# Power Enhancement in High Dimensional Cross-Sectional Tests

60 Pages Posted: 16 Oct 2013 Last revised: 17 Aug 2014

See all articles by Jianqing Fan

## Jianqing Fan

Princeton University - Bendheim Center for Finance

## Yuan Liao

Rutgers, The State University of New Jersey - New Brunswick/Piscataway

## Jiawei Yao

Princeton University

Date Written: October 14, 2013

### Abstract

We propose a novel technique to boost the power of testing a high-dimensional vector $H:\theta=0$ 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.

Keywords: sparse alternatives, thresholding, large covariance matrix estimation, Wald-test, screening, cross-sectional independence, factor pricing model

JEL Classification: C12, C33, C58

Suggested Citation

Fan, Jianqing and Liao, Yuan and Yao, Jiawei, Power Enhancement in High Dimensional Cross-Sectional Tests (October 14, 2013). Available at SSRN: https://ssrn.com/abstract=2340313 or http://dx.doi.org/10.2139/ssrn.2340313

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