Abstract :
Here, we derive optimal rank-based tests for noncausality in the sense of Granger
between two multivariate time series+ Assuming that the global process admits a
joint stationary vector autoregressive ~VAR! representation with an elliptically
symmetric innovation density, both no feedback and one direction causality hypotheses
are tested+ Using the characterization of noncausality in the VAR context,
the local asymptotic normality ~LAN! theory described in Le Cam ~1986, Asymptotic
Methods in Statistical Decision Theory! allows for constructing locally and
asymptotically optimal tests for the null hypothesis of noncausality in one or both
directions+ These tests are based on multivariate residual ranks and signs ~Hallin
and Paindaveine, 2004a, Annals of Statistics 32, 2642–2678! and are shown to be
asymptotically distribution free under elliptically symmetric innovation densities
and invariant with respect to some affine transformations+ Local powers and asymptotic
relative efficiencies are also derived+ The level, power, and robustness ~to
outliers! of the resulting tests are studied by simulation and are compared to those
of the Wald test+ Finally, the new tests are applied to Canadian money and income
data+
