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

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Chapitre 2. Estimating weak structural VARMA models 86 By the dominated convergence theorem, it follows that I(ϑ) exists and is given by whenever I(ϑ) = ∞ h=−∞ ∞ h=−∞ Recall that, for all i ∈ {1,...,k0} we have ∂lt(ϑ) ∂ϑi = Tr Σ −1 e = Tr Σ −1 e Cov ˙lt, ˙ lt−h Cov ˙lt, ˙ lt−h < ∞. (2.42) Id −et(ϑ)e ′ t (ϑ)Σ−1 ∂Σe(ϑ) e +2 ∂ϑi ∂e′ t (ϑ) ∂ϑi Id −et(ϑ)e ′ t (ϑ)Σ−1 ∞ ∂Σe(ϑ) e +2 ∂ϑi In view of the last equality, the i-th component of ˙ lt is of the form where ˙li t = ∞ d ℓ=0 k1,k2=1 ℓ=1 Σ −1 e et(ϑ) e ′ t−ℓ (ϑ)d′ ℓ Σ−1 e et(ϑ). αℓ(i,k1,k2)ek1 t(ϑ)ek2 t−ℓ(ϑ) (2.43) |αℓ(i,k1,k2)| ≤ Kρ ℓ , (2.44) uniformly in (i,k1,k2) ∈ {1,...,k0}×{1,...,d} 2 . In order to obtain (2.42), it suffices to show that ∞ Cov ˙li t, ˙ lj t−h < ∞ (2.45) h=−∞ for all (i,j) ∈ {1,...,k0} 2 . In view of (2.43) and (2.44), we have for h ≥ 0 Cov ˙li t, ˙ lj t−h ≤ c1 +c2, where and Because c1 = K c2 = K ∞ ∞ d ℓ=[h/2]+1 ℓ ′ =0 k1,...,k4=1 [h/2] ∞ d ℓ=0 ℓ ′ =0 k1,...,k4=1 ρ ℓ+ℓ′ |Cov(ek1t(ϑ)ek2t−ℓ(ϑ),ek3t−h(ϑ)ek4t−ℓ ′ −h(ϑ))| ρ ℓ+ℓ′ |Cov(ek1 t(ϑ)ek2 t−ℓ(ϑ),ek3 t−h(ϑ)ek4 t−ℓ ′ −h(ϑ))|. |Cov(ek1 t(ϑ)ek2 t−ℓ(ϑ),ek3 t−h(ϑ)ek4 t−ℓ ′ −h(ϑ))| ≤ Eet(ϑ) 4 < ∞
Chapitre 2. Estimating weak structural VARMA models 87 by Assumption A7, we have Lemma 11 entails that It follows that c2 ≤ K [h/2] ∞ ℓ=0 ℓ ′ =0 ∞ Cov ˙li t, ˙ lj t−h ≤ K h=0 c1 ≤ Kρ h . ρ ℓ+ℓ′ α ν/(2+ν) ǫ ([h/2]) ≤ Kα ν/(2+ν) ǫ ([h/2]). ∞ ρ h +K h=0 ∞ h=0 by Assumption A7. The same bounds clearly holds for 0 h=−∞ Cov ˙li t, ˙ , lj t−h which shows (2.45) and completes the proof. ✷ α ν/(2+ν) ǫ ([h/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
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
Page 55 and 56: Chapitre 1. Introduction 43 En util
Page 57 and 58: Chapitre 1. Introduction 45 I11 = 2
Page 59 and 60: Chapitre 1. Introduction 47 ceux-ci
Page 61 and 62: Chapitre 1. Introduction 49 Remarqu
Page 63 and 64: Références bibliographiques Ahn,
Page 65 and 66: Chapitre 1. Introduction 53 Hannan,
Page 67 and 68: Chapitre 2 Estimating structural VA
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Page 81 and 82: 1(2, 2) − b ^ 1(2, 2) Density Cha
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Page 131 and 132: Chapitre 4. Multivariate portmantea
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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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