Pr ( ˆ k$_{P}$ = k$_{P}$ ) → 1; note that ˆ k$_{P}$ for a polynomial constructed for A = I is the same as for any non-singular A. Proof of Lemma 6.2. Apply Lemma 6.1 to each of the estimated polyno- mials to determine with probability approaching 1 the lowest order k$_{P}$ of the non-zero homogeneous polynomial and to obtain the consistent estimators of the polynomial vector functions, ¯ G ( · ) . Substituting the limit Z provides the consistent estimator of the asymptotic distribution. Proof of Theorem 6.1. The proof follows by application of Lemma 1 to each polynomial that is estimated. References [1] Andrews, D. W. K. (1987), ‘Asymptotic results for generalized Wald tests’, Econometric Theory 3, 348–358. [2] Boudjellaba, H., Dufour, J.-M. and Roy, R. (1992), ‘Testing causal- ity between two vectors in multivariate ARMA models’, Journal of the American Statistical Association 87(420), 1082–1090. [3] Boudjellaba, H., Dufour, J.-M. and Roy, R. (1994), ‘Simplified con- ditions for non-causality between two vectors in multivariate ARMA models’, Journal of Econometrics 63, 271–287. 45