Define the GLS Estimator and Discuss Its Properties When $\Omega$ Is Known Why Is This Estimator Sometimes Called Infeasible GLS? What

question 32

Essay

Define the GLS estimator and discuss its properties when $\Omega$ is known. Why is this estimator sometimes called infeasible GLS? What happens when $\Omega$ is unknown? What would the $\Omega$ matrix look like for the case of independent sampling with heteroskedastic errors, where $\operatorname { var } \left( u _ { i } \mid X _ { i } \right) = \operatorname { ch } \left( X _ { i } \right) = \sigma ^ { 2 } X _ { 1 i } ^ { 2 } ?$
Since the inverse of the error variancecovariance matrix is needed to compute the GLS estimator, find $\Omega ^ { - 1 }$ . The textbook shows that the original model $\mathbf { Y } = \mathbf { X } \beta + \mathbf { U }$
will be transformed into $\tilde { \boldsymbol { Y } } = \tilde { \boldsymbol { X } } \boldsymbol { \beta } + \tilde { \boldsymbol { U } }$ where $\tilde { \boldsymbol { Y } } = \boldsymbol { F } \boldsymbol { Y } , \tilde { \boldsymbol { X } } = \boldsymbol { F } \boldsymbol { X } \text {, and } \tilde { \boldsymbol { U } } = \boldsymbol { F } \boldsymbol { U } \text {, and } \boldsymbol { \boldsymbol { F } ^ { \prime } \boldsymbol { F } } = \boldsymbol { \Omega } ^ { -1 }$
Find $F$ in the above case, and describe what effect the transformation has on the original data.