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

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Chapitre 2. Estimating weak structural VARMA models 62 2.4 Estimating the asymptotic variance matrix Theorem 2.2 can be used to obtain confidence intervals and significance tests for the parameters. The asymptotic variance Ω must however be estimated. The matrix J can easily be estimated by its empirical counterpart. For instance, under A9, one can take ˆJ11 ˆJ 0 = 0 J22 ˆ , ˆ J11 = 2 n n t=1 ∂ ∂ϑ (1)˜e′ t (ˆ ϑ (1) n ) ˆΣ −1 ∂ e ∂ϑ (1)′˜et( ˆ ϑ (1) n ) , and ˆ J22 = D( ˆ Σ −1 e ⊗ ˆ Σ −1 e )D′ . In the standard strong VARMA case ˆ Ω = 2 ˆ J −1 is a strongly consistent estimator of Ω. In the general weak VARMA case this estimator is not consistent when I = 2J (see Remark 2.3). So we need a consistent estimator of I. Note that I = Varas 1 √ n n t=1 Υt = +∞ h=−∞ ˆΩ HAC = ˆ J −1 Î HAC ˆ J −1 , Î HAC = 1 n Cov(Υt,Υt−h), (2.4) where Υt = ∂ logdetΣe +e ∂ϑ ′ t (ϑ(1) )Σ −1 e et(ϑ (1) ) . (2.5) ϑ=ϑ0 In the econometric literature the nonparametric kernel estimator, also called heteroscedastic autocorrelation consistent (HAC) estimator (see Newey and West, 1987, or Andrews, 1991), is widely used to estimate covariance matrices of the form I. Let ˆ Υt be the vector obtained by replacing ϑ0 by ˆ ϑn in Υt. The matrix Ω is then estimated by a "sandwich" estimator of the form n t,s=1 ω|t−s| ˆ Υt ˆ Υs, where ω0,...,ωn−1 is a sequence of weights (see Andrews, 1991, and Newey and West, 1987, for the problem of the choice of weights). Interpreting (2π) −1 I as the spectral density of the stationary process (Υt) evaluated at frequency 0 (see Brockwell and Davis, 1991, p. 459), an alternative method consists in using a parametric AR estimate of the spectral density of (Υt). This approach, which has been studied by Berk (1974) (see also den Haan and Levin, 1997), rests on the expression I = Φ −1 (1)ΣuΦ −1 (1) when (Υt) satisfies an AR(∞) representation of the form Φ(L)Υt := Υt + ∞ ΦiΥt−i = ut, (2.6) where ut is a weak white noise with variance matrix Σu. Let ˆ Φr(z) = Ik0 + r i=1 ˆ Φr,iz i , where ˆ Φr,1,··· , ˆ Φr,r denote the coefficients of the LS regression of ˆ Υt on ˆ Υt−1,··· , ˆ Υt−r. i=1
Chapitre 2. Estimating weak structural VARMA models 63 Let ûr,t be the residuals of this regression, and let ˆ Σûr be the empirical variance of ûr,1,...,ûr,n. We are now able to state the following theorem, which is an extension of a result given in Francq, Roy and Zakoïan (2005). Theorem 2.4. In addition to the assumptions of Theorem 2.2, assume that the process (Υt) defined in (2.5) admits an AR(∞) representation (2.6) in which the roots of detΦ(z) = 0 are outside the unit disk, Φi = o(i−2 ), and Σu = Var(ut) is nonsingular. Moreover we assume that ǫt8+4ν < ∞ and ∞ k=0 {αX,ǫ(k)} ν/(2+ν) < ∞ for some ν > 0, where {αX,ǫ(k)} k≥0 denotes the sequence of the strong mixing coefficients of the process (X ′ t,ǫ ′ t) ′ . Then the spectral estimator of I Î SP := ˆ Φ −1 r (1) ˆ Σûr ˆ Φ ′−1 r (1) → I in probability when r = r(n) → ∞ and r 3 /n → 0 as n → ∞. 2.5 Testing linear restrictions on the parameter It may be of interest to test s0 linear constraints on the elements of ϑ0 (in particular A0p = 0 or B0q = 0). We thus consider a null hypothesis of the form H0 : R0ϑ0 = r0 where R0 is a known s0×k0 matrix of rank s0 and r0 is a known s0-dimensional vector. The Wald, LM and LR principles are employed frequently for testing H0. The LM test is also called the score or Rao-score test. We now examine if these principles remain valid in the non standard framework of weak VARMA models. Let ˆ Ω = ˆ J−1Î ˆ J−1 , where ˆ J and Î are consistent estimator of J and I, as defined in Section 2.4. Under the assumptions of Theorems 2.2 and 2.4, and the assumption that I is invertible, the Wald statistic Wn = n(R0 ˆ ϑn −r0) ′ (R0 ˆ ΩR ′ 0) −1 (R0 ˆ ϑn −r0) asymptotically follows a χ2 s0 distribution under H0. Therefore, the standard formulation of the Wald test remains valid. More precisely, at the asymptotic level α, the Wald test consists in rejecting H0 when Wn > χ2 (1−α). It is however important to note that s0 a consistent estimator of the form ˆ Ω = ˆ J −1 Î ˆ J −1 is required. The estimator ˆ Ω = 2 ˆ J −1 , which is routinely used in the time series softwares, is only valid in the strong VARMA case. We now turn to the LM test. Let ˆ ϑ c n be the restricted QMLE of the parameter under H0. Define the Lagrangean L(ϑ,λ) = ˜ ℓn(ϑ)−λ ′ (R0ϑ−r0),
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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
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 and 70: Chapitre 2. Estimating weak structu
Page 73: Chapitre 2. Estimating weak structu
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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