Wallace and Hussain type estimators for the variance components of a one-way unbalanced panel data model.

(a) Verify the \(E\left(\widehat{q}_{1}\right)\) and \(E\left(\widehat{q}_{2}\right)\) equations given in (9.16).

(b) Verify \(E\left(\widetilde{q}_{1}\right)\) and \(E\left(\widetilde{q}_{2}\right)\) given in (9.17).

(c) Verify \(E\left(\hat{q}_{2}^{b}\right)\) given in (9.19).

\[\begin{align*}
& E\left(\widehat{q}_{1}\right)=E\left(\hat{u}_{O L S}^{\prime} Q \hat{u}_{O L S}\right)=\delta_{11} \sigma_{\mu}^{2}+\delta_{12} \sigma_{u}^{2} \\
& E\left(\widehat{q}_{2}\right)=E\left(\hat{u}_{O L S}^{\prime} P \hat{u}_{O L S}\right)=\delta_{21} \sigma_{\mu}^{2}+\delta_{22} \sigma_{u}^{2} \tag{9.16}
\end{align*}\]

\[\begin{align*}
E\left(\widetilde{q}_{1}\right)= & (n-N-K+1) \sigma_{v}^{2} \\
E\left(\widetilde{q}_{2}\right)= & \left(N-1+\operatorname{tr}\left[\left(X^{\prime} Q X\right)^{-1} X^{\prime} P X\right]-\operatorname{tr}\left[\left(X^{\prime} Q X\right)^{-1} X^{\prime} \bar{J}_{n} X\right]\right) \sigma_{u}^{2}  \tag{9.17}\\
& +\left[n-\left(\sum_{i=1}^{N} T_{i}^{2} / n\right)\right] \sigma_{\mu}^{2}
\end{align*}\]

\[\begin{equation*}
E\left(\hat{q}_{2}^{b}\right)=\left[n-\operatorname{tr}\left(\left(Z^{\prime} P Z\right)^{-1} Z^{\prime} Z_{\mu} Z_{\mu}^{\prime} Z\right)\right] \sigma_{\mu}^{2}+(N-K) \sigma_{u}^{2} \tag{9.19}
\end{equation*}\]