THÈSE Estimation, validation et identification des modèles ARMA ...

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notamment celui introduit par Akaike (AIC). Avec ce critère, on tente de donner une approximation de la distance (souvent appelée information de Kullback-Leibler) entre la vraie loi des observations (inconnue) et la loi du modèle estimé. Nous verrons que le critère corrigé (AICc) dans le cadre des modèles ARMA faibles vectoriels peut, là aussi, être très différent du cas fort. Mots-clés : AIC, autocorrelations résiduelles, estimateur HAC, estimateur spectral, information de Kullback-Leibler, modèles VARMA faibles, forme échelon, processus non linéaire, QMLE/LSE, représentation structurelle des modèles VARMA, test du Multiplicateur de La- grange, tests portmanteau de Ljung-Box et Box-Pierce, test du rapport de vraisemblance, test de Wald. iii
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 chisquared 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
Page 1 and 2: UNIVERSITÉ CHARLES DE GAULLE - LIL
Page 3: Résumé Dans cette thèse nous él
Page 7 and 8: Table des matières Résumé ii Abs
Page 9 and 10: 4.6 Limiting distribution of the po
Page 11 and 12: 4.12 Empirical size (in %) of the s
Page 13 and 14: Chapitre 1 Introduction Lorsque l
Page 15 and 16: Chapitre 1. Introduction 3 Pour les
Page 17 and 18: Chapitre 1. Introduction 5 VARMA(p0
Page 19 and 20: Chapitre 1. Introduction 7 obtenir
Page 21 and 22: Chapitre 1. Introduction 9 les outi
Page 23 and 24: Chapitre 1. Introduction 11 1.1.1 E
Page 25 and 26: Chapitre 1. Introduction 13 Propri
Page 27 and 28: Chapitre 1. Introduction 15 H7 : Po
Page 29 and 30: Chapitre 1. Introduction 17 Défini
Page 31 and 32: Chapitre 1. Introduction 19 où λ
Page 33 and 34: Chapitre 1. Introduction 21 Remarqu
Page 35 and 36: Chapitre 1. Introduction 23 où Eij
Page 37 and 38: Chapitre 1. Introduction 25 La dém
Page 39 and 40: Chapitre 1. Introduction 27 Théor
Page 41 and 42: Chapitre 1. Introduction 29 La sél
Page 43 and 44: Chapitre 1. Introduction 31 La dém
Page 45 and 46: Chapitre 1. Introduction 33 de BP d
Page 47 and 48: Chapitre 1. Introduction 35 Théor
Page 49 and 50: Chapitre 1. Introduction 37 Les exp
Page 51 and 52: Chapitre 1. Introduction 39 indispe
Page 53 and 54: Chapitre 1. Introduction 41 Lemme 1
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Chapitre 1. Introduction 43 En util
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Chapitre 1. Introduction 45 I11 = 2
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Chapitre 1. Introduction 47 ceux-ci
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Chapitre 1. Introduction 49 Remarqu
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Références bibliographiques Ahn,
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Chapitre 1. Introduction 53 Hannan,
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Chapitre 2 Estimating structural VA
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Chapitre 2. Estimating weak structu
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1(2, 2) − b ^ 1(2, 2) Density Cha
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Chapitre 3. Estimating the asymptot
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Chapitre 4. Multivariate portmantea
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Chapitre 5. Model selection of weak
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THÈSE Estimation, validation et identification des modèles ARMA ...

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