Testing Serial Correlation and ARCH Effect of High-Dimensional Time-Series Data
提出了几种检验高维数据序列相关性和ARCH效应的新方法，基于数据L1范数的样本自相关和秩自相关，适用于样本量小、维度高的情况，对重尾分布也有效。
<b>This article proposes several tests for detecting serial correlation and ARCH effect in high-dimensional data. The dimension of data</b>p=p(n)<b>may go to infinity when the sample size</b>n→∞<b>. It is shown that the sample autocorrelations and the sample rank autocorrelations (Spearman’s rank correlation) of the <i>L</i><sub>1</sub>-norm of data are asymptotically normal. Two portmanteau tests based, respectively, on the norm and its rank are shown to be asymptotically <i>χ</i><sup>2</sup>-distributed, and the corresponding weighted portmanteau tests are shown to be asymptotically distributed as a linear combination of independent <i>χ</i><sup>2</sup> random variables. These tests are dimension-free, that is, independent of <i>p</i>, 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 <i>n</i> and large <i>p</i>. We apply the tests to two real datasets. Supplementary materials for this article are available online.</b>