  This paper proposes some 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 both time series. Under regular conditions, the limiting
null distributions of our HSIC-based tests are established. Next, our
HSIC-based tests are shown to be consistent. Moreover, a residual bootstrap
method is used to obtain the critical values for our HSIC-based tests, and its
validity is justified. Compared with the existing cross-correlation-based tests
for linear dependence, our tests examine the general (including both linear and
non-linear) dependence to give investigators more complete information on the
causal relationship between two multivariate time series. The merits of our
tests are illustrated by some simulation results and a real example.
