Abstract

Testing for white noise is a fundamental problem in statistical inference, as many testing problems in linear modelling can be transformed into a white noise test. While the celebrated Box-Pierce test and its variants tests are often applied for model diagnosis, their relevance in the context of high-dimensional modeling is not well understood, as the asymptotic null distributions are established for fixed dimensions. Furthermore, those tests typically lose power when the dimension of time series is relatively large in relation to the sample size. In this talk, we introduce two new omnibus tests for high-dimensional time series.

The first method uses the maximum absolute autocorrelations and cross-correlations of the component series as the testing statistic. Based on an approximation by the L-infinity 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 departure from white noise that is not independent and identically distributed.

The second test statistic is defined as the sum of squared singular values of the first q lagged sample autocovariance matrices. Therefore, it encapsulates all the serial correlations (up to the time lag q) within and across all component series. Using the tools from random matrix theory, we derive the normal limiting distributions when both the dimension and the sample size diverge to infinity.