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

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Chapitre 4. Multivariate portmanteau test for weak structural VARMA models 134 4.9.1 Empirical size To generate the strong and weak VARMA models, we consider the bivariate model of the form X1t 0 0 X1t−1 ǫ1t = + X2t 0 a1(2,2) X2t−1 ǫ2t 0 0 ǫ1t−1 − , (4.13) b1(2,1) b1(2,2) ǫ2t−1 where (a1(2,2),b1(2,1),b1(2,2)) = (0.225,−0.313,0.750). This model is a VARMA(1,1) model in echelon form. Strong VARMA case We first consider the strong VARMA case. To generate this model, we assume that in (4.13) the innovation process (ǫt) is defined by ǫ1,t ∼ IIDN(0,I2). (4.14) ǫ2,t We simulated N = 1,000 independent trajectories of size n = 500, n = 1,000 and n = 2,000 of Model (4.13) with the strong Gaussian noise (4.14). For each of these N replications we estimated the coefficients (a1(2,2),b1(2,1),b1(2,2)) and we applied portmanteau tests to the residuals for different values of m. For the standard LB test, the VARMA(1,1) model is rejected when the statistic ˜ Pm is greater thanχ2 (4m−3) (0.95), where m is the number of residual autocorrelations used in the LB statistic. This corresponds to a nominal asymptotic levelα = 5% in the standard case. We know that the asymptotic level of the standard LB test is indeed α = 5% when (a1(2,2),b1(2,1),b1(2,2)) = (0,0,0). Note however that, even when the noise is strong, the asymmptotic level is not exactlyα = 5% when(a1(2,2),b1(2,1),b1(2,2)) = (0,0,0)). For the modified LB test, the model is rejected when the statistic ˜ Pm is greater than Sm(0.95) i.e. when the p-value P Zm( ˆ ξm) > ˜ Pm is less than the asymptotic level α = 0.05. Let A and B be the (2×2)-matrices with non zero elements a1(2,2), b1(2,1) and b1(2,2). When the roots of det(I2 − Az)det(I2 −Bz) = 0 are near the unit disk, the asymptotic distribution of ˜ Pm is likely to be far from its χ2 (4m−3) approximation. Table 4.1 displays the relative rejection frequencies of the null hypothesis H0 that the DGP follows an VARMA(1,1) model, over the N = 1,000 independent replications. As expected the observed relative rejection frequency of the standard LB test is very far from the nominal α = 5% when the number of autocorrelations used in the LB statistic is m ≤ p + q. This is in accordance with the results in the literature on the
Chapitre 4. Multivariate portmanteau test for weak structural VARMA models 135 standard VARMA models. In particular, Hosking (1980) showed that the statistic ˜ Pm has approximately the chi-squared distribution χ2 d2 (m−(p+q)) without any identifiability contraint. Thus the error of first kind is well controlled by all the tests in the strong case, except for the standard LB test when m ≤ p + q. We draw the somewhat surprising conclusion that, even in the strong VARMA case, the modified version may be preferable to the standard one. Table 4.1 – Empirical size (in %) of the standard and modified versions of the LB test in the case of the strong VARMA(1,1) model (4.13)-(4.14), with θ0 = (0.225,−0.313,0.750). m = 1 m = 2 n 500 1,000 2,000 500 1,000 2,000 modified LB 5.5 5.1 3.6 4.1 4.6 4.2 standard LB 22.0 21.3 21.7 7.1 7.9 7.5 m = 3 m = 4 m = 6 n 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 modified LB 4.2 4.4 3.6 3.0 3.9 4.2 3.3 3.9 4.1 standard LB 5.9 5.8 5.3 4.9 5.2 5.2 5.3 5.0 4.6 Table 4.2 – Empirical size (in %) of the standard and modified versions of the LB test in the case of the strong VARMA(1,1) model (4.13)-(4.14), with θ0 = (0,0,0). m = 1 m = 2 m = 3 n 500 1,000 2,000 500 1,000 2,000 500 1,000 2,000 modified LB 5.5 4.1 3.0 3.8 3.4 3.2 4.1 3.3 2.6 standard LB 18.3 18.7 16.9 6.0 6.2 4.8 5.2 4.5 3.5 Weak VARMA case We now repeat the same experiments on different weak VARMA(1,1) models. We first assume that in (4.13) the innovation process (ǫt) is an ARCH(1) (i.e. p = 0, q = 1) model ǫ1t h11 t 0 η1t = (4.15) where h 2 11t h 2 22t ǫ2t = c1 c2 0 h22 t a11 0 + a21 a22 η2t ǫ 2 1t−1 ǫ 2 2t−1 with c1 = 0.3, c2 = 0.2, a11 = 0.45, a21 = 0.4 and a22 = 0.25. As expected, Table 4.3 shows that the standard LB test poorly performs to assess the adequacy of this weak ,
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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
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Chapitre 1. Introduction 35 Théor
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Chapitre 1. Introduction 37 Les exp
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Chapitre 1. Introduction 39 indispe
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Chapitre 1. Introduction 41 Lemme 1
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Chapitre 1. Introduction 43 En util
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Chapitre 1. Introduction 45 I11 = 2
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Chapitre 1. Introduction 47 ceux-ci
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Chapitre 1. Introduction 49 Remarqu
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Références bibliographiques Ahn,
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Chapitre 1. Introduction 53 Hannan,
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Chapitre 2 Estimating structural VA
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Chapitre 2. Estimating weak structu
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1(2, 2) − b ^ 1(2, 2) Density Cha
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THÈSE Estimation, validation et identification des modèles ARMA ...

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