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

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Chapitre 4. Multivariate portmanteau test for weak structural VARMA models 132 level α, the LB test (resp. the BP test) consists in rejecting the adequacy of the weak VARMA(p,q) model when ˜Pm > Sm(1−α) (resp. Pm > Sm(1−α)) where Sm(1−α) is such that P Zm( ˆ ξm) > Sm(1−α) = α. 4.8 Implementation of the goodness-of-fit portmanteau tests LetX1,...,Xn, be observations of ad-multivariate process. For testing the adequacy of weak VARMA(p,q) model, we use the following steps to implement the modified version of the portmanteau test. 1. Compute the estimates Â1,..., Âp, ˆ B1,..., ˆ Bq by QMLE. 2. Compute the QMLE residuals êt = ˜et( ˆ θn) when p > 0 or q > 0, and let êt = et = Xt when p = q = 0. When p+q = 0, we have êt = 0 for t ≤ 0 and t > n and êt = Xt − p i=1 A −1 0 ( ˆ θn)Ai( ˆ θn) ˆ Xt−i + q i=1 A −1 0 ( ˆ θn)Bi( ˆ θn)B −1 0 ( ˆ θn)A0( ˆ θn)êt−i, for t = 1,...,n, with ˆ Xt = 0 for t ≤ 0 and ˆ Xt = Xt for t ≥ 1. 3. Compute the residual autocovariances ˆ Γe(0) = ˆ Σe0 and ˆ Γe(h) for h = 1,...,m and ˆ ˆΓe(1) ′ ′ ′ Γm = ,..., ˆΓe(m) . 4. Compute the matrix ˆ J = 2n−1n t=1 (∂ê′ t /∂θ) ˆ Σ −1 e0 (∂êt/∂θ ′ ). 5. Compute ˆ Υt = ˆΥ ′ 1 t, ˆ Υ ′ 2 t −2 ˆ J −1 (∂ê ′ t /∂θ) ˆ Σ −1 e0 êt. 6. Fit the VAR(r) model ˆΦr(L) ˆ Υt := ′ , where ˆ Υ1 t = ê ′ t−1,...,ê ′ ′ t−m ⊗ êt and ˆ Υ2 t = I d 2 m+k0 + r ˆΦr,i(L) ˆΥt = ûr,t. The VAR order r can be fixed or selected by AIC information criteria. 7. Define the estimator ˆΞ SP := ˆ Φ −1 r (1)ˆ Σûr ˆ Φ ′−1 r (1) = ˆΣγm ˆΣ ′ γm, ˆ θn i=1 ˆΣ γm, ˆ θn ˆΣˆ θn , ˆ Σûr = 1 n n t=1 ûr,tû ′ r,t .
Chapitre 4. Multivariate portmanteau test for weak structural VARMA models 133 8. Define the estimator ˆΦm = 1 n 9. Define the estimators n ê ′ t−1,...,ê ′ ′ ∂et(θ0) t−m ⊗ ∂θ ′ θ0= ˆ . θn t=1 ˆΣˆ = Γm ˆ Σγm + ˆ Φm ˆ Σˆ ˆΦ θn ′ m + ˆ Φm ˆ Σˆ + θn,γm ˆ Σ ′ ˆΦ θn,γm ˆ ′ m ˆΣˆρm = Im ⊗( ˆ Se ⊗ ˆ Se) −1 ˆΣˆ Im ⊗( Γm ˆ Se ⊗ ˆ Se) −1 10. Compute the eigenvalues ˆ ξm = ( ˆ ξ1,d2m,..., ˆ ξd2m,d2m) ′ of the matrix ˆΣˆΓm ˆΩm = Im ⊗ ˆ Σ −1/2 e0 ⊗ ˆ Σ −1/2 e0 Im ⊗ ˆ Σ −1/2 e0 ⊗ ˆ Σ −1/2 e0 . 11. Compute the portmanteau statistics Pm = nˆρ ′ m Im ⊗ ˆR −1 e (0)⊗ ˆ R −1 e (0) ˆρm and ˜Pm = n 2 m 1 (n−h) Tr ˆΓ ′ e (h) ˆ Γ −1 e (0)ˆ Γe(h) ˆ Γ −1 e (0) . h=1 12. Evaluate the p-values P Zm( ˆ ξm) > Pm and P Zm( ˆ ξm) > ˜ Pm , Zm( ˆ d ξm) = 2 m i=1 ˆξi,d 2 mZ 2 i, using the Imhof algorithm (1961). The BP test (resp. the LB test) rejects the adequacy of the weak VARMA(p,q) model when the first (resp. the second) pvalue is less than the asymptotic level α. 4.9 Numerical illustrations In this section, by means of Monte Carlo experiments, we investigate the finite sample properties of the test introduced in this paper. For illustrative purpose, we only present the results of the modified and standard versions of the LB test. The results concerning the BP test are not presented here, because they are very close to those of the LB test. The numerical illustrations of this section are made with the softwares R (see http ://cran.r-project.org/) and FORTRAN (to compute the p-values using the Imohf algorithm, 1961).
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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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1(2, 2) − b ^ 1(2, 2) Density Cha
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THÈSE Estimation, validation et identification des modèles ARMA ...

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