The conventional estimated covariance matrix for the OLS estimator o2(X'X)-1 is inappropriate; the appropriate matrix is a(X'x) QX(X'X)-.White (1980) has shown that it is still possible to obtain an appropriate covariance es- timator of the Ols estimators even the form of heteroscedasticity is unknown What is actually required is an estimate of NoX'nX I White(1980)shows that under very general conditions, the matrix where ei is the i-th least square residual, is a consistent estimator of Z. There- fore. the white estimator ()=N(XX)-s(XX)-1 can be used as an estimator of the true variance of the oLS estimator. Inference concerning B are still possible by means of OLS estimator even when the specific structure of Q2 is not specified as B is normally distributed asymptotically. more generally, White shows that tests of the general linear hypothesis RB= g, under the null hypothesis, the statistics (RB-9)R(X'X)NSo(X'X)RT(RB-)Nxm where m denote the number of restrictions imposed Reproduce the results at Table 11.1 2 Testing for Heteroscedasticity One can rarely be certain that the disturbances are heteroscedastic however and unfortunately, what form the heteroscedasticity takes if they are. As such, it is useful to be able to test for homoscedasticity and if necessary, modify our est mation procedure accordinglyThe conventional estimated covariance matrix for the OLS estimator σ 2 (X0X) −1 is inappropriate; the appropriate matrix is σ 2 (X0X) −1X0ΩX(X0X) −1 . White (1980) has shown that it is still possible to obtain an appropriate covariance es￾timator of the OLS estimators even the form of heteroscedasticity is unknown. What is actually required is an estimate of Σ = 1 N σ 2X0ΩX = 1 N X N i=1 σ 2 i xix 0 i . White (1980) shows that under very general conditions, the matrix S0 = 1 N X N i=1 e 2 i xix 0 i , where ei is the i − th least square residual, is a consistent estimator of Σ. There￾fore, the White estimator, V\ar(βˆ) = N(X0X) −1S0(X0X) −1 , can be used as an estimator of the true variance of the OLS estimator. Inference concerning β are still possible by means of OLS estimator even when the specific structure of Ω is not specified as βˆ is normally distributed asymptotically. More generally, White shows that tests of the general linear hypothesis Rβ = q, under the null hypothesis, the statistics (Rβˆ − q) 0 [R(X0X) −1NS0(X0X) −1R 0 ] −1 (Rβˆ − q) ∼ χ 2 m, where m denote the number of restrictions imposed. Exercise: Reproduce the results at Table 11.1. 2 Testing for Heteroscedasticity One can rarely be certain that the disturbances are heteroscedastic however, and unfortunately, what form the heteroscedasticity takes if they are. As such, it is useful to be able to test for homoscedasticity and if necessary, modify our esti￾mation procedure accordingly. 2