## The Annals of Statistics

### On testing for high-dimensional white noise

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 47, Number 6 (2019), 3382-3412.

Dates
Revised: September 2018
First available in Project Euclid: 31 October 2019

https://projecteuclid.org/euclid.aos/1572487397

Digital Object Identifier
doi:10.1214/18-AOS1782

Mathematical Reviews number (MathSciNet)
MR4025746

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62H15: Hypothesis testing
Secondary: 15A52

#### Citation

Li, Zeng; Lam, Clifford; Yao, Jianfeng; Yao, Qiwei. On testing for high-dimensional white noise. Ann. Statist. 47 (2019), no. 6, 3382--3412. doi:10.1214/18-AOS1782. https://projecteuclid.org/euclid.aos/1572487397

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#### Supplemental materials

• Supplement to “On testing for high-dimensional white noise”. This supplemental article contains some technical lemmas, the proof of Proposition 4.2 of the main article and some additional simulation results.