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

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.