Abstract: We propose novel one-sided omnibus tests for independence between two multivariate stationary time series. These new tests apply the Hilbert–Schmidt independence criterion (HSIC) to test the independence between the innovations of the time series. We establish the limiting null distributions of our HSIC-based tests under regular conditions. Next, our HSIC-based tests are shown to be consistent. A residual bootstrap method is used to obtain the critical values for the tests, and its validity is justified. Existing cross-correlation-based tests examine linear dependence. In contrast, our tests examine general dependence (including linear and non-linear), providing researchers with information that is more complete on the causal relationship between two multivariate time series. The merits of our tests are illustrated using simulations and a real-data example.

Key words and phrases: Hilbert-Schmidt independence criterion, multivariate time series models, non-linear dependence, residual bootstrap, testing for independence.