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December 2019 On testing for high-dimensional white noise
Zeng Li, Clifford Lam, Jianfeng Yao, Qiwei Yao
Ann. Statist. 47(6): 3382-3412 (December 2019). DOI: 10.1214/18-AOS1782


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.


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Zeng Li. Clifford Lam. Jianfeng Yao. Qiwei Yao. "On testing for high-dimensional white noise." Ann. Statist. 47 (6) 3382 - 3412, December 2019.


Received: 1 June 2017; Revised: 1 September 2018; Published: December 2019
First available in Project Euclid: 31 October 2019

Digital Object Identifier: 10.1214/18-AOS1782

Primary: 62H15, 62M10
Secondary: 15A52

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.47 • No. 6 • December 2019
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