Testing for functional white noise in high dimensions

White noise testing is a fundamental problem in time series analysis.

Yet it remains largely unsolved for high-dimensional functional time series, despite the growing attention this area has received in recent years, as existing tests are confined to either univariate functional time series or high-dimensional scalar time series.

In this paper, we develop a general error-contamination framework for testing white noise in high-dimensional functional time series.

We propose a supremum-type test statistic based on cross-autocovariance functions and develop a parametric bootstrap procedure to approximate its null distribution.

By imposing a general high-level condition, we derive a new Gaussian approximation result that ensures size control, and establish an asymptotic power guarantee.

We then apply our framework to two concrete applications: (i) white noise test for discretely observed functional time series, and (ii) residual-based goodness-of-fit test for functional factor model.

For each problem, we verify the corresponding high-level condition to ensure the theoretical validity of our proposed method.

Extensive simulations show that our proposed method achieves good finite-sample performance.

The practical utility of our proposed method is further illustrated through applications to two real datasets.