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.