Ch. 9 Heteroscedasticity Regression disturbances whose variance are not constant across observations are heteroscedastic. In the heteroscedastic model we assume that 0 0 a10 0 0 0a2 00 It will sometimes prove useful to write a2=awi. This form is an arbitrary scaling which allows us to use a normalization, tr(Q For exampl Not. This makes the classical regression with homoscedas tic disturbance a simple special cases with wi=1,i=1, 2, ..,M Ex See the residuals at Figure 11.1 1 Ordinary Least Squares Estimation We showed in Section 8.1 that in the presence of heteroscedasticity, the Ols estimator B is unbiased and consistent. However it is inefficient relative to the GLS estimator 1.1 Estimating the Appropriate Covariance Matrix for OLS EStimators If the type of heteroscedasticity is known with certainty, then the Ols estimator is undesirable; we should use the GLS instead. The precise form of the het- eroscedasticity is usually unknown, however. In that case, Gls is not usable and we may need to salvage what we can from the results of OLS estimatorsCh. 9 Heteroscedasticity Regression disturbances whose variance are not constant across observations are heteroscedastic. In the heteroscedastic model, we assume that E(εε0 ) = σ 2Ω = σ 2         ω1 0 . . . 0 0 ω2 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 . . . ωN         =         σ 2 1 0 . . . 0 0 σ 2 2 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 . . . σ 2 N         . It will sometimes prove useful to write σ 2 i = σ 2ωi . This form is an arbitrary scaling which allows us to use a normalization, tr(Ω) = X N i=1 ωi = N. (For example, σ 2 = PN i=1 σ 2 i N .) This makes the classical regression with homoscedas￾tic disturbance a simple special cases with ωi = 1, i = 1, 2, ..., N. Example: See the residuals at Figure 11.1. 1 Ordinary Least Squares Estimation We showed in Section 8.1 that in the presence of heteroscedasticity, the OLS estimator βˆ is unbiased and consistent. However it is inefficient relative to the GLS estimator. 1.1 Estimating the Appropriate Covariance Matrix for OLS Estimators If the type of heteroscedasticity is known with certainty, then the OLS estimator is undesirable; we should use the GLS instead. The precise form of the het￾eroscedasticity is usually unknown, however. In that case, GLS is not usable, and we may need to salvage what we can from the results of OLS estimators. 1