Consider Mundlak's (1978) augmented regression in (7.35) except now allow for unbalanced panel data. Show that OLS on this augmented regression yields the unbalanced Within estimator for \(\beta\) given by (9.9).

\[\begin{equation*}
\widetilde{\beta}=\left(\tilde{X}^{\prime} \tilde{X}\right)^{-1} \tilde{X}^{\prime} \tilde{y} \tag{9.9}
\end{equation*}\]

Equation 7.35:

\[y=X \beta+P X \pi+\left(Z_\mu \epsilon+u\right)\]