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 linear regression in a `large-p, large-n' framework (cf. Zhong and Chen (2011), Wang and Cui (2013)). 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 and the number of hypotheses q being tested, showing that the negative effect of dimensionality is less severe if the number of hypotheses is small. Using the recent work of Srivastava and Vershynin (2013) on high dimensional sample covariance matrices we are also able to substantially generalize previous results for non-Gaussian regressors.