Define the GLS Estimator and Discuss Its Properties When Ω $x _ { 1 i } ^ { 2 }$

Essay

Define the GLS estimator and discuss its properties when Ω is known. Why is this estimator sometimes called infeasible GLS? What happens when Ω is unknown? What would the Ω matrix look like for the case of independent sampling with heteroskedastic errors, where var(u_i | X_i)= ch(X_i)= σ²
$x _ { 1 i } ^ { 2 }$ ? Since the inverse of the error variance-covariance matrix is needed to compute the GLS estimator, find Ω^-1. The textbook shows that the original model Y = Xβ + U will be transformed into $\widetilde { Y } = \widetilde { X } \beta + \widetilde { U } , \text { where } \widetilde { Y } = F Y , \widetilde { X } = F X \text {, and } \widetilde { U }$ = FU, and $F$ F = Ω^-1. Find F in the above case, and describe what effect the transformation has on the original data.