ECONOMETRICS
BRUCE E. HANSEN ©2000, 20191
University of Wisconsin Department of Economics
This Revision: August, 2019 Comments Welcome
Copyright By PowCoder代写 加微信 powcoder
1This manuscript may be printed and reproduced for individual or instructional use, but may not be printed for commercial purposes.
Preface xv
About the Author xvi
1 Introduction 1
1.1 WhatisEconometrics? ………………………………… 1
1.2 TheProbabilityApproachtoEconometrics ……………………… 1
1.3 EconometricTermsandNotation…………………………… 2
1.4 ObservationalData ………………………………….. 3
1.5 StandardDataStructures……………………………….. 4
1.6 EconometricSoftware ………………………………… 6
1.7 Replication………………………………………. 6
1.8 DataFilesforTextbook ………………………………… 7
1.9 ReadingtheManuscript ……………………………….. 9
1.10 CommonSymbols…………………………………… 10
I Regression 11
2 Conditional Expectation and Projection 12
2.1 Introduction ……………………………………… 12
2.2 TheDistributionofWages ………………………………. 12
2.3 ConditionalExpectation ……………………………….. 14
2.4 LogDifferences*……………………………………. 16
2.5 ConditionalExpectationFunction ………………………….. 17
2.6 ContinuousVariables…………………………………. 18
2.7 LawofIteratedExpectations……………………………… 20
2.8 CEFError……………………………………….. 21
2.9 Intercept-OnlyModel…………………………………. 23
2.10 RegressionVariance………………………………….. 23
2.11 BestPredictor …………………………………….. 24
2.12 ConditionalVariance …………………………………. 24
2.13 HomoskedasticityandHeteroskedasticity………………………. 26
2.14 RegressionDerivative…………………………………. 27
2.15 LinearCEF ………………………………………. 28
2.16 LinearCEFwithNonlinearEffects ………………………….. 29
2.17 LinearCEFwithDummyVariables ………………………….. 29
2.18 BestLinearPredictor …………………………………. 32
2.19 IllustrationsofBestLinearPredictor …………………………. 36
2.20 LinearPredictorErrorVariance ……………………………. 38
2.21 RegressionCoefficients………………………………… 39
2.22 RegressionSub-Vectors………………………………… 40
CONTENTS ii
2.23 CoefficientDecomposition………………………………. 40
2.24 OmittedVariableBias…………………………………. 41
2.25 BestLinearApproximation………………………………. 42
2.26 RegressiontotheMean………………………………… 43
2.27 ReverseRegression ………………………………….. 44
2.28 LimitationsoftheBestLinearProjection ………………………. 45
2.29 RandomCoefficientModel………………………………. 46
2.30 CausalEffects …………………………………….. 47
2.31 Expectation:MathematicalDetails* …………………………. 51
2.32 MomentGeneratingandCharacteristicFunctions* …………………. 53
2.33 MomentsandCumulants*………………………………. 54
2.34 ExistenceandUniquenessoftheConditionalExpectation*. . . . . . . . . . . . . . . . . . 55
2.35 Identification* …………………………………….. 55
2.36 TechnicalProofs* …………………………………… 57
Exercises……………………………………………. 60
3 The Algebra of Least Squares 63
3.1 Introduction ……………………………………… 63
3.2 Samples………………………………………… 63
3.3 MomentEstimators………………………………….. 64
3.4 LeastSquaresEstimator ……………………………….. 65
3.5 SolvingforLeastSquareswithOneRegressor…………………….. 66
3.6 SolvingforLeastSquareswithMultipleRegressors………………….. 67
3.7 Illustration ………………………………………. 72
3.8 LeastSquaresResiduals………………………………… 73
3.9 DemeanedRegressors ………………………………… 74
3.10 ModelinMatrixNotation……………………………….. 75
3.11 ProjectionMatrix …………………………………… 76
3.12 OrthogonalProjection ………………………………… 78
3.13 EstimationofErrorVariance……………………………… 79
3.14 AnalysisofVariance………………………………….. 79
3.15 Projections ………………………………………. 80
3.16 RegressionComponents ……………………………….. 80
3.17 RegressionComponents(AlternativeDerivation)*………………….. 83
3.18 ResidualRegression………………………………….. 84
3.19 LeverageValues ……………………………………. 85
3.20 Leave-One-OutRegression………………………………. 86
3.21 InfluentialObservations ……………………………….. 88
3.22 CPSDataSet ……………………………………… 90
3.23 NumericalComputation ……………………………….. 91
3.24 CollinearityErrors…………………………………… 91
3.25 Programming……………………………………… 93
Exercises……………………………………………. 97
4 Least Squares Regression 101
4.1 Introduction ……………………………………… 101
4.2 RandomSampling…………………………………… 101
4.3 SampleMean……………………………………… 102
4.4 LinearRegressionModel ……………………………….. 102
4.5 MeanofLeast-SquaresEstimator…………………………… 103
4.6 VarianceofLeastSquaresEstimator …………………………. 105
4.7 UnconditionalMoments ……………………………….. 106
CONTENTS iii
4.8 Gauss-MarkovTheorem ……………………………….. 107
4.9 GeneralizedLeastSquares ………………………………. 108
4.10 Residuals ……………………………………….. 109
4.11 EstimationofErrorVariance……………………………… 111
4.12 Mean-SquareForecastError……………………………… 112
4.13 CovarianceMatrixEstimationUnderHomoskedasticity . . . . . . . . . . . . . . . . . . . 113
4.14 CovarianceMatrixEstimationUnderHeteroskedasticity . . . . . . . . . . . . . . . . . . . 114
4.15 StandardErrors ……………………………………. 117
4.16 CovarianceMatrixEstimationwithSparseDummyVariables . . . . . . . . . . . . . . . . 118
4.17 Computation……………………………………… 119
4.18 MeasuresofFit…………………………………….. 121
4.19 EmpiricalExample ………………………………….. 122
4.20 Multicollinearity……………………………………. 122
4.21 ClusteredSampling ………………………………….. 126
4.22 InferencewithClusteredSamples…………………………… 132
4.23 AtWhatLeveltoCluster?……………………………….. 133
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Normal Regression and Maximum Likelihood 139
5.1 Introduction ……………………………………… 139
5.2 TheNormalDistribution……………………………….. 139
5.3 Chi-SquareDistribution ……………………………….. 142
5.4 StudenttDistribution…………………………………. 143
5.5 FDistribution …………………………………….. 144
5.6 Non-CentralChi-SquareandFDistributions …………………….. 146
5.7 JointNormalityandLinearRegression………………………… 147
5.8 NormalRegressionModel ………………………………. 147
5.9 DistributionofOLSCoefficientVector ………………………… 149
5.10 DistributionofOLSResidualVector …………………………. 150
5.11 DistributionofVarianceEstimator ………………………….. 151
5.12 t-statistic ……………………………………….. 151
5.13 ConfidenceIntervalsforRegressionCoefficients…………………… 152
5.14 ConfidenceIntervalsforErrorVariance ……………………….. 154
5.15 tTest…………………………………………..154
5.16 LikelihoodRatioTest …………………………………. 156
5.17 LikelihoodProperties…………………………………. 157
5.18 InformationBoundforNormalRegression ……………………… 159
5.19 GammaFunction*…………………………………… 160
5.20 TechnicalProofs* …………………………………… 160
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Large Sample Methods 170
An Introduction to Large Sample Asymptotics 171
6.1 Introduction ……………………………………… 171
6.2 AsymptoticLimits…………………………………… 172
6.3 ConvergenceinProbability ……………………………… 173
6.4 WeakLawofLargeNumbers……………………………… 174
6.5 AlmostSureConvergenceandtheStrongLaw*……………………. 175
6.6 Vector-ValuedMoments ……………………………….. 176
6.7 ConvergenceinDistribution……………………………… 177
CONTENTS iv
6.8 CentralLimitTheorem ………………………………… 178
6.9 HigherMoments……………………………………. 181
6.10 MultivariateCentralLimitTheorem …………………………. 182
6.11 MomentsofTransformations …………………………….. 183
6.12 SmoothFunctionModel ……………………………….. 184
6.13 ContinuousMappingTheorem ……………………………. 186
6.14 DeltaMethod……………………………………… 186
6.15 AsymptoticDistributionforSmoothFunctionModel . . . . . . . . . . . . . . . . . . . . . 187
6.16 CovarianceMatrixEstimation…………………………….. 188
6.17 t-ratios………………………………………….188
6.18 StochasticOrderSymbols ………………………………. 189
6.19 UniformWLLN* ……………………………………. 190
6.20 UniformCLT* …………………………………….. 191
6.21 ConvergenceofMoments*………………………………. 192
6.22 EdgeworthExpansionfortheSampleMean* …………………….. 195
6.23 EdgeworthExpansionforSmoothFunctionModel*………………….197
6.24 Cornish-FisherExpansions* ……………………………… 199
6.25 UniformStochasticBounds*……………………………… 200
6.26 MarcinkiewiczWeakLawofLargeNumbers* …………………….. 201
6.27 SemiparametricEfficiency* ……………………………… 201
6.28 TechnicalProofs* …………………………………… 204
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7 Asymptotic Theory for Least Squares 212
7.1 Introduction ……………………………………… 212
7.2 ConsistencyofLeast-SquaresEstimator……………………….. 212
7.3 AsymptoticNormality…………………………………. 214
7.4 JointDistribution …………………………………… 217
7.5 ConsistencyofErrorVarianceEstimators ………………………. 220
7.6 HomoskedasticCovarianceMatrixEstimation ……………………. 222
7.7 HeteroskedasticCovarianceMatrixEstimation……………………. 222
7.8 SummaryofCovarianceMatrixNotation ………………………. 224
7.9 AlternativeCovarianceMatrixEstimators* ……………………… 225
7.10 FunctionsofParameters ……………………………….. 226
7.11 AsymptoticStandardErrors ……………………………… 228
7.12 t-statistic ……………………………………….. 230
7.13 ConfidenceIntervals …………………………………. 231
7.14 RegressionIntervals………………………………….. 233
7.15 ForecastIntervals …………………………………… 234
7.16 WaldStatistic……………………………………… 236
7.17 HomoskedasticWaldStatistic …………………………….. 236
7.18 ConfidenceRegions………………………………….. 237
7.19 EdgeworthExpansion* ………………………………… 238
7.20 SemiparametricEfficiencyintheProjectionModel*…………………. 239
7.21 Semiparametric Efficiency in the Homoskedastic Regression Model* . . . . . . . . . . . . 240
7.22 UniformlyConsistentResiduals* …………………………… 242
7.23 AsymptoticLeverage*…………………………………. 243
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8 Restricted Estimation 251
8.1 Introduction ……………………………………… 251
8.2 ConstrainedLeastSquares………………………………. 252
CONTENTS v
8.3 ExclusionRestriction …………………………………. 253
8.4 FiniteSampleProperties ……………………………….. 254
8.5 MinimumDistance ………………………………….. 257
8.6 AsymptoticDistribution ……………………………….. 258
8.7 VarianceEstimationandStandardErrors ………………………. 260
8.8 EfficientMinimumDistanceEstimator ……………………….. 260
8.9 ExclusionRestrictionRevisited ……………………………. 261
8.10 VarianceandStandardErrorEstimation……………………….. 263
8.11 HausmanEquality…………………………………… 263
8.12 Example:Mankiw,RomerandWeil(1992)………………………. 264
8.13 Misspecification……………………………………. 268
8.14 NonlinearConstraints ………………………………… 270
8.15 InequalityRestrictions ………………………………… 271
8.16 TechnicalProofs* …………………………………… 271
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
9 Hypothesis Testing 276
9.1 Hypotheses……………………………………….276
9.2 AcceptanceandRejection ………………………………. 277
9.3 TypeIError……………………………………….279
9.4 ttests ………………………………………….279
9.5 TypeIIErrorandPower………………………………… 281
9.6 StatisticalSignificance ………………………………… 281
9.7 P-Values…………………………………………282
9.8 t-ratiosandtheAbuseofTesting …………………………… 284
9.9 WaldTests………………………………………..285
9.10 HomoskedasticWaldTests………………………………. 287
9.11 Criterion-BasedTests…………………………………. 287
9.12 MinimumDistanceTests……………………………….. 288
9.13 MinimumDistanceTestsUnderHomoskedasticity …………………. 289
9.14 FTests………………………………………….290
9.15 HausmanTests…………………………………….. 291
9.16 ScoreTests ………………………………………. 292
9.17 ProblemswithTestsofNonlinearHypotheses ……………………. 293
9.18 MonteCarloSimulation ……………………………….. 297
9.19 ConfidenceIntervalsbyTestInversion………………………… 299
9.20 MultipleTestsandBonferroniCorrections ……………………… 300
9.21 PowerandTestConsistency ……………………………… 301
9.22 AsymptoticLocalPower ……………………………….. 302
9.23 AsymptoticLocalPower,VectorCase…………………………. 305
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
10 ResamplingMethods 314
10.1 Introduction ……………………………………… 314 10.2 Example…………………………………………314 10.3 JackknifeEstimationofVariance …………………………… 315 10.4 Example…………………………………………318 10.5 JackknifeforClusteredObservations…………………………. 319 10.6 EmpiricalDistributionFunction …………………………… 320 10.7 Quantiles………………………………………..321 10.8 TheBootstrapAlgorithm……………………………….. 323 10.9 BootstrapVarianceandStandardErrors……………………….. 325
CONTENTS vi
10.10 PercentileInterval…………………………………… 326 10.11 TheBootstrapDistribution………………………………. 327 10.12 TheDistributionoftheBootstrapObservations …………………… 328 10.13 TheDistributionoftheBootstrapSampleMean …………………… 329 10.14 BootstrapAsymptotics ………………………………… 330 10.15 ConsistencyoftheBootstrapEstimateofVariance………………….. 333 10.16 TrimmedEstimatorofBootstrapVariance………………………. 334 10.17 UnreliabilityofUntrimmedBootstrapStandardErrors . . . . . . . . . . . . . . . . . . . . 336 10.18 ConsistencyofthePercentileInterval ………………………… 336 10.19 Bias-CorrectedPercentileInterval ………………………….. 338 10.20 BCaPercentileInterval ………………………………… 340 10.21 Percentile-tInterval………………………………….. 342 10.22 Percentile-tAsymptoticRefinement …………………………. 343 10.23 BootstrapHypothesisTests………………………………. 345 10.24 Wald-TypeBootstrapTests………………………………. 347 10.25 Criterion-BasedBootstrapTests……………………………. 348 10.26 ParametricBootstrap …………………………………. 349 10.27 HowManyBootstrapReplications?………………………….. 350 10.28 SettingtheBootstrapSeed ………………………………. 351 10.29 BootstrapRegression …………………………………. 351 10.30 BootstrapRegressionAsymptoticTheory ………………………. 352 10.31 WildBootstrap…………………………………….. 354 10.32 BootstrapforClusteredObservations ………………………… 355 10.33 TechnicalProofs* …………………………………… 357 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Multiple Equation Models 367
11 MultivariateRegression 368
11.1 Introduction ……………………………………… 368 11.2 RegressionSystems ………………………………….. 368 11.3 Least-SquaresEstimator ……………………………….. 369 11.4 MeanandVarianceofSystemsLeast-Squares…………………….. 371 11.5 AsymptoticDistribution ……………………………….. 372 11.6 CovarianceMatrixEstimation…………………………….. 373 11.7 SeeminglyUnrelatedRegression …………………………… 374 11.8 EquivalenceofSURandLeast-Squares………………………… 376 11.9 MaximumLikelihoodEstimator……………………………. 377 11.10 RestrictedEstimation…………………………………. 378 11.11 ReducedRankRegression ………………………………. 378 11.12 PrincipalComponentAnalysis ……………………………. 381 11.13 PCAwithAdditionalRegressors……………………………. 383 11.14 Factor-AugmentedRegression…………………………….. 384 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
12 InstrumentalVariables 388
12.1 Introduction ……………………………………… 388 12.2 Overview………………………………………..388 12.3 Examples………………………………………..389 12.4 Instruments ……………………………………… 391 12.5 Example:CollegeProximity ……………………………… 392
CONTENTS vii
12.6 ReducedForm …………………………………….. 393
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