This exercise illustrates the transformation that is necessary to produce GLS estimates for the random effects model. It utilizes the data on investment \((I N V)\), value \((V)\) and capital \((K)\) in the data file grunfeld11. The model is

\[I N V_{i t}=\beta_{1}+\beta_{2} V_{i t}+\beta_{3} K_{i t}+u_{i}+e_{i t}\]

We assume the random effects assumptions RE1-RE5 hold.

a. Find fixed effects estimates of \(\beta_{2}\) and \(\beta_{3}\). Check that the variance estimate that you obtain is \(\hat{\sigma}_{e}^{2}=2530.042\).

b. Compute the sample means \(\overline{I N V}_{i}, \bar{V}_{i}\), and \(\bar{K}_{i}\) for each of the 11 firms. [Hint: one way to do this to regress each of the variables ( \(I N V\), then \(V\), then \(K\) ) on 11 indicator variables, 1 for each firm, and in each case save the predictions.]

c. Estimate \(\beta_{1}, \beta_{2}\), and \(\beta_{3}\) from the between regression

\[\overline{I N V}_{i}=\beta_{1}+\beta_{2} \bar{V}_{i}+\beta_{3} \bar{K}_{i}+u_{i}+\bar{e}_{i}.\]

Check that the variance estimate for \(\sigma_{*}^{2}=\operatorname{var}\left(u_{i}+\bar{e}_{i} \cdot\right)\) is \(\hat{\sigma}_{*}^{2}=6328.554\).

d. Show that

\[\hat{\alpha}=1-\sqrt{\frac{\hat{\sigma}_{e}^{2}}{T \hat{\sigma}_{*}^{2}}}=0.85862\]

e. Apply least squares to the regression model

\[I N V_{i t}^{*}=\beta_{1} x_{1}^{*}+\beta_{2} V_{i t}^{*}+\beta_{3} K_{i t}^{*}+v_{i t}^{*}\]

where the transformed variables are given by \(I N V_{i t}^{*}=I N V_{i t}-\hat{\alpha} \overline{I N V}_{i}, x_{1}^{*}=1-\hat{\alpha}, V_{i t}^{*}=V_{i t}-\hat{\alpha} \bar{V}_{i}\), and \(K_{i t}^{*}=K_{i t}-\hat{\alpha} \bar{K}_{i}\).

f. Use your software to obtain random effects estimates of the original equation. Compare those estimates with those you obtained in part (e).