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

Chapitre 4. Multivariate portmanteau test for weak structural VARMA models 144
Table 4.9 – 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).
Model Length n Level Standard Test Modified Test
m = 1 m = 2 m = 1 m = 2
α = 1% 5.7 1.9 1.1 0.9
I 500 α = 5% 22.0 7.1 5.5 4.1
α = 10% 35.8 14.7 11.1 9.0
α = 1% 6.1 1.5 1.6 0.7
I 1,000 α = 5% 21.3 7.9 5.1 4.6
α = 10% 35.5 15.0 10.5 9.6
α = 1% 4.2 1.9 0.9 0.8
I 2,000 α = 5% 21.7 7.5 3.6 4.2
α = 10% 36.2 15.5 8.6 9.2
I : Strong VARMA(1,1) model (4.13)-(4.14) with θ0 = (0.225,−0.313,0.750)
Model Length n Level Standard Test Modified Test
m = 3 m = 4 m = 6 m = 3 m = 4 m = 6
α = 1% 1.1 0.7 1.0 0.6 0.4 0.3
I 500 α = 5% 5.9 4.9 5.3 4.2 3.0 3.3
α = 10% 11.5 10.3 11.1 8.7 8.2 9.0
α = 1% 1.1 0.8 0.7 0.3 0.3 0.5
I 1,000 α = 5% 5.8 5.2 5.0 4.4 3.9 3.9
α = 10% 12.0 10.1 9.4 9.3 8.7 8.6
α = 1% 0.6 1.2 0.8 0.4 0.8 0.7
I 2,000 α = 5% 5.3 5.2 4.6 3.6 4.2 4.1
α = 10% 10.7 11.0 10.0 8.3 9.2 9.3
I : Strong VARMA(1,1) model (4.13)-(4.14) with θ0 = (0.225,−0.313,0.750)
4.11 References
Ahn, S. K. (1988) Distribution for residual autocovariances in multivariate autoregressive
models with structured parameterization. Biometrika 75, 590–93.
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance
matrix estimation. Econometrica 59, 817–858.
Bauwens, L., Laurent, S. and Rombouts, J.V.K. (2006) Multivariate GARCH
models : a survey. Journal of Applied Econometrics 21, 79–109.
Boubacar Mainassara, Y. and Francq, C. (2009) Estimating structural VARMA
models with uncorrelated but non-independent error terms. Working Papers,
http ://perso.univ-lille3.fr/ cfrancq/pub.html.