In a post last year I discussed the conditions under which the "adjusted" coefficient of determination (RA2) will increase or decrease, when regressors are deleted from (added to) a regression model. Without going over the full discussion again, here is one of the key results:
Adding a group of regressors to the model will increase (decrease) RA2 depending on whether the F-statistic for testing that their coefficients are all zero is greater (less) than one in value. RA2 is unchanged if that F-statistic is exactly equal to one.
A few days ago, "Zeba" reminded me that I had promised to post a simple proof of this result, but I still hadn't done so. Shame on me! A proof is given below. As a bonus, I've given the proof for a more general result - we don't have to be imposing "zero" restrictions on some of the coefficients - any exact linear restrictions will suffice.
Let's take a look at the proof.