Enhanced Power Enhancements for Testing Many Moment Equalities: Beyond the 2- and ∞-norm

Tests based on the 2- and ∞-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, Liao, and Yao combines these two norms to construct improved tests that are powerful against both types of alternatives. In the context of testing whether a candidate parameter satisfies a large number of moment equalities, we construct tests that harness the strength of all p-norms with p∈[2,∞]. As a result, these tests are consistent against strictly more alternatives than any test based on a single p-norm. In particular, our tests are consistent against more alternatives than tests based on the 2- and ∞-norm, which is what most implementations of the power enhancement principle target. We illustrate our general results in the linear instrumental variable model with many instruments, for which we also provide numerical results and an empirical illustration. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.