Does robust regression affect contributions to explained variance by the different variables?

Question

I've learned that in multiple linear regresion, parameter estimates as well as R$^2$ are not affected by using robust standard errors, i.e. are the same as resulting from non-robust regression.

I now have the following problem, which is closely connected to a previous question: According to non-robust regression results one variable contributes largely to explained variance and p-value is low. However, using robust standard errors the p-value for this variable is considerably increased so that I have troubles calling it significant. Thus, I assume its contribution to explained variance has become smaller, but I don't know how to quantify the change.

The actual question:

Will both R$^2$ and the respective contributions to explained variance by the different explanatory variables be unchanged by using robust standard errors?

Answer 1

Will both $R^2$ and the respective contributions to explained variance by the different explanatory variables be unchanged by using robust standard errors?

Yes, they will be unchanged. The point estimates of the regression coefficients remain the same, as you yourself noted. The geometric picture of the OLS projection is still the same, and thus also the $R^2$.

Thus, I assume its contribution to explained variance has become smaller, but I don't know how to quantify the change.

No, the contribution has not become smaller. However, it's statistical significance has decreased, which means you are less sure that the effect is due to fundamental reasons rather than pure chance. But the magnitude of the effect is still the same.