# What impact does including more predictors have on the CI for R squared?

More precisely I mean "All other things being equal, what impact does including more predictors ('IVs') have on the CI for R squared?"

Say we'll get an R squared of .30 no matter whether we include 4 predictor variables or 7 predictor variables. Would one definitely produce a more precise CI than the other, or tend to produce a more precise CI than the other? Or is the number of predictor variables unrelated to the precision of the CI?

There is an approximate formula for the standard error of $R^2$ in the CI.Rsq function in the psychometric package in R (and probably other places, too).
$(\frac{4R^2(1-R^2)^2(n-k-1)^2)}{((n^2-1)*(n+3))})^.5$
where k is the number of predictors. So, with the same sample size and $R^2$, more predictors reduces the numerator, making the SE smaller.
Intuitively, if we had $n-1$ predictors we would have a 'perfect' model and the SE would be 0; but this would be grossly overfit to the sample.