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

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Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 110 3.7 References Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858. Boubacar Mainassara, Y. and Francq, C. (2009) Estimating structural VARMA models with uncorrelated but non-independent error terms. Working Papers, http ://perso.univ-lille3.fr/ cfrancq/pub.html. Brockwell, P. J. and Davis, R. A. (1991) Time series : theory and methods. Springer Verlag, New York. Davydov, Y. A. (1968) Convergence of Distributions Generated by Stationary Stochastic Processes. Theory of Probability and Applications 13, 691–696. Dufour, J-M., and Pelletier, D. (2005) Practical methods for modelling weak VARMA processes : identification, estimation and specification with a macroeconomic application. Technical report, Département de sciences économiques and CIREQ, Université de Montréal, Montréal, Canada. Francq, C. and Raïssi, H. (2006) Multivariate Portmanteau Test for Autoregressive Models with Uncorrelated but Nonindependent Errors, Journal of Time Series Analysis 28, 454–470. Francq, C., Roy, R. and Zakoïan, J-M. (2005) Diagnostic checking in ARMA Models with Uncorrelated Errors, Journal of the American Statistical Association 100, 532–544. Francq, and Zakoïan, J-M. (2005) Recent results for linear time series models with non independent innovations. In Statistical Modeling and Analysis for Complex Data Problems, Chap. 12 (eds P. Duchesne and B. Rémillard). New York : Springer Verlag, 137–161. Hannan, E. J. and Rissanen (1982) Recursive estimation of mixed of Autoregressive Moving Average order, Biometrika 69, 81–94. Lütkepohl, H. (1993) Introduction to multiple time series analysis. Springer Verlag, Berlin. Lütkepohl, H. (2005) New introduction to multiple time series analysis. Springer Verlag, Berlin. Magnus, J.R. and H. Neudecker (1988) Matrix Differential Calculus with Application in Statistics and Econometrics. New-York, Wiley. Newey, W.K., and West, K.D. (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703–708. Reinsel, G. C. (1997) Elements of multivariate time series Analysis. Second edition. Springer Verlag, New York.
Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 111 3.8 Verification of expression of I and J on Examples The following examples illustrate how the matrices I and J depend on θ0 and terms involving the distribution of the innovations et. Example 3.1. Consider for instance a univariate ARMA(1,1) of the form Xt = aXt−1 − bǫt−1 + ǫt, with variance σ2 . Then, with our notations we have θ = (a,b) ′ , θ0 = (a0,b0) ′ , A∗ i = ai , B∗ i = bi , λi = A∗ i−1 ,B∗ i−1 i−1 2 i−1 = (a ,b ), Eǫt = σ2 , Γ(i,j) = +∞ h=−∞E(ǫtǫt−iǫt−hǫt−j−h)I4 = γ(i,j)I4 and Thus, we have vecJ = 2 vecI = 4 σ 4 0 1 1−a2 0 1 1−a0b0 λi ⊗λi = a 2(i−1) ,(ab) i−1 ,(ab) i−1 ,b 2(i−1) . i≥1 a 2(i−1) 0 ,(a0b0) i−1 ,(a0b0) i−1 ,b 2(i−1) ′ 0 +∞ i,j=1 1 1−a0b0 1 1−b2 0 and γ(i,j) a 2(i−1) 0 ,(a0b0) i−1 ,(a0b0) i−1 ,b 2(i−1) ′ 0 , where σ0 is the true value of σ. We then deduce that J = 2 and I = 4 σ4 +∞ γ(i,j) 0 i,j=1 1 1−a2 0 1 1−a0b0 Example 3.2. Now we consider a bivariate VAR(1) of the form 2 α1 0 σ11 0 Xt = Xt−1 +et, Σe = 0 α2 With our notations, we have θ = (α1,α2) ′ , θ0 = (α01,α02) ′ , A(L) = 1−α1L 0 0 1−α2L , Mij(z) = A ∗ h 1 0 α1 11,h = 0 0 0 A ∗ h 0 1 α1 12,h = 0 0 0 A ∗ h 0 0 α1 21,h = 1 0 0 A ∗ h 0 0 α1 22,h = 0 1 0 0 α h 2 0 α h 2 0 α h 2 0 α h 2 ∞ h=0 Eij 0 σ 2 22 α h 1 0 0 α h 2 h α1 0 = 0 0 h 0 α2 = 0 0 0 0 = α h 1 0 0 0 = 0 α h 2 , , , . . z h 1 1−a0b0 1 1−b2 0 . i,j = 1,2,
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UNIVERSITÉ CHARLES DE GAULLE - LIL
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Résumé Dans cette thèse nous él
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Abstract The goal of this thesis is
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Table des matières Résumé ii Abs
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4.6 Limiting distribution of the po
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4.12 Empirical size (in %) of the s
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Chapitre 1 Introduction Lorsque l
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Chapitre 1. Introduction 3 Pour les
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Chapitre 1. Introduction 5 VARMA(p0
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Chapitre 1. Introduction 7 obtenir
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Chapitre 1. Introduction 9 les outi
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Chapitre 1. Introduction 11 1.1.1 E
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Chapitre 1. Introduction 13 Propri
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Chapitre 1. Introduction 15 H7 : Po
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Chapitre 1. Introduction 17 Défini
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Chapitre 1. Introduction 19 où λ
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Chapitre 1. Introduction 21 Remarqu
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Chapitre 1. Introduction 23 où Eij
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Chapitre 1. Introduction 25 La dém
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Chapitre 1. Introduction 27 Théor
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Chapitre 1. Introduction 29 La sél
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Chapitre 1. Introduction 31 La dém
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Chapitre 1. Introduction 33 de BP d
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Chapitre 1. Introduction 35 Théor
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Chapitre 1. Introduction 37 Les exp
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Chapitre 1. Introduction 39 indispe
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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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Chapitre 5. Model selection of weak
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