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Comptes Rendus Mathématique
Volume 348, n° 15-16
pages 927-929 (août 2010)
Doi : 10.1016/j.crma.2010.07.017
Received : 22 April 2010 ;  accepted : 19 July 2010
Tests portmanteau multivariés d'adéquation de modèles VARMA faibles
Multivariate portmanteau tests of the adequacy of weak VARMA models
 

Yacouba Boubacar Mainassara
Université Lille III, EQUIPPE, BP 60 149, 59653 Villeneuve d'Ascq cedex, France 

Résumé

Dans cette Note, nous considérons les tests portmanteau, aussi appelés tests d'autocorrélation, pour tester l'adéquation de modèles ARMA multivarié (VARMA) avec innovations linéaires non corrélées mais non nécessairement indépendantes (i.e. VARMA faibles). Nous relâchons l'hypothèse standard d'indépendance pour étendre le champ d'application des modèles VARMA, ceci permettra aussi de couvrir une large classe de processus non linéaires. Dans un premier temps, nous étudions la distribution asymptotique jointe de l'estimateur du quasi-maximum de vraisemblance (QMV) et des autocovariances empiriques du bruit. Ceci nous permet ensuite d'obtenir les distributions asymptotiques des autocovariances et autocorrelations résiduelles. Enfin, nous en déduisons le comportement asymptotique des statistiques portmanteau de Ljung–Box (ou Box–Pierce) de modèles VARMA faibles. Nous proposons une méthode pour ajuster les valeurs critiques de ces tests.

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Abstract

In this Note, we consider portmanteau tests for testing the adequacy of vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. We relax the standard independence assumption to extend the range of application of the VARMA models, allowing us to treat linear representations of general nonlinear processes. We first study the joint distribution of the quasi-maximum likelihood estimator (QMLE) and the noise empirical autocovariances. We thus obtain the asymptotic distribution of residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We deduce the asymptotic distribution of the Ljung–Box (or Box–Pierce) portmanteau statistics for VARMA models with nonindependent innovations. We propose a method to adjust the critical values of the portmanteau tests.

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