Using the Model $\boldsymbol { Y } = \boldsymbol { X } \boldsymbol { \beta } + \boldsymbol { U }$

Essay

Using the model $\boldsymbol { Y } = \boldsymbol { X } \boldsymbol { \beta } + \boldsymbol { U }$ and the extended least squares assumptions, derive the OLS estimator $\hat { \beta }$ . Discuss the conditions under which $\boldsymbol { X } ^ { \prime } \boldsymbol { X }$ is invertible.