程序代写 Financial Econometrics - Slides-03: Linear Regression with Time Series Dia

Financial Econometrics – Slides-03: Linear Regression with Time Series Diagnostics Tests, Robust Inference& Model Stability

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

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Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Financial Econometrics
Slides-03: Linear Regression with Time Series

Diagnostics Tests, Robust Inference& Model Stability

School of Economics1

1©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be
removed from this material.

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Testing the CAPM: Mobil Exxon

The CAPM implies that the market rewards investors for the market risk

E(Ri)−Rf = βi [E(Rm −Rf )]
where Ri is the return on an asset i, and Rm is the return on the market index.

• To estimate the CAPM: Run an OLS regression of excess returns on asset
i, Xi,t, on the market excess return Xm,t

Xi,t = αi + βiXm,t + µt

• If the CAPM holds, the null hypothesis H0 : αi = 0 Ha : αi 6= 0
(two-tailed test)

Linear Regression Applications In Finance Review of Linear Regression model

Capital Asset Procing Model

CAPM: Application

What detremines the expected return of an asset?
Example: Mobil (a US petroleum firm), 1978:01-1987:12 with T = 120.

Topic 2. Linear Regression & Applications in Finance

School’of’Economics,’UNSW’ Slides<02,'Financial'Econometrics' 7' 78 79 80 81 82 83 84 85 86 87 MARKET RISKFREE MOBIL -.3 -.2 -.1 .0 .1 .2 Scatter Plot Dependent'Variable:'E_MOBIL ' ' Method:'Least'Squares ' ' Sample:'1978M01'1987M12 ' ' Included'observaOons:'120 ' ' Variable Coefficient Std.'Error t F̂

= 0.0046 =⇒decision: Reject the null. PythonCode

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Arbitrage Pricing Theory (APT)

What determines the expected return of an asset?

Excess returns: Xi,t = Ri,t −Rf,t and Xm,t = Rm,t −Rf,t

E(Xi,t) = αi + βiE(Xm,t)

RPi = αi + βiRPm

where RPi: risk premium for asset i, RPm: market risk premium

2 APT (Arbitrage Pricing Theory)

E(Xi,t) = RPi = αi + βiRPm + βotherRPotherfactors

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Arbitrage Pricing Theory (APT)

What determines the expected return of an asset?

• APT (Arbitrage Pricing Theory): if there are r risk factors priced in
the fiunancial market, then:

RPi = αi + βiRPm + βi,1RP1 + · · ·βi,rRPr

• RPj is the risk premium for exposure to factor j risk;, j = 1, · · · , r.
• βi,j is the sensitivity of the asset to factor j; it also measures asset i’s

exposure to the factor risk j

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

So what are the other risk factors in the APT?

A well established APT Model in the finance literature is the Fama&French
three factor model: FamaFrench

RPi = αi + βi.mRPm + βi.sRPs + βi.hRPh + βi.uRPu (1)

• RPm is the market risk premium
• RPs is the size factor risk premium (small market capitalisation)
• RPh is the value factor risk premium (high book-to-market stocks)
• RPu is the momentum risk factor premium (prior gains)
• βi.m, βi.s, βi.h and βi.u are the betas for the market risk, size factor, value

factor and momentum respectively

Dr. School of Economics Slides-03

https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Example 1: Expected Return

Based on the following data and a risk-free rate of return of 2%, compute
expected return under APT model.

Beta of each factor Factor Risk Premium
βi.m 1.2 RPm 5.1
βi.s 0.8 RPs 0.5
βi.h 0.2 RPh 0.95
βi.u -0.1 RPu 2.5

E(Ri) = Rf + αi + βi.mRPm + βi.sRPs + βi.hRPh + βi.uRPu

= 2%+ 1.2∗5.1% + 0.8∗0.5% + (0.2)∗0.95% + (−0.1)∗2.5%
E(Ri) = 8.46%

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Question 1: NIKE

Given the risk-free rate of return of 1.0%, average return of Nike (i.e: S&P 500
company with small market cap) of 15.88% p.a and the data provided in table
Q1(a) Compute the expected return of the NIKE under APT model;
Q1(b) Determine the alpha return of the NIKE;
Q1(c) Construct a portfolio comprising S&P500 index fund (market portfolio),
Wilshire 5000 index fund, Russell 1000 value index fund and US T-Bills to
replicate the expected return of Nike.

Table 1: Factor beta, returns and risk premium
Factor Beta R Risk Premium
RPm 0.7877 14.5% 13.49%
RPs 0.6701 14.65% 0.15%
RPv -0.0288 10.38% -4.12%

(i) RPm is the market risk premium, i.e: excess of S&P500 return over the risk-free rate of return;
(ii) RPs is the size factor risk premium , i.e: excess of Wilshire 5000 index returns over the S&P500 returns (iii) RPv is the value
factor risk premium , i.e: excess of Russell 1000 index returns over the S&P500 returns.

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Solution to Question 1

E(RNKE) = Rf + βNKE,m(Rm −Rf ) + βNKE,s(Rs −Rm) + βNKE,v(Rv −Rm)

(i) E(RNKE) =
1.0% + (0.7877)(13.49%) + (0.6701)(0.15%) + (−0.0288)(−4.12%) = 11.85%

(ii) αNKE = Actual return – Expected return = 15.88%− 11.85% = 4.03%

(iii) Replicating portfolios’ weights:

E(Ri) = Rf + βi,mE

+ βi,sE (Rs −Rm) + βi,vE (Rv −Rm)

= Rf (1− βi,m) + (βi,m − βi,s − βi,v)E(Rm) + βi,sE(Rs) + βi,vE(Rv)
= wi,RfRf + wi,mE(Rm) + wi,sE(Rs) + wi,vE(Rv)

wi,Rf = 1− βm = 1− 0.7877 = 0.2123
wi,m = βi,m − βi,s − βi,v = 0.7877− (0.6701)− (−0.0288) = 0.1464
ws = βi,s = 0.6701
wv = βi,v = −0.0288

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Solution to Question 1 continued

Replicating portfolio: (a) long position :21.23% in US T-Bills,
(b) long position: 14.64% in market portfolio (or S&P500 market index fund),
(c) long position: 67.01% in Wilshire 5000 index fund, and
(d) short 2.88% in Russell 1000 value index fund.

Computing the expected return of replicating portfolio:
Rf = 1.0% WRf = 0.2123
E(Rm) = 14.5% Wm = 0.1464
E(Rs) = 14.65% Ws = 0.6701
E(Rv) = 10.38% Wv = −0.0288
0.2123∗1.0% + 0.1464∗14.5% + 0.6701∗14.65%− 0.0288∗10.38%

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Estimating & Testing the APT: Exxon Example

. What determines the expected return of Exxon Mobil?

The APT extends the CAPM to allow for additional risk factors Xu,t(eg.
unexpected macro events , unexpected changes in firm profits, etc)

Xu,t = (INF,OIL) H0 : γINF = γOIL = 0,

Do we reject?

Topic 2. Linear Regression & Applications in Finance

• Applications in finance

– Arbitrage pricing theory (APT)
• What determines the expected return of an asset?

– Excess returns: L.,$ = &.,$ − &M,$ and LN,$ = &N,$ − &M,$
CAPM: L.,$ = ; + ,LN,$ + P.,$

– APT extends CAPM: L.,$ = ; + ,LN,$ + QLR,$ + P.,$ ,
to include further risk factors LR,$ (eg. unexpected macro
events, unexpected changes in firm profits, etc).

eg. Mobil,

Xu,t = INF, OIL,
01: QVWX = QYVZ = 0 ,
Do we reject?

School of Economics, UNSW Slides-02, Financial Econometrics 5

Variable Coefficient Std. Error t-Statistic Prob.

C 0.004 0.006 0.721 0.472

E_MKT 0.713 0.086 8.271 0.000

INF 0.440 0.641 0.687 0.494

OIL 0.341 0.637 0.536 0.593

Test Statistic Value df Probability

F-statistic 0.6965 (2, 116) 0.5004

1 PythonCodeAPT

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Test for Autocorrelation

. H0 : No autocorrelation in the error term µt

1 Durbin-Watson test (DW):

Reject H0 if DW is too different from 2.

2 LM test for autocorrelation (Breush-Godfrey):
• Run OLS on the original regression

Yt = β0 + β1X1t + · · ·+ βKXKt + µt (2)

and save residuals et
• Run OLS on the auxiliary regression

et = γ0 + γ1X1t + · · ·+ γKXKt (3)
+δ1et−1 + · · ·+ δqet−q + errort, (4)

and save R−squared R2a;
• Reject H0 if (T − q)R2a > χ2q−critical value.

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Test for Heteroskedasticity

. H0 : Homoskedasticity of the error term µt
1 LM test (White)

• Suppose the original regression has only two regressors.
• Run OLS on the original regression

Yt = β0 + β1X1t + β2X2t + µt (5)

and save residuals et
• Run OLS on the auxiliary regression

e2t = γ0 + γ1X1t + γ2X2t (6)

2t + δ3X1tX2t + errort, (7)

and save R−squared R2a;
• Reject H0 if TR2a > χ2m−critical value, where m is the number of

regressors in the auxiliary regression, here (m = 5)

Notice the problem of m increasing with K (m = 2K +

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Test for Heteroskedasticity

2 Alternative method for White LM test
• Run OLS on the original regression

Yt = β0 + β1X1t + · · ·+ βKXKt + µt (8)

and save residuals et, and predicted values Ŷt
• Run OLS on the auxiliary regression

e2t = γ0 + γ1Ŷt + γ2Ŷ
t + errort, (9)

and save R−squared R2a;
• Reject H0 if TR2a > χ22−critical value.

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Example:Mobil

Topic 2. Linear Regression & Applications in Finance

• Linear regression

– Diagnostic statistics
eg. CAPM: Mobil

– No evidence for AC in the error term (large p-value/Do not Reject).

– Strong evidence for heteroskedasticity (small p-value/Reject).

– Strong evidence for non-normality (small p-value/Reject).

School of Economics, UNSW Slides-02, Financial Econometrics 8

Dependent Variable: E_MOBIL

Method: Least Squares

Sample: 1978M01 1987M12

Included observations: 120

Variable Coefficient Std. Error t-Statistic Prob.

C 0.004241 0.005881 0.721087 0.4723

E_MARKET 0.714695 0.085615 8.347761 0.0000

R-squared 0.371287 Mean dependent var 0.009353

Adjusted R-squared 0.365959 S.D. dependent var 0.080468

S.E. of regression 0.064074 Akaike info criterion -2.641019

Sum squared resid 0.484452 Schwarz criterion -2.594561

Log likelihood 160.4612 F-statistic 69.68511

Durbin-Watson stat 2.087124 Prob(F-statistic) 0.000000

-0.1 -0.0 0.1 0.2 0.3

Series: Residuals
Sample 1978M01 1987M12
Observations 120

Mean 1.27e-18
Median 0.000819
Maximum 0.278652
Minimum -0.145562
Std. Dev. 0.063805
Skewness 0.788429
Kurtosis 5.152737

Jarque-Bera 35.60378
Probability 0.000000

White Heteroskedasticity Test:

F-statistic 3.587532 Probability 0.030751

Obs*R2 6.933821 Probability 0.031213

Breusch- Correlation LM Test:

F-statistic 0.229380 Probability 0.795386

Obs*R2 0.472709 Probability 0.789501

• No evidence for AC in the error term (large p-value).
• Strong evidence for heteroskedasticity (small p-value).
• Strong evidence for non-normality (small p-value).

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Robust Standard Errors

• The key assumption is E(µt|Xt) = 0
(which my be weakened to Cov(Xt, µt) = 0).

Can we test for this ’key assumption’? How would the test look like?

• Even when there is heteroskedasticity or autocorrelation in µt, the OLS
estimators are still consistent. However, the standard errors of the
estimators are incorrect and MUST be corrected.
• In practice, we should always use robust standard errors that correct the

effect of heteroskedasticity and/or autocorrelation:
1 White standard errors (correct heteroskedasticity)
2 Newey-West (HAC) standard errors (correct jointly heteroskedasticity and

autocorrelation.)

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Example:Mobil

Topic 2. Linear Regression & Applications in Finance

• Linear regression

– Robust standard errors

School of Economics, UNSW Slides-02, Financial Econometrics 10

Variable Coefficient Std. Error t-Statistic Prob.

C 0.004241 0.005881 0.721087 0.4723

E_MKT 0.714695 0.085615 8.347761 0.0000

White s.e.
Variable Coefficient Std. Error t-Statistic Prob.

C 0.004241 0.005620 0.754602 0.4520

E_MKT 0.714695 0.086243 8.287035 0.0000

Newey-West s.e.
Variable Coefficient Std. Error t-Statistic Prob.

C 0.004241 0.005130 0.826596 0.4101

E_MKT 0.714695 0.090799 7.871135 0.0000

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Miscellaneous issues

• Dynamics:the lags of Yt may be included in the RHS of the regression
eg. Mobil” Xi,t = α+ βXm,t + γXi,t−1 + µi,t
• Dummy variable

• Stock market event:

Yt = β0 + β1Xt + β2DtXt + µt, (10)

Dt = 0 pre crisis and Dt = 1 post crisis.
The effect of Xt on Yt is β1 before the crisis but becomes (β1 + β2) after
the crisis.

• Day-of-the-week effects: Frit =

0, t is not on a Friday
1, t is on a Friday

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Model Stability

Model stability: Does Its structure changes over time?

1 Recursive parameter estimates

Monitor changes in parameter estimates over time.
• Start from an initial sample of size τ , estimate the model, get β̂(τ),
• add one observation to the sample, estimate the model, get the β̂(τ + 1),
• Continue recursively until last estimate with full sample β̂(T )

– eg. Mobil: Stability of the CAPM Model Xi,t = α+ βXm,t + µi,t

Recursive estimates of the market beta β:

Topic 2. Linear Regression & Applications in Finance

• Linear regression

– Model stability: Its structure changes over time?
• Recursive parameter estimates

Monitor changes in parameter estimates over time.

!”,… , !o , !”,… , !op” , …, !”,… , !’
()(q), ()(q + 1), …, ()(/)

eg. Mobil:

L.,2 = ; + (LN,2 + P.,2
Recursive estimates of (

School of Economics, UNSW Slides-02, Financial Econometrics 12

79 80 81 82 83 84 85 86 87

Recursive C(2) Estimates ± 2 S.E.

Dr. School of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Model Stability

2 Recursive residuals
• Estimate recursively the model parameters:
β̂(τ), β̂(τ + 1), · · · , β̂(T ),

• estimate recursive residuals: eτ+1|τ = Yτ+1 −Xτ+1β̂(τ)
eτ+1|τ , eτ+2|τ+1, · · · , eT |T−1
If the model is correct (stable),

se(eτ+1|τ )
∼ N(0, 1) (11)

Mobil Recursive Residuals (CAPM)

Topic 2. Linear Regression & Applications in Finance

• Linear regression

– Model stability: Its structure changes over time?

• Recursive residuals: [op$|o = ‘op$ − *op$*+ q

‘$, … , ‘o , ‘$,… , ‘op$ , …, ‘$,… , ‘/”$
[op$|o, [op1|op$, …, [/|/”$

If the model is correct,

xy(tuvw|u)

eg. Mobil:

Recursive residuals

School of Economics, UNSW Slides-

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