In the Case When the Errors Are Homoskedastic and Normally $\hat { \boldsymbol { \beta } }$

Short Answer

In the case when the errors are homoskedastic and normally distributed, conditional on X, then a. $\hat { \boldsymbol { \beta } }$ is distributed $N \left( \boldsymbol { \beta } , \Sigma _ { \hat { \beta } \mid X } \right)$ , where $\Sigma _ { \hat { \beta } \mid X } = \sigma _ { u } ^ { 2 } \boldsymbol { I } _ { ( \mathrm { k } + 1 ) }$ .
b. $\hat { \boldsymbol { \beta } }$ is distributed $\mathrm { N } \left( \boldsymbol { \beta } , \Sigma _ { \hat { \beta } } \right)$ , where $\Sigma _ { \hat { \beta } } = \Sigma _ { \sqrt { n } ( \dot { \beta } - \beta ) } / n = \boldsymbol { Q } _ { X } ^ { - 1 } \Sigma _ { V } \boldsymbol { Q } _ { X } ^ { - 1 } / n$ .
c. $\hat { \beta }$ is distributed $N \left( \boldsymbol { \beta } , \Sigma _ { \dot { \beta } \mid X } \right)$ , where $\Sigma _ { \dot { \beta } \mid X } = \sigma _ { u } ^ { 2 } ( \boldsymbol { X } \boldsymbol { X } ) ^ { - 1 }$ .
d. $\hat { U } = \boldsymbol { P } _ { \boldsymbol { X } } \boldsymbol { Y }$ where $\boldsymbol { P } _ { \boldsymbol { X } } = \boldsymbol { X } \left( \boldsymbol { X } ^ { \prime } \boldsymbol { X } \right) ^ { - 1 } \boldsymbol { X } ^ { \prime }$ .