"Testing many moment restrictions by p-norm based tests".
Many tests used in high-dimensional testing related to gauging the distance from null hypothesis by either the 2- or supremum-norm. These tests are relatively powerful against dense and sparse deviations from the null, respectively. However, dense and sparse deviations are merely two (conceptually useful) extremes on a continuum of structures that alternatives may have. For example, many semi-sparse deviations exist between these two endpoints. Thus, it is natural to ask whether tests that are consistent against more, or at least different, alternatives can be obtained by basing these on other p-norms, p ∈ (2, ∞), than 2 or ∞. The present lack of a full understanding of the consequences of the choice of a norm on the consistency properties of the resulting test implies that currently the choice of test cannot be made on a fully informed basis. We fully characterize the alternatives that p-norm based tests are consistent against in a general family of empirically relevant testing problems. In particular, it is shown that for 2 ≤ p < q < ∞ tests based on the q-norm are consistent against no fewer, and sometimes strictly more, alternatives than tests based on the p-norm. This ranking does not extend to q = ∞. One consequence of this is that the Anderson-Rubin test, which is based on the 2-norm, can be strictly dominated in terms of consistency by using a test based on, say, the 3-norm. Although no single p-norm dominates all others, we construct a test that is consistent against a deviation from the null as soon as some p-norm based test, p ∈ [2, ∞], is consistent. Thus, this test is consistent against, dense, sparse and many semi-sparse alternatives.