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

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Chapitre 4. Multivariate portmanteau test for weak structural VARMA models 124 4.4 Joint distribution of ˆ θn and the noise empirical autocovariances Let êt = ˜et( ˆ θn) be the LS residuals 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 p q êt = Xt − i=1 A −1 0 (ˆ θn)Ai( ˆ θn) ˆ Xt−i + 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. Let, ˆ Σe0 = ˆ Γe(0) = . We denote by n −1 n t=1 êtê ′ t γ(h) = 1 n n t=h+1 ete ′ t−h and ˆ Γe(h) = 1 n n t=h+1 êtê ′ t−h the white noise "empirical" autocovariances and residual autocovariances. It should be noted that γ(h) is not a statistic (unless if p = q = 0) because it depends on the unobserved innovations et = et(θ0). For a fixed integer m ≥ 1, let and and let ˆΓm = Γ(ℓ,ℓ ′ ) = γm = {vecγ(1)} ′ ,...,{vecγ(m)} ′ ′ vecˆ ′ Γe(1) ,..., vecˆ ′ ′ Γe(m) , ∞ h=−∞ E {et−ℓ ⊗et}{et−h−ℓ ′ ⊗et−h} ′ , for (ℓ,ℓ ′ ) = (0,0). For the univariate ARMA model, Francq, Roy and Zakoïan (2005) have showed in Lemma A.1 that |Γ(ℓ,ℓ ′ )| ≤ Kmax(ℓ,ℓ ′ ) for some constant K, which is sufficient to ensure the existence of these matrices. We can generalize this result for the multivariate ARMA model. Then we obtain Γ(ℓ,ℓ ′ ) ≤ Kmax(ℓ,ℓ ′ ) for some constant K. The proof is similar to the univariate case. We are now able to state the following theorem, which is an extension of a result given in Francq, Roy and Zakoïan (2005). Theorem 4.1. Assume p > 0 or q > 0. Under Assumptions A1’-A2-A7 or A1-A4, and A6, as n → ∞, √ n(γm, ˆ θn −θ0) ′ d ⇒ N(0,Ξ) where Ξ = Σγm Σ γm, ˆ θn Σ ′ γm, ˆ θn with Σγm = {Γ(ℓ,ℓ ′ )} 1≤ℓ,ℓ ′ ≤m , Σ ′ γm, ˆ θn Varas( √ nJ −1 Yn) = J −1 IJ −1 . Σˆ θn , = Cov( √ nJ −1 Yn, √ nγm) and Σˆ θn =
Chapitre 4. Multivariate portmanteau test for weak structural VARMA models 125 4.5 Asymptotic distribution of residual empirical autocovariances and autocorrelations Let the diagonal matrices Se = Diag(σe(1),...,σe(d)) and ˆ Se = Diag(ˆσe(1),...,ˆσe(d)), where σ2 e(i) is the variance of the i-th coordinate of et and ˆσ 2 e(i) is its sample estimate, i.e. σe(i) = Ee2 it and ˆσe(i) = n−1n t=1ê2it . The theoretical and sample autocorrelations at lagℓare respectively defined byRe(ℓ) = S−1 −1 e Γe(ℓ)Se and ˆ Re(ℓ) = ˆ S−1 e ˆ Γe(ℓ) ˆ S−1 e , with Γe(ℓ) := Eete ′ t−ℓ = 0 for all ℓ = 0. Consider the vector of the first m sample autocorrelations ˆρm = vecˆ ′ Re(1) ,..., vecˆ ′ ′ Re(m) . Theorem 4.2. Under Assumptions A1-A4 and A6 or A1’, A2-A7, √ nΓm ˆ ⇒ N 0,ΣˆΓm and √ nˆρm ⇒ N (0,Σˆρm) where, ΣˆΓm = Σγm +ΦmΣˆ θn Φ ′ m +ΦmΣˆ θn,γm +Σ ′ ˆ θn,γm Φ ′ m Σˆρm = Im ⊗(Se ⊗Se) −1 ΣˆΓm Im ⊗(Se ⊗Se) −1 and Φm is given by (4.26) in the proof of this Theorem. (4.5) (4.6) 4.6 Limiting distribution of the portmanteau statistics Box and Pierce (1970) (BP hereafter) derived a goodness-of-fit test, the portmanteau test, for univariate strong ARMA models. Ljung and Box (1978) (LB hereafter) proposed a modified portmanteau test which is nowadays one of the most popular diagnostic checking tool in ARMA modeling of time series. The multivariate version of the BP portmanteau statistic was introduced by Chitturi (1974). Hosking (1981a) gave several equivalent forms of this statistic. Basic forms are
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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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