The Homoskedasticity Only F-Statistic Is Given by the Formula Where $S S R _ { \text {restricted } }$

question 40

Essay

The homoskedasticity only F-statistic is given by the formula $$F = \frac { \left( S S R _ { \text {restricted } } - S S R _ { \text {unrestricted } } \right) / q } { S S R _ { \text {unrestricted } } / \left( n - k _ { \text {unrestricted } } - 1 \right) }$$ where $S S R _ { \text {restricted } }$ is the sum of squared residuals from the restricted regression, $S S R _ { \text {unrestricted } }$ is the sum of squared residuals from the unrestricted regression, $q$ is the number of restrictions under the null hypothesis, and $k _ { \text {unrestricted } }$ is the number of regressors in the unrestricted regression. Prove that this formula is the same as the following formula based on the regression $R ^ { 2 }$ of the restricted and unrestricted regression:
$$F = \frac { \left( E S S _ { \text {wrratricted } } - E S S _ { \text {rastricted } } \right) / q } { 1 - E S S _ { \text {wurrestricted } } / \left( n - k _ { \text {umrestricted } } - 1 \right) }$$