Heteroscedasticity and Autocorrelation
Heteroscedasticity and Autocorrelation
Eco321: Econometrics
Donghwan Kim Department of Economics SUNY at Stony Brook
Spring 2005 Heteroscedasticity and Autocorrelation Outline
Spherical and Non-spherical Disturbances
Heteroscedasticity
Autocorrelation (Serial Correlation) Heteroscedasticity and Autocorrelation Spherical and Non-spherical Disturbances
The Classical Regression Model
Model
0 y = X β + with E(|X ) = 0 and E( |X ) = σ2I
2 0 0 2 E(1) E(12) ... E(1n) σ 0 ... 0 0 2 0 2 E(21) E(2) ... E(2n) 0 σ ... 0 0 0 E( ) ... 0 ... E( ) = 3 1 = . . . . . . .. . . . .. . . . . . 0 0 2 0 0 . . . σ2 E(n1) E(n2) ... E(n) Heteroscedasticity and Autocorrelation Spherical and Non-spherical Disturbances
The Model with Spherical Disturbances
Homoscedasticity 2 2 I Var(i |X ) = E(i |X ) = σ for all i = 1, 2, ..., n I The error terms have the same variance for all observations
Non-autocorrelation 0 I Cov(i , j |X ) = E(i j |X ) = 0 for i 6= j I An error term is not correlated with other error terms Heteroscedasticity and Autocorrelation Spherical and Non-spherical Disturbances
The Model with Non-spherical Disturbance
Model
0 y = X β + with E(|X ) = 0 and E( |X ) = Σ = σ2V
Properties of OLS estimator
I The OLS estimator remains unbiased and consistent 1 I The OLS estimator is inefficient (not BLUE)
I The OLS standard errors are incorrect and the test statistics based on them are incorrect
1Generalized Least Square (GLS) is BLUE Heteroscedasticity and Autocorrelation Heteroscedasticity
Heteroscedasticity
Model
0 y = X β + with E(|X ) = 0 and E( |X ) = Σ = σ2V
2 σ1 0 ... 0 0 σ2 ... 0 2 0 ... Σ = . . .. . . . . . 2 0 0 ... σn Heteroscedasticity and Autocorrelation Heteroscedasticity
Heteroscedasticity
The heteroscedastic error terms 2 2 I The error terms have different variance: E(i |X ) = σi 2 I The variance of the dependent variable y: var(y|X ) = σi I The variation of wage increases with education 2 2 Assume that σi = σ · Educationi I It is common in cross-section data Heteroscedasticity and Autocorrelation Heteroscedasticity
Test for Heteroscedasticity
The white test2
1. Regress y on X by OLS and obtain residuals ei 2 2. An auxiliary regression of ei on all regressors, squares of regressors, (and cross products of regressors). 3. Compute nR2 from step 2.
4. H0: No heteroscedasticity 2 2 nR ∼ χq where q is the number of regressors in step2 less one
2When heteroscedasticity is of unknown form Heteroscedasticity and Autocorrelation Heteroscedasticity
Possible Remedies for Heteroscedasticity
Possible Remedies
I OLS with heteroscedasticity-consistent standard errors 3 I Generalized Least Squares(GLS) estimator
3The GLS is called Weighted Least Squares(WLS) estimator when it is applied in the heteroscedasticity case. Heteroscedasticity and Autocorrelation Autocorrelation (Serial Correlation)
Autocorrelation
The serially correlated error terms 0 I E(i j |X ) 6= 0 for i 6= j I An error term is correlated with other error term
I It is common in time-series data, especially macroeconomic data
I The disturbance is often called a shock Heteroscedasticity and Autocorrelation Autocorrelation (Serial Correlation)
Autocorrelation
The 1st-order autoregressive process: AR(1) process
yt = β0 + β1xt + t t = ρt−1 + ut 2 where u ∼ iid(0, σu) and |ρ| < 1
1 ρ ρ2 . . . ρT −1 ρ 1 ρ . . . ρT −2 2 σ2 Σ = σ2 ρ ρ 1 ... where σ2 = u 2 ...... 1 − ρ . . . . . ρT −1 ρT −2 ... 1 Heteroscedasticity and Autocorrelation Autocorrelation (Serial Correlation)
Autocorrelation
Durbin-Watson test: test for AR(1) disturbances
I The test statistic
Pn (e − e )2 DW = t=2 t t−1 Pn 2 t=2(et )
I if there is no AR(1), DW is close to 2. 4 I The Lower and Upper Critical values If DW 4The Appendix E shows the critical values Heteroscedasticity and Autocorrelation Autocorrelation (Serial Correlation) Autocorrelation Possible Remedies 5 I Cochrane-Orcutt transformation (Partial/Quasi Differencing) I Generalized Least Squares (GLS) estimator 5See p. 228 in the textbook