The homoskedasticity-only estimator of the variamce of $\widehat { \beta } _ { 1 }$ is
A) $\frac { s _ { \hat { u } } ^ { 2 } } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } }$
B) $\frac{s_{\hat{u}}}{\sqrt{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}}}$

C) $\frac { s _ { \hat { u } } ^ { 2 } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } - \bar { X } }$
D) $\frac { 1 } { n } \times \frac { \frac { 1 } { n - 2 } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } \hat { u } _ { i } ^ { 2 } } { \left[ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } \right] ^ { 2 } }$

Question

The homoskedasticity-only estimator of the variamce of $\widehat { \beta } _ { 1 }$ is
A) $\frac { s _ { \hat { u } } ^ { 2 } } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } }$
B) $\frac{s_{\hat{u}}}{\sqrt{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}}}$

C) $\frac { s _ { \hat { u } } ^ { 2 } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } - \bar { X } }$
D) $\frac { 1 } { n } \times \frac { \frac { 1 } { n - 2 } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } \hat { u } _ { i } ^ { 2 } } { \left[ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } \right] ^ { 2 } }$

Quizplus · Accepted Answer

The homoskedasticity-only estimator of the variance of $\widehat{\beta}_1$ is given by $\frac{s_{\hat{u}}^2}{\sum_{i=1}^{n}(X_i - \bar{X})^2}$, which is option A. This formula is derived under the assumption of homoskedasticity, where $s_{\hat{u}}^2$ is the estimator of the variance of the error term.

The Homoskedasticity-Only Estimator of the Variamce Of β^1\widehat { \beta } _ { 1 }β​1​

The Homoskedasticity-Only Estimator of the Variamce Of $\widehat { \beta } _ { 1 }$