For testing whether a group of multiple coefficients ($\beta_{q+1}...\beta_p$) in a linear regression of the form $Y = \beta_0 + \beta_1 X_1\dots+\beta_{q}X_{q}+\beta_{q+1}X_{q+1}\dots+\beta_{p}X_{p}+\epsilon$ (with $\epsilon$ ~ $N(0,\sigma^2)$) are all simultaneously zero using the F test, we would test $H_0: {\beta_{q+1}...\beta_p = 0}$ vs the alternative where at least one of ${\beta_{q+1}...\beta_p \neq 0}$.
This is usually done with the following $F$ statistic: $\frac{(\hat\sigma_{null}^2-\hat\sigma_{full}^2)/(p-q)}{\hat\sigma_{full}^2/(n-p-1)}$ ~ $F_{p-q,n-p-1}$ where $\hat\sigma_{null}^2$ is the in sample mean sum of square residuals when fitting under the null hypothesis and $\hat\sigma_{full}^2$ is the same for the alternative.
If the null hypothesis is actually false, when trying to fit the null model (which ignores $X_{q+1}\dots X_p$), we would basically be lumping $\beta_{q+1}X_{q+1}\dots+\beta_{p}X_{p}$ with the noise and the form of this combination is not guaranteed to be Gaussian as assumed of the noise in the null model. If so, I believe that $\frac{n(\hat\sigma_{null}^2-\hat\sigma_{full}^2)}{\sigma^2}$ would no longer be $\chi_{p-q}^2$ as assumed in the $F$ statistic? If so, does this number mean anything, or does it simply serve the purpose of giving us a high number to reject the null hypothesis?
