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We formulate nonparametric and semiparametric hypothesis testing ofmultivariate stationary linear time series in a unified fashion and propose newtest statistics based on estimators of the spectral density matrix. Thelimiting distributions of these test statistics under null hypotheses arealways normal distributions, and they can be implemented easily for practicaluse. If null hypotheses are false, as the sample size goes to infinity, theydiverge to infinity and consequently are consistent tests for any alternative.The approach can be applied to various null hypotheses such as the independencebetween the component series, the equality of the autocovariance functions orthe autocorrelation functions of the component series, the separability of thecovariance matrix function and the time reversibility. Furthermore, a nullhypothesis with a nonlinear constraint like the conditional independencebetween the two series can be tested in the same way.

On nonparametric and semiparametric testing for multivariate linear time series