Tests for noncorrelation of two multivariate ARMA time series

In many situations, we want to verify the existence of a relationship between multivariate time series. In this paper, we generalize the procedure developed by Haugh (1976) for univariate time series in order to test the hypothesis of noncorrelation between two multivariate stationary ARMA series. The test statistics are based on residual cross-correlation matrices. Under the null hypothesis of noncorrelation, we show that an arbitrary vector of residual cross-correlations asymptotically follows the same distribution as the corresponding vector of cross-correlations between the two innovation series. From this result, it follows that the test statistics considered are asymptotically distributed as chi-square random variables. Two test procedures are described. The first one is based on the residual cross-correlation matrix at a particular lag, whilst the second one is based on a portmanteau type statistic that generalizes Haugh's statistic. We also discuss how the procedures for testing noncorrelation can be adapted to determine the directions of causality in the sense of Granger (1969) between the two series. An advantage of the proposed procedures is that their application does not require the estimation of a global model for the two series. The finite-sample properties of the statistics introduced were studied by simulation under the null hypothesis. It led to modified statistics whose upper quantiles are much better approximated by those of the corresponding chi-square distribution. Finally, the procedures developed are applied to two different sets of economic data. /// Dans plusieurs situations, nous voulons vérifier l'existence de relations entre des séries chronologiques multivariées. Dans cet article, nous proposons une généralisation de la procédure de Haugh (1976) afin de tester l'hypothèse de non-corrélation de deux séries stationnaires ARMA multivariées. Les statistiques de test sont basées sur les matrices de corrélations croisées résiduelles. Sous l'hypothèse nulle de non-corrélation, nous montrons qu'un vecteur quelconque de corrélations croisées résiduelles suit la même loi asymptotique que le vecteur correspondant des corrélations croisées entre les deux séries d'innovations. De ce résultat, il découle que les statistiques de test considérées suivent asymptotiquement une loi khi-deux. Deux statistiques de test sont décrites. La première est basée sur la matrice de corrélation croisée résiduelle à un délai particulier alors que la deuxième