In the random effects model, under assumptions RE1-RE5, suppose that the variance of the idiosyncratic error is \(\sigma_{e}^{2}=\operatorname{var}\left(e_{i t}\right)=1\).

a. If the variance of the individual heterogeneity is \(\sigma_{u}^{2}=1\), what is the correlation \(ho\) between \(v_{i t}=u_{i}+e_{i t}\) and \(v_{i s}=u_{i}+e_{i s}\) ?

b. If the variance of the individual heterogeneity is \(\sigma_{u}^{2}=1\), what is the value of the GLS transformation parameter \(\alpha\) if \(T=2\) ? What is the value of the GLS transformation parameter \(\alpha\) if \(T=5\) ?

c. In general, for any given values of \(\sigma_{u}^{2}\) and \(\sigma_{e}^{2}\), as the time dimension \(T\) of the panel becomes larger, the transformation parameter \(\alpha\) becomes smaller. Is this true, false, or are you uncertain? If you are uncertain, explain.

d. If \(T=2\) and \(\sigma_{e}^{2}=\operatorname{var}\left(e_{i t}\right)=1\), what value of \(\sigma_{u}^{2}\) will give the GLS transformation parameter \(\alpha=1 / 4\) ? What value of \(\sigma_{u}^{2}\) will give the GLS transformation parameter \(\alpha=1 / 2\) ? What value of \(\sigma_{u}^{2}\) will give the GLS transformation parameter \(\alpha=9 / 10\) ?

e. If we think of the random errors \(u_{i}\) and \(e_{i t}\) as noise in the regression relationship, summarize how the relative variation of these noise components, the variances of error components, affects our ability to estimate the regression parameters.