Directing Power Towards Subspaces of the Alternative Hypothesis

This paper treats two problems in high-dimensional testing that have received attention in the recent literature. The first problem is that tests based on a quadratic statistic (such as the Wald statistic) lose power against subspaces of the alternative hypothesis as the dimension of the parameter vector of interest increases. The second problem is that the Wald statistic is not defined in high-dimensional settings, as it requires an invertible sample covariance matrix. I simultaneously address these issues by generalizing the Wald statistic to a statistic that is large in a subspace of the alternative hypothesis chosen by the econometrician. The existence of the statistic depends on a restricted eigenvalue condition that is tied directly to the size of the subspace. I show that if the conventional sample covariance matrix is used, then the statistic can be computed using linear regression with a constant dependent variable, where coefficient vector is restricted to the subspace of interest. As a demonstration, I consider subspaces that contain sparse or nearly-sparse vectors. For these cases, the computation reduces to $\ell_0$-regularized regression (best subset selection) and $\ell_1$-regularized regression (Lasso), respectively.