# $R^2$ of a regression predicting noise

If you estimate a multiple regression with p predictors

y = $c_0$ + $c_1$$x_1 + c_2$$x_2$ + ... + $c_p$*$x_p$ + e

from n observations, and if the predictors and response have a multivariate normal distribution with zero correlation, what will the $R^2$ of the regression be on average, as a function of n and p? For n >> p I think the $R^2$ should approach zero, but I want to know how quickly this occurs.

• If any of the regressor variables are significant at predicting y, then $R^2$ should be approaching a value > 0 as n gets large. May 7 '18 at 3:54
• Yes, but I am assuming that the dependent variable cannot be predicted by the independent variables. May 7 '18 at 18:11
• What do you mean? Are you saying that none of the independent variables have any predictive power. If that is the case then $R^2$ will tend to 0 as n approaches infinity. May 7 '18 at 18:28
• @Michael Chernick: Maybe it will tend to zero as $n \to \infty$, but the OP asked for "as a function of $n$ and $p$, and if $p$ is large then the R-squared might well be large for practical sample sizes! May 17 '18 at 20:57