Heteroskedastic fixed effects models. This is based on problem 96.5.1 in Econometric Theory by Baltagi (1996). Consider the fixed effects model

\[y_{i t}=\alpha_{i}+u_{i t} \quad i=1,2, \ldots, N ; t=1,2, \ldots, T_{i}\]

where \(y_{i t}\) denotes output in industry \(i\) at time \(t\) and \(\alpha_{i}\) denotes the industry fixed effect. The disturbances \(u_{i t}\) are assumed to be independent with heteroskedastic variances \(\sigma_{i}^{2}\). Note that the data are unbalanced with different number of observations for each industry.

(a) Show that OLS and GLS estimates of \(\alpha_{i}\) are identical.

(b) Let \(\sigma^{2}=\sum_{i=1}^{N} T_{i} \sigma_{i}^{2} / n\) where \(n=\sum_{i=1}^{N} T_{i}\), be the average disturbance variance. Show that the GLS estimator of \(\sigma^{2}\) is unbiased, whereas the OLS estimator of \(\sigma^{2}\) is biased. Also show that this bias disappears if the data are balanced or the variances are homoskedastic.

(c) Define \(\lambda_{i}^{2}=\sigma_{i}^{2} / \sigma^{2}\) for \(i=1,2 \ldots, N\). Show that for \(\alpha^{\prime}=\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{N}ight)\)\[\begin{aligned}& E\left[\text { estimated } \operatorname{var}\left(\widehat{\alpha}_{O L S}ight)-\operatorname{true} \operatorname{var}\left(\widehat{\alpha}_{O L S}ight)ight] \\= & \sigma^{2}\left[\left(n-\sum_{i=1}^{N} \lambda_{i}^{2}ight) /(n-N)ight] \operatorname{diag}\left(1 / T_{i}ight)-\sigma^{2} \operatorname{diag}\left(\lambda_{i}^{2} / T_{i}ight)\end{aligned}\]