For the simple autoregressive model with no regressors given in (8.3)

(a) Write the first-differenced form of this equation for \(t=5\) and \(t=6\) and list the set of valid instruments for these two periods.

(b) Show that variance-covariance matrix of the first difference disturbances is given by (8.5).

(c) Verify that (8.8) is the GLS estimator of (8.7).

\[\begin{equation*}
y_{i t}=\delta y_{i, t-1}+u_{i t} \quad i=1, \ldots, N \quad t=1, \ldots, T \tag{8.3}
\end{equation*}\]

\[\begin{equation*}
E\left(\Delta u_{i} \Delta u_{i}^{\prime}\right)=\sigma_{u}^{2} G \tag{8.5}
\end{equation*}\]

\[\begin{align*}
\widehat{\delta}_{1}= & {\left[\left(\Delta y_{-1}\right)^{\prime} W\left(W^{\prime}\left(I_{N} \otimes G\right) W\right)^{-1} W^{\prime}\left(\Delta y_{-1}\right)\right]^{-1} }  \tag{8.8}\\
& \times\left[\left(\Delta y_{-1}\right)^{\prime} W\left(W^{\prime}\left(I_{N} \otimes G\right) W\right)^{-1} W^{\prime}(\Delta y)\right]
\end{align*}\]

\[\begin{equation*}
W^{\prime} \Delta y=W^{\prime}\left(\Delta y_{-1}\right) \delta+W^{\prime} \Delta u \tag{8.7}
\end{equation*}\]