For the rotating panel considered, assume that \(T=3\) and that the number of households being replaced each period is equal to \(N / 2\).

(a) Derive the variance-covariance of the disturbances \(\Omega\).

(b) Derive \(\Omega^{-1 / 2}\) and describe the transformation needed to make GLS a weighted least squares regression.

(c) How would you consistently estimate the variance components \(\sigma_{\mu}^{2}\) and \(\sigma_{u}^{2}\) ?

(d) Repeat this exercise for the case where the number of households being replaced each period is \(N / 3\). How about \(2 N / 3\) ?