  We formulate nonparametric and semiparametric hypothesis testing of
multivariate stationary linear time series in a unified fashion and propose new
test statistics based on estimators of the spectral density matrix. The
limiting distributions of these test statistics under null hypotheses are
always normal distributions, and they can be implemented easily for practical
use. If null hypotheses are false, as the sample size goes to infinity, they
diverge to infinity and consequently are consistent tests for any alternative.
The approach can be applied to various null hypotheses such as the independence
between the component series, the equality of the autocovariance functions or
the autocorrelation functions of the component series, the separability of the
covariance matrix function and the time reversibility. Furthermore, a null
hypothesis with a nonlinear constraint like the conditional independence
between the two series can be tested in the same way.
