Is Just B=[(PX)(PX)-(PX)(PY) (X)X'Q2-Y Indeed, since B is the OLs estimator of B in the transformed equation, and since the transformed equation satisfies the ideal conditions, B has all the usual de- sirable properties-it is unbiased, BLUE, efficient, consistent, and asymptotically B is the Ols of the transformed equation, but it is a generalized least square (GLS) estimator of the original regression model which take the Ols as a sub- cases when Q=I Theorem The variance-covariance of the GLS estimator B is o(X'Q2-IX) TOO Viewing B as the Ols estimator in the transformed equation, it is clearly has covariance matrix o2(PX)(PX)]-1=a2(X92-1x)-1 Theorem An unbiased, consistent, efficient, and asymptotically efficient estimator of o wheree=Y-XB Proof: Since the transformed equation satisfies the ideal conditions, the desired estimator K(PY-PXB)(PY-PXB=m 7=k(Y-x/9(Y-xB)is just β˜ = [(PX) 0 (PX)]−1 (PX) 0 (PY) = (X0Ω −1X) −1X0Ω −1Y. Indeed, since β˜ is the OLS estimator of β in the transformed equation, and since the transformed equation satisfies the ideal conditions, β˜ has all the usual de￾sirable properties–it is unbiased, BLUE, efficient, consistent, and asymptotically efficient. β˜ is the OLS of the transformed equation, but it is a generalized least square (GLS) estimator of the original regression model which take the OLS as a sub￾cases when Ω = I. Theorem: The variance -covariance of the GLS estimator β˜ is σ 2 (X0Ω−1X) −1 . Proof: Viewing β˜ as the OLS estimator in the transformed equation, it is clearly has covariance matrix σ 2 [(PX) 0 (PX)]−1 = σ 2 (X0Ω −1X) −1 . Theorem: An unbiased, consistent, efficient, and asymptotically efficient estimator of σ 2 is s˜ 2 = ˜e 0Ω−1˜e T − k , where ˜e = Y − Xβ˜. Proof: Since the transformed equation satisfies the ideal conditions, the desired estimator of σ 2 is 1 T − k (PY − PXβ˜) 0 (PY − PXβ˜) = 1 T − k (Y − Xβ˜) 0Ω −1 (Y − Xβ˜). 5