Theorem Suppose that the regression model Y=XB+E satisfy the ideal conditions except that @2 is not the identity matrix. Suppose that X9-1x lin is finite and nonsingular. Then the transformed equation PY=PXB+PE satisfies the full ideal condition Proof Since p is nonsingular and nonstochastic. PXis nonstochastic and of full rank if X is.(Condition 2 and 5). Also, for the consistency of OLS estimators -o T= lim X'n-1X lim (PX(PX T is finite and nonsingular by assumption. Therefore the transformed regressors ma- trix satisfies the required conditions, and we need consider only the transformed disturbance pe Clearly, E(PE)=0(Condition 3). Also E(PE)(PE) a2(A1/2C)(CAC)CA-12) o I(Condition 4) Finally, the normality(Condition 6) of Pe follows immediately from the nor- y of e Theorem The blue of B is just B=(X!2-1x)-1xg-1Y Proof: Since the transformed equation satisfies the full ideal conditions, the blue of 6Theorem: Suppose that the regression model Y = Xβ+ε satisfy the ideal conditions except that Ω is not the identity matrix. Suppose that lim T→∞ X0Ω−1X T is finite and nonsingular. Then the transformed equation PY = PXβ + Pε satisfies the full ideal condition. Proof: Since P is nonsingular and nonstochastic, PXis nonstochastic and of full rank if X is. (Condition 2 and 5). Also, for the consistency of OLS estimators lim T→∞ (PX) 0 (PX) T = lim T→∞ X0Ω−1X T is finite and nonsingular by assumption. Therefore the transformed regressors ma￾trix satisfies the required conditions, and we need consider only the transformed disturbance Pε. Clearly, E(Pε) = 0 (Condition 3). Also E(Pε)(Pε) 0 = σ 2PΩP0 = σ 2 (Λ −1/2C 0 )(CΛC0 )(CΛ−1/2 ) = σ 2Λ −1/2ΛΛ−1/2 = σ 2 I (Condition 4). Finally, the normality (Condition 6) of Pε follows immediately from the nor￾mality of ε. Theorem: The BLUE of β is just β˜ = (X0Ω −1X) −1X0Ω −1Y. Proof: Since the transformed equation satisfies the full ideal conditions, the BLUE of β 4