Abstract: Testing multi-dimensional white noise has been an important subject of statistical inference in time series. Such test in the high-dimensional case becomes an open problem waiting to be further investigated, 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, taking advantage of the established asymptotic 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 thoroughly study the theoretical properties of the proposed combination test, and demonstrate its advantages over some existing tests through extensive numerical results and an empirical analysis.

Key words and phrases: Asymptotic independence, Fisher’s combination test, high-dimensional white noise, hypothesis test, robustness.