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

TL;DR

This paper develops a new high-dimensional testing method that combines all p-norms between 2 and infinity, outperforming existing tests based solely on 2- or infinity-norms in detecting a wider range of alternatives.

Contribution

It introduces a test that leverages all p-norms from 2 to infinity, providing greater consistency against diverse alternatives in high-dimensional moment equality testing.

Findings

The new test is consistent against more alternatives than single p-norm tests.

It outperforms the Anderson-Rubin test and other norm-based tests in the linear IV model.

The approach generalizes the power enhancement principle to a continuum of norms.

Abstract

Contemporary testing problems in statistics are increasingly complex, i.e., high-dimensional. Tests based on the $2$- and $\infty$-norm have received considerable attention in such settings, 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 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 a test that harnesses the strength of all $p$-norms with $p\in[2, \infty]$. As a result, this test is 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 $p=2$), tests based on the $\infty$-norm and power enhancement based combinations of these in terms of consistency in the linear instrumental variable model with many instruments.