(FGLS) Suppose that $E[y_n | x_n] = x_n'\beta_0$ and $Var(y_n | x_n) = (z_n'\gamma_0)^2$ where $|z_n'\gamma_0| > a > 0$ for all possible $z_n$ ($n = 1, \dots, N$). Also suppose that conditional on $[x_n', z_n']'$ the $\{y_n\}$ are independent and normally distributed. Let $w_n(\beta) = (y_n - x_n'\beta)$. Consider the two-step FGLS estimator that regresses $|w_n(\hat{\beta}_{OLS})|$ on $z_n$ in the first step to fit $\hat{\gamma}$ and replaces $\gamma_0$ with $\hat{\gamma}$ in $\hat{\beta}_{WLS}$ in the second step.$^{35}$

(a) Show thatand find $\alpha$.

(b) Argue that the OLS regression of $|w_n(\beta_0)|$ on $z_n$ will estimate $\gamma_0$ up to a scalar factor of proportionality. Give conditions so that this is also true for $\gamma$.

(c) In general, $\tilde{\gamma}$ is an inconsistent estimator of $\gamma_0$. How does this affect the asymptotic relative efficiency of the FGLS estimator described above?

(d) Suggest a consistent estimator of the asymptotic variance of this FGLS estimator that uses OLS software output.**

E(B x 1-azy