How do joint test, r-squared behave when using autocorrelation / heteroskedasticity robust std. errors?

Question

Recently we discussed on SO how to update a standard linear regression summary with NeweyWest standard errors. I used coeftestfrom the sandwich package. It was told to use unclass to update my already existing summary like this:

library(sandwich)
library(lmtest)
temp.lm <- lm(runif(100) ~ rnorm(100))
temp.summ <- summary(temp.lm)
temp.summ$coefficients <- unclass(coeftest(temp.lm, vcov. = NeweyWest)

Now I wonder whether the joint parameters shown in the summary aren't affected at all when using a NeweyWest VC matrix? I mean with this code they are not affected obviously – but is this correct? Note this is not a syntax but a stats question :) Stuff like

Residual standard error: 1.177 on 83 degrees of freedom  
Multiple R-squared: 0.7265, Adjusted R-squared:  0.71 
F-statistic:  44.1 on 5 and 83 DF,  p-value: < 2.2e-16 

remains the same. Are there any cases that need adjustment as well?

Answer 1

The $R^2$ is the sum of squared residuals divided by the total variation in your outcome; neither of these statistics change when robust/HAC standard errors are applied, so the $R^2$ doesn't change either. Adjusted $R^2$ alters the formula somewhat, but only based upon the number of observations and the number of predictors in your model, which don't change under robust standard errors, so this value remains unchanged as well.

The F statistic can't incorporate heteroskedasticity or autocorrelation---it requires homoskedasticity and no correlations among the errors (actually, it requires that the errors be distributed according to identical normal distributions conditional on your predictors). This statistic can't be corrected. Instead, to perform robust joint tests, you need to use a Wald test that follows the $\chi^2$ distribution under the null hypothesis. So, yes, the F statistic doesn't change when you apply the robust/HAC standard errors, but that's because it isn't robust.