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

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Chapitre 2. Estimating weak structural VARMA models 58 2.2 Model and assumptions Consider a d-dimensional stationary process (Xt) satisfying a structural VARMA(p,q) representation of the form p q A00Xt − A0iXt−i = B00ǫt − B0iǫt−i, (ǫt) ∼ WN(0,Σ0), (2.2) i=1 where Σ0 is non singular and t ∈ Z = {0,±1,...}. The standard VARMA(p,q) form, which is sometimes called the reduced form, is obtained for A00 = B00 = Id. The structural forms are mainly used in econometrics to identify structural economic shocks and to allow instantaneous relationships between economic variables. Of course, constraints are necessary for the identifiability of the (p+q+3)d 2 elements of the matrices involved in the VARMA equation (2.2). We thus assume that these matrices are parameterized by a vector ϑ0 of lower dimension. We then write A0i = Ai(ϑ0) and B0j = Bj(ϑ0) for i = 0,...,p and j = 0,...,q, and Σ0 = Σ(ϑ0), where ϑ0 belongs to the parameter space Θ ⊂ Rk0 , and k0 is the number of unknown parameters, which is typically much smaller that (p + q + 3)d2 . The parametrization is often linear (see Example 2.1 below), and thus satisfies the following smoothness conditions. A1 : The applications ϑ ↦→ Ai(ϑ) i = 0,...,p, ϑ ↦→ Bj(ϑ) j = 0,...,q and ϑ ↦→ Σ(ϑ) admit continuous third order derivatives for all ϑ ∈ Θ. For simplicity we now write Ai, Bj and Σ instead of Ai(ϑ), Bj(ϑ) and Σ(ϑ). Let Aϑ(z) = A0 − p i=1Aiz i and Bϑ(z) = B0 − q i=1Bizi . We assume that Θ corresponds to stable and invertible representations, namely A2 : for all ϑ ∈ Θ, we have detAϑ(z)detBϑ(z) = 0 for all |z| ≤ 1. To show the strong consistency of the QMLE, we will use the following assumptions. A3 : We have ϑ0 ∈ Θ, where Θ is compact. i=1 A4 : The process (ǫt) is stationary and ergodic. A5 : For all ϑ ∈ Θ such that ϑ = ϑ0, either the transfer functions for some z ∈ C, or A −1 −1 0 B0B ϑ (z)Aϑ(z) = A −1 00 B00B −1 ϑ0 (z)Aϑ0(z) A −1 0 B0ΣB ′ 0A −1′ 0 = A −1 00B00Σ0B ′ 00A −1′ 00 . Remark 2.1. The previous identifiability assumption is satisfied when the parameter space Θ is sufficiently constrained. Note that the last condition in A5 can be dropped for the standard reduced forms in which A0 = B0 = Id, but may be important for structural VARMA forms (see Example 2.1 below). The identifiability of VARMA processes has been studied in particular by Hannan (1976) who gave several procedures ensuring identifiability. In particular A5 is satisfied when we impose A0 = B0 = Id, A2, the common left divisors of Aϑ(L) and Bϑ(L) are unimodular (i.e. with nonzero constant determinant), and the matrix [Ap : Bq] is of full rank.
Chapitre 2. Estimating weak structural VARMA models 59 The structural form (2.2) allows to handle seasonal models, instantaneous economic relationships, VARMA in the so-called echelon form representation, and many other constrained VARMA representations. Example 2.1. Assume that income (Inc) and consumption (Cons) variables are related by the equations Inct = c1 + α01 Inct−1 + α02 Const−1 + ǫ1t and Const = c2 + α03 Inct + α04 Inct−1 + α05 Const−1 + ǫ2t. In the stationary case the process Xt = {Inct −E(Inct),Const −E(Const)} ′ satisfies a structural VAR(1) equation 1 0 α01 α02 Xt = Xt−1 +ǫt. −α03 1 α04 α05 We also assume that the two components of ǫt correspond to uncorrelated structural economic shocks, with respective variances σ2 01 and σ2 02 . We thus have α1α3 +α4 α2α3 +α5 ϑ ′ 0 = (α01,α02,α03,α04,α05,σ 2 01,σ 2 02). Note that the identifiability condition A5 is satisfied because for all ϑ = (α1,α2,α3,α4,α5,σ 2 1 ,σ2 2 )′ = ϑ0 we have α1 α2 α01 α02 I2 − z = I2 − z for some z ∈ C, or 2 σ1 σ2 1α3 σ2 1α3 σ2 1α2 3 +σ2 2 = α01α03 +α04 α02α03 +α05 σ2 01 σ2 01α03 σ2 01α03 σ2 01α2 03 +σ2 02 2.3 Quasi-maximum likelihood estimation Let X1,...,Xn be observations of a process satisfying the VARMA representation (2.2). Note that from A2 the matrices A00 and B00 are invertible. Introducing the innovation process et = A −1 00B00ǫt, the structural representation Aϑ0(L)Xt = Bϑ0(L)ǫt can be rewritten as the reduced VARMA representation Xt − p i=1 A −1 00 A0iXt−i = et − For all ϑ ∈ Θ, we recursively define ˜et(ϑ) for t = 1,...,n by ˜et(ϑ) = Xt − p i=1 A −1 0 AiXt−i + q i=1 q i=1 . A −1 −1 00B0iB00 A00et−i. (2.3) A −1 0 BiB −1 0 A0˜et−i(ϑ),
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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
Page 19 and 20: Chapitre 1. Introduction 7 obtenir
Page 21 and 22: Chapitre 1. Introduction 9 les outi
Page 23 and 24: Chapitre 1. Introduction 11 1.1.1 E
Page 25 and 26: Chapitre 1. Introduction 13 Propri
Page 27 and 28: Chapitre 1. Introduction 15 H7 : Po
Page 29 and 30: Chapitre 1. Introduction 17 Défini
Page 31 and 32: Chapitre 1. Introduction 19 où λ
Page 33 and 34: Chapitre 1. Introduction 21 Remarqu
Page 35 and 36: Chapitre 1. Introduction 23 où Eij
Page 37 and 38: 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
Page 69: Chapitre 2. Estimating weak structu
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Page 81 and 82: 1(2, 2) − b ^ 1(2, 2) Density Cha
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Chapitre 3. Estimating the asymptot
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Chapitre 4. Multivariate portmantea
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Chapitre 5. Model selection of weak
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