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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.

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  • $\begingroup$ If any of the regressor variables are significant at predicting y, then $R^2$ should be approaching a value > 0 as n gets large. $\endgroup$ May 7 '18 at 3:54
  • $\begingroup$ Yes, but I am assuming that the dependent variable cannot be predicted by the independent variables. $\endgroup$
    – Fortranner
    May 7 '18 at 18:11
  • $\begingroup$ 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. $\endgroup$ May 7 '18 at 18:28
  • $\begingroup$ @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! $\endgroup$ May 17 '18 at 20:57

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