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

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Chapitre 2. Estimating weak structural VARMA models 70 0.0 0.5 1.0 1.5 0.0 0.4 0.8 1.2 Strong VARMA: estimator 2J −1 (a) (b) (c) Estimates of diag(Ω) Weak VARMA: estimator 2J −1 (a) (b) (c) Estimates of diag(Ω) 0.0 0.5 1.0 1.5 0.0 0.4 0.8 1.2 Strong VARMA: sandwich estimator of Ω (a) (b) (c) Estimates of diag(Ω) Weak VARMA: sandwich estimator of Ω (a) (b) (c) Estimates of diag(Ω) Figure 2.2 – Comparison of standard and modified estimates of the asymptotic variance Ω of the QMLE, on the simulated models presented in Figure 2.1. The diamond symbols represent the mean, over the N = 1,000 replications, of the standardized squared errors n{â1(2,2)−0.95} 2 for (a) (0.02 2 in the strong and weak cases), n ˆb1(2,1)−2 for (b) (1.02 in the strong case and 1.01 in the strong 2 case) and n ˆb1(2,2) for (c) (0.94 in the strong case and 0.43 in the weak case).
Chapitre 2. Estimating weak structural VARMA models 71 2.7 Appendix of technical proofs We begin with a lemma useful to show the identifiability of ϑ0. Lemma 1. Assume that Σ0 is non singular and that A5 holds true. If A −1 −1 0 B0Bϑ (L)Aϑ(L)Xt = A −1 00B00ǫt with probability one and A −1 0 B0ΣB ′ 0A−1′ 0 = A −1 00B00Σ0B ′ 00A −1′ 00 , then ϑ = ϑ0. Proof : Let ϑ = ϑ0. Assumption A5 implies that either A −1 0 B0ΣB ′ 0A−1′ 0 = A −1 00B00Σ0B ′ 00A −1′ 00 or there exist matrices Ci such that Ci0 = 0 for some i0 > 0 and A −1 0 B0B −1 ϑ −1 −1 (z)Aϑ(z)−A 00B00Bϑ0 (z)Aϑ0(z) = By contradiction, assume that A −1 0 B0B −1 ∞ Ciz i . 00B00ǫt = A −1 −1 00B00Bϑ0 (L)Aϑ0(L)Xt with probability one. This implies that there exists λ = 0 such that λ ′ Xt−i0 is almost surely a linear combination of the components of Xt−i,i > i0. By stationarity, it follows that λ ′ Xt is almost surely a linear combination of the components of Xt−i,i > 0. Thus λ ′ ǫt = 0 almost surely, which is impossible when the variance Σ0 of ǫt is positive definite. ✷ i=i0 ϑ (L)Aϑ(L)Xt = A −1 Proof of Theorem 2.1 : Note that, due to the initial conditions, {˜et(ϑ)} is not stationary, but can be approximated by the stationary ergodic process et(ϑ) = A −1 −1 0 B0Bϑ (L)Aϑ(L)Xt. (2.12) From an extension of Lemma 1 in FZ, it is easy to show that sup ϑ∈Θ˜et(ϑ)−et(ϑ) → 0 almost surely at an exponential rate, as t → ∞. We thus have where ˜ℓn(ϑ) oP(1) = ℓn(ϑ) := 1 n n lt(ϑ) as n → ∞, t=1 lt(ϑ) = dlog(2π)+logdetΣe +e ′ t(ϑ)Σ −1 e et(ϑ). Now the ergodic theorem shows that almost surely ℓn(ϑ) → dlog(2π)+Q(ϑ), where Q(ϑ) = logdetΣe +Ee ′ 1 (ϑ)Σ−1 e e1(ϑ). We have Q(ϑ) = E{e1(ϑ0)} ′ Σ −1 e {e1(ϑ0)}+logdetΣe +E{e1(ϑ)−e1(ϑ0)} ′ Σ −1 e {e1(ϑ)−e1(ϑ0)} +2E{e1(ϑ)−e1(ϑ0)} ′ Σ −1 e e1(ϑ0).
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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
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
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Page 81: 1(2, 2) − b ^ 1(2, 2) Density Cha
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Chapitre 4. Multivariate portmantea
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THÈSE Estimation, validation et identification des modèles ARMA ...

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