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Lecture Advanced Econometrics (Part II) - Chapter 7: Greneralized linear regression model

Lecture "Advanced Econometrics (Part II) - Chapter 7: Greneralized linear regression model" presentation of content: Model, properties of ols estimators, white's heteroscedascity consistent estimator, greneralized least squares estimation. | Advanced Econometrics Chapter 7 Generalized Linear Regression Model Chapter 7 GENERALIZED LINEAR REGRESSION MODEL I. MODEL k-i Our basic model Y X.p s with s N 0 CT2 31 We will now generalize the specification of the error term. E s 0 E sS CT2 Q L nxn This allows for one or both of 1. Heteroskedasticity. 2. Autocorrelation. The model now is 1 Y X p s Y X p n k n k 2 Xis non-stochastic and Rank X k . 3 E s 0 nx1 4 E sS L X Q nxn nxn Heteroskedasticity case - 0 0 0 0 L 0 0 Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 7 Generalized Linear Regression Model Autocorrelation case L 3 1 P1 P1 1 Pn-1 Pn-2 Pn-1 Pn-2 1 Pị Corr st st_i correlation between errors that are i periods apart. II. PROPERTIES OF OLS ESTIMATORS 1. p XX -1X Y XX -1X Xp s p p X X -1X s E 3 p X X -1X E s p P is still an unbiased estimator 2. VarCov P E P - p p - 3 E X X -1X s XX -1X s E X X -1X ss X XX -1 X X -1X E ep X XX -1 XX -1X 3 Q X X X -1 ơ2 X X -1 so standard formula fof 3. no longer holds and 33 is a biased estimator of true ỉĩ p . p N P ơ2 XX -1XQ X XX -1 so the usual OLS output will be misleading the std error t-statistics etc will be based on 3ệ2 X X -1 not on the correct formula. 3. OLS estimators are no longer best inefficient . Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam Advanced Econometrics Chapter 7 Generalized Linear Regression Model Note for non-stochastic X we care about the efficient of 3. Because we know if n Var 3 ị plim 3 p 3 is consistent. 4. If X is stochastic - OLS estimators are still consistent when E s X 0. - IV estimators are still consistent when E s X 0 . - The usual covariance matrix estimator of VarCov 3 which is 2 X X 1 will be . . 1 T7- z f inconsistent n for the true VarCov 3 . We need to know how to deal with these issues. This will lead us to some generalized estimator. III. WHITE S HETEROSCEDASCITY CONSISTENT ESTIMATOR OF VarCov 3 . irx