Testing for high-dimensional white noise

TL;DR

This paper introduces a robust high-dimensional white noise test that combines max-type and sum-type statistics using Fisher's method, effectively handling sparse serial correlation structures in large-dimensional time series.

Contribution

The paper proposes a novel Fisher's combination test that integrates max-type and sum-type statistics, addressing robustness issues in high-dimensional white noise testing.

Findings

The proposed test outperforms existing methods in simulations.

It effectively detects various forms of departure from white noise.

Empirical analysis confirms its practical advantages.

Abstract

Testing for multi-dimensional white noise is an important subject in statistical inference. Such test in the high-dimensional case becomes an open problem waiting to be solved, especially when the dimension of a time series is comparable to or even greater than the sample size. To detect an arbitrary form of departure from high-dimensional white noise, a few tests have been developed. Some of these tests are based on max-type statistics, while others are based on sum-type ones. Despite the progress, an urgent issue awaits to be resolved: none of these tests is robust to the sparsity of the serial correlation structure. Motivated by this, we propose a Fisher's combination test by combining the max-type and the sum-type statistics, based on the established asymptotically independence between them. This combination test can achieve robustness to the sparsity of the serial correlation structure,and combine the advantages of the two types of tests. We demonstrate the advantages of the proposed test over some existing tests through extensive numerical results and an empirical analysis.