and, in general, unbiased. Therefore, using the OLs residual from the model with only a single overall constant, we have plim spooled= plim nt-k This provides the two estimators needed for the variance components; the second Examp Example 13.4 of greene at p. 299 2.3 Husman's specification Test for the Random Effect Model Fixed Effect Model--Costly in terms of degree of freedom lost Random Effect Model-little justification for treating the individual effects as un- correlated with other regressors The specification test developed by Hausman(1978) is used to test for or thogonality of the random effect and the regressors. Under the null hypothesis of no correlation, both Ols in the lsdv B model and gls in the random effect B model are consistent, but OLS is inefficient, whereas under the alternative OLS is consistent, but Gls is not. Therefore, under the null hypothesis, the two estimates should not different systematically, and a test can be based on the difference The essential ingredient for the test is the covariance matrix of the difference vector,3-月 Var[B-B=Var[B+VarB-Cou[B, B-CoulB, B Hausman's essential results is that the covariance of an efficient estimator with its difference from an inefficient estimator is zero, which implies Co(-B),月=Co,月-Var]=0 i Referring to the glS matrix weighted average given earlier, we see that the efficient weight Ises 6 where Ols sets 0=1and, in general, unbiased. Therefore, using the OLS residual from the model with only a single overall constant, we have plim s 2 Pooled = plim e 0e NT − k − 1 = σ 2 ε + σ 2 u . This provides the two estimators needed for the variance components; the second would be σˆ 2 u = s 2 Pooled − s 2 LSDV . Example: Example 13.4 of Greene at p.299. 2.3 Husman’s Specification Test for the Random Effect Model Fixed Effect Model–Costly in terms of degree of freedom lost. Random Effect Model–little justification for treating the individual effects as un￾correlated with other regressors. The specification test developed by Hausman (1978) is used to test for or￾thogonality of the random effect and the regressors . Under the null hypothesis of no correlation, both OLS in the LSDV βˆ model and GLS in the random effect β˜ model are consistent, but OLS is inefficient,1 whereas under the alternative, OLS is consistent, but GLS is not. Therefore, under the null hypothesis, the two estimates should not different systematically, and a test can be based on the difference. The essential ingredient for the test is the covariance matrix of the difference vector, [βˆ − β˜]: V ar[βˆ − β˜] = V ar[βˆ] + V ar[β˜] − Cov[βˆ, β˜] − Cov[βˆ, β˜] 0 . Hausman’s essential results is that the covariance of an efficient estimator with its difference from an inefficient estimator is zero, which implies Cov[(βˆ − β˜), β˜] = Cov[βˆ, β˜] − V ar[β˜] = 0 (8) 1Referring to the GLS matrix weighted average given earlier, we see that the efficient weight uses θ, where OLS sets θ = 1 14