(a) Using (5.6) and (5.7), verify that \(\Omega \Omega^{-1}=I\) and \(\Omega^{-1 / 2} \Omega^{-1 / 2}=\Omega^{-1}\).

(b) Show that \(y^{*}=\Omega^{-1 / 2} y\) has a typical element \(y_{i t}^{*}=\left(y_{i t}-\theta_{i} \bar{y}_{i}ight) / w_{i}\) where \(\theta_{i}=1-\left(w_{i} / \tau_{i}ight)\) and \(\tau_{i}^{2}=T \sigma_{\mu}^{2}+w_{i}^{2}\) for \(i=1, \ldots, N\).

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= diag[(77)] JT +diag[(w/)'] ET (5.6)