Adjusted $R^2$ & F test are not shown in regression with robust standard errors in Stata

Question

The adjusted $R^2$ is not shown when a regression with robust standard errors is calculated in Stata. This is surprising to me since the value of the $R^2$ is unaffected in regressions with robust standard errors.

Is there any statistical reason for not quoting the adjusted $R^2$ when using robust standard errors in regression?

Furthemore if I add more variables the F test disappears.(e.g. with 9 variables it shows but not with 13), Is this for the same reason? How can we report such results and deal with this issue?

Answer 1

If you specify the robust option, you are telling Stata that you don't really believe the errors are homoskedastic. There are several implications:

• The sums of squared residuals are still used to drive the estimation (you minimize them).

• As every observation has its own variance, the sum of the squared errors is no longer distributed as a $\chi^2$, and the ratios of the model and residual sums of squares are no longer distributed as an $F$.

• Since different observations contribute different amount of information, there isn't any way now to correct $R^2$ for the prognostic value the way $R^2_{\rm adj}$ does: one observation $\neq$ one degree of freedom.

Essentially, as the notion of variance is not quite applicable to the dependent variable (do you want to talk about the conditional variance of $Y$, i.e., the variance of the error terms, which, as we said, is not constant any more; or do you want to talk about the total variance, which, inconveniently, depends on the distribution of the explanatory variables), the whole concept of $R^2$ starts breaking down in its meaning.