In the case when the errors are homoskedastic and normally distributed, conditional on X, then A) $\hat \beta$ is distributed N(?, $\sum _ { \hat { \beta } \mid X }$ ),where $\sum _ { \hat { \beta } \mid X }$ = $\sigma _ { u } ^ { 2 }$ I_(k₊₁₎. B) $\hat \beta$ is distributed N(?, $\sum _ { \hat { \beta } }$ ), where $\sum _ { \hat { \beta } }$ = $\sum _ { \sqrt { n } \left( \hat { \boldsymbol { \beta } } _ { - } \boldsymbol { \beta } \right) }$ /n = $Q _ { X } ^ { - 1 }$ $\sum _ { V }$ $Q _ { X } ^ { - 1 }$ /n. C) $\hat \beta$ is distributed N(?, $\sum _ { \hat { \beta } \mid X }$ ),where $\sum _ { \hat { \beta } \mid X }$ = $\sigma _ { u } ^ { 2 }$ ( $X ^ { \prime }$ X)^-¹. D) $\hat { u }$ = P_XY where P_X = X( $X ^ { \prime }$ X)^-¹ $X ^ { \prime }$

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Quizplus · Accepted Answer

When errors are homoskedastic and normally distributed conditional on X, the distribution of $\hat \beta$ is indeed normally distributed with a variance-covariance matrix that can be represented as $\sigma_u^2(X'X)^{-1}$, which matches the description in option C. This reflects the standard formula for the variance of the OLS estimator in a classical linear regression model.

In the Case When the Errors Are Homoskedastic and Normally β^\hat \betaβ^​

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