Automatic tests for serial correlation and ARCH effect of high-dimensional time series∗

Abstract

This paper proposes a norm-rank-based automatic test for detecting serial correlation and ARCH effect in high-dimensional time series (HDTS). The proposed automatic test is based on the Spearman’s rank autocorrelations of the Lu-norm of the HDTS up to lag m, where the values of u and m are chosen by a completely data-driven method. The asymptotic null distribution of this automatic test is established without assuming any moment condition of HDTS, so this automatic test has a large application scope covering the commonly observed heavy-tailed data. To account for the possible scaling effect, another standardized norm-rank-based automatic test is further proposed. Simulations and one real example are given to demonstrate the advantage of these two automatic tests over the portmanteau tests, which seek the rejection evidence only from the L1-norm of HDTS, perform unstably across the user-chosen value of m, and have unsatisfactory power for small sample sizes.