Suppose I am building a multiple regression model, perhaps with 5 explanatory variables. Suppose the residuals are not normally distributed (based on a Q-Q plot and a D'Agostino test, due to kurtosis). Suppose I have a sample size in the neighborhood of 100: small enough that I can trust the D'Agostino test to be telling me about a significant departure from normality, but large enough that I can lean on the Central Limit Theorem.
Can I trust the ANOVA F-statistic and the corresponding p-value given by R? In other words, is there a theoretical reason that, with a sufficiently large sample size (much bigger than 100 if you want), that the ANOVA F-statistic should be F-distributed even if the residuals are not normal?
This is claimed by Wooldridge on page 176 of his Introductory Econometrics book (4th edition). Wooldridge is usually fairly careful and gives theoretical justifications, but does not in this case. I am a mathematician, so I'd like to understand why the F-statistic should be F-distributed in this setting. I know that the t-statistics for the individual explanatory variables (assuming no multicollinearity) are approximately normally distributed by the Central Limit Theorem. I know that I can do a randomization based test and build an empirical distribution for the F-statistics, and it looks like an F-distribution. I just don't have a theoretical reason to believe it is one, and my students keep asking me this question. Thanks!