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 verifier l'existence de relations entre des series chronologiques multivariees. Dans cet article, nous proposons une generalisation de la procedure de Haugh (1976) afin de tester l'hypothese de non-correlation de deux series stationnaires ARMA multivariees. Les statistiques de test sont basees sur les matrices de correlations croisees residuelles. Sous l'hypothese nulle de non-correlation, nous montrons qu'un vecteur quelconque de correlations croisees residuelles suit la měme loi asymptotique que le vecteur correspondant des correlations croisees entre les deux series d'innovations. De ce resultat, il decoule que les statistiques de test considerees suivent asymptotiquement une loi khi-deux. Deux statistiques de test sont decrites. La premiere est basee sur la matrice de correlation croisee residuelle a un delai particulier alors que la deuxieme est basee sur une statistique de type portemanteau qui generalise la statistique de Haugh. Nous discutons aussi comment les procedures pour tester la non-correlation peuvent ětre adaptees afin de determiner les directions de causalite au sens de Granger (1969) entre les deux series. Un avantage des procedures proposees est que leur utilisation ne necessite pas l'estimation d'un modele global pour les deux series. Les proprietes des statistiques introduites pour des echantillons finis furent etudiees par simulation sous l'hypothese nulle. Cela a conduit a la definition de statistiques modifiees dont les quantites d'ordre superieure sont beaucoup mieux approximes par ceux de la loi khi-deux correspondante. Finalement, la methodologie developpee est appliquee a deux jeux de donnees economiques.
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