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

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Chapitre 2. Estimating weak structural VARMA models 78 2.9 Appendix of additional example Example 2.2. Denoting by a0i(k,ℓ) and b0i(k,ℓ) the generic elements of the matrices A0i and B0i, the Kronecker indices are defined by pk = max{i : a0i(k,ℓ) = 0 or b0i(k,ℓ) = 0 for some ℓ = 1,...,d}. To ensure relatively parsimonious parameterizations, one can specify an echelon form depending on the Kronecker indices(p1,...,pd). The reader is refereed to Lütkepohl (1993) for details about the echelon form. For instance, a 3-variate ARMA process with Kronecker indices (1,2,0) admits the echelon form = ⎛ ⎝ ⎛ ⎝ 1 0 0 0 1 0 × × 1 1 0 0 0 1 0 × × 1 ⎞ ⎛ ⎠Xt −⎝ ⎞ ⎛ ⎠ǫt −⎝ × × 0 0 × 0 0 0 0 × × × × × × 0 0 0 ⎞ ⎛ ⎠Xt−1 −⎝ ⎞ ⎛ ⎠ǫt−1 −⎝ 0 0 0 × × 0 0 0 0 0 0 0 × × × 0 0 0 ⎞ ⎠Xt−2 ⎞ ⎠ǫt−2 where × denotes an unconstrained element. The variance of ǫt is defined by 6 additional parameters. This echelon form thus corresponds to a parametrization by a vector ϑ of size k0 = 24. 2.9.1 Verification of Assumption A8 on Example 2.1 Thus In this example, we have Mϑ0 = α01 α02 σ 2 01 σ 2 01 α03 α01α03 +α04 α02α03 +α05 σ 2 01 α03 σ 2 01 α2 03 +σ2 02 Mϑ0= is of full rank k0 = 7. ⎛ ⎜ ⎝ 1 α03 0 0 0 0 0 0 0 0 1 α03 0 0 0 0 0 α01 0 α02 0 σ 2 01 σ2 01 2α03σ 2 01 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 α03 α03 α 2 03 0 0 0 0 0 0 0 1 ⎞ ⎟ ⎠ .
Chapitre 2. Estimating weak structural VARMA models 79 2.9.2 Details on the proof of Theorem 2.1 Lemma 6. Under the assumptions of Theorem 2.1, we have sup˜et(ϑ)−et(ϑ) ≤ Kρ ϑ∈Θ t , where ρ is a constant belonging to [0,1), and K > 0 is measurable with respect to the σ-field generated by {Xu,u ≤ 0}. and Proof of Lemma 6 : We have p et(ϑ) = Xt − ˜et(ϑ) = Xt − p i=1 i=1 A −1 0 AiXt−i + A −1 0 AiXt−i + q i=1 q i=1 A −1 0 BiB −1 0 A0et−i(ϑ) ∀t ∈ Z, (2.26) A −1 −1 0 BiB0 A0˜et−i(ϑ) t = 1,...,n (2.27) with the initial values ˜e0(ϑ) = ··· = ˜e1−q(ϑ) = X0 = ··· = X1−p = 0. Let ⎛ ⎞ et(ϑ) ⎜ et−1(ϑ) ⎟ et(ϑ) = ⎜ ⎟ ⎝ . ⎠ et−q+1(ϑ) , ˜e ⎛ ⎞ ˜et(ϑ) ⎜ ˜et−1(ϑ) ⎟ t(ϑ) = ⎜ ⎟ ⎝ . ⎠ ˜et−q+1(ϑ) . From (2.26) and (2.27), we have and where C = e t(ϑ) = b t +Ce t−1(ϑ) ∀t ∈ Z, ˜e t(ϑ) = ˜ b t +C˜e t−1(ϑ) t = 1,...,n, A −1 0 B1B −1 0 A0 ··· A −1 0 BqB −1 0 A0 I(q−1)d 0(q−1)d , b t = ˜bt = bt for t > p, ˜bt = 0qd for t ≤ 0, and ⎛ Xt − ⎜ ˜ ⎜ bt = ⎜ ⎝ t−1 i=1A−1 ⎞ 0 AiXt−i ⎟ 0d ⎟ for t = 1,...,p. . ⎠ 0d ⎛ ⎜ ⎝ Xt − p i=1 A−1 0 AiXt−i 0d . 0d ⎞ ⎟ ⎠ ,
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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
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
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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Chapitre 4. Multivariate portmantea
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Chapitre 5. Model selection of weak
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