Testing for high-dimensional white noise using maximum cross-correlations

Abstract

SUMMARYWe propose a new omnibus test for vector white noise using the maximum absolute autocorrelations and cross-correlations of the component series. Based on an approximation by the L ∞ -norm of a normal random vector, the critical value of the test can be evaluated by bootstrapping from a multivariate normal distribution. In contrast to the conventional white noise test, the new method is proved to be valid for testing the departure from white noise that is not independent and identically distributed. We illustrate the accuracy and the power of the proposed test by simulation, which also shows that the new test outperforms several commonly used methods including, for example, the Lagrange multiplier test and the multivariate Box-Pierce portmanteau tests, especially when the dimension of time series is high in relation to the sample size. The numerical results also indicate that the performance of the new test can be further enhanced when it is applied to pre-transformed data obtained via the time series principal component analysis proposed by Chang, Yao (arXiv:1410.2323). The proposed procedures have been implemented in an R package.

On testing for high-dimensional white noise

“…In this section, we compare our test statistics with some others in recent literature. Chang, Yao and Zhou [11] proposed an omnibus test for vector white noise using the maximum absolute autocorrelations and cross-correlations of the component series. LetΓ…”

“…Designed via Frobenius norm of sample autocovariance matrices, the strength of our test statistics are fully demonstrated in such VAR(1) settings. While T n and T * n are more adapted to settings where majority coordinates of the test sequence x t or their linear transformations remain to be white noise, see the model settings in Chang, Yao and Zhou [11]. Moreover, it can be seen that test size of T n is a little biased when p = 20, T = 100.…”

“…Similarly as in Section 3.4, we further compare our test statistics with others, i.e. T n and T * n in Chang, Yao and Zhou [11] under the VMA(1) settings. Here we fix p = 20, T = 100 and let x t = A 0 z t + A 1 z t 1 where A 1 follows model (V) or (VI) and z t is either Gaussian or Non-Gaussian.…”

“…Whenever possible, comparison is made with the popular Hosking and Li-McLeod tests. Numerical evidence also indicates that the new test is more powerful than that of [11]. Section 4 collects all the technical proofs.…”

“…On the other hand, testing for high-dimensional time series is still in an infancy stage. To our best knowledge, the only available methods are [11] and [38].…”