GLS is more efficient than Within. Using the \(\operatorname{var}\left(\widehat{\beta}_{G L S}ight)\) expression below (2.30) and \(\operatorname{var}\left(\widetilde{\beta}_{W i \text { thin }}ight)=\sigma_{v}^{2} W_{X X}^{-1}\), show that
\[\left(\operatorname{var}\left(\widehat{\beta}_{G L S}ight)ight)^{-1}\left(\operatorname{var}\left(\widetilde{\beta}_{\text {Within }}ight)ight)^{-1}=\phi^{2} B_{X X} / \sigma_{v}^{2}\]
which is positive semi-definite. Conclude that \(\operatorname{var}\left(\widetilde{\beta}_{\text {Within }}ight)-\operatorname{var}\left(\widehat{\beta}_{G L S}ight)\) is positive semi-definite.

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BGLS = [(X'QX/o2) + X'(P - INT)X/o1[(X'Qy/o) + X'(P - INT)Y/}] = [Wxx + Bxx][Wxy + Bxy] (2.30)