therefore, Var(B)= EL6-B)(6-B)y] El((XMDX)XMDE)((XMDXXMDE) EI(X'MDX)XMDEE X(X'MDX) g((XMDX)-XMDINTMDX(XMDX)- a(X'MD X) X(XM X)-1 0(X'MDX) With the above results, the appropriate estimator of Var(B)is therefore Est(Var(B))=s2(X'MpX)-l where the disturbance variance estimator is s2 p - Da)(y-xB- Dc给 Show that Var(a)=+xar()风 1.1 Testing the Significance of the Group Effects Consider the null hypothesis that Ho: 01=02 a. Under this null hypothesis, the efficient estimator is the pooled least squares. The F ration used for the test would be FN-LNt-N-k (REsDy- Pooled)/(N-1) (1-BsD)/(T-N-k) where RisDy indicates the R2 from the dummy variables model and Pooled in- dicates the R2 from the pooled or restricted model with only a single overalltherefore, V ar(βˆ) = E[(βˆ − β)(βˆ − β) 0 ] = E[((X0MDX) −1X0MDε)((X0MDX) −1X0MDε) 0 ] = E[(X0MDX) −1X0MDεε 0MDX(X0MDX) −1 ] = σ 2 [(X0MDX) −1X0MDINTMDX(X0MDX) −1 ] = σ 2 [(X0MDX) −1X0MDX(X0MDX) −1 ] = σ 2 (X0MDX) −1 . With the above results, the appropriate estimator of V ar(βˆ) is therefore Est(V ar(βˆ)) = s 2 (X0MDX) −1 , where the disturbance variance estimator is s 2 s 2 = (y − Xβˆ − Dαˆ) 0 (y − Xβˆ − Dαˆ) NT − N − K = PN i=1 PT t=1(yit − x 0 itβˆ − αˆi) 2 NT − N − k . Exercise: Show that V ar(αˆi) = σ 2 T + x¯ 0 iV ar(βˆ)x¯i . 1.1 Testing the Significance of the Group Effects Consider the null hypothesis that H0 : α1 = α2 = ... = αN = α. Under this null hypothesis, the efficient estimator is the pooled least squares. The F ration used for the test would be FN−1,NT −N−k = (R2 LSDV − R2 Pooled)/(N − 1) (1 − R2 LSDV )/(NT − N − k) , where R 2 LSDV indicates the R 2 from the dummy variables model and R 2 Pooled in￾dicates the R2 from the pooled or restricted model with only a single overall 7