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Theses

Estimation, validation et identification des modèles ARMA faibles multivariés

Abstract : The goal of this thesis is to study the vector autoregressive moving-average (V)ARMA models with uncorrelated but non-independent error terms. These models are called weak VARMA by opposition to the standard VARMA models, also called strong VARMA models, in which the error terms are supposed to be iid. We relax the standard independence assumption, and even the martingale difference assumption, on the error term in order to be able to cover VARMA representations of general nonlinear models. The problems that are considered here concern the statistical analysis. More precisely, we concentrate on the estimation and validation steps. We study the asymptotic properties of the quasi-maximum likelihood (QMLE) and/or least squares estimators (LSE) of weak VARMA models. Conditions are given for the consistency and asymptotic normality of the QMLE/LSE. A particular attention is given to the estimation of the asymptotic variance matrix, which may be very different from that obtained in the standard framework. After identification and estimation of the vector autoregressive moving-average processes, the next important step in the VARMA modeling is the validation stage. The validity of the different steps of the traditional methodology of Box and Jenkins, identification, estimation and validation, depends on the noise properties. Several validation methods are studied. This validation stage is not only based on portmanteau tests, but also on the examination of the autocorrelation function of the residuals and on tests of linear restrictions on the parameters. Modified versions of the Wald, Lagrange Multiplier and Likelihood Ratio tests are proposed for testing linear restrictions on the parameters. We studied the joint distribution of the QMLE/LSE and of the noise empirical autocovariances. We then derive 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. In the standard framework (i.e. under the assumption of an iid noise), it is shown that the asymptotic distribution of the portmanteau tests is that of a weighted sum of independent chi-squared random variables. The asymptotic distribution can be quite different when the independence assumption is relaxed. Consequently, the usual chi-squared distribution does not provide an adequate approximation to the distribution of the Box-Pierce goodness-of fit portmanteau test. Hence we propose a method to adjust the critical values of the portmanteau tests. Finally, we considered the problem of orders selection of weak VARMA models by means of information criteria. We propose a modified Akaike information criterion (AIC).
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Submitted on : Monday, February 1, 2010 - 1:13:04 PM
Last modification on : Thursday, February 21, 2019 - 11:02:54 AM
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Yacouba Boubacar Mainassara. Estimation, validation et identification des modèles ARMA faibles multivariés. Mathématiques [math]. Université Charles de Gaulle - Lille III, 2009. Français. ⟨tel-00452032⟩

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