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 }

Consider the Multiple Regression Model from Chapter 5, Where K β\betaβ

Essay