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 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 tests that harness the strength of <i>all p</i>-norms with p∈[2,∞]. As a result, these tests are consistent against strictly more alternatives than <i>any</i> test based on a single <i>p</i>-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.

基于2-范数（2-norm）与∞-范数（∞-norm）的检验方法在高维检验问题中受到了广泛关注，因其分别对稠密备择假设与稀疏备择假设具有优异的检验功效。Fan等（2015）提出的功效增强原理将这两种范数相结合，构建出对两类备择假设均具备良好检验功效的改进型检验方法。在检验某候选参数是否满足大量矩等式的场景下，我们构建了可利用所有p∈[2,∞]的p-范数（p-norm）优势的检验方法。据此，相较于仅基于单一种p-范数的检验方法，本研究提出的检验对严格更多类型的备择假设均具有相合性。具体而言，相较于基于2-范数与∞-范数的检验方法（这也是多数功效增强原理实现版本的目标对象），本检验对更多类型的备择假设均具备相合性。我们在含多工具变量的线性工具变量模型中对一般性研究结果进行了演示，同时针对该模型提供了数值结果与实证案例。