  We propose a new omnibus test for vector white noise using the maximum
absolute auto-correlations and cross-correlations of the component series.
Based on the newly established approximation by the $L_\infty$-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
non-IID white noise. 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 the pre-transformed data obtained via the time series principal
component analysis proposed by Chang, Guo and Yao (2014). The proposed
procedures have been implemented in an R-package HDtest and is available online
at CRAN.
