Bounds for \(s^{2}\) in a one-way random effects model. For the random one-way error component model given in (2.1) and (2.2), consider the OLS estimator of \(\operatorname{var}\left(u_{i t}ight)=\sigma^{2}\), which is given by \(s^{2}=\widehat{u}_{O L S}^{\prime} \widehat{u}_{O L S} /\left(n-K^{\prime}ight)\), where \(n=N T\) and \(K^{\prime}=K+1\).

(a) Show that \(E\left(s^{2}ight)=\sigma^{2}+\sigma_{\mu}^{2}\left[K^{\prime}-\operatorname{tr}\left(I_{N} \otimes J_{T}ight) P_{x}ight] /\left(n-K^{\prime}ight)\).

(b) Consider the inequalities given by Kiviet and Krämer (1992) which state that

\[\begin{aligned}0 & \leqslant \text { mean of }\left(n-K^{\prime}ight) \text { smallest roots of } \Omega \leqslant E\left(s^{2}ight) \\& \leqslant \text { mean of }\left(n-K^{\prime}ight) \text { largest roots of } \Omega \leqslant \operatorname{tr}(\Omega) /\left(n-K^{\prime}ight)\end{aligned}\]

where \(\Omega=E\left(u u^{\prime}ight)\). Show that for the one-way error component model, these bounds are

\[\begin{aligned}0 & \leqslant \sigma_{v}^{2}+\left(n-T K^{\prime}ight) \sigma_{\mu}^{2} /\left(n-K^{\prime}ight) \leqslant E\left(s^{2}ight) \leqslant \sigma_{v}^{2}+n \sigma_{\mu}^{2} /\left(n-K^{\prime}ight) \\& \leqslant n \sigma^{2} /\left(n-K^{\prime}ight)\end{aligned}\]

As \(n ightarrow \infty\), both bounds tend to \(\sigma^{2}\), and \(s^{2}\) is asymptotically unbiased, irrespective of the particular evolution of \(X\). See Baltagi and Krämer (1994) for a proof of this result.