if var(ut) = 0. Another way to see this is to imagine that Yt starts out at zero—that is, Y0 = 0. Then Y1 = u1, Y2 = u1 + u2, and so forth so that Yt = u1 + u2 + g + ut. Because ut is serially uncorrelated, var(Yt) = var(u1 + u2 + g + ut) = ts2u. Thus the variance of Yt depends on t; in fact, it increases as t increases. Because the variance of Yt depends on t, its distribution depends on t; that is, it is nonstationary.
Because the variance of a random walk increases without bound, its popula- tion autocorrelations are not defined. (The first autocovariance and variance are infinite, and the ratio of the two is not well defined.) However, a feature of a
sists of random “steps” u , that path is a “random walk.” The conditional mean of
Y , c) = 0, E(Y 0 Y t , Y , c) = Y . In other words, if Y follows a ran- t-2 tt-1t-2 t-1 t
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random walk is that its sample autocorrelations tend to be very close to 1; in fact,
the jth sample autocorrelation of a random walk converges to 1 in probability.
Stochastic trends, autoregressive models, and a unit root. The random walk model is a special case of the AR(1) model [Equation (14.8)] in which b1 = 1. In other words, if Yt follows an AR(1) with b1 = 1, then Yt contains a stochastic trend and is nonstationary. If, however, b1 6 1 and ut is stationary, then the joint distribution of Yt and its lags does not depend on t (a result shown in Appen- dix 14.2), so Yt is stationary.
The analogous condition for an AR(p) to be stationary is more complicated
than the condition b1 6 1 for an AR(1). Its formal statement involves the roots
of the polynomial, 1 – b1z – b2z2 – b3z3 – g – bpzp. (The roots of this poly-
nomial are the values of z that satisfy 1 – b1z – b2z2 – b3z3 – g- bpzp = 0.)
For an AR(p) to be stationary, the roots of this polynomial must all be greater
zthatsolves1 – b z = 0,soitsrootisz = 1>b .Thusthestatementthattheroot 11
be greater than 1 in absolute value is equivalent to b1 6 1.
If an AR(p) has a root that equals 1, the series is said to have a unit autoregressive
root or, more simply, a unit root. If Yt has a unit root, then it contains a stochastic trend. If Yt is stationary (and thus does not have a unit root), it does not contain a stochastic trend. For this reason, we will use the terms stochastic trend and unit root interchangeably.
Problems Caused by Stochastic Trends
If a regressor has a stochastic trend (that is, has a unit root), then the OLS estimator of its coefficient and its OLS t-statistic can have nonstandard (that is, nonnormal) distributions, even in large samples. We discuss three specific aspects of this problem: (1) The estimator of the autoregressive coefficient in an AR(1) is biased toward 0 if its true value is 1; (2) the t-statistic on a regressor with a stochastic trend can have a nonnormal distribution, even in large samples; and (3) an extreme example of the risks posed by stochastic trends is that two series that are independent will, with high probability, misleadingly appear to be related if they both have stochastic trends, a situation known as spurious regression.
Problem #1: Autoregressive coefficients that are biased toward zero. Suppose that Yt follows the random walk in Equation (14.25), but this is unknown to the econometrician, who instead estimates the AR(1) model in Equation (14.8). Because Yt is nonstationary, the least squares assumptions for time series
than 1 in absolute value. In the special case of an AR(1), the root is the value of
regression in Key Concept 14.6 do not hold, so as a general matter, we cannot rely on estimators and test statistics having their usual large-sample normal distribu- tions. In fact, in this example, the OLS estimator of the autoregressive coefficient, bn1, is consistent, but it has a nonnormal distribution, even in large samples: The asymptotic distribution of bn1 is shifted toward zero. The expected value of bn1 is
n1
approximately E(b ) = 1 – 5.3>T. This results in a large bias in sample sizes
typically encountered in economic applications. For example, 20 years of quar- terly data contain 80 observations, in which case the expected value of bn1 is
n1
E(b ) = 1 – 5.3>80 = 0.934. Moreover, this distribution has a long left tail: The
n1
5% percentile of b is approximately 1 – 14.1>T, which, for T = 80, corresponds
to 0.824, so 5% of the time bn1 6 0.824.
One implication of this bias toward zero is that if Yt follows a random walk,
then forecasts based on the AR(1) model can perform substantially worse than those based on the random walk model, which imposes the true value b1 = 1. This conclusion also applies to higher-order autoregressions, in which there are forecasting gains from imposing a unit root (that is, from estimating the autoregression in first differences instead of in levels) when in fact the series con- tains a unit root.
Problem#2:Nonnormaldistributionsoft-statistics. Ifaregressorhasastochas- tic trend, then its usual OLS t-statistic can have a nonnormal distribution under the null hypothesis, even in large samples. This nonnormal distribution means that conventional confidence intervals are not valid, and hypothesis tests cannot be conducted as usual. In general, the distribution of this t-statistic is not readily tabulated because the distribution depends on the relationship between the regressor in question and the other regressors. An important example of this problem arises in regressions that attempt to forecast stock returns using regres- sors that could have stochastic trends (see the box in Section 14.7, “Can You Beat the Market? Part II”).
One important case in which it is possible to tabulate the distribution of the t-statistic when the regressor has a stochastic trend is in the context of an autoregression with a unit root. We return to this special case when we take up the problem of testing whether a time series contains a stochastic trend.
Problem #3: Spurious regression. Stochastic trends can lead two time series to appear related when they are not, a problem called spurious regression.
For example, the U.S. unemployment rate was steadily rising from the mid- 1960s through the early 1980s, and at the same time Japanese industrial produc- tion (plotted in logarithms in Figure 14.2c) was steadily rising. These two trends
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conspire to produce a regression that appears to be “significant” using conven- tional measures. Estimated by OLS using data from 1962 through 1985, this regression is
U.S.UnemploymentRatet =-2.37+2.22*ln(JapaneseIPt),R2 =0.34.
(1 .19) (0 .32) (14.27)
The t-statistic on the slope coefficient is 7, which by usual standards indicates a strong positive relationship between the two series, and the R 2 is moderately high. However, running this regression using data from 1986 through 2012 yields
U. S. Unemployment Ratet = 41.78 – 7.78 * ln(Japanese IPt), R2 = 0.15. 18.052 11.752 (14.28)
The regressions in Equations (14.27) and (14.28) could hardly be more different. Interpreted literally, Equation (14.27) indicates a strong positive relationship, while Equation (14.28) indicates an even stronger negative relationship.
The source of these conflicting results is that both series have stochastic trends. These trends happened to align from 1962 through 1985 but were reversed from 1986 through 2012. There is, in fact, no compelling economic or political reason to think that the trends in these two series are related. In short, these regressions are spurious.
The regressions in Equations (14.27) and (14.28) illustrate empirically the theoretical point that OLS can be misleading when the series contain stochastic trends. (See Exercise 14.6 for a computer simulation that demonstrates this result.) One special case in which certain regression-based methods are reliable is when the trend component of the two series is the same—that is, when the series contain a common stochastic trend; in such a case, the series are said to be cointegrated. Econometric methods for detecting and analyzing cointegrated economic time series are discussed in Section 16.4.
Detecting Stochastic Trends: Testing for a Unit AR Root
Trends in time series data can be detected using informal and formal methods. The informal methods involve inspecting a time series plot of the data and com- puting the autocorrelation coefficients, as we did in Section 14.2. Because the first autocorrelation coefficient will be near 1 if the series has a stochastic trend, at least in large samples, a small first autocorrelation coefficient combined with a time series plot that has no apparent trend suggests that the series does not have
a trend. If doubt remains, however, formal statistical procedures can be used to test the hypothesis that there is a stochastic trend in the series against the alterna- tive that there is no trend.
In this section, we use the Dickey–Fuller test (named after its inventors David Dickey and Wayne Fuller, 1979) to test for a stochastic trend. Although the Dickey–Fuller test is not the only test for a stochastic trend (another test is dis- cussed in Section 16.3), it is the most commonly used test in practice and is one of the most reliable.
The Dickey–Fuller test in the AR(1) model. The starting point for the Dickey– Fuller test is the autoregressive model. As discussed earlier, the random walk in Equation (14.26) is a special case of the AR(1) model with b1 = 1. If b1 = 1, Yt is nonstationary and contains a (stochastic) trend. Thus, within the AR(1) model, the hypothesis that Yt has a trend can be tested by testing
H0:b1 = 1vs.H1: b1 6 1inYt = b0 + b1Yt-1 + ut. (14.29)
If b1 = 1, the AR(1) has an autoregressive root of 1, so the null hypothesis in Equation (14.29) is that the AR(1) has a unit root, and the alternative is that it is stationary.
This test is most easily implemented by estimating a modified version of Equation (14.29), obtained by subtracting Yt – 1 from both sides. Let d = b1 – 1; then Equation (14.29) becomes
H0:d = 0vs.H1:d 6 0in∆Yt = b0 + dYt-1 + ut. (14.30)
The OLS t-statistic testing d = 0 in Equation (14.30) is called the Dickey–Fuller statistic. The formulation in Equation (14.30) is convenient because regression software automatically prints out the t-statistic testing d = 0. Note that the Dickey–Fuller test is one-sided because the relevant alternative is that Yt is stationary, so b1 6 1 or, equivalently, d 6 0. The Dickey–Fuller statistic is computed using “nonrobust” standard errors—that is, the “homoskedasticity-only” standard errors presented in Appendix 5.1 [Equation (5.29) for the case of a single regressor and in Section 18.4 for the multiple regression model].3
3Under the null hypothesis of a unit root, the usual “nonrobust” standard errors produce a t-statistic that is in fact robust to heteroskedasticity, a surprising and special result.
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The Dickey–Fuller test in the AR(p) model. The Dickey–Fuller statistic presented in the context of Equation (14.30) applies only to an AR(1). As discussed in Sec- tion 14.3, for some series the AR(1) model does not capture all the serial correla- tion in Yt, in which case a higher-order autoregression is more appropriate.
The extension of the Dickey–Fuller test to the AR(p) model is summarized in Key Concept 14.8. Under the null hypothesis, d = 0 and ∆Yt is a stationary AR(p). Under the alternative hypothesis, d 6 0 so that Yt is stationary. Because the regression used to compute this version of the Dickey–Fuller statistic is aug- mented by lags of ∆Yt, the resulting t-statistic is referred to as the augmented Dickey–Fuller (ADF) statistic.
In general, the lag length p is unknown, but it can be estimated using an infor- mation criterion applied to regressions of the form in Equation (14.31) for various values of p. Studies of the ADF statistic suggest that it is better to have too many lags than too few, so it is recommended to use the AIC instead of the BIC to estimate p for the ADF statistic.4
Testing against the alternative of stationarity around a linear deterministic time trend. The discussion so far has considered the null hypothesis that a series has a unit root and the alternative hypothesis that it is stationary. This alternative hypothesis of stationarity is appropriate for series, such as the unemployment rate, that do not exhibit long-term growth. But other economic time series, such as U.S. GDP, exhibit long-run growth, and for such series the alternative of sta- tionarity without a trend is inappropriate. Instead, a commonly used alternative is that the series are stationary around a deterministic time trend—that is, a trend that is a deterministic function of time.
One specific formulation of this alternative hypothesis is that the time trend is linear; that is, the trend is a linear function of t. Thus the null hypothesis is that the series has a unit root, and the alternative is that it does not have a unit root but does have a deterministic time trend. The Dickey–Fuller regression must be modified to test the null hypothesis of a unit root against the alternative that it is stationary around a linear time trend. As summarized in Equation (14.32) in Key Concept 14.8, this is accomplished by adding a time trend (the regressor Xt = t) to the regression.
A linear time trend is not the only way to specify a deterministic time trend; for example, the deterministic time trend could be quadratic, or it could be linear but have breaks (that is, be linear with slopes that differ in two parts of the
4See Stock (1994) and Haldrup and Jansson (2006) for reviews of simulation studies of the finite- sample properties of the Dickey–Fuller and other unit root test statistics.
14.6 Nonstationarity I: Trends 559
The augmented Dickey–Fuller Test for a Unit autoregressive Root
Key ConCept
14.8
The augmented Dickey–Fuller (ADF) test for a unit autoregressive root tests the null hypothesis H0 : d = 0 against the one-sided alternative H1 : d 6 0 in the regression
∆Yt = b0 + dYt-1 + g1∆Yt-1 + g2∆Yt-2 + g+ gp∆Yt-p + ut. (14.31)
Under the null hypothesis, Yt has a stochastic trend; under the alternative hypoth- esis, Yt is stationary. The ADF statistic is the OLS t-statistic testing d = 0 in Equation (14.31).
If instead the alternative hypothesis is that Yt is stationary around a deter- ministic linear time trend, then this trend, “t” (the observation number), must be added as an additional regressor, in which case the Dickey–Fuller regression becomes
∆Yt = b0 + at + dYt-1 + g1∆Yt-1 + g2∆Yt-2 + g+ gp∆Yt-p + ut, (14.32)
where a is an unknown coefficient and the ADF statistic is the OLS t-statistic testing d = 0 in Equation (14.32).
The lag length p can be estimated using the BIC or AIC. When p = 0, lags of ∆Yt are not included as regressors in Equations (14.31) and (14.32), and the ADF test simplifies to the Dickey–Fuller test in the AR(1) model. The ADF statistic does not have a normal distribution, even in large samples. Critical values for the one-sided ADF test depend on whether the test is based on Equation (14.31) or (14.32) and are given in Table 14.4.
sample). The use of alternatives like these with nonlinear deterministic trends should be motivated by economic theory. For a discussion of unit root tests against stationarity around nonlinear deterministic trends, see Maddala and Kim (1998, Chapter 13).
Critical values for the ADF statistic. Under the null hypothesis of a unit root, the ADF statistic does not have a normal distribution, even in large samples. Because its distribution is nonstandard, the usual critical values from the normal
560 ChapTeR 14 Introduction to Time Series Regression and Forecasting
taBLe 14.4
Large-Sample Critical Values of the augmented Dickey–Fuller Statistic
distribution cannot be used when using the ADF statistic to test for a unit root; a special set of critical values, based on the distribution of the ADF statistic under the null hypothesis, must be used instead.
The critical values for the ADF test are given in Table 14.4. Because the alternative hypothesis of stationarity implies that d 6 0 in Equations (14.31) and (14.32), the ADF test is one-sided. For example, if the regression does not include a time trend, then the hypothesis of a unit root is rejected at the 5% significance level if the ADF statistic is less than -2.86. If a time trend is included in the regression, the critical value is instead – 3.41.
The critical values in Table 14.4 are substantially larger (more negative) than the one-sided critical values of – 1.28 (at the 10% level) and −1.64 (at the 5% level) from the standard normal distribution. The nonstandard distribution of the ADF statistic is an example of how OLS t-statistics for regressors with stochastic trends can have nonnormal distributions. Why the large-sample distribution of the ADF statistic is nonstandard is discussed further in Section 16.3.
Does U.S. GDP have a stochastic trend? The null hypothesis that the logarithm of U.S. GDP has a stochastic trend can be tested against the alternative that it is stationary by performing the ADF test for a unit autoregressive root. The ADF regression with two lags of Δln(GDPt) is
∆ln(GDPt) = 0.244 + 0.0002t – 0.030ln(GDPt-1) (0 .109) (0 .0001) (0 .014)
+ 0.269∆ln(GDPt-1) + 0.178∆ln(GDPt-2).
(0 .069) (0 .070) (14.33)
The ADF t-statistic is the t-statistic testing the hypothesis that the coefficient on ln(GDPt−1) is zero; this is t = -2.18. From Table 14.4, the 10% critical value is −3.12. Because the ADF statistic of −2.18 is less negative than −3.12, the test does not reject the null hypothesis at the 10% significance level. Based on the regres- sion in Equation (14.33), we therefore cannot reject (at the 10% significance level) the null hypothesis that the logarithm of GDP has a unit autoregressive
Deterministic regressors
Intercept only
Intercept and time trend
10%
-2.57
-3.12
5%
-2.86
-3.41
1%
-3.43
-3.96
root—that is, that ln(GDP) has a stochastic trend—against the alternative that it is stationary around a linear trend.
The ADF regression in Equation (14.33) includes two lags of Δ ln(GDPt) to compute the ADF statistic. When the number of lags is estimated using the AIC, where 0 … p … 5, the AIC estimator of the lag length is, however, one. When one lag is used (that is, when Δ ln(GDPt−1) is included as a regressor), the ADF statis- tic is −1.84, which is less negative than −3.12. Thus, when the number of lags in the ADF regression is chosen by AIC, the hypothesis that the logarithm of U.S. GDP contains a stochastic trend is not rejected at the 10% significance level.
Avoiding the Problems Caused by Stochastic Trends
The most reliable way to handle a trend in a series is to transform the series so that it does not have the trend. If the series has a stochastic trend—that is, if the series has a unit root—then the first difference of the series does not have a trend. For example, if Yt follows a random walk so that Yt = b0 + Yt-1 + ut, then ∆Yt = b0 + ut is stationary. Thus using first differences eliminates random walk trends in a series.
In practice, you can rarely be sure whether a series has a stochastic trend. Recall that, as a general point, failure to reject the null hypothesis does not necessarily mean that the null hypothesis is true; rather, it simply means that you have insufficient evidence to conclude that it is false. Thus failure to reject the null hypothesis of a unit root using the ADF test does not mean that the series actually has a unit root. For example, in an AR(1) model, the true coefficient b1 might be very close to 1, say 0.98, in which case the ADF test would have low power—that is, a low probability of cor- rectly rejecting the null hypothesis in samples the size of our GDP series. Even though failure to reject the null hypothesis of a unit root does not mean the series has a unit root, it still can be reasonable to approximate the true autoregressive root as equaling 1 and therefore to use differences of the series rather than its levels.5
14.7 Nonstationarity II: Breaks
A second type of nonstationarity arises when the population regression function changes over the course of the sample. In economics, this can occur for a variety of reasons, such as changes in economic policy, changes in the structure of the economy, or changes in a specific industry due to an invention. If such changes,
5For additional discussion of stochastic trends in economic time series variables and of the problems they pose for regression analysis, see Stock and Watson (1988).
14.7 Nonstationarity II: Breaks 561
562 ChapTeR 14 Introduction to Time Series Regression and Forecasting
or “breaks,” occur, then a regression model that neglects those changes can pro- vide a misleading basis for inference and forecasting.
This section presents two strategies for checking for breaks in a time series regression function over time. The first strategy looks for potential breaks from the perspective of hypothesis testing and entails testing for changes in the regres- sion coefficients using F-statistics. The second strategy looks for potential breaks from the perspective of forecasting: You pretend that your sample ends sooner than it actually does and evaluate the forecasts you would have made had this been so. Breaks are detected when the forecasting performance is substantially poorer than expected.
What Is a Break?
Breaks can arise either from a discrete change in the population regression coef- ficients at a distinct date or from a gradual evolution of the coefficients over a longer period of time.
One source of discrete breaks in macroeconomic data is a major change in macroeconomic policy. For example, the breakdown of the Bretton Woods sys- tem of fixed exchange rates in 1972 produced the break in the time series behavior of the $ > £ exchange rate that is evident in Figure 14.2b. Prior to 1972, the exchange rate was essentially constant, with the exception of a single devaluation in 1968, in which the official value of the pound, relative to the dollar, was decreased. In contrast, since 1972 the exchange rate has fluctuated over a very wide range.
Breaks also can occur more slowly, as the population regression evolves over time. For example, such changes can arise because of slow evolution of economic policy and ongoing changes in the structure of the economy. The methods for detecting breaks described in this section can detect both types of breaks: distinct changes and slow evolution.
Problems caused by breaks. If a break occurs in the population regression func- tion during the sample, then the OLS regression estimates over the full sample will estimate a relationship that holds “on average,” in the sense that the estimate combines the two different periods. Depending on the location and the size of the break, the “average” regression function can be quite different from the true regression function at the end of the sample, and this leads to poor forecasts.
Testing for Breaks
One way to detect breaks is to test for discrete changes, or breaks, in the regres- sion coefficients. How this is done depends on whether the date of the suspected break (the break date) is known.
Testingforabreakataknowndate. Insomeapplications,youmightsuspectthat there is a break at a known date. For example, if you are studying international trade relationships using data from the 1970s, you might hypothesize that there is a break in the population regression function of interest in 1972, when the Bretton Woods system of fixed exchange rates was abandoned in favor of floating exchange rates.
If the date of the hypothesized break in the coefficients is known, then the null hypothesis of no break can be tested using a binary variable interaction regression of the type discussed in Chapter 8 (Key Concept 8.4). To keep things simple, consider an ADL(1,1) model, so there is an intercept, a single lag of Yt, and a single lag of Xt. Let t denote the hypothesized break date and let Dt(t) be a binary variable that equals 0 before the break date and 1 after, so Dt(t) = 0 if t … t and Dt(t) = 1 if t 7 t. Then the regression including the binary break indi- cator and all interaction terms is
Y =b +bY +dX +gD(t)+g3D(t)*Y 4 t01t-11t-10t1tt-1
+g3D(t)*X 4+u. 2tt-1t
(14.34)
If there is not a break, then the population regression function is the same over both parts of the sample, so the terms involving the break binary variable Dt(t) do not enter Equation (14.34). That is, under the null hypothesis of no break, g0 = g1 = g2 = 0. Under the alternative hypothesis that there is a break, the population regression function is different before and after the break date t, in which case at least one of the g’s is nonzero. Thus the hypothesis of a break can be tested using the F-statistic that tests the hypothesis that g0 = g1 = g2 = 0 against the hypothesis that at least one of the g’s is nonzero. This is often called a Chow test for a break at a known break date, named for its inventor, Gregory Chow (1960).
If there are multiple predictors or more lags, then this test can be extended by constructing binary variable interaction variables for all the regressors and testing the hypothesis that all the coefficients on terms involving Dt(t) are zero.
This approach can be modified to check for a break in a subset of the coeffi- cients by including only the binary variable interactions for that subset of regres- sors of interest.
Testingforabreakatanunknownbreakdate. Oftenthedateofapossiblebreak is unknown or known only within a range. Suppose, for example, that you suspect that a break occurred sometime between two dates, t0 and t1. The Chow test can be modified to handle this by testing for breaks at all possible dates t in between
14.7 Nonstationarity II: Break 563
564 ChapTeR 14 Introduction to Time Series Regression and Forecasting
t0 and t1 and then using the largest of the resulting F-statistics to test for a break at an unknown date. This modified Chow test is variously called the Quandt like- lihood ratio (QLR) statistic (Quandt, 1960) (the term we shall use) or, more obscurely, the sup-Wald statistic.
Because the QLR statistic is the largest of many F-statistics, its distribution is
not the same as an individual F-statistic. Instead, the critical values for the QLR
statistic must be obtained from a special distribution. Like the F-statistic, this dis-
tribution depends on the number of restrictions being tested, q—that is, the number
of coefficients (including the intercept) that are being allowed to break, or change,
on t >T and t >T, that is, on the endpoints, t and t , of the subsample over which 0101
under the alternative hypothesis. The distribution of the QLR statistic also depends
the F-statistics are computed, expressed as a fraction of the total sample size.
For the large-sample approximation to the distribution of the QLR statistic
to be a good one, the subsample endpoints, t0 and t1, cannot be too close to the
beginning or the end of the sample. For this reason, in practice the QLR statistic
is computed over a “trimmed” range, or subset, of the sample. A common choice
is to use 15% trimming, that is, to set for t = 0.15T and t = 0.85T (rounded to 01
the nearest integer). With 15% trimming, the F-statistic is computed for break dates in the central 70% of the sample.
The critical values for the QLR statistic, computed with 15% trimming, are given in Table 14.5. Comparing these critical values with those of the Fq, ∞ distri- bution (Appendix Table 4) shows that the critical values for the QLR statistics are larger. This reflects the fact that the QLR statistic looks at the largest of many individual F-statistics. By examining F-statistics at many possible break dates, the QLR statistic has many opportunities to reject the null hypothesis, leading to QLR critical values that are larger than the individual F-statistic critical values.
Like the Chow test, the QLR test can be used to focus on the possibility that there are breaks in only some of the regression coefficients. This is done by first computing the Chow tests at different break dates, using binary variable interac- tions only for the variables with the suspect coefficients, and then computing the maximum of those Chow tests over the range t0 … t … t1. The critical values for this version of the QLR test are also taken from Table 14.5, where the number of restrictions (q) is the number of restrictions tested by the constituent F-statistics.
If there is a discrete break at a date within the range tested, then the QLR statistic will reject with high probability in large samples. Moreover, the date at which the constituent F-statistic is at its maximum, tn, is an estimate of the break
tions, tn>T ¡ t>T; that is, the fraction of the way through the sample at which the break occurs is estimated consistently.
date t .This estimate is a good one in the sense that, under certain technical condi- p
14.7 Nonstationarity II: Break 565 taBLe 14.5 Critical Values of the QLR Statistic with 15% Trimming
number of restrictions (q)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
10% 5%
7.12 8.68
5.00 5.86
4.09 4.71
3.59 4.09
3.26 3.66
3.02 3.37
2.84 3.15
2.69 2.98
2.58 2.84
2.48 2.71
2.40 2.62
2.33 2.54
2.27 2.46
2.21 2.40
2.16 2.34
2.12 2.29
2.08 2.25
2.05 2.20
2.01 2.17
1.99 2.13
1%
12.16
7.78
6.02
5.12
4.53
4.12
3.82
3.57
3.38
3.23
3.09
2.97
2.87
2.78
2.71
2.64
2.58
2.53
2.48
2.43
Note: These critical values apply when t0
puted for all potential break dates in the central 70% of the sample. The number of restrictions q is the number of restrictions tested by each individual F-statistic. Critical values for other trimming percentages are given in Andrews (2003).
The QLR statistic also rejects the null hypothesis with high probability in large samples when there are multiple discrete breaks or when the break comes in the form of a slow evolution of the regression function. This means that the QLR statistic detects forms of instability other than a single discrete break. As a result, if the QLR statistic rejects the null hypothesis, it can mean that there is a single discrete break, that there are multiple discrete breaks, or that there is slow evolution of the regression function.
The QLR statistic is summarized in Key Concept 14.9.
=
0.15T and t1 =
0.85T (rounded to the nearest integer), so the F-statistic is com-
566 ChapTeR 14 Introduction to Time Series Regression and Forecasting
The QLR Test for Coefficient Stability
14.9
Key ConCept
Let F(t) denote the F-statistic testing the hypothesis of a break in the regression coefficients at date t; in the regression in Equation (14.34), for example, this is the F-statistic testing the null hypothesis that g0 = g1 = g2 = 0. The QLR (or sup-Wald) test statistic is the largest of statistics in the range t0 … t … t1:
QLR = max3F(t ), F(t + 1), c, F(t )4 . (14.35) 001
1. Like the F-statistic, the QLR statistic can be used to test for a break in all or just some of the regression coefficients.
2. In large samples, the distribution of the QLR statistic under the null hypoth-
esis depends on the number of restrictions being tested, q, and on the end-
points t0 and t1 as a fraction of T. Critical values are given in Table 14.5 for
15% trimming (t = 0.15T and t = 0.85T, rounded to the nearest integer). 01
3. The QLR test can detect a single discrete break, multiple discrete breaks, and/or slow evolution of the regression function.
4. If there is a distinct break in the regression function, the date at which the largest Chow statistic occurs is an estimator of the break date.
Warning: You probably don’t know the break date even if you think you do.
Sometimes an expert might believe that he or she knows the date of a possible break so that the Chow test can be used instead of the QLR test. But if this knowl- edge is based on the expert’s knowledge of the series being analyzed, then in fact this date was estimated using the data, albeit in an informal way. Preliminary estimation of the break date means that the usual F critical values cannot be used for the Chow test for a break at that date. Thus it remains appropriate to use the QLR statistic in this circumstance.
Application: Has the predictive power of the term spread been stable? The QLR test provides a way to check whether the GDP–term spread relation has been stable from 1962 to 2012. Specifically, we focus on whether there have been changes in the coefficients on the lagged values of the term spread and the inter- cept in the ADL(2,2) specification in Equation (14.16) containing two lags each, of GDPGRt and TSpreadt.
Figure 14.5
7
6
5
4
3
2
1
The Chow F-statistics testing the hypothesis that the intercept and the coef- ficients on TSpreadt-1, TSpreadt-2, and the intercept in Equation (14.16) are con- stant against the alternative that they break at a given date are plotted in Figure 14.5 for breaks in the central 70% of the sample. For example, the F-statistic testing for a break in 1975:Q1 is 1.93, the value plotted at that date in the figure. Each F-statistic tests three restrictions (no change in the intercept and in the two coefficients on lags of the term spread), so q = 3. The largest of these F-statistics is 6.39, which occurs in 1980:Q4; this is the QLR statistic. Comparing 6.39 to the critical values for q = 3 in Table 14.5 indicates that the hypothesis that these coef- ficients are stable is rejected at the 1% significance level. (The 1% critical value is 6.02.) Thus, there is statistically significant evidence that at least one of these coefficients changed over the sample.
Pseudo Out-of-Sample Forecasting
The ultimate test of a forecasting model is its out-of-sample performance—that is, its forecasting performance in “real time,” after the model has been estimated. Pseudo out-of-sample forecasting is a method for simulating the real-time perfor- mance of a forecasting model. The idea of pseudo out-of-sample forecasting is simple: Pick a date near the end of the sample, estimate your forecasting model using data up to that date, and then use that estimated model to make a forecast. Performing this exercise for multiple dates near the end of your sample yields a
F-Statistics Testing for a Break in equation (14.16) at Different Dates
QLR Statistic = 6.39
1% Critical value 5% Critical value
14.7 Nonstationarity II: Break 567
0
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
At a given break date, the F-statistic plotted here tests the null hypothesis of a break in at least one of the coefficients on TSpreadt−1, TSpreadt−2, or the intercept in Equation (14.16). For example, the F-statistic testing for a break in 1975:Q1 is 1.93. The QLR statistic is the largest of these F-statistics, which is 6.39. This exceeds the 1% critical value of 6.02.
568 ChapTeR 14 Introduction to Time Series Regression and Forecasting
pseudo Out-of-Sample Forecasts
14.10
Key ConCept
Pseudo out-of-sample forecasts are computed using the following steps:
1. Choose a number of observations, P, for which you will generate pseudo out- of-sample forecasts; for example, P might be 10% or 20% of the sample size. Let s = T – P.
2. Estimate the forecasting regression using the shortened data set for t = 1,c,s.
3. Compute the forecast for the first period beyond this shortened sample, s + 1; ∼
∼
5. Repeatsteps2through4fortheremainingdates,s = T – P + 1toT – 1
call this Ys + 1s .
∼
4. Compute the forecast error, u s + 1 = Ys + 1 – Ys + 1s .
(re-estimate the regression at each date). The pseudo out-of-sample forecasts
errors are 5 u , s = T – P, c, T – 16 . ∼s+1
∼
are{Ys+1s,s = T – P,c,T – 1},andthepseudoout-of-sampleforecast
series of pseudo forecasts and thus pseudo forecast errors. The pseudo forecast errors can then be examined to see whether they are representative of what you would expect if the forecasting relationship were stationary.
The reason this is called “pseudo” out-of-sample forecasting is that it is not true out-of-sample forecasting. True out-of-sample forecasting occurs in real time; that is, you make your forecast without the benefit of knowing the future values of the series. In pseudo out-of-sample forecasting, you simulate real-time forecasting using your model, but you have the “future” data against which to assess those simulated, or pseudo, forecasts. Pseudo out-of-sample forecasting mimics the forecasting pro- cess that would occur in real time, but without having to wait for new data to arrive.
Pseudo out-of-sample forecasting gives a forecaster a sense of how well the model has been forecasting at the end of the sample. This can provide valuable information, either bolstering confidence that the model has been forecasting well or suggesting that the model has gone off track in the recent past. The methodol- ogy of pseudo out-of-sample forecasting is summarized in Key Concept 14.10.
Other uses of pseudo out-of-sample forecasting. A second use of pseudo out-of- sample forecasting is to estimate the RMSFE. Because the pseudo out-of-sample
forecasts are computed using only data prior to the forecast date, the pseudo out- of-sample forecast errors reflect both the uncertainty associated with future val- ues of the error term and the uncertainty arising because the regression coefficients were estimated; that is, the pseudo out-of-sample forecast errors include both sources of error in Equation (14.20). Thus the sample standard deviation of the pseudo out-of-sample forecast errors is an estimator of the RMSFE. As discussed in Section 14.4, this estimator of the RMSFE can be used to quantify forecast uncertainty and to construct forecast intervals.
A third use of pseudo out-of-sample forecasting is to compare two or more candidate forecasting models. Two models that appear to fit the data equally well can perform quite differently in a pseudo out-of-sample forecasting exer- cise. When the models are different, such as when they include different pre- dictors, pseudo out-of-sample forecasting provides a convenient way to compare the two models that focuses on their potential to provide reliable forecasts.
Application: Did the predictive power of the term spread change during the 2000s? UsingtheQLRstatistic,werejectedthenullhypothesisthatthepredictive power of the term spread has been stable against the alternative of a break at the 1% significance level; that is, we rejected the joint hypothesis of no change in the intercept and the coefficients on the term spread in the ADL(2,2) model (see Fig- ure 14.5). The maximal F-statistic occurred in 1980:Q4, indicating that a break occurred in the early 1980s. This suggests that a forecaster using lagged values of the term spread to forecast the growth rate of GDP should use an estimation sam- ple starting after the break in 1980:Q4. Even so, a question remains: Does the ADL(2,2) model provide a stable forecasting model subsequent to the 1980:Q4 break?
If the coefficients of the ADL(2,2) model changed some time during the 1981:Q1–2012:Q4 period, then pseudo out-of-sample forecasts computed using data starting in 1981:Q1 should deteriorate. The pseudo out-of-sample forecasts of the growth rate of GDP for the period 2003:Q1–20012:Q4, computed using the ADL(2,2) model estimated with data starting in 1981:Q1, are plotted in Figure 14.6, along with the actual values of the growth rate of GDP. For example, the forecast of the growth rate of GDP for 2003:Q1 was computed by regressing GDPGRt on GDPGRt−1, GDPGRt−2, TSpreadt−1, and TSpreadt−2 with an intercept using the data through 2002:Q4, then computing the forecast GDPGR2003:Q12002:Q4 using these estimated coefficients and the data through 2002:Q4. This entire procedure was repeated using data through 2003:Q1 to compute the fore- cast GDPGR2003:Q22003:Q1.Doingthisforall40quartersfrom2003:Q1to2012:Q4
14.7 Nonstationarity II: Break 569
570 Chapter 14 Introduction to Time Series Regression and Forecasting Can You Beat the Market? Part II
Perhaps you have heard the advice that you should buy a stock when its earnings are high relative to its price. Buying a stock is, in effect, buying the stream of future dividends paid by that company out of its earnings. If the dividend stream is unusually large relative to the price of the company’s stock, then the company could be considered undervalued. If cur- rent dividends are an indicator of future dividends, then the dividend yield—the ratio of current dividends to the stock price—might forecast future excess stock returns. If the dividend yield is high, the stock is under- valued, and returns would be forecasted to go up.
This reasoning suggests examining autoregressive distributed lag models of excess returns, where the predictor variable is the dividend yield. But a difficulty arises with this approach: The dividend yield is highly persistent and might even contain a stochastic trend. Using monthly data from 1960:M1 to 2002:M12 on the logarithm of the dividend–price ratio for the CRSP value-weighted index (the data are described in Appendix 14.1), a Dickey–Fuller unit root test including an intercept fails to reject the null hypothesis of a unit root at the 10% signifi- cance level. As always, this failure to reject the null hypothesis does not mean that the null hypothesis is true, but it does underscore that the dividend yield is a highly persistent regressor. Following the logic of Section 14.6, this result suggests that we should use the first difference of the log dividend yield as a regressor, not the level of the log dividend yield.
Table 14.6 presents ADL models of excess returns on the CRSP value-weighted index. In columns (1) and (2), the dividend yield appears in first differences, and the individual t-statistics and joint F-statistics fail to reject the null hypothesis of no predictability. But while these specifications accord with the modeling recommendations of Section 14.6, they do not corre-
spond to the economic reasoning in the introductory paragraph, which relates returns to the level of the dividend yield. Column (3) of Table 14.6 therefore reports an ADL(1,1) model of excess returns using the log dividend yield, estimated through 1992:M12. The t-statistic is 2.25, which exceeds the usual 5% critical value of 1.96. However, because the regressor is highly persistent, the distribution of this t-statistic is suspect, and the 1.96 critical value may be inap- propriate. (The F-statistic for this regression is not reported because it does not necessarily have a chi- squared distribution, even in large samples, because of the persistence of the regressor.)
One way to evaluate the apparent predictability found in column (3) of Table 14.6 is to conduct a pseudo out-of-sample forecasting analysis. Doing so over the out-of-sample period 1993:M1–2002:M12 provides a sample root mean squared forecast error of 4.08%. In contrast, the sample RMSFEs of always forecasting excess returns to be zero is 4.00%, and the sample RMSFE of a “constant forecast” (in which the recursively estimated forecasting model includes only an intercept) is 3.98%. The pseudo out-of-sample forecast based on the ADL(1,1) model with the log dividend yield does worse than forecasts in which there are no predictors!
This lack of predictability is consistent with the strong form of the efficient markets hypothesis, which holds that all publicly available information is incorporated into stock prices so that returns should not be predictable using publicly available information. (The weak form concerns forecasts based on past returns only.) The core message that excess returns are not easily predicted makes sense: If they were, the prices of stocks would be driven up to the point that no expected excess returns would exist.
taBLe 14.6 autoregressive Distributed Lag Models of Monthly excess Stock Returns
14.7 Nonstationarity II: Break 571
The interpretation of results like those in Table 14.6 is a matter of heated debate among financial econo- mists. Some consider the lack of predictability in pre- dictive regressions to be a vindication of the efficient markets hypothesis (see, for example, Goyal and Welch, 2003). Others say that regressions over lon- ger time periods and longer horizons, when analyzed using tools that are specifically designed to handle persistent regressors, show evidence of predictabil- ity (see Campbell and Yogo, 2006). This predictabil- ity might arise from rational economic behavior, in which investor attitudes toward risk change over the
business cycle (Campbell, 2003), or it might reflect “irrational exuberance” (Shiller, 2005).
The results in Table 14.6 concern monthly returns, but some financial econometricians have focused on ever-shorter horizons. The theory of “market microstructure”—the minute-to-minute movements of the stock market—suggests that there can be fleeting periods of predictability and that money can be made by the clever and nimble. But doing so requires nerve, plus lots of computing power—and a staff of talented econo- metricians.
Dependent variable: excess returns on the CrSp value-weighted index
Specification
Estimation period
Regressors
(1)
ADL(1,1)
1960:M1–2002:M12
0.059 (0.158)
0.009 (0.157)
0.0031 (0.0020)
0.501 (0.606)
–0.0014
(2)
ADL(2,2)
1960:M1–2002:M12
0.042 (0.162)
– 0.213 (0.193)
– 0.012 (0.163)
– 0.161 (0.185)
0.0037 (0.0021)
0.843 (0.497)
–0.0008
(3)
ADL(1,1)
1960:M1–1992:M12
0.078 (0.057)
a 0.026
(0.012)
0.090a (0.039)
0.0134
∆ln1dividend yield ∆ln1dividend yield ln1dividend yield
Intercept
t-1
t-2
t-1
2 2 2
excess returnt – 1
excess returnt – 2
F-statistic on all coefficients (p- value)
R2
Note: The data are described in Appendix 14.1. Entries in the regressor rows are coefficients, with standard errors in paren- theses. The final two rows report the F-statistic testing the hypothesis that all the coefficients in the regression are zero, with its p-value in parentheses, and the adjusted R2.
a t 7 1 .96 .
572 Chapter 14 Introduction to Time Series Regression and Forecasting Figure 14.6 U.S. GDp Growth rates and pseudo Out-of-Sample Forecasts
The pseudo out-of- 7.5 sample forecasts made 5.0
Forecast growth rate of GDP
using the ADL(2,2)
model of the form in
Equation (14.16) gen-
erally track the actual
growth rate of GDP
from 2003 to 2012 –5.0 but fail to forecast the –7.5 sharp decline in GDP –10.0 following the financial
2.5 0 –2.5
Actual growth rate of GDP
Forecast errors
crisis of 2008.
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013
creates 40 pseudo out-of-sample forecasts, which are plotted in Figure 14.6. The pseudo out-of-sample forecast errors are the differences between the actual growth rate of GDP and its pseudo out-of-sample forecast—that is, the differ- ences between the two lines in Figure 14.6. For example, in 2006:Q4, the growth rate of GDP was 3.1 percentage points (at an annual rate), but the pseudo out-of- sample forecast of GDPGR2006:Q4 was 1.6 percentage points, so the pseudo out- of-sample forecast error was GDPGR2006:Q4 – GDPGR2006:Q42006:Q3 = 1.5 percentage points. In other words, a forecaster using the ADL(2,2), estimated through 2006:Q3, would have forecasted GDP growth of 1.6 percentage points in 2006:Q4, whereas in reality GDP grew by 3.1 percentage points.
How do the mean and standard deviation of the pseudo out-of-sample forecast errors compare with the in-sample fit of the model? The standard error of the regres- sion of the ADL(2,2) model fit using data from 1981:Q1 through 2002:Q4 is 2.39, so based on the in-sample fit, we would expect the out-of-sample forecast errors to have mean zero and root mean squared forecast error of 2.39. In fact, over the 2003:Q1–2012:Q4 pseudo out-of-sample forecast period, the average forecast error is −0.73 and the t-statistic testing the hypothesis that the mean forecast error equals zero is −1.87; thus the hypothesis that the forecasts have mean zero is rejected at the 10% significance level but not at the 5% significance level. In addition, the RMSFE over the pseudo out-of-sample forecast period is 2.54, somewhat higher than the value of 2.39 for the standard error of the regression for the 1981:Q1–2002:Q4 period. Examination of Figure 14.6 shows that the pseudo out-of-sample forecasts track the actual values’ GDP growth reasonably well except during late 2008 and early 2009, the period of steepest decline of GDP during the Great Recession. Excluding the single quarter 2008:Q4 lowers the pseudo out-of-sample RMSFE from 2.54 to 1.93.
According to the pseudo out-of-sample forecasting exercise, the performance of the ADL(2,2) forecasting model during the pseudo out-of-sample period 2003:Q1–2012:Q4 was, with the exception of the sharp decline in GDP in late 2008 following the financial crisis, comparable to its performance during the in-sample period of 1981:Q1–2002:Q4.6 Although the QLR test points to instability in the ALD(2,2) model in the early 1980s, this pseudo out-of-sample analysis suggests that, after the early 1980s break, the forecasting model has been stable.
Avoiding the Problems Caused by Breaks
The best way to adjust for a break in the population regression function depends on the source of that break. If a distinct break occurs at a specific date, that break will be detected with high probability by the QLR statistic, and the break date can be estimated. Thus the regression function can be estimated using a binary variable indicating the two subsamples associated with this break, interacted with the other regressors as needed. If all the coefficients break, then this regression takes the form of Equation (14.34), where t is replaced by the estimated break date, tn, while if only some of the coefficients break, only the relevant interaction terms appear in the regression. If there is in fact a distinct break, then inference on the regression coefficients can proceed as usual—for example, using the usual normal critical values for hypothesis tests based on t-statistics. In addition, forecasts can be pro- duced using the estimated regression function that applies to the end of the sample.
If the break is not distinct but rather arises from a slow, ongoing change in the parameters, the remedy is more difficult and goes beyond the scope of this book.7
14.8 Conclusion
In time series data, a variable generally is correlated from one observation, or date, to the next. A consequence of this correlation is that linear regression can be used to forecast future values of a time series, based on its current and past values. The starting point for time series regression is an autoregression, in which
6The ADL(2,2) was not alone in this failing to forecast GDP growth in 2008:Q4. Researchers at the Federal Reserve Bank of Philadelphia surveyed 47 professional forecasters in the third quarter of 2008 and asked for their forecasts of the growth rate of GDP in the fourth quarter. The median of the 47 forecasts was 0.7%, similar to the ADL(2,2) forecast of 1.0. The actual growth rate of GDP in 2008:Q4 was −8.7%.
7For additional discussion of estimation and testing in the presence of discrete breaks, see Hansen (2001). For an advanced discussion of estimation and forecasting when there are slowly evolving coef- ficients, see Hamilton (1994, Chapter 13).
14.8 Conclusion 573
574 ChapTeR 14 Introduction to Time Series Regression and Forecasting
the regressors are lagged values of the dependent variable. If additional predictors are available, then their lags can be added to the regression.
This chapter has considered several technical issues that arise when estimat- ing and using regressions with time series data. One such issue is determining the number of lags to include in the regressions. As discussed in Section 14.5, if the number of lags is chosen to minimize the BIC, then the estimated lag length is consistent for the true lag length.
Another of these issues concerns whether the series being analyzed are sta- tionary. If the series are stationary, then the usual methods of statistical inference (such as comparing t-statistics to normal critical values) can be used, and because the population regression function is stable over time, regressions estimated using historical data can be used reliably for forecasting. If, however, the series are nonstationary, then things become more complicated, and the specific complica- tion depends on the nature of the nonstationarity. For example, if the series is nonstationary because it has a stochastic trend, then the OLS estimator and t-statistic can have nonstandard (nonnormal) distributions, even in large samples, and forecast performance can be improved by specifying the regression in first differences. A test for detecting this type of nonstationarity—the augmented Dickey–Fuller test for a unit root—was introduced in Section 14.6. Alternatively, if the population regression function has a break, then neglecting this break results in estimating an average version of the population regression function that in turn can lead to biased and/or imprecise forecasts. Procedures for detecting a break in the population regression function were introduced in Section 14.7.
In this chapter, the methods of time series regression were applied to eco- nomic forecasting, and the coefficients in these forecasting models were not given a causal interpretation. You do not need a causal relationship to forecast, and ignoring causal interpretations liberates the quest for good forecasts. In some applications, however, the task is not to develop a forecasting model but rather to estimate causal relationships among time series variables—that is, to estimate the dynamic causal effect on Y over time of a change in X. Under the right conditions, the methods of this chapter, or closely related methods, can be used to estimate dynamic causal effects, and that is the topic of the next chapter.
Summary
1. Regression models used for forecasting need not have a causal interpretation.
2. A time series variable generally is correlated with one or more of its lagged
values; that is, it is serially correlated.
3. An autoregression of order p is a linear multiple regression model in which the regressors are the first p lags of the dependent variable. The coefficients of an AR(p) can be estimated by OLS, and the estimated regression func- tion can be used for forecasting. The lag order p can be estimated using an information criterion such as the BIC.
4. Adding other variables and their lags to an autoregression can improve fore- casting performance. Under the least squares assumptions for time series regression (Key Concept 14.6), the OLS estimators have normal distribu- tions in large samples, and statistical inference proceeds the same way as for cross-sectional data.
5. Using forecast intervals is one way to quantify forecast uncertainty. If the errors are normally distributed, an approximate 68% forecast interval can be constructed as the forecast plus or minus an estimate of the root mean squared forecast error.
6. A series that contains a stochastic trend is nonstationary, violating the sec- ond least squares assumption in Key Concept 14.6. The OLS estimator and t-statistic for the coefficient of a regressor with a stochastic trend can have a nonstandard distribution, potentially leading to biased estimators, ineffi- cient forecasts, and misleading inferences. The ADF statistic can be used to test for a stochastic trend. A random walk stochastic trend can be eliminated by using first differences of the series.
7. If the population regression function changes over time, then OLS estimates neglecting this instability are unreliable for statistical inference or forecast- ing. The QLR statistic can be used to test for a break, and, if a discrete break is found, the regression function can be re-estimated in a way that allows for the break.
8. Pseudo out-of-sample forecasts can be used to assess model stability toward the end of the sample, to estimate the root mean squared forecast error, and to compare different forecasting models.
Key Terms
gross domestic product (GDP) (524) first lag (526)
jth lag (526)
first difference (526) autocorrelation (528)
serial correlation (528)
autocorrelation coefficient (529) jth autocovariance (529) autoregression (531)
forecast error (532)
root mean squared forecast error (RMSFE) (533)
14.8 Key Terms 575
576 Chapter 14 Introduction to Time Series Regression and Forecasting
pth order autoregressive model [AR(p)] (534)
term spread (538)
autoregressive distributed lag (ADL)
model (539) ADL(p,q) (540)
stationarity (541)
weak dependence (543)
Granger causality statistic (543) Granger causality test (544) forecast interval (545)
Bayes information criterion (BIC)
(548)
Akaike information criterion (AIC)
(549) trend (551)
deterministic trend (552) stochastic trend (552) random walk (552)
random walk with drift (553) unit root (554)
spurious regression (555) Dickey–Fuller test (557) Dickey–Fuller statistic (557) augmented Dickey–Fuller (ADF)
statistic (558) breaks (562)
break date (562)
Quandt likelihood ratio (QLR)
statistic (564) pseudoout-of-sample forecasting
(567)
MyEconLab Can Help You Get a Better Grade
MyEconLab If your exam were tomorrow, would you be ready? For each chapter, MyeconLab Practice Tests and Study Plan help you prepare for your exams. You can also find the Exercises and all Review the Concepts Questions available now in MyeconLab.
To see how it works, turn to the MyeconLab spread on the inside front cover of this book and then go to www.myeconlab.com.
For additional Empirical Exercises and Data Sets, log on to the Companion Website at www.pearsonhighered.com/stock_watson.
Review the Concepts
14.1 Look at the plot of the logarithm of the index of industrial production for Japan in Figure 14.2c. Does this time series appear to be stationary? Explain. Suppose that you calculated the first difference of this series. Would it appear to be stationary? Explain.
14.2 Many financial economists believe that the random walk model is a good description of the logarithm of stock prices. It implies that the percentage changes in stock prices are unforecastable. A financial analyst claims to have a new model that makes better predictions than the random walk model. Explain how you would examine the analyst’s claim that his model is superior.
14.3 A researcher estimates an AR(1) with an intercept and finds that the OLS estimate of b1 is 0.95, with a standard error of 0.02. Does a 95% confidence interval include b1 = 1? Explain.
14.4 Suppose that you suspected that the intercept in Equation (14.16) changed in 1992:Q1. How would you modify the equation to incorporate this change? How would you test for a change in the intercept? How would you test for a change in the intercept if you did not know the date of the change?
Exercises
14.1 Consider the AR(1) model Yt = b0 + b1Yt – 1 + ut. Suppose that the pro- cess is stationary.
a. Show that E(Yt) = E(Yt – 1). (Hint: Read Key Concept 14.5.)
b. Show that E(Y ) = b >(1 – b ).
14.2 The index of industrial production (IPt) is a monthly time series that mea-
sures the quantity of industrial commodities produced in a given month.
t01
This problem uses data on this index for the United States. All regressions
ary 1986 through December 2013). Let Y = 1200 * ln(IP >IP ). ttt-1
a. The forecaster states that Yt shows the monthly percentage change in IP, measured in percentage points per annum. Is this correct? Why?
b. Suppose that a forecaster estimates the following AR(4) model for Yt: Ynt = 0.787 + 0.052Yt-1 + 0.185Yt-2 + 0.234Yt-3 + 0.164Yt-4.
(0.539) (0.093) (0.053) (0.078) (0.066)
Use this AR(4) to forecast the value of Yt in January 2014, using the follow-
ing values of IP for July 2013 through December 2013:
Date 2013:M7 2013:M8 2013:M9 2013:M10 2013:M11 2013:M12
IP 99.016 99.561 100.196 100.374 101.034 101.359
c. Worried about potential seasonal fluctuations in production, the fore- caster adds Yt – 12 to the autoregression. The estimated coefficient on Yt – 12 is −0.063 with a standard error of 0.045. Is this coefficient statis- tically significant?
are estimated over the sample period 1986:M1 to 2013:M12 (that is, Janu-
Exercises 577
578 ChapTeR 14 Introduction to Time Series Regression and Forecasting
d. Worried about a potential break, she computes a QLR test (with 15% trimming) on the constant and AR coefficients in the AR(4) model. The resulting QLR statistic was 3.94. Is there evidence of a break? Explain.
e. Worried that she might have included too few or too many lags in the model, the forecaster estimates AR(p) models for p = 0, 1, c, 6 over the same sample period. The sum of squared residuals from each of these estimated models is shown in the table. Use the BIC to estimate the number of lags that should be included in the autoregression. Do the results differ if you use the AIC?
arorder 0 1 2 3 4 5 6
SSR 19,533 18,643 17,377 16,285 15,842 15,824 15,824 14.3. Using the same data as in Exercise 14.2, a researcher tests for a stochastic
trend in ln(IPt), using the following regression:
∆ln(IPt) = 0 .030 + 0 .000014t – 0 .0085 ln(IPt – 1) + 0 .050∆ln(IPt – 1)
(0 .015) (0 .000009) (0 .0044) (0 .054)
+ 0.186∆ln(IPt-2) + 0.240∆ln(IPt-3) + 0.173∆ln(IPt-4),
10.0532 10.0532 10.0542
where the standard errors shown in parentheses are computed using the
homoskedasticity-only formula and the regressor t is a linear time trend.
a. Use the ADF statistic to test for a stochastic trend (unit root) in ln(IP).
b. Do these results support the specification used in Exercise 14.2? Explain.
14.4 The forecaster in Exercise 14.2 augments her AR(4) model for IP growth to include four lagged values of ∆Rt, where Rt is the interest rate on 3-month U.S. Treasury bills (measured in percentage points at an annual rate).
a. The Granger-causality F-statistic on the four lags of ∆Rt is 4.16. Do interest rates help predict IP growth? Explain.
b. The researcher also regresses ∆Rt on a constant, four lags of ∆Rt and four lags of IP growth. The resulting Granger-causality F-statistic
on the four lags of IP growth is 1.52. Does IP growth help to predict interest rates? Explain.
Exercises 579 14.5 Prove the following results about conditional means, forecasts, and forecast
b. Consider the problem of forecasting Yt, using data on Yt – 1, Yt – 2, c .
errors:
a. Let W be a random variable with mean mW and variance s2w and let c
2 2w W 2 be a constant. Show that E3(W – c) 4 = s + (m – c) .
Let ft – 1 denote some forecast of Yt, where the subscript t – 1 on ft – 1
E3(Y – f ) 0 Y , Y , c4 be the conditional mean squared error t t-12 t-1 t-2
indicates that the forecast is a function of data through date t – 1. Let
of the forecast ft – 1, conditional on values of Y observed through date
mizedwhen f = Y ,whereY = E(Y 0Y ,Y ,c).(Hint: t-1 t0t-1 t0t-1 t t-1 t-2
Review Exercise 2.27.)
c. Let ut denote the error in Equation (14.13). Show that cov(ut, ut – j) = 0
for j ≠ 0. [Hint: Use Equation (2.27).]
14.6 In this exercise you will conduct a Monte Carlo experiment to study the phenomenon of spurious regression discussed in Section 14.6. In a Monte Carlo study, artificial data are generated using a computer, and then those artificial data are used to calculate the statistics being studied. This makes it possible to compute the distribution of statistics for known models when mathematical expressions for those distributions are complicated (as they are here) or even unknown. In this exercise, you will generate data so that two series, Yt and Xt, are independently distributed random walks. The specific steps are as follows:
i. Use your computer to generate a sequence of T = 100 i.i.d. standard normal random variables. Call these variables e1, e2, c, e100. Set
Y1 = e1 and Yt = Yt – 1 + et for t = 2, 3, c, 100.
ii. Use your computer to generate a new sequence, a1, a2, c, a100, of T = 100 i.i.d. standard normal random variables. Set X1 = a1 and Xt = Xt-1 + at fort = 2,3,c,100.
iii. Regress Yt onto a constant and Xt. Compute the OLS estimator, the regression R2, and the (homoskedastic-only) t-statistic testing the null hypothesis that b1 (the coefficient on Xt) is zero.
Use this algorithm to answer the following questions:
a. Run the algorithm (i) through (iii) once. Use the t-statistic from (iii) to test the null hypothesis that b1 = 0, using the usual 5% critical value of 1.96. What is the R2 of your regression?
t – 1. Show that the conditional mean squared forecast error is mini-
580 ChapTeR 14 Introduction to Time Series Regression and Forecasting
b. Repeat (a) 1000 times, saving each value of R2 and the t-statistic. Construct a histogram of the R2 and t-statistic. What are the 5%, 50%, and 95% percentiles of the distributions of the R2 and the t-statistic? In what fraction of your 1000 simulated data sets does the t-statistic exceed 1.96 in absolute value?
c. Repeat (b) for different numbers of observations, such as T = 50 and T = 200. As the sample size increases, does the fraction of times that you reject the null hypothesis approach 5%, as it should because you have generated Y and X to be independently distributed? Does this fraction seem to approach some other limit as T gets large? What is that limit?
14.7 Suppose that Yt follows the stationary AR(1) model Yt = 2 .5 + 0 .7Yt – 1 + ut, where u is i.i.d. with E1u 2 = 0 and var1u 2 = 9.
ttt
a. Compute the mean and variance of Yt. (Hint: See Exercise 14.1.)
b. Compute the first two autocovariances of Yt. (Hint: Read Appendix 14.2.)
c. Compute the first two autocorrelations of Yt.
d. SupposethatY = 102.3.ComputeY = E(Y 0Y ,Y ,c). T T+10T T+1 T t-1
14.8 Suppose that Yt is the monthly value of the number of new home construc- tion projects started in the United States. Because of the weather, Yt has a pronounced seasonal pattern; for example, housing starts are low in Janu- ary and high in June. Let mJan denote the average value of housing starts in January and let mFeb, mMar, c, mDec denote the average values in the other months. Show that the values of mJan, mFeb, c, mDec can be estimated from theOLSregressionYt = b0 + b1Febt + b2Mart + g+b11Dect + ut,where Febt is a binary variable equal to 1 if t is February, Mart is a binary variable equal to 1 if t is March, and so forth. (Hint: Show that b0 + b2 = mMar, and so forth.)
14.9 The moving average model of order q has the form
Yt =b0 +et +b1et-1 +b2et-2 + g+bqet-q,
where et is a serially uncorrelated random variable with mean 0 and vari- 2
ance se.
a. Show that E1Y 2 = b .
t0
b. Show that the variance of Yt is var(Yt) = s2e(1 + b21 + b2 + g+ b2q).
c. Show that rj = 0 for j > q.
d. Suppose that q = 1. Derive the autocovariances for Y.
14.10 A researcher carries out a QLR test using 25% trimming, and there are q = 5 restrictions. Answer the following questions, using the values in Table 14.5 (“Critical Values of the QLR Statistic with 15% Trimming”) and Appendix Table 4 (“Critical Values of the Fm, ∞ Distribution”).
a. The QLR F-statistic is 4.2. Should the researcher reject the null hypothesis at the 5% level?
b. The QLR F-statistic is 2.1. Should the researcher reject the null hypothesis at the 5% level?
c. The QLR F-statistic is 3.5. Should the researcher reject the null hypothesis at the 5% level?
14.11 Suppose that ∆Yt follows the AR(1) model ∆Yt = b0 + b1∆Yt – 1 + ut.
a. Show that Yt follows an AR(2) model.
b. Derive the AR(2) coefficients for Yt as a function of b0 and b1.
Empirical Exercises
E14.1 On the text website, http://www.pearsonhighered.com/stock_watson, you will find the data file USMacro_Quarterly, which contains quarterly data on several macroeconomic series for the United States; the data are described in the file USMacro_Description. The variable PCEP is the price index for personal consumption expenditures from the U.S. National Income and Product Accounts. In this exercise you will construct forecast- ing models for the rate of inflation, based on PCEP. For this analysis, use the sample period 1963:Q1–2012:Q4 (where data before 1963 may be used, as necessary, as initial values for lags in regressions).
a.
i. Compute the inflation rate, Infl = 400 * 3 ln (PCEP ) – ln (PCEP )4. t t-1
What are the units of Infl? (Is Infl measured in dollars, percentage points, percentage per quarter, percentage per year, or something else? Explain.)
ii. Plot the value of Infl from 1963:Q1 through 2012:Q4. Based on the plot, do you think that Infl has a stochastic trend? Explain.
b. i. Compute the first four autocorrelations of ∆Infl.
ii. Plot the value of ∆Infl from 1963:Q1 through 2012:Q4. The plot should look “choppy” or “jagged.” Explain why this behavior is consistent with the first autocorrelation that you computed in part (i).
Empirical Exercises 581
582 ChapTeR 14 Introduction to Time Series Regression and Forecasting
c
i. Run an OLS regression of ∆Inflt on ∆Inflt-1. Does knowing the change in inflation this quarter help predict the change in inflation next quarter? Explain.
ii. Estimate an AR(2) model for ∆Infl. Is the AR(2) model better than an AR(1) model? Explain.
iii. Estimate an AR(p) model for p = 0, c, 8. What lag length is chosen by BIC? What lag length is chosen by AIC?
iv. Use the AR(2) model to predict the change in inflation from 2012:Q4 to 2013:Q1—that is, predict the value of ∆Infl2013:Q1.
v. Use the AR(2) model to predict the level of the inflation rate in 2013:Q1—that is, Infl2013:Q1.
i. Use the ADF test for the regression in Equation (14.31) with two lags of ΔInfl to test for a stochastic trend in Infl.
ii. Is the ADF test based on Equation (14.31) preferred to the test based on Equation (14.32) for testing for stochastic trend in Infl? Explain.
iii. In (i) you used two lags of ΔInfl. Should you use more lags? Fewer lags? Explain.
iv. Based on the test you carried out in (i), does the AR model for Infl contain a unit root? Explain carefully. (Hint: Does the failure to reject a null hypothesis mean that the null hypothesis is true?)
Use the QLR test with 15% trimming to test the stability of the coeffi- cients in the AR(2) model for ΔInfl. Is the AR(2) model stable? Explain.
i. Using the AR(2) model for ΔInfl with a sample period that begins in 1963:Q1, compute pseudo out-of-sample fore- casts for the change in inflation beginning in 2003:Q1 and going through 2012:Q4. (That is, compute ∆Infl2003:Q12002:Q4, ∆Infl2003:Q22003:Q1, c, ∆Infl2012:Q42012:Q3.)
ii. Are the pseudo out-of-sample forecasts biased? That is, do the forecast errors have a nonzero mean?
iii. How large is the RMSFE of the pseudo out-of-sample forecasts? Is this consistent with the AR(2) model for ΔInfl estimated over the 1963:Q1–2002:Q4 sample period?
iv. There is a large outlier in 2008:Q4. Why did inflation fall so much in 2008:Q4? (Hint: Collect some data on oil prices. What happened to oil prices during 2008?)
d.
e. f.
Appendix
E14.2 Read the boxes “Can You Beat the Market? Part I” and “Can You Beat the Market? Part II” in this chapter. Next, go to the course website, where you will find an extended version of the data set described in the boxes; the data are in the file Stock_Returns_1931_2002 and are described in the file Stock_Returns_1931_2002_Description.
a. Repeat the calculations reported in Table 14.2, using regressions estimated over the 1932:M1–2002:M12 sample period.
b. Repeat the calculations reported in Table 14.6, using regressions esti- mated over the 1932:M1–2002:M12 sample period.
c. Is the variable ln(dividend yield) highly persistent? Explain.
d. Construct pseudo out-of-sample forecasts of excess returns over the 1983:M1–2002:M12 period, using regressions that begin in 1932:M1.
e. Do the results in (a) through (d) suggest any important changes to the conclusions reached in the boxes? Explain.
Time Series Data Used in Chapter 14 583
14.1 Time Series Data Used in Chapter 14
Macroeconomic time series data for the United States are collected and published by various government agencies. The Bureau of Economic Analysis in the Department of Commerce publishes the National Income and Product Accounts, which include the GDP data used in this chapter. The unemployment rate is computed from the Bureau of Labor Statistics’s Current Population Survey (see Appendix 3.1). The quarterly data used here were computed by averaging the monthly values. The 10-year Treasury bond rate, 3-month Treasury bill rate, and the dollar/pound exchange rate data are quarterly averages of daily rates, as reported by the Federal Reserve. The index of industrial production for Japan is published by the Organisation for Economic Co-operation and Development (OECD). The daily percentage change in the Wilshire 5000 stock price index was computed as 100Δln(W5000t), where W5000t is the daily value of the index; because the stock exchange is not open on weekends and holidays, the time period of analysis is a business day. We obtained all these data series from the Federal Reserve Economic Data (FRED) website at the Federal Reserve Bank of St. Louis. There you can find times series data on thou- sands of macroeconomic variables.
The regressions in Table 14.2 and 14.6 use monthly financial data for the United States. Stock prices (Pt) are measured by the broad-based (NYSE and AMEX) value-weighted
584 ChapTeR 14 Introduction to Time Series Regression and Forecasting
index of stock prices constructed by the Center for Research in Security Prices (CRSP). The
appenDix
monthly percentage excess return is 100 * 5ln3(P + Div ) > P 46 – ln(TBill )6, where ttt-1t
Divt is the dividends paid on the stocks in the CRSP index and TBillt is the gross return (1 plus the interest rate) on a 30-day Treasury bill during month t. The dividend–price ratio is constructed as the dividends over the past 12 months, divided by the price in the current month. We thank Motohiro Yogo for his help and for providing these data.
14.2 Stationarity in the AR(1) Model
This appendix shows that if b1 6 1 and ut is stationary, then Yt is stationary. Recall from Key Concept 14.5 that the time series variable Yt is stationary if the joint distribution of (Ys + 1, c, Ys + T) does not depend on s, regardless of the value of T. To streamline the argument, we show this formally for T = 2 under the simplifying assumptions that b = 0 and 5u 6 are i.i.d. N(0,s ).
0 t 2u
The first step is deriving an expression for Yt in terms of the ut’s. Because b0 = 0,
Equation (14.8) implies that Yt = b1Yt – 1 + ut. Substituting Yt – 1 = b1Yt – 2 + ut – 1 into this expression yields Yt = b1(b1Yt-2 + ut-1) + ut = b21Yt-2 + b1ut-1 + ut. Continuing this substitution another step yields Yt = b31Yt-3 + b21ut-2 + b1ut-1 + ut, and continuing indefinitely yields
Y = u + b u + b2u + b3u + g= biu . (14.36) t t 1 t – 1 1 t – 2 1 t – 3 a∞ 1 t – i
i=0
Thus Yt is a weighted average of current and past ut’s. Because the ut’s are normally dis- tributed and because the weighted average of normal random variables is normal (Section 2.4), Ys + 1 and Ys + 2 have a bivariate normal distribution. Recall from Section 2.4 that the bivariate normal distribution is completely determined by the means of the two variables, their variances, and their covariance. Thus, to show that Yt is stationary, we need to show that the means, variances, and covariance of (Ys + 1, Ys + 2) do not depend on s. An extension of the argument used below can be used to show that the distribution of (Ys + 1, Ys + 2, c, Ys + T) does not depend on s.
The means and variances of Ys + 1 and Ys + 2 can be computed using Equation (14.36), with the subscript s + 1 or s + 2 replacing t. First, because E(ut) = 0 for all t,
∞i ∞i
E(Y) = E(g b u ) = g b E(u ) = 0, so the means of Y and Y are both
zero and in particular do not depend on s. Second, var(Y) = var(g b u ) = t i=0 1 t-i
t i=01t-i i=01 t-i s+1 s+2 ∞
i
appenDix
Lag Operator Notation 585 g (b ) var(u ) = s g (b ) = s >(1 – b ),wherethefinalequalityfollowsfromthe
i=0 1 t-i
fact that if a 6 1, g a = 1 > (1 – a); thus var(Y ) = var(Y ) = s > (1 – b ),
∞i22∞i222
u i=0 1 u 1 ∞i22
i=0 s+1 s+2 u 1 which does not depend on s as long as b1 6 1. Finally, because Ys+2 = b1Ys+1 + us+2,
cov(Y Y )=E(Y Y )=E3Y (bY +u )4=b var(Y )+cov(Y ,u ) s+1, s+2 s+1, s+2 s+1 1 s+1 s+2 1 s+1 s+1 s+2
= b var(Y ) = b s >(1 – b ). 1 s+1 12u 21
The covariance does not depend on s, so Ys + 1 and Ys + 2 have a joint probability distri- bution that does not depend on s; that is, their joint distribution is stationary. If b1 Ú 1, this calculation breaks down because the infinite sum in Equation (14.36) does not converge, and the variance of Yt is infinite. Thus Yt is stationary if b1 6 1 but not if b1 Ú 1.
The preceding argument was made under the assumptions that b0 = 0 and ut is nor-
Y are b >(1 – b ), and Equation (14.36) must be modified for this nonzero mean. The s+201
mally distributed. If b0 ≠ 0, the argument is similar except that the means of Ys + 1 and
assumption that ut is i.i.d. normal can be replaced with the assumption that ut is stationary with a finite variance because, by Equation (14.36), Yt can still be expressed as a function of current and past ut’s, so the distribution of Yt is stationary, as long as the distribution of ut is stationary and the infinite sum expression in Equation (14.36) is meaningful in the sense that it converges, which requires that b1 6 1.
14.3 Lag Operator Notation
The notation in this and the next two chapters is streamlined considerably by adopting what is known as lag operator notation. Let L denote the lag operator, which has the property that it transforms a variable into its lag. That is, the lag operator L has the prop- erty LYt = Yt-1. By applying the lag operator twice, one obtains the second lag: L2Yt = L(LYt) = LYt – 1 = Yt – 2. More generally, by applying the lag operator j times, one obtains the jth lag. In summary, the lag operator has the property that
LYt = Yt-1, L2Yt = Yt-2, and LjYt = Yt-j. (14.37) The lag operator notation permits us to define the lag polynomial, which is a polynomial
in the lag operator:
a(L) = a0 + a1L + a2L2 + g +apLp =
ap j=0
ajLj, (14.38)
586 ChapTeR 14 Introduction to Time Series Regression and Forecasting
where a0, c, ap are the coefficients of the lag polynomial and L0 = 1. The degree of the
lag polynomial a(L) in Equation (14.38) is p. Multiplying Yt by a(L) yields jj
a(L)Yt = aap ajL bYt = ap aj(LYt)= ap ajYt-j = a0Yt + a1Yt-1 + g + apYt-p. (14.39) j=0 j=0 j=0
The expression in Equation (14.39) implies that the AR(p) model in Equation (14.13) can be written compactly as
a(L)Yt = b0 + ut, (14.40)
where a0 = 1 and aj = -bj, for j = 1, c, p. Similarly, an ADL(p, q) model can be written
a(L)Yt = b0 + c(L)Xt-1 + ut, (14.41)
where a(L) is a lag polynomial of degree p (with a0 = 1) and c(L) is a lag polynomial of degree q – 1.
appenDix
14.4 ARMA Models
The autoregressive–moving average (ARMA) model extends the autoregressive model by modeling ut as serially correlated, specifically as being a distributed lag (or “moving aver- age”) of another unobserved error term. In the lag operator notation of Appendix 14.3, let ut = b(L)et, where b(L) is a lag polynomial of degree q with b0 = 1 and et is a serially uncorrelated, unobserved random variable. Then the ARMA(p,q) model is
a(L)Yt = b0 + b(L)et, (14.42)
where a(L) is a lag polynomial of degree p with a0 = 1.
Both the AR and ARMA models can be thought of as ways to approximate the auto-
covariances of Yt. The reason for this is that any stationary time series Yt with a finite variance can be written either as an AR or as a MA with a serially uncorrelated error term, although the AR or MA models might need to have an infinite order. The second of these results, that a stationary process can be written in moving average form, is known as the Wold decomposition theorem and is one of the fundamental results underlying the theory of stationary time series analysis.
Appendix
Consistency of the BIC Lag Length Estimator 587
As a theoretical matter, the families of AR, MA, and ARMA models are equally rich, as long as the lag polynomials have a sufficiently high degree. Still, in some cases the auto- covariances can be better approximated using an ARMA(p,q) model with small p and q than by a pure AR model with only a few lags. As a practical matter, however, the estima- tion of ARMA models is more difficult than the estimation of AR models, and ARMA models are more difficult to extend to additional regressors than are AR models.
14.5 Consistency of the BIC Lag Length Estimator
This appendix summarizes the argument that the BIC estimator of the lag length, pn, in an autoregression is correct in large samples; that is, Pr( pn = p) S 1. This is not true for the AIC estimator, which can overestimate p even in large samples.
BIC
First consider the special case that the BIC is used to choose among autoregressions with zero, one, or two lags, when the true lag length is one. It is shown below that (i) Pr(pn = 0)S0 and (ii) Pr(pn = 2)S0, from which it follows that Pr(pn = 1)S1. The extension of this argument to the general case of searching over 0 … p … pmax entails showing that Pr(pn 6 p) S 0 and Pr(pn 7 p) S 0; the strategy for showing these is the same as used in (i) and (ii) below.
Proof of (i) and (ii)
Proof of (i). To choose pn = 0 it must be the case that BIC(0) 6 BIC(1); that is, BIC(0) – BIC(1) 6 0. Now BIC(0) – BIC(1) = 3ln(SSR(0)>T) + (lnT)>T4 – 3ln(SSR(1)>T)4 + 2(lnT)>T4 = ln(SSR(0)>T) – ln(SSR(1)>T) – (lnT)>T. Now SSR(0)>T = 3(T – 1)>T4sY2 ¡p sY2,SSR(1)>T ¡p s2u,and(lnT)>T ¡ 0;putting these pieces together, BIC(0) – BIC(1) ¡p lnsY2 – lns2u 7 0 because s2Y 7 s2u. It follows that Pr3BIC(0) 6 BIC(1)4 S 0, so Pr(pn = 0) ¡ 0.
Proof of (ii). To choose pn = 2, it must be the case that BIC(2) 6 BIC(1) or BIC(2) – BIC(1) 6 0. Now T3BIC(2) – BIC(1)4 = T53ln(SSR(2)>T) + 3(lnT)>T] – 3ln(SSR(1)>T) + 2(lnT)>T46 = T ln3SSR(2)>SSR(1)4 + lnT = -Tln31 + F>(T – 2)] + lnT, where F = 3SSR(1) – SSR(2)4 > 3SSR(2) > (T – 2)4 is the homoskedasticity-only F-statistic [Equation (7.13)] testing the null hypothesis that b2 = 0 in the AR(2). If ut is
588 ChapTeR 14 Introduction to Time Series Regression and Forecasting
homoskedastic, then F has a x21 asymptotic distribution; if not, it has some other asymp- totic distribution. Thus Pr3BIC(2) – BIC(1) 6 04 = Pr5T3BIC(2) – BIC(1)4 6 06
= Pr5-Tln31 + F>(T – 2)4 + (lnT) 6 06 = Pr5Tln31 + F>(T – 2)4 7 lnT6. p
As T increases, T ln[1 + F>(T – 2)] – F ¡ 0 [a consequence of the logarithmic approximation ln(1 + a) ≅ a, which becomes exact as a ¡ 0]. Thus Pr3BIC(2) – BIC(1) 6 04 ¡ Pr(F 7 lnT) ¡ 0, so Pr(pn = 2) ¡ 0.
AIC
In the special case of an AR(1) when zero, one, or two lags are considered, (i) applies to
the AIC where the term lnT is replaced by 2, so Pr(pn = 0) ¡ 0. All the steps in the
by 2; thus Pr3AIC(2) – AIC(1) 6 04 ¡ Pr(F 7 2) 7 0. If u is homoskedastic, then t
Pr(F 7 2) ¡ Pr(x21 7 2) = 0.16, so Pr(pn = 2) ¡ 0.16. In general, when pn is chosen using the AIC, Pr(pn 6 p) ¡ 0, but Pr(pn 7 p) tends to a positive number, so Pr(pn = p) does not tend to 1.
proof of (ii) for the BIC also apply to the AIC, with the modification that lnT is replaced
CHAPTER
15
Estimation of Dynamic Causal Effects
In the 1983 movie Trading Places, the characters played by Dan Aykroyd and Eddie Murphy used inside information on how well Florida oranges had fared over the winter to make millions in the orange juice concentrate futures market, a market for contracts to buy or sell large quantities of orange juice concentrate at a specified price on a future date. In real life, traders in orange juice futures in fact do pay close attention to the weather in Florida: Freezes in Florida kill Florida oranges, the source of almost all frozen orange juice concentrate made in the United States, so its sup- ply falls and the price rises. But precisely how much does the price rise when the weather in Florida turns sour? Does the price rise all at once, or are there delays; if so, for how long? These are questions that real-life traders in orange juice futures need to answer if they want to succeed.
This chapter takes up the problem of estimating the effect on Y now and in the future of a change in X, that is, the dynamic causal effect on Y of a change in X. What, for example, is the effect on the path of orange juice prices over time of a freezing spell in Florida? The starting point for modeling and estimating dynamic causal effects is the so-called distributed lag regression model, in which Yt is expressed as a function of current and past values of Xt. Section 15.1 introduces the distributed lag model in the context of estimating the effect of cold weather in Florida on the price of orange juice concentrate over time. Section 15.2 takes a closer look at what, precisely, is meant by a dynamic causal effect.
One way to estimate dynamic causal effects is to estimate the coefficients of the distributed lag regression model using OLS. As discussed in Section 15.3, this estimator is consistent if the regression error has a conditional mean of zero given current and past values of X, a condition that (as in Chapter 12) is referred to as exo- geneity. Because the omitted determinants of Yt are correlated over time—that is, because they are serially correlated—the error term in the distributed lag model can be serially correlated. This possibility in turn requires “heteroskedasticity- and autocorrelation-consistent” (HAC) standard errors, the topic of Section 15.4.
A second way to estimate dynamic causal effects, discussed in Section 15.5, is to model the serial correlation in the error term as an autoregression and then to use this autoregressive model to derive an autoregressive distributed lag (ADL) model. Alternatively, the coefficients of the original distributed lag model can be estimated
589
590
CHAPTER 15
Estimation of Dynamic Causal Effects
15.1
An Initial Taste of the Orange Juice Data
Orlando, the historical center of Florida’s orange-growing region, is normally sunny and warm. But now and then there is a cold snap, and if temperatures drop below freezing for too long, the trees drop many of their oranges. If the cold snap is severe, the trees freeze. Following a freeze, the supply of orange juice concen- trate falls and its price rises. The timing of the price increases is rather complicated, however. Orange juice concentrate is a “durable,” or storable, commodity; that is, it can be stored in its frozen state, albeit at some cost (to run the freezer). Thus the price of orange juice concentrate depends not only on current supply but also on expectations of future supply. A freeze today means that future supplies of con- centrate will be low, but because concentrate currently in storage can be used to meet either current or future demand, the price of existing concentrate rises today. But precisely how much does the price of concentrate rise when there is a freeze? The answer to this question is of interest not just to orange juice traders but more generally to economists interested in studying the operations of modern commod- ity markets. To learn how the price of orange juice changes in response to weather conditions, we must analyze data on orange juice prices and the weather.
Monthly data on the price of frozen orange juice concentrate, its monthly percentage change, and temperatures in the orange-growing region of Florida from January 1950 to December 2000 are plotted in Figure 15.1. The price, plot- ted in Figure 15.1a, is a measure of the average real price of frozen orange juice concentrate paid by wholesalers. This price was deflated by the overall producer price index for finished goods to eliminate the effects of overall price inflation.
by generalized least squares (GLS). Both the ADL and GLS methods, however, require a stronger version of exogeneity than we have used so far: strict exogeneity, under which the regression errors have a conditional mean of zero given past, present, and future values of X.
Section 15.6 provides a more complete analysis of the relationship between orange juice prices and the weather. In this application, the weather is beyond human control and thus is exogenous (although, as discussed in Section 15.6, economic theory suggests that it is not necessarily strictly exogenous). Because exogeneity is necessary for estimating dynamic causal effects, Section 15.7 examines this assumption in several applications taken from macroeconomics and finance.
This chapter builds on the material in Sections 14.1 through 14.4 but, with the exception of a subsection (that can be skipped) of the empirical analysis in Section 15.6, does not require the material in Sections 14.5 through 14.7.
15.1 An Initial Taste of the Orange Juice Data 591 FIGURE 15.1 Orange Juice Prices and Florida Weather, 1950–2000
Price index
Percent
50 40 30 20 10
0
-10
-20
-30
-40
1950
250 200 150 100
50 0
1960
1970
1980
1990
2000
1950
1960
1970
1980
1990
2000
Year
Year
(a) Price Index for Frozen Concentrated Orange Juice
(b) Percent Change in the Price of Frozen Concentrated Orange Juice
Freezing degree days
40 35 30 25 20 15 10
5 0
1950 1960 1970
1980 1990 2000
Year
(c) Monthly Freezing Degree Days in Orlando, Florida
There have been large month-to-month changes in the price of frozen concentrated orange juice. Many of the large movements coincide with freezing weather in Orlando, home of many orange groves.
The percentage price change plotted in Figure 15.1b is the percent change in the price over the month. The temperature data plotted in Figure 15.1c are the number of “freezing degree days” at the Orlando, Florida, airport, calculated as the sum of the number of degrees Fahrenheit that the minimum temperature falls below freezing in a given day over all days in the month; for example, in November 1950 the airport temperature dropped below freezing twice, on the 25th (31°) and on the 29th (29°), for a total of 4 freezing degree days 3(32 – 31) + (32 – 29) = 44. (The data are described in more detail in Appen- dix 15.1.) As you can see by comparing the panels in Figure 15.1, the price of orange juice concentrate has large swings, some of which appear to be associ- ated with cold weather in Florida.
592 CHAPTER 15
Estimation of Dynamic Causal Effects
We begin our quantitative analysis of the relationship between orange juice
price and the weather by using a regression to estimate the amount by which
orange juice prices rise when the weather turns cold. The dependent variable is
the percentage change in the price over that month [%ChgPt, where %ChgPt =
100 * ∆ln(POJ) and POJ is the real price of orange juice]. The regressor is the tt
number of freezing degree days during that month (FDDt). This regression is estimated using monthly data from January 1950 to December 2000 (as are all regressions in this chapter), for a total of T = 612 observations:
%ChgPt = – 0.40 + 0.47 FDDt. (15.1) (0.22) (0.13)
The standard errors reported in this section are not the usual OLS standard errors, but rather are heteroskedasticity- and autocorrelation-consistent (HAC) standard errors that are appropriate when the error term and regressors are auto- correlated. HAC standard errors are discussed in Section 15.4, and for now they are used without further explanation.
According to this regression, an additional freezing degree day during a month increases the price of orange juice concentrate over that month by 0.47%. In a month with 4 freezing degree days, such as November 1950, the price of orange juice concentrate is estimated to have increased by 1.88% (4 * 0.47% = 1.88%), relative to a month with no days below freezing.
Because the regression in Equation (15.1) includes only a contemporaneous measure of the weather, it does not capture any lingering effects of the cold snap on the orange juice price over the coming months. To capture these we need to consider the effect on prices of both contemporaneous and lagged values of FDD, which in turn can be done by augmenting the regression in Equation (15.1) with, for example, lagged values of FDD over the previous 6 months:
%ChgPt = – 0.65 + 0.47FDDt + 0.14FDDt – 1 + 0.06FDDt – 2 (0.23) (0.14) (0.08) (0.06)
+ 0.07FDDt-3 + 0.03FDDt-4 + 0.05FDDt-5 + 0.05FDDt-6. (15.2) (0.05) (0.03) (0.03) (0.04)
Equation (15.2) is a distributed lag regression. The coefficient on FDDt in Equa- tion (15.2) estimates the percentage increase in prices over the course of the month in which the freeze occurs; an additional freezing degree day is estimated to increase prices that month by 0.47%. The coefficient on the first lag of FDDt, FDDt – 1, estimates the percentage increase in prices arising from a freezing degree
day in the preceding month, the coefficient on the second lag estimates the effect of a freezing degree day 2 months ago, and so forth. Equivalently, the coefficient on the first lag of FDD estimates the effect of a unit increase in FDD 1 month after the freeze occurs. Thus the estimated coefficients in Equation (15.2) are estimates of the effect of a unit increase in FDDt on current and future values of %ChgP; that is, they are estimates of the dynamic effect of FDDt on %ChgPt. For example, the 4 freezing degree days in November 1950 are estimated to have increased orange juice prices by 1.88% during November 1950, by an additional 0.56%(= 4 * 0.14) in December 1950, by an additional 0.24%(= 4 * 0.06) in January 1951, and so forth.
15.2
Dynamic Causal Effects
Before learning more about the tools for estimating dynamic causal effects, we should spend a moment thinking about what, precisely, is meant by a dynamic causal effect. Having a clear idea about what a dynamic causal effect is leads to a clearer understanding of the conditions under which it can be estimated.
Causal Effects and Time Series Data
Section 1.2 defined a causal effect as the outcome of an ideal randomized con- trolled experiment: When a horticulturalist randomly applies fertilizer to some tomato plots but not others and then measures the yield, the expected difference in yield between the fertilized and unfertilized plots is the causal effect on tomato yield of the fertilizer. This concept of an experiment, however, is one in which there are multiple subjects (multiple tomato plots or multiple people), so the data are either cross-sectional (the tomato yield at the end of the harvest) or panel data (individual incomes before and after an experimental job training program). By having multiple subjects, it is possible to have both treatment and control groups and thereby to estimate the causal effect of the treatment.
In time series applications, this definition of causal effects in terms of an ideal randomized controlled experiment needs to be modified. To be concrete, consider an important problem of macroeconomics: estimating the effect of an unanticipated change in the short-term interest rate on the current and future economic activity in a given country, as measured by GDP. Taken literally, the randomized controlled experiment of Section 1.2 would entail randomly assigning different economies to treatment and control groups. The central banks in the treatment group would apply the treatment of a random interest rate change, while those in the control
15.2 Dynamic Causal Effects 593
594 CHAPTER 15
Estimation of Dynamic Causal Effects
group would apply no such random changes; for both groups, economic activity (for example, GDP) would be measured over the next few years. But what if we are interested in estimating this effect for a specific country, say the United States? Then this experiment would entail having different “clones” of the United States as subjects and assigning some clone economies to the treatment group and some to the control group. Obviously, this “parallel universes” experiment is infeasible.
Instead, in time series data it is useful to think of a randomized controlled experiment consisting of the same subject (e.g., the U.S. economy) being given dif- ferent treatments (randomly chosen changes in interest rates) at different points in time (the 1970s, the 1980s, and so forth). In this framework, the single subject at different times plays the role of both treatment and control group: Sometimes the Fed changes the interest rate, while at other times it does not. Because data are collected over time, it is possible to estimate the dynamic causal effect, that is, the time path of the effect on the outcome of interest of the treatment. For example, a surprise increase in the short-term interest rate of two percentage points, sustained for one quarter, might initially have a negligible effect on output; after two quarters GDP growth might slow, with the greatest slowdown after 112 years; then over the next 2 years, GDP growth might return to normal. This time path of causal effects is the dynamic causal effect on GDP growth of a surprise change in the interest rate.
As a second example, consider the causal effect on orange juice price changes of a freezing degree day. It is possible to imagine a variety of hypothetical experi- ments, each yielding a different causal effect. One experiment would be to change the weather in the Florida orange groves, holding weather constant elsewhere—for example, holding weather constant in the Texas grapefruit groves and in other citrus fruit regions. This experiment would measure a partial effect, holding other weather constant. A second experiment might change the weather in all the regions, where the “treatment” is application of overall weather patterns. If weather is correlated across regions for competing crops, then these two dynamic causal effects differ. In this chapter, we consider the causal effect in the latter experiment, that is, the causal effect of applying general weather patterns. This corresponds to measuring the dynamic effect on prices of a change in Florida weather, not holding weather constant in other agricultural regions.
Dynamic effects and the distributed lag model. Because dynamic effects neces- sarily occur over time, the econometric model used to estimate dynamic causal effects needs to incorporate lags. To do so, Yt can be expressed as a distributed lag of current and r past values of Xt:
Yt = b0 + b1Xt + b2Xt-1 + b3Xt-2 + g+ br+1Xt-r + ut, (15.3)
where ut is an error term that includes measurement error in Yt and the effect of omitted determinants of Yt. The model in Equation (15.3) is called the distributed lag model relating Xt, and r of its lags, to Yt.
As an illustration of Equation (15.3), consider a modified version of the tomato/fertilizer experiment: Because fertilizer applied today might remain in the ground in future years, the horticulturalist wants to determine the effect on tomato yield over time of applying fertilizer. Accordingly, she designs a 3-year experiment and randomly divides her plots into four groups: The first is fertilized in only the first year; the second is fertilized in only the second year; the third is fertilized in only the third year; and the fourth, the control group, is never fertilized. Tomatoes are grown annually in each plot, and the third-year harvest is weighed. The three treatment groups are denoted by the binary variables Xt – 2, Xt – 1, and Xt, where t represents the third year (the year in which the harvest is weighed), Xt – 2 = 1 if the plot is in the first group (fertilized two years earlier), Xt – 1 = 1 if the plot was fertilized 1 year earlier, and Xt = 1 if the plot was fertilized in the final year. In the context of Equation (15.3) (which applies to a single plot), the effect of being fertil- ized in the final year is b1, the effect of being fertilized 1 year earlier is b2, and the effect of being fertilized 2 years earlier is b3. If the effect of fertilizer is greatest in the year it is applied, then b1 would be larger than b2 and b3.
More generally, the coefficient on the contemporaneous value of Xt, b1, is the contemporaneous or immediate effect of a unit change in Xt on Yt. The coefficient on Xt – 1, b2, is the effect on Yt of a unit change in Xt – 1 or, equivalently, the effect on Yt + 1 of a unit change in Xt; that is, b2 is the effect of a unit change in X on Y one period later. In general, the coefficient on Xt – h is the effect of a unit change in X on Y after h periods. The dynamic causal effect is the effect of a change in Xt on Yt, Yt + 1, Yt + 2, and so forth; that is, it is the sequence of causal effects on cur- rent and future values of Y. Thus, in the context of the distributed lag model in Equation (15.3), the dynamic causal effect is the sequence of coefficients b1, b2, c, br + 1.
Implicationsforempiricaltimeseriesanalysis. Thisformulationofdynamiccausal effects in time series data as the expected outcome of an experiment in which dif- ferent treatment levels are repeatedly applied to the same subject has two implica- tions for empirical attempts to measure the dynamic causal effect with observational time series data. The first implication is that the dynamic causal effect should not change over the sample on which we have data. This in turn is implied by the data being jointly stationary (Key Concept 14.5). As discussed in Section 14.7, the hypothesis that a population regression function is stable over time can be tested using the QLR test for a break, and it is possible to estimate the dynamic causal
15.2 Dynamic Causal Effects 595
596 CHAPTER 15
Estimation of Dynamic Causal Effects
effect in different subsamples. The second implication is that X must be uncorre- lated with the error term, and it is to this implication that we now turn.
Two Types of Exogeneity
Section 12.1 defined an “exogenous” variable as a variable that is uncorrelated with the regression error term and an “endogenous” variable as a variable that is correlated with the error term. This terminology traces to models with multiple equations, in which an “endogenous” variable is determined within the model while an “exogenous” variable is determined outside the model. Loosely speak- ing, if we are to estimate dynamic causal effects using the distributed lag model in Equation (15.3), the regressors (the X’s) must be uncorrelated with the error term. Thus X must be exogenous. Because we are working with time series data, however, we need to refine the definitions of exogeneity. In fact, there are two different concepts of exogeneity that we use here.
The first concept of exogeneity is that the error term has a conditional mean
of zero given current and all past values of Xt, that is, that E(ut0Xt,Xt-1,
Xt – 2, c) = 0. This modifies the standard conditional mean assumption for mul-
tiple regression with cross-sectional data (Assumption #1 in Key Concept 6.4),
which requires only that u has a conditional mean of zero given the included t
regressors, that is, E(u 0 X , X , c, X ) = 0. Including all lagged values of X ttt-1 t-r t
in the conditional expectation implies that all the more distant causal effects—all the causal effects beyond lag r—are zero. Thus, under this assumption, the r dis- tributed lag coefficients in Equation (15.3) constitute all the nonzero dynamic causal effects. We can refer to this assumption—that E(ut 0 Xt, Xt – 1, c) = 0—as past and present exogeneity, but because of the similarity of this definition and the definition of exogeneity in Chapter 12, we just use the term exogeneity.
The second concept of exogeneity is that the error term has mean zero, given all past, present, and future values of Xt, that is, that E(ut 0 c, Xt + 2, Xt + 1, Xt, Xt – 1, Xt – 2, c) = 0. This is called strict exogeneity; for clarity, we also call it past, present, and future exogeneity. The reason for introducing the concept of strict exogeneity is that, when X is strictly exogenous, there are more efficient estima- tors of dynamic causal effects than the OLS estimators of the coefficients of the distributed lag regression in Equation (15.3).
The difference between exogeneity (past and present) and strict exogeneity (past, present, and future) is that strict exogeneity includes future values of X in the conditional expectation. Thus strict exogeneity implies exogeneity, but not the reverse. One way to understand the difference between the two concepts is to con- sider the implications of these definitions for correlations between X and u. If X is
15.3 Estimation of Dynamic Causal Effects with Exogenous Regressors 597
(past and present) exogenous, then ut is uncorrelated with current and past values of Xt. If X is strictly exogenous, then in addition ut is uncorrelated with future values of Xt. For example, if a change in Yt causes future values of Xt to change, then Xt is not strictly exogenous even though it might be (past and present) exogenous.
As an illustration, consider the hypothetical multiyear tomato/fertilizer experiment described following Equation (15.3). Because the fertilizer is ran- domly applied in the hypothetical experiment, it is exogenous. Because tomato yield today does not depend on the amount of fertilizer applied in the future, the fertilizer time series is also strictly exogenous.
As a second illustration, consider the orange juice price example, in which Yt is the monthly percentage change in orange juice prices and Xt is the number of freezing degree days in that month. From the perspective of orange juice markets, we can think of the weather—the number of freezing degree days—as if it were randomly assigned, in the sense that the weather is outside human control. If the effect of FDD is linear and if it has no effect on prices after r months, then it fol- lows that the weather is exogenous. But is the weather strictly exogenous? If the conditional mean of ut given future FDD is nonzero, then FDD is not strictly exogenous. Answering this question requires thinking carefully about what, pre- cisely, is contained in ut. In particular, if OJ market participants use forecasts of FDD when they decide how much they will buy or sell at a given price, then OJ prices, and thus the error term ut, could incorporate information about future FDD that would make ut a useful predictor of FDD. This means that ut will be correlated with future values of FDDt. According to this logic, because ut includes forecasts of future Florida weather, FDD would be (past and present) exogenous but not strictly exogenous. The difference between this and the tomato/fertilizer example is that, while tomato plants are unaffected by future fertilization, OJ market participants are influenced by forecasts of future Florida weather. We return to the question of whether FDD is strictly exogenous when we analyze the orange juice price data in more detail in Section 15.6.
The two definitions of exogeneity are summarized in Key Concept 15.1.
15.3
Estimation of Dynamic Causal Effects with Exogenous Regressors
If X is exogenous, then its dynamic causal effect on Y can be estimated by OLS estimation of the distributed lag regression in Equation (15.4). This section sum- marizes the conditions under which these OLS estimators lead to valid statistical inferences and introduces dynamic multipliers and cumulative dynamic multipliers.
598 CHAPTER 15 Estimation of Dynamic Causal Effects
The Distributed Lag Model and Exogeneity
15.1
KEY CONCEPT
In the distributed lag model
Yt = b0 + b1Xt + b2Xt-1 + b3Xt-2 + g+ br+1Xt-r + ut, (15.4)
there are two different types of exogeneity, that is, two different exogeneity conditions: Past and present exogeneity (exogeneity):
E(ut0Xt,Xt-1,Xt-2,c) = 0; (15.5) Past, present, and future exogeneity (strict exogeneity):
E(ut0c,Xt+2,Xt+1,Xt,Xt-1,Xt-2,c) = 0. (15.6)
If X is strictly exogenous, it is exogenous, but exogeneity does not imply strict exogeneity.
The Distributed Lag Model Assumptions
The four assumptions of the distributed lag regression model are similar to the four assumptions for the cross-sectional multiple regression model (Key Concept 6.4), modified for time series data.
The first assumption is that X is exogenous, which extends the zero condi- tional mean assumption for cross-sectional data to include all lagged values of X. As discussed in Section 15.2, this assumption implies that the r distributed lag coefficients in Equation (15.3) constitute all the nonzero dynamic causal effects. In this sense, the population regression function summarizes the entire dynamic effect on Y of a change in X.
The second assumption has two parts: Part (a) requires that the variables have a stationary distribution, and part (b) requires that they become indepen- dently distributed when the amount of time separating them becomes large. This assumption is the same as the corresponding assumption for the ADL model (the second assumption in Key Concept 14.6), and the discussion of this assumption in Section 14.4 applies here as well.
The third assumption is that large outliers are unlikely, made mathematically precise by assuming that the variables have more than eight nonzero, finite moments.
15.3 Estimation of Dynamic Causal Effects with Exogenous Regressors 599
The Distributed Lag Model Assumptions
KEY CONCEPT
15.2
The distributed lag model is given in Key Concept 15.1 [Equation (15.4)], where
1. X is exogenous, that is, E(ut 0 Xt, Xt – 1, Xt – 2, c) = 0;
2. (a) The random variables Yt and Xt have a stationary distribution, and
(b) (Yt, Xt) and (Yt – j, Xt – j) become independent as j gets large;
3. Large outliers are unlikely: Yt and Xt have more than eight nonzero, finite
moments; and
4. There is no perfect multicollinearity.
This is stronger than the assumption of four finite moments that is used elsewhere in this book. As discussed in Section 15.4, this stronger assumption is used in the mathematics behind the HAC variance estimator.
The fourth assumption, which is the same as in the cross-sectional multiple regression model, is that there is no perfect multicollinearity.
The distributed lag regression model and assumptions are summarized in Key Concept 15.2.
Extension to additional X’s. The distributed lag model extends directly to multi- ple X’s: The additional X’s and their lags are simply included as regressors in the distributed lag regression, and the assumptions in Key Concept 15.2 are modified to include these additional regressors. Although the extension to multiple X’s is conceptually straightforward, it complicates the notation, obscuring the main ideas of estimation and inference in the distributed lag model. For this reason, the case of multiple X’s is not treated explicitly in this chapter but is left as a straight- forward extension of the distributed lag model with a single X.
Autocorrelated ut, Standard Errors, and Inference
In the distributed lag regression model, the error term ut can be autocorrelated; that is, ut can be correlated with its lagged values. This autocorrelation arises because, in time series data, the omitted factors included in ut can themselves be serially correlated. For example, suppose that the demand for orange juice also depends on income, so one factor that influences the price of orange juice is income, spe- cifically, the aggregate income of potential orange juice consumers. Then aggre- gate income is an omitted variable in the distributed lag regression of orange juice
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price changes against freezing degree days. Aggregate income, however, is serially correlated: Income tends to fall in recessions and rise in expansions. Thus, income is serially correlated, and, because it is part of the error term, ut will be serially corre- lated. This example is typical: Because omitted determinants of Y are themselves serially correlated, in general ut in the distributed lag model will be serially correlated.
The autocorrelation of ut does not affect the consistency of OLS, nor does it introduce bias. If, however, the errors are autocorrelated, then in general the usual OLS standard errors are inconsistent and a different formula must be used. Thus serial correlation of the errors is analogous to heteroskedasticity: The homoskedasticity-only standard errors are “wrong” when the errors are in fact heteroskedastic, in the sense that using homoskedasticity-only standard errors results in misleading statistical inferences when the errors are heteroskedastic. Similarly, when the errors are serially correlated, standard errors predicated upon i.i.d. errors are “wrong” in the sense that they result in misleading statistical infer- ences. The solution to this problem is to use heteroskedasticity- and autocorrelation- consistent (HAC) standard errors, the topic of Section 15.4.
Dynamic Multipliers and Cumulative
Dynamic Multipliers
Another name for the dynamic causal effect is the dynamic multiplier. The cumulative dynamic multipliers are the cumulative causal effects, up to a given lag; thus the cumu- lative dynamic multipliers measure the cumulative effect on Y of a change in X.
Dynamic multipliers. The effect of a unit change in X on Y after h periods, which is bh + 1 in Equation (15.4), is called the h-period dynamic multiplier. Thus the dynamic multipliers relating X to Y are the coefficients on Xt and its lags in Equa- tion (15.4). For example, b2 is the one-period dynamic multiplier, b3 is the two- period dynamic multiplier, and so forth. In this terminology, the zero-period (or contemporaneous) dynamic multiplier, or impact effect, is b1, the effect on Y of a change in X in the same period.
Because the dynamic multipliers are estimated by the OLS regression coef- ficients, their standard errors are the HAC standard errors of the OLS regression coefficients.
Cumulative dynamic multipliers. The h-period cumulative dynamic multiplier is the cumulative effect of a unit change in X on Y over the next h periods. Thus the cumulative dynamic multipliers are the cumulative sum of the dynamic multipliers. In terms of the coefficients of the distributed lag regression in Equation (15.4),
15.4 Heteroskedasticity- and Autocorrelation-Consistent Standard Errors 601
the zero-period cumulative multiplier is b1, the one-period cumulative multiplier is b1 + b2, and the h-period cumulative dynamic multiplier is b1 + b2 + g + bh+1.Thesumofalltheindividualdynamicmultipliers,b1 +b2 +g+br+1,is the cumulative long-run effect on Y of a change in X and is called the long-run cumulative dynamic multiplier.
For example, consider the regression in Equation (15.2). The immediate effect of an additional freezing degree day is that the price of orange juice con- centrate rises by 0.47%. The cumulative effect of a price change over the next month is the sum of the impact effect and the dynamic effect one month ahead; thus the cumulative effect on prices is the initial increase of 0.47% plus the sub- sequent smaller increase of 0.14% for a total of 0.61%. Similarly, the cumulative dynamic multiplier over 2 months is 0.47% + 0.14% + 0.06% = 0.67%.
The cumulative dynamic multipliers can be estimated directly using a modifica- tion of the distributed lag regression in Equation (15.4). This modified regression is
Yt = d0 + d1∆Xt + d2∆Xt-1 + d3∆Xt-2 + g+ dr∆Xt-r+1 + dr+1Xt-r + ut. (15.7)
The coefficients in Equation (15.7), d1, d2, c, dr + 1, are in fact the cumulative dynamic multipliers. This can be shown by a bit of algebra (Exercise 15.5), which demonstrates that the population regressions in Equations (15.7) and (15.4) are equiv- alent,whered0 = b0,d1 = b1,d2 = b1 + b2,d3 = b1 + b2 + b3,andsoforth.The coefficient on Xt – r, dr + 1, is the long-run cumulative dynamic multiplier; that is, dr+1 = b1 + b2 + b3 + g+ br+1. Moreover, the OLS estimators of the coeffi- cients in Equation (15.7) are the same as the corresponding cumulative sum of the OLS estimators in Equation (15.4). For example, dn2 = bn1 + bn2. The main benefit of estimating the cumulative dynamic multipliers using the specification in Equation (15.7) is that, because the OLS estimators of the regression coefficients are estimators of the cumulative dynamic multipliers, the HAC standard errors of the coefficients in Equation (15.7) are the HAC standard errors of the cumulative dynamic multipliers.
15.4
Heteroskedasticity- and Autocorrelation- Consistent Standard Errors
If the error term ut is autocorrelated, then OLS coefficient estimators are consistent, but in general the usual OLS standard errors for cross-sectional data are not. This means that conventional statistical inferences—hypothesis tests and confidence intervals—based on the usual OLS standard errors will, in general, be misleading.
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For example, confidence intervals constructed as the OLS estimator { 1 .96 conven- tional standard errors need not contain the true value in 95% of repeated samples, even if the sample size is large. This section begins with a derivation of the correct formula for the variance of the OLS estimator with autocorrelated errors, then turns to heteroskedasticity- and autocorrelation-consistent (HAC) standard errors.
This section covers HAC standard errors for regression with time series data. Chapter 10 introduced a type of HAC standard errors, clustered standard errors, which are appropriate for panel data. Although clustered standard errors for panel data and HAC standard errors for time series data have the same goal, the different data structures lead to different formulas. This section is self-contained, and Chapter 10 is not a prerequisite.
Distribution of the OLS Estimator
with Autocorrelated Errors
To keep things simple, consider the OLS estimator bn1 in the distributed lag regres- sion model with no lags, that is, the linear regression model with a single regressorXt:
Yt = b0 + b1Xt + ut, (15.8)
where the assumptions of Key Concept 15.2 are satisfied. This section shows that the variance of bn1 can be written as the product of two terms: the expression for var(bn1), applicable if ut is not serially correlated, multiplied by a correction factor that arises from the autocorrelation in ut or, more precisely, the autocorrelation in (Xt – mX)ut.
As shown in Appendix 4.3, the formula for the OLS estimator bn1 in Key Con- cept 4.2 can be rewritten as
bn1 = b1 +
1T
T a(Xt – X)ut
t=1 ,
1T
T a(Xt – X)2
t=1
(15.9)
where Equation (15.9) is Equation (4.30) with a change of notation so that i and n are replaced by t and T. Because X ¡p mX and T1 gTt=1 (Xt – X)2 ¡p sX2 , in large samples bn1 – b1 is approximately given by
1T 1T
Ta(Xt – mX)ut
bn1 – b1 ≅ t=1
sX2
=
Tavt t=1 =
sX2
v sX2
, (15.10)
15.4 Heteroskedasticity- and Autocorrelation-Consistent Standard Errors 603 where vt = (Xt – mX)ut and v = T1 gTt=1vt. Thus
var(bn1) = vara v b = var(v). (15.11) s X2 ( s X2 ) 2
If vt is i.i.d.—as assumed for cross-sectional data in Key Concept 4.3—then var(v) = var(vt)>T and the formula for the variance of bn1 from Key Concept 4.4 applies. If, however, ut and Xt are not independently distributed over time, then in general vt will be serially correlated, so var(v) ≠ var(vt)>T and Key Concept 4.4 does not apply. Instead, if vt is serially correlated, the variance of v is given by
where
var(v) = var[(v1 + v2 + g+ vT)>T]
= [var(v1) + cov(v1, v2) + g + cov(v1, vT)
+ cov(v2, v1) + var(v2) + g + var(vT)]>T2 = [Tvar(vt) + 2(T – 1)cov(vt, vt – 1)
+ 2(T – 2)cov(vt, vt-2) + g + 2cov(vt, vt-T+1)]>T2
= s2vfT, T
(15.12)
T-1
f =1+2 aT-jbr, T aTj j=1
(15.13) where rj = corr(vt,vt-j). In large samples, fT tends to the limit, fT¡f∞ =
1 + 2g∞ r. j=1 j
Combining the expressions in Equation (15.10) for bn1 and Equation (15.12) for var(v) gives the formula for the variance of bn1 when vt is autocorrelated:
var(bn1)=c1 s2v dfT, (15.14) T (s2X)2
where fT is given in Equation (15.13).
Equation (15.14) expresses the variance of bn1 as the product of two terms. The
first, in square brackets, is the formula for the variance of bn1 given in Key Concept 4.4, which applies in the absence of serial correlation. The second is the factor fT, which adjusts this formula for serial correlation. Because of this additional factor
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fT in Equation (15.14), the usual OLS standard error computed using Equation (5.4) is incorrect if the errors are serially correlated: If vt = (Xt – mX)ut is serially correlated, the estimator of the variance is off by the factor fT.
HAC Standard Errors
If the factor fT, defined in Equation (15.13), was known, then the variance of bn1 could be estimated by multiplying the usual cross-sectional estimator of the vari- ance by fT. This factor, however, depends on the unknown autocorrelations of vt, so it must be estimated. The estimator of the variance of bn1 that incorporates this adjustment is consistent whether or not there is heteroskedasticity and whether or not vt is autocorrelated. Accordingly, this estimator is called the heteroskedasticity- and autocorrelation-consistent (HAC) estimator of the variance of bn1, and the square root of the HAC variance estimator is the HAC standard error of bn1.
TheHACvarianceformula. Theheteroskedasticity-andautocorrelation-consistent estimator of the variance of bn1 is
s∼2 =sn2fn, (15.15) bn 1 bn 1 T
where sn 2 is the estimator of the variance of bn bn1 n1
in the absence of serial correlation, given in Equation (5.4), and where fT is an estimator of the factor fT in Equation
(15.13).
The task of constructing a consistent estimator fnT is challenging. To see why,
consider two extremes. At one extreme, given the formula in Equation (15.13), it
might seem natural to replace the population autocorrelations rj with the sample
autocorrelations rnj [defined in Equation (14.6)], yielding the estimator
1 + 2gT-1(T – j)rn.Butthisestimatorcontainssomanyestimatedautocorrelations j=1Tj
that it is inconsistent. Intuitively, because each of the estimated autocorrelations con- tains an estimation error, by estimating so many autocorrelations the estimation error in this estimator of fT remains large even in large samples. At the other extreme, one could imagine using only a few sample autocorrelations, for example, only the first sample autocorrelation, and ignoring all the higher autocorrelations. Although this estimator eliminates the problem of estimating too many autocorrelations, it has a different problem: It is inconsistent because it ignores the additional autocorrelations that appear in Equation (15.13). In short, using too many sample autocorrelations makes the estimator have a large variance, but using too few autocorrelations ignores the autocorrelations at higher lags, so in either of these extreme cases the estimator is inconsistent.
15.4 Heteroskedasticity- and Autocorrelation-Consistent Standard Errors 605
Estimators of fT used in practice strike a balance between these two extreme cases by choosing the number of autocorrelations to include in a way that depends on the sample size T. If the sample size is small, only a few autocorrelations are used, but if the sample size is large, more autocorrelations are included (but still far fewer than T ). Specifically, let fnT be given by
m-1
fnT =1+2aam-jb∼rj, (15.16)
j=1 m
where ∼r = gT vn vn >gT vn2, where vn = (X – X)un (as in the definition of
jt=j+1tt-jt=1t ttt
sn 2 ). The parameter m in Equation (15.16) is called the truncation parameter of
bn 1
the HAC estimator because the sum of autocorrelations is shortened, or truncated,
to include only m – 1 autocorrelations instead of the T – 1 autocorrelations appearing in the population formula in Equation (15.13).
For fnT to be consistent, m must be chosen so that it is large in large samples, although still much less than T. One guideline for choosing m in practice is to use the formula
m = 0.75T1>3, (15.17)
rounded to an integer. This formula, which is based on the assumption that there is a moderate amount of autocorrelation in vt, gives a benchmark rule for deter- mining m as a function of the number of observations in the regression.1
The value of the truncation parameter m resulting from Equation (15.17) can be modified using your knowledge of the series at hand. On the one hand, if there is a great deal of serial correlation in vt, then you could increase m beyond the value from Equation (15.17). On the other hand, if vt has little serial correlation, you could decrease m. Because of the ambiguity associated with the choice of m, it is good practice to try one or two alternative values of m for at least one speci- fication to make sure your results are not sensitive to m.
The HAC estimator in Equation (15.15), with fnT given in Equation (15.16), is called the Newey–West variance estimator, after the econometricians Whitney Newey and Kenneth West, who proposed it. They showed that, when used along with a rule like that in Equation (15.17), under general assumptions this estimator is a consistent estimator of the variance of bn1 (Newey and West, 1987). Their
1Equation (15.17) gives the “best” choice of m if ut and Xt are first-order autoregressive processes with first autocorrelation coefficients 0.5, where “best” means the estimator that minimizes E(s∼2 – s2)2.
bn1 Equation (15.17) is based on a more general formula derived by Andrews [1991, Equation (5.3)].
bn1
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15.5
Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors
When Xt is strictly exogenous, two alternative estimators of dynamic causal effects are available. The first such estimator involves estimating an autoregressive dis- tributed lag (ADL) model instead of a distributed lag model and calculating the dynamic multipliers from the estimated ADL coefficients. This method can entail estimating fewer coefficients than OLS estimation of the distributed lag model, thus potentially reducing estimation error. The second method is to estimate the coefficients of the distributed lag model, using generalized least squares (GLS)
proofs (and those in Andrews, 1991) assume that vt has more than four moments, which in turn is implied by Xt and ut having more than eight moments, and this is the reason that the third assumption in Key Concept 15.2 is that Xt and ut have more than eight moments.
OtherHACestimators. TheNewey–WestvarianceestimatorisnottheonlyHAC estimator. For example, the weights (m – j)>m in Equation (15.16) can be replaced by different weights. If different weights are used, then the rule for choosing the truncation parameter in Equation (15.17) no longer applies and a different rule, developed for those weights, should be used instead. Discussion of HAC estimators using other weights goes beyond the scope of this book. For more information on this topic, see Hayashi (2000, Section 6.6).
Extension to multiple regression. All the issues discussed in this section general- ize to the distributed lag regression model in Key Concept 15.1 with multiple lags and, more generally, to the multiple regression model with serially correlated errors. In particular, if the error term is serially correlated, then the usual OLS standard errors are an unreliable basis for inference and HAC standard errors should be used instead. If the HAC variance estimator used is the Newey–West estimator [the HAC variance estimator based on the weights (m – j)>m4, then the truncation parameter m can be chosen according to the rule in Equation (15.17) whether there is a single regressor or multiple regressors. The formula for HAC standard errors in multiple regression is incorporated into modern regres- sion software designed for use with time series data. Because this formula involves matrix algebra, we omit it here and instead refer the reader to Hayashi (2000, Section 6.6) for the mathematical details.
HAC standard errors are summarized in Key Concept 15.3.
15.5 Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors 607
HAC Standard Errors
KEY CONCEPT
15.3
The problem: The error term ut in the distributed lag regression model in Key Concept 15.1 can be serially correlated. If so, the OLS coefficient estimators are consistent but in general the usual OLS standard errors are not, resulting in misleading hypothesis tests and confidence intervals.
The solution: Standard errors should be computed using a heteroskedasticity- and autocorrelation-consistent (HAC) estimator of the variance. The HAC estimator involves estimates of m – 1 autocovariances as well as the variance; in the case of a single regressor, the relevant formulas are given in Equations (15.15) and (15.16).
In practice, using HAC standard errors entails choosing the truncation parameter m. To do so, use the formula in Equation (15.17) as a benchmark, then increase or decrease m depending on whether your regressors and errors have high or low serial correlation.
instead of OLS. Although the same number of coefficients in the distributed lag model are estimated by GLS as by OLS, the GLS estimator has a smaller variance. To keep the exposition simple, these two estimation methods are initially laid out and discussed in the context of a distributed lag model with a single lag and AR(1) errors. The potential advantages of these two estimators are greatest, however, when many lags appear in the distributed lag model, so these estimators are then extended to the general distributed lag model with higher-order autoregressive errors.
The Distributed Lag Model with AR(1) Errors
Suppose that the causal effect on Y of a change in X lasts for only two periods; that is, it has an initial impact effect b1 and an effect in the next period of b2, but no effect thereafter. Then the appropriate distributed lag regression model is the distributed lag model with only current and past values of Xt – 1:
Yt = b0 + b1Xt + b2Xt-1 + ut. (15.18)
As discussed in Section 15.2, in general the error term ut in Equation (15.18) is serially correlated. One consequence of this serial correlation is that, if the distrib- uted lag coefficients are estimated by OLS, then inference based on the usual OLS standard errors can be misleading. For this reason, Sections 15.3 and 15.4
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emphasized the use of HAC standard errors when b1 and b2 in Equation (15.18) are estimated by OLS.
In this section, we take a different approach toward the serial correlation in ut. This approach, which is possible if Xt is strictly exogenous, involves adopting an autoregressive model for the serial correlation in ut, then using this AR model to derive some estimators that can be more efficient than the OLS estimator in the distributed lag model.
Specifically, suppose that ut follows the AR(1) model
ut = f1ut-1 + ∼ut, (15.19)
where f1 is the autoregressive parameter, ∼ut is serially uncorrelated, and no intercept isneededbecauseE(ut) = 0.Equations(15.18)and(15.19)implythatthedistributed lag model with a serially correlated error can be rewritten as an autoregressive distributed lag model with a serially uncorrelated error. To do so, lag each side of Equation (15.18) and subtract f1 multiplied by this lag from each side:
Yt – f1Yt-1 = (b0 + b1Xt + b2Xt-1 + ut) – f1(b0 + b1Xt-1 + b2Xt-2 + ut-1)
= b0 + b1Xt + b2Xt-1 – f1b0 – f1b1Xt-1 – f1b2Xt-2 + ∼ut, (15.20)
where the second equality uses ∼ut = ut – f1ut – 1. Collecting terms in Equation (15.20), we have that
where
Yt = a0 + f1Yt-1 + d0Xt + d1Xt-1 + d2Xt-2 + ∼ut, (15.21) a0 = b0(1 – f1),d0 = b1,d1 = b2 – f1b1,andd2 = -f1b2, (15.22)
where b0, b1, and b2 are the coefficients in Equation (15.18) and f1 is the autocor- relation coefficient in Equation (15.19).
Equation (15.21) is an ADL model that includes a contemporaneous value of X and two of its lags. We will refer to Equation (15.21) as the ADL representation of the distributed lag model with autoregressive errors given in Equations (15.18) and (15.19).
The terms in Equation (15.20) can be reorganized differently to obtain an expressionthatisequivalenttoEquations(15.21)and(15.22).LetY∼ = Y – f Y
tt1t-1 be the quasi-difference of Yt (“quasi” because it is not the first difference, the
difference between Yt and Yt – 1; rather, it is the difference between Yt and f1Yt – 1).
15.5 Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors 609 Similarly, let X∼ = X – f X be the quasi-difference of X . Then Equation
tt1t-1 t (15.20) can be written
Y∼ = a + b X∼ + b X∼ + ∼u . (15.23) t01t2t-1t
We will refer to Equation (15.23) as the quasi-difference representation of the dis- tributed lag model with autoregressive errors given in Equations (15.18) and (15.19). The ADL model Equation (15.21) [with the parameter restrictions in Equation (15.22)] and the quasi-difference model in Equation (15.23) are equivalent. In both models, the error term, ∼ut, is serially uncorrelated. The two representations, however, suggest different estimation strategies. But before discussing those strategies, we turn to the assumptions under which they yield
consistent estimators of the dynamic multipliers, b1 and b2.
The conditional mean zero assumption in the ADL(1,2) and quasi-difference models. Because Equations (15.21) [with the restrictions in Equation (15.22)] and (15.23) are equivalent, the conditions for their estimation are the same, so for convenience we consider Equation (15.23).
The quasi-difference model in Equation (15.23) is a distributed lag model
involving the quasi-differenced variables with a serially uncorrelated error. Accord-
ingly, the conditions for OLS estimation of the coefficients in Equation (15.23) are
the least squares assumptions for the distributed lag model in Key Concept 15.2,
expressed in terms of ∼u and X∼ . The critical assumption here is the first assumption, tt∼
which, applied to Equation (15.23), is that Xt is exogenous; that is,
E(∼u0X∼,X∼ ,c)=0, (15.24)
t t t-1
where letting the conditional expectation depend on distant lags of X∼ ensures that
∼t
no additional lags of Xt, other than those appearing in Equation (15.23), enter the
population regression function.
BecauseX∼ = X – f X ,soX = X∼ + f X ,conditioningonX∼ andall
tt1t-1tt1t-1 t
of its lags is equivalent to conditioning on Xt and all of its lags. Thus the conditional
expectation condition in Equation (15.24) is equivalent to the condition that E(∼ut 0 Xt, Xt – 1, c) = 0. Furthermore, because ∼ut = ut – f1ut – 1, this condition in turn implies that
0 = E ( ∼u t 0 X t , X t – 1 , c )
= E(ut – f1ut-10Xt,Xt-1,c)
= E(ut0Xt,Xt-1,c) – f1E(ut-10Xt,Xt-1,c). (15.25)
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For the equality in Equation (15.25) to hold for general values of f1, it must bethecasethatbothE(ut0Xt,Xt-1,c) = 0andE(ut-10Xt,Xt-1,c) = 0.By shifting the time subscripts forward one time period, the condition that E(ut – 1 0 Xt, Xt – 1, c) = 0 can be rewritten as
E(ut0Xt+1,Xt,Xt-1,c) = 0, (15.26)
which (by the law of iterated expectations) implies that E(ut 0 Xt, Xt – 1, c) = 0. In summary, having the zero conditional mean assumption in Equation (15.24) hold for general values of f1 is equivalent to having the condition in Equation (15.26) hold.
The condition in Equation (15.26) is implied by Xt being strictly exogenous, but it is not implied by Xt being (past and present) exogenous. Thus the least squares assumptions for estimation of the distributed lag model in Equation (15.23) hold if Xt is strictly exogenous, but it is not enough that Xt be (past and present) exogenous.
Because the ADL representation [Equations (15.21) and (15.22)] is equivalent to the quasi-differenced representation [Equation (15.23)], the conditional mean assumption needed to estimate the coefficients of the quasi-differenced represen- tation [that E(ut 0 Xt + 1, Xt, Xt – 1, c) = 0] is also the conditional mean assumption for consistent estimation of the coefficients of the ADL representation.
We now turn to the two estimation strategies suggested by these two repre- sentations: estimation of the ADL coefficients and estimation of the coefficients of the quasi-difference model.
OLS Estimation of the ADL Model
The first strategy is to use OLS to estimate the coefficients in the ADL model in Equation (15.21). As the derivation leading to Equation (15.21) shows, including the lag of Y and the extra lag of X as regressors makes the error term serially uncorrelated (under the assumption that the error follows a first order autoregression). Thus the usual OLS standard errors can be used; that is, HAC standard errors are not needed when the ADL model coefficients in Equation (15.21) are estimated by OLS.
The estimated ADL coefficients are not themselves estimates of the dynamic multipliers, but the dynamic multipliers can be computed from the ADL coeffi- cients. A general way to compute the dynamic multipliers is to express the esti- mated regression function as a function of current and past values of Xt, that is, to eliminate Yt from the estimated regression function. To do so, repeatedly substitute
15.5 Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors 611 expressions for lagged values of Yt into the estimated regression function. Specifi-
cally, consider the estimated regression function
Ynt = fn1Yt-1 + dn0Xt + dn1Xt-1 + dn2Xt-2, (15.27)
where the estimated intercept has been omitted because it does not enter any expression for the dynamic multipliers. Lagging both sides of Equation (15.27) yields Ynt-1 = fn1Yt-2 + dn0Xt-1 + dn1Xt-2 + dn2Xt-3, so replacing Ynt-1 in Equa- tion (15.27) by this expression for Yn t – 1 and collecting terms yields
Ynt = fn1(fn1Yt-2 + dn0Xt-1 + dn1Xt-2 + dn2Xt-3) + dn0Xt + dn1Xt-1 + dn2Xt-2
= dn0Xt + (dn1 + fn1dn0)Xt – 1 + (dn2 + fn1dn1)Xt – 2 + fn1dn2Xt – 3 + fn21Yt-2. (15.28)
Repeating this process by repeatedly substituting expressions for Yt – 2, Yt – 3, and so forth yields
Ynt = dn0Xt + (dn1 + fn1dn0)Xt-1 + (dn2 + fn1dn1 + fn21dn0)Xt-2
+ fn1(dn2 + fn1dn1 + fn21dn0)Xt – 3 + fn21(dn2 + fn1dn1 + fn21dn0)Xt – 4 + g. (15.29)
The coefficients in Equation (15.29) are the estimators of the dynamic multipliers, computed from the OLS estimators of the coefficients in the ADL model in Equa- tion (15.21). If the restrictions on the coefficients in Equation (15.22) were to hold exactly for the estimated coefficients, then the dynamic multipliers beyond the second (that is, the coefficients on Xt – 2, Xt – 3, and so forth) would all be zero.2 However, under this estimation strategy those restrictions will not hold exactly, so the estimated multipliers beyond the second in Equation (15.29) will generally be nonzero.
GLS Estimation
The second strategy for estimating the dynamic multipliers when Xt is strictly exog- enous is to use generalized least squares (GLS), which entails estimating Equation (15.23). To describe the GLS estimator, we initially assume that f1 is known. Because in practice it is unknown, this estimator is infeasible, so it is called the infeasible GLS estimator. The infeasible GLS estimator, however, can be modified using an estimator of f1, which yields a feasible version of the GLS estimator.
2Substitute the equalities in Equation (15.22) to show that, if those equalities hold, then d2 + f1d1 + f 21 d 0 = 0 .
612 CHAPTER 15
Estimation of Dynamic Causal Effects
InfeasibleGLS. Supposethatf1wereknown;thenthequasi-differencedvariables X∼ and Y∼ could be computed directly. As discussed in the context of Equations
tt
(15.24) and (15.26), if X is strictly exogenous, then E(∼u 0 X∼ , X∼ , c) = 0. Thus,
t t t t-1
if Xt is strictly exogenous and if f1 is known, the coefficients a0, b1, and b2 in
Equation (15.23) can be estimated by the OLS regression of Y∼ on X∼ and X∼ ttt-1
(including an intercept). The resulting estimator of b1 and b2—that is, the OLS
estimator of the slope coefficients in Equation (15.23) when f1 is known—is the
infeasible GLS estimator. This estimator is infeasible because f1 is unknown, so
X∼ and Y∼ cannot be computed and thus these OLS estimators cannot actually be tt
computed.
Feasible GLS. The feasible GLS estimator modifies the infeasible GLS estimator
by using a preliminary estimator of f1, fn1, to compute the estimated quasi-
differences. Specifically, the feasible GLS estimators of b1 and b2 are the OLS
estimators of b1 and b2 in Equation (15.23), computed by regressing Y∼nt on X∼nt ∼n ∼n n ∼n n
and Xt-1 (with an intercept), where Xt = Xt – f1Xt-1 and Yt = Yt – f1Yt-1. The preliminary estimator, fn1, can be computed by first estimating the dis- tributed lag regression in Equation (15.18) by OLS, then using OLS to estimate f1 in Equation (15.19) with the OLS residuals unt replacing the unobserved regres- sion errors ut. This version of the GLS estimator is called the Cochrane–Orcutt
(1949) estimator.
An extension of the Cochrane–Orcutt method is to continue this process
iteratively: Use the GLS estimator of b1 and b2 to compute revised estimators of ut; use these new residuals to re-estimate f1; use this revised estimator of f1 to compute revised estimated quasi-differences; use these revised estimated quasi-differences to re-estimate b1 and b2; and continue this process until the estimators of b1 and b2 converge. This is referred to as the iterated Cochrane– Orcutt estimator.
A nonlinear least squares interpretation of the GLS estimator. An equivalent interpretation of the GLS estimator is that it estimates the ADL model in Equa- tion (15.21), imposing the parameter restrictions in Equation (15.22). These restrictions are nonlinear functions of the original parameters b0, b1, b2, and f1, so this estimation cannot be performed using OLS. Instead, the parameters can be estimated by nonlinear least squares (NLLS). As discussed in Appendix 8.1, NLLS minimizes the sum of squared mistakes made by the estimated regression function, recognizing that the regression function is a nonlinear function of the parameters being estimated. In general, NLLS estimation can require sophisti- cated algorithms for minimizing nonlinear functions of unknown parameters.
15.5 Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors 613
In the special case at hand, however, those sophisticated algorithms are not needed; rather, the NLLS estimator can be computed using the algorithm described previously for the iterated Cochrane–Orcutt estimator. Thus the iter- ated Cochrane–Orcutt GLS estimator is in fact the NLLS estimator of the ADL coefficients, subject to the nonlinear constraints in Equation (15.22).
Efficiency of GLS. The virtue of the GLS estimator is that when X is strictly exog-
enous and the transformed errors ∼ut are homoskedastic, it is efficient among lin-
ear estimators, at least in large samples. To see this, first consider the infeasible
GLS estimator. If ∼u is homoskedastic, if f is known (so that X∼ and Y∼ can be t1tt
treated as if they are observed), and if Xt is strictly exogenous, then the Gauss–
Markov theorem implies that the OLS estimator of a0, b1, and b2 in Equation
(15.23) is efficient among all linear conditionally unbiased estimators based on X∼ ∼t
and Yt, for t = 2, c, T, where the first observation (t = 1) is lost because of quasi- differencing. That is, the OLS estimator of the coefficients in Equation (15.23) is the best linear unbiased estimator, or BLUE (Section 5.5). Because the OLS estimator of Equation (15.23) is the infeasible GLS estimator, this means that the infeasible GLS estimator is BLUE. The feasible GLS estimator is similar to the infeasible GLS estimator, except that f1 is estimated. Because the estimator of f1 is consistent and its variance is inversely proportional to T, the feasible and infeasible GLS estimators have the same variances in large samples, and the loss of information from the first observation (t = 1) is negligible when T is large. In this sense, if X is strictly exoge- nous, then the feasible GLS estimator is BLUE in large samples. In particular, if X is strictly exogenous, then GLS is more efficient than the OLS estimator of the dis- tributed lag coefficients discussed in Section 15.3.
The Cochrane–Orcutt and iterated Cochrane–Orcutt estimators presented here are special cases of GLS estimation. In general, GLS estimation involves transform- ing the regression model so that the errors are homoskedastic and serially uncorre- lated, then estimating the coefficients of the transformed regression model by OLS. In general, the GLS estimator is consistent and BLUE in large samples if X is strictly exogenous, but is not consistent if X is only (past and present) exogenous. The math- ematics of GLS involve matrix algebra, so they are postponed to Section 18.6.
The Distributed Lag Model with Additional
Lags and AR(p) Errors
The foregoing discussion of the distributed lag model in Equations (15.18) and (15.19), which has a single lag of Xt and an AR(1) error term, carries over to the general distributed lag model with multiple lags and an AR(p) error term.
614 CHAPTER 15
Estimation of Dynamic Causal Effects
The general distributed lag model with autoregressive errors. The general distributed lag model with r lags and an AR(p) error term is
Yt = b0 + b1Xt + b2Xt-1 + g+ br+1Xt-r + ut, (15.30) ut = f1ut-1 + f2ut-2 + g+ fput-p + ∼ut, (15.31)
where b1, c, br + 1 are the dynamic multipliers and f1, c, fp are the autoregressive coefficients of the error term. Under the AR(p) model for the errors, ∼ut is serially uncorrelated.
Algebra of the sort that led to the ADL model in Equation (15.21) shows that Equations (15.30) and (15.31) imply that Yt can be written in ADL form:
Yt = a0 + f1Yt-1 + g+ fpYt-p + d0Xt + d1Xt-1 + g+ dqXt-q + ∼ut, (15.32)
where q = r + p and d0, c, dq are functions of the b’s and f’s in Equations (15.30) and (15.31). Equivalently, the model of Equations (15.30) and (15.31) can be written in quasi-difference form as
Y∼ = a + b X∼ + b X∼
t 0 1t 2t-1
+ g + b X∼ + ∼u , (15.33) r+1t-r t
.
ConditionsforestimationoftheADLcoefficients. Theforegoingdiscussionofthe conditions for consistent estimation of the ADL coefficients in the AR(1) case extends to the general model with AR(p) errors. The conditional mean zero assumption for Equation (15.33) is that
E(∼u0X∼,X∼ ,c)=0. (15.34) t t t-1
∼∼
Because ut = ut – f1ut – 1 – f2ut – 2 – g – fput – p and Xt = Xt – f1Xt – 1 – g –
whereY∼ =Y -fY -g-fY tt1t-1pt-ptt1t-1pt-p
fpXt – p, this condition is equivalent to
E(ut0Xt,Xt-1,c) – f1E(ut-10Xt,Xt-1,c)
andX∼ =X -fX -g-fX
– g – fpE(ut-p0Xt,Xt-1,c) = 0. (15.35)
For Equation (15.35) to hold for general values of f1, c, fp, it must be the case that each of the conditional expectations in Equation (15.35) is zero; equivalently, it must be the case that
E(ut0Xt+p,Xt+p-1,Xt+p-2,c) = 0. (15.36)
15.5 Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors 615
This condition is not implied by Xt being (past and present) exogenous, but it is implied by Xt being strictly exogenous. In fact, in the limit when p is infinite (so that the error term in the distributed lag model follows an infinite-order autoregression), the condition in Equation (15.36) becomes the condition in Key Concept 15.1 for strict exogeneity.
EstimationoftheADLmodelbyOLS. Asinthedistributedlagmodelwithasingle lag and an AR(1) error term, the dynamic multipliers can be estimated from the OLS estimators of the ADL coefficients in Equation (15.32). The general formu- las are similar to, but more complicated than, those in Equation (15.29) and are best expressed using lag multiplier notation; these formulas are given in Appendix 15.2. In practice, modern regression software designed for time series regression analysis does these computations for you.
Estimation by GLS. Alternatively, the dynamic multipliers can be estimated by (feasible) GLS. This entails OLS estimation of the coefficients of the quasi- differenced specification in Equation (15.33), using estimated quasi-differences. The estimated quasi-differences can be computed using preliminary estimators of the autoregressive coefficients f1, c, fp, as in the AR(1) case. The GLS estimator is asymptotically BLUE, in the sense discussed earlier for the AR(1) case.
Estimation of dynamic multipliers under strict exogeneity is summarized in Key Concept 15.4.
Which to use: ADL or GLS? The two estimation options, OLS estimation of the ADL coefficients and GLS estimation of the distributed lag coefficients, have both advantages and disadvantages.
The advantage of the ADL approach is that it can reduce the number of parameters needed for estimating the dynamic multipliers, compared to OLS esti- mation of the distributed lag model. For example, the estimated ADL model in Equation (15.27) led to the infinitely long estimated distributed lag representation in Equation (15.29). To the extent that a distributed lag model with only r lags is really an approximation to a longer-lagged distributed lag model, the ADL model can provide a simple way to estimate those many longer lags using only a few unknown parameters. Thus in practice it might be possible to estimate the ADL model in Equation (15.39) with values of p and q much smaller than the value of r needed for OLS estimation of the distributed lag coefficients in Equation (15.37). In other words, the ADL specification can provide a compact, or parsi- monious, summary of a long and complex distributed lag (see Appendix 15.2 for additional discussion).
616 CHAPTER 15 Estimation of Dynamic Causal Effects
Estimation of Dynamic Multipliers Under Strict Exogeneity
15.4
KEY CONCEPT
The general distributed lag model with r lags and AR(p) error term is
Yt = b0 + b1Xt + b2Xt-1 + g+ br+1Xt-r + ut (15.37)
ut = f1ut-1 + f2ut-2 + g+ fput-p + ∼ut. (15.38) If Xt is strictly exogenous, then the dynamic multipliers b1, c, br + 1 can be
estimated by first using OLS to estimate the coefficients of the ADL model
Yt =a0 +f1Yt-1 +g+fpYt-p +d0Xt +d1Xt-1 +g+dqXt-q +∼ut, (15.39)
where q = r + p, and then computing the dynamic multipliers using regression software. Alternatively, the dynamic multipliers can be estimated by estimating the distributed lag coefficients in Equation (15.37) by GLS.
The advantage of the GLS estimator is that, for a given lag length r in the distributed lag model, the GLS estimator of the distributed lag coefficients is more efficient than the ADL estimator, at least in large samples. In practice, then, the advantage of using the ADL approach arises because the ADL specification can permit estimating fewer parameters than are estimated by GLS.
15.6
Orange Juice Prices and Cold Weather
This section uses the tools of time series regression to squeeze additional insights from our data on Florida temperatures and orange juice prices. First, how long lasting is the effect of a freeze on the price? Second, has this dynamic effect been stable or has it changed over the 51 years spanned by the data and, if so, how?
We begin this analysis by estimating the dynamic causal effects using the method of Section 15.3, that is, by OLS estimation of the coefficients of a distributed lag regression of the percentage change in prices (%ChgPt) on the number of freez- ing degree days in that month (FDDt) and its lagged values. For the distributed lag
15.6 Orange Juice Prices and Cold Weather 617
estimator to be consistent, FDD must be (past and present) exogenous. As dis- cussed in Section 15.2, this assumption is reasonable here. Humans cannot influence the weather, so treating the weather as if it were randomly assigned experimentally is appropriate. Because FDD is exogenous, we can estimate the dynamic causal effects by OLS estimation of the coefficients in the distributed lag model of Equation (15.4) in Key Concept 15.1.
As discussed in Sections 15.3 and 15.4, the error term can be serially corre- lated in distributed lag regressions, so it is important to use HAC standard errors, which adjust for this serial correlation. For the initial results, the truncation parameter for the Newey–West standard errors (m in the notation of Section 15.4) was chosen using the rule in Equation (15.17): Because there are 612 monthly observations,accordingtothatrulem = 0.75T1>3 = 0.75 * 6121>3 = 6.37,but because m must be an integer, this was rounded up to m = 7; the sensitivity of the standard errors to this choice of truncation parameter is investigated below.
The results of OLS estimation of the distributed lag regression of %ChgPt on FDDt, FDDt – 1, c, FDDt – 18 are summarized in column (1) of Table 15.1. The coefficients of this regression (only some of which are reported in the table) are estimates of the dynamic causal effect on orange juice price changes (in percent) for the first 18 months following a unit increase in the number of freezing degree days in a month. For example, a single freezing degree day is estimated to increase prices by 0.50% over the month in which the freezing degree day occurs. The subsequent effect on price in later months of a freezing degree day is less: After 1 month the estimated effect is to increase the price by a further 0.17%, and after 2 months the estimated effect is to increase the price by an additional 0.07%. The R2 from this regression is 0.12, indicating that much of the monthly variation in orange juice prices is not explained by current and past values of FDD.
Plots of dynamic multipliers can convey information more effectively than tables such as Table 15.1. The dynamic multipliers from column (1) of Table 15.1 are plotted in Figure 15.2a along with their 95% confidence intervals, computed as the estimated coefficient {1.96 HAC standard errors. After the initial sharp price rise, subsequent price rises are less, although prices are estimated to rise slightly in each of the first 6 months after the freeze. As can be seen from Figure 15.2a, for months other than the first the dynamic multipliers are not statistically significantly different from zero at the 5% significance level, although they are estimated to be positive through the seventh month.
Column (2) of Table 15.1 contains the cumulative dynamic multipliers for this specification, that is, the cumulative sum of the dynamic multipliers reported in
618 CHAPTER 15 Estimation of Dynamic Causal Effects
TABLE 15.1
Lag Number
0 1
2
3
4
5
6 . . .
The Dynamic Effect of a Freezing Degree Day (FDD) on the Price of Orange Juice: Selected Estimated Dynamic Multipliers and Cumulative Dynamic Multipliers
(1)
Dynamic Multipliers
0.50 (0.14)
0.17 (0.09)
0.07 (0.06)
0.07 (0.04)
0.02 (0.03)
0.03 (0.03)
0.03 (0.05)
(2)
Cumulative Multipliers
0.50 (0.14)
0.67 (0.14)
0.74 (0.17)
0.81 (0.18)
0.84 (0.19)
0.87 (0.19)
0.90 (0.20)
(3)
Cumulative Multipliers
0.50 (0.14)
0.67 (0.13)
0.74 (0.16)
0.81 (0.18)
0.84 (0.19)
0.87 (0.19)
0.90 (0.21)
(4)
Cumulative Multipliers
0.51 (0.15)
0.70 (0.15)
0.76 (0.18)
0.84 (0.19)
0.87 (0.20)
0.89 (0.20)
0.91 (0.21)
12 -0.14 0.54 0.54 0.54
. (0.08) .
.
18 0.00 (0.02)
Monthly indicators? No
HAC standard 7 error truncation
parameter (m)
(0.27) (0.28)
(0.28)
0.37 0.37 0.37
(0.30) (0.31)
(0.30)
No No Yes
F = 1.01
(p = 0.43)
7 14 7
All regressions were estimated by OLS using monthly data (described in Appendix 15.1) from January 1950 to December 2000, for a total of T = 612 monthly observations. The dependent variable is the monthly percentage change in the price of orange juice (%ChgPt). Regression (1) is the distributed lag regression with the monthly number of freezing degree days and 18 of its lagged values, that is, FDDt, FDDt – 1, c, FDDt – 18, and the reported coefficients are the OLS estimates of the dynamic multipli- ers. The cumulative multipliers are the cumulative sum of estimated dynamic multipliers. All regressions include an intercept, which is not reported. Newey–West HAC standard errors, computed using the truncation number given in the final row, are reported in parentheses.
15.6 Orange Juice Prices and Cold Weather 619 FIGURE 15.2 The Dynamic Effect of a Freezing Degree Day (FDD) on the Price of Orange Juice
Multiplier
1.0 0.8 0.6 0.4 0.2
-0.0
-0.2
-0.4
(a) Estimated Dynamic Multipliers and 95% Confidence Interval Multiplier
1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
-0.2
Estimated multiplier
95% Confidence interval
0 2 4 6 8 10 12 14 16 18 20
Lag (in months)
Estimated multiplier
95% Confidence interval
-0.40 2 4 6 8 10 12 14 16 18 20
Lag (in months)
(b) Estimated Cumulative Dynamic Multipliers and 95% Confidence Interval
The estimated dynamic multipliers show that a freeze leads to an immediate increase in prices. Future price rises are much smaller than the initial impact. The cumulative multiplier shows that freezes have a persistent effect on the level of orange juice prices, with prices peaking seven months after the freeze.
620 CHAPTER 15
Estimation of Dynamic Causal Effects
column (1). These dynamic multipliers are plotted in Figure 15.2b along with their 95% confidence intervals. After 1 month, the cumulative effect of the freezing degree day is to increase prices by 0.67%, after 2 months the price is estimated to have risen by 0.74%, and after 6 months the price is estimated to have risen by 0.90%. As can be seen in Figure 15.2b, these cumulative multipliers increase through the seventh month, because the individual dynamic multipliers are posi- tive for the first 7 months. In the eighth month, the dynamic multiplier is negative, so the price of orange juice begins to fall slowly from its peak. After 18 months, the cumulative increase in prices is only 0.37%; that is, the long-run cumulative dynamic multiplier is only 0.37%. This long-run cumulative dynamic multiplier is not statistically significantly different from zero at the 10% significance level (t = 0.37>0.30 = 1.23).
Sensitivityanalysis. Asinanyempiricalanalysis,itisimportanttocheckwhether these results are sensitive to changes in the details of the empirical analysis. We therefore examine three aspects of this analysis: sensitivity to the computation of the HAC standard errors; an alternative specification that investigates poten- tial omitted variable bias; and an analysis of the stability over time of the esti- mated multipliers.
First, we investigate whether the standard errors reported in the second col- umn of Table 15.1 are sensitive to different choices of the HAC truncation param- eter m. In column (3), results are reported for m = 14, twice the value used in column (2). The regression specification is the same as in column (2), so the esti- mated coefficients and dynamic multipliers are identical; only the standard errors differ but, as it happens, not by much. We conclude that the results are insensitive to changes in the HAC truncation parameter.
Second, we investigate a possible source of omitted variable bias. Freezes in Florida are not randomly assigned throughout the year, but rather occur in the winter (of course). If demand for orange juice is seasonal (is demand for orange juice greater in the winter than the summer?), then the seasonal patterns in orange juice demand could be correlated with FDD, resulting in omitted variable bias. The quantity of oranges sold for juice is endogenous: Prices and quantities are simultaneously determined by the forces of supply and demand. Thus, as dis- cussed in Section 9.2, including quantity would lead to simultaneity bias. Never- theless, the seasonal component of demand can be captured by including seasonal variables as regressors. The specification in column (4) of Table 15.1 therefore includes 11 monthly binary variables, one indicating whether the month is Janu- ary, one indicating February, and so forth (as usual one binary variable must be omitted to prevent perfect multicollinearity with the intercept). These monthly
15.6 Orange Juice Prices and Cold Weather 621
indicator variables are not jointly statistically significant at the 10% level (p = 0.43), and the estimated cumulative dynamic multipliers are essentially the same as for the specifications excluding the monthly indicators. In summary, sea- sonal fluctuations in demand are not an important source of omitted variable bias.
Have the dynamic multipliers been stable over time?3 To assess the stability of the dynamic multipliers, we need to check whether the distributed lag regression coefficients have been stable over time. Because we do not have a specific break date in mind, we test for instability in the regression coefficients using the Quandt likelihood ratio (QLR) statistic (Key Concept 14.9). The QLR statistic (with 15% trimming and HAC variance estimator), computed for the regression of column (1) with all coefficients interacted, has a value of 21.19, with q = 20 degrees of freedom (the coefficients on FDDt, its 18 lags, and the intercept). The 1% critical value in Table 14.5 is 2.43, so the QLR statistic rejects at the 1% significance level. These QLR regressions have 40 regressors, a large number; recomputing them for six lags only (so that there are 16 regressors and q = 8) also results in rejection at the 1% level. Thus the hypothesis that the dynamic multipliers are stable is rejected at the 1% significance level.
One way to see how the dynamic multipliers have changed over time is to compute them for different parts of the sample. Figure 15.3 plots the estimated cumulative dynamic multipliers for the first third (1950–1966), middle third (1967–1983), and final third (1984–2000) of the sample, computed by running separate regressions on each subsample. These estimates show an interesting and noticeable pattern. In the 1950s and early 1960s, a freezing degree day had a large and persistent effect on the price. The magnitude of the effect on price of a freez- ing degree day diminished in the 1970s, although it remained highly persistent. In the late 1980s and 1990s, the short-run effect of a freezing degree day was the same as in the 1970s, but it became much less persistent and was essentially elim- inated after a year. These estimates suggest that the dynamic causal effect on orange juice prices of a Florida freeze became smaller and less persistent over the second half of the twentieth century. The box “Orange Trees on the March” dis- cusses one possible explanation for the instability of the dynamic causal effects.
ADL and GLS estimates. As discussed in Section 15.5, if the error term in the distributed lag regression is serially correlated and FDD is strictly exogenous, it is possible to estimate the dynamic multipliers more efficiently than by OLS
3The discussion of stability in this subsection draws on material from Section 14.7 and can be skipped if that material has not been covered.
622 CHAPTER 15 Estimation of Dynamic Causal Effects
FIGURE 15.3 Estimated Cumulative Dynamic Multipliers from Different Sample Periods
The dynamic effect on orange juice prices of freezes changed significantly over the second half of the twentieth century.
A freeze had a larger impact on prices dur- ing 1950–1966 than later, and the effect
of a freeze was less persistent during 1984–2000 than earlier.
Multiplier
2.0 1.5 1.0 0.5 0.0
-0.50 2 4 6 8 10 12 14 16 18 20 Lag (in months)
1950 –1966
1967–1983
1984–2000
estimation of the distributed lag coefficients. Before using either the GLS esti- mator or the estimator based on the ADL model, however, we need to consider whether FDD is in fact strictly exogenous. True, humans cannot affect the daily weather, but does that mean that the weather is strictly exogenous? Does the error term ut in the distributed lag regression have conditional mean zero, given past, present, and future values of FDD?
The error term in the population counterpart of the distributed lag regression in column (1) of Table 15.1 is the discrepancy between the price and its population prediction based on the past 18 months of weather. This discrepancy might arise for many reasons, one of which is that traders use forecasts of the weather in Orlando. For example, if an especially cold winter is forecasted, then traders would incorporate this into the price, so the price would be above its predicted value based on the population regression; that is, the error term would be positive. If this forecast is accurate, then in fact future weather would turn out to be cold. Thus future freezing degree days would be positive (Xt + 1 7 0) when the current price is unusually high (ut 7 0), so corr(Xt + 1, ut) is positive. Stated more simply, although orange juice traders cannot influence the weather, they can—and do— predict it (see the box). Consequently, the error term in the price/weather regression
15.6 Orange Juice Prices and Cold Weather 623
Orange Trees on the March
Why do the dynamic multipliers in Figure 15.3 vary over time? One possible explanation is changes in markets, but another is that the trees moved south.
According to the Florida Department of Citrus, the severe freezes in the 1980s, which are visible in Figure 15.1(c), spurred citrus growers to seek a warmer climate. As shown in Figure 15.4, the num- ber of acres of orange trees in the more frost-prone northern and western counties fell from 232,000 acres in 1981 to 53,000 acres in 1985, and orange acreage in southern and central counties subse- quently increased from 413,000 in 1985 to 588,000 in
1993. With the groves farther south, northern frosts damage a smaller fraction of the crop, and—as indi- cated by the dynamic multipliers in Figure 15.3— price becomes less sensitive to temperatures in the more northern city of Orlando.
OK, the orange trees themselves might not have been on the march—that can be left to MacBeth— but southern migration of the orange groves does give new meaning to the term “nonstationarity.”4
4We are grateful to Professor James Cobbe of Florida State University for telling us about the southern movement of the orange groves.
FIGURE 15.4 Orange Grove Acreage in Regions of Florida Acres (thousands)
800
700
600
500
400
300
200
100
0
1965 1970
1975 1980 1985
1990 1995
2000 2005 Year
Southern and central counties
Northern and western counties
624
CHAPTER 15
Estimation of Dynamic Causal Effects
15.7
Is Exogeneity Plausible? Some Examples
As in regression with cross-sectional data, the interpretation of the coefficients in a distributed lag regression as causal dynamic effects hinges on the assumption that X is exogenous. If Xt or its lagged values are correlated with ut, then the conditional mean of ut will depend on Xt or its lags, in which case X is not (past and present) exogenous. Regressors can be correlated with the error term for several reasons, but with economic time series data a particularly important concern is that there could be simultaneous causality, which (as discussed in Sections 9.2 and 12.1) results in endogenous regressors. In Section 15.6, we discussed the assumptions of exogeneity and strict exogeneity of freezing degree days in detail. In this section, we examine the assumption of exogeneity in four other economic applications.
U.S. Income and Australian Exports
The United States is an important source of demand for Australian exports. Pre- cisely how sensitive Australian exports are to fluctuations in U.S. aggregate income could be investigated by regressing Australian exports to the United States against a measure of U.S. income. Strictly speaking, because the world economy is inte- grated, there is simultaneous causality in this relationship: A decline in Australian exports reduces Australian income, which reduces demand for imports from the United States, which reduces U.S. income. As a practical matter, however, this effect is very small because the Australian economy is much smaller than the U.S. econ- omy. Thus U.S. income plausibly can be treated as exogenous in this regression.
In contrast, in a regression of European Union exports to the United States against U.S. income, the argument for treating U.S. income as exogenous is less convincing because demand by residents of the European Union for U.S. exports constitutes a substantial fraction of the total demand for U.S. exports. Thus a decline in U.S. demand for EU exports would decrease EU income, which in turn would decrease demand for U.S. exports and thus decrease U.S. income. Because of these linkages through international trade, EU exports to the United States and U.S. income are simultaneously determined, so in this regression U.S. income arguably is not exogenous. This example illustrates a more general point that
is correlated with future weather. In other words, FDD is exogenous, but if this reasoning is true, it is not strictly exogenous, and the GLS and ADL estimators will not be consistent estimators of the dynamic multipliers. These estimators therefore are not used in this application.
15.7 Is Exogeneity Plausible? Some Examples 625 NEWS FLASH: Commodity Traders Send Shivers Through Disney World
Although the weather at Disney World in Orlando, Florida, is usually pleasant, now and then a cold spell can settle in. If you are visiting Disney World on a winter evening, should you bring a warm coat? Some people might check the weather forecast on TV, but those in the know can do better: They can check that day’s closing price on the New York orange juice futures market!
The financial economist Richard Roll under- took a detailed study of the relationship between orange juice prices and the weather. Roll (1984) examined the effect on prices of cold weather in Orlando, but he also studied the “effect” of changes in the price of an orange juice futures con- tract (a contract to buy frozen orange juice con- centrate at a specified date in the future) on the weather. Roll used daily data from 1975 to 1981 on the prices of OJ futures contracts traded at the New York Cotton Exchange and on daily and overnight temperatures in Orlando. He found that a rise in the price of the futures contract during the trading day in New York predicted cold weather, in particular a freezing spell, in Orlando over the following night. In fact, the market was so effective
in predicting cold weather in Florida that a price rise during the trading day actually predicted fore- cast errors in the official U.S. government weather forecasts for that night.
Roll’s study is also interesting for what he did not find: Although his detailed weather data explained some of the variation in daily OJ futures prices, most of the daily movements in OJ prices remained unex- plained. He therefore suggested that the OJ futures market exhibits “excess volatility,” that is, more vol- atility than can be attributed to movements in funda- mentals. Understanding why (and if) there is excess volatility in financial markets is now an important area of research in financial economics.
Roll’s finding also illustrates the difference between forecasting and estimating dynamic causal effects. Price changes on the OJ futures market are a useful predictor of cold weather, but that does not mean that commodity traders are so powerful that they can cause the temperature to fall. Visitors to Disney World might shiver after an OJ futures con- tract price rise, but they are not shivering because of the price rise—unless, of course, they went short in the OJ futures market.
whether a variable is exogenous depends on the context: U.S. income is plausibly exogenous in a regression explaining Australian exports, but not in a regression explaining EU exports.
Oil Prices and Inflation
Ever since the oil price increases of the 1970s, macroeconomists have been inter- ested in estimating the dynamic effect of an increase in the international price of crude oil on the U.S. rate of inflation. Because oil prices are set in world markets in large part by foreign oil-producing countries, initially one might think that oil
626 CHAPTER 15
Estimation of Dynamic Causal Effects
prices are exogenous. But oil prices are not like the weather: Members of OPEC set oil production levels strategically, taking many factors, including the state of the world economy, into account. To the extent that oil prices (or quantities) are set based on an assessment of current and future world economic conditions, including inflation in the United States, oil prices are endogenous.
Monetary Policy and Inflation
The central bankers in charge of monetary policy need to know the effect on infla- tion of monetary policy. Because an important tool of monetary policy is the short-term interest rate (the “short rate”), they need to know the dynamic causal effect on inflation of a change in the short rate. Although the short rate is deter- mined by the central bank, it is not set by the central bankers at random (as it would be in an ideal randomized experiment) but rather is set endogenously: The central bank determines the short rate based on an assessment of the current and future states of the economy, especially including the current and future rates of inflation. The rate of inflation in turn depends on the interest rate (higher interest rates reduce aggregate demand), but the interest rate depends on the rate of infla- tion, its past value, and its (expected) future value. Thus the short rate is endog- enous, and the causal dynamic effect of a change in the short rate on future inflation cannot be consistently estimated by an OLS regression of the rate of inflation on current and past interest rates.
The Growth Rate of GDP and the Term Spread
In Chapter 14 lagged values of the term spread were used to forecast future values of the growth rate of GDP. Because lags of the term spread happened in the past, one might initially think that there cannot be feedback from current growth rates of GDP to past values of the term spread, so past values of the term spread can be treated as exogenous. But past values of the term spread were not randomly assigned in an experiment; instead, the past term spread was simultaneously determined with past values of the growth rate of GDP. Because GDP and the interest rates making up the term spread are simultaneously determined, the other factors that determine the growth rate of GDP contained in ut are corre- lated with past values of the term spread; that is, the term spread is not exogenous. It follows that the term spread is not strictly exogenous, so the dynamic multipli- ers computed using an ADL model [for example, the ADL model in Equation (14.17)] are not consistent estimates of the dynamic causal effect on the growth rate of GDP of a change in the term spread.
15.8
Conclusion
Time series data provide the opportunity to estimate the time path of the effect on Y of a change in X, that is, the dynamic causal effect on Y of a change in X. To estimate dynamic causal effects using a distributed lag regression, however, X must be exogenous, as it would be if it were set randomly in an ideal randomized exper- iment. If X is not just exogenous but is strictly exogenous, then the dynamic causal effects can be estimated using an autoregressive distributed lag model or by GLS.
In some applications, such as estimating the dynamic causal effect on the price of orange juice of freezing weather in Florida, a convincing case can be made that the regressor (freezing degree days) is exogenous; thus the dynamic causal effect can be estimated by OLS estimation of the distributed lag coeffi- cients. Even in this application, however, economic theory suggests that the weather is not strictly exogenous, so the ADL or GLS methods are inappropri- ate. Moreover, in many relations of interest to econometricians, there is simulta- neous causality, so the regressor in these specifications are not exogenous, strictly or otherwise. Ascertaining whether the regressor is exogenous (or strictly exog- enous) ultimately requires combining economic theory, institutional knowledge, and careful judgment.
Summary
1. Dynamic causal effects in time series are defined in the context of a random- ized experiment, where the same subject (entity) receives different randomly assigned treatments at different times. The coefficients in a distributed lag regression of Y on X and its lags can be interpreted as the dynamic causal effects when the time path of X is determined randomly and independently of other factors that influence Y.
2. The variable X is (past and present) exogenous if the conditional mean of the error ut in the distributed lag regression of Y on current and past values of X does not depend on current and past values of X. If in addition the con- ditional mean of ut does not depend on future values of X, then X is strictly exogenous.
3. If X is exogenous, then the OLS estimators of the coefficients in a distributed lag regression of Y on current and past values of X are consistent estima- tors of the dynamic causal effects. In general, the error ut in this regression is serially correlated, so conventional standard errors are misleading, and HAC standard errors must be used instead.
Summary 627
628 CHAPTER 15
Estimation of Dynamic Causal Effects
4. If X is strictly exogenous, then the dynamic multipliers can be estimated using OLS estimation of an ADL model or using GLS.
5. Exogeneity is a strong assumption that often fails to hold in economic time
series data because of simultaneous exogeneity is even stronger.
Key Terms
dynamic causal effect (589) distributed lag model (595) exogeneity (596)
strict exogeneity (596)
dynamic multiplier (600)
impact effect (600)
cumulative dynamic multiplier (600) long-run cumulative dynamic
multiplier (601)
causality, and the assumption of strict
heteroskedasticity- and autocorrelation- consistent (HAC) standard error (604)
truncation parameter (605) Newey–West variance estimator (605) generalized least squares (GLS) (606) quasi-difference (608)
infeasible GLS estimator (612) feasible GLS estimator (612)
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Review the Concepts
15.1 In the 1970s a common practice was to estimate a distributed lag model relating changes in nominal gross domestic product (Y) to current and past changes in the money supply (X). Under what assumptions will this regression estimate the causal effects of money on nominal GDP? Are these assumptions likely to be satisfied in a modern economy like that of the United States?
15.2 Suppose that X is strictly exogenous. A researcher estimates an ADL(1,1) model, calculates the regression residual, and finds the residual to be highly serially correlated. Should the researcher estimate a new ADL model with
Exercises 629 additional lags or simply use HAC standard errors for the ADL(1,1) esti-
mated coefficients?
15.3 Suppose that a distributed lag regression is estimated, where the dependent variable is ∆Yt instead of Yt. Explain how you would compute the dynamic multipliers of Xt on Yt.
15.4 Suppose that you added FDDt + 1 as an additional regressor in Equation (15.2). If FDD is strictly exogenous, would you expect the coefficient on FDDt + 1 to be zero or nonzero? Would your answer change if FDD is exog- enous but not strictly exogenous?
Exercises
15.1 Increases in oil prices have been blamed for several recessions in developed countries. To quantify the effect of oil prices on real economic activity, researchers have run regressions like those discussed in this chapter. Let GDPt denote the value of quarterly gross domestic product in the United States and let Yt = 100ln(GDPt>GDPt – 1) be the quarterly percentage change in GDP. James Hamilton, an econometrician and macroeconomist, has suggested that oil prices adversely affect that economy only when they jump above their values in the recent past. Specifically, let Ot equal the greater of zero or the percentage point difference between oil prices at date t and their maximum value during the past 3 years. A distributed lag regression relating Yt and Ot, estimated over 1960:Q1–2013:Q4, is
Ynt = 1.0 – 0.007Ot – 0.015Ot-1 – 0.019Ot-2 – 0.024Ot-3 – 0.037Ot-4 (0.1) (0.013) (0.011) (0.011) (0.010) (0.012)
-0.012Ot-5 + 0.005Ot-6 – 0.008Ot-7 + 0.006Ot-8. (0.008) (0.010) (0.008) (0.008)
a. Suppose that oil prices jump 25% above their previous peak value and stay at this new higher level (so that Ot = 25 and Ot+1 = Ot+2 = g = 0). What is the predicted effect on output growth for each quarter over the next 2 years?
b. Construct a 95% confidence interval for your answers in (a).
c. What is the predicted cumulative change in GDP growth over 8 quarters?
d. The HAC F-statistic testing whether the coefficients on Ot and its lags are zero is 5.79. Are the coefficients significantly different from zero?
630 CHAPTER 15
Estimation of Dynamic Causal Effects
15.2 Macroeconomists have also noticed that interest rates change following oil price jumps. Let Rt denote the interest rate on 3-month Treasury bills (in percentage points at an annual rate). The distributed lag regression relat- ing the change in Rt (∆Rt) to Ot estimated over 1960:Q1–2013:Q4 is
∆Rt = 0.03 + 0.013Ot + 0.013Ot-1 – 0.004Ot-2 – 0.024Ot-3 – 0.000Ot-4 (0.05) (0.010) (0.010) (0.008) (0.015) (0.010)
+ 0.006Ot-5 – 0.005Ot-6 – 0.018Ot-7 – 0.004Ot-8. (0.015) (0.015) (0.010) (0.006)
a. Suppose that oil prices jump 25% above their previous peak value and stay at this new higher level (so that Ot = 25 and Ot+1 =
Ot + 2 = g = 0) . What is the predicted change in interest rates for each quarter over the next 2 years?
b. Construct 95% confidence intervals for your answers to (a).
c. What is the effect of this change in oil prices on the level of interest rates
in period t + 8? How is your answer related to the cumulative multiplier?
d. The HAC F-statistic testing whether the coefficients on Ot and its lags
are zero is 1.93. Are the coefficients significantly different from zero?
15.3 Consider two different randomized experiments. In experiment A, oil prices are set randomly, and the central bank reacts according to its usual policy rules in response to economic conditions, including changes in the oil price. In experiment B, oil prices are set randomly, and the central bank holds interest rates constant and in particular does not respond to the oil price changes. In both experiments, GDP growth is observed. Now suppose that oil prices are exogenous in the regression in Exercise 15.1. To which experiment, A or B, does the dynamic causal effect estimated in Exercise 15.1 correspond?
15.4 Suppose that oil prices are strictly exogenous. Discuss how you could improve on the estimates of the dynamic multipliers in Exercise 15.1.
15.5 Derive Equation (15.7) from Equation (15.4) and show that d0 = b0, d1 =b1,d2 =b1 +b2,d3 =b1 +b2 +b3 (etc.).(Hint:NotethatXt = ∆Xt + ∆Xt-1 + g + ∆Xt-p+1 + Xt-p.)
15.6 Consider the regression model Yt = b0 + b1Xt + ut, where ut follows the stationary AR(1) model ut = f1ut – 1 + ∼ut with u∼t i.i.d. with mean 0 and
2
variance s∼ and 0 f 0 6 1; the regressorX follows the stationary AR(1)
u1t
model Xt = g1Xt – 1 + et with et i.i.d. with mean 0 and variance s2e and
0g0 6 1; and et is independent of ∼ui for all t and i.
Exercises 631 s2
c. Show that corr(ut, ut – j) = f1j and corr(Xt, Xt – j) = gj. 1
d. Consider the terms s2v and ƒT in Equation (15.14).
i. Show that s2v = s2Xs2u, where s2X is the variance of X and s2u is the
variance of u.
ii. Derive an expression for f ∞ .
15.7 Consider the regression model Yt = b0 + b1Xt + ut, where ut follows the stationary AR(1) model ut = f1ut – 1 + ∼ut with ∼ut i.i.d. with mean 0 and
2
su∼ and var(Xt) =
e . 1 – g2
a. Show that var(ut) =
b. Show that cov(ut, ut – j) = f j1var(ut) and cov(Xt, Xt – j) = gjvar(Xt).
1
2
variances∼ and 0f 0 6 1.
1 – f2 11
u1
a. Suppose that Xt is independent of ∼uj for all t and j. Is Xt exogenous
(past and present)? Is Xt strictly exogenous (past, present, and future)? b. Suppose that Xt = ∼ut + 1. Is Xt exogenous? Is Xt strictly exogenous?
15.8 Consider the model in Exercise 15.7 with Xt = ∼ut + 1.
a. Is the OLS estimator of b1 consistent? Explain.
b. Explain why the GLS estimator of b1 is not consistent.
c. Show that the infeasible GLS estimator bnGLS ¡p b – f1
1 11+f2
.
1 [Hint: Use the omitted variable formula (6.1) applied to the quasi-
differenced regression in Equation (15.23)].
15.9 Consider the “constant-term-only” regression model Yt = b0 + ut, where
ut follows the stationary AR(1) model ut = f1ut – 1 + ∼ut with ∼ut i.i.d. with 2
mean 0 and variance s∼ and 0 f 0 6 1. u1
a. Show that the OLS estimator is bn0 = T-1gTt=1Yt.
b. Show that the (infeasible) GLS estimator is bnGLS =
0
(1 – f1)-1(T – 1)-1gTt=2(Yt – f1Yt-1). [Hint: The GLS estimator
of b0 is (1 – f1)-1 multiplied by the OLS estimator of a0 in Equation (15.23). Why?]
c. Show that bnGLS can be written as bnGLS = (T – 1)-1gT-1Y + 00t=2t
(1 – f1)-1(T – 1)-1(YT – f1Y1). [Hint: Rearrange the formula in (b).]
d. Derive the difference bn – bnGLS and discuss why it is likely to be 00
small when T is large.
632 CHAPTER 15
Estimation of Dynamic Causal Effects
15.10 Consider the ADL model Yt = 3.1 + 0.4Yt-1 + 2.0Xt – 0.8Xt-1 + ∼ut, where Xt is strictly exogenous.
a. Derive the impact effect of X on Y.
b. Derive the first five dynamic multipliers.
c. Derive the first five cumulative multipliers.
d. Derive the long-run cumulative dynamic multiplier.
15.11 Suppose that a(L) = (1 – fL), with 0f10 6 1, and b(L) = 1 + fL + f2L2 + f3L3c.
a. Show that the product b(L)a(L) = 1, so that b(L) = a(L) – 1. b. Why is the restriction 0 f1 0 6 1 important?
Empirical Exercises
(Only two empirical exercises for this chapter are given in the text, but you can find more on the text website, http://www.pearsonhighered.com/stock_watson/.)
E15.1 In this exercise you will estimate the effect of oil prices on macroeconomic activity, using monthly data on the Index of Industrial Production (IP) and the monthly measure of Ot described in Exercise 15.1. The data can be found on the textbook website, http://www.pearsonhighered.com/stock_ watson, in the file USMacro_Monthly.
a. Compute the monthly growth rate in IP, expressed in percentage points, ip_growtht = 100 * ln(IPt > IPt – 1). What are the mean and standard deviation of ip_growth over the 1960:M1–2012:M12 sample period? What are the units for ip_growth (percent, percent per annum, percent per month, or something else)?
b. Plot the value of Ot. Why are so many values of Ot equal to zero? Why aren’t some values of Ot negative?
c. Estimate a distributed lag model by regressing ip_growth onto the cur- rent value and 18 lagged values of Ot , including an intercept. What value of the HAC standard truncation parameter m did you choose? Why?
d. Taken as a group, are the coefficients on Ot statistically significantly different from zero?
e. Construct graphs like those in Figure 15.2, showing the estimated dynamic multipliers, cumulative multipliers, and 95% confidence intervals. Comment on the real-world size of the multipliers.
f. Suppose that high demand in the United States (evidenced by large values of ip_growth) leads to increases in oil prices. Is Ot exogenous? Are the estimated multipliers shown in the graphs in (e) reliable? Explain.
E15.2 In the data file USMacro_Quarterly, you will find data on two aggregate
price series for the United States: the price index for personal consump-
tion expenditures (PCEP) that you used in Empirical Exercise 14.1 and
the Consumer Price Index (CPI). These series are alternative measures
of consumer prices in the United States. The CPI prices a basket of
goods whose composition is updated every 5–10 years. PCEP uses chain-
weighting to price a basket of goods whose composition changes from month
to month. Economists have argued that the CPI will overstate inflation
because it does not take into account the substitution that occurs when rel-
ative prices change. If this substitution bias is important, then average CPI
inflation should be systematically higher than PCEP inflation. Let pCPI = t
400 × [ln(CPI ) − ln(CPI )], and pPCEP = 400 × [ln(PCEP ) − ln(PCEP )], t t−1 t t t−1
and Y = pCPI − pPCEP, so pCPI is the quarterly rate of price inflation (mea- tttt
sured in percentage points at an annual rate) based on the CPI, pPCEP is t
the quarterly rate of price inflation from the PCEP, and Yt is their differ- ence. Using data from 1963:Q1 through 2012:Q4, carry out the following exercises.
a. Compute the sample means of pCPI and pPCED. Are these point tt
estimates consistent with the presence of economically significant substitution bias in the CPI?
b. Compute the sample mean of Yt. Explain why it is numerically equal to the difference in the means computed in (a).
c. Show that the population mean of Y is equal to the difference of the population means of the two inflation rates.
d. Consider the “constant-term-only” regression: Yt = b0 + ut. Show that b0 = E(Y). Do you think that ut is serially correlated? Explain.
e. Construct a 95% confidence interval for b0. What value of the HAC standard truncation parameter m did you choose? Why?
f. Is there statistically significant evidence that the mean inflation rate for the CPI is greater than the rate for the PCEP?
g. Is there evidence of instability in b0? Carry out a QLR test. (Hint: Make sure you use HAC standard errors for the regressions in the QLR procedure.)
Empirical Exercises 633
634 CHAPTER 15
Estimation of Dynamic Causal Effects
APPENDIX
15.1
The Orange Juice Data Set
The orange juice price data are the frozen orange juice component of processed foods and feeds group of the Producer Price Index (PPI), collected by the U.S. Bureau of Labor Statistics (BLS series wpu02420301). The orange juice price series was divided by the over- all PPI for finished goods to adjust for general price inflation. The freezing degree days series was constructed from daily minimum temperatures recorded at Orlando-area air- ports, obtained from the National Oceanic and Atmospheric Administration (NOAA) of the U.S. Department of Commerce. The FDD series was constructed so that its timing and the timing of the orange juice price data were approximately aligned. Specifically, the frozen orange juice price data are collected by surveying a sample of producers in the middle of every month, although the exact date varies from month to month. Accordingly, the FDD series was constructed to be the number of freezing degree days from the 11th of one month to the 10th of the next month; that is, FDD is the maximum of zero and 32 minus the minimum daily temperature, summed over all days from the 11th to the 10th. Thus %ChgPt for February is the percentage change in real orange juice prices from mid- January to mid-February, and FDDt for February is the number of freezing degree days from January 11 to February 10.
APPENDIX
15.2
The ADL Model and Generalized Least Squares in Lag Operator Notation
This appendix presents the distributed lag model in lag operator notation, derives the ADL and quasi-differenced representations of the distributed lag model, and discusses the condi- tions under which the ADL model can have fewer parameters than the original distributed lag model.
The Distributed Lag, ADL, and Quasi-Difference
Models, in Lag Operator Notation
As defined in Appendix 14.3, the lag operator, L, has the property that LjXt = Xt – j, and the distributed lag b1Xt + b2Xt – 1 + g +br + 1Xt – r can be expressed as b(L)Xt, where b(L) = g rj = 0 bj + 1Lj, where L0 = 1. Thus the distributed lag model in Key Concept 15.1 [Equation (15.4)] can be written in lag operator notation as
where
The ADL Model and Generalized Least Squares in Lag Operator Notation
635
Yt = b0 + b(L)Xt + ut.
In addition, if the error term ut follows an AR(p), then it can be written as
f(L)ut = ∼ut,
(15.40)
(15.41)
where f(L) = g pj = 0 fjLj, where f0 = 1 and ∼ut is serially uncorrelated [note that f1, c, fp as defined here are the negatives of f1, c, fp in the notation of Equation (15.31)].
To derive the ADL model, premultiply each side of Equation (15.40) by f(L) so that f(L)Yt = f(L)3b0 + b(L)Xt + ut4 = a0 + d(L)Xt + ∼ut, (15.42)
p
a0 = f(1)b0 and d(L) = f(L)b(L), where f(1) = a fj . (15.43)
j=0
To derive the quasi-differenced model, note that f(L)b(L)X = b(L)f(L)X = b(L)X∼ ,
where X∼ = f(L)X . Thus rearranging Equation (15.42) yields tt
Y∼ = a + b(L)X∼ + ∼u , t0 tt
∼∼
where Yt is the quasi-difference of Yt; that is, Yt = f(L)Yt.
The Inverse of a Lag Polynomial
ttt
Let a(x) = gpj=0ajxj denote a polynomial of order p. The inverse of a(x), say b(x), is a
function that satisfies b(x)a(x) = 1. If the roots of the polynomial a(x) are greater than 1 in
absolute value, then b(x) is a polynomial in nonnegative powers of x: b(x) = g ∞ b x j.
Because b(x) is the inverse of a(x), it is denoted as a(x)−1 or as 1>a(x).
The inverse of a lag polynomial a(L) is defined analogously: a(L)-1 = 1>a(L) =
b(L) = g∞ bLj, where b(L)a(L) = 1. For example, if a(L) = (1 – fL), with 0f0 6 1, j=0 j
you can verify that a(L)-1 = 1 + fL + f2L2 + f3L3c = g ∞ fjLj. (See Exercise 15.11.) j=0
The ADL and GLS Estimators
The OLS estimator of the ADL coefficients is obtained by OLS estimation of Equation (15.42). The original distributed lag coefficients are b(L), which, in terms of the estimated coefficients, is b(L) = f(L)-1d(L); that is, the coefficients in b(L) satisfy the restrictions
(15.44)
j=0 j
636 CHAPTER 15
Estimation of Dynamic Causal Effects
implied by f(L)b(L) = d(L). Thus the estimator of the dynamic multipliers based on the OLS estimators of the coefficients of the ADL model, dn(L) and fn(L), is
bnADL(L) = fn(L)-1dn(L). (15.45)
The expressions for the coefficients in Equation (15.29) in the text are obtained as a special case of Equation (15.45) when r = 1 and p = 1.
The feasible GLS estimator is computed by obtaining a preliminary estimator of f(L), computing estimated quasi-differences, estimating b(L) in Equation (15.44) using these estimated quasi-differences, and (if desired) iterating until convergence. The iterated GLS estimator is the NLLS estimator computed by NLLS estimation of the ADL model in Equation (15.42), subject to the nonlinear restrictions on the parameters contained in Equation (15.43).
As stressed in the discussion surrounding Equation (15.36) in the text, it is not enough for Xt to be (past and present) exogenous to use either of these estimation methods, for exogeneity alone does not ensure that Equation (15.36) holds. If, however, X is strictly exogenous, then Equation (15.36) does hold, and assuming that Assumptions 2 through 4 of Key Concept 14.6 hold, these estimators are consistent and asymptotically normal. Moreover, the usual (cross-sectional heteroskedasticity-robust) OLS standard errors pro- vide a valid basis for statistical inference.
Parameter reduction using the ADL model. Suppose that the distributed lag polynomial b(L) can be written as a ratio of lag polynomials, u2(L)-1u1(L), where u1(L) and u2(L) are both lag polynomials of a low degree. Then f(L)b(L) in Equation (15.43) is f(L)b(L) = f(L)3u2(L)-1u1(L)4 = 3f(L)u2(L)-14u1(L). If it so happens that f(L) = u2(L), then d(L) = f(L)b(L) = u1(L). If the degree of u1(L) is low, then q, the number of lags of Xt in the ADL model, can be much less than r. Thus, under these assumptions, estimation of the ADL model entails estimating potentially many fewer parameters than the original distributed lag model. It is in this sense that the ADL model can achieve more parsimoni- ous parameterizations (that is, use fewer unknown parameters) than the distributed lag model.
As developed here, the assumption that f(L) and u2(L) happen to be the same seems like a coincidence that would not occur in an application. However, the ADL model is able to capture a large number of shapes of dynamic multipliers with only a few coefficients.
ADL or GLS: Bias versus variance. A good way to think about whether to estimate dynamic multipliers by first estimating an ADL model and then computing the dynamic multipliers from the ADL coefficients or, alternatively, by estimating the distributed lag model directly using GLS is to view the decision in terms of a trade-off between bias and variance. Estimating the dynamic multipliers using an approximate ADL model
The ADL Model and Generalized Least Squares in Lag Operator Notation 637
introduces bias; however, because there are few coefficients, the variance of the estima- tor of the dynamic multipliers can be small. In contrast, estimating a long distributed lag model using GLS produces less bias in the multipliers; however, because there are so many coefficients, their variance can be large. If the ADL approximation to the dynamic multipliers is a good one, then the bias of the implied dynamic multipliers will be small, so the ADL approach will have a smaller variance than the GLS approach with only a small increase in the bias. For this reason, unrestricted estimation of an ADL model with small number of lags of Y and X is an attractive way to approximate a long distrib- uted lag when X is strictly exogenous.
Additional Topics in Time Series Regression
T his chapter takes up some further topics in time series regression, starting with forecasting. Chapter 14 considered forecasting a single variable. In practice,
however, you might want to forecast two or more variables, such as the growth rate of GDP and the rate of inflation. Section 16.1 introduces a model for forecasting multiple variables, vector autoregressions (VARs), in which lagged values of two or more variables are used to forecast future values of those variables. Chapter 14 also focused on making forecasts one period (e.g., one quarter) into the future, but making forecasts two, three, or more periods into the future is important as well. Methods for making multiperiod forecasts are discussed in Section 16.2.
Sections 16.3 and 16.4 return to the topic of Section 14.6, stochastic trends. Section 16.3 introduces additional models of stochastic trends and an alternative test for a unit autoregressive root. Section 16.4 introduces the concept of cointegration, which arises when two variables share a common stochastic trend—that is, when each variable contains a stochastic trend, but a weighted difference of the two variables does not.
In some time series data, especially financial data, the variance changes over time: Sometimes the series exhibits high volatility, while at other times the volatility is low, so the data exhibit clusters of volatility. Section 16.5 discusses volatility cluster- ing and introduces models in which the variance of the forecast error changes over time, that is, models in which the forecast error is conditionally heteroskedastic. Mod- els of conditional heteroskedasticity have several applications. One application is computing forecast intervals, where the width of the interval changes over time to reflect periods of high or low uncertainty. Another application is forecasting the uncertainty of returns on an asset, such as a stock, which in turn can be useful in assessing the risk of owning that asset.
16.1
Chapter
16
638
Vector Autoregressions
Chapter 14 focused on forecasting the growth rate of GDP, but in reality eco- nomic forecasters are in the business of forecasting other key macroeconomic variables as well, such as the rate of inflation, the unemployment rate, and interest rates. One approach is to develop a separate forecasting model for each variable,
16.1 Vector Autoregressions 639
Vector Autoregressions
Key ConCept
16.1
A vector autoregression (VAR) is a set of k time series regressions, in which the regressors are lagged values of all k series. A VAR extends the univariate autoregression to a list, or “vector,” of time series variables. When the number of lags in each of the equations is the same and is equal to p, the system of equations is called a VAR(p).
In the case of two time series variables, Yt and Xt, the VAR(p) consists of the two equations
Yt = b10 + b11Yt-1 + g+ b1pYt-p + g11Xt-1 + g+ g1pXt-p + u1t (16.1)
Xt = b20 + b21Yt-1 + g+ b2pYt-p + g21Xt-1 + g+ g2pXt-p + u2t, (16.2)
where the b’s and the g’s are unknown coefficients and u1t and u2t are error terms. The VAR assumptions are the time series regression assumptions of Key Concept 14.6, applied to each equation. The coefficients of a VAR are estimated
by estimating each equation by OLS.
using the methods of Section 14.4. Another approach is to develop a single model that can forecast all the variables, which can help to make the forecasts mutually consistent. One way to forecast several variables with a single model is to use a vector autoregression (VAR). A VAR extends the univariate autoregression to multiple time series variables, that is, it extends the univariate autoregression to a “vector” of time series variables.
The VAR Model
A vector autoregression (VAR) with two time series variables, Yt and Xt, consists of two equations: In one, the dependent variable is Yt; in the other, the dependent variable is Xt. The regressors in both equations are lagged values of both vari- ables. More generally, a VAR with k time series variables consists of k equations, one for each of the variables, where the regressors in all equations are lagged values of all the variables. The coefficients of the VAR are estimated by estimat- ing each of the equations by OLS.
VARs are summarized in Key Concept 16.1.
640 ChApTeR 16 Additional Topics in Time Series Regression
Inference in VARs. Under the VAR assumptions, the OLS estimators are consis- tent and have a joint normal distribution in large samples. Accordingly, statistical inference proceeds in the usual manner; for example, 95% confidence intervals on coefficients can be constructed as the estimated coefficient {1.96 standard errors.
One new aspect of hypothesis testing arises in VARs because a VAR with k variables is a collection, or system, of k equations. Thus it is possible to test joint hypotheses that involve restrictions across multiple equations.
For example, in the two-variable VAR(p) in Equations (16.1) and (16.2), you could ask whether the correct lag length is p or p – 1; that is, you could ask whether the coefficients on Yt – p and Xt−p are zero in these two equations. The null hypothesis that these coefficients are zero is
H0: b1p = 0, b2p = 0, g1p = 0, and g2p = 0. (16.3)
The alternative hypothesis is that at least one of these four coefficients is nonzero. Thus the null hypothesis involves coefficients from both of the equations, two from each equation.
Because the estimated coefficients have a jointly normal distribution in large samples, it is possible to test restrictions on these coefficients by computing an F-statistic. The precise formula for this statistic is complicated because the nota- tion must handle multiple equations, so we omit it. In practice, most modern software packages have automated procedures for testing hypotheses on coeffi- cients in systems of multiple equations.
How many variables should be included in a VAR? The number of coefficients in each equation of a VAR is proportional to the number of variables in the VAR. For example, a VAR with 5 variables and 4 lags will have 21 coefficients (4 lags each of 5 variables, plus the intercept) in each of the 5 equations, for a total of 105 coefficients! Estimating all these coefficients increases the amount of estimation error entering a forecast, which can result in deterioration of the accuracy of the forecast.
The practical implication is that one needs to keep the number of variables in a VAR small and, especially, to make sure the variables are plausibly related to each other so that they will be useful for forecasting one another. For example, we know from a combination of empirical evidence (such as that discussed in Chapter 14) and economic theory that the growth rate of GDP, the term spread, and the rate of inflation are related to one another, suggesting that these variables could help forecast one another in a VAR. Including an unrelated variable in a
16.1 Vector Autoregressions 641 VAR, however, introduces estimation error without adding predictive content,
thereby reducing forecast accuracy.
Determining lag lengths in VARs. Lag lengths in a VAR can be determined using either F-tests or information criteria.
The information criterion for a system of equations extends the single-equation
information criterion in Section 14.5. To define this information criterion, we
need to adopt matrix notation. Let Σu be the k * k covariance matrix of the VAR n
of Σ is g un
nu T1 Tt=1 it jt it
errors and let Σu be the estimate of the covariance matrix where the i, j element
th the OLS residual from the jth equation. The BIC for the VAR is
un , where un
BIC(p) = ln[det(Σnu)] + k(kp + 1)ln(T), (16.4)
The expression for the BIC for the k equations in the VAR in Equation (16.4) extends the expression for a single equation given in Section 14.5. When there is a single equation, the first term simplifies to ln[SSR(p)>T]. The second term in Equation (16.4) is the penalty for adding additional regressors; k(kp + 1) is the total number of regression coefficients in the VAR. (There are k equations, each of which has an intercept and p lags of each of the k time series variables.)
Lag length estimation in a VAR using the BIC proceeds analogously to the single equation case: Among a set of candidate values of p, the estimated lag length pn is the value of p that minimizes BIC(p).
UsingVARsforcausalanalysis. ThediscussionsofarhasfocusedonusingVARsfor forecasting. Another use of VAR models is for analyzing causal relationships among economic time series variables; indeed, it was for this purpose that VARs were first introduced to economics by the econometrician and macroeconomist Christopher Sims (1980). (See the box “Nobel Laureates in Time Series Econometrics” at the end of this chapter.) The use of VARs for causal inference is known as structural VAR modeling, “structural” because in this application VARs are used to model the underlying structure of the economy. Structural VAR analysis uses the techniques introduced in this section in the context of forecasting, plus some additional tools. The biggest conceptual difference between using VARs for forecasting and using them for structural modeling, however, is that structural modeling requires very
is
where det(Σu) is the determinant of the matrix Σu. The AIC is computed using Equation (16.4), modified by replacing the term “ln(T)” with “2.”
nn
is the OLS residual from the i
equation and un
jt
T
642 ChApTeR 16 Additional Topics in Time Series Regression
specific assumptions, derived from economic theory and institutional knowledge, of what is exogenous and what is not. The discussion of structural VARs is best under- taken in the context of estimation of systems of simultaneous equations, which goes beyond the scope of this book. For an introduction to using VARs for forecasting and policy analysis, see Stock and Watson (2001). For additional mathematical detail on structural VAR modeling, see Hamilton (1994) or Watson (1994).
A VAR Model of the Growth Rate
of GDP and the Term Spread
As an illustration, consider a two-variable VAR for the growth rate of GDP, GDPGRt, and the term spread, TSpreadt. The VAR for GDPGRt and TSpreadt consists of two equations: one in which GDPGRt is the dependent variable and one in which TSpreadt is the dependent variable. The regressors in both equations are lagged values of GDPGRt and TSpreadt. Because of the apparent break in the relation in the early 1980s found in Section 14.7 using the QLR test, the VAR is estimated using data from 1981:Q1 to 2012:Q4.
The first equation of the VAR is the GDP growth rate equation:
GDPGRt = 0.52 + 0.29GDPGRt-1 + 0.22GDPGRt-2 (0.52) (0.11) (0.09)
-0.90TSpreadt-1 + 1.33TSpreadt-2. (0.36) (0.39)
(16.5)
The adjusted R2 is R 2 = 0.29.
The second equation of the VAR is the term spread equation, in which the
regressors are the same as in the GDPGR equation, but the dependent variable is the term spread:
TSpreadt = 0.46 + 0.01GDPGRt-1 – 0.06GDPGRt-2 (0.12) (0.02) (0.03)
+ 1.06TSpreadt-1 – 0.22TSpreadt-2. (0.10) (0.11)
(16.6)
The adjusted R2 is R 2 = 0.83.
Equations (16.5) and (16.6), taken together, are a VAR(2) model of the
growth rate of GDP, GDPGRt, and the term spread, TSpreadt.
These VAR equations can be used to perform Granger causality tests. The F-statistic testing the null hypothesis that the coefficients on TSpreadt−1 and TSpreadt−2 are zero in the GDP growth rate equation [Equation (16.5)] is 5.91, which has a p-value less than 0.001. Thus the null hypothesis is rejected, so we can conclude that the term spread is a useful predictor of the growth rate of GDP, given lags in the growth rate of GDP (that is, the term spread rate Granger-causes the growth rate of GDP). The F-statistic testing the hypothesis that the coeffi- cients on the two lags of GDPGRt are zero in the term spread equation [Equation (16.6)] is 3.48, which has a p-value of 0.03. Thus the growth rate of GDP Granger- causes the term spread at the 5% significance level.
Forecasts of the growth rate of GDP and the term spread one period ahead are obtained exactly as discussed in Section 14.4. The forecast of the growth rate of GDP for 2013:Q1, based on Equation (16.5), is GDP2013:Q102012:Q4 = 1.7 percentage point. A similar calculation using Equation (16.6) gives a forecast of the term spread 2013:Q1, based on data through 2012:Q4 of TSpread2013:Q102012:Q4 = 1.7%. The actual values for 2013:Q1 are GDPGR2013:Q1 = 1.1% and TSpread2013:Q1 = 1.9%.
16.2
Multiperiod Forecasts
The discussion of forecasting so far has focused on making forecasts one period in advance. Often, however, forecasters are called upon to make forecasts further into the future. This section describes two methods for making multiperiod fore- casts. The usual method is to construct “iterated” forecasts, in which a one-period- ahead model is iterated forward one period at a time, in a way that is made precise in this section. The second method is to make “direct” forecasts by using a regres- sion in which the dependent variable is the multiperiod variable that one wants to forecast. For reasons discussed at the end of this section, in most applications, the iterated method is recommended over the direct method.
Iterated Multiperiod Forecasts
The essential idea of an iterated forecast is that a forecasting model is used to make a forecast one period ahead, for period T + 1, using data through period T. The model then is used to make a forecast for date T + 2, given the data through date T, where the forecasted value for date T + 1 is treated as data for the pur- pose of making the forecast for period T + 2. Thus the one-period-ahead forecast (which is also referred to as a one-step-ahead forecast) is used as an intermediate
16.2 Multiperiod Forecasts 643
644 ChApTeR 16 Additional Topics in Time Series Regression
step to make the two-period-ahead forecast. This process repeats, or iterates, until
the forecast is made for the desired forecast horizon h.
The iterated AR forecast method: AR(1). An iterated AR(1) forecast uses an AR(1) for the one-period-ahead model. For example, consider the first-order autoregression for GDPGR [Equation (14.7)]:
GDPGRt = 1.99 + 0.34GDPGRt-1. (16.7) (0.35) (0.08)
The first step in computing the two-quarter-ahead forecast of GDPGR2013:Q2
based on Equation (16.7) using data through 2012:Q4 is to compute the one-
quarter-ahead forecast of GDPGR based on data through 2012:Q4:
GDPGR = 1.99 + 0.34GDPGR = 1.99 + 0.34 * 0.15 = 2.0. 2013:Q102012:Q4 2013:Q1 2012:Q4
The second step is to substitute this forecast into Equation (16.7) so that GDPGR2013:Q202012:Q4 = 1.99 + 0.34GDPGR2013:Q102012:Q4 = 1.99 + 0.34 * 2.0 = 2.7. Thus, based on information through the fourth quarter of 2012, this forecast states that the growth rate of GDP will be 2.7% in the second quarter of 2013.
The iterated AR forecast method: AR(p). The iterated AR(1) strategy is extended to an AR(p) by replacing YT + 1 with its forecast, YnT + 10T, and then treating that forecast as data for the AR(p) forecast of YT + 2. For example, consider the iter- ated two-period-ahead forecast of the growth rate of GDP based on the AR(2) model from Section 14.3 [Equation (14.13)]:
GDPGRt = 1.63 + 0.28GDPGRt-1 + 0.18GDPGRt-2. (16.8) (0.40) (0.08) (0.08)
The forecast of GDPGR based on data through 2012:Q4 using this AR(2), 2013:Q1
2013:Q1 0 2012:Q4
computed in Section 14.3, is GDPGR
ahead iterated forecast based on the AR(2) is GDPGR2013:Q202012:Q4 = 1.63+0.28GDPGR2013:Q102012:Q4 +0.18GDPGR2012:Q4 =1.63+0.28*2.1+0.18 × 0.15 = 2.2. According to this iterated AR(2) forecast, based on data through the fourth quarter of 2012, the growth rate of GDP is predicted to be 2.2 percent- age points in the second quarter of 2013.
Iterated multivariate forecasts using an iterated VAR. Iterated multivariate fore- casts can be computed using a VAR in much the same way as iterated univariate forecasts are computed using an autoregression. The main new feature of an
= 2.1. Thus the two-quarter-
iterated multivariate forecast is that the two-step-ahead (period T + 2) forecast of one variable depends on the forecasts of all variables in the VAR in period T + 1. For example, to compute the forecast of the growth rate of GDP in period T + 2 using a VAR with the variables GDPGRt and TSpreadt, one must forecast both GDPGRT+1 and TSpreadT+1, using data through period T as an intermediate step in forecasting GDPGRT+2. More generally, to compute multiperiod iterated VAR forecasts h periods ahead, it is necessary to compute forecasts of all vari- ables for all intervening periods between T and T + h.
As an example, we will compute the iterated VAR forecast of GDPGR2013:Q2 based on data through 2012:Q4, using the VAR(2) for GDPGRt and TSpreadt in Section 16.1 [Equations (16.5) and (16.6)]. The first step is to compute the one- quarter-ahead forecasts GDPGR2013:Q1∙2012:Q4 and TSpread2013:Q1∙2012:Q4 from that VAR. These one-period-ahead forecasts were computed in Section 16.1 based on Equations (16.5) and (16.6). The forecasts were GDPGR2013:Q1∙2012:Q4 = 1.7 and TSpread2013:Q1∙2012:Q4 = 1.7. In the second step, these forecasts are substituted into Equations (16.5) and (16.6) to produce the two-quarter-ahead forecast:
GDPGR2013:Q2∙2012:Q4 = 0.52 + 0.29 GDPGR2013:Q1∙2012:Q4 + 0.22GDPGR2012:Q4 – 0.90 TSpread2013:Q1∙2012:Q4 + 1.33TSpread2012:Q4
= 0.52 + 0.30 * 1.7 + 0.22 * 0.15
– 0.90 * 1.7 + 1.33 * 1.6 = 1.7. (16.9)
Thus the iterated VAR(2) forecast, based on data through the fourth quarter of 2012, is that the growth rate of GDP will be 1.7% in the second quarter of 2013.
Iterated multiperiod forecasts are summarized in Key Concept 16.2.
Direct Multiperiod Forecasts
Direct multiperiod forecasts are computed without iterating by using a single regression in which the dependent variable is the multiperiod-ahead variable to be forecasted and the regressors are the predictor variables. Forecasts computed this way are called direct forecasts because the regression coefficients can be used directly to make the multiperiod forecast.
The direct multiperiod forecasting method. Suppose that you want to make a forecast of YT + 2 using data through time T. The direct multivariate method takes the ADL model as its starting point but lags the predictor variables by an addi- tional time period. For example, if two lags of the predictors are used, then the
16.2 Multiperiod Forecasts 645
646 ChApTeR 16 Additional Topics in Time Series Regression
Iterated Multiperiod Forecasts
16.2
Key ConCept
The iterated multiperiod AR forecast is computed in steps: First compute the one-period-ahead forecast, then use that to compute the two-period-ahead fore- cast, and so forth. The two- and three-period-ahead iterated forecasts based on an AR(p) are
Y =b +bY +bY +bY +g+bY (16.10) nT+2∙T n0 n1 nT+1∙T n2 T n3 T-1 np T-p+2
Y =b +bY +bY +bY +g+bY nT+3∙T n0 n1 nT+2∙T n2 nT+1∙T n3 T np T-p+3
, (16.11)
where the bn’s are the OLS estimates of the AR(p) coefficients. Continuing this process (“iterating”) produces forecasts further into the future.
The iterated multiperiod VAR forecast is also computed in steps: First com- pute the one-period-ahead forecast of all the variables in the VAR, then use those forecasts to compute the two-period-ahead forecasts, and continue this process iteratively to the desired forecast horizon. The two-period-ahead iterated forecast of YT + 2, based on the two-variable VAR(p) in Key Concept 16.1, is
Y =b+bY +bY+bY +g+bY nT+2∙T n10 n11 nT+1∙T n12 T n13 T-1 n1p T-p+2
+gn X +gn X +gn X +g+gn X
11 nT+1∙T 12 T 13 T-1 1p T-p+2
, (16.12)
where the coefficients in Equation (16.12) are the OLS estimates of the VAR coefficients. Iterating produces forecasts further into the future.
dependent variable is Yt and the regressors are Yt – 2, Yt – 3, Xt−2, and Xt−3. The coefficients from this regression can be used directly to compute the forecast of YT + 2 using data on YT, YT – 1, XT, and XT−1, without the need for any iteration. More generally, in a direct h-period-ahead forecasting regression, all predictors are lagged h periods to produce the h-period-ahead forecast.
For example, the forecast of GDPGRt two quarters ahead using two lags each of GDPGRt−2 and TSpreadt−2 is computed by first estimating the regression:
GDPGRt∙t-2 = 0.57 + 0.34GDPGRt-2 + 0.03GDPGRt-3 (0.67) (0.07) (0.10)
+ 0.62TSpreadt – 2 – 0.01TSpreadt – 3. (16.13) (0.47) (0.46)
The two-quarter-ahead forecast of the growth rate of GDP in 2013:Q2 based on data through 2012:Q4 is computed by substituting the values of GDPGR2012:Q4, GDPGR2012:Q3, TSpread2012:Q4, and TSpread2012:Q3 into Equation (16.13); this yields
GDPGR2013:Q2∙2012:Q4 = 0.57 + 0.34GDPGR2012:Q4 + 0.03GDPGR2012:Q3 + 0.62TSpread2012:Q4- 0.01TSpread2012:Q3 = 1.68.
(16.14)
The three-quarter-ahead direct forecast ofGDPGRT+3 is computed by lagging all the regressors in Equation (16.13) by one additional quarter, estimating that regression, and then computing the forecast. The h-quarter-ahead direct forecast of GDPGRT+h is computed by using GPDGRt as the dependent variable and the regressors GPDGRt−h and TSpreadt−h, plus additional lags of GPDGRt−h and TSpreadt−h, as desired.
Standard errors in direct multiperiod regressions. Because the dependent vari- able in a multiperiod regression occurs two or more periods into the future, the error term in a multiperiod regression is serially correlated. To see this, consider the two-period-ahead forecast of the growth rate of GDP and suppose that a surprise jump in oil prices occurs in the next quarter. Today’s two-period-ahead forecast of the growth rate of GDP will be too low because it does not incorporate this unexpected event. Because the oil price rise was also unknown in the previous quarter, the two-period-ahead forecast made last quarter will also be too low. Thus the surprise oil price jump next quarter means that both last quarter’s and this quarter’s two-period-ahead forecasts are too low. Because of such intervening events, the error term in a multiperiod regression is serially correlated.
As discussed in Section 15.4, if the error term is serially correlated, the usual OLS standard errors are incorrect or, more precisely, they are not a reliable basis for inference. Therefore, heteroskedasticity- and autocorrelation-consistent (HAC) standard errors must be used with direct multiperiod regressions. The standard errors reported in Equation (16.13) for direct multiperiod regressions therefore are Newey–West HAC standard errors, where the truncation parameter m is set according to Equation (15.17); for these data (for which T = 128), Equa- tion (15.17) yields m = 4. For longer forecast horizons, the amount of overlap— and thus the degree of serial correlation in the error—increases: In general, the first h – 1 autocorrelation coefficients of the errors in an h-period-ahead regres- sion are nonzero. Thus larger values of m than indicated by Equation (15.17) are appropriate for multiperiod regressions with long forecast horizons.
Direct multiperiod forecasts are summarized in Key Concept 16.3.
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648 ChApTeR 16 Additional Topics in Time Series Regression
Direct Multiperiod Forecasts
16.3
Key ConCept
The direct multiperiod forecast h periods into the future based on p lags each of Yt and an additional predictor Xt is computed by first estimating the regression
Yt = d0 + d1Yt-h + g + dpYt-p-h+1 + dp+1Xt-h
+ g+ d2pXt-p-h+1 + ut, (16.15)
and then using the estimated coefficients directly to make the forecast of YT + h using data through period T.
Which Method Should You Use?
In most applications, the iterated method is the recommended procedure for multiperiod forecasting, for two reasons. First, from a theoretical perspective, if the underlying one-period-ahead model (the AR or VAR that is used to compute the iterated forecast) is specified correctly, then the coefficients are estimated more efficiently if they are estimated by a one-period-ahead regres- sion (and then iterated) than by a multiperiod-ahead regression. Second, from a practical perspective, forecasters are usually interested in forecasts not just at a single horizon but at multiple horizons. Because they are produced using the same model, iterated forecasts tend to have time paths that are less erratic across horizons than do direct forecasts. Because a different model is used at every horizon for direct forecasts, sampling error in the estimated coefficients can add random fluctuations to the time paths of a sequence of direct multi- period forecasts.
Under some circumstances, however, direct forecasts are preferable to iter- ated forecasts. One such circumstance is when you have reason to believe that the one-period-ahead model (the AR or VAR) is not specified correctly. For exam- ple, you might believe that the equation for the variable you are trying to forecast in a VAR is specified correctly, but that one or more of the other equations in the VAR is specified incorrectly, perhaps because of neglected nonlinear terms. If the one-step-ahead model is specified incorrectly, then in general the iterated multi- period forecast will be biased, and the MSFE of the iterated forecast can exceed the MSFE of the direct forecast, even though the direct forecast has a larger vari- ance. A second circumstance in which a direct forecast might be desirable arises
16.3 Orders of Integration and the DF-GLS Unit Root Test 649
in multivariate forecasting models with many predictors, in which case a VAR specified in terms of all the variables could be unreliable because it would have very many estimated coefficients.
16.3
Orders of Integration and the DF-GLS Unit Root Test
This section extends the treatment of stochastic trends in Section 14.6 by address- ing two further topics. First, the trends of some time series are not well described by the random walk model, so we introduce an extension of that model and dis- cuss its implications for regression modeling of such series. Second, we continue the discussion of testing for a unit root in time series data and, among other things, introduce a second test for a unit root, the DF-GLS test.
Other Models of Trends and Orders of Integration
Recall that the random walk model for a trend, introduced in Section 14.6, speci- fies that the trend at date t equals the trend at date t – 1, plus a random error term. If Yt follows a random walk with drift b0, then
Yt = b0 + Yt-1 + ut, (16.16)
where ut is serially uncorrelated. Also recall from Section 14.6 that, if a series has a random walk trend, then it has an autoregressive root that equals 1.
Although the random walk model of a trend describes the long-run move- ments of many economic time series, some economic time series have trends that are smoother—that is, vary less from one period to the next—than is implied by Equation (16.16). A different model is needed to describe the trends of such series.
One model of a smooth trend makes the first difference of the trend follow a random walk—that is,
∆Yt = b0 + ∆Yt-1 + ut, (16.17)
where ut is serially uncorrelated. Thus, if Yt follows Equation (16.17), ∆Yt follows a random walk, so ∆Yt – ∆Yt – 1 is stationary. The difference of the first differences, ∆Yt – ∆Yt-1, is called the second difference of Yt and is denoted ∆2Yt = ∆Yt – ∆Yt – 1. In this terminology, if Yt follows Equation (16.17), then its second
650 ChApTeR 16 Additional Topics in Time Series Regression
Orders of Integration, Differencing, and Stationarity
16.4
Key ConCept
• IfYtisintegratedoforderone—thatis,ifYtisI(1)—thenYthasaunitautoregressive root and its first difference, ∆Yt, is stationary.
• If Yt is integrated of order two—that is, if Yt is I(2)—then ∆Yt has a unit autoregressive root and its second difference, ∆2Yt, is stationary.
• If Yt is integrated of order d—that is, if Yt is I(d )—then Yt must be differ- enced d times to eliminate its stochastic trend; that is, ∆dYt is stationary.
difference is stationary. If a series has a trend of the form in Equation (16.17), then the first difference of the series has an autoregressive root that equals 1.
“Orders of integration” terminology. Some additional terminology is useful for distinguishing between these two models of trends. A series that has a random walk trend is said to be integrated of order one, or I(1). A series that has a trend of the form in Equation (16.17) is said to be integrated of order two, or I(2). A series that does not have a stochastic trend and is stationary is said to be inte- grated of order zero, or I(0).
The order of integration in the I(1) and I(2) terminology is the number of times that the series needs to be differenced for it to be stationary: If Yt is I(1), then the first difference of Yt, ∆Yt, is stationary, and if Yt is I(2), then the second difference of Yt, ∆2Yt, is stationary. If Yt is I(0), then Yt is stationary.
Orders of integration are summarized in Key Concept 16.4.
How to test whether a series is I(2) or I(1). If Yt is I(2), then ∆Yt is I(1), so ∆Yt has an autoregressive root that equals 1. If, however, Yt is I(1), then ∆Yt is stationary. Thus the null hypothesis that Yt is I(2) can be tested against the alternative hypothesis that Yt is I(1) by testing whether ∆Yt has a unit autoregressive root. If the hypothesis that ∆Yt has a unit autoregressive root is rejected, then the hypoth- esis that Yt is I(2) is rejected in favor of the alternative that Yt is I(1).
Examples of I(2) and I(1) series: The price level and the rate of inflation. The rate of inflation is the growth rate of the price level. Recall from Section 14.2 that the growth rate of a time series Xt can be computed as the first difference of the loga- rithm of Xt; that is Δln(Xt) is the growth rate of Xt (expressed as fraction). If Pt is
16.3 Orders of Integration and the DF-GLS Unit Root Test 651
a time series for the price level measured quarterly, then Δln(Pt) is its growth rate, and Inflt = 400 × Δln(Pt) is the quarterly rate of inflation, measured in percentage points at an annual rate. As in the expression for the growth of GDP, GDPGR in Equation (14.2), the factor 400 arises from converting fractional changes to per- centage changes (multiplying by 100) and converting quarterly percentages to an annual rate (multiplying by 4).
In Empirical Exercise 14.1, you analyzed the rate of inflation, Inflt, computed using the price index for personal consumption expenditures in the United States as Pt. In that exercise you concluded that the rate of inflation in the United States plausibly has a random walk stochastic trend—that is, that the rate of inflation is I(1). If inflation is I(1), then its stochastic trend is removed by first differencing, so ∆Inft is stationary. But treating inflation as I(1) is equivalent to treating Δln(Pt) as I(1), but this in turn is equivalent to treating the logarithm of the price level, ln(Pt), as I(2).
The logarithm of the price level and the rate of inflation are plotted in Fig- ure 16.1. The long-run trend of the logarithm of the price level (Figure 16.1a) varies more smoothly than the long-run trend in the rate of inflation (Fig- ure 16.1b). The smoothly varying trend in the logarithm of the price level is typical of I(2) series.
The DF-GLS Test for a Unit Root
This section continues the discussion of Section 14.6 regarding testing for a unit autoregressive root. We first describe another test for a unit autoregressive root, the so-called DF-GLS test. Next, in an optional mathematical section, we discuss why unit root test statistics do not have normal distributions, even in large samples.
The DF-GLS test. The ADF test was the first test developed for testing the null hypothesis of a unit root and is the most commonly used test in practice. Other tests subsequently have been proposed, however, many of which have higher power (Key Concept 3.5) than the ADF test. A test with higher power than the ADF test is more likely to reject the null hypothesis of a unit root against the sta- tionary alternative when the alternative is true; thus a more powerful test is better able to distinguish between a unit AR root and a root that is large but less than 1.
This section discusses one such test, the DF-GLS test developed by Elliott, Rothenberg, and Stock (1996). The test is introduced for the case that, under the null hypothesis, Yt has a random walk trend, possibly with drift, and under the alternative Yt is stationary around a linear time trend.
652 ChApTeR 16 Additional Topics in Time Series Regression
Figure 16.1
Logarithm
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004 2008 2012
(a) Logarithm of the United States PCE Price Index Percent per annum
12
10
8
6
4
2
0
–2
–4
–6
1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004 2008 2012
(b) United States PCE price inflation
The trend in the logarithm of prices (Figure 16.1a) is much smoother than the trend in inflation (Figure 16.1b).
The Logarithm of the price Level and the Inflation Rate in the United States, 1960–2012
16.3 Orders of Integration and the DF-GLS Unit Root Test 653
The DF-GLS test is computed in two steps. In the first step, the intercept and trend are estimated by generalized least squares (GLS; see Section 15.5). The GLS estimation is performed by computing three new variables, Vt, X1t, and X2t, whereV1 = Y1 andVt = Yt – a*Yt-1,t = 2, c,T,X11 = 1andX1t = 1 – a*,
= t – a*1t – 12,wherea*iscomputedusing t 1t2t
t = 2,c,T,andX
the formula a* = 1 – 13.5>T. Then V is regressed against X and X ; that is,
= 1andX
OLS is used to estimate the coefficients of the population regression equation
21
2t
Vt = d0X1t + d1X2t + et, (16.18) using the observations t = 1, c, T, where et is the error term. Note that there is
dnn are then used to compute a “detrended” version of Y , Y = Y – 1d + d t2.
no intercept in the regression in Equation (16.18). The OLS estimators dn0 and dn1
ttt01
In the second step, the Dickey–Fuller test is used to test for a unit autoregressive
root in Ydt , where the Dickey–Fuller regression does not include an intercept or a time trend. That is, ∆Ydt is regressed against Ydt – 1 and ∆Ydt – 1, c, ∆Ydt – p, where the number of lags p is determined, as usual, either by expert knowl- edge or by using a data-based method such as the AIC or BIC, as discussed in Section 14.5.
If the alternative hypothesis is that Yt is stationary with a mean that might be
a* is computed using the formula a* = 1 – 7>T, X is omitted from the regres- 2t
nonzero but without a time trend, the preceding steps are modified. Specifically,
sion in Equation (16.18), and the series Yd is computed as Yd = Y – dn . t tt0
The GLS regression in the first step of the DF-GLS test makes this test more complicated than the conventional ADF test, but it is also what improves its abil- ity to discriminate between the null hypothesis of a unit autoregressive root and the alternative that Yt is stationary. This improvement can be substantial. For example, suppose that Yt is in fact a stationary AR(1) with autoregressive coef- ficient b1 = 0.95, that there are T = 200 observations, and that the unit root tests are computed without a time trend [that is, t is excluded from the Dickey–Fuller regression, and X2t is omitted from Equation (16.18)]. Then the probability that the ADF test correctly rejects the null hypothesis at the 5% significance level is approximately 31% compared to 75% for the DF-GLS test.
CriticalvaluesforDF-GLStest. Becausethecoefficientsonthedeterministicterms are estimated differently in the ADF and DF-GLS tests, the tests have different critical values. The critical values for the DF-GLS test are given in Table 16.1. If the DF-GLS test statistic (the t-statistic on Ydt – 1 in the regression in the second
654 ChApTeR 16 Additional Topics in Time Series Regression taBLe 16.1 Critical Values of the DF-GLS Test
Deterministic regressors [regressors in equation (16.18)]
Intercept only (X1t only)
Intercept and time trend (X1t and X2t)
10%
−1.62
−2.57
5%
−1.95
−2.89
1%
−2.58
−3.48
Source: Fuller (1976) and Elliott, Rothenberg, and Stock (1996, Table 1).
step) is less than the critical value (that is, it is more negative than the critical value), then the null hypothesis that Yt has a unit root is rejected. Like the critical values for the Dickey–Fuller test, the appropriate critical value depends on which version of the test is used—that is, on whether or not a time trend is included [whether or not X2t is included in Equation (16.18)].
Application to the logarithm of GDP. The DF-GLS statistic, computed for the logarithm of GDP, ln(GDPt), over the period 1962:Q1 to 2012:Q4 with an inter- cept and time trend, is −2.85 when two lags of ∆Ydt are included in the Dickey– Fuller regression in the second stage, where the choice of two lags was based on the AIC (out of a maximum of six lags). This value is greater than the 5% critical value in Table 16.1, −2.89, so the DF-GLS test does not reject the null hypothesis of a unit root at the 5% significance level.
Why Do Unit Root Tests Have
Nonnormal Distributions?
In Section 14.6, it was stressed that the large-sample normal distribution on which regression analysis relies so heavily does not apply if the regressors are nonstationary. Under the null hypothesis that the regression contains a unit root, the regressor Yt – 1 in the Dickey–Fuller regression (and the regressor Ydt – 1 in the modified Dickey– Fuller regression in the second step of the DF-GLS test) is nonstationary. The non- normal distribution of the unit root test statistics is a consequence of this nonstationarity.
To gain some mathematical insight into this nonnormality, consider the sim-
plest possible Dickey–Fuller regression, in which ∆Yt is regressed against the
14.8, the OLS estimator in this regression is d = g Y ∆Y >g Y , so t=1 t-1 t t=1 t-1
single regressor Yt – 1 and the intercept is excluded. In the notation of Key Concept nT T2
16.3 Orders of Integration and the DF-GLS Unit Root Test 655
1 aT
T Yt-1∆Yt Tdn= t=1
. (16.19) Consider the numerator in Equation (16.19). Under the additional assumption
12aT 2 T Yt-1
t=1
that Y0 = 0, a bit of algebra (Exercise 16.5) shows that
t=1 2T t=1
Under the null hypothesis, ∆Yt = ut, which is serially uncorrelated and has a
1 aT Yt-1∆Yt = 1ca YT b2 – 1 aT 1∆Yt22d. (16.20) T2T
tion(16.20)canbewrittenY >2T = 2 g ∆Y = 2 g u,whichinturn
T Tt=1tTdt=1t
obeys the central limit theorem; that is, Y > 2T ¡ N10, s 2. Thus
g (∆Y ) ¡ s . Under the assumption that Y = 0, the first term in Equa- Tt=1tu1T01T
finite variance, so the second term in Equation (16.20) has the probability limit 1T2p2
Tu
1Y >2T2 – g 1∆Y2 ¡ s (Z – 1),whereZisastandardnormal
random variable. Recall, however, that the square of a standard normal distribu- tion has a chi-squared distribution with 1 degree of freedom. It therefore follows from Equation (16.20) that, under the null hypothesis, the numerator in Equation (16.19) has the limiting distribution
1aT Yt-1∆Yt ¡d s2u1x21 – 12. (16.21) Tt=1 2
The large-sample distribution in Equation (16.21) is different than the usual large- sample normal distribution when the regressor is stationary. Instead, the numera- tor of the OLS estimator of the coefficient on Yt in this Dickey–Fuller regression has a distribution that is proportional to a chi-squared distribution with 1 degree of freedom minus 1.
21T 2d22 TTt=1tu
This discussion has considered only the numerator of Tdn. The denominator also behaves unusually under the null hypothesis: Because Yt follows a random
1T2
walk under the null hypothesis, g Y does not converge in probability to a
T t=1 t-1
constant. Instead, the denominator in Equation (16.19) is a random variable, even
1T2
in large samples: Under the null hypothesis, g Y converges in distribution
T2 t=1 t-1
jointly with the numerator. The unusual distributions of the numerator and
denominator in Equation (16.19) are the source of the nonstandard distribution of the Dickey–Fuller test statistic and the reason that the ADF statistic has its own special table of critical values.
2
656
Chapter 16 Additional Topics in Time Series Regression
16.4
Cointegration
Sometimes two or more series have the same stochastic trend in common. In this special case, referred to as cointegration, regression analysis can reveal long-run relationships among time series variables, but some new methods are needed.
Cointegration and Error Correction
Two or more time series with stochastic trends can move together so closely over the long run that they appear to have the same trend component; that is, they appear to have a common trend. For example, Figure 16.2 reproduces the plot of the 10-year and 3-month interest rates from Figure 14.3. The interest rates exhibit the same long-run tendencies or trends: Both were low in the 1960s, both rose through the 1970s to peaks in the early 1980s, then both fell through the 1990s. However, the difference between the long-term and short-term interest rates, the term spread, does not appear to have a trend. That is, subtracting the short-term rate from the long-term rate appears to eliminate the trends in both of the
10-Year Interest rate, 3-Month Interest rate, and the term Spread
Figure 16.2
Percent per annum
20 15 10
5
0 –5
1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004 2008 2012
10-year and 3-month interest rates share a common stochastic trend. The term spread, or the difference, between the two rates does not exhibit a trend. These two interest rates appear to be cointegrated.
Interest rate spread
10-year rates
3-month rates
Interest rate spread
16.4 Cointegration 657
Cointegration
Key ConCept
16.5
Suppose that Xt and Yt are integrated of order one. If, for some coefficient u, Yt – uXt is integrated of order zero, then Xt and Yt are said to be cointegrated. The coefficient u is called the cointegrating coefficient.
If Xt and Yt are cointegrated, then they have the same, or common, stochastic trend. Computing the difference Yt – uXt eliminates this common stochastic trend.
individual rates. Said differently, although the two interest rates differ, they appear to share a common stochastic trend: Because the trend in each individual series is eliminated by subtracting one series from the other, the two series must have the same trend; that is, they must have a common stochastic trend.
Two or more series that have a common stochastic trend are said to be coin- tegrated. The formal definition of cointegration (due to the econometrician Clive Granger, 1983; see the box “Nobel Laureates in Time Series Econometrics”) is given in Key Concept 16.5. In this section, we introduce a test for whether cointe- gration is present, discuss estimation of the coefficients of regressions relating cointegrated variables, and illustrate the use of the cointegrating relationship for forecasting. The discussion initially focuses on the case that there are only two variables, Xt and Yt.
Vectorerrorcorrectionmodel. Untilnow,wehaveeliminatedthestochastictrend in an I(1) variable Yt by computing its first difference, ∆Yt; the problems created by stochastic trends were then avoided by using ∆Yt instead of Yt in time series regressions. If Xt and Yt are cointegrated, however, another way to eliminate the trend is to compute Yt – uXt, where u is chosen to eliminate the common trend from the difference. Because the term Yt – uXt is stationary, it too can be used in regression analysis.
In fact, if Xt and Yt are cointegrated, the first differences of Xt and Yt can be mod- eled using a VAR, augmented by including Yt – 1 – uXt – 1 as an additional regressor:
(16.22) (16.23)
∆Y = b + b ∆Y + g + b ∆Y + g ∆X
t 10 11 t-1 1p t-p 11 t-1
+ g+ g ∆X + a 1Y – uX 2 + u 1p t-p 1 t-1 t-1 1t
∆X = b + b ∆Y + g + b ∆Y + g ∆X
t 20 21 t-1 2p t-p 21 t-1
+ g+ g ∆X + a 1Y – uX 2 + u . 2p t-p 2 t-1 t-1 2t
658 Chapter 16 Additional Topics in Time Series Regression
The term Yt – uXt is called the error correction term. The combined model in Equations (16.22) and (16.23) is called a vector error correction model (VECM). In a VECM, past values of Yt – uXt help to predict future values of ∆Yt and/or ∆Xt.
How Can You Tell Whether Two Variables
Are Cointegrated?
There are three ways to determine whether two variables can plausibly be mod- eled as cointegrated: Use expert knowledge and economic theory, graph the series and see whether they appear to have a common stochastic trend, and perform statistical tests for cointegration. All three methods should be used in practice.
First, you must use your expert knowledge of these variables to decide whether cointegration is in fact plausible. For example, the two interest rates in Figure 16.2 are linked together by the so-called expectations theory of the term structure of interest rates. According to this theory, the interest rate on January 1 on the 10-year Treasury bond is the average of the interest rate on a 3-month Treasury bill for the first quarter of the year and the expected interest rates on future 3-month Treasury bills issued in the subsequent 39 quarters, for total of 40 quarters, or 10 years. If this was not the case, then investors could expect to make money by holding either the 10-year Treasury note or a sequence of forty 3-month Treasury bills, and they would bid up prices until the expected returns were equalized. If the 3-month interest rate has a random walk stochastic trend, this theory implies that this stochastic trend is inherited by the 10-year interest rate and that the difference between the two rates—that is, the term spread—is stationary. Thus the expectations theory of the term structure implies that if the interest rates are I(1), then they will be cointegrated with a cointegrating coeffi- cient of u = 1 (Exercise 16.2).
Second, visual inspection of the series helps to identify cases in which cointe- gration is plausible. For example, the graph of the two interest rates in Figure 16.2 shows that each of the series appears to be I(1) but that the term spread appears to be I(0), so the two series appear to be cointegrated.
Third, the unit root testing procedures introduced so far can be extended to tests for cointegration. The insight on which these tests are based is that if Yt and Xt are cointegrated with cointegrating coefficient u, then Yt – uXt is stationary; otherwise, Yt – uXt is nonstationary [is I(1)]. The hypothesis that Yt and Xt are not cointegrated [that is, that Yt – uXt is I(1)] therefore can be tested by testing the null hypothesis that Yt – uXt has a unit root; if this hypothesis is rejected, then Yt and Xt can be modeled as cointegrated. The details of this test depend on whether the cointegrating coefficient u is known.
16.4 Cointegration 659 taBLe 16.2 Critical Values for the engle–Granger ADF Statistic
number of X’s in equation (16.24)
1
2
3
4
10% 5%
-3.12 -3.41
-3.52 -3.80
-3.84 -4.16
-4.20 -4.49
1%
-3.96
-4.36
-4.73
-5.07
Testing for cointegration when θ is known. In many cases expert knowledge or economic theory suggests a value for u. When u is known, the Dickey–Fuller and DF-GLS unit root tests can be used to test for cointegration by first constructing the series zt = Yt – uXt and then testing the null hypothesis that zt has a unit autoregressive root.
Testing for cointegration when θ is unknown. If the cointegrating coefficient u is unknown, then it must be estimated prior to testing for a unit root in the error correction term. This preliminary step makes it necessary to use different critical values for the subsequent unit root test.
Specifically, in the first step the cointegrating coefficient u is estimated by OLS estimation of the regression
Yt =a+uXt +zt. (16.24)
In the second step, a Dickey–Fuller t-test (with an intercept but no time trend) is used to test for a unit root in the residual from this regression, znt. This two-step procedure is called the Engle–Granger Augmented Dickey–Fuller test for coin- tegration, or EG-ADF test (Engle and Granger, 1987).
Critical values of the EG-ADF statistic are given in Table 16.2.1 The critical values in the first row apply when there is a single regressor in Equation (16.26), so there are two cointegrated variables (Xt and Yt). The subsequent rows apply to the case of multiple cointegrated variables, which is discussed at the end of this section.
Estimation of Cointegrating Coefficients
If Xt and Yt are cointegrated, then the OLS estimator of the coefficient in the coin- tegrating regression in Equation (16.24) is consistent. However, in general the OLS
1The critical values in Table 16.2 are taken from Fuller (1976) and Phillips and Ouliaris (1990). Fol- lowing a suggestion by Hansen (1992), the critical values in Table 16.2 are chosen so that they apply whether or not Xt and Yt have drift components.
660 ChApTeR 16 Additional Topics in Time Series Regression
estimator has a nonnormal distribution, and inferences based on its t-statistics can be misleading whether or not those t-statistics are computed using HAC standard errors. Because of these drawbacks of the OLS estimator of u, econometricians have developed a number of other estimators of the cointegrating coefficient.
One such estimator of u that is simple to use in practice is the dynamic OLS (DOLS) estimator (Stock and Watson, 1993). The DOLS estimator is based on a modified version of Equation (16.24) that includes past, present, and future values of the change in Xt:
ap j=-p
Thus, in Equation (16.25), the regressors are Xt, ∆Xt + p, c, ∆Xt – p. The DOLS estimator of u is the OLS estimator of u in the regression of Equation (16.25).
If Xt and Yt are cointegrated, then the DOLS estimator is efficient in large samples. Moreover, statistical inferences about u and the d’s in Equation (16.25) based on HAC standard errors are valid. For example, the t-statistic constructed using the DOLS estimator with HAC standard errors has a standard normal dis- tribution in large samples.
One way to interpret Equation (16.25) is to recall from Section 15.3 that cumula- tive dynamic multipliers can be computed by modifying the distributed lag regression of Yt on Xt and its lags. Specifically, in Equation (15.7), the cumulative dynamic multipliers were computed by regressing Yt on ∆Xt, lags of ∆Xt, and Xt – r; the coef- ficient on Xt – r in that specification is the long-run cumulative dynamic multiplier. Similarly, if Xt were strictly exogenous, then in Equation (16.25) the coefficient on Xt, u would be the long-run cumulative multiplier—that is, the long-run effect on Y of a change in X. If Xt is not strictly exogenous, then the coefficients do not have this interpretation. Nevertheless, because Xt and Yt have a common stochastic trend if they are cointegrated, the DOLS estimator is consistent even if Xt is endogenous.
The DOLS estimator is not the only efficient estimator of the cointegrating coefficient. The first such estimator was developed by Søren Johansen (Johansen, 1988). For a discussion of Johansen’s method and of other ways to estimate the cointegrating coefficient, see Hamilton (1994, Chapter 20).
Even if economic theory does not suggest a specific value of the cointegrating coefficient, it is important to check whether the estimated cointegrating relation- ship makes sense in practice. Because cointegration tests can be misleading (they can improperly reject the null hypothesis of no cointegration more frequently than they should, and frequently they improperly fail to reject the null hypothe- sis), it is especially important to rely on economic theory, institutional knowledge, and common sense when estimating and using cointegrating relationships.
Yt = b0 + uXt +
dj∆Xt-j + ut. (16.25)
16.4 Cointegration 661 Extension to Multiple Cointegrated Variables
The concepts, tests, and estimators discussed here extend to more than two vari- ables. For example, if there are three variables, Yt, X1t, and X2t, each of which is I(1), then they are cointegrated with cointegrating coefficients u1 and u2 if Yt – u1X1t – u2X2t is stationary. When there are three or more variables, there can be multiple cointegrating relationships. For example, consider modeling the relationship among three interest rates: the 3-month rate (R3m), the 1-year (R1y) rate, and the 10-year rate (R10y). If they are I(1), then the expectations theory of the term structure of interest rates suggests that they will all be cointegrated. One cointegrating relationship suggested by the theory is R10yt − R3mt, and a second relationship is R1yt − R3mt. (The relationship R10yt − R1yt is also a cointegrat- ing relationship, but it contains no additional information beyond that in the other relationships because it is perfectly multicollinear with the other two cointegrat- ing relationships.)
The EG-ADF procedure for testing for a single cointegrating relationship among multiple variables is the same as for the case of two variables, except that the regression in Equation (16.24) is modified so that both X1t and X2t are regres- sors; the critical values for the EG-ADF test are given in Table 16.2, where the appropriate row depends on the number of regressors in the first-stage OLS cointegrating regression. The DOLS estimator of a single cointegrating relation- ship among multiple X’s involves including the level of each X along with leads and lags of the first difference of each X. Tests for multiple cointegrating rela- tionships can be performed using system methods, such as Johansen’s (1988) method, and the DOLS estimator can be extended to multiple cointegrating rela- tionships by estimating multiple equations, one for each cointegrating relation- ship. For additional discussion of cointegration methods for multiple variables, see Hamilton (1994).
A cautionary note. If two or more variables are cointegrated, then the error correction term can help to forecast these variables and, possibly, other related variables. However, cointegration requires the variables to have the same sto- chastic trends. Trends in economic variables typically arise from complex inter- actions of disparate forces, and closely related series can have different trends for subtle reasons. If variables that are not cointegrated are incorrectly modeled using a VECM, then the error correction term will be I(1); this introduces a trend into the forecast that can result in poor out-of-sample forecast perfor- mance. Thus forecasting using a VECM must be based on a combination of compelling theoretical arguments in favor of cointegration and careful empirical analysis.
662 ChApTeR 16 Additional Topics in Time Series Regression Application to Interest Rates
As discussed earlier, the expectations theory of the term structure of interest rates implies that if two interest rates of different maturities are I(1), then they will be cointegrated with a cointegrating coefficient of u = 1; that is, the spread between the two rates will be stationary. Inspection of Figure 16.2 provides qualitative support for the hypothesis that the 10-year and 3-month interest rates are cointe- grated. We first use unit root and cointegration test statistics to provide more formal evidence on this hypothesis, then estimate a vector error correction model for these two interest rates.
Unit root and cointegration tests. Various unit root and cointegration test sta- tistics for these two series are reported in Table 16.3. The unit root test statistics in the first two rows examine the hypothesis that the two interest rates, the 3-month rate (R3m) and the 10-year rate (R10y), individually have a unit root. The ADF and DF-GLS test statistics are larger than the 10% critical values, so the null hypothesis of a unit root is not rejected for either series at the 10% significance level. Thus, these results suggest that the interest rates are plausibly modeled as I(1).
The unit root statistics for the term spread, R10yt − R3mt, test the further hypothesis that these variables are not cointegrated against the alternative hypothesis that they are. The null hypothesis that the term spread contains a unit root is rejected at the 1% level, using both unit root tests. Thus we reject the hypothesis that the series are not cointegrated against the alternative that they are, with a cointegrating coefficient u = 1. Taken together, the evidence in the first three rows of Table 16.3 suggests that these variables plausibly can be mod- eled as cointegrated with u = 1.
taBLe 16.3
Series
R3m
R10y
R10y − R3m
R10y − 0.814 × R3m
Unit Root and Cointegration Test Statistics for Two Interest Rates
aDF Statistic
– 2.17
– 1.03
-3.97**
-3.15
DF-gLS Statistic
– 1.84
– 0.96
-3.92**
—
R3m is the interest rate on 3-month U.S. Treasury bills, and R10y is the interest rate on 10-year U.S. Treasury bonds. Regressions were estimated using quarterly data over the period 1962:Q1–2012:Q4. The number of lags in the unit root test statistic regressions were chosen by AIC (six lags maximum). Unit root test statistics are significant at the *5% or **1% significance level.
Because in this application economic theory suggests a value for u (the expectations theory of the term structure suggests that u = 1) and because the error correction term is I(0) when this value is imposed (the spread is station- ary), in principle it is not necessary to use the EG-ADF test, in which u is estimated. Nevertheless, we compute the test as an illustration. The first step in the EG-ADF test is to estimate u by the OLS regression of one variable on the other; the result is
R10yt = 2.46 + 0.81R3mt,R2 = 0.83. (16.26)
The second step is to compute the ADF statistic for the residual from this regression, znt. The result, given in the final row of Table 16.3, is −3.15. This value is smaller than the 10% critical value (which is -3.12) but not smaller than the 5% critical value (−3.41), so the null hypothesis of no cointegration is rejected at the 10% significance level but not the 5% significance level. An interpretation of this result is that the EG-ADF test, which uses an estimated value of u, is less powerful than the test that uses what is arguably the correct value of u = 1.
A vector error correction model of the two interest rates. If Yt and Xt are cointe- grated, then forecasts of ∆Yt and ∆Xt can be improved by augmenting a VAR of ∆Yt and ∆Xt by the lagged value of the error correction term—that is, by comput- ing forecasts using the VECM in Equations (16.22) and (16.23). If u is known, then the unknown coefficients of the VECM can be estimated by OLS, including zt-1 = Yt-1 – uXt-1 asanadditionalregressor.Ifuisunknown,thentheVECM can be estimated using znt – 1 as a regressor, where znt = Yt – unXt, and where un is an estimator of u.
In the application to the two interest rates, theory suggests that u = 1, and the unit root tests support modeling the two interest rates as cointegrated with a cointegrating coefficient of 1. We therefore specify the VECM using the theo- retically suggested value of u = 1—that is, by adding the lagged value of the term spread, R10y −R3m, to a VAR in ∆R10yt and ∆R3mt. Specified with two lags of first differences, the resulting VECM is
∆R3mt = -0.06 + 0.24∆R3mt-1 – 0.16∆R3mt-2 + 0.11∆R10yt-1 (0.12) (0.13) (0.18) (0.20)
-0.15∆R10yt-2 + 0.03(R10yt-1 – R3mt-1) (16.27) (0.15) (0.05)
16.4 Cointegration 663
664
ChApTeR 16 Additional Topics in Time Series Regression
∆R10yt = 0.12 – 0.00∆R3mt-1 – 0.07∆R3mt-2 + 0.22∆R10yt-1
(0.06) (0.09) (0.07) (0.11)
-0.07∆R10yt-2 – 0.09(R10yt-1 – R3mt-1). (16.28) (0.09) (0.03)
In Equation (16.27), none of the coefficients is individually significant at the 5% level, and the coefficients on the lagged first differences of the interest rates are not jointly significant at the 5% level. In Equation (16.28), the coefficients on the lagged first differences are not jointly significant, but the coefficient on the lagged spread (the error correction term), which is estimated to be −0.09, has a t-statistic of −2.74, so it is statistically significant at the 1% level. Although lagged values of the first difference of the interest rates are not useful for predicting future inter- est rates, the lagged spread does help predict the change in the 10-year Treasury bond rate. When the 10-year rate exceeds the 3-month rate, the 10-year rate is forecasted to fall in the future.
16.5
Volatility Clustering and Autoregressive Conditional Heteroskedasticity
The phenomenon that some times are tranquil while others are not—that is, that volatility comes in clusters—shows up in many economic time series. This section presents a pair of models for quantifying volatility clustering or, as it is also known, conditional heteroskedasticity.
Volatility Clustering
The volatility of many financial and macroeconomic variables changes over time. For example, daily percentage changes in the Wilshire 5000 stock price index, shown in Figure 16.3, exhibit periods of high volatility, such as in 2001 and 2008, and other periods of low volatility, such as in 2004. A series with some periods of low volatility and some periods of high volatility is said to exhibit volatility clustering. Because the volatility appears in clusters, the variance of the daily percentage price change in the Wilshire 5000 index can be forecasted, even though the daily price change itself is very difficult to forecast.
Forecasting the variance of a series is of interest for several reasons. First, the variance of an asset price is a measure of the risk of owning that asset: The larger
Figure 16.3
Percent
12.5 10.0 7.5 5.0 2.5 0.0 –2.5 –5.0 –7.5
16.5 Volatility Clustering and Autoregressive Conditional Heteroskedasticity 665 Daily percentage Changes in the Wilshire Index, 1990–2013
–10.01990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2013 Daily percentage price changes in the Wilshire 5000 index exhibit volatility clustering, in which there are some periods
of high volatility, such as in 2008, and other periods of relative tranquility, such as in 2004.
the variance of daily stock price changes, the more a stock market participant stands to gain—or lose—on a typical day. An investor who is worried about risk would be less tolerant of participating in the stock market during a period of high—rather than low—volatility.
Second, the value of some financial derivatives, such as options, depends on the variance of the underlying asset. An options trader wants the best available forecasts of future volatility to help him or her know the price at which to buy or sell options.
Third, forecasting variances makes it possible to have accurate forecast inter- vals. Suppose that you are forecasting the rate of inflation. If the variance of the forecast error is constant, then an approximate forecast confidence interval can be constructed along the lines discussed in Section 14.4—that is, as the forecast plus or minus a multiple of the SER. If, however, the variance of the forecast error changes over time, then the width of the forecast interval should change over time: At periods when inflation is subject to particularly large disturbances or shocks, the interval should be wide; during periods of relative tranquility, the interval should be tighter.
666 ChApTeR 16 Additional Topics in Time Series Regression
Volatility clustering can be thought of as clustering of the variance of the error term over time: If the regression error has a small variance in one period, its variance tends to be small in the next period, too. In other words, volatility clus- tering implies that the error exhibits time-varying heteroskedasticity.
Autoregressive Conditional Heteroskedasticity
Two models of volatility clustering are the autoregressive conditional heteroskedas- ticity (ARCH) model and its extension, the generalized ARCH (GARCH) model.
ARCH. Consider the ADL(1,1) regression
Yt = b0 + b1Yt-1 + g1Xt-1 + ut. (16.29)
In the ARCH model, which was developed by the econometrician Robert Engle (Engle, 1982; see the box “Nobel Laureates in Time Series Econometrics”), the error ut is modeled as being normally distributed with mean zero and variance s2t , where s2t depends on past squared values ut. Specifically, the ARCH model of order p, denoted ARCH(p), is
s2t = a0 + a1u2t-1 + a2u2t-2 + g+ apu2t-p, (16.30)
where a0, a1, c, ap are unknown coefficients. If these coefficients are positive, then if recent squared errors are large, the ARCH model predicts that the current squared error will be large in magnitude, in the sense that its variance, s2t , is large.
Although it is described here for the ADL(1,1) model in Equation (16.29), the ARCH model can be applied to the error variance of any time series regres- sion model with an error that has a conditional mean of zero, including higher- order ADL models, autoregressions, and time series regressions with multiple predictors.
GARCH. ThegeneralizedARCH(GARCH)model,developedbytheeconometri- cianTimBollerslev(Bollerslev,1986),extendstheARCHmodeltolets2t depend on its own lags as well as lags of the squared error. The GARCH(p,q) model is
s2t = a0 + a1u2t-1 + g + apu2t-p + f1s2t-1 + g+ fqs2t-q, (16.31) where a0, a1, c, ap, f1, c, fq are unknown coefficients.
16.5 Volatility Clustering and Autoregressive Conditional Heteroskedasticity 667
The ARCH model is analogous to a distributed lag model, and the GARCH model is analogous to an ADL model. As discussed in Appendix 15.2, the ADL model (when appropriate) can provide a more parsimonious model of dynamic multipliers than can the distributed lag model. Similarly, by incorporating lags of s2t , the GARCH model can capture slowly changing variances with fewer param- eters than the ARCH model.
An important application of ARCH and GARCH models is to measuring and forecasting the time-varying volatility of returns on financial assets, particularly assets observed at high sampling frequencies such as the daily stock returns in Figure 16.3. In such applications, the return itself is often modeled as unpredict- able, so the regression in Equation (16.29) only includes the intercept.
Estimation and inference. ARCH and GARCH models are estimated by the method of maximum likelihood (Appendix 11.2). The estimators of the ARCH and GARCH coefficients are normally distributed in large samples, so in large samples, t-statistics have standard normal distributions, and confidence inter- vals can be constructed as the maximum likelihood estimate {1.96 standard errors.
Application to Stock Price Volatility
A GARCH(1,1) model of the Wilshire daily percentage stock price changes, Rt, estimated using data on all trading days from January 2, 1990, through December 31, 2013, is
Rt = 0.057 (0.010)
sn 2t = 0.011 + 0.082 u2t – 1 (0.002) (0.007)
+ 0.908s2t – 1. (0.008)
(16.32)
(16.33)
n
No lagged predictors appear in Equation (16.32) because daily Wilshire 5000 per- centage price changes are essentially unpredictable.
The two coefficients in the GARCH model (the coefficients on u2t – 1 and s2t – 1) are both individually statistically significant at the 5% significance level. One measure of the persistence of movements in the variance is the sum of the coef- ficients on u2t – 1 and s2t – 1 in the GARCH model (Exercise 16.9). This sum (0.99) is large, indicating that changes in the conditional variance are persistent. Said
668 Chapter 16 Additional Topics in Time Series Regression
Figure 16.4
Percent
12.5 10.0 7.5 5.0 2.5 0.0 –2.5 –5.0 –7.5 –10.0
Daily percentage Changes in the Wilshire 5000 Index and GarCh(1,1) Bands
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012 2013
The GARCH(1,1) bands, which are {snt, where snt is computed using Equation (16.33), are narrow when the conditional variance is small and wide when it is large. The conditional volatility of stock price changes varies considerably over the 1990–2013 period.
differently, the estimated GARCH model implies that periods of high volatility in stock prices will be long-lasting. This implication is consistent with the long periods of volatility clustering seen in Figure 16.3.
The estimated conditional variance at date t, sn 2t , can be computed using the residuals from Equation (16.32) and the coefficients in Equation (16.33). Figure 16.4 plots bands of plus or minus one conditional standard deviation (that is, {snt), based on the GARCH(1,1) model, along with deviations of the percentage price change series from its mean. The conditional standard devi- ation bands quantify the time-varying volatility of the daily price changes. During the mid-1990s, the conditional standard deviation bands are tight, indicating lower levels of risk for investors holding a portfolio of stocks mak- ing up the Wilshire index. In contrast, during 2008, these conditional standard deviation bands are wide, indicating a period of greater daily stock price volatility.
16.5 Volatility Clustering and Autoregressive Conditional Heteroskedasticity 669 Nobel Laureates in Time Series Econometrics
In 2003 Robert Engle and Clive Granger won the Nobel Prize in economics for fundamental theo- retical research in time series econometrics. Engle’s work was motivated by the volatility clustering evident in plots like Figure 16.3. Engle wondered whether series like these could be stationary and whether econometric models could be developed to explain and predict their time-varying volatility. Engle’s answer was to develop the autoregressive conditional heteroskedasticity (ARCH) model, described in Section 16.5. The ARCH model and
spurious? Granger discovered that when variables shared common trends—in his terminology, were “co-integrated”—meaningful relationships could be uncovered by regression analysis using a vector error correction model. The methods of cointegra- tion analysis are now a staple in modern macro- econometrics.
In 2011, Thomas Sargent and Christopher Sims won the Nobel Prize for their empirical research on cause and effect in the macroeconomy. Sargent was recognized for developing models that featured the important role that expecta-
tions about the future play
in disentangling cause and
effect. Sims was recognized for
developing structural VAR
(SVAR) models. Sims’s key
insight concerned the forecast
errors in a VAR model—the
ut errors in Equations (16.1)
and (16.2). These errors, he
realized, arose because of
unforeseen “shocks” that buf-
feted the macroeconomy, and
in many cases, these shocks
had well defined sources like
OPEC (oil price shocks), the
Fed (interest rate shocks), or Congress (tax shocks). By disentangling the various sources of shocks that comprise the VAR errors, Sims was able to estimate the dynamic causal effect of these shocks on the vari- ables appearing in the VAR. This disentangling of shocks is never without controversy, but SVARs are now a standard tool for estimating dynamic causal effects in macroeconomics.
continued on next page
Clive W. J. Granger
its extensions proved espe- cially useful for modeling the volatility of asset returns, and the resulting volatility forecasts are used to price financial derivatives and to assess changes over time in the risk of holding financial assets. Today, measures and forecasts of volatility are a core component of finan- cial econometrics, and the ARCH model and its descen- dants are the workhorse tools for modeling volatility.
Christopher A. Sims
Granger’s work focused on how to handle stochastic trends in economic time series data. From his ear- lier work, he knew that two unrelated series with stochastic trends could, by the usual statistical measures of t-statistics and regression R2’s, falsely appear to be meaningfully related; this is the “spuri- ous regression” problem exemplified by the regres- sions in Equations (14.28) and (14.29). But are all regressions involving stochastic trending variables
Lars Peter Hansen
Robert F. Engle
670 Chapter 16 Additional Topics in Time Series Regression
In 2013, Eugene Fama, Lars Peter Hansen, and Robert Shiller won the Nobel Prize for their empiri- cal analysis of asset prices. The work in the two “Can You Beat the Market” boxes in Chapter 14 and the box “Commodity Traders Send Shivers Through Disney World” in Chapter 15 was motivated in part by the “efficient markets” (unpredictability) work of Fama and the “irrational exuberance” (unexplained volatility) work of Shiller. Hansen was honored for developing “Generalized Method of Moments” (GMM) methods to investigate whether asset returns are consistent with expected utility theory. Microeconomics says that investors should equate the marginal cost of an investment (today’s foregone
utility from investing rather than consuming) with its marginal benefit (tomorrow’s boost in utility from consumption financed by the investment’s return). But a simple test of this proposition is complicated because marginal utility is difficult to measure, asset returns are uncertain, and the argument should hold across all asset returns. Hansen developed GMM methods to test asset-pricing models. As it turned out, Hansen’s GMM methods had applications well beyond finance and are now widely used in econo- metrics. Section 18.7 introduces GMM.
For more information on these and other Nobel laureates in economics, visit the Nobel Foundation website, http://www.nobel.se/economics.
16.6
Conclusion
This part of the book has covered some of the most frequently used tools and concepts of time series regression. Many other tools for analyzing economic time series have been developed for specific applications. If you are interested in learn- ing more about economic forecasting, see the introductory textbooks by Diebold (2007) and Enders (2009). For an advanced treatment of econometrics with time series data, see Hamilton (1994) and Hayashi (2000).
Summary
1. Vector autoregressions model a “vector” of k time series variables as each depends on its own lags and the lags of the k – 1 other series. The forecasts of each of the time series produced by a VAR are mutually consistent, in the sense that they are based on the same information.
2. Forecasts two or more periods ahead can be computed either by iterating forward a one-step-ahead model (an AR or a VAR) or by estimating a multiperiod-ahead regression.
3. Two series that share a common stochastic trend are cointegrated; that is, Yt and Xt are cointegrated if Yt and Xt are I(1) but Yt – uXt is I(0). If Yt and
Xt are cointegrated, the error correction term Yt – uXt can help predict ∆Yt and/or ∆Xt. A vector error correction model is a VAR model of ∆Yt and ∆Xt, augmented to include the lagged error correction term.
4. Volatility clustering—in which the variance of a series is high in some peri- ods and low in others—is common in economic time series, especially finan- cial time series.
5. The ARCH model of volatility clustering expresses the conditional variance of the regression error as a function of recent squared regression errors. The GARCH model augments the ARCH model to include lagged conditional variances as well. Estimated ARCH and GARCH models produce forecast intervals with widths that depend on the volatility of the most recent regres- sion residuals.
Key Terms
vector autoregression (VAR) (639) iterated multiperiod AR forecast (646) iterated multiperiod VAR forecast
(646)
direct multiperiod forecast (648) integrated of order d[I(d)] (650) second difference (649)
integrated of order zero [I(0)], one
[I(1)], or two [I(2)] (650) order of integration (650) DF-GLS test (651)
common trend (656)
cointegration (657) cointegrating coefficient (657) error correction term (658) vector error correction model
(VECM) (658)
EG-ADF test (659)
dynamic OLS (DOLS) estimator
(660)
volatility clustering (664) autoregressive conditional
heteroskedasticity (ARCH) (666) generalized ARCH (GARCH) (666)
Key Terms 671
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672 ChApTeR 16 Additional Topics in Time Series Regression Review the Concepts
16.1 A macroeconomist wants to construct forecasts for the following macroeconomic variables: GDP, consumption, investment, government purchases, exports, imports, short-term interest rates, long-term interest rates, and the rate of price inflation. He has quarterly time series for each of these variables from 1970 to 2014. Should he estimate a VAR for these variables and use this for forecasting? Why or why not? Can you suggest an alternative approach?
16.2 Suppose that Yt follows a stationary AR(1) model with b0 = 0 and b1 = 0.7. If Yt = 5, what is your forecast of Yt+2 (that is, what is Yt+2∙t)? What is Yt + h∙t for h = 30? Does this forecast for h = 30 seem reasonable to you?
16.3 A version of the permanent income theory of consumption implies that the logarithm of real GDP (Y) and the logarithm of real consumption (C) are cointegrated with a cointegrating coefficient equal to 1. Explain how you would investigate this implication by (a) plotting the data and (b) using a statistical test.
16.4 Consider the ARCH model, s2t = 1.0 + 0.8 u2t – 1. Explain why this will lead to volatility clustering. (Hint: What happens when u2t – 1 is unusually large?)
16.5 The DF-GLS test for a unit root has higher power than the Dickey–Fuller test. Why should you use a more powerful test?
Exercises
16.1 Suppose that Yt follows a stationary AR(1) model, Yt = b0 + b1Yt – 1 + ut. a. Show that the h-period-ahead forecast of Yt is given by
h
Y = m + b (Y – m ), where m = b >11 – b 2.
b. Suppose that X is related to Y by X = g d Y , where ∙d∙< 1. t t t i=0 t+i∙t
t+h∙tY1tY Y0 1 ∞i
Show that X = [m >(1 – d)] + [(Y – m )>(1 – b d)]. tYtY1
16.2 One version of the expectations theory of the term structure of interest
rates holds that a long-term rate equals the average of the expected values
of short-term interest rates into the future, plus a term premium that is
I(0). Specifically, let Rkt denote a k-period interest rate, let R1t denote
Rk = g R1 + e,whereR1 istheforecastmadeatdatetofthe t k i=0 t+i∙t t t+i∙t
a one-period interest rate, and let et denote an I(0) term premium. Then 1 k-1
Exercises 673 value of R1 at date t + i. Suppose that R1t follows a random walk so that
R1t = R1t-1 + ut.
a. Show that Rkt = R1t + et.
b. Show that Rkt and R1t are cointegrated. What is the cointegrating coefficient?
c. Now suppose that ∆R1t = 0.5∆R1t – 1 + ut. How does your answer to (b) change?
d. Now suppose that R1t = 0.5R1t – 1 + ut. How does your answer to (b) change?
16.3 Suppose that ut follows the ARCH process, s2t = 1.0 + 0.5 u2t – 1.
a. Let E(u2t ) = var(ut) be the unconditional variance of ut. Show
E1u 2 = E3E1u ∙u 24.) 2t 2t t-1
that var(ut) = 2. (Hint: Use the law of iterated expectations,
b. Suppose that the distribution of ut conditional on lagged values of ut is N(0, s ). If u = 0.2, what is Pr1-3 … u … 32? If u = 2.0,
2tt-1 tt-1 what is Pr1-3 … u … 32?
t
16.4 Suppose that Yt follows the AR(p) model Yt = b0 + b1Yt-1 + g+
bpYt-p + ut, where E(ut∙Yt-1, Yt-2, c) = 0 Let Yt+h∙t = E(Yt+h∙Yt,
Yt-1, c). Show that Yt+h∙t = b0 + b1Yt-1+h∙t + g + bpYt-p+h∙t for h > p.
22 16.5 Verify Equation (16.20). [Hint: Use g Y = g 1Y + ∆Y 2 to
T
for g Y ∆Y.4
t=1tt=1t-1t T2T2TT2
t=1 t-1 t
16.6 A regression of Yt onto current, past, and future values of Xt yields
a. b.
Yt = 3.0 + 1.7Xt+1 + 0.8Xt – 0.2Xt-1 + ut.
Rearrange the regression so that it has the form shown in Equation
(16.25). What are the values of u, d – 1, d0, and d1?
i. Suppose that Xt is I(1) and ut is I(1). Are Y and X cointegrated?
ii. Suppose that Xt is I(0) and ut is I(1). Are Y and X cointegrated?
iii. Suppose that Xt is I(1) and ut is I(0). Are Y and X cointegrated?
16.7 Suppose that ∆Yt = ut, where ut is i.i.d. N(0, 1), and consider the regression Yt = bXt + error, where Xt = ∆Yt + 1 and error is the regression error.
TT
∆Y + g ∆Y and solve t=1t t=1t-1 t=1t-1 t t=1 t
show that g Y = g Y + 2g Y
674 ChApTeR 16 Additional Topics in Time Series Regression
Show that bn ¡d 121×21 – 12. [Hint: Analyze the numerator of bn using analysis like that in Equation (16.21). Analyze the denominator using the law of large numbers.]
16.8 Consider the following two-variable VAR model with one lag and no intercept:
a.
b. 16.9 a.
b. c. d. e.
Yt = b11Yt-1 + g11Xt-1 + u1t Xt = b21Yt-1 + g21Xt-1 + u2t.
Show that the iterated two-period-ahead forecast for Y can be written as Yt∙t-2 = d1Yt-2 + d2Xt-2 and derive values for d1 and d2 in terms of the coefficients in the VAR.
In light of your answer to (a), do iterated multiperiod forecasts differ from direct multiperiod forecasts? Explain.
Suppose that E(u∙u ,u ,c) = 0, that var1u∙u ,u ,c2 t t-1 t-2 t t-1 t-2
follows the ARCH(1) model s2t = a0 + a1u2t – 1, and that the process for u is stationary. Show that var1u 2 = a >11 – a 2. (Hint: Use the
t t01 law of iterated expectations E(u2t ) = E[E(u2t ∙ ut – 1)].)
Extend the result in (a) to the ARCH(p) model.
Show that g pi = 1 ai 6 1 for a stationary ARCH(p) model. Extend the result in (a) to the GARCH(1,1) model.
Show that a1 + f1 6 1 for a stationary GARCH(1,1) model.
16.10 Consider the cointegrated model Yt = uXt + v1t and Xt = Xt – 1 + v2t,
with E1v v 2 = 0 for all t and j. Derive the vector error correction model 1t 2j
[Equations (16.22) and (16.23)] for X and Y. Empirical Exercises
(Only two empirical exercises for this chapter are given in the text, but you can find more on the text website, http://www.pearsonhighered.com/stock_watson/.)
E16.1 This exercise is an extension of Empirical Exercise 14.1. On the text web- site, http://www.pearsonhighered.com/stock_watson, you will find the data file USMacro_Quarterly, which contains quarterly data on several macroeconomic series for the United States; the data are described in the file USMacro_Description. Compute inflation, Infl, using the price index
where v1t and v2t are mean zero serially uncorrelated random variables
for personal consumption expenditures. For all regressions use the sample period 1963:Q1–2012:Q4 (where data before 1963 may be used as initial values for lags in regressions).
a.
Using the data on inflation through 2012:Q4 and an estimated AR(2) model:
i. Forecast ΔInfl2013:Q1, the change in inflation from 2012:Q4 to 2013:Q1.
ii. Forecast ΔInfl2013:Q2, the change in inflation from 2013:Q1 to 2013:Q2. (Use an iterated forecast.)
iii. Forecast Infl2013:Q2 − Infl2012:Q4, the change in inflation from 2012:Q4 to 2013:Q2.
iv. Forecast Infl2013:Q2, the level of inflation in 2013:Q2.
Repeat (a) using the direct forecasting method.
In Exercise 14.1 you carried out an ADF test for a unit root in the autoregression for Infl. Now carry out the unit root test using the DF-GLS test. Are the conclusions based on the DF-GLS test the same as you reached using the ADF test? Explain.
b. c.
E16.2 On the text website, http://www.pearsonhighered.com/stock_watson, you will find the data file USMacro_Quarterly, which contains quarterly data on real GDP, measured in $1996. Compute GDPGRt = 400 × [ln(GDPt) − ln(GDPt−1)], the growth rate of GDP.
a. Using data on GDPGRt from 1960:1 to 2012:4, estimate an AR(2) model with GARCH(1,1) errors.
b. Plot the residuals from the AR(2) model along with {snt bands as in Figure 16.4.
c. Some macroeconomists have claimed that there was a sharp drop in the variability of the growth rate of GDP around 1983, which they call the “Great Moderation.” Is this Great Moderation evident in the plot that you formed in (b)?
Empirical Exercises 675
CHAPTER
17
The Theory of Linear Regression with One Regressor
W hy should an applied econometrician bother learning any econometric the- ory? There are several reasons. Learning econometric theory turns your sta-
tistical software from a “black box” into a flexible tool kit from which you are able to select the right tool for the job at hand. Understanding econometric theory helps you appreciate why these tools work and what assumptions are required for each tool to work properly. Perhaps most importantly, knowing econometric theory helps you recognize when a tool will not work well in an application and when you should look for a different econometric approach.
This chapter provides an introduction to the econometric theory of linear regression with a single regressor. This introduction is intended to supplement— not replace—the material in Chapters 4 and 5, which should be read first.
This chapter extends Chapters 4 and 5 in two ways.
First, it provides a mathematical treatment of the sampling distribution of the OLS estimator and t-statistic, both in large samples under the three least squares assumptions of Key Concept 4.3 and in finite samples under the two additional assumptions of homoskedasticity and normal errors. These five extended least squares assumptions are laid out in Section 17.1. Sections 17.2 and 17.3, augmented by Appendix 17.2, mathematically develop the large-sample normal distributions of the OLS estimator and t-statistic under the first three assumptions (the least squares assumptions of Key Concept 4.3). Section 17.4 derives the exact distributions of the OLS estimator and t-statistic under the two additional assumptions of homoskedasticity and normally distributed errors.
Second, this chapter extends Chapters 4 and 5 by providing an alternative method for handling heteroskedasticity. The approach of Chapters 4 and 5 is to use heteroskedasticity-robust standard errors to ensure that statistical inference is valid even if the errors are heteroskedastic. This method comes with a cost, however: If the errors are heteroskedastic, then in theory a more efficient estima- tor than OLS is available. This estimator, called weighted least squares, is pre- sented in Section 17.5. Weighted least squares requires a great deal of prior knowledge about the precise nature of the heteroskedasticity—that is, about the conditional variance of u given X. When such knowledge is available, weighted least squares improves upon OLS. In most applications, however, such knowledge
676
17.1 The Extended Least Squares Assumptions and the OLS Estimator 677 is unavailable; in those cases, using OLS with heteroskedasticity-robust standard
errors is the preferred method.
17.1
The Extended Least Squares Assumptions and the OLS Estimator
This section introduces a set of assumptions that extend and strengthen the three least squares assumptions of Chapter 4. These stronger assumptions are used in subsequent sections to derive stronger theoretical results about the OLS estimator than are possible under the weaker (but more realistic) assumptions of Chapter 4.
The Extended Least Squares Assumptions
Extended least squares Assumptions #1, #2, and #3. The first three extended least squares assumptions are the three assumptions given in Key Concept 4.3: that the conditional mean of ui, given Xi, is zero; that (Xi,Yi), i = 1, c, n, are i.i.d. draws from their joint distribution; and that Xi and ui have four moments.
Under these three assumptions, the OLS estimator is unbiased, is consistent, and has an asymptotically normal sampling distribution. If these three assump- tions hold, then the methods for inference introduced in Chapter 4—hypothesis testing using the t-statistic and construction of 95% confidence intervals as { 1.96 standard errors—are justified when the sample size is large. To develop a theory of efficient estimation using OLS or to characterize the exact sampling distribution of the OLS estimator, however, requires stronger assumptions.
Extended least squares Assumption #4. The fourth extended least squares assumption is that ui is homoskedastic; that is, var(ui 0 Xi) = s2u, where s2u is a constant. As seen in Section 5.5, if this additional assumption holds, then the OLS estimator is efficient among all linear estimators that are unbiased, conditional on X1, c, Xn.
Extended least squares Assumption #5. The fifth extended least squares assump- tion is that the conditional distribution of ui, given Xi, is normal.
Under least squares Assumptions #1 and #2 and the extended least squares Assumptions #4 and #5, ui is i.i.d. N(0, s2u), and ui and Xi are independently dis- tributed. To see this, note that the fifth extended least squares assumption states that the conditional distribution of ui 0 Xi is N(0, var(ui 0 Xi)), where the distribution has mean zero by the first extended least squares assumption. By the fourth least
678 CHAPTER 17 The Theory of Linear Regression with One Regressor
KEY CONCEPT
17.1
The Extended Least Squares Assumptions for Regression with a Single Regressor
The linear regression model with a single regressor is
Yi = b0 + b1Xi + ui,i = 1,c,n.
The extended least squares assumptions are 1. E(ui 0 Xi) = 0 (conditional mean zero);
(17.1)
2. (Xi, Yi), i = 1, c, n, are independent and identically distributed (i.i.d.) draws from their joint distribution;
3. (Xi, ui) have nonzero finite fourth moments;
4. var(uiXi) = s2u (homoskedasticity); and
5. The conditional distribution of ui given Xi is normal (normal errors).
squares assumption, however, var(ui 0 Xi) = s2u, so the conditional distribution of ui 0 Xi is N(0, s2u). Because this conditional distribution does not depend on Xi, ui and Xi are independently distributed. By the second least squares assumption, ui is distributed independently of uj for all j ≠ i. It follows that, under the extended least squares Assumptions #1, #2, #4, and #5, ui and Xi are independently distrib- uted and ui is i.i.d. N(0, s2u).
It is shown in Section 17.4 that, if all five extended least squares assumptions hold, the OLS estimator has an exact normal sampling distribution and the homoskedasticity- only t-statistic has an exact Student t distribution.
The fourth and fifth extended least squares assumptions are much more restrictive than the first three. Although it might be reasonable to assume that the first three assumptions hold in an application, the final two assumptions are less realistic. Even though these final two assumptions might not hold in practice, they are of theoretical interest because if one or both of them hold, then the OLS esti- mator has additional properties beyond those discussed in Chapters 4 and 5. Thus we can enhance our understanding of the OLS estimator and the theory of estima- tion in the linear regression model by exploring estimation under these stronger assumptions.
The five extended least squares assumptions for the single-regressor model are summarized in Key Concept 17.1.
17.2 Fundamentals of Asymptotic Distribution Theory
679
The OLS Estimator
For easy reference, we restate the OLS estimators of b0 and b1 here:
n
a(Xi – X)(Yi – Y)
bn = i=1 1n
a(Xi – X)2 i=1
bn0 = Y – bn1X. Equations (17.2) and (17.3) are derived in Appendix 4.2.
17.2
(17.2)
(17.3)
Fundamentals of Asymptotic Distribution Theory
Asymptotic distribution theory is the theory of the distribution of statistics—esti- mators, test statistics, and confidence intervals—when the sample size is large. Formally, this theory involves characterizing the behavior of the sampling distribu- tion of a statistic along a sequence of ever-larger samples. The theory is asymptotic in the sense that it characterizes the behavior of the statistic in the limit as n S ∞ .
Even though sample sizes are, of course, never infinite, asymptotic distribu- tion theory plays a central role in econometrics and statistics for two reasons. First, if the number of observations used in an empirical application is large, then the asymptotic limit can provide a high-quality approximation to the finite sample distribution. Second, asymptotic sampling distributions typically are much sim- pler, and thus easier to use in practice, than exact finite-sample distributions. Taken together, these two reasons mean that reliable and straightforward meth- ods for statistical inference—tests using t-statistics and 95% confidence intervals calculated as { 1.96 standard errors—can be based on approximate sampling dis- tributions derived from asymptotic theory.
The two cornerstones of asymptotic distribution theory are the law of large num- bers and the central limit theorem, both introduced in Section 2.6. We begin this sec- tion by continuing the discussion of the law of large numbers and the central limit theorem, including a proof of the law of large numbers. We then introduce two more tools, Slutsky’s theorem and the continuous mapping theorem, that extend the useful- ness of the law of large numbers and the central limit theorem. As an illustration, these tools are then used to prove that the distribution of the t-statistic based on Y testing the hypothesis E(Y) = m0 has a standard normal distribution under the null hypothesis.
680 CHAPTER 17
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Convergence in Probability and
the Law of Large Numbers
The concepts of convergence in probability and the law of large numbers were intro- duced in Section 2.6. Here we provide a precise mathematical definition of conver- gence in probability, followed by a statement and proof of the law of large numbers.
Consistencyandconvergenceinprobability. LetS1,S2,c,Sn,cbeasequence of random variables. For example, Sn could be the sample average Y of a sample of n observations of the random variable Y. The sequence of random variables {Sn} is said to converge in probability to a limit, m (that is, Sn ¡p m), if the prob- ability that Sn is within {d of m tends to 1 as n S ∞, as long as the constant d is positive. That is,
S ¡p m if and only if Pr( 0 S – m 0 Ú d) ¡ 0 (17.4) nn
asnS ∞ foreveryd 7 0.IfSn ¡p mthenSn issaidtobeaconsistentestimator of m.
The law of large numbers. The law of large numbers says that, under certain con- ditions on Y1, c, Yn, the sample average Y converges in probability to the pop- ulation mean. Probability theorists have developed many versions of the law of large numbers, corresponding to various conditions on Y1, c, Yn. The version of the law of large numbers used in this book is that Y1, c, Yn are i.i.d. draws from a distribution with finite variance. This law of large numbers (also stated in Key Concept 2.6) is
if Y1, c, Yn are i.i.d., E(Yi) = mY, and var(Yi) 6 ∞, then Y ¡p mY. (17.5)
The idea of the law of large numbers can be seen in Figure 2.8: As the sample size increases, the sampling distribution of Y concentrates around the population mean, mY. One feature of the sampling distribution is that the variance of Y decreases as the sample size increases; another feature is that the probability that Y falls outside {d of mY vanishes as n increases. These two features of the sam- pling distribution are in fact linked, and the proof of the law of large numbers exploits this link.
Proof of the law of large numbers. The link between the variance of Y and the probability that Y is within { d of mY is provided by Chebychev’s inequality, which
17.2 Fundamentals of Asymptotic Distribution Theory 681 is stated and proven in Appendix 17.2 [see Equation (17.42)]. Written in terms of
Y, Chebychev’s inequality is
Pr(0Y – m 0 Ú d) … var(Y), (17.6)
d2
Y
for any positive constant d. Because Y1, c, Yn are i.i.d. with variance s2Y, var(Y) = s2Y>n; thus, for any d 7 0, var(Y)>d2 = s2Y>(d2n) ¡ 0. It follows fromEquation(17.6)thatPr(0Y – mY0 Ú d) ¡ 0foreveryd 7 0,provingthe law of large numbers.
Some examples. Consistency is a fundamental concept in asymptotic distribu-
tion theory, so we present some examples of consistent and inconsistent estima-
tors of the population mean, mY. Suppose that Yi, i = 1, c, n are i.i.d. with
variance s2Y that is positive and finite. Consider the following three estimators of mY:
(1)m =Y;(2)m =(1 – an)-1gn ai-1Y,where06a61;and(3)m =Y+ a1b1-ai=1i c
1>n. Are these estimators consistent?
The first estimator, ma, is just the first observation, so E(ma) = E(Y1) = mY
and ma is unbiased. However, ma is not consistent: Pr( 0 ma – mY 0 Ú d) = Pr(0Y1 – mY0 Ú d),whichmustbepositiveforsufficientlysmalld(becauses2Y 7 0), so Pr(0ma – mY0 Ú d) does not tend to zero as nS ∞, so ma is not consistent. This inconsistency should not be surprising: Because ma uses the information in only one observation, its distribution cannot concentrate around mY as the sample size increases.
The second estimator, mb, is unbiased but is not consistent. It is unbiased because
n -1 n n -1 n
E(mb)=Eca1-ab aai-1Yid=a1-ab aai-1mY=mY, 1-a i=1 1-a i=1
n∞n since aai-1 = a1 – anbaai = 1 – a .
i=1 i=0 1-a
n-2n 2n2n
The variance of mb is
var(m)=a1-ab a2(i-1)s2 =s2(1-a )(1-a) =s2(1+a)(1-a),
b 1 – a a Y Y (1 – a2)(1 – an)2 Y (1 – an)(1 + a) i-1
which has the limit var(mb) S s2Y(1 – a)>(1 + a) as n S ∞. Thus the variance of this estimator does not tend to zero, the distribution does not concentrate around mY, and the estimator, although unbiased, is not consistent. This is perhaps
682 CHAPTER 17
The Theory of Linear Regression with One Regressor
surprising, because all the observations enter this estimator. But most of the obser- vations receive very small weight (the weight of the ith observation is proportional to ai – 1, a very small number when i is large), and for this reason there is an insuf- ficient amount of cancellation of sampling errors for the estimator to be consistent.
The third estimator, mc, is biased but consistent. Its bias is 1/n: E(mc) = E(Y + 1/n) = mY + 1>n, so the bias tends to zero as the sample size increases. Toseewhymcisconsistent:Pr(0mc – mY0 Ú d) = Pr(0Y + 1>n – mY0 Ú d).Now, from Equation (17.43) in Appendix 17.2, a generalization of Chebychev’s inequal- ity implies that for any random variable W,Pr(0W0 Ú d) … E(W2)>d2 for any positive constant d. Thus. Pr(0Y + 1>n – mY0 Ú d) … E[(Y + 1>n – mY)2]>d2. ButE3(Y+1/n-mY)24=var(Y)+1/n2 =s2/n+1>n2 ¡0asngrows large. It follows that Pr(0Y + 1>n – mY0 Ú d)¡ 0, and mc is consistent. This example illustrates the general point that an estimator can be biased in finite sam- ples but, if that bias vanishes as the sample size gets large, the estimator can still be consistent (Exercise 17.10).
The Central Limit Theorem and
Convergence in Distribution
If the distributions of a sequence of random variables converge to a limit as n S ∞ , then the sequence of random variables is said to converge in distribution. The central limit theorem says that, under general conditions, the standardized sample average converges in distribution to a normal random variable.
Convergence in distribution. Let F1, F2, c, Fn, cbe a sequence of cumula- tive distribution functions corresponding to a sequence of random variables, S1, S2, c, Sn, c. For example, Sn might be the standardized sample average, (Y – mY)>sY. Then the sequence of random variables Sn is said to converge in distribution to S (denoted Sn ¡d S) if the distribution functions {Fn} converge to F, the distribution of S. That is,
Sn ¡d S if and only if lim Fn(t) = F(t), (17.7) nS ∞
where the limit holds at all points t at which the limiting distribution F is continu- ous. The distribution F is called the asymptotic distribution of Sn.
It is useful to contrast the concepts of convergence in probability (¡p ) and convergence in distribution ( ¡d ). If Sn ¡p m, then Sn becomes close to m with high probability as n increases. In contrast, if Sn ¡d S, then the distribution of Sn becomes close to the distribution of S as n increases.
17.2 Fundamentals of Asymptotic Distribution Theory 683
The central limit theorem. We now restate the central limit theorem using the
concept of convergence in distribution. The central limit theorem in Key Concept
2.7 states that if Y1, c,Yn are i.i.d. and 0 6 s2Y 6 ∞, then the asymptotic distri-
bution of (Y – mY)>sY is N(0, 1). Because sY = sY > 2n, (Y – mY)>sY =
2n(Y – m ) >s . Thus the central limit theorem can be restated as 2n(Y – m ) ¡d YYY
sYZ, where Z is a standard normal random variable. This means that the distribution of2n(Y – mY)convergestoN(0,s2Y)asn ¡ ∞.Conventionalshorthandforthis limit is
2n(Y – mY) ¡d N(0, s2Y). (17.8) That is, if Y1, c, Yn are i.i.d. and 0 6 s2Y 6 ∞ , then the distribution of
2n(Y – mY) converges to a normal distribution with mean zero and variance s2Y.
Extensions to time series data. The law of large numbers and central limit theorem stated in Section 2.6 apply to i.i.d. observations. As discussed in Chapter 14, the i.i.d. assumption is inappropriate for time series data, and these theorems need to be extended before they can be applied to time series observations. Those extensions are technical in nature, in the sense that the conclusion is the same—versions of the law of large numbers and the central limit theorem apply to time series data—but the conditions under which they apply are different. This is discussed briefly in Section 16.4, but a mathemati- cal treatment of asymptotic distribution theory for time series variables is beyond the scope of this book and interested readers are referred to Hayashi (2000, Chapter 2).
Slutsky’s Theorem and the Continuous Mapping Theorem
Slutsky’s theorem combines consistency and convergence in distribution. Suppose that an ¡p a, where a is a constant, and Sn ¡d S. Then
a +S ¡d a+S,aS ¡d aS,and,ifa≠0,S>a ¡d S>a. (17.9) nnnnnn
These three results are together called Slutsky’s theorem.
The continuous mapping theorem concerns the asymptotic properties of a con-
tinuous function, g, of a sequence of random variables, Sn. The theorem has two parts. The first is that if Sn converges in probability to the constant a, then g(Sn)
684 CHAPTER 17
The Theory of Linear Regression with One Regressor
converges in probability to g(a); the second is that if Sn converges in distribution to S, then g(Sn) converges in distribution to g(S). That is, if g is a continuous func- tion, then
(i) if Sn ¡p a, then g(Sn) ¡p g(a), and
(ii) if Sn ¡d S, then g(Sn) ¡d g(S). (17.10)
Asanexampleof(i),ifs2Y ¡p s2Y,then2s2Y = sY ¡p sY.Asanexampleof(ii),
suppose that Sn ¡d Z, where Z is a standard normal random variable, and let
g(Sn) = S2n. Because g is continuous, the continuous mapping theorem applies
and g(Sn) ¡d g(Z); that is, S2n ¡d Z2. In other words, the distribution of S2n
converges to the distribution of a squared standard normal random variable,
which in turn has a x2 distribution; that is, S2 ¡d x2. 1n1
Application to the t-Statistic Based
on the Sample Mean
We now use the central limit theorem, the law of large numbers, and Slutsky’s theorem to prove that, under the null hypothesis, the t-statistic based on Y has a standard normal distribution when Y1, c, Yn are i.i.d. and 0 6 E(Y4i ) 6 ∞ .
The t-statistic for testing the null hypothesis that E(Yi) = m0 based on the sample average Y is given in Equations (3.8) and (3.11), and can be written
t=Y-m0 =2n(Y-m0),sY, (17.11) sY>2n sY sY
where the second equality uses the trick of dividing both the numerator and the denominator by sY.
Because Y1, c, Yn have two moments (which is implied by their having four
moments; see Exercise 17.5), and because Y1, c, Yn are i.i.d., the first term after
the final equality in Equation (17.11) obeys the central limit theorem: Under the
null hypothesis, 2n(Y – m0)>sY ¡d N(0, 1). In addition, s2Y ¡p s2Y (as
proven in Appendix 3.3), so s2Y > s2Y ¡p 1 and the ratio in the second term in
Equation (17.11) tends to 1 (Exercise 17.4). Thus the expression after the final
equality in Equation (17.11) has the form of the final expression in Equation
(17.9), where [in the notation of Equation (17.9)] S = 2n(Y – m )>s ¡d n0Y
N(0, 1) and a = s >s ¡p 1. It follows by applying Slutsky’s theorem that nYY
t¡d N(0,1).
17.3
Asymptotic Distribution of the OLS Estimator and t-Statistic
Recall from Chapter 4 that, under the assumptions of Key Concept 4.3 (the first three assumptions of Key Concept 17.1), the OLS estimator bn1 is consistent and 2n(bn1 -b1)hasanasymptoticnormaldistribution.Moreover,thet-statistictest- ing the null hypothesis b1 = b1,0 has an asymptotic standard normal distribution under the null hypothesis. This section summarizes these results and provides
additional details of their proofs.
Consistency and Asymptotic Normality
of the OLS Estimators
The large-sample distribution of bn1, originally stated in Key Concept 4.4, is
2n(bn – b ) ¡d Na0, var(vi) b, (17.12) 1 1 3var(Xi)42
where vi = (Xi – mX)ui. The proof of this result was sketched in Appendix 4.3, but that proof omitted some details and involved an approximation that was not formally shown. The missing steps in that proof are left as Exercise 17.3.
An implication of Equation (17.12) is that bn1 is consistent (Exercise 17.4).
Consistency of Heteroskedasticity-Robust
Standard Errors
Under the first three least squares assumptions, the heteroskedasticity-robust standard error for bn1 forms the basis for valid statistical inferences. Specifically,
where s2 n
2 = var(v )>{n[var(X )] }
and sn2 is square of the heteroskedasticity- n
17.3 Asymptotic Distribution of the OLS Estimator and t-Statistic 685
b1 i i b1
robust standard error defined in Equation (5.4); that is,
sn 2
bn1 ¡p 1,
s2 bn1
(17.13)
1n
1 n – 2 a ( X i – X ) 2 un 2i
sn 2 = i=1 . bnnn 2
(17.14)
1 c 1 (X – X)2 d nai
i=1
686 CHAPTER 17
The Theory of Linear Regression with One Regressor
To show the result in Equation (17.13), first use the definitions of s2 and sn 2 bn1 bn1
to rewrite the ratio in Equation (17.13) as
1n1n2 sn2 a(Xi – X)2un2i a(Xi – X)2
bn1 =J n RDni=1 T,Dni=1 T. (17.15) s2 n – 2 var(v) var(X)
bn1 i i
We need to show that each of the three terms in brackets on the right-hand side of Equation (17.15) converge in probability to 1. Clearly the first term converges to 1, and by the consistency of the sample variance (Appendix 3.3) the final term converges in probability to 1. Thus all that remains is to show that the second term converges in probability to 1, that is, that n1 g ni = 1(Xi – X)2un 2i ¡p var(vi).
The proof that n1 gni=1(Xi – X)2un2i ¡p var(vi) proceeds in two steps. The first shows that n1 gni=1v2i ¡p var(vi); the second shows that n1 gni=1(Xi – X)2un2i – n1gni=1v2i¡p 0.
For the moment, suppose that Xi and ui have eight moments [that is, E(X8i) 6 ∞ and E(u8i) 6 ∞4, which is a stronger assumption than the four moments required by the third least squares assumption. To show the first step, we must show that n1 gni=1v2i obeys the law of large numbers in Equation (17.5). To do so,v2i mustbei.i.d.(whichitisbythesecondleastsquaresassumption)andvar(v2i) must be finite. To show that var(v2i ) 6 ∞ , apply the Cauchy–Schwarz inequality (Appendix17.2):var(v2i ) … E(v4i )= E3(Xi – mX)4u4i 4 … 5E3(Xi – mX)84E(u8i )61>2. Thus, if Xi and ui have eight moments, then v2i has a finite variance and thus satis- fies the law of large numbers in Equation (17.5).
The second step is to prove that n1 gni = 1(Xi – X)2un2i – n1 gni = 1v2i ¡p 0. Because vi = (Xi – mX)ui, this second step is the same as showing that
n1 gni = 13(Xi – X)2un2i – (Xi – mX)2u2i 4 ¡p 0. (17.16)
Showing this result entails setting uni = ui – (bn0 – b0) – (bn1 – b1)Xi, expanding the term in Equation (17.16) in brackets, repeatedly applying the Cauchy–Schwarz inequality, and using the consistency of bn0 and bn1. The details of the algebra are left as Exercise 17.9.
The preceding argument supposes that Xi and ui have eight moments. This is not necessary, however, and the result n1gni=1(Xi – X)2un2i ¡p var(vi) can be proven under the weaker assumption that Xi and ui have four moments, as stated in the third least squares assumption. That proof, however, is beyond the scope of this textbook; see Hayashi (2000, Section 2.5) for details.
17.4 Exact Sampling Distributions When the Errors Are Normally Distributed 687
Asymptotic Normality of the
Heteroskedasticity-Robust t-Statistic
We now show that, under the null hypothesis, the heteroskedasticity-robust OLS t-statistic testing the hypothesis b1 = b1,0 has an asymptotic standard normal dis- tribution if least squares Assumptions #1, #2, and #3 hold.
The t-statistic constructed using the heteroskedasticity-robust standard error SE(bn1) = snbn1 [defined in Equation (17.14)] is
bn – b
2 n ( bn – b ) sn 2
1,0=
It follows from Equation (17.12) and the definition of s2 that first term after the
t= 1
1
(17.17)
bn 1
second equality in Equation (17.17) converges in distribution to a standard normal
17.4
random variable. In addition, because the heteroskedasticity-robust standard error is consistent [Equation (17.13)], 2sn 2 >s2 ¡p 1 (Exercise 17.4). It follows
1
1,0 , bn1. B s 2
sn bn
2
2nsn b1
bn1
d bn 1 from Slutsky’s theorem that t ¡ N(0, 1).
bn 1
Exact Sampling Distributions When the Errors Are Normally Distributed
In small samples, the distribution of the OLS estimator and t-statistic depends on the distribution of the regression error and typically is complicated. As discussed in Section 5.6, however, if the regression errors are homoskedastic and normally distributed, then these distributions are simple. Specifically, if all five extended least squares assumptions in Key Concept 17.1 hold, then the OLS estimator has a normal sampling distribution, conditional on X1, c, Xn. Moreover, the t-statistic has a Student t distribution. We present these results here for bn1.
Distribution of bn1 with Normal Errors
If the errors are i.i.d. normally distributed and independent of the regressors, then
the distribution of bn , conditional on X , c, X , is N(b , s2
), where
1
1 n 1bn10X
s2
bn10X = n
s2u .
(17.18)
a(Xi – X)2 i=1
688 CHAPTER 17
The Theory of Linear Regression with One Regressor
The derivation of the normal distribution N(b , s2 ), conditional on 1 bn10X
X1, c, Xn, entails (i) establishing that the distribution is normal; (ii) showing that E(bn1 X1 , c, Xn) = b1; and (iii) verifying Equation (17.18).
n
1n
na(Xi – X)ui
To show (i), note that, conditional on X1, c, Xn, b1 – b1 is a weighted aver- age of u1, c, un:
bn=b+ i=1 111n
. (17.19)
na(Xi – X)2 i=1
This equation was derived in Appendix 4.3 [Equation (4.30) and is restated here for convenience]. By extended least squares Assumptions #1, #2, #4, and #5, ui is i.i.d. N(0, s2u), and ui and Xi are independently distributed. Because weighted averages of normally distributed variables are themselves normally distributed, it follows that bn1 is normally distributed, conditional on X1, c, Xn.
To show (ii), take conditional expectations of both sides of Equation (17.19): E[(bn1 – b1)0X1,c,Xn)] = E[gni=1(Xi – X)ui>gni=1(Xi – X)20X1,c,Xn] = [gni=1(Xi-X)E(ui0X1,c,Xn)]>[gni=1(Xi -X)2]=0,wherethefinalequality follows because E(ui 0 X1, X2, c, Xn) = E(ui 0 Xi) = 0. Thus bn1 is conditionally unbiased; that is,
E(bn1 0 X1, c, Xn) = b1. (17.20) To show (iii), use that the errors are independently distributed, conditional on
X1 , c, Xn, to calculate the conditional variance of bn1 using Equation (17.19): n
a(Xi – X)ui
v a r ( bn X , c , X ) = v a r ≥ i = 1 X , c , X ¥
11nn1n a(Xi – X)2
i=1 n
a(Xi -X)2var(uiX1,c,Xn)
n
a ( X i – X ) 2 s 2u = i=1 .
= i=1
ca(Xi -X)2d
n2 i=1
n2
(17.21)
ca(Xi -X)2d i=1
17.4 Exact Sampling Distributions When the Errors Are Normally Distributed 689 Canceling the term in the numerator in the final expression in Equation
(17.21) yields the formula for the conditional variance in Equation (17.18).
Distribution of the Homoskedasticity-Only t-Statistic
The homoskedasticity-only t-statistic testing the null hypothesis b1 = b1,0 is
t = bn1 – b1,0, (17.22) SE(bn1)
where SE(bn1) is computed using the homoskedasticity-only standard error of bn1. Substituting the formula for SE(bn1) [Equation (5.29) of Appendix 5.1] into Equa- tion (17.22) and rearranging yields
bn – b bn – b s2 t= 1 1,0 = 1 1,0 , un
n n Bs2
2 22uai2 s un > a ( X i – X ) s > ( X – X )
u
an N(b , s2 ) distribution conditional on X , c, X , so the distribution of the 1,0 bn10X 1 n
numerator in the final expression in Equation (17.23) is N(0, 1). It is shown in Section 18.4 that W has a chi-squared distribution with n – 2 degrees of freedom and moreover that W is distributed independently of the standardized OLS esti- mator in the numerator of Equation (17.23). It follows from the definition of the Student t distribution (Appendix 17.1) that, under the five extended least squares assumptions, the homoskedasticity-only t-statistic has a Student t distribution with n – 2 degrees of freedom.
B i=1 B i=1 = (bn1 – b1,0)>sbn1X,
(17.23) where s2 = 1 gn un2 and W = gn un2>s2. Under the null hypothesis, bn has
2W>(n – 2)
un n – 2 i=1 i i=1 i u 1
Where does the degrees of freedom adjustment fit in? The degrees of freedom adjustmentins2un ensuresthats2un isanunbiasedestimatorofs2uandthatthet-statistic has a Student t distribution when the errors are normally distributed.
Because W = gni=1 un2i >s2u is a chi-squared random variable with n – 2 degrees of
freedom,itsmeanisE(W) = n – 2.ThusE3W>(n – 2)4 = (n – 2)>(n – 2) = 1.
Rearranging the definition of W, we have that E( 1 gn un2) = s2. Thus the n – 2 i=1 i u
degreesoffreedomcorrectionmakess2un anunbiasedestimatorofs2u.Also,bydivid- ing by n – 2 rather than n, the term in the denominator of the final expression of
690
CHAPTER 17
The Theory of Linear Regression with One Regressor
17.5
Weighted Least Squares
Under the first four extended least squares assumptions, the OLS estimator is efficient among the class of linear (in Y1, c, Yn), conditionally (on X1, c, Xn) unbiased estimators; that is, the OLS estimator is BLUE. This result is the Gauss– Markov theorem, which was discussed in Section 5.5 and proven in Appendix 5.2. The Gauss–Markov theorem provides a theoretical justification for using the OLS estimator. A major limitation of the Gauss–Markov theorem is that it requires homoskedastic errors. If, as is often encountered in practice, the errors are heteroskedastic, the Gauss–Markov theorem does not hold and the OLS estimator is not BLUE.
This section presents a modification of the OLS estimator, called weighted least squares (WLS), which is more efficient than OLS when the errors are heteroskedastic.
WLS requires knowing quite a bit about the conditional variance function, var(ui Xi). We consider two cases. In the first case, var(ui Xi) is known up to a factor of proportionality, and WLS is BLUE. In the second case, the functional form of var(ui Xi) is known, but this functional form has some unknown param- eters that can be estimated. Under some additional conditions, the asymptotic distribution of WLS in the second case is the same as if the parameters of the conditional variance function were in fact known, and in this sense the WLS esti- mator is asymptotically BLUE. The section concludes with a discussion of the practical advantages and disadvantages of handling heteroskedasticity using WLS or, alternatively, heteroskedasticity-robust standard errors.
WLS with Known Heteroskedasticity
Suppose that the conditional variance var(ui 0 Xi) is known up to a factor of pro- portionality; that is,
var(ui0Xi) = lh(Xi), (17.24)
where l is a constant and h is a known function. In this case, the WLS estimator is the estimator obtained by first dividing the dependent variable and regressor
Equation (17.23) matches the definition of a random variable with a Student t distribution given in Appendix 17.1. That is, by using the degrees of freedom adjustment to calculate the standard error, the t-statistic has the Student t distribu- tion when the errors are normally distributed.
by the square root of h and then regressing this modified dependent variable on the modified regressor using OLS. Specifically, divide both sides of the single- variable regressor model by 2h(Xi) to obtain
Y∼ =bX∼ +bX∼ +∼u, (17.25) i 00i 11i i
whereY∼ = Y>2h(X),X∼ = 1>2h(X),X∼ = X>2h(X),and∼u = u>2h(X). i i i 0i i 1i i i i i i
The WLS estimator is the OLS estimator of b1 in Equation (17.25); that is, it is the estimator obtained by the OLS regression of Y∼ on X∼ and X∼ ,
∼ i0i1i where the coefficient on X0i takes the place of the intercept in the unweighted
regression.
Under the first three least squares assumptions in Key Concept 17.1 plus the
known heteroskedasticity assumption in Equation (17.24), WLS is BLUE. The reason that the WLS estimator is BLUE is that weighting the variables has made the error term ∼ui in the weighted regression homoskedastic. That is,
var(∼u 0X) = varc ui 0X d = var(uiXi) = lh(Xi) = l, (17.26) i i 1h(Xi) i h(Xi) h(Xi)
so the conditional variance of ∼ui, var(∼ui Xi), is constant. Thus the first four least
squares assumptions apply to Equation (17.25). Strictly speaking, the Gauss–Markov
theorem was proven in Appendix 5.2 for Equation (17.1), which includes the
intercept b0, so it does not apply to Equation (17.25), in which the intercept is
replaced by b X∼ . However, the extension of the Gauss–Markov theorem for 0 0i
multiple regression (Section 18.5) does apply to estimation of b1 in the weighted population regression, Equation (17.25). Accordingly, the OLS estimator of b1 in Equation (17.25)—that is, the WLS estimators of b1:is BLUE.
In practice, the function h typically is unknown, so neither the weighted vari- ables in Equation (17.25) nor the WLS estimator can be computed. For this rea- son, the WLS estimator described here is sometimes called the infeasible WLS estimator. To implement WLS in practice, the function h must be estimated, the topic to which we now turn.
WLS with Heteroskedasticity
of Known Functional Form
If the heteroskedasticity has a known functional form, then the heteroskedasticity function h can be estimated and the WLS estimator can be calculated using this estimated function.
17.5 Weighted Least Squares 691
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The Theory of Linear Regression with One Regressor
Example#1:ThevarianceofuisquadraticinX. Supposethattheconditionalvari- ance is known to be the quadratic function
var(ui0Xi) = u0 + u1X2i, (17.27)
where u0 and u1 are unknown parameters, u0 7 0, and u1 Ú 0.
Because u0 and u1 are unknown, it is not possible to construct the weighted
variables Y∼ , X∼ , and X∼ . It is, however, possible to estimate u and u , and to use i0i 1i 0 1
those estimates to compute estimates of var(ui Xi). Let un0 and un1 be estimators of
∼n
u andu,andletvar(u0X)=u +uX.DefinetheweightedregressorsY=
nn2 01ii01ii
∼n ∼n
Y>2var(u 0X),X = 1>2var(u 0X),andX = X >2var(u 0X).TheWLSesti-
iii0i ii1i1iii∼n∼n mator is the OLS estimator of the coefficients in the regression of Yi on X0i and
∼n ∼n
X1i (where b0X0i takes the place of the intercept b0).
Implementation of this estimator requires estimating the conditional variance function, that is, estimating u0 and u1 in Equation (17.27). One way to estimate u0 and u1 consistently is to regress un2i on X2i using OLS, where un2i is the square of the ith OLS residual.
Suppose that the conditional variance has the form in Equation (17.27) and that un0 and un1 are consistent estimators of u0 and u1. Under Assumptions #1 through #3 of Key Concept 17.1, plus additional moment conditions that arise because u0 and u1 are estimated, the asymptotic distribution of the WLS estimator is the same as if u0 and u1 were known. Thus the WLS estimator with u0 and u1 estimated has the same asymptotic distribution as the infeasible WLS estimator and is in this sense asymptotically BLUE.
Because this method of WLS can be implemented by estimating unknown parameters of the conditional variance function, this method is sometimes called feasible WLS or estimated WLS.
Example #2: The variance depends on a third variable. WLS also can be used when the conditional variance depends on a third variable, Wi, which does not appear in the regression function. Specifically, suppose that data are collected on three variables, Yi, Xi, and Wi, i = 1, c, n; the population regression function depends on Xi but not Wi; and the conditional variance depends on Wi but not Xi. That is, the population regression function is E(Yi Xi, Wi) = b0 + b1Xi and the conditional variance is var(ui Xi, Wi) = lh(Wi), where l is a constant and h is a function that must be estimated.
For example, suppose that a researcher is interested in modeling the relation- ship between the unemployment rate in a state and a state economic policy vari- able (Xi). The measured unemployment rate (Yi), however, is a survey-based
estimate of the true unemployment rate (Y*i ). Thus Yi measures Y*i with error, where the source of the error is random survey error, so Yi = Y*i + vi, where vi is the measurement error arising from the survey. In this example, it is plausible that the survey sample size, Wi, is not itself a determinant of the true state unemploy- ment rate. Thus the population regression function does not depend on Wi; that is, E(Y*i Xi,Wi) = b0 + b1Xi. We therefore have the two equations
Y*i = b0 + b1Xi + u*i and (17.28) Yi = Y*i + vi, (17.29)
where Equation (17.28) models the relationship between the state economic pol- icy variable and the true state unemployment rate and Equation (17.29) repre- sents the relationship between the measured unemployment rate Yi and the true unemployment rate Y*i .
The model in Equations (17.28) and (17.29) can lead to a population regres-
sion in which the conditional variance of the error depends on Wi but not on Xi.
The error term u*i in Equation (17.28) represents other factors omitted from this
regression, while the error term vi in Equation (17.29) represents measurement
error arising from the unemployment rate survey. If u*i is homoskedastic, then
var(u*i 0 Xi, Wi) = s2u* is constant. The survey error variance, however, depends
inversely on the survey sample size Wi; that is, var(vi 0 Xi, Wi) = a > Wi where a is a
constant. Because v is random survey error, it is safely assumed to be uncorrelated i
**2
with u , so var(u + v 0 X , W ) = s + a>W Thus, substituting Equation (17.28)
i i i i i u* i
into Equation (17.29) leads to the regression model with heteroskedasticity
17.5 Weighted Least Squares 693
Yi = b0 + b1Xi + ui, (17.30) var(uX,W)=u +ua1b, (17.31)
i i i 0 1 Wi
whereui = u*i + vi,u0 = s2u*,u1 = a,andE(uiXi,Wi) = 0.
If u0 and u1 were known, then the conditional variance function in Equation
(17.31) could be used to estimate b0 and b1 by WLS. In this example, u0 and u1 are unknown, but they can be estimated by regressing the squared OLS residual [from OLS estimation of Equation (17.30)] on 1>Wi. Then the estimated conditional variance function can be used to construct the weights in feasible WLS.
It should be stressed that it is critical that E(ui 0 Xi, Wi) = 0; if not, the weighted errors will have nonzero conditional mean and WLS will be inconsistent. Said differently, if Wi is in fact a determinant of Yi, then Equation (17.30) should be a multiple regression equation that includes both Xi and Wi.
694 CHAPTER 17
The Theory of Linear Regression with One Regressor
General method of feasible WLS. In general, feasible WLS proceeds in five steps:
1. Regress Yi on Xi by OLS and obtain the OLS residuals uni, i = 1, c, n.
2. Estimate a model of the conditional variance function var(ui 0 Xi). For example, if the conditional variance function has the form in Equation (17.27), this entails regressing un2i on X2i . In general, this step entails estimating a function for the
conditional variance, var(ui Xi).
3. Use the estimated function to compute predicted values of the conditional
variance function, var(ui 0 Xi).
4. Weight the dependent variable and regressor (including the intercept) by the
inverse of the square root of the estimated conditional variance function.
5. Estimate the coefficients of the weighted regression by OLS; the resulting
estimators are the WLS estimators.
Regression software packages typically include optional weighted least squares commands that automate the fourth and fifth of these steps.
Heteroskedasticity-Robust Standard Errors or WLS?
There are two ways to handle heteroskedasticity: estimating b0 and b1 by WLS or estimating b0 and b1 by OLS and using heteroskedasticity-robust standard errors. Deciding which approach to use in practice requires weighing the advantages and disadvantages of each.
The advantage of WLS is that it is more efficient than the OLS estimator of the coefficients in the original regressors, at least asymptotically. The disadvan- tage of WLS is that it requires knowing the conditional variance function and estimating its parameters. If the conditional variance function has the quadratic form in Equation (17.27), this is easily done. In practice, however, the functional form of the conditional variance function is rarely known. Moreover, if the func- tional form is incorrect, then the standard errors computed by WLS regression routines are invalid in the sense that they lead to incorrect statistical inferences (tests have the wrong size).
The advantage of using heteroskedasticity-robust standard errors is that they produce asymptotically valid inferences even if you do not know the form of the conditional variance function. An additional advantage is that heteroskedasticity- robust standard errors are readily computed as an option in modern regression packages, so no additional effort is needed to safeguard against this threat. The disadvantage of heteroskedasticity-robust standard errors is that the OLS estima- tor will have a larger variance than the WLS estimator (based on the true condi- tional variance function).
In practice, the functional form of var(ui 0 Xi) is rarely if ever known, which poses a problem for using WLS in real-world applications. This problem is difficult enough with a single regressor, but in applications with multiple regressors it is even more difficult to know the functional form of the conditional variance. For this rea- son, practical use of WLS confronts imposing challenges. In contrast, in modern statistical packages it is simple to use heteroskedasticity-robust standard errors, and the resulting inferences are reliable under very general conditions; in particular, heteroskedasticity-robust standard errors can be used without needing to specify a functional form for the conditional variance. For these reasons, it is our opinion that, despite the theoretical appeal of WLS, heteroskedasticity-robust standard errors provide a better way to handle potential heteroskedasticity in most applications.
Summary
1. The asymptotic normality of the OLS estimator, combined with the consistency of heteroskedasticity-robust standard errors, implies that, if the first three least squares assumptions in Key Concept 17.1 hold, then the heteroskedasticity- robust t-statistic has an asymptotic standard normal distribution under the null hypothesis.
2. If the regression errors are i.i.d. and normally distributed, conditional on the regressors, then bn1 has an exact normal sampling distribution, conditional on the regressors. In addition, the homoskedasticity-only t-statistic has an exact Student tn–2 sampling distribution under the null hypothesis.
3. The weighted least squares (WLS) estimator is OLS applied to a weighted regres- sion, where all variables are weighted by the square root of the inverse of the conditional variance, var(ui 0 Xi), or its estimate. Although the WLS estimator is asymptotically more efficient than OLS, to implement WLS you must know the functional form of the conditional variance function, which usually is a tall order.
Key Terms
convergence in probability (680) consistent estimator (680) convergence in distribution (682) asymptotic distribution (682) Slutsky’s theorem (683) continuous mapping theorem (683)
weighted least squares (WLS) (690) WLS estimator (691)
infeasible WLS (691)
feasible WLS (692)
normal p.d.f. (701)
bivariate normal p.d.f. (702)
17.5 Weighted Least Squares 695
696 CHAPTER 17
The Theory of Linear Regression with One Regressor
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Review the Concepts
17.1 Suppose that Assumption #4 in Key Concept 17.1 is true, but you construct a 95% confidence interval for b1 using the heteroskedastic- robust standard error in a large sample. Would this confidence interval be valid asymptotically in the sense that it contained the true value of b1 in 95% of all repeated samples for large n? Suppose instead that Assumption #4 in Key Concept 17.1 is false, but you construct a 95% confidence interval for b1 using the homoskedasticity-only standard error formula in a large sample. Would this confidence interval be valid asymptotically?
17.2 Suppose that An is a sequence of random variables that converges in probability to 3. Suppose that Bn is a sequence of random variables that converges in distribution to a standard normal. What is the asymptotic dis- tribution of AnBn? Use this asymptotic distribution to compute an approxi- mate value of Pr(AnBn < 2).
17.3 Suppose that Y and X are related by the regression Y = 1.0 + 2.0X + u. A researcher has observations on Y and X, where 0 ... X ... 20, where the conditional variance is var(ui 0Xi = x) = 1 for 0 ... x ... 10 and var(ui 0 Xi = x) = 16 for 10 6 x ... 20. Draw a hypothetical scatterplot of the observations (Xi, Yi), i = 1, c, n. Does WLS put more weight on observations with x ... 10 or x 7 10? Why?
17.4 Instead of using WLS, the researcher in the previous problem decides to compute the OLS estimator using only the observations for which x ... 10, then using only the observations for which x 7 10, and then using the average the two OLS of estimators. Is this estimator more efficient than WLS?
Exercises
17.1 Consider the regression model without an intercept term, Yi = b1Xi + ui (so the true value of the intercept, b0, is zero).
a. Derive the least squares estimator of b1 for the restricted regression
model Yi = b1Xi + ui. This is called the restricted least squares esti-
mator (bnRLS) of b because it is estimated under a restriction, which in 11
this case is b0 = 0.
b. Derive the asymptotic distribution of bnRLS under Assumptions #1
through #3 of Key Concept 17.1.
c. Show that bnRLS is linear [Equation (5.24)] and, under Assumptions #1 1
and #2 of Key Concept 17.1, conditionally unbiased [Equation (5.25)].
d. Derive the conditional variance of bnRLS under the Gauss–Markov 1
conditions (Assumptions #1 through #4 of Key Concept 17.1).
e. Compare the conditional variance of bnRLS in (d) to the conditional 1
variance of the OLS estimator bn1 (from the regression including an intercept) under the Gauss–Markov conditions. Which estimator is more efficient? Use the formulas for the variances to explain why.
f. Derive the exact sampling distribution of bnRLS under Assumptions #1 1
through #5 of Key Concept 17.1.
g. Now consider the estimator b∼ = gn Y > gn X . Derive an
1i=1ii=1i
expression for var( b∼ X , c, X ) – var(bnRLS 0 X , c, X ) under
11n11n
the Gauss–Markov conditions and use this expression to show that
var(b∼10X1,c,Xn)Úvar(bnR1LS0X1,c,Xn).
17.2 Suppose that (Xi,Yi) are i.i.d. with finite fourth moments. Prove that the sample covariance is a consistent estimator of the population covariance— that is, sXY ¡p sXY, where sXY is defined in Equation (3.24). (Hint: Use the strategy outlined in Appendix 3.3 and the Cauchy–Schwarz inequality.)
17.3. This exercise fills in the details of the derivation of the asymptotic distribu- tion of bn1 given in Appendix 4.3.
a. Use Equation (17.19) to derive the expression
1n 1n
Anavi (X – mX)Anaui 2n(bn-b)= i=1 – i=1 ,
17.5 Weighted Least Squares 697
1
111n 1n
na(Xi – X)2 na(Xi – X)2
i=1 i=1 where vi = (Xi – mX)ui.
698 CHAPTER 17
The Theory of Linear Regression with One Regressor
b. Use the central limit theorem, the law of large numbers, and Slutsky’s theorem to show that the final term in the equation converges in probability to zero.
c. Use the Cauchy–Schwarz inequality and the third least squares assumption in Key Concept 17.1 to prove that var(vi) 6 ∞. Does the term 2n1 g ni = 1 vi > sv satisfy the central limit theorem?
d. Apply the central limit theorem and Slutsky’s theorem to obtain the result in Equation (17.12).
17.4 Show the following results:
a. Show that 2n(bn1 – b1) ¡d N(0, a2), where a2 is a constant, implies
that bn1 is consistent. (Hint: Use Slutsky’s theorem.)
b. Show that s2u>s2u ¡p 1 implies that su>su ¡p 1.
17.5 Suppose that W is a random variable with E(W4) 6 ∞. Show that E(W2) 6 ∞.
17.6 Show that if bn1 is conditionally unbiased, then it is unbiased; that is, show that if E(bn1 0 X1, c, Xn) = b1, then E(bn1) = b1.
17.7 Suppose that X and u are continuous random variables and (Xi, ui), i = 1, c, n, are i.i.d.
a. Show that the joint probability density function (p.d.f.) of (ui, uj, Xi, Xj) can be written as f(ui, Xi)f(uj, Xj) for i ≠ j, where f(ui, Xi) is the joint p.d.f. of ui and Xi.
b. Show that E(uiuj 0 Xi, Xj) = E(ui 0 Xi) E(uj 0 Xj) for i ≠ j.
c. Show that E(ui 0 X1, c, Xn) = E(ui 0 Xi).
d. Show that E(uiuj 0 X1, X2, c, Xn) = E(ui 0 Xi) E(uj 0 Xj) for i ≠ j.
17.8 Consider the regression model in Key Concept 17.1 and suppose that Assumptions #1, #2, #3, and #5 hold. Suppose that Assumption #4 is replaced by the assumption that var(ui0Xi) = u0 + u10Xi0, where 0Xi0 is the absolute value of Xi, u0 7 0, and u1 Ú 0.
a. Is the OLS estimator of b1 BLUE?
b. Suppose that u0 and u1 are known. What is the BLUE estimator of b1?
c. Derive the exact sampling distribution of the OLS estimator, bn1, con- ditional on X1, c, Xn.
d. Derive the exact sampling distribution of the WLS estimator (treating u0 and u1 as known) of b1, conditional on X1, c, Xn.
17.5 Weighted Least Squares 699
17.9 Prove Equation (17.16) under Assumptions #1 and #2 of Key Concept 17.1
plus the assumption that Xi and ui have eight moments.
17.10 Let un be an estimator of the parameter u, where un might be biased. Show that if E3(un – u)24 ¡ 0 as n ¡ ∞ (that is, the mean squared error of un tends to zero), then un ¡p u. [Hint: Use Equation (17.43) with W = un – u.4
17.11 Suppose that X and Y are distributed bivariate normal with density given in Equation (17.38).
a. Show that the density of Y given X = x can be written as
f (y) = 1 expc – 1ay – mYXb2d
YX=x
sYX22p
2 sYX
where sYX = 2s2Y(1 – r2XY) and mYX = mY – (sXY>s2X)(x – mX). [Hint: Use the definition of the conditional probability density
fY 0 X = x(y) = 3gX, Y(x, y)4 > 3fX(x)4, where gX,Y is the joint density of X and Y, and ƒX is the marginal density of X.]
b. Use the result in (a) to show that Y 0 X = x ∼ N(mY 0 X, s2Y 0 X).
c. Use the result in (b) to show that E(Y0X = x) = a + bx for suitably
chosen constants a and b.
17.12 a. Suppose that u ∼ N(0, s2u). Show that E(eu) = e12su2
b. Suppose that the conditional distribution of u given X = x is N(0, a + bx2), where a and b are positive constants. Show that E(eu0X = x) = e12(a+bx2).
17.13 Consider the heterogeneous regression model Yi = b0i + b1iXi + ui, where b0i and b1i are random variables that differ from one observation to the next. Suppose that E(ui 0 Xi) = 0 and (b0i, b1i) are distributed independently of Xi.
a. Let bnOLS denote the OLS estimator of b given in Equation (17.2). 11
Show that bnOLS ¡p E(b ), where E(b ) is the average value of b in 1 1 1 1i
the population. [Hint: See Equation (13.10).]
b. Suppose that var(ui 0 Xi) = u0 + u1X2i , where u0 and u1 are known posi-
tive constants. Let bnWLS denote the weighted least squares estimator. 1
Does bnWLS ¡p E(b )? Explain. 11
17.14 Suppose that Yi, i = 1, 2, c, n, are i.i.d. with E(Yi) = m, var(Yi) = s2, and finite fourth moment. Show the following:
700 CHAPTER 17
The Theory of Linear Regression with One Regressor
APPENDIX
17.1
a. E(Y2i ) = m2 + s2 b.Y¡p μ
1np
c. naY2i ¡m2 +s2
i=1
1n 1n
d. na(Yi – Y)2 = naY2i – Y2
i=1 i=1
1np
e. na(Yi – Y)2 ¡ s2
i=1
f. s2 = n – 1a(Yi – Y)2 ¡ s2
1np i=1
17.15 Z is distributed N (0,1), W is distributed x2n, and V is distributed x2m. Show, as n S ∞ and m is fixed, that:
a.W>n¡p 1.
b. Z ¡d N(0,1). Use the result to explain why the t
distribution is
distribution is
the same as the x2m>m distribution.
The Normal and Related Distributions and
Moments of Continuous Random Variables
This appendix defines and discusses the normal and related distributions. The definitions of the chi-squared, F, and Student t distributions, given in Section 2.4, are restated here for convenient reference. We begin by presenting definitions of probabilities and moments involving continuous random variables.
Probabilities and Moments of Continuous
Random Variables
As discussed in Section 2.1, if Y is a continuous random variable, then its probability is summarized by its probability density function (p.d.f.). The probability that Y falls between two values is the area under its p.d.f. between those two values. Because Y is continuous, however, the mathematical expressions for its probabilities involve integrals rather than the summations that are appropriate for discrete random variables.
1W>n
the same as the standard normal distribution.
∞
c. V>m ¡d x2 >m. Use the result to explain why the F W>n m m,∞
17.5 Weighted Least Squares 701 Let fY denote the probability density function of Y. Because probabilities cannot be
negative, fY(y) Ú 0 for all y. The probability that Y falls between a and b (where a < b) is b
Pr(a ... Y ... b) =
Because Y must take on some value on the real line, Pr(- ∞ ... Y ... ∞) = 1, which implies
that ∞ f (y)dy = 1. 1-∞ Y
Expected values and moments of continuous random variables, like those of discrete random variables, are probability-weighted averages of their values, except that summa- tions [for example, the summation in Equation (2.3)] are replaced by integrals. Accord- ingly, the expected value of Y is
E(Y) = mY = LyfY(y)dy, (17.33)
where the range of integration is the set of values for which fY is nonzero. The variance is the expected value of (Y - mY)2, the rth moment of a random variable is the expected value of Yr, and the rth central moment is the expected value of (Y - mY)r. Thus
(17.34) (17.35)
and similarly for the rth central moment, E(Y - mY)r. The Normal Distribution
The normal distribution for a single variable. The probability density function of a nor- mally distributed random variable (the normal p.d.f.) is
fY(y) = 1 expc- 1ay - mb2 d, (17.36) s22p 2 s
La
fY(y)dy. (17.32)
var(Y) = E(Y - mY)2 = L(y - mY)2 fY(y)dy,
E(Yr) = LyrfY(y)dy,
where exp(x) is the exponential function of x. The factor 1>(s22p) in Equation (17.36) ensuresthatPr(-∞ … Y … ∞) = ∞f (y)dy = 1.
1-∞ Y
The mean of the normal distribution is m, and its variance is s2. The normal distribu-
tion is symmetric, so all odd central moments of order three and greater are zero. The fourth central moment is 3s4. In general, if Y is distributed N(m, s2), then its even central moments are given by
702 CHAPTER 17
The Theory of Linear Regression with One Regressor
E(Y – m)k = k! sk (k even). (17.37) 2k>2(k>2)!
When m = 0 and s2 = 1, the normal distribution is called the standard normal distribu- tion. The standard normal p.d.f. is denoted f, and the standard normal c.d.f. is denoted Φ.
1y2 y
Thus the standard normal density is f(y) = 22p exp (- 2 ) and Φ(y) = 1-∞ f(s)ds.
The bivariate normal distribution. The bivariate normal p.d.f. for the two random vari- ables X and Y is
g (x,y)= X,Y
1 2psXsY21 – r2XY
*expe
2
1 cax-mXb –
-2(1 – r2XY) sX
2r
XY
ax – mXbay – mYb + ay – mYb2df, (17.38) sX sY sY
where rXY is the correlation between X and Y.
When X and Y are uncorrelated (rXY = 0), gX,Y(x, y) = fX(x)fY(y), where f is the
normal density given in Equation (17.36). This proves that if X and Y are jointly normally distributed and are uncorrelated, then they are independently distributed. This is a special feature of the normal distribution that is typically not true for other distributions.
The multivariate normal distribution extends the bivariate normal distribution to handle more than two random variables. This distribution is most conveniently stated using matrices and is presented in Appendix 18.1.
The conditional normal distribution. Suppose that X and Y are jointly normally distrib- uted. Then the conditional distribution of Y given X is N(mYX, s2YX), with mean mYX = mY + (sXY>s2X)(X – mX) and variance s2YX = (1 – r2XY)s2Y. The mean of this conditional distribution, conditional on X = x, is a linear function of x, and the variance does not depend on x (Exercise 17.11).
Related Distributions
The chi-squared distribution. Let Z1, Z2, c, Zn be n i.i.d. standard normal random vari- ables. The random variable
n
W = aZ2i (17.39)
i=1
has a chi-squared distribution with n degrees of freedom. This distribution is denoted x2n. Because E(Z2i ) = 1 and E(Z4i ) = 3, E(W) = n and var(W) = 2n.
17.5 Weighted Least Squares 703 The Student t distribution. Let Z have a standard normal distribution, let W have a x2m
distribution, and let Z and W be independently distributed. Then the random variable t= Z (17.40)
2W>m
has a Student t distribution with m degrees of freedom, denoted tm. The t ∞ distribution is
the standard normal distribution. (See Exercise 17.15.)
The F distribution. Let W1 and W2 be independent random variables with chi-squared
distributions with respective degrees of freedom n1 and n2. Then the random variable
W1>n1
F = W >n (17.41)
22
has an F distribution with (n1, n2) degrees of freedom. This distribution is denoted Fn1,n2. The F distribution depends on the numerator degrees of freedom n1 and the denomi- nator degrees of freedom n2. As number of degrees of freedom in the denominator gets large, the Fn1,n2 distribution is well approximated by a x2n1 distribution, divided by n1. In the limit, the Fn1, ∞ distribution is the same as the x2n1 distribution, divided by n1; that is, it is the
17.2
same as the x2n1>n1 distribution. (See Exercise 17.15.)
APPENDIX
Two Inequalities
This appendix states and proves Chebychev’s inequality and the Cauchy–Schwarz inequality.
Chebychev’s Inequality
Chebychev’s inequality uses the variance of the random variable V to bound the probabil- ity that V is farther than {d from its mean, where d is a positive constant:
Pr(0V – m 0 Ú d) … var(V) (Chebychev’sinequality). (17.42)
d2
V
To prove Equation (17.42), let W = V – mV, let f be the p.d.f. of W, and let d be any positive number. Now
704 CHAPTER 17
The Theory of Linear Regression with One Regressor
E(W2) = =
∞
L- ∞
-d d ∞
Ú
Ú d2c
w2f(w)dw + w2f(w)dw Ld
L- ∞
w2f(w)dw
w2f(w)dw + w2f(w)dw + L- d
w2f(w)dw Ld
L- ∞
-d ∞
L- ∞
-d ∞
Ld
f(w)dwd
f(w)dw + = d2Pr(W Ú d),
where the first equality is the definition of E(W2), the second equality holds because the ranges of integration divides up the real line, the first inequality holds because the term that was dropped is nonnegative, the second inequality holds because w2 Ú d2 over the range of integration, and the final equality holds by the definition of Pr( W Ú d). Substi- tutingW = V – mv intothefinalexpression,notingthatE(W2) = E3(V – mV)24 = var(V), and rearranging yields the inequality given in Equation (17.42). If V is discrete, this proof applies with summations replacing integrals.
The Cauchy–Schwarz Inequality
The Cauchy–Schwarz inequality is an extension of the correlation inequality, rXY … 1, to incorporate nonzero means. The Cauchy–Schwarz inequality is
E(XY) … 2E(X2)E(Y2) (Cauchy9Schwarz inequality). (17.44)
The proof of Equation (17.44) is similar to the proof of the correlation inequality in Appendix 2.1. Let W = Y + bX, where b is a constant. Then E(W2) = E(Y2) + 2bE(XY) + b2E(X2). Now let b = -E(XY)>E(X2) so that (after simplification) the expression becomesE(W2) = E(Y2) – 3E(XY)42>E(X2).BecauseE(W2) Ú 0(sinceW2 Ú 0),itmust bethecasethat3E(XY)42 …E(X2)E(Y2),andtheCauchy–Schwarzinequalityfollowsby taking the square root.
(17.43)
Chapter
The Theory
of Multiple Regression
This chapter provides an introduction to the theory of multiple regression analy- sis. The chapter has four objectives. The first is to present the multiple regression model in matrix form, which leads to compact formulas for the OLS estimator and test statistics. The second objective is to characterize the sampling distribution of the OLS estimator, both in large samples (using asymptotic theory) and in small samples (if the errors are homoskedastic and normally distributed). The third objective is to study the theory of efficient estimation of the coefficients of the multiple regression model and to describe generalized least squares (GLS), a method for estimating the regression coefficients efficiently when the errors are heteroskedastic and/or corre- lated across observations. The fourth objective is to provide a concise treatment of the asymptotic distribution theory of instrumental variables (IV) regression in the linear model, including an introduction to generalized method of moments (GMM) estimation in the linear IV regression model with heteroskedastic errors.
The chapter begins by laying out the multiple regression model and the OLS estimator in matrix form in Section 18.1. This section also presents the extended least squares assumptions for the multiple regression model. The first four of these assumptions are the same as the least squares assumptions of Key Concept 6.4 and underlie the asymptotic distributions used to justify the procedures described in Chapters 6 and 7. The remaining two extended least squares assumptions are stronger and permit us to explore in more detail the theoretical properties of the OLS estimator in the multiple regression model.
The next three sections examine the sampling distribution of the OLS estimator and test statistics. Section 18.2 presents the asymptotic distributions of the OLS estimator and t-statistic under the least squares assumptions of Key Concept 6.4. Section 18.3 unifies and generalizes the tests of hypotheses involving multiple coef- ficients presented in Sections 7.2 and 7.3, and provides the asymptotic distribution of the resulting F-statistic. In Section 18.4, we examine the exact sampling distributions of the OLS estimator and test statistics in the special case that the errors are homo- skedastic and normally distributed. Although the assumption of homoskedastic normal errors is implausible in most econometric applications, the exact sampling distributions are of theoretical interest, and p-values computed using these distri- butions often appear in the output of regression software.
18
705
706
ChapTeR 18 The Theory of Multiple Regression
The next two sections turn to the theory of efficient estimation of the coefficients of the multiple regression model. Section 18.5 generalizes the Gauss–Markov theorem to multiple regression. Section 18.6 develops the method of generalized least squares (GLS).
The final section takes up IV estimation in the general IV regression model when the instruments are valid and strong. This section derives the asymptotic distribution of the TSLS estimator when the errors are heteroskedastic and provides expressions for the standard error of the TSLS estimator. The TSLS estimator is one of many possible GMM estimators, and this section provides an introduction to GMM estimation in the linear IV regression model. It is shown that the TSLS estimator is the efficient GMM estimator if the errors are homoskedastic.
Mathematical prerequisite. The treatment of the linear model in this chapter uses matrix notation and the basic tools of linear algebra and assumes that the reader has taken an introductory course in linear algebra. Appendix 18.1 reviews vectors, matrices, and the matrix operations used in this chapter. In addition, multivariate calculus is used in Section 18.1 to derive the OLS estimator.
18.1 The Linear Multiple Regression Model and OLS Estimator in Matrix Form
The linear multiple regression model and the OLS estimator can each be repre- sented compactly using matrix notation.
The Multiple Regression Model in Matrix Notation
The population multiple regression model (Key Concept 6.2) is
Yi = b0 + b1X1i + b2X2i +g+bkXki + ui,i = 1,c,n. (18.1)
To write the multiple regression model in matrix form, define the following vectors and matrices:
(18.2)
Y2 u2 1 X12 g Xk2 X2′ b1
Y = ± Y1 ≤ , U = ± u1 ≤ , X = ± 1 X11 g Xk1 ≤ = ± X1′ ≤ , and B = ± b0 ≤ ,
ffffffff
Y u 1XgX X′ b n n 1n kn n k
18.1 The Linear Multiple Regression Model and OLS Estimator in Matrix Form 707 so Y is n * 1, X is n * (k + 1), U is n * 1, and B is (k + 1) * 1. Throughout we
denote matrices and vectors by bold type. In this notation,
• Y is the n * 1 dimensional vector of n observations on the dependent variable.
• X is the n * (k + 1) dimensional matrix of n observations on the k + 1 regressors (including the “constant” regressor for the intercept).
• The (k + 1) * 1 dimensional column vector Xi is the ith observation on the k + 1 regressors; that is, Xi′ = (1 X1i cXki), where Xi′ denotes the transpose of Xi.
• U is the n * 1 dimensional vector of the n error terms.
• B is the (k + 1) * 1 dimensional vector of the k + 1 unknown regression
coefficients.
The multiple regression model in Equation (18.1) for the ith observation, writ-
ten using the vectors B and Xi, is
Yi = Xi′B + ui,i = 1,c,n. (18.3)
The extended Least Squares assumptions in the Multiple Regression Model
The linear regression model with multiple regressors is Yi = Xi′B + ui,i = 1,c,n.
The extended least squares assumptions are
1. E(ui Xi) = 0 (ui has conditional mean zero);
Key ConCept
18.1
2. (Xi, Yi), i = 1, c, n, are independently and identically distributed (i.i.d.) draws from their joint distribution;
3. Xi and ui have nonzero finite fourth moments;
4. X has full column rank (there is no perfect multicollinearity);
5. var(ui Xi) = s2u (homoskedasticity); and
6. The conditional distribution of ui given Xi is normal (normal errors).
(18.4)
708 ChapTeR 18 The Theory of Multiple Regression
In Equation (18.3), the first regressor is the “constant” regressor that always equals 1, and its coefficient is the intercept. Thus the intercept does not appear separately in Equation (18.3); rather, it is the first element of the coefficient vector B.
Stacking all n observations in Equation (18.3) yields the multiple regression model in matrix form:
Y = XB + U. (18.5) The Extended Least Squares Assumptions
The extended least squares assumptions for the multiple regressor model are the four least squares assumptions for the multiple regression model in Key Concept 6.4, plus the two additional assumptions of homoskedasticity and normally distrib- uted errors. The assumption of homoskedasticity is used when we study the effi- ciency of the OLS estimator, and the assumption of normality is used when we study the exact sampling distribution of the OLS estimator and test statistics.
The extended least squares assumptions are summarized in Key Concept 18.1.
Except for notational differences, the first three assumptions in Key Concept 18.1 are identical to the first three assumptions in Key Concept 6.4.
The fourth assumption in Key Concepts 6.4 and 18.1 might appear different, but in fact they are the same: They are simply different ways of saying that there cannot be perfect multicollinearity. Recall that perfect multicollinearity arises when one regressor can be written as a perfect linear combination of the others. In the matrix notation of Equation (18.2), perfect multicollinearity means that one column of X is a perfect linear combination of the other columns of X, but if this is true, then X does not have full column rank. Thus saying that X has rank k + 1, that is, rank equal to the number of columns of X, is just another way to say that the regressors are not perfectly multicollinear.
The fifth least squares assumption in Key Concept 18.1 is that the error term is conditionally homoskedastic, and the sixth assumption is that the conditional distribution of ui, given Xi, is normal. These two assumptions are the same as the final two assumptions in Key Concept 17.1, except that they are now stated for multiple regressors.
Implications for the mean vector and covariance matrix of U. The least squares assumptions in Key Concept 18.1 imply simple expressions for the mean vector and covariance matrix of the conditional distribution of U given the matrix of regressors X. (The mean vector and covariance matrix of a vector of random
18.1 The Linear Multiple Regression Model and OLS Estimator in Matrix Form 709
variables are defined in Appendix 18.2.) Specifically, the first and second assump-
cov(u,u0X)=E(uu0X)=E(uu0X,X)=E(u0X)E(u0X)=0 for i≠j ijijijijiijj
tions in Key Concept 18.1 imply that E(uiX) = E(uiXi) = 0 and that
E(u 0 X) = E(u 0 X ) =
2i 2ii2u
(Exercise 17.7). The first, second, and fifth assumptions imply that
s . Combining these results, we have that
under Assumptions #1 and #2, E(U 0 X) = 0n, and (18.6) under Assumptions #1, #2, and #5, E(UU′ 0 X) = s2uIn, (18.7)
where 0n is the n-dimensional vector of zeros and In is the n * n identity matrix. Similarly, the first, second, fifth, and sixth assumptions in Key Concept 18.1 imply that the conditional distribution of the n-dimensional random vector U, conditional on X, is the multivariate normal distribution (defined in Appen-
dix 18.2). That is,
under Assumptions #1, #2, #5, and #6, the
conditional distribution of U given X is N(0n, s2uIn). (18.8) The OLS Estimator
The OLS estimator minimizes the sum of squared prediction mistakes, g (Y – b – b X – g- b X ) [Equation(6.8)].TheformulafortheOLS
ni=1i 0 11i kki2
estimator is obtained by taking the derivative of the sum of squared prediction mistakes with respect to each element of the coefficient vector, setting these derivatives to zero, and solving for the estimator Bn.
The derivative of the sum of squared prediction mistakes with respect to the jth regression coefficient, bj, is
0 an
0bj i=1(Yi – b0 – b1X1i – g- bkXki)2
an i=1
= -2
Xji(Yi – b0 – b1X1i – g- bkXki)
(18.9)
for j = 0, c, k, where, for j = 0, X0i = 1 for all i. The derivative on the right- hand side of Equation (18.9) is the jth element of the k + 1 dimensional vector, -2X′(Y – Xb), where b is the k + 1 dimensional vector consisting of b0, c, bk. There are k + 1 such derivatives, each corresponding to an element of b. Com- bined, these yield the system of k + 1 equations that, when set to zero, constitute
710 ChapTeR 18 The Theory of Multiple Regression
the first order conditions for the OLS estimator Bn. That is, Bn solves the system of
k + 1 equations
n
form:
n -1
B = (X′X ) X′Y, (18.11)
where (X′X)-1 is the inverse of the matrix X′X.
The role of “no perfect multicollinearity.” The fourth least squares assumption in Key Concept 18.1 states that X has full column rank. In turn, this implies that the matrix X′X has full rank, that is, X′X is nonsingular. Because X′X is nonsingular, it is invertible. Thus the assumption that there is no perfect multicollinearity ensures that (X′X)−1 exists, so Equation (18.10) has a unique solution and the formula in Equation (18.11) for the OLS estimator can actually be computed. Said differently, if X does not have full column rank, there is not a unique solution to Equation (18.10) and X′X is singular. Therefore, (X′X)−1 cannot be computed and thus Bn cannot be computed from Equation (18.11).
18.2 Asymptotic Distribution of the OLS Estimator and t-Statistic
If the sample size is large and the first four assumptions of Key Concept 18.1 are satisfied, then the OLS estimator has an asymptotic joint normal distribution, the heteroskedasticity-robust estimator of the covariance matrix is consistent, and the heteroskedasticity-robust OLS t-statistic has an asymptotic standard normal dis- tribution. These results make use of the multivariate normal distribution (Appen- dix 18.2) and a multivariate extension of the central limit theorem.
The Multivariate Central Limit Theorem
The central limit theorem of Key Concept 2.7 applies to a one-dimensional random variable. To derive the joint asymptotic distribution of the elements of Bn, we need a multivariate central limit theorem that applies to vector-valued random variables.
X′(Y – XB) = 0k+1, (18.10) n
or, equivalently, X′Y = X′XB.
Solving the system of equations (18.10) yields the OLS estimator Bn in matrix
18.2 Asymptotic Distribution of the OLS Estimator and t-Statistic 711
The Multivariate Central Limit Theorem
Key ConCept
18.2
= 𝚺 , where 𝚺 is positive definite and finite. Let W = ng W. Then 2n(W – m ) ¡
tor E(W ) = m and covariance matrix E3(W – m )(W – m )′4 iW iWiWW
Suppose that W1, c, Wn are i.i.d. m-dimensional random variables with mean vec-
W 1 ni=1 i W d N(0m, 𝚺W).
The multivariate central limit theorem extends the univariate central limit theorem to averages of observations on a vector-valued random variable, W, where W is m-dimensional. The difference between the central limit theorems for a scalar as opposed to a vector-valued random variable is the conditions on the variances. In the scalar case in Key Concept 2.7, the requirement is that the vari- ance is both nonzero and finite. In the vector case, the requirement is that the covariance matrix is both positive definite and finite. If the vector-valued random variable W has a finite positive definite covariance matrix, then 0 6 var(c′W) 6 ∞ for all nonzero m-dimensional vectors c (Exercise 18.3).
The multivariate central limit theorem that we will use is stated in Key Con- cept 18.2.
Asymptotic Normality of βn
In large samples, the OLS estimator has the multivariate normal asymptotic dis-
d
1n(B – B) ¡ N(0
1n(B – B)
1n(B – B)
tribution
n
, 𝚺
n
), where 𝚺
n
-1 -1
= Q 𝚺 Q , (18.12)
k + 1
X V X
where QX is the (k + 1) * (k + 1)-dimensional matrix of second moments of the
Written in terms of B rather than 1n(B – B), the normal approximation in Equation (18.12) is
regressors, that is, QX = E(XiXi′), and 𝚺V is the (k + 1) * (k + 1)-dimensional
covariance matrix of V = X u , that is, 𝚺 = E(V V′). Note that the second least iiiVii
squares assumption in Key Concept 18.1 implies that Vi, i = 1, c, n, are i.i.d. nn
-1 -1
n where𝚺Bn=𝚺1n(Bn-B)>n=QX 𝚺VQX >n. (18.13)
B, in large samples, is approximately distributed N(B, 𝚺Bn)
712 ChapTeR 18 The Theory of Multiple Regression
The covariance matrix 𝚺Bn in Equation (18.13) is the covariance matrix of the
1n(B – B)
the covariance matrix of the asymptotic normal distribution of 2n(B – B).
These two covariance matrices differ by a factor of n, depending on whether the OLS estimator is scaled by 2n. n
approximate normal distribution of B, whereas 𝚺 n in Equation (18.12) is
Derivation of Equation (18.12). To derive Equation (18.12), first use Equations
(18.4) and (18.11) to write B = (X′X) n
X′Y = (X′X) -1
X′(XB + U) so that (18.14)
n
X′X -1 X′U -12n(B-B)=a n b a1nb.
Thus B – B = (X′X) X′U, so n
n
-1
B = B + (X′X) X′U.
-1
(18.15) The derivation of Equation (18.12) involves arguing first that the “denominator”
“numerator” matrix, X′U>1n, obeys the multivariate central limit theorem in Key Concept 18.2. The details are given in Appendix 18.3.
matrix in Equation (18.15), X′X>n, is consistent for Q and second that the X
The heteroskedasticity-robust estimator of 𝚺 n is obtained by replacing the 1n(B – B)
n
Heteroskedasticity-Robust Standard Errors
2n(B – B) is
𝚺n1n(Bn-B) =aX′Xb-1𝚺nVn aX′Xb-1, where𝚺nVn = 1 an XiXi′un2i, (18.16)
population moments in its definition [Equation (18.12)] by sample moments. Accordingly, the heteroskedasticity-robust estimator of the covariance matrix of
n
n n n – k – 1i=1
The estimator 𝚺n Vn incorporates the same degrees-of-freedom adjustment that is in the SER for the multiple regression model (Section 6.4) to adjust for potential downward bias because of estimation of k + 1 regression coefficients.
The proof that 𝚺n n ¡p 𝚺 n is conceptually similar to the proof, 1n(B – B) 1n(B – B)
presented in Section 17.3, of the consistency of heteroskedasticity-robust standard errors for the single-regressor model.
Heteroskedasticity-robust standard errors. The heteroskedasticity-robust esti- mator of the covariance matrix of B, 𝚺Bn is
𝚺n n = n-1𝚺n n . (18.17) B n 1n(B-B)
The heteroskedasticity-robust standard error for the j regression coefficient is the square root of the jth diagonal element of 𝚺n Bn. That is, the heteroskedasticity- robust standard error of the jth coefficient is
where (𝚺nBn)jj is the (j, j) element of 𝚺nBn. Confidence Intervals for Predicted Effects
Section 8.1 describes two methods for computing the standard error of predicted effects that involve changes in two or more regressors. There are compact matrix expressions for these standard errors and thus for confidence intervals for pre- dicted effects.
Consider a change in the value of the regressors for the ith observation from some initial value, say Xi,0, to some new value, X i, 0 + d, so that the change in Xi is ∆Xi = d, where d is a k + 1 dimensional vector. This change in X can involve multiple regressors (that is, multiple elements of Xi). For example, if two of the regressors are the value of an independent variable and its square, then d is the difference between the subsequent and initial values of these two variables.
The expected effect of this change in Xi is d′B, and the estimator of this effect is d′B. Because linear combinations of normally distributed random variables are
themselves normally distributed, 2n(d′B – d′B) = d′1n(B – B) ¡
n 1n(B – B) B A 95% confidence interval for this predicted effect is
n d1>2 N(0, d′g n d). Thus the standard error of this predicted effect is (d′𝚺nd) .
(18.19)
Asymptotic Distribution of the t-Statistic
The t-statistic testing the null hypothesis that bj = bj,0, constructed using the heteroskedasticity-robust standard error in Equation (18.18), is given in Key Concept 7.1. The argument that this t-statistic has an asymptotic standard normal distribution parallels the argument given in Section 17.3 for the single-regressor model.
18.3 Tests of Joint Hypotheses
Section 7.2 considers tests of joint hypotheses that involve multiple restrictions, where each restriction involves a single coefficient, and Section 7.3 considers tests of a single restriction involving two or more coefficients. The matrix setup of
nnB d′B { 1.962d′𝚺nd.
18.3 Tests of Joint Hypotheses 713
nj nB,
SE(B) = 2(𝚺n)jj th (18.18)
nn
714 ChapTeR 18 The Theory of Multiple Regression
Section 18.1 permits a unified representation of these two types of hypotheses as linear restrictions on the coefficient vector, where each restriction can involve multiple coefficients. Under the first four least squares assumptions in Key Con- cept 18.1, the heteroskedasticity-robust OLS F-statistic testing these hypotheses has an Fq,∞ asymptotic distribution under the null hypothesis.
Joint Hypotheses in Matrix Notation
Consider a joint hypothesis that is linear in the coefficients and imposes q restric- tions, where q … k + 1. Each of these q restrictions can involve one or more of the regression coefficients. This joint null hypothesis can be written in matrix notation as
RB = r, (18.20)
where R is a q * (k + 1) nonrandom matrix with full row rank and r is a nonrandom q * 1 vector. The number of rows of R is q, which is the number of restrictions being imposed under the null hypothesis.
The null hypothesis in Equation (18.20) subsumes all the null hypotheses considered in Sections 7.2 and 7.3. For example, a joint hypothesis of the type consideredinSection7.2isthatb0 = 0,b1 = 0,c,bq-1 = 0.Towritethisjoint hypothesis in the form of Equation (18.20), set R = [Iq 0q * (k + 1- q)] and r = 0q.
The formulation in Equation (18.20) also captures the restrictions of Section 7.3 involving multiple regression coefficients. For example, if k = 2, then the hypoth- esis that b1 + b2 = 1 can be written in the form of Equation (18.20) by setting R = [0 1 1], r = 1, and q = 1.
Asymptotic Distribution of the F-Statistic
The heteroskedasticity-robust F-statistic testing the joint hypothesis in Equa-
tion (18.20) is
If the first four assumptions in Key Concept 18.1 hold, then under the null hypothesis
d
F ¡ Fq,∞. (18.22)
F = (RBn – r)′3R𝚺nBnR′4-1(RBn – r)>q. (18.21)
18.3 Tests of Joint Hypotheses 715 This result follows by combining the asymptotic normality of Bn with the con-
sistency of the heteroskedasticity-robust estimator 𝚺n n of the covariance 1n(B – B)
matrix. Specifically, first note that Equation (18.12) and Equation (18.74) in Appendix 18.2 imply that, under the null hypothesis, 1n(RB – r) =
1nR(B – B) ¡ N(0, R𝚺 n 1n(B – B)
under the null hypothesis, (RB – r)′[R𝚺n R′] (RB – r) = [1nR(B – B)]′
-1nd2 np [1nR (B – B)] ¡ x . However, because 𝚺 n n
𝚺 n , it follows from Slutsky’s theorem that 31nR(B – B)4′ 1n(B – B)
[R𝚺 n R′]
1n(B-B) d q 1n(B-B) ¡
n
R′). It follows from Equation (18.77) that, n B -1 n n
n -1 n d 2 n [R𝚺 n R′] 31nR (B – B)4 ¡ x . or, equivalently (because 𝚺 =
1n(B – B) q B nd2
𝚺 n >n), that F ¡ x >q, which is in turn distributed F . 1n(B – B) q q, ∞
n
As discussed in Section 7.4, an asymptotically valid confidence set for two or more elements of B can be constructed as the set of values that, when taken as the null hypothesis, are not rejected by the F-statistic. In principle, this set could be computed by repeatedly evaluating the F-statistic for many values of B, but, as is the case with a confidence interval for a single coefficient, it is simpler to manipulate the formula for the test statistic to obtain an explicit formula for the confidence set.
Here is the procedure for constructing a confidence set for two or more of the
elements of B. Let D denote the q-dimensional vector consisting of the coefficients
for which we wish to construct a confidence set. For example, if we are construct-
ing a confidence set for the regression coefficients b1 and b2, then q = 2 and
Confidence Sets for Multiple Coefficients
D = (b1 b2)′. In general, we can write D = RB, where the matrix R consists of
0 n0nB-1n0
the hypothesis that D = D is F = (D – D)′[R𝚺nR′] (D – D)>q, where
zeros and ones [as discussed following Equation (18.20)]. The F-statistic testing
nn
D = RB. A 95% confidence set for D is the set of values D0 that are not rejected by the F-statistic. That is, when D = RB, a 95% confidence set for D is
n nB-1n
5D: (D – D)′[R𝚺n R′] (D – D)>q … c6, (18.23)
where c is the 95th percentile (the 5% critical value) of the Fq, ∞ distribution. The set in Equation (18.23) consists of all the points contained inside the ellipse determined when the inequality in Equation (18.23) is an equality (this is an ellipsoid when q 7 2). Thus the confidence set for d can be computed by solv-
ing Equation (18.23) for the boundary ellipse.
716 ChapTeR 18 The Theory of Multiple Regression
18.4 Distribution of Regression Statistics
with Normal Errors
The distributions presented in Sections 18.2 and 18.3, which were justified by appealing to the law of large numbers and the central limit theorem, apply when the sample size is large. If, however, the errors are homoskedastic and normally distributed, conditional on X, then the OLS estimator has a multivariate normal distri- bution in finite sample, conditional on X. In addition, the finite sample distribu- tion of the square of the standard error of the regression is proportional to the chi-squareddistributionwithn – k – 1degreesoffreedom,thehomoskedasticity- only OLS t-statistic has a Student t distribution with n – k – 1 degrees of free- dom, and the homoskedasticity-only F-statistic has an Fq, n – k – 1 distribution. The arguments in this section employ some specialized matrix formulas for OLS regression statistics, which are presented first.
Matrix Representations of OLS Regression Statistics
The OLS predicted values, residuals, and sum of squared residuals have compact matrix representations. These representations make use of two matrices, PX and MX.
The matrices PX and MX. The algebra of OLS in the multivariate model relies on the two symmetric n * n matrices, PX and MX:
PX = X(X′X)-1X′ and (18.24)
MX = In – PX. (18.25)
AmatrixCisidempotentifCissquareandCC = C(seeAppendix18.1).Because PX = PXPX and MX = MXMX (Exercise 18.5), and because PX and MX are symmetric, PX and MX are symmetric idempotent matrices.
The matrices PX and MX have some additional useful properties (Exercise 18.5), which follow directly from the definitions in Equations (18.24) and (18.25):
PXX = X and MXX = 0n*(k+1);
rank(PX) = k + 1andrank(MX) = n – k – 1, (18.26)
where rank(PX) is the rank of PX.
18.4 Distribution of Regression Statistics with Normal Errors 717
The matrices PX and MX can be used to decompose an n-dimensional vector Z into two parts: a part that is spanned by the columns of X and a part orthogonal to the columns of X. In other words, PXZ is the projection of Z onto the space spanned by the columns of X, MXZ is the part of Z orthogonal to the columns of X,andZ = PXZ + MXZ.
OLS predicted values and residuals. The matrices PX and MX provide some sim- ple expressions for OLS predicted values and residuals. The OLS predicted val- ues, Yn = XBn, and the OLS residuals, Un = Y – Yn, can be expressed as follows (Exercise 18.5):
Yn = PXY and (18.27)
Un = MXY = MXU. (18.28)
The expressions in Equations (18.27) and (18.28) provide a simple proof that
the OLS residuals and predicted values are orthogonal, that is, Equation (4.37)
holds: Yn′Un = Y′P′M Y = 0, where the second equality follows from XX
PX′ MX = 0n * n, which in turn follows from MXX = 0n * (k + 1) in Equation (18.26). The standard error of the regression. The SER, defined in Section 4.3, is sun,
where
2 1an2 1nn1
sun =n-k-1 uni =n-k-1U′U=n-k-1U′MXU, (18.29)
i=1
wherethefinalequalityfollowsbecauseUn′Un = (MXU)′(MXU) = U′MXMXU =
U′MXU (because MX is symmetric and idempotent). Distribution of βn with Normal Errors
Because Bn = B + (X′X)-1X′U [Equation (18.14)] and because the distribu-
tion of U conditional on X is, by assumption, N(0n, s2uIn) [Equation (18.8)], the
covariance matrix of B, conditional on X, is 𝚺 n = E[(B – B)(B – B)′ 0 X] = n B0Xnn
conditional distribution of Bn given X is multivariate normal with mean B. The
E[(X′X)-1 X′UU′X(X′X)-1X] = (X′X)-1X′(su2In)X(X′X)-1 = s2u(X′X)-1.
718 ChapTeR 18 The Theory of Multiple Regression
Accordingly, under all six assumptions in Key Concept 18.1, the finite-sample
conditional distribution of Bn given X is
Bn ∼ N(B, 𝚺Bn 0X ), where 𝚺Bn 0X = s2u(X′X )- 1. (18.30)
Distribution of sun2
If all six assumptions in Key Concept 18.1 hold, then s2un has an exact sampling distribution that is proportional to a chi-squared distribution with n – k – 1 degrees of freedom:
s2 ∼ s2u * x2 (18.31) un n-k-1 n-k-1
The proof of Equation (18.31) starts with Equation (18.29). Because U is normally
the quadratic form U′M U>s has an exact chi-squared distribution with degrees X 2u
distributed conditional on X and because MX is a symmetric idempotent matrix,
of freedom equal to the rank of MX [Equation (18.78) in Appendix 18.2]. From Equation (18.26), the rank of M is n – k – 1. Thus U′M U>s has an exact
X X2u x2n – k – 1 distribution, from which Equation (18.31) follows.
The degrees-of-freedom adjustment ensures that s2 is unbiased. The expecta- 2 un
tion of a random variable with a xn-k-1 distribution is n – k – 1; thus E(U′M U) = (n – k – 1)s2, so E(s2) = s2.
X uunu Homoskedasticity-Only Standard Errors
The homoskedasticity-only estimator 𝚺∼Bn of the covariance matrix of Bn, condi-
tional on X, is obtained by substituting the sample variance s2 for the population
variance s in the expression for 𝚺 n in Equation (18.30). Accordingly, 2u B 0 X un
∼ 2 -1
𝚺Bn = sun(X′X) (homoskedasticity@only). (18.32)
The estimator of the variance of the normal conditional distribution of bnj, given X, is the (j, j) element of 𝚺∼Bn. Thus the homoskedasticity-only standard error of bnj is the square root of the jth diagonal element of 𝚺∼Bn. That is, the homoskedasticity-only standard error of bnj is
SE(b) = 2(𝚺n) (homoskedasticity-only). (18.33) j b jj
∼n∼
18.4 Distribution of Regression Statistics with Normal Errors 719 Distribution of the t-Statistic
(18.34)
Let ∼t be the t-statistic testing the hypothesis bj = bj,0, constructed using the homoskedasticity-only standard error; that is, let
2(𝚺n) b jj
n
∼t = bj – bj,0 . ∼
Under all six of the extended least squares assumptions in Key Concept 18.1, the exact sampling distribution of ∼t is the Student t distribution with n – k – 1 degrees of freedom; that is,
∼t ∼ tn-k-1. (18.35) The proof of Equation (18.35) is given in Appendix 18.4.
Distribution of the F-Statistic
If all six least squares assumptions in Key Concept 18.1 hold, then the F-statistic testing the hypothesis in Equation (18.20), constructed using the homoskedasticity- only estimator of the covariance matrix, has an exact Fq, n – k – 1 distribution under the null hypothesis.
The homoskedasticity-only F-statistic. The homoskedasticity-only F-statistic is
similar to the heteroskedasticity-robust F-statistic in Equation (18.21), except that
the homoskedasticity-only estimator 𝚺∼Bn is used instead of the heteroskedasticity-
robust estimator 𝚺∼ n. Substituting the expression 𝚺∼ n = s2(X′X)-1 into the expres- B Bun
sion for the F-statistic in Equation (18.21) yields the homoskedasticity-only F-statistic testing the null hypothesis in Equation (18.20):
(RB – r)′3R(X′X) R′4 (RB – r)>q
F∼=n -1-1n.(18.36) s2
un
If all six assumptions in Key Concept 18.1 hold, then under the null hypothesis
∼
F ∼ Fq,n-k-1. (18.37)
The proof of Equation (18.37) is given in Appendix 18.4.
720 ChapTeR 18 The Theory of Multiple Regression
The F-statistic in Equation (18.36) is called the Wald version of the F-statistic (named after the statistician Abraham Wald). Although the formula for the homoskedastic-only F-statistic given in Equation (7.13) appears quite different from the formula for the Wald statistic in Equation (18.36), the homoskedastic- only F-statistic and the Wald F-statistic are two versions of the same statistic. That is, the two expressions are equivalent, a result shown in Exercise 18.13.
18.5 Efficiency of the OLS Estimator with Homoskedastic Errors
Under the Gauss–Markov conditions for multiple regression, the OLS estimator of B is efficient among all linear conditionally unbiased estimators; that is, the OLS estimator is BLUE.
The Gauss–Markov Conditions for Multiple Regression
The Gauss–Markov conditions for multiple regression are
(i)E(U0X) = 0 , n
(ii) E(UU′0X) = s I , and 2u n
(iii) X has full column rank.
The Gauss–Markov conditions for multiple regression in turn are implied by the first five assumptions in Key Concept 18.1 [see Equations (18.6) and (18.7)]. The conditions in Equation (18.38) generalize the Gauss–Markov conditions for a sin- gle regressor model to multiple regression. [By using matrix notation, the second and third Gauss–Markov conditions in Equation (5.31) are collected into the sin- gle condition (ii) in Equation (18.38).]
Linear Conditionally Unbiased Estimators
We start by describing the class of linear unbiased estimators and by showing that OLS is in that class.
The class of linear conditionally unbiased estimators. An estimator of B is said to be linear if it is a linear function of Y1, c, Yn. Accordingly, the estimator B∼ is linear in Y if it can be written in the form
∼
B = A′Y, (18.39)
(18.38)
18.5 Efficiency of the OLS Estimator with Homoskedastic Errors 721
where A is an n * (k + 1) dimensional matrix of weights that may depend on X and on nonrandom constants, but not on Y.
distribution, given X, is B. That is, B is conditionally unbiased if E(B 0 X) = B. The OLS estimator is linear and conditionally unbiased. Comparison of Equa-
An estimator is conditionally unbiased if the mean of its conditional sampling
∼∼
tions (18.11) and (18.39) shows that the OLS estimator is linear in Y; specifically, nnn-1 n
B = A′Y, where A = X(X′X) . To show that B is conditionally unbiased, recall n -1
tationofbothsidesofthisexpressionyields,E(B0X) = B +E[(X′X) X′U0X] = -1
from Equation (18.14) that B = B + (X′X) X′U. Taking the conditional expec- n -1
B + (X′X) X′E(U 0 X) = B,wherethefinalequalityfollowsbecauseE(U0X) = 0 by the first Gauss–Markov condition.
The Gauss–Markov Theorem for Multiple Regression
The Gauss–Markov theorem for multiple regression provides conditions under which the OLS estimator is efficient among the class of linear conditionally unbiased estimators. A subtle point arises, however, because Bn is a vector and its “variance” is a covariance matrix. When the “variance” of an estimator is a matrix, just what does it mean to say that one estimator has a smaller variance than another?
The Gauss–Markov theorem handles this problem by comparing the variance of a candidate estimator of a linear combination of the elements of B to the variance of the corresponding linear combination of Bn. Specifically, let c be a k + 1 dimensional vector and consider the problem of estimating the linear combination c′B using the candidate estimator c′B∼ (where B∼ is a linear conditionally unbiased estimator) on the one hand and c′Bn on the other hand. Because c′B∼ and c′Bn are both scalars and are both linear conditionally unbiased estimators of c′B, it now makes sense to compare their variances.
The Gauss–Markov theorem for multiple regression says that the OLS esti- mator of c′B is efficient; that is, the OLS estimator c′Bn has the smallest conditional variance of all linear conditionally unbiased estimators c′B∼. Remarkably, this is true no matter what the linear combination is. It is in this sense that the OLS estimator is BLUE in multiple regression.
The Gauss–Markov theorem is stated in Key Concept 18.3 and proven in Appendix 18.5.
722 ChapTeR 18 The Theory of Multiple Regression
Gauss–Markov Theorem for Multiple Regression
18.3
Key ConCept
Suppose that the Gauss–Markov conditions for multiple regression in Equation
(18.38) hold. Then the OLS estimator Bn is BLUE. That is, let B∼ be a linear con-
∼
vector. Then var(c′B0X) … var(c′B0X) for every nonzero vector c, where the
ditionally unbiased estimator of B and let c be a nonrandom k + 1 dimensional
n
∼
inequality holds with equality for all c only if B = B.
n
18.6 Generalized Least Squares1
The assumption of i.i.d. sampling fits many applications. For example, suppose that
Yi and Xi correspond to information about individuals, such as their earnings, edu-
cation, and personal characteristics, where the individuals are selected from a
population by simple random sampling. In this case, because of the simple random
sampling scheme, (Xi,Yi) are necessarily i.i.d. Because (Xi,Yi) and (Xj,Yj) are inde-
Some sampling schemes encountered in econometrics do not, however, result in independent observations and instead can lead to error terms ui that are cor- related from one observation to the next. The leading example is when the data are sampled over time for the same entity, that is, when the data are time series data. As discussed in Section 15.3, in regressions involving time series data, many omitted factors are correlated from one period to the next, and this can result in regression error terms (which represent those omitted factors) that are correlated from one period of observation to the next. In other words, the error term in one period will not, in general, be distributed independently of the error term in the
1The GLS estimator was introduced in Section 15.5 in the context of distributed lag time series regres- sion. This presentation here is a self-contained mathematical treatment of GLS that can be read inde- pendently of Section 15.5, but reading that section first will help to make these ideas more concrete.
pendently distributed for i ≠ j, u and u are independently distributed for i ≠ j. ij
This in turn implies that ui and uj are uncorrelated for i ≠ j. In the context of the Gauss–Markov assumptions, the assumption that E(UU ′ 0 X ) is diagonal therefore is appropriate if the data are collected in a way that makes the observations inde- pendently distributed.
next period. Instead, the error term in one period could be correlated with the error term in the next period.
The presence of correlated error terms creates two problems for inference based on OLS. First, neither the heteroskedasticity-robust nor the homoskedasticity-only standard errors produced by OLS provide a valid basis for inference. The solution to this problem is to use standard errors that are robust to both heteroskedasticity and correlation of the error terms across observations. This topic—heteroskedasticity- and autocorrelation-consistent (HAC) covariance matrix estimation—is the subject of Section 15.4 and we do not pursue it further here.
Second, if the error term is correlated across observations, then E(UU′X) is not diagonal, the second Gauss–Markov condition in Equation (18.38) does not hold, and OLS is not BLUE. In this section we study an estimator, generalized least squares (GLS), that is BLUE (at least asymptotically) when the condi- tional covariance matrix of the errors is no longer proportional to the identity matrix. A special case of GLS is weighted least squares, discussed in Section 17.5, in which the conditional covariance matrix is diagonal and the ith diagonal ele- ment is a function of Xi. Like WLS, GLS transforms the regression model so that the errors of the transformed model satisfy the Gauss–Markov conditions. The GLS estimator is the OLS estimator of the coefficients in the transformed model.
The GLS Assumptions
There are four assumptions under which GLS is valid. The first GLS assumption is that u has a mean of zero, conditional on X , c, X ; that is,
i1n E(U0X) = 0 .
(18.40) This assumption is implied by the first two least squares assumptions in Key Concept
18.1; that is, if E(u 0X) = 0 and (X,Y), i = 1,c, n, are i.i.d., then E(U0X) = 0 . iiiin
In GLS, however, we will not want to maintain the i.i.d. assumption; after all, one purpose of GLS is to handle errors that are correlated across observations. We dis- cuss the significance of the assumption in Equation (18.40) after introducing the GLS estimator.
The second GLS assumption is that the conditional covariance matrix of U given X is some function of X:E(UU′ 0 X ) = 𝛀(X ), (18.41)
where 𝛀(X) is an n * n positive definite matrix-valued function of X.
18.6 Generalized Least Squares 723
n
724 ChapTeR 18 The Theory of Multiple Regression
The GLS assumptions
18.4
Key ConCept
In the linear regression model Y = XB + U, the GLS assumptions are 1. E(U0X) = 0 ;
n
2. E(UU′ 0 X) = 𝛀(X), where 𝛀(X) is an n * n positive definite matrix that can
depend on X;
3. Xi and ui satisfy suitable moment conditions; and
4. X has full column rank (there is no perfect multicollinearity).
There are two main applications of GLS that are covered by this assumption. The first is independent sampling with heteroskedastic errors, in which case 𝛀(X) is a diagonal matrix with diagonal element lh(Xi), where l is a constant and h is a function. In this case, discussed in Section 17.5, GLS is WLS.
The second application is to homoskedastic errors that are serially correlated. In practice, in this case a model is developed for the serial correlation. For exam- ple, one model is that the error term is correlated with only its neighbor, so corr(ui, ui – 1) = r ≠ 0 but corr(ui, uj) = 0 if i – j Ú 2. In this case, 𝛀(X ) has s2u as its diagonal element, rs2u in the first off-diagonal, and zeros elsewhere. Thus 𝛀(X) does not depend on X, 𝛀ii = s2u, 𝛀ij = rs2u for i – j = 1, and 𝛀ij = 0 for i – j 7 1. Other models for serial correlation, including the first order autoregressive model, are discussed further in the context of GLS in Section 15.5 (also see Exercise 18.8).
One assumption that has appeared on all previous lists of least squares assump- tions for cross-sectional data is that Xi and ui have nonzero, finite fourth moments. In the case of GLS, the specific moment assumptions needed to prove asymptotic results depend on the nature of the function 𝛀(X), whether 𝛀(X) is known or estimated, and the statistic under consideration (the GLS estimator, t-statistic, etc.). Because the assumptions are case- and model-specific, we do not present specific moment assumptions here, and the discussion of the large-sample properties of GLS assumes that such moment conditions apply for the relevant case at hand. For completeness, as the third GLS assumption, Xi and ui are simply assumed to satisfy suitable moment conditions.
The fourth GLS assumption is that X has full column rank; that is, the regres- sors are not perfectly multicollinear.
The GLS assumptions are summarized in Key Concept 18.4.
We consider GLS estimation in two cases. In the first case, 𝛀(X) is known. In the second case, the functional form of 𝛀(X) is known up to some parameters that can be estimated. To simplify notation, we refer to the function 𝛀(X) as the matrix 𝛀, so the dependence of 𝛀 on X is implicit.
GLS When Ω Is Known
When 𝛀 is known, the GLS estimator uses 𝛀 to transform the regression model to one with errors that satisfy the Gauss–Markov conditions. Specifically, let F be a matrix square root of 𝛀-1; that is, let F be a matrix that satisfies F′F = 𝛀-1 (see Appendix 18.1). A property of F is that F𝛀F′ = In. Now premultiply both sides of Equation (18.4) by F to obtain
∼∼∼
of zero and a covariance matrix that equals the identity matrix. To show this
n the first GLS assumption [Equation (18.40)]. In addition, E(UU′0X) =
∼∼
∼∼∼ E[(FU)(FU)′0FX] = FE(UU′0FX)F′ = F𝛀F′ = I ,wherethesecondequality
follows because (FU)′ = U′F′ and the final equality follows from the definition of F. It follows that the transformed regression model in Equation (18.42) satisfies the Gauss–Markov conditions in Key Concept 18.3.
The GLS estimator, B∼GLS, is the OLS estimator of B in Equation (18.42); ∼GLS ∼∼-1 ∼∼
that is, B = (X′X) (X′Y). Because the transformed regression model satis- fies the Gauss–Markov conditions, the GLS estimator is the best conditionally unbiased estimator that is linear in Y∼. But because Y∼ = FY and F is (here) assumed to be known, and because F is invertible (because 𝛀 is positive defi- nite), the class of estimators that are linear in Y∼ is the same as the class of estimators that are linear in Y. Thus the OLS estimator of B in Equation (18.42) is also the best conditionally unbiased estimator among estimators that are lin- ear in Y. In other words, under the GLS assumptions, the GLS estimator is BLUE.
18.6 Generalized Least Squares 725
Y = XB + U, (18.42)
where∼Y =FY,X∼=FX,andU∼=FU.
The key insight of GLS is that, under the four GLS assumptions, the Gauss–
Markov assumptions hold for the transformed regression in Equation (18.42).
That is, by transforming all the variables by the inverse of the matrix square root
of 𝛀, the regression errors in the transformed regression have a conditional mean
mathematically, first note that E(U0X) = E(FU0FX) = FE(U0FX) = 0 by
n
726 ChapTeR 18 The Theory of Multiple Regression
The GLS estimator can be expressed directly in terms of 𝛀, so in principle
(18.43)
In practice, 𝛀 is typically unknown, so the GLS estimator in Equation (18.43) typically cannot be computed and thus is sometimes called the infeasible GLS estimator. If, however, 𝛀 has a known functional form but the parameters of that function are unknown, then 𝛀 can be estimated and a feasible version of the GLS estimator can be computed.
GLS When Ω Contains Unknown Parameters
If 𝛀 is a known function of some parameters that in turn can be estimated, then
these estimated parameters can be used to calculate an estimator of the covari-
there is no need to compute the square root matrix F. Because X∼ = FX and ∼∼GLS-1 -1
Y = FY, B = (X′F′FX) (X′F′FY). But F′F = 𝛀 , so ∼GLS -1-1-1
B = (X′𝛀 X) (X′𝛀 Y).
ance matrix 𝛀. For example, consider the time series application discussed fol-
rs for0i-j0=1,and𝛀 =0for0i-j071.Then𝛀hastwounknown 2u ij
parameters, s2u and r. These parameters can be estimated using the residuals from a preliminary OLS regression; specifically, s2u can be estimated by su2 and r can
be estimated by the sample correlation between all neighboring pairs of OLS residuals. These estimated parameters can in turn be used to compute an estima- tor of 𝛀, 𝛀n .
In general, suppose that you have an estimator 𝛀n of 𝛀. Then the GLS esti- mator based on 𝛀n is
BnGLS = (X′𝛀n-1X)-1(X′𝛀n-1Y). (18.44)
The GLS estimator in Equation (18.44) is sometimes called the feasible GLS estimator because it can be computed if the covariance matrix contains some unknown parameters that can be estimated.
lowing Equation (18.41), in which 𝛀(X) does not depend on X, 𝛀ii = s2u, 𝛀ij =
The Zero Conditional Mean Assumption and GLS
hold; that is, E(u 0 X ) must be zero. In contrast, the first GLS assumption is that
for the ith observation has a conditional mean of zero, given the values of the regressors for that observation, whereas the first GLS assumption is that ui has a conditional mean of zero, given the values of the regressors for all observations.
For the OLS estimator to be consistent, the first least squares assumption must
E(u 0 X , c, X )i = i 0. In other words, the first OLS assumption is that the error i1n
n
18.6 Generalized Least Squares 727 As discussed in Section 18.1, the assumptions that E(u 0 X ) = 0 and that sam-
ii
pling is i.i.d. together imply that E(u 0 X , c, X ) = 0. Thus, when sampling is
i1n
i.i.d. so that GLS is WLS, the first GLS assumption is implied by the first least
squares assumption in Key Concept 18.1.
When sampling is not i.i.d., however, the first GLS assumption is not implied
by the assumption that E(u 0 X ) = 0; that is, the first GLS assumption is stronger. ii
Although the distinction between these two conditions might seem slight, it can
be very important in applications to time series data. This distinction is discussed
enous or “strictly” exogenous; the assumption that E(u 0 X , c, X ) = 0 corre- i1n
sponds to strict exogeneity. Here, we discuss this distinction at a more general level using matrix notation. To do so, we focus on the case that U is homoskedastic, 𝛀 is known, and 𝛀 has nonzero off-diagonal elements.
The role of the first GLS assumption. To see the source of the difference between these assumptions, it is useful to contrast the consistency arguments for GLS and OLS.
in Section 15.5 in the context of whether the regressor is “past and present” exog-
We first sketch the argument for the consistency of the GLS estimator in Equa- tion(18.43).SubstitutingEquation(18.4)intoEquation(18.43),wehaveBGLS =
-1 -1 -1 -1
B + (X′𝛀 X>n) (X′𝛀 U>n). Under the first GLS assumption, E(X′𝛀∼U) =
-1n -1
E3X′𝛀 E(U 0 X )4 = 0 . If in addition the variance of X′𝛀 U>n tends to zero
-1 p ∼ ∼ ∼GLS p
and X′𝛀 X>n ¡ Q, where Q is some invertible matrix, then B ¡ B.
Critically, when 𝛀 has off-diagonal elements, the term X′𝛀-1U = g g X(𝛀 ) u involvesproductsofX andu fordifferenti,j,where(𝛀 )
ni=1 nj=1i -1ijj i j -1ij denotes the (i, j) element of 𝛀-1. Thus, for X′𝛀-1U to have a mean of zero, it is
not enough that E(u 0X) = 0; rather E(u 0X) must equal zero for all i, j pairs ii ij
corresponding to nonzero values of (𝛀-1)ij. Depending on the covariance structure
of the errors, only some of or all the elements of (𝛀-1)ij might be nonzero. For
the only nonzero elements (𝛀 ) are those for which 0 i – j 0 … 1. In general, however,
example, if ui follows a first order autoregression (as discussed in Section 15.5),
p
all the elements of 𝛀 can be nonzero, so in general for X′Ω U>n ¡ 0
-1 ij
-1 -1
(k + 1)*1 (andthusforB tobeconsistent)weneedthatE(U0X) = 0 ;thatis,thefirstGLS
∼GLS assumption must hold.
n
Equation (18.14) as B = B + (X′X>n) ng 1n
In contrast, recall the argument that the OLS estimator is consistent. Rewrite
n
Xu. If E(u X) = 0, then the i=1ii ii
i=1 i i
converges in probability to zero. If in addition X′X>n ¡ Q , then B ¡ B.
-11 n
term n g X u has mean zero, and if this term has a variance that tends to zero, it
IsthefirstGLSassumptionrestrictive? ThefirstGLSassumptionrequiresthatthe errors for the ith observation be uncorrelated with the regressors for all other
pnp X
728 ChapTeR 18 The Theory of Multiple Regression
observations. This assumption is dubious in some time series applications. This issue is discussed in Section 15.6 in the context of an empirical example, the rela- tionship between the change in the price of a contract for future delivery of frozen orange concentrate and the weather in Florida. As explained there, the error term in the regression of price changes on the weather is plausibly uncorrelated with current and past values of the weather, so the first OLS assumption holds. How- ever, this error term is plausibly correlated with future values of the weather, so the first GLS assumption does not hold.
This example illustrates a general phenomenon in economic time series data that arises when the value of a variable today is set in part based on expectations of the future: Those future expectations typically imply that the error term today depends on a forecast of the regressor tomorrow, which in turn is correlated with the actual value of the regressor tomorrow. For this reason, the first GLS assump- tion is in fact much stronger than the first OLS assumption. Accordingly, in some applications with economic time series data the GLS estimator is not consistent even though the OLS estimator is.
18.7 Instrumental Variables and Generalized Method of Moments Estimation
This section provides an introduction to the theory of instrumental variables (IV) estimation and the asymptotic distribution of IV estimators. It is assumed through- out that the IV regression assumptions in Key Concepts 12.3 and 12.4 hold and, moreover, that the instruments are strong. These assumptions apply to cross- sectional data with i.i.d. observations. Under certain conditions the results derived in this section are applicable to time series data as well, and the extension to time series data is briefly discussed at the end of this section. All asymptotic results in this section are developed under the assumption of strong instruments.
This section begins by presenting the IV regression model, the two stage least squares (TSLS) estimator, and its asymptotic distribution in the general case of heteroskedasticity, all in matrix form. It is next shown that, in the special case of homoskedasticity, the TSLS estimator is asymptotically efficient among the class of IV estimators in which the instruments are linear combinations of the exoge- nous variables. Moreover, the J-statistic has an asymptotic chi-squared distribu- tion in which the degrees of freedom equal the number of overidentifying restrictions. This section concludes with a discussion of efficient IV estimation and the test of overidentifying restrictions when the errors are heteroskedastic—a situation in which the efficient IV estimator is known as the efficient generalized method of moments (GMM) estimator.
18.7 Instrumental Variables and Generalized Method of Moments Estimation 729 The IV Estimator in Matrix Form
In this section, we let X denote the n * (k + r + 1) matrix of the regressors in the equation of interest, so X contains the included endogenous regressors (the X’s in Key Concept 12.1) and the included exogenous regressors (the W’s in Key Concept 12.1). That is, in the notation of Key Concept 12.1, the ith row of X is X′i = (1 X1i X2i . . . Xki W1i W2i . . . Wri). Also, let Z denote the n * (m + r + 1) matrix of all the exogenous regressors, both those included in the equation of interest (the W’s) and those excluded from the equation of interest (the instruments). That is, in the notation of Key Concept 12.1, the ith rowofZisZ′i = (1 Z1i Z2i … Zmi W1i W2i … Wri).
With this notation, the IV regression model of Key Concept 12.1, written in matrix form, is
Y = XB + U, (18.45)
where U is the n * 1 vector of errors in the equation of interest, with ith element ui. The matrix Z consists of all the exogenous regressors, so under the IV regres-
sion assumptions in Key Concept 12.4,
E(Ziui) = 0 (instrument exogeneity). (18.46)
Because there are k included endogenous regressors, the first stage regression consists of k equations.
The TSLS estimator. The TSLS estimator is the instrumental variables estimator in which the instruments are the predicted values of X based on OLS estimation of the first stage regression. Let Xn denote this matrix of predicted values so that the ith row of Xn is (Xn1i Xn2i c Xnki W1i W2i c Wri), where Xn1i is the predicted value from the regression of X1i on Z, and so forth. Because the W’s are contained in Z, the predicted value from a regression of W1i on Z is just W1i, and so forth, so Xn = PZX, where PZ = Z(Z′Z)−1Z′ [see Equation (18.27)]. Accordingly, the TSLS estimator is
nTSLS n n -1 n
B = (X′X) X′Y. (18.47)
nTSLS -1
B = (X′PZX) X′PZY. (18.48)
nnnn
Because X = PZX, X′X = X′PZX, and X′Y = X′PZY, the TSLS estimator can be rewritten as
730 ChapTeR 18 The Theory of Multiple Regression
Asymptotic Distribution of the TSLS Estimator
Substituting Equation (18.45) into Equation (18.48), rearranging, and multiplying by 2n yields the expression for the centered and scaled TSLS estimator:
n TSLS
2n(B – B) = a
b
-1 2n 2n
X′PZX X′PZU
X′Z Z′Z Z′X X′Z Z′Z Z′U =cnab-1 d-1cab-1 d,(18.49)
nnnnn
assumptions, X′Z>n ¡ Q and Z′Z>n ¡ Q , where Q = E(X Z′) and XZ ZZXZii
where the second equality uses the definition of PZ. Under the IV regression pp
with mean zero [Equation (18.46)] and a nonzero finite variance, so its sum, divided by 2n, satisfies the conditions of the central limit theorem and
2
Q = E(Z Z′). In addition, under the IV regression assumptions, Z u is i.i.d. ZZ ii ii
d
Z′U> 2n ¡ 𝚿 , where 𝚿 ∼ N(0, H), H = E(Z Z′u ) (18.50)
ZU ZU iii
and 𝚿ZU is (m + r + 1) * 1.
Application of Equation (18.50) and of the limits X′Z>n ¡ Q and
p
Z′Z>n ¡ Q to Equation (18.49) yields the result that, under the IV regres-
2n(B -B)¡(Q Q Q ) Q Q Ψ ∼N(0,𝚺 ), (18.51) XZ ZZ ZX XZ ZZ ZU
p
XZ
ZZ
sion assumptions, the TSLS estimator is asymptotically normally distributed: nTSLS d -1 -1 -1 TSLS
where
𝚺TSLS = (Q Q-1 Q )-1Q Q-1 HQ-1 Q (Q Q-1 Q )-1, (18.52)
XZ ZZ ZX XZ ZZ ZZ ZX XZ ZZ ZX where H is defined in Equation (18.50).
StandarderrorsforTSLS. TheformulainEquation(18.52)isdaunting.Neverthe- less, it provides a way to estimate 𝚺TSLS by substituting sample moments for the population moments. The resulting variance estimator is
𝚺n TSLS = (Qn Qn-1 Qn )-1Qn Qn-1 HnQn-1 Qn (Qn Qn-1 Qn )-1, (18.53) XZ ZZ ZX XZ ZZ ZZ ZX XZ ZZ ZX
nXZ nZZ nZX
where Q = X′Z>n, Q = Z′Z>n, Q = Z′X>n, and
Hn = n1 an ZiZi un2i , where Un = Y – XBnTSLS (18.54) i=1
18.7 Instrumental Variables and Generalized Method of Moments Estimation 731
so that Un is the vector of TSLS residuals and where uni is the ith element of that vector (the TSLS residual for the ith observation).
n TSLS
𝚺 >n.
The TSLS standard errors are the square roots of the diagonal elements of
Properties of TSLS When the Errors Are Homoskedastic
If the errors are homoskedastic, then the TSLS estimator is asymptotically effi- cient among the class of IV estimators in which the instruments are linear combi- nations of the rows of Z. This result is the IV counterpart to the Gauss–Markov theorem and constitutes an important justification for using TSLS.
TheTSLSdistributionunderhomoskedasticity. Iftheerrorsarehomoskedastic,thatis, if E(u 0 Z ) = s , then H = E(Z Z′u ) = E[E(Z Z′u Z )] = E[Z Z′E(u Z )] =
2ii2u ii2iii2iiii2ii QZZs2u. In this case, the variance of the asymptotic distribution of the TSLS estimator in Equation (18.52) simplifies to
𝚺TSLS = (Q Q-1 Q )-1s2 (homoskedasticity only). XZ ZZ ZX u
The homoskedasticity-only estimator of the TSLS variance matrix is
(18.55)
nn 𝚺∼TSLS = (Qn Qn-1 Qn )-1sn2, where sn2 = U′U
XZ ZZ ZX u u n-k-r-1 (homoskedasticity only),
(18.56) and the homoskedasticity-only TSLS standard errors are the square root of the
∼ TSLS diagonal elements of 𝚺 >n.
The class of IV estimators that use linear combinations of Z. The class of IV estimators that use linear combinations of Z as instruments can be generated in two equivalent ways. Both start with the same moment equation: Under the assumption of instrument exogeneity, the errors U = Y – XB are uncorrelated with the exogenous regressors; that is, at the true value of B, Equation (18.46) implies that
E[(Y – XB)′Z] = 0. (18.57)
Equation (18.57) constitutes a system of m + r + 1 equations involving the k + r + 1 unknown elements of B. When m 7 k, these equations are redundant,
732 ChapTeR 18 The Theory of Multiple Regression
in the sense that all are satisfied at the true value of B. When these population moments are replaced by their sample moments, the system of equations (Y – Xb)′Z = 0 can be solved for b when there is exact identification (m = k). This value of b is the IV estimator of B. However, when there is overidentification (m 7 k), the system of equations typically cannot all be satisfied by the same value of b because of sampling variation—there are more equations than unknowns—and in general this system does not have a solution.
The first approach to the problem of estimating B when there is overidentifica-
tion is to trade off the desire to satisfy each equation by minimizing a quadratic form
involving all the equations. Specifically, let A be an (m + r + 1) * (m + r + 1)
symmetric positive semidefinite weight matrix and let BnIV denote the estimator that A
minimizes
minb(Y – Xb)′ZAZ′(Y – Xb). (18.58)
The solution to this minimization problem is found by taking the derivative of the
objective function with respect to b, setting the result equal to zero, and rearrang-
ing. Doing so yields BnIV, the IV estimator based on the weight matrix A: A
BnIV = (X′ZAZ′X)-1X′ZAZ′Y. (18.59) A
Comparison of Equations (18.59) and (18.48) shows that TSLS is the IV estimator with A = (Z′Z)-1. That is, TSLS is the solution of the minimization problem in Equation (18.58) with A = (Z′Z)-1.
The calculations leading to Equations (18.51) and (18.52), applied to BA , show that
nIV d IV
2n(B – B) ¡ N(0, 𝚺 ), where nIV
AA
𝚺IV = (Q AQ )-1Q AHAQ (Q AQ )-1.
(18.60)
A XZ ZX XZ ZX XZ ZX
The second way to generate the class of IV estimators that use linear combinations of Z is to consider IV estimators in which the instruments are ZB, where B is an (m + r + 1) * (k + r + 1)matrixwithfullrowrank.Thenthesystemof(k + r + 1) equations, (Y − Xb)′ZB = 0, can be solved uniquely for the (k + r + 1) unknown elements of b. Solving these equations for b yields BnIV = (B′Z′X)−1(B′Z′Y), and substitution of B = AZ′X into this expression yields Equation (18.59). Thus the two approaches to defining IV estimators that are linear combinations of the instruments yield the same family of IV estimators. It is conventional to work with the first approach, in which the IV estimator solves the quadratic minimization problem in Equation (18.58), and that is the approach taken here.
18.7 Instrumental Variables and Generalized Method of Moments Estimation 733 Asymptotic efficiency of TSLS under homoskedasticity. If the errors are homo-
skedastic, then H = Q s2 and the expression for 𝚺IV in Equation (18.60) ZZ u A
becomes
𝚺IV = (Q AQ )-1Q AQ AQ (Q AQ )-1s2. (18.61) A XZ ZX XZ ZZ ZX XZ ZX u
To show that TSLS is asymptotically efficient among the class of estimators that are linear combinations of Z when the errors are homoskedastic, we need to show that, under homoskedasticity,
c′𝚺IVc Ú c′𝚺TSLSc (18.62) A
for all positive semidefinite matrices A and all (k + r + 1) * 1 vectors c, where 𝚺TSLS = (Q Q-1 Q )-1s2 [Equation (18.55)]. The inequality (18.62), which is
XZ ZZ ZX u
proven in Appendix 18.6, is the same efficiency criterion as is used in the multi-
variate Gauss–Markov theorem in Key Concept 18.3. Consequently, TSLS is the efficient IV estimator under homoskedasticity, among the class of estimators in which the instruments are linear combinations of Z.
The J-statistic under homoskedasticity. The J-statistic (Key Concept 12.6) tests the null hypothesis that all the overidentifying restrictions hold against the alter- native that some or all of them do not hold.
The idea of the J-statistic is that, if the overidentifying restrictions hold, ui will be uncorrelated with the instruments and thus a regression of U on Z will have population regression coefficients that all equal zero. In practice, U is not observed, but it can be estimated by the TSLS residuals Un , so a regression of Un on Z should yield statistically insignificant coefficients. Accordingly, the TSLS J-statistic is the homoskedasticity-only F-statistic testing the hypothesis that the coefficients on Z are all zero, in the regression of Un on Z, multiplied by (m + r + 1) so that the F-statistic is in its asymptotic chi-squared form.
An explicit formula for the J-statistic can be obtained using Equation (7.13)
for the homoskedasticity-only F-statistic. The unrestricted regression is the regres-
sion of Un on the m + r + 1 regressors Z, and the restricted regression has no nn
regressors. Thus, in the notation of Equation (7.13), SSRunrestricted = U′MZU and nn nnnnnn
SSRrestricted = U′U, so SSRrestricted – SSRunrestricted = U′U – U′MZU = U′PZU and the J-statistic is
nZn
U′M U>(n – m – r – 1)
J = Un′PZUn . (18.63)
734 ChapTeR 18 The Theory of Multiple Regression
The method for computing the J-statistic described in Key Concept 12.6 entails testing only the hypothesis that the coefficients on the excluded instru- ments are zero. Although these two methods have different computational steps, they produce identical J-statistics (Exercise 18.14).
It is shown in Appendix 18.6 that, under the null hypothesis that E(uiZi) = 0, J¡d x2 . (18.64)
Generalized Method of Moments Estimation
in Linear Models
If the errors are heteroskedastic, then the TSLS estimator is no longer efficient among the class of IV estimators that use linear combinations of Z as instruments. The efficient estimator in this case is known as the efficient generalized method of moments (GMM) estimator. In addition, if the errors are heteroskedastic, then the J-statistic as defined in Equation (18.63) no longer has a chi-squared distribution. However, an alternative formulation of the J-statistic, constructed using the efficient GMM estimator, does have a chi-squared distribution with m − k degrees of freedom.
These results parallel the results for the estimation of the usual regression model with exogenous regressors and heteroskedastic errors: If the errors are heteroskedastic, then the OLS estimator is not efficient among estimators that are linear in Y (the Gauss–Markov conditions are not satisfied) and the homoskedasticity- only F-statistic no longer has an F distribution, even in large samples. In the regres- sion model with exogenous regressors and heteroskedasticity, the efficient estimator is weighted least squares; in the IV regression model with heteroskedasticity, the efficient estimator uses a different weighting matrix than TSLS, and the resulting estimator is the efficient GMM estimator.
GMM estimation. Generalized method of moments (GMM) estimation is a gen- eral method for the estimation of the parameters of linear or nonlinear models, in which the parameters are chosen to provide the best fit to multiple equations, each of which sets a sample moment to zero. These equations, which in the con- text of GMM are called moment conditions, typically cannot all be satisfied simultaneously. The GMM estimator trades off the desire to satisfy each of the equations by minimizing a quadratic objective function.
In the linear IV regression model with exogenous variables Z, the class of GMM estimators consists of all the estimators that are solutions to the quadratic minimization problem in Equation (18.58). Thus the class of GMM estimators based on the full set of instruments Z with different-weight matrices A is the same as the class of IV estimators in which the instruments are linear combinations of Z.
m-k
18.7 Instrumental Variables and Generalized Method of Moments Estimation 735 In the linear IV regression model, GMM is just another name for the class of
estimators we have been studying—that is, estimators that solve Equation (18.58).
TheasymptoticallyefficientGMMestimator. AmongtheclassofGMMestimators, the efficient GMM estimator is the GMM estimator with the smallest asymptotic variance matrix [where the smallest variance matrix is defined as in Equation (18.62)]. Thus the result in Equation (18.62) can be restated as saying that TSLS is the efficient GMM estimator in the linear model when the errors are homoskedastic.
To motivate the expression for the efficient GMM estimator when the errors are heteroskedastic, recall that when the errors are homoskedastic, H [the vari- ance matrix of Ziui; see Equation (18.50)] equals QZZs2u, and the asymptotically efficient weight matrix is obtained by setting A = (Z′Z)-1, which yields the TSLS estimator. In large samples, using the weight matrix A = (Z′Z)-1 is equivalent to using A = (QZZs2u)-1 = H -1. This interpretation of the TSLS estimator suggests that, by analogy, the efficient IV estimator under heteroskedasticity can be obtained by setting A = H -1 and solving
minb(Y – Xb)′ZH-1Z′(Y – Xb). (18.65) This analogy is correct: The solution to the minimization problem in Equation
∼Eff.GMM
(18.65) is the efficient GMM estimator. Let B denote the solution to the
minimization problem in Equation (18.65). By Equation (18.59), this estimator is ∼Eff.GMM -1 -1 -1
B = (X′ZH Z′X) X′ZH Z′Y. (18.66) The asymptotic distribution of BEff.GMM is obtained by substituting A = H -1 into
Equation (18.60) and simplifying; thus
2n(B ∼ -B)¡N(0,𝚺 where 𝚺Eff.GMM = (QXZH-1QZX)-1.
∼Eff.GMM d Eff.GMM
),
The result that B is the efficient GMM estimator is proven by showing that
(18.67) c′𝚺IVc Ú c′𝚺Eff.GMMc for all vectors c, where 𝚺IV is given in Equation (18.60).
∼Eff.GMM AA
The proof of this result is given in Appendix 18.6.
Feasible efficient GMM estimation. The GMM estimator defined in Equation (18.66) is not a feasible estimator because it depends on the unknown variance matrix H. However, a feasible efficient GMM estimator can be computed by sub- stituting a consistent estimator of H into the minimization problem of Equation (18.65) or, equivalently, by substituting a consistent estimator of H into the for- mula for BnEff.GMM in Equation (18.66).
736 ChapTeR 18 The Theory of Multiple Regression
The efficient GMM estimator can be computed in two steps. In the first step, estimate B using any consistent estimator. Use this estimator of B to compute the residuals from the equation of interest, and then use these residuals to compute an estimator of H. In the second step, use this estimator of H to estimate the optimal weight matrix H−1 and to compute the efficient GMM estimator. To be concrete, in the linear IV regression model, it is natural to use the TSLS estimator in the first step and to use the TSLS residuals to estimate H. If TSLS is used in the first step, then the feasible efficient GMM estimator computed in the second step is
Because H ¡ H,2n(B
n Eff.GMM
n 2n (B
BnEff.GMM = (X′ZHn -1Z′X)-1X′ZHn -1Z′Y, (18.68)
where H is given in Equation (18.54). ∼
n p nEff.GMM Eff.GMM p
– B ) ¡ 0 (Exercise 18.12), and d Eff.GMM
– B) ¡ N(0, 𝚺 ), (18.69)
where 𝚺Eff.GMM = (QXZH -1QZX)-1 [Equation (18.67)]. That is, the feasible two- step estimator BnEff.GMM in Equation (18.68) is, asymptotically, the efficient GMM estimator.
The heteroskedasticity-robust J-statistic. The heteroskedasticity-robust J-statistic, also known as the GMM J-statistic, is the counterpart of the TSLS- based J-statistic, computed using the efficient GMM estimator and weight function. That is, the GMM J-statistic is given by
JGMM = (Z′UnGMM)′Hn -1(Z′UnGMM)>n, (18.70) where Un GMM = Y – XBnEff.GMM are the residuals from the equation of interest,
estimated by (feasible) efficient GMM, and Hn -1 is the weight matrix used to com- pute BnEff.GMM.
Under the null hypothesis E(Ziui) = 0, JGMM ¡d x2m-k (see Appendix 18.6).
GMMwithtimeseriesdata. TheresultsinthissectionwerederivedundertheIV regression assumptions for cross–sectional data. In many applications, however, these results extend to time series applications of IV regression and GMM. Although a formal mathematical treatment of GMM with time series data is beyond the scope of this book (for such a treatment, see Hayashi, 2000, Chapter 6), we nevertheless will summarize the key ideas of GMM estimation with time series data. This summary assumes familiarity with the material in Chapters 14 and 15. For this discussion, it is assumed that the variables are stationary.
It is useful to distinguish between two types of applications: applications in which the error term ut is serially correlated and applications in which ut is serially uncorrelated. If the error term ut is serially correlated, then the asymptotic distri- bution of the GMM estimator continues to be normally distributed, but the for- mula for H in Equation (18.50) is no longer correct. Instead, the correct expression for H depends on the autocovariances of Ztut and is analogous to the formula given in Equation (15.14) for the variance of the OLS estimator when the error term is serially correlated. The efficient GMM estimator is still constructed using a consistent estimator of H; however, that consistent estimator must be computed using the HAC methods discussed in Chapter 15.
If the error term ut is not serially correlated, then HAC estimation of H is unnecessary and the formulas presented in this section all extend to time series GMM applications. In modern applications to finance and macroeconometrics, it is common to encounter models in which the error term represents an unexpected or unforecastable disturbance, in which case the model implies that ut is serially uncorrelated. For example, consider a model with a single included endogenous variable and no included exogenous variables so that the equation of interest is Yt = b0 + b1Xt + ut. Suppose that an economic theory implies that ut is unpre- dictable given past information. Then the theory implies the moment condition
E(u0Y ,X ,Z ,Y ,X ,Z ,c) = 0, (18.71) t t-1 t-1 t-1 t-2 t-2 t-2
where Zt−1 is the lagged value of some other variable. The moment condition in
Equation (18.71) implies that all the lagged variables Yt – 1, Xt – 1, Zt – 1, Yt – 2, Xt – 2,
Zt – 2, care candidates for being valid instruments (they satisfy the exogeneity
tion in Equation (18.71) is equivalent to E(u0u ,X ,Z ,u ,X , t t-1 t-1 t-1 t-2 t-2
condition). Moreover, because ut – 1 = Yt – 1 – b0 – b1Xt – 1, the moment condi-
Zt – 2, c) = 0. Because ut is serially uncorrelated, HAC estimation of H is unnecessary. The theory of GMM presented in this section, including efficient GMM estimation and the GMM J-statistic, therefore applies directly to time series applications with moment conditions of the form in Equation (18.71), under the hypothesis that the moment condition in Equation (18.71) is, in fact, correct.
Summary
1. The linear multiple regression model in matrix form is Y = XB + U, where Y is the n * 1 vector of observations on the dependent variable, X is the n * (k + 1) matrix of n observations on the k + 1 regressors (including a constant), B is the k + 1 vector of unknown parameters, and U is the n * 1 vector of error terms.
Summary 737
738 ChapTeR 18 The Theory of Multiple Regression
n -1
2. The OLS estimator is B = (X′X) X′Y. Under the first four least squares
assumptions in Key Concept 18.1, Bn is consistent and asymptotically nor-
mally distributed. If in addition the errors are homoskedastic, then the con- n n 2 -1
ditional variance of B is var(B X) = su(X′X) .
3. General linear restrictions on B can be written as the q equations RB = r,
and this formulation can be used to test joint hypotheses involving multiple
coefficients or to construct confidence sets for elements of B.
4. When the regression errors are i.i.d. and normally distributed, condi- tional on X, B has an exact normal distribution and the homoskedasticity- only t- and F-statistics have exact tn-k-1and Fq,n-k-1 distributions,
respectively.
5. The Gauss–Markov theorem says that, if the errors are homoskedastic and
conditionally uncorrelated across observations and if E(ui|X) = 0, the OLS estimator is efficient among linear conditionally unbiased estimators (that is, OLS is BLUE).
6. If the error covariance matrix 𝛀 is not proportional to the identity matrix, and if 𝛀 is known or can be estimated, then the GLS estimator is asymp- totically more efficient than OLS. However, GLS requires that, in general, ui be uncorrelated with all observations on the regressors, not just with Xi, as is required by OLS, an assumption that must be evaluated carefully in applications.
7. The TSLS estimator is a member of the class of GMM estimators of the linear model. In GMM, the coefficients are estimated by mak- ing the sample covariance between the regression error and the exogenous variables as small as possible—specifically, by solving
min 3(Y – Xb)′Z4A3Z′(Y – Xb)4, where A is a weight matrix. The
b
asymptotically efficient GMM estimator sets A = 3E(Z Z u )4 . When the errors are homoskedastic, the asymptotically efficient GMM estima- tor in the linear IV regression model is TSLS.
Key Terms
Gauss–Markov conditions for multiple regression (720)
Gauss–Markov theorem for multiple regression (721)
generalized least squares (GLS) (723)
infeasible GLS (726)
feasible GLS (726)
generalized method of moments
(GMM) (734) efficient GMM (735)
i i′ 2i -1
heteroskedasticity-robust mean vector (750) J-statistic (736) covariance matrix (750)
GMM J-statistic (736)
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Review the Concepts
18.1 A researcher studying the relationship between earnings and gen- der for a group of workers specifies the regression model Yi = b0 + X1ib1 + X2ib2 + ui, where X1i is a binary variable that equals 1 if the ith person is a female and X2i is a binary variable that equals 1 if the ith person is a male. Write the model in the matrix form of Equation (18.2) for a hypothetical set of n = 5 observations. Show that the columns of X are linearly dependent so that X does not have full rank. Explain how you would respecifiy the model to eliminate the perfect multicollinearity.
18.2 You are analyzing a linear regression model with 500 observations and one regressor. Explain how you would construct a confidence interval for b1 if:
a. Assumptions #1 through #4 in Key Concept 18.1 are true, but you think Assumption #5 or #6 might not be true.
b. Assumptions #1 through #5 are true, but you think Assumption #6 might not be true. (Give two ways to construct the confidence interval.)
c. Assumptions #1 through #6 are true.
18.3 Suppose that Assumptions #1 through #5 in Key Concept 18.1 are true but that Assumption #6 is not. Does the result in Equation (18.31) hold? Explain.
18.4 Can you compute the BLUE estimator of B if Equation (18.41) holds and you do not know 𝛀? What if you know 𝛀?
18.5 Construct an example of a regression model that satisfies the assumption
E(ui Xi) = 0 but for which E(U X) ≠ 0n.
Review the Concepts 739
740 ChapTeR 18 The Theory of Multiple Regression Exercises
18.1 Consider the population regression of test scores against income and the square of income in Equation (8.1).
a. Write the regression in Equation (8.1) in the matrix form of Equation (18.5). Define Y, X, U, and B.
b. Explain how to test the null hypothesis that the relationship between test scores and income is linear against the alternative that it is qua- dratic. Write the null hypothesis in the form of Equation (18.20). What are R, r, and q?
18.2 Suppose that a sample of n = 20 households has the sample means and sample covariances below for a dependent variable and two regressors:
Sample Means
Y 6.39 X1 7.24 X2 4.00
Sample Covariances
Y X1 X2
0.26
0.22 0.32
0.80 0.28
2.40
a. Calculate the OLS estimates of b0, b1, and b2. Calculate s2un. Calculate the R2 of the regression.
b. Suppose that all six assumptions in Key Concept 18.1 hold. Test the hypothesis that b1 = 0 at the 5% significance level.
18.3 Let W be an m * 1 vector with covariance matrix 𝚺W, where 𝚺W is finite and positive definite. Let c be a nonrandom m * 1 vector and let Q = c′W.
a. Show that var(Q) = c′𝚺W c.
b. Suppose that c ≠ 0m. Show that 0 < var(Q) 6 ∞.
18.4 Consider the regression model Yi = b0 + b1Xi + ui from Chapter 4 and assume that the least squares assumptions in Key Concept 4.3 hold.
a. Write the model in the matrix form given in Equations (18.2) and (18.4).
b. Show that Assumptions #1 through #4 in Key Concept 18.1 are satisfied.
c. Use the general formula for Bn in Equation (18.11) to derive the expressions for bn0 and bn1 given in Key Concept 4.2.
d. Show that the (1, 1) element of 𝚺Bn in Equation (18.13) is equal to the
expression for s2 given in Key Concept 4.4. bn 0
Exercises 741
18.5 Let PX and MX be as defined in Equations (18.24) and (18.25).
a. Prove that PXMX = 0n * n and that PX and MX are idempotent.
b. Derive Equations (18.27) and (18.28).
c. Show that rank(PX) = k + 1 and rank(MX) = n − k − 1. [Hint: First solve Exercise 18.10 and then use the fact that trace(AB) = trace(BA) for conformable matrices A and B.]
18.6 Consider the regression model in matrix form, Y = XB + WG + U, whereXisann * k1 matrixofregressorsandWisann * k2 matrixof regressors. Then, as shown in Exercise 18.17, the OLS estimator Bn can be expressed
Bn = (X′MWX)-1(X′MWY).
Now let bnBV be the “binary variable” fixed effects estimator computed
by estimating Equation (10.11) by OLS and let bnDM be the “de-meaning” 1
fixed effects estimator computed by estimating Equation (10.14) by OLS,
in which the entity-specific sample means have been subtracted from X
and Y. Use the expression for Bn given above to prove that bnBV = bnDM. 11
[Hint: Write Equation (10.11) using a full set of fixed effects, D1i, D2i, . . . , Dni and no constant term. Include all of the fixed effects in W. Write out the matrix MWX.]
18.7 Consider the regression model Yi = b1Xi + b2Wi + ui, where for simplicity the intercept is omitted and all variables are assumed to have a mean of zero. Suppose that Xi is distributed independently of (Wi, ui) but Wi and ui might be correlated and let bn1 and bn2 be the OLS estimators for this model. Show that
np
a. Whether or not Wi and ui are correlated, b1 ¡ b1.
b. If Wi and ui are correlated, then bn2 is inconsistent.
c. Let bnr1 be the OLS estimator from the regression of Y on X (the restricted regression that excludes W). Will bn1 have a smaller asymp- totic variance than bnr1, allowing for the possibility that Wi and ui are correlated? Explain.
18.8 Consider the regression model Yi = b0 + b1Xi + ui, where u1 = ∼u1 and
ui = 0.5ui-1 + ∼ui for i = 2, 3, . . . , n. Suppose that ∼ui are i.i.d. with mean 0 and variance 1 and are distributed independently of Xj for all i and j.
a. Derive an expression for E(UU′) = 𝛀.
1
742 ChapTeR 18 The Theory of Multiple Regression
b. Explain how to estimate the model by GLS without explicitly invert- ing the matrix 𝛀. (Hint: Transform the model so that the regression errors are ∼u1, ∼u2, c, ∼un.)
18.9 This exercise shows that the OLS estimator of a subset of the regres-
sion coefficients is consistent under the conditional mean independence
assumption stated in Appendix 7.2. Consider the multiple regression
modelinmatrixformY = XB + WG + U,whereXandWare,respec-
tively, n * k and n * k matrices of regressors. Let X ′ and W ′ denote the 12ii
ith rows of X and W [as in Equation (18.3)]. Assume that (i) E(ui|Xi, Wi) = W′D,whereDisak * 1vectorofunknownparameters;(ii)(X,W,Y)
i2iii are i.i.d.; (iii) (Xi, Wi, ui) have four finite, nonzero moments;
and (iv) there is no perfect multicollinearity. These are Assumptions #1 through #4 of Key Concept 18.1, with the conditional mean inde- pendence assumption (i) replacing the usual conditional mean zero assumption.
a. Use the expression for Bn given in Exercise 18.6 to write Bn - B = (n-1X′MWX )-1(n-1X′MWU ).
b. Show that n-1X′M X ¡p 𝚺 W
XX
- 𝚺 𝚺-1 𝚺 , where 𝚺 = XW WW WX p XX
E(XiXi′), 𝚺XW = E(XiWi′), and so forth. [The matrix An ¡ A if p
An,ij ¡ Aij for all i, j, where An,ij and Aij are the (i, j) elements of An and A.]
c. Show that assumptions (i) and (ii) imply that E(U|X, W) = WD.
d. Use (c) and the law of iterated expectations to show that
-1 p
n X′MWU ¡ 0k1*1.
e. Use (a) through (d) to conclude that, under conditions (i) through (iv), Bn¡p B.
18.10 Let C be a symmetric idempotent matrix.
a. Show that the eigenvalues of C are either 0 or 1. (Hint: Note that Cq = gq implies 0 = Cq - gq = CCq - gq = gCq - gq = g2q - gq and solve for G.)
b. Show that trace(C) = rank(C).
c. Let d be an n * 1 vector. Show that d′Cd Ú 0.
18.11 Suppose that C is an n * n symmetric idempotent matrix with rank r and let V ∼ N(0n, In).
a.
b. c.
18.12 a.
b. c.
Show that C = A A′, where A is n * r with A′A = Ir. (Hint: C is positive semidefinite and can be written as Q𝚲Q′, as explained in Appendix 18.1.)
Show that A′V N10 I 2. ∼ r, r
Show that V′CV x2 ∼Eff.GMM
Show that 2n1B∼ r. - B 2 ¡ 0. Show that JGMM ¡d x2 .
Show that B is the efficient GMM estimator—that is, that ∼Eff.GMM
B in Equation (18.66) is the solution to Equation (18.65). n Eff.GMM ∼Eff.GMM p
m-k
18.13 Consider the problem of minimizing the sum of squared residuals, subject to the constraint that Rb = r, where R is q * (k + 1) with rank q. Let B∼ be the value of b that solves the constrained minimization problem.
a. Show that the Lagrangian for the minimization problem is
L(b, G) = (Y − Xb)′ (Y − Xb) + G′(Rb − r), where G is a q * 1 vector of Lagrange multipliers.
∼n-1 -1-1n
b. Show that B = B - (X′X) R′[R(X′X) R′] (RB - r).
∼∼nn
c. Show that (Y - XB)′(Y - XB) - (Y - XB)(Y - XB) =
(RBn - r)′[R(X′X)-1R′]-1(RBn - r).
d. Show that F∼ in Equation (18.36) is equivalent to the homoskedasticity-
only F-statistic in Equation (7.13).
18.14 Consider the regression model Y = XB + U. Partition X as [X1 X2] and B as [B1′ B2′]′, where X1 has k1 columns and X2 has k2 columns. Suppose that X2′Y = 0k2 *1. Let R = [Ik1 0k1 *k2].
a. Show that Bn′(X′X)Bn = (RBn)′[R(X′X)-1R]-1(RBn).
b. Consider the regression described in Equation (12.17). Let W =
[1 W1 W2 c Wr], where 1 is an n * 1 vector of ones, W1 is the n * 1 vector with ith element W1i, and so forth. Let Un TSLS denote the vector of two-stage least squares residuals.
i. Show that W′Un TSLS = 0.
ii. Show that the method for computing the J-statistic described in Key Concept 12.6 (using a homoskedasticity-only F-statistic) and the formula in Equation (18.63) produce the same value for the J-statistic. [Hint: Use the results in (a), (b, i), and Exercise 18.13.]
Exercises 743
744 ChapTeR 18 The Theory of Multiple Regression
18.15 (Consistencyofclusteredstandarderrors.)ConsiderthepaneldatamodelYit =
bXit + ai + uit, where all variables are scalars. Assume that Assumptions #1, #2,
and #4 in Key Concept 10.3 hold and strengthen Assumption #3 so that Xit and
uit have eight nonzero finite moments. Let M = IT - T -1II′, where I is a T * 1
vector of ones. Also let Yi = (Yi1 Yi2 g YiT)′, Xi = (Xi1 Xi2 g XiT)′, ∼∼∼
ui = (ui1 ui2 g uiT)′,Yi = MYi,Xi = MXi, and ui = Mui. For the asymptoticcalculationsinthisproblem,supposethatTisfixedandn ¡ ∞.
a. Show that the fixed effects estimator of b from Section 10.3 can be
n∼∼-1 n∼∼ written as b = (g X′X ) g X′Y .
n
n n ∼∼-1 n ∼
iii h i 3i=1i h e. Use your answers to (b) through (d) to prove Equation (10.25); that
hX
f. Let sh,clustered be the infeasible clustered variance estimator,
i=1 i i i=1 i i
b. Show that b - b = (g X′X ) g X′u . (Hint: M is idempotent.)
∼21nd2 d. Leth=X′u>2Tands =var(h).Showthat ng h¡N(0,s).
c.LetQ=TE(X′X)andQ= g g X.ShowthatQ∼¡Q. X ii XnTi=1t=1it X X
i=1 i i i=1 i i
∼ -1 ∼∼ n∼ 1 n T ∼2 n p ∼
nd22 is, show that 2nT(b – b) ¡ N(0, s >Q∼ ).
∼2
g (X′u ) . Show that s ¡ s .
∼n ∼ n∼ n2
g. Let u = Y – bX and s =
2 ∼2 p
that snh, clustered – sh, clustered ¡ 0 and then use your answer
to (f).]
18.16 This exercise takes up the problem of missing data discussed in Section 9.2. Consider the regression model Yi = Xib + ui, i = 1, c, n, where all variables are scalars and the constant term/intercept is omitted for convenience.
a. Suppose that the least squares assumptions in Key Concept 4.3 are satisfied. Show that the least squares estimator of b is unbiased and consistent.
b. Now suppose that some of the observations are missing. Let Ii denote a binary random variable that indicates the nonmissing observations; that is, Ii = 1 if observation i is not missing and Ii = 0 if observation i is missing. Assume that {Ii, Xi, ui} are i.i.d.
computed using the true errors instead of the residuals so that ∼2 1n∼2 ∼2 p2
s = h,clustered
i=1 i
g (X u ) [this is i=1 i i p
nT
i
h,clustered
n 1 n ∼=∼n2
h
h, clustered
[Hint: Use an argument like that used in Equation (17.16) to show
1 i i
Equation (10.27) in matrix form]. Show that sn2h, clustered ¡ s2h.
n – 1 nT
=c
X′X X′W
W′X W′W d.
i. Show that the OLS estimator can be written as
bn = a an IiXiXi′b-1a an IiXiYib = b + a an IiXiXi′b-1a an IiXiuib.
i=1 i=1 i=1 i=1
ii. Suppose that data are missing, “completely at random,” in the
b. Show that
(X′MWX)-1 -(W′W)-1W′X(X′MWX)-1
-1
– (X′MWX)-1X′W(W′W)-1
(W′W)-1 + (W′W)-1W′X(X′MWX)-1X′W(W′W)-1
sense that Pr(I = 10X,u) = p, where p is a constant. Show that b iii
is unbiased and consistent.
iii. Suppose that the probability that the ith observation is missing
depends of X, but not on u; that is, Pr(I = 10X, u) = p(X). iiiiii
Show that bn is unbiased and consistent.
iv. Suppose that the probability that the ith observation is missing
depends on both X and u; that is, Pr(I = 10X, u) = p(X, u). Is iiiiiii
bn unbiased? Is bn consistent? Explain.
c. Suppose that b = 1 and that Xi and ui are mutually independent standard normal random variables [so that both Xi and ui are dis- tributed N(0, 1)]. Suppose that Ii = 1 when Yi Ú 0, but Ii = 0 when Yi 6 0. Is bn unbiased? Is bn consistent? Explain.
18.17 Consider the regression model in matrix form Y = XB + WG + U, where X and W are matrices of regressors and B and G are vectors of
unknown regression coefficients. Let X = MWX and Y = MWY, where MW = I – W(W′W)-1W.
Bn X′X X′W -1 X′Y JR=J RJR Gn JW′X W′WR W′Y
a. Show that the OLS estimators of B and G can be written as
(Hint: Show that the product of the two matrices is equal to the iden- tity matrix.)
∼∼
Exercises 745
n
746 ChapTeR 18 The Theory of Multiple Regression
n -1
c. Show that B = (X′MWX) X′MWY.
d. The Frisch–Waugh theorem (Appendix 6.2) says that Bn =
theorem.
∼ ∼ -1 ∼ ∼
(X′X) X′Y. Use the result in (c) to prove the Frisch–Waugh
appendix
18.1 Summary of Matrix Algebra
This appendix summarizes vectors, matrices, and the elements of matrix algebra used in Chapter 1. The purpose of this appendix is to review some concepts and definitions from a course in linear algebra, not to replace such a course.
Definitions of Vectors and Matrices
A vector is a collection of n numbers or elements, collected either in a column (a column vector) or in a row (a row vector). The n-dimensional column vector b and the n-dimensional row vector c are
b2
b= ≥b1¥andc=3c c g c4,
bn
where b1 is the first element of b and in general bi is the ith element of b.
Throughout, a boldface symbol denotes a vector or matrix.
A matrix is a collection, or an array, of numbers or elements in which the elements are
laid out in columns and rows. The dimension of a matrix is n * m, where n is the number of rows and m is the number of columns. The n * m matrix A is
f
12n
≥ a11 a12 g a1m ¥ A= a21 a22 g a2m ,
ffff an1 an2 g anm
where aij is the (i, j) element of A, that is, aij is the element that appears in the ith row and jth column. An n * m matrix consists of n row vectors or, alternatively, of m column vectors. To distinguish one-dimensional numbers from vectors and matrices, a one-dimensional
number is called a scalar.
Types of Matrices
Square, symmetric, and diagonal matrices. A matrix is said to be square if the number of rows equals the number of columns. A square matrix is said to be symmetric if its (i, j) ele- ment equals its (j, i) element. A diagonal matrix is a square matrix in which all the off- diagonal elements equal zero; that is, if the square matrix A is diagonal, then aij = 0 for i ≠ j.
Special matrices. An important matrix is the identity matrix, In, which is an n * n diago- nal matrix with ones on the diagonal. The null matrix, 0n * m, is the n * m matrix with all elements equal to zero.
The transpose. The transpose of a matrix switches the rows and the columns. That is, the transpose of a matrix turns the n * m matrix A into the m * n matrix, which is denoted by A′, where the (i, j) element of A becomes the (j, i) element of A′; said differently, the transpose of the matrix A turns the rows of A into the columns of A′. If aij is the (i, j) element of A, then A′ (the transpose of A) is
≥ a11 a21 g an1 ¥ A′= a12 a22 g an2 .
ffff a1m a2m g anm
The transpose of a vector is a special case of the transpose of a matrix. Thus the transpose of a vector turns a column vector into a row vector; that is, if b is an n * 1 column vector, then its transpose is the 1 * n row vector
b′=3b1 b2 g bn4. The transpose of a row vector is a column vector.
Elements of Matrix Algebra: Addition and Multiplication
Matrix addition. Two matrices A and B that have the same dimensions (for example, that are both n * m) can be added together. The sum of two matrices is the sum of their ele- ments; that is, if C = A + B, then cij = aij + bij. A special case of matrix addition is vec- toraddition:Ifaandbarebothn * 1columnvectors,thentheirsumc = a + bisthe element-wise sum; that is, ci = ai + bi.
n
product of the transpose of a (which is itself a row vector) with b is a′b = g a b . Apply-
Vector and matrix multiplication. Let a and b be two n * 1 column vectors. Then the
ing this definition with b = a yields a′a = gni = 1 a2i .
Similarly, the matrices A and B can be multiplied together if they are conformable—
that is, if the number of columns of A equals the number of rows of B. Specifically, suppose
Summary of Matrix Algebra 747
i=1ii
748 ChapTeR 18 The Theory of Multiple Regression
that A has dimension n * m and B has dimension m * r. Then the product of A and B is
m
an n * r matrix, C; that is, C = AB, where the (i, j) element of C is c = g a b . Said
ij k=1 ik kj differently, the (i, j) element of AB is the product of multiplying the row vector that is the
ith row of A with the column vector that is the j th column of B.
The product of a scalar d with the matrix A has the (i, j) element daij; that is, each
element of A is multiplied by the scalar d.
Some useful properties of matrix addition and multiplication.
Then:
a. A+B=B+A;
b. (A+B)+C=A+(B+C);
c. (A+B)′=A′+B′;
d. If A is n * m, then AIm = A and InA = A;
e. A(BC) = (AB)C;
f. (A + B)C = AC + BC; and
g. (AB)′ = B′A′.
Let A and B be matrices.
In general, matrix multiplication does not commute; that is, in general AB ≠ BA, although there are some special cases in which matrix multiplication commutes; for exam- ple, if A and B are both n * n diagonal matrices, then AB = BA.
Matrix Inverse, Matrix Square Roots, and Related Topics
The matrix inverse. Let A be a square matrix. Assuming that it exists, the inverse of the matrix A is defined as the matrix for which A−1A = In. If in fact the inverse matrix A−1 exists, then A is said to be invertible or nonsingular. If both A and B are invertible, then (AB)−1 = B−1A−1.
Positive definite and positive semidefinite matrices. Let V be an n * n square matrix. Then V is positive definite if c′Vc 7 0 for all nonzero n * 1 vectors c. Similarly, V is positive semidefinite if c′Vc Ú 0 for all nonzero n * 1 vectors c. If V is positive definite, then it is invertible.
Linear independence. The n * 1 vectors a1 and a2 are linearly independent if there do not exist nonzero scalars c1 and c2 such that c1a1 + c2a2 = 0n * 1. More generally, the set of k vectors a1, a2, c, ak are linearly independent if there do not exist nonzero scalars c1, c2, c, ck such thatc1a1 + c2a2 + g+ ckak = 0n*1.
The rank of a matrix. The rank of the n * m matrix A is the number of linearly independ- ent columns of A. The rank of A is denoted rank(A). If the rank of A equals the number of columns of A, then A is said to have full column rank. If the n * m matrix A has full column rank, then there does not exist a nonzero m * 1 vector c such that Ac = 0n * 1. If Aisn * nwithrank(A) = n,thenAisnonsingular.Ifthen * mmatrixAhasfullcolumn rank, then A′A is nonsingular.
The matrix square root. Let V be an n * n square symmetric positive definite matrix. The matrix square root of V is defined to be an n * n matrix F such that F′F = V. The matrix square root of a positive definite matrix will always exist, but it is not unique. The matrix square root has the property that FV -1F′ = In. In addition, the matrix square root of a positive definite matrix is invertible, so F′-1VF -1 = In.
Eigenvalues and eigenvectors. Let A be an n * n matrix. If the n * 1 vector q and the scalar l satisfy Aq = lq, where q′q = 1, then l is an eigenvalue of A, and q is the eigen- vector of A associated with that eigenvalue. An n * n matrix has n eigenvalues, which need not take on distinct values, and n eigenvectors.
If V is an n * n symmetric positive definite matrix, then all the eigenvalues of V are positive real numbers, and all the eigenvectors of V are real. Also, V can be written in terms of its eigenvalues and eigenvectors as V = Q𝚲Q′, where 𝚲 is a diagonal n * n matrix with diagonal elements that equal the eigenvalues of V, and Q is an n * n matrix consisting of the eigenvectors of V, arranged so that the ith column of Q is the eigenvector corresponding to the eigenvalue that is the ith diagonal element of 𝚲. The eigenvectors are orthonormal, so Q′Q = In.
Idempotent matrices. A matrix C is idempotent if C is square and CC = C. If C is an n * n idempotent matrix that is also symmetric, then C is positive semidefinite and C has r eigenvalues that equal 1 and n − r eigenvalues that equal 0, where r = rank(C) (Exercise 18.10).
18.2
appendix
Multivariate Distributions 749
Multivariate Distributions
This appendix collects various definitions and facts about distributions of vectors of ran- dom variables. We start by defining the mean and covariance matrix of the n-dimensional random variable V. Next we present the multivariate normal distribution. Finally, we sum- marize some facts about the distributions of linear and quadratic functions of jointly nor- mally distributed random variables.
750 ChapTeR 18 The Theory of Multiple Regression
The Mean Vector and Covariance Matrix
The first and second moments of an m * 1 vector of random variables, V = (V1 V2 g Vm)′, are summarized by its mean vector and covariance matrix.
Because V is a vector, the vector of its means—that is, its mean vector—is E(V) = MV. The ith element of the mean vector is the mean of the ith element of V.
. (18.72)
The m * 1 vector random variable V has a multivariate normal distribution with mean
ThecovariancematrixofVisthematrixconsistingofthevariancevar(Vi),i = 1,…,m, along the diagonal and the (i, j) off-diagonal elements cov(Vi, Vj). In matrix form, the covariance matrix 𝚺V is
VVVCfS cov(Vm, V1) g var(Vm)
var(V1) g cov(V1, Vm) 𝚺 =E[(V-M )(V-M )′]= f f
The Multivariate Normal Distribution
and covariance matrix 𝚺 if it has the joint probability density function VV
vector M
where det(𝚺V) is the determinant of the matrix 𝚺V. The multivariate normal distribution is denoted N(MV, 𝚺V).
An important fact about the multivariate normal distribution is that if two jointly normally distributed random variables are uncorrelated (equivalently, have a block-diagonal covariance matrix), then they are independently distributed. That is, let V1 and V2 be jointly normally distributed random variables with respective dimensions m1 * 1 and m2 * 1. Then if cov(V1,V2) = E[(V1 – MV1)(V2 – MV2)′] = 0m1*m2, V1 and V2 are independent.
If {Vi} are i.i.d. N(0, s2v), then 𝚺V = s2v Im, and the multivariate normal distribution simplifies to the product of m univariate normal densities.
Distributions of Linear Combinations and Quadratic
Forms of Normal Random Variables
Linear combinations of multivariate normal random variables are themselves normally distributed, and certain quadratic forms of multivariate normal random variables have a chi-squared distribution. Let V be an m * 1 random variable distributed N(MV, 𝚺V), let A
2(2p) det(𝚺
m V) 2
f(V)= 1 expc-1(V-M )′𝚺-1(V-M )d,
(18.73)
VVV
appendix
18.3
Derivation of the Asymptotic Distribution of βn
This appendix provides the derivation of the asymptotic normal distribution of 2n(B – B)
Derivation of the Asymptotic Distribution of bn and B be nonrandom a * m and b * m matrices, and let d be a nonrandom a *
751
1 vector.
(18.74) (18.75) (18.76) (18.77)
Then
d + AV is distributed N(d + AMV, A𝚺VA′);
cov (AV, BV) = A𝚺VB′;
if A𝚺VB′ = 0a * b, then AV and BV are independently distributed; and
(V – M )′𝚺-1(V – M ) is distributed x2 . VVVm
Let U be an m-dimensional multivariate standard normal random variable with distribu- tion N(0, Im). If C is symmetric and idempotent, then
U′CU has a x2r distribution, where r = rank(C). Equation (18.78) is proven as Exercise 18.11.
(18.78)
Firstconsiderthe“denominator”matrixX′X>n = ng XX′inEquation(18.15).The
given in Equation (18.12). An implication of this result is that Bn ¡p B. 1n
i=1 i i
(j, l) element of this matrix is n g X X . By the second assumption in Key Concept 18.1,
1n
i=1 ji li
Xi is i.i.d., so XjiXli is i.i.d. By the third assumption in Key Concept 18.1, each element of Xi has four moments, so, by the Cauchy–Schwarz inequality (Appendix 17.2), XjiXli has two
1n
moments. Because X X is i.i.d. with two moments, n g X X obeys the law of large
numbers, so ng X X ¡ E(X X ). This is true for all the elements of X′X>n, so
1n
i=1 i
p
Next consider the “numerator” matrix in Equation (18.15), X′U > 2n = 2n g V ,
ji li i=1 ji li 1np
i=1 ji li ji li X′X>n ¡ E(XX′) = Q .
iiX
222244
E[(c′V) ] = E[(c′Xu) ] = E[(c′X) (u) ] … 2E[(c′X) ]E(u ), which is finite by the
where Vi = Xiui. By the first assumption in Key Concept 18.1 and the law of iterated
expectations, E(Vi) = E[XiE(ui|Xi)] = 0k+1. By the second least squares assumption,
Vi is i.i.d. Let c be a finite k + 1 dimensional vector. By the Cauchy–Schwarz inequality,
Concept 18.2 applies to 2n g V = i=1 i
n
iiiiiii
third least squares assumption. This is true for every such vector c, so E(ViV′i) = 𝚺V is
1 n2n 1
2n
finite and, we assume, positive definite. Thus the multivariate central limit theorem of Key
X′U; that is,
1 X′U ¡d N(0k+1,𝚺V). (18.79)
752 ChapTeR 18 The Theory of Multiple Regression
The result in Equation (18.12) follows from Equations (18.15) and (18.79), the consist-
ency of X′X > n, the fourth least squares assumption (which ensures that (X′X) and Slutsky’s theorem.
18.4
-1
Derivations of Exact Distributions of OLS Test Statistics with Normal Errors
exists),
appendix
This appendix presents the proofs of the distributions under the null hypothesis of the homoskedasticity-only t-statistic in Equation (18.35) and the homoskedasticity-only F-statistic in Equation (18.37), assuming that all six assumptions in Key Concept 18.1 hold.
If (i) Z has a standard normal distribution, (ii) W has a x2m distribution, and (iii) Z and W are independently distributed, then the random variable Z > 2W > m has the t-distribution
∼
Proof of Equation (18.35)
n bn
2u
= (s >s )𝚺
(b-b)>2(𝚺n ) nj j,0 B 0X jj
with m degrees of freedom (Appendix 17.1). To put t in this form, notice that
nu2
whereW = (n–k–1)(s >s ), andletZ = (b – b )>2(𝚺n ) andm = n−k−1.
𝚺
. Then rewrite Equation (18.34) as
Bn X
∼t = , (18.80) 2W>(n – k – 1)
∼
With these definitions, t = Z> 2W>m. Thus, to prove the result in Equation (18.35), we must show (i) through (iii) for these definitions of Z, W, and m.
22n
unu jj,0 B0Xjj
i. An implication of Equation (18.30) is that, under the null hypothesis, Z = (b – b )> 2(𝚺 n ) has an exact standard normal distribution, which shows (i).
nj j,0 B 0X jj
ii. From Equation (18.31), W is distributed as x2n – k – 1, which shows (ii).
iii. To show (iii), it must be shown that bn and s2 are independently distributed. j un
n -1 2
From Equations (18.14) and (18.29), B – B = (X′X) X′U and s = (M U)′(M U)>
un X X
(n – k – 1). Thus B – B and sun are independent if (X′X) X′U and MXU are independ-
n2 -1
ent. Both (X′X)-1X′U and MXU are linear combinations of U, which has an N(0n * 1, s2uIn) distribution, conditional on X. But because MXX(X′X)-1 = 0n * (k + 1) [Equation (18.26)], it follows that (X′X)-1X′U and MXU are independently distributed [Equation (18.76)]. Con- sequently, under all six assumptions in Key Concept 18.1,
Bn and s2 are independently distributed, (18.81) un
which shows (iii) and thus proves Equation (18.35).
Proof of the Gauss–Markov Theorem for Multiple Regression 753 Proof of Equation (18.37)
The Fn1, n2 distribution is the distribution of (W1>n1)>(W2>n2), where (i) W1 is distributed x2n1; (ii) W2 is distributed x2n2; and (iii) W1 and W2 are independently distributed (Appendix
∼
17.1). To express F in this form, let W1 = (RB – r)′[R(X′X) R′su] (RB – r) and
n -12-1n
W = (n – k – 1)s >s Substitution of these definitions into Equation (18.36) shows that
∼2unu ∼
F = (W >q)>[W2>(n2 – k – 1)]. Thus, by the definition of the F distribution, F has an
12
Fq, n−k−1 distribution if (i) through (iii) hold with n1 = q and n2 = n − k − 1.
i. Under the null hypothesis, RB – r = R(B – B). Because B has the conditional normal distribution in Equation (18.30) and because R is a nonrandom matrix,
n -12
R(B – B) is distributed N(0q * 1, R(X′X) R′su), conditional on X. Thus, by
n 2-1n
Equation (18.77) in Appendix 18.2, (RB – r)′[R(X′X)R′su] (RB – r) is dis-
tributed x2q, proving (i).
ii. Requirement (ii) is shown in Equation (18.31).
n2
iii. It has already been shown that B – B and sun are independently distributed [Equa-
n2
tion (18.81)]. It follows that RB – r and sun are independently distributed, which
in turn implies that W1 and W2 are independently distributed, proving (iii) and completing the proof.
18.5
appendix
nnn
Proof of the Gauss–Markov Theorem for Multiple Regression
This appendix proves the Gauss–Markov theorem (Key Concept 18.3) for the multiple
∼∼ regression model. Let B be a linear conditionally unbiased estimator of B so that B = A′Y
∼
and E(B X) = B, where A is an n * (k + 1) matrix that can depend on X and nonran-
dom constants. We show that var(c′Bn) … var(c′B∼) for all k + 1 dimensional vectors c,
where the inequality holds with equality only if B = B. ∼∼
∼ BythefirstGauss–Markovcondition,E(U0X) = 0 ,soE(B0X) = (A′X)B,butbecause
Because B is linear, it can be written as B = A′Y = A′(XB + U) = (A′X)B + A′U.
n*1
B is conditionally unbiased, E(B X) = B = (A′X )B, which implies that A′X = Ik + 1.
Thus B = B + A′U, so var(B 0X) = var(A′U0X) = E(A′UU′A0X) = A′E(UU′0X)A = S2uA′A, where the third equality follows because A can depend on X but not U, and the final equality follows from the second Gauss–Markov condition. That is, if B∼ is linear and unbiased, then under the Gauss–Markov conditions,
∼∼ ∼∼
∼
n
A′X = I andvar(B0X) = s A′A. (18.82) k+1 ∼ 2u
n n -1 -1 The results in Equation (18.82) also apply to B with A = A = X(X′X) , where (X′X)
exists by the third Gauss–Markov condition.
754 ChapTeR 18 The Theory of Multiple Regression
Now let A = An + D so that D is the difference between the matrices A and An .
n-1-1 nn Note that A′A = (X′X) X′A = (X′X) [by Equation (18.82)] and A′A =
-1 -1 -1nnnnnn
(X′X) X′X(X′X) = (X′X) ,so A′D = A′(A – A) = A′A – A′A = 0(k+ 1)*(k+ 1).
Substituting A = An + D into the formula for the conditional variance in Equation (18.82) yields
∼
var(B0X) = s (A + D)′(A + D)
2un n 2nnnn
= su[A′A + A′D + D′A + D′D] 2-12
= su(X′X) + suD′D, (18.83) nn -1 n
where the final equality uses the facts A′A = (X′X) and A′D = 0(k + 1) * (k + 1). n 2 -1
var(B 0 X) – var(B 0 X) = s D′D. The difference between the variances of the two estima- tors of the linear combination c′B thus is
Because var(B X ) = su(X′ X ) , Equations (18.82) and (18.83) imply that ∼ n 2u
∼
var (c′B0X) – var(c′B0X) = s c′D′Dc Ú 0. (18.84)
n 2u
The inequality in Equation (18.84) holds for all linear combinations c′B, and the inequality
holds with equality for all nonzero c only if D = 0n * (k+1)—that is, if A = An or, equiva- ∼nn
lently, B = B. Thus c′B has the smallest variance of all linear conditionally unbiased esti- mators of c′B; that is, the OLS estimator is BLUE.
appendix
18.6
Proof of Selected Results for IV and GMM Estimation
The Efficiency of TSLS Under Homoskedasticity
[Proof of Equation (18.62)]
When the errors u are homoskedastic, the difference between 𝚺IV [Equation (18.61)] and iA
𝚺TSLS [Equation (18.55)] is given by
𝚺IV -𝚺TSLS =(Q AQ )-1Q AQ AQ (Q AQ )-1s2 -(Q Q-1Q )-1s2
A XZ ZX XZ ZZ ZX XZ ZX u XZ ZZ ZX u =(Q AQ )-1Q A[Q -Q (Q Q-1Q )-1Q ]AQ (Q AQ )-1s2, (18.85)
XZ ZX XZ ZZ ZX XZ ZZ ZX XZ ZX XZ ZX u
where the second term in brackets in the second equality follows from
(QXZAQZX)-1QXZAQZX = I(k + r + 1). Let F be the matrix square root of QZZ, so QZZ = F′F
and Q-1 = F -1F -1′. [The latter equality follows from noting that (F′F)-1 = F -1F′-1 and ZZ
F′-1 = F -1′.] Then the final expression in Equation (18.85) can be rewritten to yield
𝚺IV – 𝚺TSLS = (Q AQ )-1Q AF′[I – F -1′Q (Q F-1F-1′Q )-1Q F-1] A XZ ZX XZ ZX XZ ZX XZ
* FAQZX(QXZAQZX)-1s2u,
(18.86)
Proof of Selected Results for IV and GMM Estimation 755 where the second expression in brackets uses F′F -1′ = I. Thus
c′(𝚺IV – 𝚺TSLS)c = d′[I – D(D′D)-1D′]ds2, (18.87) Au
where d = FAQZX(QXZAQZX)-1c and D = F -1′QZX. Now I – D(D′D)-1D′ is a symmetric
idempotent matrix (Exercise 18.5). As a result, I – D(D′D)-1D′ has eigenvalues that are
either0or1andd′[I – D(D′D)-1D′]d Ú 0(Exercise18.10).Thusc′(𝚺IV – 𝚺TSLS)c Ú 0, A
proving that TSLS is efficient under homoskedasticity.
Asymptotic Distribution of the J-Statistic Under Homoskedasticity
The J-statistic is defined in Equation (18.63). First note that
Thus
Un = Y – X Bn T S L S
= Y – X(X′PZX)-1X′PZY
= (XB + U) – X(X′PZX)-1X′PZ(XB + U) = U – X(X′PZX)-1 X′PZU
= [I – X(X′PZX)-1X′PZ]U.
UnPZUn = U′[I – PZX(X′PZX)-1X′]PZ[I – X(X′PZX)-1X′PZ]U = U′[PZ – PZX(X′PZX)-1X′PZ]U,
(18.88)
(18.89)
where the second equality follows by simplifying the preceding expression. Because Z′Z is symmetric and positive definite, it can be written in terms of its matrix square root,
1>2′ 1>2 -1 Z′Z = (Z′Z) (Z′Z) , and this matrix square root is invertible, so (Z′Z) =
-1 -1>2 final expression in Equation (18.89) yields
nn -1
U′PZU = U′[BB′ – BB′X(X′BB′X) X′BB′]U
= U′B [I – B′X(X′BB′X)-1X′B]B′U = U′BMB′XB′U,
null hypothesis. Under the null hypothesis that E(Z u ) = 0, Z′U > 2n has mean zero and
d
(Z′Z)-1>2(Z′Z)-1>2′, where (Z′Z)-1>2 = [(Z′Z)1>2]-1. Thus P can be written as P =
ZZ Z(Z′Z) Z′ = BB′, where B = Z(Z′Z) . Substituting this expression for PZ into the
where MB′X = I – B′X(X′BB′X)-1X′B is a symmetric idempotent matrix.
the central limit theorem applies, so Z′U > 2n ¡ N(0, Q s ). In addition, ZZ 2u
The asymptotic null distribution of Un ′PZUn is found by computing the limits in probability
and in distribution of the various terms in the final expression in Equation (18.90) under the
ii
(18.90)
ZZ
(Z′U> 2n) ¡ s z where z is distributed N(0 , I ). In addition, B′X > 2n =
756 ChapTeR 18 The Theory of Multiple Regression Z′Z>n ¡p Q andX′Z>n ¡p
Q .ThusB′U = (Z′Z)-1>2′Z′U = (Z′Z/n)-1>2′ XZ
(Z′Z>n)
QXZQZZ = MQZZ QZX
-1>2
ZZ ZX B′X ZZ ZX XZ ZZ ZZ ZX
-1>2′ -1>2′
-1>2
nnd -1>22
-1>2′ -1>2 -1 (18.91)
d
(Z′X>n) ¡ Q Q , so M ¡ I – Q Q (Q Q Q Q )
-1>2
. Thus
u
m+r+1 m+r+1
pp
U′PZU ¡ (z′MQXZ QZZ z)su.
Under the null hypothesis, the TSLS estimator is consistent and the coefficients in the regression of Un on Z converge in probability to zero [an implication of Equation (18.91)], so the denominator in the definition of the J-statistic is a consistent estimator of s2u:
n Zn 2u
U′M U>(n – m – r – 1) ¡p s . (18.92)
From the definition of the J-statistic and Equations (18.91) and (18.92), it follows that
U n ′ P Z U n
U′M U>(n – m – r – 1)
J =
Because z is a standard normal random vector and MQZZ QZX
z. (18.93) is a symmetric idempotent
-1>2
the rank of MQ-1>2Q [Equation (18.78)]. Because QZZ QZX is (m + r + 1) * (k + r + 1)
ZZ ZX
and m > k, the rank of M
-1>2
-1>2 QZZ QZX
result stated in Equation (18.64).
d 2 ism−k[Exercise18.5].Thus J ¡ x
,whichisthe
The Efficiency of the Efficient GMM Estimator
∼Eff.GMM
The infeasible efficient GMM estimator, B , is defined in Equation (18.66). The
∼Eff.GMM IV proof that B is efficient entails showing that c′(𝚺A – 𝚺
Eff.GMM
Distribution of the GMM J-Statistic
The GMM J-statistic is given in Equation (18.70). The proof that, under the null hypoth-
esis, JGMM ¡d x2 closely parallels the corresponding proof for the TSLS J-statistic m-k
under homoskedasticity.
)c Ú 0 for all vectors c. The proof closely parallels the proof of the efficiency of the TSLS estimator in the first section of this appendix, with the sole modification that H−1 replaces QZZs2u in Equation
(18.85) and subsequently.
¡d z ′ M Q
-1>2
ZZ XZ
Q
n Zn
matrix, J is distributed as a chi-squared random variable with degrees of freedom that equal
m-k
Appendix
TABLE 1 The Cumulative Standard Normal Distribution Function, 𝚽(z) = Pr(Z ” z)
Area = Pr(Z < z) –
0z Second Decimal Value of z
z0123456789
–2.9 0.0019 –2.8 0.0026 –2.7 0.0035 –2.6 0.0047 –2.5 0.0062 –2.4 0.0082 –2.3 0.0107 –2.2 0.0139 –2.1 0.0179 –2.0 0.0228 –1.9 0.0287 –1.8 0.0359 –1.7 0.0446 –1.6 0.0548 –1.5 0.0668 –1.4 0.0808 –1.3 0.0968 –1.2 0.1151 –1.1 0.1357 –1.0 0.1587 –0.9 0.1841
0.0018 0.0018 0.0025 0.0024 0.0034 0.0033 0.0045 0.0044 0.0060 0.0059 0.0080 0.0078 0.0104 0.0102 0.0136 0.0132 0.0174 0.0170 0.0222 0.0217 0.0281 0.0274 0.0351 0.0344 0.0436 0.0427 0.0537 0.0526 0.0655 0.0643 0.0793 0.0778 0.0951 0.0934 0.1131 0.1112 0.1335 0.1314 0.1562 0.1539 0.1814 0.1788
0.0017 0.0016 0.0016 0.0015 0.0023 0.0023 0.0022 0.0021 0.0032 0.0031 0.0030 0.0029 0.0043 0.0041 0.0040 0.0039 0.0057 0.0055 0.0054 0.0052 0.0075 0.0073 0.0071 0.0069 0.0099 0.0096 0.0094 0.0091 0.0129 0.0125 0.0122 0.0119 0.0166 0.0162 0.0158 0.0154 0.0212 0.0207 0.0202 0.0197 0.0268 0.0262 0.0256 0.0250 0.0336 0.0329 0.0322 0.0314 0.0418 0.0409 0.0401 0.0392 0.0516 0.0505 0.0495 0.0485 0.0630 0.0618 0.0606 0.0594 0.0764 0.0749 0.0735 0.0721 0.0918 0.0901 0.0885 0.0869 0.1093 0.1075 0.1056 0.1038 0.1292 0.1271 0.1251 0.1230 0.1515 0.1492 0.1469 0.1446 0.1762 0.1736 0.1711 0.1685
0.0015 0.0014 0.0021 0.0020 0.0028 0.0027 0.0038 0.0037 0.0051 0.0049 0.0068 0.0066 0.0089 0.0087 0.0116 0.0113 0.0150 0.0146 0.0192 0.0188 0.0244 0.0239 0.0307 0.0301 0.0384 0.0375 0.0475 0.0465 0.0582 0.0571 0.0708 0.0694 0.0853 0.0838 0.1020 0.1003 0.1210 0.1190 0.1423 0.1401 0.1660 0.1635
0.0014 0.0019 0.0026 0.0036 0.0048 0.0064 0.0084 0.0110 0.0143 0.0183 0.0233 0.0294 0.0367 0.0455 0.0559 0.0681 0.0823 0.0985 0.1170 0.1379 0.1611
(Table 1 continued)
757
758 Appendix (Table 1 continued)
Second Decimal Value of z z0123456789
–0.8 0.2119 –0.7 0.2420 –0.6 0.2743 –0.5 0.3085 –0.4 0.3446 –0.3 0.3821 –0.2 0.4207 –0.1 0.4602 –0.0 0.5000
0.0 0.5000
0.1 0.5398
0.2 0.5793
0.3 0.6179
0.4 0.6554
0.5 0.6915
0.6 0.7257
0.7 0.7580
0.8 0.7881
0.9 0.8159
1.0 0.8413
1.1 0.8643
1.2 0.8849
1.3 0.9032
1.4 0.9192
1.5 0.9332
1.6 0.9452
1.7 0.9554
1.8 0.9641
1.9 0.9713
2.0 0.9772
2.1 0.9821
2.2 0.9861
2.3 0.9893
2.4 0.9918
2.5 0.9938
2.6 0.9953
2.7 0.9965
2.8 0.9974
2.9 0.9981
0.2090 0.2061 0.2389 0.2358 0.2709 0.2676 0.3050 0.3015 0.3409 0.3372 0.3783 0.3745 0.4168 0.4129 0.4562 0.4522 0.4960 0.4920 0.5040 0.5080 0.5438 0.5478 0.5832 0.5871 0.6217 0.6255 0.6591 0.6628 0.6950 0.6985 0.7291 0.7324 0.7611 0.7642 0.7910 0.7939 0.8186 0.8212 0.8438 0.8461 0.8665 0.8686 0.8869 0.8888 0.9049 0.9066 0.9207 0.9222 0.9345 0.9357 0.9463 0.9474 0.9564 0.9573 0.9649 0.9656 0.9719 0.9726 0.9778 0.9783 0.9826 0.9830 0.9864 0.9868 0.9896 0.9898 0.9920 0.9922 0.9940 0.9941 0.9955 0.9956 0.9966 0.9967 0.9975 0.9976 0.9982 0.9982
0.2033 0.2005 0.2327 0.2296 0.2643 0.2611 0.2981 0.2946 0.3336 0.3300 0.3707 0.3669 0.4090 0.4052 0.4483 0.4443 0.4880 0.4840 0.5120 0.5160 0.5517 0.5557 0.5910 0.5948 0.6293 0.6331 0.6664 0.6700 0.7019 0.7054 0.7357 0.7389 0.7673 0.7704 0.7967 0.7995 0.8238 0.8264 0.8485 0.8508 0.8708 0.8729 0.8907 0.8925 0.9082 0.9099 0.9236 0.9251 0.9370 0.9382 0.9484 0.9495 0.9582 0.9591 0.9664 0.9671 0.9732 0.9738 0.9788 0.9793 0.9834 0.9838 0.9871 0.9875 0.9901 0.9904 0.9925 0.9927 0.9943 0.9945 0.9957 0.9959 0.9968 0.9969 0.9977 0.9977 0.9983 0.9984
0.1977 0.1949 0.2266 0.2236 0.2578 0.2546 0.2912 0.2877 0.3264 0.3228 0.3632 0.3594 0.4013 0.3974 0.4404 0.4364 0.4801 0.4761 0.5199 0.5239 0.5596 0.5636 0.5987 0.6026 0.6368 0.6406 0.6736 0.6772 0.7088 0.7123 0.7422 0.7454 0.7734 0.7764 0.8023 0.8051 0.8289 0.8315 0.8531 0.8554 0.8749 0.8770 0.8944 0.8962 0.9115 0.9131 0.9265 0.9279 0.9394 0.9406 0.9505 0.9515 0.9599 0.9608 0.9678 0.9686 0.9744 0.9750 0.9798 0.9803 0.9842 0.9846 0.9878 0.9881 0.9906 0.9909 0.9929 0.9931 0.9946 0.9948 0.9960 0.9961 0.9970 0.9971 0.9978 0.9979 0.9984 0.9985
0.1922 0.1894 0.2206 0.2177 0.2514 0.2483 0.2843 0.2810 0.3192 0.3156 0.3557 0.3520 0.3936 0.3897 0.4325 0.4286 0.4721 0.4681 0.5279 0.5319 0.5675 0.5714 0.6064 0.6103 0.6443 0.6480 0.6808 0.6844 0.7157 0.7190 0.7486 0.7517 0.7794 0.7823 0.8078 0.8106 0.8340 0.8365 0.8577 0.8599 0.8790 0.8810 0.8980 0.8997 0.9147 0.9162 0.9292 0.9306 0.9418 0.9429 0.9525 0.9535 0.9616 0.9625 0.9693 0.9699 0.9756 0.9761 0.9808 0.9812 0.9850 0.9854 0.9884 0.9887 0.9911 0.9913 0.9932 0.9934 0.9949 0.9951 0.9962 0.9963 0.9972 0.9973 0.9979 0.9980 0.9985 0.9986
0.1867 0.2148 0.2451 0.2776 0.3121 0.3483 0.3859 0.4247 0.4641 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986
This table can be used to calculate Pr(Z ... z) where Z is is 0.8790, which is the table entry for the row labeled 1.1
a standard normal variable. For example, when z = 1.17, this probability and the column labeled 7.
TABLE 2
Degrees of Freedom
1 2 3 4 5 6 7 8 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
60
90
120
∞
Appendix 759 Critical Values for Two-Sided and One-Sided Tests Using the Student t Distribution
20% (2-Sided) 10% (1-Sided)
3.08 1.89 1.64 1.53 1.48 1.44 1.41 1.40 1.38 1.37 1.36 1.36 1.35 1.35 1.34 1.34 1.33 1.33 1.33 1.33 1.32 1.32 1.32 1.32 1.32 1.32 1.31 1.31 1.31 1.31 1.30 1.29 1.29 1.28
10% (2-Sided) 5% (1-Sided)
6.31 2.92 2.35 2.13 2.02 1.94 1.89 1.86 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.75 1.74 1.73 1.73 1.72 1.72 1.72 1.71 1.71 1.71 1.71 1.70 1.70 1.70 1.70 1.67 1.66 1.66 1.64
Significance Level
5% (2-Sided) 2.5% (1-Sided)
12.71 4.30 3.18 2.78 2.57 2.45 2.36 2.31 2.26 2.23 2.20 2.18 2.16 2.14 2.13 2.12 2.11 2.10 2.09 2.09 2.08 2.07 2.07 2.06 2.06 2.06 2.05 2.05 2.05 2.04 2.00 1.99 1.98 1.96
2% (2-Sided) 1% (1-Sided)
31.82 6.96 4.54 3.75 3.36 3.14 3.00 2.90 2.82 2.76 2.72 2.68 2.65 2.62 2.60 2.58 2.57 2.55 2.54 2.53 2.52 2.51 2.50 2.49 2.49 2.48 2.47 2.47 2.46 2.46 2.39 2.37 2.36 2.33
1% (2-Sided) 0.5% (1-Sided)
63.66 9.92 5.84 4.60 4.03 3.71 3.50 3.36 3.25 3.17 3.11 3.05 3.01 2.98 2.95 2.92 2.90 2.88 2.86 2.85 2.83 2.82 2.81 2.80 2.79 2.78 2.77 2.76 2.76 2.75 2.66 2.63 2.62 2.58
Values are shown for the critical values for two-sided (≠) and one-sided (7) alternative hypotheses. The critical value for the one-sided ( 6 ) test is the negative of the one-sided ( 7 ) critical value shown in the table. For example, 2.13 is the critical value for a two-sided test with a significance level of 5% using the Student t distribution with 15 degrees of freedom.
760
Appendix
TABLE 3 Critical Values for the x2 Distribution
Degrees of Freedom
1 2 3 4 5 6 7 8 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Significance Level
10% 5%
2.71 3.84 4.61 5.99 6.25 7.81 7.78 9.49 9.24 11.07
10.64 12.59 12.02 14.07 13.36 15.51 14.68 16.92 15.99 18.31 17.28 19.68 18.55 21.03 19.81 22.36 21.06 23.68 22.31 25.00 23.54 26.30 24.77 27.59 25.99 28.87 27.20 30.14 28.41 31.41 29.62 32.67 30.81 33.92 32.01 35.17 33.20 36.41 34.38 37.65 35.56 38.89 36.74 40.11 37.92 41.34 39.09 42.56 40.26 43.77
1%
6.63
9.21 11.34 13.28 15.09 16.81 18.48 20.09 21.67 23.21 24.72 26.22 27.69 29.14 30.58 32.00 33.41 34.81 36.19 37.57 38.93 40.29 41.64 42.98 44.31 45.64 46.96 48.28 49.59 50.89
This table contains the 90th, 95th, and 99th percentiles of the x2 distribution. These serve as critical values for tests with significance levels of 10%, 5%, and 1%.
Appendix 761
TABLE 4 Critical Values for the Fm, ∞
Distribution
Degrees of Freedom
1 2
3 4 5 6 7 8 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
10%
2.71 2.30
2.08 1.94 1.85 1.77 1.72 1.67 1.63 1.60 1.57 1.55 1.52 1.50 1.49 1.47 1.46 1.44 1.43 1.42 1.41 1.40 1.39 1.38 1.38 1.37 1.36 1.35 1.35 1.34
0
Area = Significance Level
Critical Value
Significance Level
5% 1%
3.84 6.63 3.00 4.61
2.60 3.78 2.37 3.32 2.21 3.02 2.10 2.80 2.01 2.64 1.94 2.51 1.88 2.41 1.83 2.32 1.79 2.25 1.75 2.18 1.72 2.13 1.69 2.08 1.67 2.04 1.64 2.00 1.62 1.97 1.60 1.93 1.59 1.90 1.57 1.88 1.56 1.85 1.54 1.83 1.53 1.81 1.52 1.79 1.51 1.77 1.50 1.76 1.49 1.74 1.48 1.72 1.47 1.71 1.46 1.70
This table contains the 90th, 95th, and 99th percentiles of the Fm, ∞ distribution. These serve as critical values for tests with significance levels of 10%, 5%, and 1%.
762 Appendix
TABLE 5A Critical Values for the Fn1 ,n2 Distribution—10% Significance Level
Denominator
Degrees of
Freedom (n2) 1 2 3 4 5 6 7 8 9 10
Numerator Degrees of Freedom (n1)
1 39.86
2 8.53
3 5.54
4 4.54
5 4.06
6 3.78
7 3.59
8 3.46
9 3.36
10 3.29
11 3.23
12 3.18
13 3.14
14 3.10
15 3.07
16 3.05
17 3.03
18 3.01
19 2.99
20 2.97
21 2.96
22 2.95
23 2.94
24 2.93
25 2.92
26 2.91
27 2.90
28 2.89
29 2.89
30 2.88
60 2.79 90 2.76
120 2.75
H 2.71
49.50 53.59 55.83 9.00 9.16 9.24 5.46 5.39 5.34 4.32 4.19 4.11 3.78 3.62 3.52 3.46 3.29 3.18 3.26 3.07 2.96 3.11 2.92 2.81 3.01 2.81 2.69 2.92 2.73 2.61 2.86 2.66 2.54 2.81 2.61 2.48 2.76 2.56 2.43 2.73 2.52 2.39 2.70 2.49 2.36 2.67 2.46 2.33 2.64 2.44 2.31 2.62 2.42 2.29 2.61 2.40 2.27 2.59 2.38 2.25 2.57 2.36 2.23 2.56 2.35 2.22 2.55 2.34 2.21 2.54 2.33 2.19 2.53 2.32 2.18 2.52 2.31 2.17 2.51 2.30 2.17 2.50 2.29 2.16 2.50 2.28 2.15 2.49 2.28 2.14 2.39 2.18 2.04 2.36 2.15 2.01 2.35 2.13 1.99 2.30 2.08 1.94
57.24 58.20 58.90 59.44 9.29 9.33 9.35 9.37 5.31 5.28 5.27 5.25 4.05 4.01 3.98 3.95 3.45 3.40 3.37 3.34 3.11 3.05 3.01 2.98 2.88 2.83 2.78 2.75 2.73 2.67 2.62 2.59 2.61 2.55 2.51 2.47 2.52 2.46 2.41 2.38 2.45 2.39 2.34 2.30 2.39 2.33 2.28 2.24 2.35 2.28 2.23 2.20 2.31 2.24 2.19 2.15 2.27 2.21 2.16 2.12 2.24 2.18 2.13 2.09 2.22 2.15 2.10 2.06 2.20 2.13 2.08 2.04 2.18 2.11 2.06 2.02 2.16 2.09 2.04 2.00 2.14 2.08 2.02 1.98 2.13 2.06 2.01 1.97 2.11 2.05 1.99 1.95 2.10 2.04 1.98 1.94 2.09 2.02 1.97 1.93 2.08 2.01 1.96 1.92 2.07 2.00 1.95 1.91 2.06 2.00 1.94 1.90 2.06 1.99 1.93 1.89 2.05 1.98 1.93 1.88 1.95 1.87 1.82 1.77 1.91 1.84 1.78 1.74 1.90 1.82 1.77 1.72 1.85 1.77 1.72 1.67
59.86 60.20 9.38 9.39 5.24 5.23 3.94 3.92 3.32 3.30 2.96 2.94 2.72 2.70 2.56 2.54 2.44 2.42 2.35 2.32 2.27 2.25 2.21 2.19 2.16 2.14 2.12 2.10 2.09 2.06 2.06 2.03 2.03 2.00 2.00 1.98 1.98 1.96 1.96 1.94 1.95 1.92 1.93 1.90 1.92 1.89 1.91 1.88 1.89 1.87 1.88 1.86 1.87 1.85 1.87 1.84 1.86 1.83 1.85 1.82 1.74 1.71 1.70 1.67 1.68 1.65 1.63 1.60
Thistablecontainsthe90thpercentileoftheFn1,n2 distribution,whichservesasthecriticalvaluesforatestwitha10%significance level.
Appendix 763
TABLE 5B Critical Values for the Fn1, n2 Distribution—5% Significance Level
Denominator
Degrees of
Freedom (n2) 1 2 3 4 5 6 7 8 9 10
Numerator Degrees of Freedom (n1)
1 161.40 199.50
2 18.51 19.00
3 10.13 9.55
4 7.71 6.94
5 6.61 5.79
6 5.99 5.14
7 5.59 4.74
8 5.32 4.46
9 5.12 4.26
10 4.96 4.10
11 4.84 3.98
12 4.75 3.89
13 4.67 3.81
14 4.60 3.74
15 4.54 3.68
16 4.49 3.63
17 4.45 3.59
18 4.41 3.55
19 4.38 3.52
20 4.35 3.49
21 4.32 3.47
22 4.30 3.44
23 4.28 3.42
24 4.26 3.40
25 4.24 3.39
26 4.23 3.37
27 4.21 3.35
28 4.20 3.34
29 4.18 3.33
30 4.17 3.32
60 4.00 3.15 90 3.95 3.10
120 3.92 3.07
H 3.84 3.00
215.70 224.60 230.20 234.00 19.16 19.25 19.30 19.33 9.28 9.12 9.01 8.94 6.59 6.39 6.26 6.16 5.41 5.19 5.05 4.95 4.76 4.53 4.39 4.28 4.35 4.12 3.97 3.87 4.07 3.84 3.69 3.58 3.86 3.63 3.48 3.37 3.71 3.48 3.33 3.22 3.59 3.36 3.20 3.09 3.49 3.26 3.11 3.00 3.41 3.18 3.03 2.92 3.34 3.11 2.96 2.85 3.29 3.06 2.90 2.79 3.24 3.01 2.85 2.74 3.20 2.96 2.81 2.70 3.16 2.93 2.77 2.66 3.13 2.90 2.74 2.63 3.10 2.87 2.71 2.60 3.07 2.84 2.68 2.57 3.05 2.82 2.66 2.55 3.03 2.80 2.64 2.53 3.01 2.78 2.62 2.51 2.99 2.76 2.60 2.49 2.98 2.74 2.59 2.47 2.96 2.73 2.57 2.46 2.95 2.71 2.56 2.45 2.93 2.70 2.55 2.43 2.92 2.69 2.53 2.42 2.76 2.53 2.37 2.25 2.71 2.47 2.32 2.20 2.68 2.45 2.29 2.18 2.60 2.37 2.21 2.10
236.80 238.90 240.50 19.35 19.37 19.39 8.89 8.85 8.81 6.09 6.04 6.00 4.88 4.82 4.77 4.21 4.15 4.10 3.79 3.73 3.68 3.50 3.44 3.39 3.29 3.23 3.18 3.14 3.07 3.02 3.01 2.95 2.90 2.91 2.85 2.80 2.83 2.77 2.71 2.76 2.70 2.65 2.71 2.64 2.59 2.66 2.59 2.54 2.61 2.55 2.49 2.58 2.51 2.46 2.54 2.48 2.42 2.51 2.45 2.39 2.49 2.42 2.37 2.46 2.40 2.34 2.44 2.37 2.32 2.42 2.36 2.30 2.40 2.34 2.28 2.39 2.32 2.27 2.37 2.31 2.25 2.36 2.29 2.24 2.35 2.28 2.22 2.33 2.27 2.21 2.17 2.10 2.04 2.11 2.04 1.99 2.09 2.02 1.96 2.01 1.94 1.88
241.90 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.32 2.30 2.27 2.25 2.24 2.22 2.20 2.19 2.18 2.16 1.99 1.94 1.91 1.83
Thistablecontainsthe95thpercentileofthedistributionFn1,n2 whichservesasthecriticalvaluesforatestwitha5%significance level.
764 Appendix
TABLE 5C Critical Values for the Fn1, n2 Distribution—1% Significance Level
Denominator
Degrees of
Freedom (n2) 1 2 3 4 5 6 7 8 9 10
Numerator Degrees of Freedom (n1)
1 4052.00
2 98.50
3 34.12
4 21.20
5 16.26
6 13.75
7 12.25
8 11.26
9 10.56
10 10.04
11 9.65
12 9.33
13 9.07
14 8.86
15 8.68
16 8.53
17 8.40
18 8.29
19 8.18
20 8.10
21 8.02
22 7.95
23 7.88
24 7.82
25 7.77
26 7.72
27 7.68
28 7.64
29 7.60
30 7.56
60 7.08 90 6.93
120 6.85
H 6.63
4999.00 5403.00 5624.00 99.00 99.17 99.25 30.82 29.46 28.71 18.00 16.69 15.98 13.27 12.06 11.39 10.92 9.78 9.15
9.55 8.45 7.85 8.65 7.59 7.01 8.02 6.99 6.42 7.56 6.55 5.99 7.21 6.22 5.67 6.93 5.95 5.41 6.70 5.74 5.21 6.51 5.56 5.04 6.36 5.42 4.89 6.23 5.29 4.77 6.11 5.18 4.67 6.01 5.09 4.58 5.93 5.01 4.50 5.85 4.94 4.43 5.78 4.87 4.37 5.72 4.82 4.31 5.66 4.76 4.26 5.61 4.72 4.22 5.57 4.68 4.18 5.53 4.64 4.14 5.49 4.60 4.11 5.45 4.57 4.07 5.42 4.54 4.04 5.39 4.51 4.02 4.98 4.13 3.65 4.85 4.01 3.53 4.79 3.95 3.48 4.61 3.78 3.32
5763.00 5859.00 5928.00 99.30 99.33 99.36 28.24 27.91 27.67 15.52 15.21 14.98 10.97 10.67 10.46
8.75 8.47 8.26 7.46 7.19 6.99 6.63 6.37 6.18 6.06 5.80 5.61 5.64 5.39 5.20 5.32 5.07 4.89 5.06 4.82 4.64 4.86 4.62 4.44 4.69 4.46 4.28 4.56 4.32 4.14 4.44 4.20 4.03 4.34 4.10 3.93 4.25 4.01 3.84 4.17 3.94 3.77 4.10 3.87 3.70 4.04 3.81 3.64 3.99 3.76 3.59 3.94 3.71 3.54 3.90 3.67 3.50 3.85 3.63 3.46 3.82 3.59 3.42 3.78 3.56 3.39 3.75 3.53 3.36 3.73 3.50 3.33 3.70 3.47 3.30 3.34 3.12 2.95 3.23 3.01 2.84 3.17 2.96 2.79 3.02 2.80 2.64
5981.00 6022.00 6055.00 99.37 99.39 99.40 27.49 27.35 27.23 14.80 14.66 14.55 10.29 10.16 10.05
8.10 7.98 7.87 6.84 6.72 6.62 6.03 5.91 5.81 5.47 5.35 5.26 5.06 4.94 4.85 4.74 4.63 4.54 4.50 4.39 4.30 4.30 4.19 4.10 4.14 4.03 3.94 4.00 3.89 3.80 3.89 3.78 3.69 3.79 3.68 3.59 3.71 3.60 3.51 3.63 3.52 3.43 3.56 3.46 3.37 3.51 3.40 3.31 3.45 3.35 3.26 3.41 3.30 3.21 3.36 3.26 3.17 3.32 3.22 3.13 3.29 3.18 3.09 3.26 3.15 3.06 3.23 3.12 3.03 3.20 3.09 3.00 3.17 3.07 2.98 2.82 2.72 2.63 2.72 2.61 2.52 2.66 2.56 2.47 2.51 2.41 2.32
Thistablecontainsthe99thpercentileoftheFn1,n2 distribution,whichservesasthecriticalvaluesforatestwitha1%significance level.
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Glossary
Acceptance region: The set of values of a test statis- tic for which the null hypothesis is accepted (is not rejected).
Adjusted R2(R2): A modified version of R2 that does not necessarily increase when a new regressor is added to the regression.
ADL(p,q): See autoregressive distributed lag (ADL) model.
AIC: See information criterion.
Akaike information criterion (AIC): See information
criterion.
Alternative hypothesis: The hypothesis that is assumed to be true if the null hypothesis is false. The alternative hypothesis is often denoted H1.
ARCH: Seeautoregressiveconditionalheteroskedasticity.
AR(p): See autoregression.
Asymptotic distribution: The approximate sampling distribution of a random variable computed using a large sample. For example, the asymptotic distri- bution of the sample average is normal.
Asymptotic normal distribution: A normal distribu- tion that approximates the sampling distribution of a statistic computed using a large sample.
Attrition: The loss of subjects from a study after assignment to the treatment or control group.
Augmented Dickey–Fuller (ADF) test: A regres- sion-based test for a unit root in an AR(p) model.
Autocorrelation: The correlation between a time series variable and its lagged value. The jth autocorrelation of Y is the correlation between Yt and Yt–j.
Autocovariance: The covariance between a time series variable and its lagged value. The jth autocovariance of Y is the covariance between Yt and Yt–j.
Autoregression: A linear regression model that relates a time series variable to its past (that is, lagged) values. An autoregression with p lagged values as regressors is denoted AR(p).
Autoregressive conditional heteroskedasticity (ARCH): A time series model of conditional heteroskedasticity.
Autoregressive distributed lag (ADL) model: A linear regression model in which the time series variable Yt is expressed as a function of lags of Yt and of another variable, Xt. The model is denoted ADL(p,q), where p denotes the number of lags of Yt and q denotes the number of lags of Xt.
Average causal effect: The population average of the individual causal effects in a heterogeneous popu- lation. Also called the average treatment effect.
Balanced panel: A panel data set with no missing observations, that is, in which the variables are observed for each entity and each time period.
Base specification: A baseline or benchmark regression specification that includes a set of regressors chosen using a combination of expert judgment, economic theory, and knowledge of how the data were collected.
Bayes information criterion: See information criterion. Bernoulli distribution: The probability distribution
of a Bernoulli random variable.
Bernoulli random variable: A random variable that takes on one of two values, 0 and 1. Also known as a binary random variable.
Best linear unbiased estimator (BLUE): An estima- tor that has the smallest variance of any estimator that is a linear function of the sample values Y and is unbiased. Under the Gauss–Markov conditions, the ordinary least squares estimator is the best linear unbiased estimator of the regression coeffi- cients conditional on the values of the regressors.
Bias: The expected value of the difference between an estimator and the parameter that it is estimat- ing. If mnY is an estimator of mY, then the bias of mnY is E(mnY) - mY.
BIC: See information criterion.
Binary variable: A variable that is either 0 or 1. A
binary variable is used to indicate a binary out- come. For example, X is a binary (or indicator, or dummy) variable for a person’s sex if X = 1 if the person is female and X = 0 if the person is male.
Bivariate normal distribution: A generalization of the normal distribution to describe the joint distri- bution of two random variables.
BLUE: See best linear unbiased estimator (BLUE). Break date: The date of a discrete change in popula-
tion time series regression coefficient(s).
Causal effect: The expected effect of a given inter- vention or treatment as measured in an ideal ran- domized controlled experiment.
Central limit theorem: A result in mathematical sta- tistics that says that, under general conditions, the sampling distribution of the standardized sample average is well approximated by a standard nor- mal distribution when the sample size is large.
771
772 Glossary
Chi-squared distribution: The distribution of the sum of m squared independent standard normal random variables. The parameter m is called the degrees of the freedom of the chi-squared distri- bution.
Chow test: A test for a break in a time series regression at a known break date.
Clustered standard errors: A method of computing standard errors that is appropriate for panel data.
Coefficient of determination: See R2.
Cointegration: When two or more time series variables
share a common stochastic trend.
Common trend: A trend shared by two or more time series.
Conditional distribution: The probability distribution of one random variable given that another random variable takes on a particular value.
Conditional expectation: The expected value of one random value given that another random variable takes on a particular value.
Conditional heteroskedasticity: The variance, usually of an error term, depends on other variables.
Conditional mean: The mean of a conditional distri- bution. See conditional expectation.
Conditional mean independence: The conditional expectation of the regression error ui, given the regressors, depends on some but not all of the regressors.
Conditional variance: The variance of a conditional distribution.
Confidence interval (confidence set): An interval (or set) that contains the true value of a population parameter with a prespecified probability when computed over repeated samples.
Confidence level: The prespecified probability that a confidence interval (or set) contains the true value of the parameter.
Consistency: The property that an estimator is con- sistent. See consistent estimator.
Consistent estimator: An estimator that converges in probability to the parameter that it is estimating.
Constant regressor: The regressor associated with the regression intercept; this regressor is always equal to 1.
Constant term: The regression intercept. Continuous random variable: A random variable
that can take on a continuum of values.
Control group: The group that does not receive the treatment or intervention in an experiment.
Control variable: A regressor that controls for an omitted factor that determines the dependent variable.
Convergence in distribution: When a sequence of distributions converges to a limit; a precise defini- tion is given in Section 17.2.
Converge in probability: When a sequence of ran- dom variables converges to a specific value; for example, when the sample average becomes close to the population mean as the sample size increases; see Key Concept 2.6 and Section 17.2.
Correlation: A unit-free measure of the extent to which two random variables move, or vary, together. The correlation (or correlation coef- ficient) between X and Y is sXY>sXsY and is denoted corr(X, Y).
Correlation coefficient: See correlation.
Covariance: A measure of the extent to which two random variables move together. The covariance between X and Y is the expected value E[(X – μX)(Y – μY)] and is denoted cov(X, Y) or by sXY.
Covariance matrix: A matrix composed of the variances and covariances of a vector of random variables.
Critical value: The value of a test statistic for which the test just rejects the null hypothesis at the given significance level.
Cross-sectional data: Data collected for different entities in a single time period.
Cubic regression model: A nonlinear regression function that includes X, X2, and X3 as regressors.
Cumulativedistributionfunction(c.d.f.): Seecumulative probability distribution.
Cumulative dynamic multiplier: The cumulative effect of a unit change in the time series vari- able X on Y. The h-period cumulative dynamic multiplier is the effect of a unit change in Xt on Yt + Yt+1 + g+ Yt+h.
Cumulative probability distribution: A function showing the probability that a random variable is less than or equal to a given number.
Dependent variable: The variable to be explained in a regression or other statistical model; the variable appearing on the left-hand side in a regression.
Deterministic trend: A persistent long-term move- ment of a variable over time that can be repre- sented as a nonrandom function of time.
Dickey–Fuller test: A method for testing for a unit root in a first order autoregression [AR(1)].
Differences estimator: An estimator of the causal effect constructed as the difference in the sample average outcomes between the treatment and con- trol groups.
Differences-in-differences estimator: The average change in Y for those in the treatment group minus the average change in Y for those in the control group.
Discrete random variable: A random variable that takes on discrete values.
Distributed lag model: A regression model in which the regressors are current and lagged values of X.
Dummy variable: See binary variable.
Dummy variable trap: A problem caused by includ- ing a full set of binary variables in a regression together with a constant regressor (intercept), leading to perfect multicollinearity.
Dynamic causal effect: The causal effect of one variable on current and future values of another variable.
Dynamic multiplier: The h-period dynamic multiplier is the effect of a unit change in the time series variable Xt on Yt+h.
Endogenous variable: A variable that is correlated with the error term.
Errors-in-variables bias: The bias in an estimator of a regression coefficient that arises from measurement errors in the regressors.
Error term: The difference between Y and the population regression function, denoted u in this textbook.
Estimate: The numerical value of an estimator computed from data in a specific sample.
Estimators: A function of a sample of data to be drawn randomly from a population. An estimator is a procedure for using sample data to compute an educated guess of the value of a population parameter, such as the population mean.
Exact (finite-sample) distribution: The exact prob- ability distribution of a random variable.
Exact identification: When the number of instrumen- tal variables equals the number of endogenous regressors.
Exogenous variable: A variable that is uncorrelated with the regression error term.
Expectation: See expected value.
Expected value: The long-run average value of a ran- dom variable over many repeated trials or occur- rences. It is the probability-weighted average of all
possible values that the random variable can take on. The expected value of Y is denoted E(Y) and is also called the expectation of Y.
Experimental data: Data obtained from an experi- ment designed to evaluate a treatment or policy or to investigate a causal effect.
Explained sum of squares (ESS): The sum of squared deviations of the predicted values of Yi, Yni from their average; see Equation (4.14).
Explanatory variable: See regressor.
External validity: Inferences and conclusions from a statistical study are externally valid if they can be generalized from the population and the setting studied to other populations and settings.
Feasible GLS estimator: A version of the generalized least squares (GLS) estimator that uses an estima- tor of the conditional variance of the regression errors and covariance between the regression errors at different observations.
Feasible WLS: A version of the weighted least squares (WLS) estimator that uses an estimator of the conditional variance of the regression errors.
First difference: The first difference of a time series variable Yt is Yt – Yt–1, denoted ΔYt.
First-stage regression: The regression of an included endogenous variable on the included exogenous variables, if any, and the instrumental variable(s) in two stage least squares.
Fitted value: See predicted values.
Fixed effects: Binary variables indicating the entity
or time period in a panel data regression.
Fixed effects regression model: A panel data regression that includes entity fixed effects.
Fm,n distribution: The distribution of a ratio of inde- pendent random variables, where the numerator is a chi-squared random variable with m degrees of freedom divided by m and the denominator is a chi-squared random variable with n degrees of freedom divided by n.
Fm,H distribution: The distribution of a random variable with a chi-squared distribution with m degrees of freedom divided by m.
Forecast error: The difference between the value of the variable that actually occurs and its forecasted value.
Forecast interval: An interval that contains the future value of a time series variable with a pre- specified probability.
F-statistic: A statistic used to a test joint hypothesis concerning more than one of the regression coefficients.
Glossary 773
774 Glossary
Functional form misspecification: When the form of the estimated regression function does not match the form of the population regression function; for example, when a linear specification is used but the true population regression function is quadratic.
GARCH: See generalized autoregressive conditional heteroskedasticity.
Gauss–Markov theorem: Mathematical result stating that, under certain conditions, the ordinary least squares estimator is the best linear unbiased estimator of the regression coefficients conditional on the values of the regressors.
Generalized autoregressive conditional heteroskedas- ticity (GARCH): A time series model for condi- tional heteroskedasticity.
Generalized least squares (GLS): A generalization of ordinary least squares that is appropriate when the regression errors have a known form of heteroske- dasticity (in which case GLS is also referred to as weighted least squares, WLS) or a known form of serial correlation.
Generalized method of moments (GMM): A method for estimating parameters by fitting sample moments to population moments that are func- tions of the unknown parameters. Instrumental variables estimators are an important special case.
GMM: See generalized method of moments.
Granger causality test: A procedure for testing whether current and lagged values of one time series help predict future values of another time series.
HAC standard errors: See heteroskedasticity- and autocorrelation-consistent (HAC) standard errors.
Hawthorne effect: See experimental effect.
Heteroskedasticity: The situation in which the vari- ance of the regression error term ui, conditional on the regressors, is not constant.
Heteroskedasticity- and autocorrelation-consistent (HAC) standard errors: Standard errors for ordi- nary least squares estimators that are consistent whether or not the regression errors are hetero- skedastic and autocorrelated.
Heteroskedasticity-robust standard error: A stan- dard error for the ordinary least squares estimator that is appropriate whether or not the error term is homoskedastic or heteroskedastic.
Heteroskedasticity-robust t-statistic: A t-statistic constructed using a heteroskedasticity-robust stan- dard error.
Homoskedasticity: The variance of the error term ui, conditional on the regressors, is constant.
Homoskedasticity-only F-statistic: A form of the F-statistic that is valid only when the regression errors are homoskedastic.
Homoskedasticity-only standard errors: Standard errors for the ordinary least squares estimator that are appropriate only when the error term is homo- skedastic.
Hypothesis test: A procedure for using sample evi- dence to help determine if a specific hypothesis about a population is true or false.
I(0), I(1), and I(2): See order of integration. Identically distributed: When two or more random
variables have the same distribution.
Impact effect: The contemporaneous, or immediate, effect of a unit change in the time series variable Xt on Yt.
Imperfect multicollinearity: The condition in which two or more regressors are highly correlated.
Included endogenous variables: Regressors that are correlated with the error term (usually in the context of instrumental variable regression).
Included exogenous variables: Regressors that are uncorrelated with the error term (usually in the context of instrumental variable regression).
Independence: When knowing the value of one ran- dom variable provides no information about the value of another random variable. Two random variables are independent if their joint distribution is the product of their marginal distributions.
Independently and identically distributed
(i.i.d.): When two or more independent random variables have the same distribution.
Indicator variable: See binary variable.
Information criterion: A statistic used to estimate the number of lagged variables to include in an autoregression or a distributed lag model. Lead- ing examples are the Akaike information criterion (AIC) and the Bayes information criterion (BIC).
Instrument: See instrumental variable.
Instrumental variable: A variable that is correlated with an endogenous regressor (instrument rel- evance) and is uncorrelated with the regression error (instrument exogeneity).
Instrumental variables (IV) regression: A way to obtain a consistent estimator of the unknown coefficients of the population regression function when the regressor, X, is correlated with the error term, u.
Interaction term: A regressor that is formed as the product of two other regressors, such as X1i * X2i.
Intercept: The value of b0 in the linear regression model.
Internal validity: When inferences about causal effects in a statistical study are valid for the popu- lation being studied.
Joint hypothesis: A hypothesis consisting of two or more individual hypotheses, that is, involving more than one restriction on the parameters of a model.
Joint probability distribution: The probability distri- bution determining the probabilities of outcomes involving two or more random variables.
J-statistic: A statistic for testing overidentifying restrictions in instrumental variables regression.
Kurtosis: A measure of how much mass is contained in the tails of a probability distribution.
Lags: The value of a time series variable in a previous time period. The jth lag of Yt is Yt–j.
Law of iterated expectations: A result in probability theory that says that the expected value of Y is the expected value of its conditional expectation given X, that is, that E(Y) = E3E(Y X)4
Law of large numbers: According to this result from probability theory, under general conditions the sample average will be close to the population mean with very high probability when the sample size is large.
Least squares assumptions: The assumptions for the linear regression model listed in Key Concept 4.3 (single variable regression) and Key Concept 6.4 (multiple regression model).
Least squares estimator: An estimator formed by minimizing the sum of squared residuals.
Limited dependent variable: A dependent variable that can take on only a limited set of values. For example, the variable might be a 0–1 binary vari- able or arise from one of the models described in Appendix 11.3.
Linear-log model: A nonlinear regression function in which the dependent variable is Y and the inde- pendent variable is ln(X).
Linear probability model: A regression model in which Y is a binary variable.
Linear regression function: A regression function with a constant slope.
Local average treatment effect: A weighted average treatment effect estimated, for example, by two stage least squares.
Logarithm: A mathematical function defined for a positive argument; its slope is always positive but tends to zero. The natural logarithm is the inverse of the exponential function; that is, X = ln(eX).
Logit regression: A nonlinear regression model for a binary dependent variable in which the population regression function is modeled using the cumula- tive logistic distribution function.
Log-linear model: A nonlinear regression function in which the dependent variable is ln(Y) and the independent variable is X.
Log-log model: A nonlinear regression function in which the dependent variable is ln(Y) and the independent variable is ln(X).
Longitudinal data: See panel data.
Long-run cumulative dynamic multiplier: The cumu- lative long-run effect on the time series variable Y of a change in X.
Marginal probability distribution: Another name for the probability distribution of a random variable Y, which distinguishes the distribution of Y alone (the marginal distribution) from the joint distribu- tion of Y and another random variable.
Maximum likelihood estimator (MLE): An estimator of unknown parameters that is obtained by maxi- mizing the likelihood function; see Appendix 11.2.
Mean: The expected value of a random variable. The mean of Y is denoted μY.
Moments of a distribution: The expected value of a random variable raised to different powers. The rth moment of the random variable Y is E(Yr).
Multicollinearity: See perfect multicollinearity and imperfect multicollinearity.
Multiple regression model: An extension of the single variable regression model that allows Y to depend on k regressors.
Natural experiment: See quasi-experiment.
Natural logarithm: See logarithm.
95% confidence set: A confidence set with a 95% confidence level. See confidence interval.
Nonlinear least squares: The analog of ordinary least squares that applies when the regression function is a nonlinear function of the unknown parameters.
Nonlinear least squares estimator: The estimator obtained by minimizing the sum of squared residu- als when the regression function is nonlinear in the parameters.
Nonlinear regression function: A regression function with a slope that is not constant.
Nonstationary: When the joint distribution of a time series variable and its lags changes over time.
Normal distribution: A commonly used bell-shaped distribution of a continuous random variable.
Glossary 775
776 Glossary
Null hypothesis: The hypothesis being tested in a
hypothesis test, often denoted H0.
Observational data: Data based on observing, or measuring, actual behavior outside an experimen- tal setting.
Observation number: The unique identifier assigned to each entity in a data set.
OLS estimator: See ordinary least squares estimator.
OLS regression line: The regression line with popu- lation coefficients replaced by the ordinary least squares estimators.
OLS residual: The difference between Yi and the ordinary least squares regression line, denoted uni in this textbook.
Omitted variables bias: The bias in an estimator that arises because a variable that is a determinant of Y and is correlated with a regressor has been omitted from the regression.
One-sided alternative hypothesis: The parameter of interest is on one side of the value given by the null hypothesis.
Order of integration: The number of times that a time series variable must be differenced to make it stationary. A time series variable that is inte- grated of order d must be differenced d times and is denoted I(d).
Ordinaryleastsquares(OLS)estimators: Theestimators of the regression intercept and slope(s) that minimizes the sum of squared residuals.
Outlier: An exceptionally large or small value of a random variable.
Overidentification: When the number of instru- mental variables exceeds the number of included endogenous regressors.
Panel data: Data for multiple entities where each entity is observed in two or more time periods.
Parameters: Constants that determine a character- istic of a probability distribution or population regression function.
Partial compliance: The failure of some participants to follow the treatment protocol in a randomized experiment.
Partial effect: The effect on Y of changing one of the regressors, holding the other regressors constant.
Perfect multicollinearity: A situation in which one of the regressors is an exact linear function of the other regressors.
Polynomial regression model: A nonlinear regres- sion function that includes X, X2, . . . and Xr as regressors, where r is an integer.
Population: The group of entities—such as people, companies, or school districts—being studied.
Population coefficients: See population intercept and slope.
Population intercept and slope: The true, or popu- lation, values of b0 (the intercept) and b1 (the slope) in a single variable regression. In a multiple regression, there are multiple slope coefficients (b1, b2, . . . , bk), one for each regressor.
Population multiple regression model: The multiple regression model in Key Concept 6.2.
Population regression line: In a single variable regression, the population regression line is b0 + b1Xi, and in a multiple regression, it is b0 + b1X1i + b2X2i + g + bkXki.
Potential outcomes: The set of outcomes that might occur to an individual (treatment unit) after receiving, or not receiving, an experimental treatment.
Power: The probability that a test correctly rejects the null hypothesis when the alternative is true.
Predicted value: The value of Yi that is predicted by the ordinary least squares regression line, denoted Yni in this textbook.
Price elasticity of demand: The percentage change in the quantity demanded resulting from a 1% increase in price.
Probability: Theproportionofthetimethatanoutcome (or event) from a random experiment will occur in the long run.
Probability density function (p.d.f.): For a continu- ous random variable, the area under the probabil- ity density function between any two points is the probability that the random variable falls between those two points.
Probability distribution: For a discrete random vari- able, a list of all values that a random variable can take on and the probability associated with each of these values.
Probit regression: A nonlinear regression model for a binary dependent variable in which the popu- lation regression function is modeled using the cumulative standard normal distribution function.
Program evaluation: The field of study concerned with estimating the effect of a program, policy, or some other intervention or “treatment.”
Pseudo out-of-sample forecast: A forecast computed over part of the sample using a procedure that is “as if” these sample data have not yet been realized.
p-value (significance pobability): The probability of drawing a statistic at least as adverse to the null hypothesis as the one actually computed,
assuming the null hypothesis is correct. Also called the marginal significance probability, the p-value is the smallest significance level at which the null hypothesis can be rejected.
Quadratic regression model: A nonlinear regression function that includes X and X2 as regressors.
Quasi-experiment: A circumstance in which random- ness is introduced by variations in individual circum- stances that make it appear “as if ” the treatment is randomly assigned.
R2: In a regression, the fraction of the sample vari- ance of the dependent variable that is explained by the regressors.
R 2: See adjusted R2.
Randomized controlled experiment: An experiment in which participants are randomly assigned to a control group, which receives no treatment, or to a treatment group, which receives a treatment.
Random walk: A time series process in which the value of the variable equals its value in the previous period plus an unpredictable error term.
Random walk with drift: A generalization of the random walk in which the change in the variable has a nonzero mean but is otherwise unpredictable.
Regressand: See dependent variable.
Regression discontinuity: A regression involving a quasi-experiment in which treatment depends on whether an observable variable crosses a threshold.
Regression specification: A description of a regres- sion that includes the set of regressors and any nonlinear transformation that has been applied.
Regressor: A variable appearing on the right-hand side of a regression; an independent variable in a regression.
Rejection region: The set of values of a test statistic for which the test rejects the null hypothesis.
Repeated cross-sectional data: A collection of cross- sectional data sets, where each cross-sectional data set corresponds to a different time period.
Restricted regression: A regression in which the coef- ficients are restricted to satisfy some condition. For example, when computing the homoskedasticity- only F-statistic, it is the regression with coefficients restricted to satisfy the null hypothesis.
Rootmeansquaredforecasterror(RMSFE): The square root of the mean of the squared forecast error.
Sample correlation coefficient (sample correlation):
An estimator of the correlation between two random variables.
Glossary 777 Sample covariance: An estimator of the covariance
between two random variables.
Sample selection bias: The bias in an estimator of a regression coefficient that arises when a selection process influences the availability of data and that process is related to the dependent variable. This bias induces correlation between one or more regressors and the regression error.
Sample standard deviation: An estimator of the standard deviation of a random variable.
Sample variance: An estimator of the variance of a random variable.
Sampling distribution: The distribution of a statistic over all possible samples; the distribution arising from repeatedly evaluating the statistic using a series of randomly drawn samples from the same population.
Scatterplot: A plot of n observations on Xi and Yi, in which each observation is represented by the point (Xi, Yi).
Serial correlation: See autocorrelation.
Serially uncorrelated: A time series variable with all
autocorrelations equal to zero.
Significance level: The prespecified rejection prob- ability of a statistical hypothesis test when the null hypothesis is true.
Simple random sampling: When entities are chosen independently from a population using a method that ensures that each entity is equally likely to be chosen.
Simultaneous causality: When, in addition to the causal link of interest from X to Y, there is a causal link from Y to X. Simultaneous causality makes X correlated with the error term in the population regression of interest.
Simultaneous equations: See simultaneous causality.
Size of a test: The probability that a test incorrectly rejects the null hypothesis when the null hypothesis is true.
Skewness: A measure of the aysmmetry of a prob- ability distribution.
Standard deviation: The square root of the variance. The standard deviation of the random variable Y, denoted sY, has the units of Y and is a measure of the spread of the distribution of Y around its mean.
Standard error of an estimator: An estimator of the standard deviation of the estimator.
Standard error of the regression (SER): An estima- tor of the standard deviation of the regression error u.
778 Glossary
Standardizing a random variable: An operation accom- plished by subtracting the mean and dividing by the standard deviation, which produces a random vari- able with a mean of 0 and a standard deviation of 1. The standardized value of Y is (Y – mY)>sY.
Standard normal distribution: The normal distribu- tion with mean equal to 0 and variance equal to 1, denoted N(0, 1).
Stationarity: When the joint distribution of a time series variable and its lagged values does not change over time.
Statistically insignificant: The null hypothesis (typically, that a regression coefficient is zero) cannot be rejected at a given significance level.
Statistically significant: The null hypothesis (typically, that a regression coefficient is zero) is rejected at a given significance level.
Stochastic trend: A persistent but random long-term movement of a variable over time.
Strict exogeneity: The requirement that the regression error has a mean of zero conditional on current, future, and past values of the regressor in a distrib- uted lag model.
Student t distribution: The Student t distribution with m degrees of freedom is the distribution of the ratio of a standard normal random variable, divided by the square root of an independently distributed chi-squared random variable with m degrees of freedom divided by m. As m gets large, the Student t distribution converges to the standard normal distribution.
Sum of squared residuals (SSR): The sum of the squared ordinary least squares residuals.
t-distribution: See Student t distribution.
Test for the difference between two means: A proce- dure for testing whether two populations have the same mean.
Time and entity fixed effects regression model: A panel data regression that includes both entity fixed effects and time fixed effects.
Time effects: Binary variables indicating the time period in a panel data regression.
Time fixed effects: See time effects.
Time series data: Data for the same entity for mul-
tiple time periods.
Total sum of squares (TSS): The sum of squared
deviations of Yi from its average.
t-ratio: See t-statistic.
Treatment effect: The causal effect in an experiment or a quasi-experiment. See causal effect.
Treatment group: The group that receives the treat- ment or intervention in an experiment.
TSLS: See two stage least squares.
t-statistic: A statistic used for hypothesis testing. See
Key Concept 5.1.
Two-sided alternative hypothesis: When, under the alternative hypothesis, the parameter of interest is not equal to the value given by the null hypothesis.
Two stage least squares: An instrumental variable estimator, described in Key Concept 12.2.
Type I error: In hypothesis testing, the error made when the null hypothesis is true but is rejected.
Type II error: In hypothesis testing, the error made when the null hypothesis is false but is not rejected.
Unbalanced panel: A panel data set in which some data are missing.
Unbiased estimator: An estimator with a bias that is equal to zero.
Uncorrelated: Two random variables are uncorre- lated if their correlation is zero.
Underidentification: When the number of instru- mental variables is less than the number of endog- enous regressors.
Unit root: An autoregression with a largest root equal to 1.
Unrestricted regression: When computing the homo- skedasticity-only F-statistic, it is the regression that applies under the alternative hypothesis so that the coefficients are not restricted to satisfy the null hypothesis.
VAR: See vector autoregression.
Variance: The expected value of the squared differ-
ence between a random variable and its mean; the variance of Y is denoted s2Y.
Vector autoregression: A model of k time series vari- ables consisting of k equations, one for each vari- able, in which the regressors in all equations are lagged values of all the variables.
Volatility clustering: When a time series variable exhibits some clustered periods of high variance and other clustered periods of low variance.
Weak instruments: Instrumental variables that have a low correlation with the endogenous regressor(s).
Weighted least squares (WLS): An alternative to ordinary least squares that can be used when the regression error is heteroskedastic and the form of the heteroskedasticity is known or can be esti- mated.
Index
Page numbers followed by f indicate figures; those followed by t indicate tables.
A
Acceptance region, 78
ADF statistic. See Augmented Dickey–Fuller
statistic
ADF test. See Augmented Dickey–Fuller
Augmented Dickey–Fuller statistic, 557–558, 651
critical values for, 559–560, 560t Augmented Dickey–Fuller test, 557–560,
651–654
Autocorrelated errors, OLS estimator
errors-in-variable, 322–325, 340
in estimators, 67, 68, 70
in OLS estimator, 182–189, 327–328,
720–721
omitted variable. See Omitted variable
bias
sample selection. See Sample selection
bias
simultaneous causality and, 326–329,
340–341
simultaneous equations, 328–329 survivorship, 327
BIC. See Bayes information criterion Binary regressors, 155–157
Binary variables
dependent variable in regression, 385–423
applications of, 402–409 linear probability model and,
388–391
logit model and, 396–402
maximum likelihood estimator and,
400–401, 418–420
measures of fit and, 401–402 nonlinear least squares estimator
and, 399
overview of, 386–388
probit model and, 391–396, 398–402 regression R2 and, 389
interaction between, 280
interaction with continuous variable,
282–286, 283f
Bivariate normal distribution, 38–41, 702 The Black Swan (Taleb), 40
BLUE (Best Linear Unbiased
Estimator), 69 GLS estimator as, 613
OLS estimator as, 164–165 Bonferroni test, 251–253 Bound, John, 446
Break
definition of, 561–562 F-statistic testing for, 567f problems caused by, 562 testing for, 562–567
Break date, 562–567 known, 563 unknown, 563–566
Bureau of Labor Statistics, 71
C
Capital asset pricing model, 120 Card, David, 497
Cardiac catheterization outcomes,
456–458, 495, 508–509
test
Adjusted R2, 196–198, 237–238 distribution and, 602–604
ADL model, 539–540, 569–572, 571t, 634–637
applications of, 621–624 coefficient estimation for, 610–611,
614–615
GLS estimation of, 615–616 OLS estimation of, 608, 615–616
ADL(p, q) model, 539–540
Akaike information criterion (AIC), 549
lag length estimator, consistency of, 588 Alcohol taxes and drunk driving, 352–354,
353f Alternative hypothesis
one-sided, 79–80
two-sided, 71, 79 Anderson–Rubin test, 472–473 Angrist, Joshua, 446, 495, 503, 504 Approximations, large-sample. See
Large-sample approximations AR(1) errors, distribution lag model and,
607–610
AR(1) forecast, iterated, 644–645 AR(1) model
Dickey–Fuller test in, 557
stationarity in, 584–585
ARCH model. See Autoregressive condi-
Autocorrelation, 366–367, 528–529 Autocovariance, 528–530 Autoregression, 532–537. See also Time
series regression bias in, 554
definition of, 531
first order, 531–534
forecast errors and, 533
lag length of, 535
order of, 547–550
pth order, 534–537
vector. See Vector autoregression
Autoregressive conditional heteroskedas- ticity (ARCH) model, 669
volatility clustering and, 666–667 Autoregressive distributed lag model,
539–540, 569–572, 570t Autoregressive distributed lag regressors,
437–438
Autoregressive errors, distributed lag
model and, 614
Autoregressive model, of stock market,
536–537, 536t, 570–572 Autoregressive-moving average (ARMA)
model, 586–587 Average causal effect, 477
Average treatment effect, 477, 506–509 ARMA model. See Autoregressive-moving B
tional heteroskedasticity model AR forecast, iterated multiperiod, 646
average model AR(p) model, 534–537
Asymptotic distribution, 47
central limit theorem and, 682–683 consistency and, 680–682, 685–686 continuous mapping theorem and,
683–684
convergence and, 680–682 definition of, 682
normal, 52
of OLS estimator, 685, 710–713 Slutsky’s theorem and, 683–684 of TSLS estimator, 730–731
of t-statistic, 687
Asymptotic normality, 711–712 Attrition
in quasi-experiments, 503
in randomized controlled experiments,
481
Balanced panel, 351
Bank of England, 546
Bayes information criterion (BIC),
548–549, 549t
lag length estimator, consistency of,
587–588
“Before and after” analysis, 354–356, 360 Behavioral economics, 90
Bernoulli distribution, 17
Bernoulli random variable, 17
maximum likelihood estimator for, 419
variance of, 22
Best Linear Unbiased Estimator. See
BLUE (Best Linear Unbiased
Estimator) “Beta” of stock, 120
Bias
in autoregression, 554 consistency and, 67–68, 680–682
779
780 Index Cauchy–Schwarz inequality, 704
Causal effect. See also Treatment effect average, 477
definition of, 6, 85, 504, 593 difference-of-means estimation of,
84–85
dynamic, 589–636. See also Dynamic
causal effects
estimation of, 6–7, 84–85, 478–479 heterogenous, 504–509, 518–519 simultaneous causality and, 326–329 time series data and, 593–594
c.d.f. See Cumulative distribution function Censored regression models, 421–422 Central limit theorem, 50–52, 51f, 130–131,
682–683, 684 multivariate, 710–711
Chebychev’s inequality, 680–681, 703–704 Chi-squared distribution, 41, 702
Chow test, 564–566
Cigarette taxes and smoking
instrumental variables and, 433–435, 441–442, 448–453
panel data on, 11–12, 467
Classical measurement error model, 323 Class size. See Student–teacher ratio CLR test, 472–473
Clustered standard errors, 367–368 Cochrane–Orcutt estimator, 612–613 Coefficient, 146–153
ADL, 610–611, 614–615
with binary variables, 281 cointegrating, 657, 659–660 confidence interval for, 153–155,
219–220
confidence sets for, 219–220, 231–232,
715
distributed lag, 611–613, 615–616 exactly identified, 435
in general IV regression model, 435 interpretation of in linear regression,
155–157
joint hypothesis testing for, 217–219,
222–231, 713–715
of linear regression model, 112,
114–116 multiple, 189
confidence sets for, 231–232, 232f
single restrictions involving, 229–230 in nonlinear regression, 265, 269 overidentified, 435
population, 73–74, 112, 114–116
of probit regression model, 395–396 probit regression model and, 395–396 single restrictions involving, 229–231 slope, 189–190
stability, QRL test for, 563–565 underidentified, 435
vector autoregression and, 640–641
Cointegrating coefficient, 657 estimation of, 659–660
Cointegration, 656–664, 669 applications of, 662–664 definition of, 656, 657
error correction and, 656–658, 661 for multiple variables, 661–662 stochastic trends and, 656–664 testing for, 658–660
time series regression and, 656–664 Cold weather and orange juice prices,
590–593, 591f data set for, 634
time series regression and, 616–624 College graduates, earnings of
gender and, 33–34, 33t, 34t, 86–87, 162, 162f, 279–281, 287–288
return to school and, 287–288, 446 Common stochastic trend, 556, 656 Compliance, partial, 480–481, 502–503 Conditional distribution, 27–28, 28t
regression line and, 124–125, 125f Conditional expectations, 28–29
difference of, 85
Conditional likelihood ratio test, 472–473 Conditional mean, 28–29
zero conditional mean assumption, GLS estimator and, 726–728
correlation and, 32, 126
in fixed effects regression model, 365 independence, 235, 253–255
in quasi-difference model, 609–610 in time series regression, 541
Conditional variance, 30 Confidence ellipse, 231 Confidence interval
difference-of-means, 84
vs. forecast interval, 545
in multiple regression, 231–232
for OLS estimator, 153–155
for population means, 80–82
for predicted effects, 713
for regression coefficient, 153–155 for regression with binary regressor,
156–157
for single coefficient, 219–220 for slope, 153–155
Confidence level, 80, 153 Confidence sets, 80
Anderson–Rubin, 472–473
for multiple coefficients, 231–232, 715 for single coefficient, 219–220
weak instruments and, 472–473
Consistency, 48
asymptotic distribution and, 680–682,
685–686
bias and, 67–68, 680–682
of estimators, 67, 68
of heteroskedasticity-robust standard
error, 685–686
law of large numbers and, 48–50,
680–682
of OLS estimators, 131, 685
in probability, 680
of sample variance, 75–76, 108 Consistent estimators, 67, 68, 131, 680–682,
685
Constant regressor, in multiple regression,
191
Constant term, in multiple regression, 191 Contemporaneous dynamic multiplier, 600 Continuous mapping theorem, 683–684 Continuous random variables, 15
conditional, 701
distribution of, 700–703 expected value of, 21
interaction between, 286–290 interaction with binary variables,
282–286, 283f normal, 702–703 bivariate, 702
probabilities and moments of, 701 Control group, 6
Control variables
in instrumental variables regression, 435–436
in multiple regression, 189, 233–236
TSLS with, 473–474 Convergence
in distribution, 682–683
in probability, 48, 680–682 Correlation. See also Autocorrelation
conditional mean and, 32, 126 error term, across observations,
329–330, 341
between random variables, 32 sample, 92–95
serial, 366–367, 528–529
serial error term, 606
Count data, 423 Covariance, 32–35 definition of, 31
matrix, 750 sample, 92–95, 432
consistency of, 93–95
Coverage probability, 82
Crime rates and incarceration, 454–455 Critical value, 78, 80
Cross-sectional data, 8–9, 9t
repeated, 499
Cubic regression model, 267–269, 278 Cumulative distribution function (c.d.f.),
17
Cumulative dynamic multiplier, 600–601,
617–620, 621t long-run, 601
Cumulative probability distribution.
See also Probability distributions of continuous random variable, 18f, 19
definition of, 16–17
of discrete random variable, 16–17 normal, 37
Current Population Survey, 71 Curves, logistic, 309–310
D
Data
California student–teacher ratio and
test scores, 8–9, 9t, 141, 183, 333,
337–339, 338t
for cold weather and orange juice
prices, 634 count, 423
cross-sectional, 8–9, 9t
discrete choice, 423
for drunk driving, 380
experimental, 7–8
internal validity and, 325–326 Massachusetts student–teacher ratio
and test scores, 333, 334t, 335t,
336t, 337–339, 338t, 349 mortgage lending and race, 418 observational, 7–8, 491–493, 504
panel. See Panel data
Project STAR, 486–491, 518
in randomized controlled experiments,
478–479, 486–491, 486t, 488t, 490t,
493t, 520–521
repeated cross-sectional, 499
sources and types of, 7–12 Tennessee student–teacher ratio and
test scores, 7–8
time series. See Time series data
Degrees of freedom, 75, 689–690 Demand elasticity
definition of, 427
price and, 3–4, 269, 427–430, 433–434 Density function, 18f, 19
Dependent variable, 112
binary, in regression, 385–423 applications of, 402–409
linear probability model and, 388–391 logit model and, 396–402
maximum likelihood estimator and,
400–401, 418–420
measures of fit and, 401–402 nonlinear least squares estimator
and, 399
overview of, 386–388
probit model and, 391–396, 398–402 regression R2 and, 389
Deterministic trend, 552 DF-GLS test, 651–652, 654t
applications of, 654
critical values for, 653, 654t
vs. Dickey–Fuller test, 653, 654
DF statistic. See Dickey–Fuller statistic DF test. See Dickey–Fuller test Dickey, David, 557
Dickey–Fuller regression, 556–560
nonnormal distributions and, 654–655 Dickey–Fuller (DF) statistic, 557 augmented, 558–560, 560t, 651
Dickey–Fuller (DF) test. See also DF-GLS test
augmented, 558–560, 651–653
vs. DF-GLS test, 653, 654
for stochastic trends, 557–559 Difference of means
confidence interval for, 84
in estimating causal effects, 84–85 hypothesis testing for, 82–84 t-statistic testing for, 88–90
Differences estimator, 478 Differences-in-differences estimator,
496–499, 498f
with additional regressors, 498–499,
498f
repeated cross-sectional data for, 499
Direct multiperiod forecasts, 645–647 vs. iterated multiperiod forecasts,
648–649 Discrete choice data, 423
Discrete random variable, 15 probability distribution of, 16, 16t, 17f
Distributed lag coefficients, GLS, 611–613, 615–616
Distributed lag model
with additional lags and AR(p) errors,
613–616
applications of, 616–624
with AR(1) errors, 607–610 assumptions of, 598–600 autocorrelation and, 599–600
with autoregressive errors, 614 dynamic causal effects and, 594–596 exogeneity and, 596–598
extension to multiple regression, 606 generalized least squares and, 606–607 inference and, 599–600
ordinary least squares and, 606–607 standard error and, 599–600
Distribution
asymptotic. See Asymptotic distribution Bernoulli, 17
chi-squared, 41, 702
conditional, 27–28, 28t
regression line and, 124–125, 125f continuous random, 19, 700–703 convergence in, 682–683
cumulative. See Cumulative probability
distribution exact, 47
F, 42, 703
finite-sample, 47
of F-statistic, 719–720, 752–753
of GMM J-statistic, 756
identical, 44
independent, 31, 44
independent and identical (i.i.d.), 44,
237–238
joint probability, 26–27, 26t, 28t
likelihood function and, 400 kurtosis of, 24f, 25 large-sample normal, 130–132
nonstationarity and, 654–655 leptokurtic, 25
Index 781 marginal probability, 26t, 27
moments of, 23–25 multivariate, 38–41, 749–751 normal, 36–37, 36f, 38f
approximate, 47
asymptotic, 52
bivariate, 38–41, 702 conditional, 701
large-sample, 130–132, 654–655 multivariate, 38–41
of OLS estimator, 201–202
population, variance of, p-value and, 73–74 probability. See Probability distributions of regression statistics with normal errors,
716–720
sampling. See Sampling distribution skewness of, 23–25, 24f
student t, 41–42, 703
of t-statistic, 76, 687, 719, 752–753
District income and test scores, 258–261, 263–264, 277–278, 312–313
Diversification, 46
Dollar/pound exchange rate, 530 DOLS estimator, 660
for multiple variables, 661 Double-blind experiments, 482
Dow Jones Industrial Average, 39–40 Drift, random walk with, 553
Drunk driving
data set for, 380
fixed effects regression and, 352–354,
353f, 361, 364
panel data for, 352–354, 353f, 361 regression analysis, 369f
traffic deaths and, 368–372
Dummy variable, 155
Dummy variable trap, 204 Dynamic causal effects, 589–636
applications of, 616–624
assumption of exogeneity and, 624–626 distributed lag model and, 592–600.
See also Distributed lag model estimation with exogenous regressors,
597–601
estimation with strictly exogenous
regressors, 606–616
exogeneity and, 596–598
GLS estimation of, 606–607, 611–613,
615–616
HAC standard errors and, 601–606 measurement of, 595–596
OLS estimation of, 615–616
in time series data, 593–595
Dynamic multipliers
applications of, 616–624, 621t cumulative, 600–601
definition of, 600
long-run cumulative, 601
stability of, 621, 623
zero period (contemporaneous), 600
Dynamic OLS estimator, 660
782 Index E
Economic journals, demand for, 290–292 Economics, behavioral, 90
Economic time series. See under Time
series
Efficiency of estimators, 67–69, 160, 163,
165–166, 720–722, 754–756 Efficient capital market theory, 537 Efficient GMM estimator, 734–737 EG-ADF test, 659
applications of, 663–664
for multiple variables, 661–662 Eigenvalues, 749
Eigenvectors, 749
Elasticity
of demand, 3–4, 269, 427–430, 433–434 from nonlinear regression function,
313–314
of supply, 427–430
Employment rates and minimum wage, 497 Endogenous variables, 426
definition of, 425
in general IV regression model, 437–438 weak instruments and, 443–445, 446,
471–473 Engle, Robert, 669
Engle–Granger ADF test, 659
Entity and time fixed effects regression,
363–364
Entity fixed effects regression, 357, 362 Error correction
cointegration and, 656–658
vector, 657–658, 663–664, 669 Error correction term, 656–658 Errors-in-variable bias, 322–325, 340 Error term, 112, 113
in AR(p) model, 533–535 correlation across observations,
329–330, 341 heteroskedastic/homoskedastic,
158–163, 191 serially correlated, 606
Estimates, definition of, 67 Estimator
AIC, 588
AIC lag length, 588
Best Linear Unbiased, 69, 164–165, 613 BIC, 587–588
Cochrane–Orcutt, 612–613
consistent, 67, 68, 131, 680–682, 685 definition of, 67
differences, 478 differences-in-differences, 496–499, 498f DOLS, 660, 661
efficiency of, 67–69, 160, 163, 165–166,
720–722, 754–756
feasible GLS, 612, 726
feasible GMM, 734–737
feasible WLS, 692
generalized least squares. See Generalized
least squares estimator
HAC, 604–606
infeasible GLS, 612, 726
infeasible WLS, 691
instrumental variables. See Instrumental
variables estimator lag length, 587–588
least absolute deviations, 166
least squares. See Least squares estimator LIML, 473
linear conditionally unbiased, 164–165,
720–721
maximum likelihood, 400–402, 418–420 Newey–West variance, 605–606 nonlinear least squares, 312–313, 399 omitted variable bias in, 182–189,
319–321
ordinary least squares. See Ordinary
least squares estimator
pooled variance, 88–89
properties of, 66–68
in quasi-experiments, 496–500 regression discontinuity, 500–502, 501f sample average as, 68–70
standard error as, 75
standard error of regression as, 122 TSLS, 426–427, 437–442
unbiased, 67, 164–165
variance of, 67
weighted, 165–166
European Central Bank, 4 Events
definition of, 15
probability of, 16
Exact distribution, 47
Exactly identified coefficients, 435 Exchange rate, dollar/pound, 530, 530f Exogeneity
plausibility of, 624–626
strict, 596–598
Exogenous instruments, weak, 443–445,
446, 471–473
Exogenous regressors, in dynamic
causal effect estimation, 597–601,
606–616 Exogenous variables
definition of, 425
included, 435–437
in IV regression, 425–426, 434–437,
445–448
test of overriding restrictions and, 447–448
Expectations conditional, 28–29, 85 iterated, law of, 29–30 of random variable, 19
Expectations theory of the term structure of interest rates, 658–659
Expected value
of Bernoulli random variable, 21 of continuous random variable, 21 definition of, 19
of random variable, 19–20, 25
Experimental data, 7–8. See also Data Experiments. See Quasi-experiments; Ran-
domized controlled experiments Explained sum of squares, 121–122 Exponential function, natural, 269–270 Exponential growth, 310–311, 312f Extended least squares assumptions,
677–678, 707 External validity
assessment of, 318
definition of, 316 experimental design and, 318 in forecasting, 331–332 population of interest and, 316 population studied and, 315
in quasi-experiments, 504 threats to, 317–318, 483–484
F
Fama, Eugene, 670
F distribution, 42, 703
Feasible GLS estimator, 612, 726 Feasible GMM estimator, 734–737 Feasible WLS estimator, 692 Federal Reserve Bank of Boston, 3 Federal Reserve Board, 4 Financial diversification, 46 Finite-sample distribution, 47
First difference, in time series data,
525–528 First lag, 526
First order autoregression, 531–534 First-stage F-statistic, 444 First-stage regression, 438
Fixed effects
entity, 357
time, 362
Fixed effects regression
assumptions of, 365–368
autocorrelation in, 366–367
“before and after” analysis and, 354–356 conditional mean and, 365
definition of, 359
entity, 357, 362
entity and time, 363–364
estimation and inference and, 359–361 large outliers and, 366
multicollinearity and, 366
OLS estimator and, 359–360
panel data in, 357–361
sampling distribution in, 360
serial correlation in, 366–367
standard errors in, 360, 367–368,
380–384 time, 361–364
Forecast error
in AR(p) model, 533–535 pseudo, 567–573
root mean squared, 533, 544
pseudo out-of-sample forecasting and, 567–568
Forecasting
AR(1), iterated, 644–645 autoregression in, 532–538
causality and, 7
direct multiperiod, 645–649
forecast errors and, 533
of inflation, 4–5, 546
interest rates, 537–540 internal/external validity and, 331–332 interval, 545–547
iterated multiperiod, 643–646, 648–649 momentum, 536–537
multiperiod, 643–649
multiple regression in, 331–332
vs. predicted value, 533
pseudo out-of-sample, 567–573 regression models in, 331–332, 523–524 of stock returns, 536–537
time series data and, 524–532 uncertainty, 544–545
vector autoregression in, 638–643
Fraction correctly predicted, 401–402 Frisch–Waugh theorem, 215–216 F-statistic
application of, 226–227
break testing, 567f
confidence sets for multiple coefficients
and, 231–232, 715 definition of, 224
distribution of, 719–720, 752–753 first-stage, 444 heteroskedasticity-robust, 225 homoskedasticity-only, 227–229,
719–720
for joint hypothesis testing, 224–226,
714–715
lag length selection and, 547–548,
550–551
order of autoregression and, 547–548 overall regression, 226
p-value of, 225–226
QLR statistic and, 563–565
Wald, 719–720
Fuller, Wayne, 557
Functional form misspecification, 321–322 Fuzzy regression discontinuity, designs,
501–502
G
GARCH model, 666–667, 668f Gauss–Markov conditions, 720–722 Gauss–Markov theorem, 163–166,
178–181, 721–722 proof of, 753–754
GDP
quarterly Japanese, 530f, 531 U.S. growth rate, 525f, 527t
Gender gap. See College graduates, earnings of
General IV regression, 426, 434 definition of, 436
instrument exogeneity and relevance in, 438–439
regression coefficients in, 435 terminology for, 436
TSLS estimator in, 437–438
Generalized ARCH model (GARCH), 666–667
Generalized least squares (GLS) estimator, 606–607, 611–613,
634–637, 722–728
advantages and disadvantages of, 615–616 applications of, 621–624
assumptions for, 723–725 Cochrane–Orcutt method and, 612 efficiency of, 613
feasible, 612, 726
infeasible, 612, 726
nonlinear least squares interpretation
of, 612–613
vs. OLS estimator, 615–616
when Ω contains unknown parameters,
726
when Ω is known, 725–726
zero conditional mean assumption and,
726–728
Generalized method of moments (GMM)
estimator, 670, 734–737 efficiency of, 756
GLS estimator. See Generalized least squares estimator
GMM. See Generalized method of mo- ments estimator
GMM J-statistic, 736 distribution of, 756
Granger, Clive, 669
Granger causality, 543–544
Gross domestic product. See GDP Growth rates, time series regression and,
525–528
H
HAC estimator, 604–605 weights for, 606
HAC standard error, 367–368, 592, 604–606, 607
in direct multiperiod regression, 647 HAC truncation parameter, 605, 620 HAC variance formula, 604–606 Hansen, Lars Peter, 670
Hawthorne effect, 482
Heart attack treatment outcomes, 456–458,
495, 508–509 Heckman, James, 410
Heterogeneous populations, experimental/ quasi-experimental estimates in,
504–509, 518–519 Heteroskedasticity
autoregressive conditional, 666–667, 669 definition of, 158
error term correlation and, 329–330,
341–342
Index 783 inconsistent standard error and, 329
internal validity and, 329–330, 341–342 of known functional form, 691–694 mathematical implications of, 160
in multiple regression, 157–163, 158f,
177, 191
weighted least squares and, 690–695
Heteroskedasticity and autocorrelation- consistent estimator. See HAC
estimator
Heteroskedasticity and autocorrelation-
consistent standard error. See
HAC standard error Heteroskedasticity-consistent standard
error, 601, 604–606 Heteroskedasticity-robust F-statistic, 225 Heteroskedasticity-robust J-statistic, 736 Heteroskedasticity-robust standard error,
177, 329, 330
asymptotic distribution and, 685–686 consistency of, 685–686
in multiple regression, 712–713
vs. weighted least squares, 694–695
High school graduates, earnings of, 33–34, 33t, 34t
Homoskedasticity, 157–163, 177–178 definition of, 159
J-statistic under, 733, 754–756 mathematical implications of, 160
in multiple regression, 191, 731–734 TSLS estimator and, 730–734, 755–756
Homoskedasticity-only F-statistic, 227–229, 719–720
Homoskedasticity-only standard error, 160, 163, 177–178, 718
Homoskedasticity-only variance formula, 160, 177–178
Homoskedastic normal regression assump- tions, 166
Hypothesis
alternative, 71, 79–80
joint, Bonferroni test of, 251–253 null. See Null hypothesis one-sided
alternative, 79–80
for slope, 150–152 two-sided
alternative, 71, 79
for slope, 147–150 Hypothesis testing, 71
acceptance region in, 78 confidence intervals in, 84 coverage probability and, 82 critical value in, 78
degrees of freedom and, 74 difference-of-means, 82–84
for instrumental variables regression,
472–473
for intercept, 152–153
for joint hypotheses, 217–219, 222–231, 713–715
784 Index
Hypothesis testing (continued)
for multiple coefficients, 222–231 for population means, 147–148 power of test in, 78
with prespecified significance level,
77–79
p-value and, 72–74, 76–77
for regression with binary regressor,
156–157
rejection region in, 78, 80
sample standard deviation and, 74 sample variance and, 74–75 significance level in, 78–79
for single coefficient, 217–219
size of test in, 78
standard error and, 74–76 terminology of, 78
t-statistic in, 76–77, 80, 87–91
type I/II errors in, 78
weak instruments and, 472–473
I
I(0), I(1), and I(2), 650
Ideal randomized controlled experiments,
6–7
Idempotent matrices, 749
Identical distribution, 44
i.i.d. See Independent and identical
distributed
Immigration and labor market, 494, 504 Impact effect, 600
Imperfect multicollinearity, 205–206 Incarceration and crime reduction,
454–455
Included exogenous variables, 435–437 Independent and identical distributed
(i.i.d.), 44, 237–238, 722 Independent variables, 31, 112
Indicator variable, 155
Infeasible GLS estimator, 612, 726 Infeasible WLS estimator, 691 Inflation
forecasting, 4–5, 546 monetary policy and, 626 oil prices and, 625–626 price levels and, 651, 652f
Inflation Report, 546 Information criterion Akaike, 549, 588
Bayes, 548–549, 549t, 587–588 calculation of, 549
for lag length selection, 548–549, 549t,
551–552 multiple-equation, 641 Schwartz, 548, 549t single-equation, 548–551
Instrumental variables (IV) estimator, 728–734
with heterogenous causal effects, 506–509
in matrix form, 729
in quasi-experiments, 499–500
using linear combinations of Z, 731–733 Instrumental variables (IV) regression
applications of, 427–431, 433–435, 448–453, 506–509
assumptions of, 425–426, 439–440 causal effect in heterogeneous popula-
tions and, 518–519
confidence set for, 472–473 development of, 428
endogenous and exogenous variables
and, 425–426
exogeneity of, 426, 434, 438–439
in general IV regression model, 426,
434–439
general model for. See General IV
regression
hypothesis testing for, 472–473 instrument exogeneity and, 426, 434,
439
instrument relevance and, 426, 434, 439,
443–445
instrument validity and, 426, 434,
439–448
randomly assigned, 503–504
rationale for, 427
regression, 424–475
relevance of, 426, 434, 438–439
with single regressor and single instru-
ment, 425–435 sources of, 453–458
in treatment effect estimation, 481 TSLS estimator in, 426–427, 437–442. See also Two stage least squares
validity of, 433, 438–439
variables in, 435–442
weak, 443–445, 446, 471–473
weak instruments and, 443–445, 446,
471–473
Instrument exogeneity condition, 426, 434,
439, 445–448
Instrument relevance condition, 426, 434,
439, 443–445
Instrument validity, 426, 433–434, 439
assessment of, 442–448
in quasi-experiments, 503–504 sources of, 453–458
Integrated of order d, 650 Interacted regressor, 280 Interaction terms, 280 Intercept, 112, 113
hypothesis testing for, 152–153
in multiple regression, 189 Interest rates
cointegration tests and, 662–664 forecasting, 537–540
term structure of, expectations theory
of, 658–659
unit root tests and, 662–664
vector error correction model and,
663–664
Internal validity, 315–343
definition of, 316
errors-in-variable bias and, 322–325 in forecasting, 331–332
functional form misspecification and, 321–322
heteroskedasticity and, 329–330, 341–342
measurement error and, 322–325 missing data and, 325–326
omitted variable bias and, 319–321 population of interest and, 316 population studied and, 315
in quasi-experiments, 502–504 sample selection bias and, 326 threats to, 316–317, 319–330, 479–483
Iterated Cochrane–Orcutt estimator, 612–613
Iterated multiperiod forecasts, 643–645 AR, 646
vs. direct multiperiod forecasts, 648–649
VAR, 646
IV estimator. See Instrumental variables
estimator
J
Japanese quarterly GDP, 530f, 531 Johansen, Soren, 660, 661
Joint hypotheses
Bonferroni test of, 251–253 definition of, 223
in matrix form, 714
null, 222–223
testing of, 713
applications of, 226–227
with F-statistic, 224–226, 714–715 joint null hypothesis and, 222–223 on multiple coefficients, 222–224,
229–231
one-at-a-time approach for, 223–224 for single restriction with multiple
coefficients, 229–231 with t-statistic, 223–224
Joint probability distribution, 26–27, 26t, 28t likelihood function and, 400
Journals, economic, demand for, 290–292 J-statistic, 447–448
applications of, 452, 508
asymptotic distribution of, 733–734,
755–756 GMM, 736
heteroskedasticity-robust, 736 under homoskedasticity, 733–734,
755–756
jth autocovariance, 529
jth lag, 526 K
Klein, Joseph, 428 Krueger, Alan, 446, 497 Kurtosis, 24f, 25, 37
L
Lag length
of autoregression, 535 estimators, 587–588 selection
F-statistic approach in, 547–548, 550–551
information criteria for, 548–549, 549t, 551–552
in time series regression with multiple predictors, 550–551
in vector autoregression, 641
Lag operator notation, 585–586, 634–637 Lag polynomial, 585–586
Lags, 525–528
in ADL model, 540
Landon, Alf, 70, 326
Large numbers law. See Law of large
numbers Large-sample approximations
central limit theorem and, 50–52
law of large numbers and, 47–50 Large-sample distribution, of TSLS
estimator, 432–433, 468–471 Large-sample normal distribution, 130–132
nonstationarity and, 654–655 Law of iterated expectations, 29–30 Law of large numbers, 47, 48f
asymptotic distribution and, 680–682, 684
consistency and, 48–50, 680–682
proof of, 680–681
Least absolute deviations estimators, 166 Least squares assumptions, 233–234
conditional distribution has zero mean, 124–126, 199
extended, 677–678, 708–709
in multiple regression, 199–201
no perfect multicollinearity, 200 outliers are unlikely, 126–127, 199–200 regressors are independently and
identically distributed, 126–127, 199 in single-regressor model, 124–129
Least squares estimator, 69–70, 107.
See also Generalized least squares estimator; Ordinary least squares estimator
nonlinear, 311–312, 399
weighted, 690–695
Leptokurtic distribution, 25 Likelihood function, 400
Limited dependent variable, 386 Limited dependent variable models
binary dependent variable, 385–421 censored models, 421–422
count data and, 423
discrete choice data and, 423
logit regression model, 396–402 ordered responses and, 423 probit regression model, 391–396,
sample selection models, 422
truncated models, 421–422
Limited information maximum likelihood
(LIML) estimator, 473
Linear conditionally unbiased estimators,
164–165, 720–721
Linear function of random variable, mean
and variance of, 22–23
Linear-log model, 271–272, 273f, 275–276,
312f
vs. cubic model, 278
Linear probability model, 388 applications of, 389–390 definition of, 389
limitations of, 390–391, 398
vs. probit and logit models, 398
Linear regression. See also Regression coefficients in, 112, 114–116 independent and identical distribution
and, 127–128
least squares assumptions for, 124–129 measures of fit and, 121–124
multiple, 182–207. See also Multiple
regression
OLS estimator and, 116–121 omitted variable bias and, 182–189,
195
outliers in, 127–128
single-regressor, 109–169, 676–704. See also Single-regressor linear regression
terminology of, 112, 113
Local average treatment effect, 506–509 Logarithmic regression, 271–278
linear-log, 271–272, 273f, 275–276, 312f
log-linear, 272–273, 275–276, 275f log-log, 274–276, 275f
selection of, 275–276
Logarithms
natural, 269–270
in nonlinear regression, 269–278 percentages and, 270–271
in time series regression, 525–528
Logistic (logit) regression, 309–310, 396–398
applications of, 397
definition of, 396
estimation and inference in, 398–402 vs. linear probability model, 398 maximum likelihood estimator for,
400–401, 420
measures of fit in, 401–402 multinomial, 423
nonlinear least squares estimator and,
399
vs. probit model, 396–397, 397f
Log-linear model, 272–273, 275–276, 275f Longitudinal data. See Panel data Long-run cumulative dynamic multiplier,
601
M
Madrian, Brigitte, 90
Marginal probability distribution, 26t, 27 Matrix
algebra, 746–749 covariance, 750 idempotent, 749 inverse, 748–750 square roots, 748–750
Matrix form
for IV estimator, 729
for joint hypotheses, 714
for multiple regression model, 706–708 for OLS estimator, 709–710, 716–717 for standard error, 717
for TSLS estimator, 730–734
Maximum likelihood estimator, 402, 418–420
for i.i.d. Bernoulli random variables, 419
for logit model, 400–401, 420
for probit model, 400–401, 419–420 McFadden, Daniel, 410
Mean
conditional, 28–29, 32, 126, 235, 253–255, 365, 541, 609–610, 726–728
definition of, 19
population. See Population means of random variable, 19, 22–23, 25,
28–29, 32–35 sample, 44–47
of sample average, 44–47, 68–70
vector, 750
Measurement error, 322–325
classical, 323
in Y vs. X, 323 Measures of fit, 121–124
in logit and probit models, 401–402
in multiple regression, 196–198 Military service and civilian earnings,
494–495, 504
Minimum wage and employment rates, 497 Model specification, 232–238
alternative, 236, 238–239
base, 236, 238–239
Moments of distribution, 23–25 Momentum forecasts, 536–537 Mortgage lending and race, 3, 386–388,
402–409 data set for, 418
linear probability model and, 389–390 logit model and, 397
probit model and, 394–397
Mostellar, Frederick, 428 Mozart effect, 186 Multicollinearity
dummy variable trap and, 204
in fixed effects regression model, 366 imperfect, 205–206
perfect, 200, 203–205
solutions to, 204–205
398–402
Index 785
786 Index
Multinomial logit and probit models, 423
Multiperiod forecasts
direct, 645–649
iterated, 643–645, 646, 648–649 method selection for, 648–649
Multiple choice variable, 423 Multiple coefficients
confidence sets for, 231–232, 232f
single restrictions involving, 229–231 Multiple regression, 5, 182–207
applications of, 194–195, 198, 238–243 coefficients in, 189
constant regressor in, 191
constant term in, 191
controlling for X in, 190
control variables in, 189, 233–236 definition of, 182, 192
extended least squares assumptions for,
708–709
in forecasting, 331–332
Gauss–Markov conditions for, 720–722 Gauss–Markov theorem for, 163–166,
178–181, 721–722, 753–754 GLS estimator and, 722–728 heteroskedasticity/homoskedasticity
and, 191, 731–734 hypothesis testing and
for multiple coefficients, 222–229
for single coefficient, 219–220
joint hypothesis and, 222–229
least squares assumptions of, 199–201,
234–235
in matrix form, 706–708
measures of fit in, 196–198 multicollinearity and imperfect, 205–206 nonlinear. See Nonlinear regression OLS estimator in. See Ordinary least
squares estimator
omitted variable bias in, 182–189,
233–236, 238–243, 319–321 panel data in, 350–373. See also Panel
data
partial effect in, 190
perfect, 202–205
population, 190–192
population regression line in, 189, 192 predicted value and, 193
regression R2 in, 196–198, 237–238 regression specification and, 232–238
alternative, 236, 238–239
base, 236, 238–239
restrictions on, 223, 227–230
scale of variables and, 239–240 single restrictions involving, 229–230 slope coefficients in, 189–190, 192 standard error of the regression and,
196
tabular presentation of results and,
240–243
threats to internal validity in, 319–330
Multivariate central limit theory, 710–711
Multivariate distribution, 749–751 normal, 38–41
Mutual funds. See also Stock market mutual funds survivorship bias
survivorship bias and, 326, 327
N
O
Observational data, 7–8. See also Data vs. experimental data, 491–493
in quasi-experiments, 504
Observation number, 9
OLS estimator. See Ordinary least squares
estimator
OLS regression and, 127–128
Omitted variable bias, 195, 339–340, 620–621
applications of, 186, 238–243 definition of, 183, 319
examples of, 183–184
formula for, 185–187, 214 internal validity and, 319–321 least squares assumption and, 184 Mozart effect and, 186
in multiple regression, 182–189, 233–236, 238–243, 319–321
number of variables and, 321
solutions to, 319–321 One-sided hypothesis
alternative, 79–80
for slope, 150–152
OPEC, 669
Orange juice prices and cold weather,
590–593, 591f, 625 data set for, 634
time series regression and, 616–624 Ordered probit model, 423
Ordered responses, 423
Orders of integration, 649–651 Ordinary least squares (OLS) estimator
for ADL model, 610–611
algebraic facts about, 144–145 asymptotic distribution of, 710–713 bias in, 182–189, 327–328, 720–721.
See also Bias
as BLUE estimator, 164–165 confidence interval for, 153–155 consistency of, 131, 685
definition of, 116
derivation of, 141–142
distribution lag model and, 606–607 distribution of, 201–202, 214–215,
602–604 dynamic, 660
efficiency of, 160, 163
estimator, 116–121, 679
fixed effects regression and, 359–360 formulas, 116–117, 120
Gauss–Markov conditions and, 720–722 Gauss–Markov theorem and, 164–166 vs. GLS estimator, 615–616
with heterogeneous causal effects,
505–506
imperfect multicollinearity and, 205–206 least squares assumptions and, 124–129,
677–679
in matrix form, 709–710, 716–717 in multiple regression, 192–206
Natural experiments. See Quasi-experiments Natural logarithms, exponential function
and, 269–270
Negative exponential growth function,
310–311
Newey–West variance estimator, 605–606 Newey, Whitney, 605
95% confidence sets, 231–232, 232f Nobel Prize in economics, 410, 669–670 Nonlinear least squares estimator,
311–312, 399
Nonlinear least squares method, 311–312 Nonlinear regression, 256–299, 309–314
coefficients in, 265
cubic, 267–269, 278
effect of Y on change in X, 261–265 general strategy for, 258–266 interactions between two variables and,
278–290 logarithmic, 269–278
logit, 396–402
multiple regression and, 266
natural logarithms in, 269–270 polynomial, 267–269
predicted value and, 264, 276–277 probit, 391–396, 398–402
for test scores, 293–298, 294t, 296f, 297f
Nonlinear regression function, 262 exponential, 269–270
as linear function of unknown parameters,
309–311
polynomials in, 267–269
of single independent variable, 266–278 slope and elasticity of, 313–314
Nonstationarity
breaks and, 561–573
large-sample normal distribution and,
551, 655 trends and, 551–561
Normal distributions, 36–37, 36f, 38f approximate, 47
asymptotic, 52
bivariate, 38–41, 702
conditional, 701
large-sample, 130–132, 654–655 multivariate, 38–41
Normal p.d.f., 701 bivariate, 702
Null hypothesis, 71, 148 failure to reject, 561 joint, 222–223
Bonferroni test of, 251–253
J-statistic and, 447–448, 452, 508, 733–734 of unit root, 561, 654
nonlinear, 311–312
notation and terminology for, 116–117 omitted variable bias in, 182–189 p-value for, 149
rationale for using, 119–121
sampling distribution of
in multiple regression, 201–202
in single-regressor model, 129–132,
142–145, 687–690
standard error of, 148, 157–163, 177–178,
217–218. See also Heteroskedasticity;
Homoskedasticity
theoretical foundations of, 163–166 t-statistic for, 148
variance of, 205–206
weighted least squares and, 690–695
Ordinary least squares predicted value, 116–117
vs. forecast, 533
in multiple regression, 193, 194
in nonlinear regression, 264, 276–277
Ordinary least squares regression line, 116 in multiple regression, 193, 194
in nonlinear regression, 285–286 outliers in, 127–128
Ordinary least squares residual, 116–117 in multiple regression, 193, 194
Outcomes
definition of, 15
probability of, 15. See also Probability
Outliers, 25
in fixed effects regression model, 366 law of large numbers and, 48–50
in least squares assumptions, 126–127,
199–200
in linear regression, 127–128
Overidentified coefficients, 435 Overidentifying restrictions test, 447–448
applications of, 452, 508
P
Panel data, 11–12, 11t, 350–373 “before and after” analysis and,
354–356, 360 definition of, 351
for drunk driving, 352–354, 353f
in fixed effects regression, 357–361 notation for, 351
panel balance and, 351
sampling distribution and, 360
Parameters, of linear regression model, 112 Partial compliance, 480–481
in quasi-experiments, 502–503 Percentages, logarithms and, 270–271 Perfect multicollinearity, 200, 203–205
dummy variable trap and, 204
solutions to, 204–205
Perfect multiple regression, 202–205 Political polls, 70, 326
Polynomial regression model, 267–269,
277–278
Pooled standard error, 88–89
Pooled variance estimator, 88–89 Population, in random sampling, 43–44 Population coefficients, 112, 114–116
variance of, p-value and, 73–74 Population intercept, 112, 113 Population means
comparing, 82–86. See also Difference of means
confidence intervals for, 80–82 estimation of, 66–71
hypothesis testing for, 71–80, 147–148
Population multiple regression model, 190–192
Population of interest, 316
Population regression function, 112, 113
breaks and, 561
estimation of, 262–263
Population regression line, 112, 113, 192
conditional probability, distributions and, 124–125, 125f
in multiple regression, 189
in nonlinear regression, 285–286 Population studied, 315–343
Potential outcomes, 476–477
Potential outcomes framework, 520–521 Power of test, 78
Predicted value, OLS. See Ordinary least squares predicted value
Price elasticity of demand, 3–4, 269 Price levels, inflation and, 651, 652f Probability, 14–54
consistency in, 680–682 convergence in, 48, 680–682 coverage, 82
definition of, 15
of event, 16
outcomes and, 15 significance. See p-value
Probability density function (p.d.f.), 18f, 19 bivariate normal, 702
normal, 36–41, 36f, 38f, 702
Probability distributions. See also Distribution
of continuous random variable, 19 cumulative, 16–18, 16t, 17f, 37 definition of, 16
of discrete random variable, 16–17 normal, 36–41, 36f, 38f
of sample average, 44–47 Probit regression model, 391–392
applications of, 394–395
definition of, 394
effect of change in X, 393
estimation of coefficients and, 395–396 vs. linear probability model, 398
vs. logit model, 396–397, 397f, 398 maximum likelihood estimator for,
400–401, 419–420 measures of fit in, 401–402 multinomial, 423
Index 787 with multiple regressors, 393
nonlinear least squares estimator and, 399
ordered, 423
Program evaluation, 475
Project STAR, data set for, 486–491, 518 Pseudo out-of-sample forecasting, 567–573 Pseudo-R2, 402, 420
pth order autoregression, 534–537
p-value
calculation of, 73–74, 74f, 76–77, 80 definition of, 72
of F-statistic, 225–226
for OLS estimator, 149
Q
QLR. See Quandt likelihood ratio statistic Quadratic regression, 259–261
Quandt likelihood ratio (QLR) statistic,
564–566, 565t Quasi-difference model, 608
conditional mean zero in, 609–610 Quasi-experiments
attrition in, 503
definition of, 85, 493 differences-in-differences estimator
and, 496–499, 498f examples of, 86–87, 494–496 experimental effects and, 503 external validity in, 504
failure to randomize in, 502 heterogeneous populations and,
504–509, 518–519
instrumental variables estimator and,
499–500
instrument validity in, 503–504 internal validity in, 502–504
partial compliance in, 502–503 potential problems with, 502–504 regression discontinuity estimator and,
500–502, 501f R
R2. See Regression R2
Race and mortgage lending. See Mortgage
lending and race Randomization based on covariates, 479 Randomization failure
in quasi-experiments, 502
in randomized controlled experiments,
480
Randomized controlled experiments, 6–7
attrition and, 481
average causal effect and, 477
on class size reduction, 491–493
data analysis in, 478–479, 486–491, 486t,
488t, 490t, 493t, 520–521 double-blind, 482
example of, 484–493
experimental design, 485–486 experimental vs. observational estimates
and, 491–493
788 Index Randomized controlled
experiments (continued) external validity of, 483–484 failure to randomize and, 480 general equilibrium effects in, 484 Hawthorne effect in, 482 heterogeneous populations and,
504–509, 518–519
internal validity of, 479–483 nonrepresentative program/policy in, 484 nonrepresentative sample in, 483 observational vs. experimental esti-
mates and, 491–493
partial compliance and, 480–481 potential outcomes in, 476–477 randomization based on covariates
and, 479 sample size in, 483
Random sampling
in estimation, 70–71 i.i.d. draws in, 44 population in, 43–44 simple, 44
Random variables
Bernoulli. See Bernoulli random variable conditional expectation of, 28–29 continuous. See Continuous random
variables
correlation between, 32
covariance between, 31
discrete, 15, 16, 16t, 17f
expected value of, 19–20 independent, 31
linear function of, mean and variance
of, 22–23
mean of, 19, 22–23, 25, 28–29, 32–35 moments of, 23–25
probability distribution of, 16–18, 16t, 17f rth moment of, 25
standard deviation for, 21–22
sums of, mean and variance of, 32–35 variance of, conditional, 30
Random walk, 552, 561, 649–650 with drift, 553
Reduced form equation, 437 Regression
binary dependent variable in, 385–423 applications of, 402–409
linear probability model and, 388–391 logit model and, 396–402
maximum likelihood estimator and, 400–401, 418–420
measures of fit and, 401–402 nonlinear least squares estimator
and, 399
overview of, 386–388
probit model and, 391–396, 398–402 regression R2 and, 389
with binary regressor, 155–157 censored, 421–422 coefficients. See Coefficient
Dickey–Fuller, 556–560, 654–655 discontinuity designs
fuzzy, 501–502
sharp, 500–501
drunk driving laws and traffic deaths, 369f entity fixed effects, 357, 362
error. See Error term
estimators. See Estimator
first-stage, 438
fixed effects. See Fixed effects regression in forecasting, 331–332, 523–524.
See also Forecasting
instrumental variables. See Instrumental
variables regression interaction, 280, 285
line
conditional distribution and,
124–125, 125f
ordinary least squares. See Ordinary
least squares regression line population. See Population regression
line
linear. See Linear regression logarithmic, 271–278, 273f, 275f, 312f logistic. See Logistic (logit) regression multiple. See Multiple regression negative exponential growth, 310–311,
312f
nonlinear, 256–299, 309–314. See also
Nonlinear regression quadratic, 259–261
quasi-difference, 608 restricted, 227–228
single restrictions involving multiple coefficients and, 229–230
sample selection, 422 specification, 232–238
alternative, 236, 238–239
base, 236, 238–239 spurious, 555–556 standard error of, 122–123 time fixed effects, 361–364 time series, 522–588
tobit, 422
truncated, 421–422 unrestricted, 227–228
Regression R2
adjusted, 196–198, 237–238
binary dependent variables and, 389
in multiple regression, 196–198, 237–238 pseudo-R2 and, 402, 420
in single-regressor model, 121–122
Regressors, 112. See also Variables autoregressive distributed lag, 437–438 binary, 155–157
constant, 191 differences-in-differences estimator
with, 498–499, 498f exogenous, 597–601, 606–616
in instrument variables, 425–435 interacted, 280
least squares, 126–127, 199
in probit regression model, 393 scale of variables for, 239–240 single endogenous, 437 single-regressor model, 109–169,
676–704 Rejection region, 78, 80
Repeated cross-sectional data, 499 Residual
OLS
in multiple regression, 193, 194
in single-regressor model, 116–117
sum of squared, 122
Restricted regression, 227–230 Retirement savings, 90
River of blood, 546
Roll, Richard, 625
Roosevelt, Franklin D., 70
Root mean squared forecast error, 533,
544
pseudo out-of-sample forecasting and,
567–568
rth moment, of random variable, 25
S
Sample average (mean) definition of, 44
as estimator, 68–70
mean and variance of, 45–46 sampling distribution of, 44–47
Sample correlation, 92 consistency of, 93–95
Sample covariance, 92, 432 consistency of, 93–95
Sample regression function, 116 Sample regression line, 116 Sample selection bias, 340
mutual funds and, 70, 326
in polling, 70, 326
Sample selection models, 422 Sample size, 483
Sample space, 15
Sample standard deviation, 74–75 Sample variance, 74
consistency of, 75–76, 108 Sampling
non-i.i.d., 126–127
random, 43–44
Sampling distribution, 43. See also
Distribution asymptotic, 47
central limit theorem and, 50–52, 51f exact, 47
finite-sample, 47
in fixed effects regression, 360
in instrumental variables regression, 431–433
instrument relevance and, 443 large-sample approximations to, 47 law of large numbers and, 46, 47–50 of OLS estimators, 129–132, 142–145
of sample average, 44–47
of TSLS estimator, 431–433, 440, 468–471 Sargent, Thomas, 669
Scatterplots, 91–92, 92f, 94f
Schwartz information criteria (SIC), 548, 549t Second difference, 650
Second-stage regression, 438
Serial correlation, 366–367, 528–529 Sharp regression discontinuity designs,
500–501 Shea, Dennis, 90
Shiller, Robert, 670
SIC. See Schwartz information criteria Significance level, 78
Significance probability. See p-value Simple random sampling, 44
Sims, Christopher, 669
Simultaneous causality, 326–329, 340–341 Simultaneous equations bias, 328–329 Single-regressor linear regression, 109–169,
676–704
asymptotic distribution and, 679–687.
See also Asymptotic distribution confidence intervals and, 146–181
error distribution and, 687–689 extended least squares assumptions and,
677–678 heteroskedasticity-only t-statistic in,
689–690
hypothesis tests and, 146–181
Single-regressor model, 109–169, 676–704 Single restrictions involving multiple coef-
ficients, 229–231 Size of test, 78
Skewness, 23–25, 24f
Slope coefficient, in multiple regression,
189–190
Slope of linear regression function, 112, 113
confidence interval for, 153–155
of nonlinear regression function, 313–314 one-sided hypotheses for, 150–152 two-sided hypotheses for, 147–150
Slutsky’s theorem, 683–684 Smoking. See under Cigarette Smooth trend, 649–650 Spurious regression, 555–556 Standard deviation
definition of, 21 sample, 74–75 variance and, 21–22
Standard error, 76
AR(p), 613–616 autocorrelated, 602–604 of b1, 148
clustered, 367–368
definition of, 75
in direct multiperiod regression, 647
in distributed lag model, 599–600
in fixed effects regression, 360, 367–368,
380–384
HAC, 367–368, 592, 604–606, 607, 647
heteroskedasticity-consistent, 601, 604–606 heteroskedasticity-robust, 177, 329, 330,
685–686, 694–695, 712–713 homoskedasticity-only, 160, 163,
177–178, 718 inconsistent, 329–330
in matrix form, 717
in multiple regression, 196
in nonlinear regression, 264–265
for OLS estimator, 148, 157–163, 177,
217–218, 329–330, 602–604
for predicted effects, 713
for predicted probabilities, 421
in single-regressor linear regression,
687–689
in single-regressor model, 122–124 for TSLS estimator, 441, 730–731
Standardization of variable, 36–37 Standard normal distribution, 36, 36f, 38f Stationarity, 540–541
in AR(1) model, 584–585 Statistics review, 65–96 Stochastic trends, 552
autoregression and, 554 cointegration and, 656–664 common, 556, 656
detection of, 556–560 Dickey–Fuller test for, 557–559 nonnormal distribution of t-statistic
and, 555
orders of integration and, 649–651 problems caused by, 554–556, 561 spurious regression and, 555–556 unit root and, 554
Stock market
autoregressive model of, 536–537, 536t,
570–572
“beta” of stock and, 120
capital asset pricing model and, 120 diversification and, 46
forecasting returns and, 536–537
mutual funds survivorship bias and, 326, 327 percent change in value, 39–40
volatility clustering and, 664–668, 664f,
668f
Strict exogeneity, 596–598
Structural VAR modeling, 641–642, 669 Student t distribution, 41–42
in hypothesis testing, 87–91 in practice, 89–91
t-statistic and, 166–167
Student–teacher ratio and test scores, 2 California data, 8–9, 9t, 141, 183, 333,
337–339, 338t
district income and, 258–261 experimental estimates for, 484–491 external validity and, 318, 332–339 internal validity and, 339–341
linear regression and, 118–119, 123–128 Massachusetts data, 333, 334t, 335t,
336t, 337–339, 338t, 349
Index 789 non-English speakers and, 278, 281–282,
289–290
nonlinear effects on, 293–298
OLS estimate of, 118–119, 194–195 omitted variable bias and, 182–189,
238–243
regression analysis of, 168–169, 194–195,
198, 238–243 Tennessee data, 7–8
Tennessee experiment, 484–493, 486t, 488t, 490t, 493t
Stylometrics, 428
Sum of squared residuals, 122 Sum of squares
explained, 121–122
total, 121–122
Supply elasticity, 427–430 Sup-Wald statistic, 564, 566 Survivorship bias, 327
T
Taleb, Nassim, 40
Taxes. See Alcohol taxes; Cigarette taxes
t distribution, 41–42
Telephone polls, 70, 326
Term spread, 5, 537–540, 566–569, 642–643 Test for random receipt of treatment, 480 Test of overidentifying restrictions, 447–448
applications of, 452, 508
Test scores, student–teacher ratio and. See
Student–teacher ratio and test
scores Test statistic, 76
critical value of, 78
Time fixed effects, regression, 361–364 Time series data, 9–10, 10t, 524–532,
583–584, 722 applications of, 524–526, 525f autocorrelation and, 366–367 autocovariance and, 528–530 causal effect and, 593–594 examples of, 530–532
first differences and, 525–528 growth rates and, 525–528 lags and, 525–528
logarithms and, 525–528
Time series regression, 522–588. See also Autoregression
applications of, 616–624
AR(p) model and, 532–534 assumptions of, 541–543 autoregressive conditional heteroske-
dasticity and, 666–667, 669 autoregressive distributed lag model
and, 539–540 breaks in, 561–573
cointegration and, 656–664
conditional mean and, 541
data for. See Time series data
first order autoregression and, 532–534 in forecasting. See Forecasting
790 Index
Time series regression (continued)
Granger causality and, 543–544
with multiple predictors, 541–544 orders of integration and, 649–651
pth order autoregression and, 532–534 stationarity and, 540–541
trends in, 551–561
unit root tests and, 651–656
vector autoregressions and, 638–643 volatile clustering and, 664–665 weak dependence and, 543
Tobin, James, 422
Tobit regression model, 422
Total sum of squares, 121–122
Traffic deaths, 368–372. See also Drunk
driving
regression analysis, 369f
t-ratio. See t-statistic Treatment
partial compliance with, 480–481
random receipt of, 480
Treatment effect, 85. See also Causal effect
average, 506–509
instrumental variables estimation of, 481 local average, 506–509
Treatment group, 6 Trend
common, 556, 656
definition of, 551
deterministic, 552
orders of integration and, 649–651 random walk model of, 552–553, 561,
649–650 smooth, 649–650
stochastic. See Stochastic trends Truncated regression models, 421–422 Truncation parameter for HAC, 605, 620 TSLS. See Two stage least squares t-statistic, 80, 149. See also Test statistic
asymptotic, 687, 713
based on sample mean, 684 definition of, 76
distribution of, 719, 752–753 general form of, 147 homoskedastic-only, 166–167
in joint hypothesis testing, 223–224 large-sample distribution of, 76–77 nonnormal distribution of, 555
for OLS estimator, 148
pooled, 88–89
small-sample distribution of, 87–91,
166–167
student t-distribution and, 166–167 for testing the mean, 87
in time series regression, 555
Two-sided hypothesis alternative, 71, 148 for slope, 147–150
Two stage least squares (TSLS) applications of, 427–431, 433–435,
assumptions of, 439–440 asymptotic distribution of, 730–731 autoregressive distributed lag
regressors, 437–438
with control variables, 473–474 efficiency under homoskedasticity,
755–756
estimator, 426–427, 437–442 first-stage regression and, 438 formula for, 431, 467–468 homoskedasticity and, 730–734 inference using, 440–441 instrumental validity and, 433–435 local average treatment effect and,
506–509
in matrix form, 730–734
sampling distribution of, 431–433, 440,
468–471
second-stage regression and, 438
with single endogenous regressor, 437 standard error calculation for, 441, 730–731 weak instruments and, 443–445, 446,
471–473 Type I error, 78
Type II error, 78
U
Unbalanced panel, 351 Uncertainty, forecast, 544–545 Uncorrelated variables, 32 Underidentified coefficients, 435 Unit root, 554
null hypothesis of, 561, 654 Unit root tests
applications of, 662–664
augmented Dickey–Fuller, 558–560,
651–654
DF-GLS test for, 651–654, 654t Dickey–Fuller, 556–560, 651–654 nonnormal distributions and, 654–655
Unrestricted regression, 227–228
U.S. Current Population Survey, 71, 106 U.S. federal funds rate, 530
U.S. stock market. See Stock market
V
Validity. See External validity; Instrument validity; Internal validity
Value, expected. See Expected value Variables. See also Regressors
Bernoulli. See Bernoulli random variable binary, 280, 282–286, 283f
binary dependent, 385–423
continuous random. See Continuous
random variables
control, 189, 234–236, 435–436, 473–474 dependent, 112, 113
discrete choice, 423
dummy, 155
endogenous, 425–426, 437–438, 443–445,
446, 471–473
exogenous, 425–426, 434–437, 445–448 independent, 112, 113
indicator, 155
instrumental. See Instrumental variables interaction between, 278–290
of interest, vs. control variables, 234–236 limited dependent, 386. See also Limited dependent variable
models multiple choice, 423
random. See Random variables standardization of, 36–37 uncorrelated, 32
Variance
conditional, 30
definition of, 21–22
of estimators, 67
of random variable, 21–23 sample, 74–76
of sample average, 45–46 standard deviation and, 21–22 of summed variables, 32–35
VAR model, 639–643 Vector, mean, 750 Vector autoregression
applications of, 642–643
for causal analysis, 641–642
for forecasting, 638–642
iterated, 644–645
lag lengths in, 641
number of coefficients and, 640–641
Vector error correction model, 657–658, 663–664, 669
Volatility clustering, 664, 664f ARCH model for, 666–667 GARCH model for, 666–667, 668f
W
Wald, Abraham, 720
Wald statistic, 720
Wallace, David, 428
Weak dependence, 543
Weak instruments, 443–445, 446,
471–473
Weighted least squares estimator (WLS),
690–695 definition of, 691
feasible, 692
vs. heteroskedasticity-robust standard
error, 694–695 infeasible, 691
Weighted regression estimators, 165–166 West, Kenneth, 605
WLS estimator. See Weighted least
squares estimator Wright, Philip G., 427, 428, 433
Wright, Sewall, 427, 428
Z
Zero conditional mean assumption, GLS estimator and, 726–728
448–453, 451t
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Large-Sample Critical Values for the t-statistic from the Standard Normal Distibution
2-Sided Test ( 3 )
Reject if |t| is greater than
1-Sided Test ( + )
Reject if t is greater than
1-Sided Test ( * ) Reject if t is less than
Significance Level
10% 5% 1%
1.64 1.96 2.58
1.28 1.64 2.33
–1.28 –1.64 –2.33
Large-Sample Critical Values for the F-statistic from the Fm, ∞ Distribution Reject if F + Critical Value
Significance Level
Degrees of Freedom (m) 10% 5% 1%
1 2.71 3.84 6.63 2 2.30 3.00 4.61 3 2.08 2.60 3.78 4 1.94 2.37 3.32 5 1.85 2.21 3.02 6 1.77 2.10 2.80 7 1.72 2.01 2.64 8 1.67 1.94 2.51 9 1.63 1.88 2.41
10 1.60 1.83 2.32 11 1.57 1.79 2.25 12 1.55 1.75 2.18 13 1.52 1.72 2.13 14 1.50 1.69 2.08 15 1.49 1.67 2.04 16 1.47 1.64 2.00 17 1.46 1.62 1.97 18 1.44 1.60 1.93 19 1.43 1.59 1.90 20 1.42 1.57 1.88 21 1.41 1.56 1.85 22 1.40 1.54 1.83 23 1.39 1.53 1.81 24 1.38 1.52 1.79 25 1.38 1.51 1.77 26 1.37 1.50 1.76 27 1.36 1.49 1.74 28 1.35 1.48 1.72 29 1.35 1.47 1.71 30 1.34 1.46 1.70