Power Enhancement in High Dimensional Cross-Sectional Tests

TL;DR

This paper introduces a new testing method that significantly improves power for high-dimensional sparse hypothesis testing by combining a power enhancement component with a pivotal statistic, avoiding strict regularity conditions.

Contribution

It proposes a novel power enhancement technique for high-dimensional tests that improves detection of sparse alternatives without requiring stringent regularity conditions.

Findings

The method effectively boosts power under sparse alternatives.

The null distribution remains simple and does not need bootstrap.

It can identify which components violate the null hypothesis.

Abstract

We propose a novel technique to boost the power of testing a high-dimensional vector $H:\btheta=0$ against sparse alternatives where the null hypothesis is violated only by a couple of components. Existing tests based on quadratic forms such as the Wald statistic often suffer from low powers due to the accumulation of errors in estimating high-dimensional parameters. More powerful tests for sparse alternatives such as thresholding and extreme-value tests, on the other hand, require either stringent conditions or bootstrap to derive the null distribution and often suffer from size distortions due to the slow convergence. Based on a screening technique, we introduce a "power enhancement component", which is zero under the null hypothesis with high probability, but diverges quickly under sparse alternatives. The proposed test statistic combines the power enhancement component with an asymptotically pivotal statistic, and strengthens the power under sparse alternatives. The null distribution does not require stringent regularity conditions, and is completely determined by that of the pivotal statistic. As a byproduct, the power enhancement component also consistently identifies the elements that violate the null hypothesis. As specific applications, the proposed methods are applied to testing the factor pricing models and validating the cross-sectional independence in panel data models.