A dependent variable is regressed on $K$ independent variables, using $n$ sets of sample observations. We denote SSE as the error sum of squares and $R^2$ as the coefficient of determination for this estimated regression. We want to test the null hypothesis that $K_1$ of these independent variables, taken together, do not linearly affect the dependent variable, given that the other $\left(K-K_1\right)$ independent variables are also to be used. Suppose that the regression is reestimated with the $K_1$ independent variables of interest excluded. Let $S S E^*$ denote the error sum of squares and $R^{* 2}$, the coefficient of determination for this regression. Show that the statistic for testing our null hypothesis, introduced in Section 12.5, can be expressed as follows:
$$
\frac{\left(S S E^*-S S E\right) / K_1}{S S E /(n-K-1)}=\frac{R^2-R^{* 2}}{1-R^2} \cdot \frac{n-K-1}{K_1}
$$