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:
$$F = \frac { \left( E S S _ { \text {wrratricted } } - E S S _ { \text {rastricted } } \right) / q } { 1 - E S S _ { \text {wurrestricted } } / \left( n - k _ { \text {umrestricted } } - 1 \right) }$$

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Definitions:

Z-value

A uniform measure displaying the number of standard deviations a data point lies away from the average.

Binomial Proportions

The ratios that represent the outcomes of processes that follow binomial distribution, often involving two possible outcomes.

Population Mean

The average value of a population attribute, representing the central point of a data set for the entire population.

Random Sample

A subset of a statistical population in which each member has an equal chance of being chosen.