Prediction in the spatially autocorrelated error component model. This is based on problem 99.2.4 in Econometric Theory by Baltagi and Li (1999). Consider the panel data regression model described in (13.1) with random country effects and spatially autocorrelated remainder disturbances described by (13.2) and (13.3). Using the Goldberger (1962) best linear unbiased prediction results, Eq. (2.37), derive the BLUP of \(y_{i, T+S}\) for the ith country at period \(T+S\) for this spatial panel model. 

\[\begin{equation*}
y_{t i}=X_{t i}^{\prime} \beta+u_{t i}, i=1, . ., N ; t=1, \cdots, T \text {, } \tag{13.1}
\end{equation*}\]

\[\begin{equation*}
u_{t}=\mu+\epsilon_{t} \tag{13.2}
\end{equation*}\]

\[\begin{equation*}
\epsilon_{t}=\lambda W_{N} \epsilon_{t}+u_{t} \tag{13.3}
\end{equation*}\]

\[\widehat{y}_{i, T+S}=Z_{i, T+S}^{\prime} \widehat{\delta}_{G L S}+w^{\prime} \Omega^{-1} \widehat{u}_{G L S} \quad \text { for } s \geqslant 1\]