Robust Optimal Tests for Causality in Multivariate Time Series ∗

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 autore-gressive (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)) 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) 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. Finally, the level, power and robustness (to outliers) of the resulting tests are studied by simulation and are compared to those of Wald test