(a) For (Delta_{1}) and (Delta_{2}) defined in (9.28), verify that (Delta_{N} equiv Delta_{1}^{prime} Delta_{1}=operatorname{diag}left[T_{i} ight]) and (Delta_{T}...

(a) For \(\Delta_{1}\) and \(\Delta_{2}\) defined in (9.28), verify that \(\Delta_{N} \equiv \Delta_{1}^{\prime} \Delta_{1}=\operatorname{diag}\left[T_{i}\right]\) and \(\Delta_{T} \equiv \Delta_{2}^{\prime} \Delta_{2}=\) \(\operatorname{diag}\left[N_{t}\right]\). Show that for the complete panel data case \(\Delta_{1}=\iota_{T} \otimes I_{N}, \Delta_{2}=\) \(I_{T} \otimes \iota_{N}, \Delta_{N}=T I_{N}\), and \(\Delta_{T}=N I_{T}\).

(b) Under the complete panel data case, verify that \(\Delta_{T N} \equiv \Delta_{2}^{\prime} \Delta_{1}\) is \(J_{T N}\) and \(Q=E_{T} \otimes E_{N}\),

(c) Let \(X=\left(X_{1}, X_{2}\right)\) with \(\left|I+X X^{\prime}\right| eq 0\). Using the result that \(\left[I_{n}+X X^{\prime}\right]^{-1}=\) \(I_{n}-X\left(I+X^{\prime} X\right)^{-1} X^{\prime}\), apply the partitioned inverse formula for matrices to show that \(\left(I+X X^{\prime}\right)^{-1}=\widetilde{Q}_{\left[X_{2}\right]}-\widetilde{Q}_{\left[X_{2}\right]} X_{1} S^{-1} X_{1}^{\prime} \widetilde{Q}_{\left[X_{2}\right]}\) where \(\widetilde{Q}_{\left[X_{2}\right]}=\) \(I-X_{2}\left(I+X_{2}^{\prime} X_{2}\right)^{-1} X_{2}^{\prime}=\left(I+X_{2} X_{2}^{\prime}\right)^{-1}\) and \(S=I+X_{1}^{\prime} \widetilde{Q}_{\left[X_{2}\right]} X_{1}\). This is lemma 2 of Davis (2002).

(d) Apply the results in part (c) using \(X=\left(\frac{\sigma_{\mu}}{\sigma_{u}} \Delta_{1}, \frac{\sigma_{\lambda}}{\sigma_{u}} \Delta_{2}\right)\) to verify \(\Sigma^{-1}\) given in (9.33).

(e) Derive \(E\left(q_{W}\right), E\left(q_{N}\right)\), and \(E\left(q_{T}\right)\) given in (9.34), (9.35), and (9.36).

\[\Delta=\left(\Delta_{1}, \Delta_{2}\right) \equiv\left[\begin{array}{cccc}
D_{1} & D_{1} \iota_{N} & &  \tag{9.28}\\
\vdots & \ddots & \\
D_{T} & & & D_{T} \iota_{N}
\end{array}\right]\]

\[\begin{equation*}
\Sigma^{-1}=V-V \Delta_{2} \widetilde{P}^{-1} \Delta_{2}^{\prime} V \tag{9.33}
\end{equation*}\]

\[\begin{align*}
& q_{W}=e^{\prime} Q_{[\Delta]} e  \tag{9.34}\\
& q_{N}=e^{\prime} \Delta_{2} \Delta_{T}^{-1} \Delta_{2}^{\prime} e=e^{\prime} P_{\left[\Delta_{2}\right]} e  \tag{9.35}\\
& q_{T}=e^{\prime} \Delta_{1} \Delta_{N}^{-1} \Delta_{1}^{\prime} e=e^{\prime} P_{\left[\Delta_{1}\right]} e \tag{9.36}
\end{align*}\]