An Automatic Multi‐Scale Test for Serial Correlation of High‐Dimensional Time Series

This article proposes an automatic multi‐scale test for detecting serial correlation of high‐dimensional time series (HDTS) from the perspective of time‐frequency analysis. Three theoretical tools fuel the construction and implementation of the test, including the L2$$ {L}_2 $$‐norm, the maximum overlap discrete wavelet transform (MODWT) and a Bayesian Information Criterion (BIC)‐like penalty term. The three accomplish, in turn, data dimensionality reduction, scale‐by‐scale correlation time‐frequency analysis and data‐driven selection of the optimal number of scales, thus completing the implementation of our testing principle. Under some mild conditions, the limiting null distribution of the proposed test is proved to be chi‐square with degrees of freedom 1, and the testing power of our test is analyzed in theory under a general alternative hypothesis. The finite‐sample performance of the automatic multi‐scale test is demonstrated through a series of simulations. A study of stock returns in different trading sectors is implemented as a practical application.