Enhanced power enhancements for testing many moment equalities: Beyond the $2$- and $\infty$-norm

Tests based on the $2$- and $\infty$-norm have received considerable attention in high-dimensional testing problems, as they are powerful against dense and sparse alternatives, respectively. The power enhancement principle of Fan et al. (2015) combines these two norms to construct tests that are powerful against both types of alternatives. Nevertheless, the $2$- and $\infty$-norm are just two out of the whole spectrum of $p$-norms that one can base a test on. In the context of testing whether a candidate parameter satisfies a large number of moment equalities, we construct a test that harnesses the strength of all $p$-norms with $p\in[2, \infty]$. As a result, this test consistent against strictly more alternatives than any test based on a single $p$-norm. In particular, our test is consistent against more alternatives than tests based on the $2$- and $\infty$-norm, which is what most implementations of the power enhancement principle target. We illustrate the scope of our general results by using them to construct a test that simultaneously dominates the Anderson-Rubin test (based on $p=2$) and tests based on the $\infty$-norm in terms of consistency in the linear instrumental variable model with many (weak) instruments.