The relative effects of dimensionality and multiplicity of hypotheses on the $F$-test in linear regression

Recently, several authors have re-examined the power of the classical $F$-test in a non-Gaussian linear regression under a “large-$p$, large-$n$” framework [e.g. 29, 27]. They highlight the loss of power as the number of regressors $p$ increases relative to sample size $n$. These papers essentially focus only on the overall test of the null hypothesis that all $p$ slope coefficients are equal to zero. Here, we consider the general case of testing $q$ linear hypotheses on the $p+1$-dimensional regression parameter vector that includes $p$ slope coefficients and an intercept parameter. In the case of Gaussian design, we describe the dependence of the local asymptotic power function on both the relative number of parameters $p/n$ and the relative number of hypotheses $q/n$ being tested, showing that the negative effect of dimensionality is less severe if the number of hypotheses is small. Using the recent work of [23] on high-dimensional sample covariance matrices we are also able to substantially generalize previous results for non-Gaussian regressors.