For \(T=4\) and the simple autoregressive model considered in (8.3)

(a) What are the moment restrictions given by (8.25)? Compare with Problem 8.3.

(b) What are the additional moment restrictions given by (8.26)?

(c) Write down the system of equations to be estimated by 3SLS using these additional restrictions and list the matrix of instruments for each equation.

\[\begin{equation*}
E\left(y_{i s} \Delta u_{i t}\right)=0 \quad t=2, \ldots, T \quad s=0, \ldots, t-2 \tag{8.25}
\end{equation*}\]

Data From Problem 3:

For \(T=5\), list the moment restrictions available for the simple autoregressive model given in (8.3). What overidentifying restrictions are being tested by Sargan's statistic given below (8.11)?

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

\[\begin{align*}
& E\left[\left(y_{i, 1}\left(u_{i, 3}-u_{i, 2}\right)\right]=0\right.  \tag{8.11a}\\
& E\left[\left(y_{i, 1}\left(u_{i, 4}-u_{i, 3}\right)\right]=0\right.  \tag{8.11b}\\
& E\left[\left(y_{i, 2}\left(u_{i, 4}-u_{i, 3}\right)\right]=0\right. \tag{8.11c}
\end{align*}\]

\[\begin{equation*}
E\left(u_{i T} \Delta u_{i t}\right)=0 \quad t=2, \ldots, T-1 \tag{8.26}
\end{equation*}\]