Tests for white noise via asymptotically independent U-statistics in high-dimensions

TL;DR

This paper introduces a high-dimensional white noise test using U-statistics that detects serial correlations across components without assuming cross-sectional independence, supported by theoretical and simulation results.

Contribution

It develops a novel high-dimensional white noise test based on U-statistics that handles cross-correlations without specifying an alternative model.

Findings

The test achieves asymptotic normality under the null hypothesis.

Simulation studies show reliable size control and good power.

The method works for large p and T with spectral conditions on covariance.

Abstract

We propose a high-dimensional white noise test that captures serial correlations within and across component series without specifying an alternative model. The test statistic is a U-statistic based on sample autocovariances. Under the null, asymptotic normality is established as $p, T \to \infty$ jointly using martingale difference theory. Our approach imposes no cross-sectional independence assumption, requiring only spectral conditions on $\Sigma_0$. Theoretically, we link cross-sectional correlations to a graph structure, integrating algebraic and geometric analyses to facilitate the derivation. Simulations confirm reliable size control and satisfactory power across various $(p, T)$ settings.