(a) Show that the variancecovariance matrix of the disturbances in (9.1) is given by (9.2).

(b) Show that the two nonzero block matrices in (9.2) can be written as in (9.3).

(c) Show that \(\sigma_{u} \Omega_{j}^{-1 / 2} y_{j}\) has a typical element \(\left(y_{j t}-\theta_{j} \bar{y}_{j}\right.\). , where \(\theta_{j}=\) \(1-\sigma_{u} / \omega_{j}\) and \(\omega_{j}^{2}=T_{j} \sigma_{\mu}^{2}+\sigma_{u}^{2}\).

\[\left(\begin{array}{l}
y_{1}  \tag{9.1}\\
y_{2}
\end{array}\right)=\left(\begin{array}{l}
X_{1} \\
X_{2}
\end{array}\right) \beta+\left(\begin{array}{l}
u_{1} \\
u_{2}
\end{array}\right)\]

\[\Omega=\left[\begin{array}{ccc}
\sigma_{u}^{2} I_{n_{1}}+\sigma_{\mu}^{2} J_{n_{1} n_{1}} & 0 & 0  \tag{9.2}\\
0 & \sigma_{u}^{2} I_{n_{1}}+\sigma_{\mu}^{2} J_{n_{1} n_{1}} & \sigma_{\mu}^{2} J_{n_{1} n_{2}} \\
0 & \sigma_{\mu}^{2} J_{n_{2} n_{1}} & \sigma_{u}^{2} I_{n_{2}}+\sigma_{\mu}^{2} J_{n_{2} n_{2}}
\end{array}\right]\]

\[\begin{equation*}
\Omega_{j}=\left(T_{j} \sigma_{\mu}^{2}+\sigma_{u}^{2}\right) \bar{J}_{T_{j}}+\sigma_{u}^{2} E_{T_{j}} \tag{9.3}
\end{equation*}\]