ON TESTING FOR HIGH-DIMENSIONAL WHITE NOISE

• 成果类型: Article
• 署名作者: Li, Zeng; Lam, Clifford; Yao, Jianfeng; Yao, Qiwei
• 署名单位: University of London; London School Economics & Political Science; Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Pennsylvania State University - University Park; University of London; London School Economics & Political Science; University of Hong Kong
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1782
• 发表日期: 2019
• 页码: 3382-3412
• 关键词: dynamic-factor model; number

摘要

Testing for white noise is a classical yet important problem in statistics, especially for diagnostic checks in time series modeling and linear regression. For high-dimensional time series in the sense that the dimension p is large in relation to the sample size T, the popular omnibus tests including the multivariate Hosking and Li-McLeod tests are extremely conservative, leading to substantial power loss. To develop more relevant tests for high-dimensional cases, we propose a portmanteau-type test statistic which is the sum of squared singular values of the first q lagged sample autocovariance matrices. It, therefore, encapsulates all the serial correlations (up to the time lag q) within and across all component series. Using the tools from random matrix theory and assuming both p and T diverge to infinity, we derive the asymptotic normality of the test statistic under both the null and a specific VMA(1) alternative hypothesis. As the actual implementation of the test requires the knowledge of three characteristic constants of the population cross-sectional covariance matrix and the value of the fourth moment of the standardized innovations, nontrivial estimations are proposed for these parameters and their integration leads to a practically usable test. Extensive simulation confirms the excellent finite-sample performance of the new test with accurate size and satisfactory power for a large range of finite (p, T) combinations, therefore, ensuring wide applicability in practice. In particular, the new tests are consistently superior to the traditional Hosking and Li-McLeod tests.