Consider the fixed-effects panel data model \(Y_{j t}=\alpha_{j}+u_{j t}\) for \(j=1, \ldots, k\) and \(t=1, \ldots, T\). Assume that \(u_{j t}\) is i.i.d. across entities \(j\) and over time \(t\) with \(E\left(u_{j t}\right)=0\) and \(\operatorname{var}\left(u_{j t}\right)=\sigma_{u}^{2}\).

a. The OLS estimator of \(\alpha_{j}\) is the value of \(a_{j}\) that makes the sum of squared residuals \(\sum_{j=1}^{k} \Sigma_{t=1}^{T}\left(Y_{j t}-a_{j}\right)^{2}\) as small as possible. Show that the OLS estimator is \(\hat{\alpha}_{j}=\bar{Y}_{j}=\frac{1}{T} \sum_{t=1}^{T} Y_{j t}\).

b. Show that i. \(\hat{\alpha}_{j}\) is an unbiased estimator of \(\alpha_{j}\).

ii. \(\operatorname{var}\left(\hat{\alpha}_{j}\right)=\sigma_{u}^{2} / T\).

iii. \(\operatorname{cov}\left(\hat{\alpha}_{i}, \hat{\alpha}_{j}\right)=0\) for \(i eq j\).

c. You are interested in predicting an out-of-sample value for entity \(j-\) that is, for \(Y_{j, T+1}-\) and use \(\hat{\alpha}_{j}\) as the predictor. Show that MSPE \(=\sigma_{u}^{2}+\sigma_{u}^{2} / T\).

d. You are interested in predicting an out-of-sample value for a randomly selected entity - that is, for \(Y_{j, T+1}\), where \(j\) is selected at random. You again use \(\hat{\alpha}_{j}\) as the predictor. Show the MSPE \(=\sigma_{u}^{2}+\sigma_{u}^{2} / T\).

e. The total number of in-sample observations is \(n=k T\). Show that in both (c) and (d) MSPE \(=\sigma_{u}^{2}(1+k / n)\).