  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.
