(a) Verify that GLS on (8.19) yields (8.20).

(b) For the error component model with \(\widetilde{\Omega}=\widetilde{\sigma}_{u}^{2} I_{T}+\widetilde{\sigma}_{\mu}^{2} J_{T}\) and \(\widetilde{\sigma}_{u}^{2}\) and \(\widetilde{\sigma}_{\mu}^{2}\) denoting consistent estimates of \(\sigma_{v}^{2}\) and \(\sigma_{\mu}^{2}\), respectively, show that \(\widehat{\eta}\) in (8.20) can be written as \[\begin{aligned}
\widehat{\eta}= & {\left[\sum_{i=1}^{N} W_{i}^{\prime}\left(I_{T}-\bar{J}_{T}\right) W_{i}+\widetilde{\theta}^{2} T \sum_{i=1}^{N} \bar{w}_{i} m_{i}^{\prime}\left(\sum_{i=1}^{N} m_{i} m_{i}^{\prime}\right)^{-1} \sum_{i=1}^{N} m_{i} \bar{w}_{i}^{\prime}\right]^{-1} } \\
& \times\left[\sum_{i=1}^{N} W_{i}^{\prime}\left(I_{T}-\bar{J}_{T}\right) y_{i}+\widetilde{\theta}^{2} T \sum_{i=1}^{N} \bar{w}_{i} m_{i}^{\prime}\left(\sum_{i=1}^{N} m_{i} m_{i}^{\prime}\right)^{-1} \sum_{i=1}^{N} m_{i} \bar{y}_{i}\right]
\end{aligned}\]
where \(\bar{w}_{i}=W_{i}^{\prime} \iota_{T} / T\) and \(\tilde{\theta}^{2}=\widetilde{\sigma}_{u}^{2} /\left(T \widetilde{\sigma}_{\mu}^{2}+\widetilde{\sigma}_{u}^{2}\right)\). These are the familiar expressions for the HT, AM, and BMS estimators for the corresponding choices of \(m_{i}\). (Hint: see the proof in the Appendix of Arellano and Bover (1995)).

\[\begin{equation*}
M^{\prime} \bar{H} y=M^{\prime} \bar{H} W \eta+M^{\prime} \bar{H} u \tag{8.19}
\end{equation*}\]

\[\begin{equation*}
\widehat{\eta}=\left[W^{\prime} \bar{H}^{\prime} M\left(M^{\prime} \bar{H} \bar{\Omega} \bar{H}^{\prime} M\right)^{-1} M^{\prime} \bar{H} W\right]^{-1} W^{\prime} \bar{H}^{\prime} M\left(M^{\prime} \bar{H} \bar{\Omega} \bar{H}^{\prime} M\right)^{-1} M^{\prime} \bar{H} y \tag{8.20}
\end{equation*}\]