Testing Serial Correlation and ARCH Effect of High-Dimensional Time-Series Data
This article proposes several tests for detecting serial correlation and ARCH effect in high-dimensional data. The dimension of datamay go to infinity when the sample size. It is shown that the sample autocorrelations and the sample rank autocorrelations (Spearman’s rank correlation) of the L1-norm of data are asymptotically normal. Two portmanteau tests based, respectively, on the norm and its rank are shown to be asymptotically χ2-distributed, and the corresponding weighted portmanteau tests are shown to be asymptotically distributed as a linear combination of independent χ2 random variables. These tests are dimension-free, that is, independent of p, and the norm rank-based portmanteau test and its weighted counterpart can be used for heavy-tailed time series. We further discuss two standardized norm-based tests. Simulation results show that the proposed test statistics have satisfactory sizes and are powerful even for the case of small n and large p. We apply the tests to two real datasets. Supplementary materials for this article are available online.