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

Chapitre 4. Multivariate portmanteau test for weak structural VARMA models 146
Table 4.11 – Empirical size (in %) of the standard and modified versions of the LB test in
the case of the weak VARMA(1,1) model (4.13)-(4.15).
Model Length n Level Standard Test Modified Test
m = 1 m = 2 m = 1 m = 2
α = 1% 25.7 18.3 2.0 1.4
II 500 α = 5% 48.3 33.1 6.7 4.6
α = 10% 62.7 44.6 11.3 9.1
α = 1% 27.5 20.6 2.4 1.5
II 1,000 α = 5% 50.0 36.5 7.5 4.1
α = 10% 64.8 47.9 10.9 10.3
α = 1% 29.9 22.6 2.7 1.8
II 2,000 α = 5% 50.3 39.4 8.5 6.5
α = 10% 63.7 50.0 14.1 12.9
II : Weak VARMA(1,1) model (4.13)-(4.15) 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% 14.0 11.6 10.7 1.1 1.1 1.3
II 500 α = 5% 28.2 24.5 22.0 4.4 2.8 4.0
α = 10% 38.7 34.4 30.1 7.3 5.9 9.7
α = 1% 17.1 15.0 11.5 1.1 1.0 0.7
II 1,000 α = 5% 31.3 28.1 22.9 4.4 4.1 3.3
α = 10% 42.0 38.8 32.6 9.6 8.7 7.4
α = 1% 20.4 17.9 14.8 1.2 1.4 1.3
II 2,000 α = 5% 35.4 32.3 27.8 5.4 5.5 4.9
α = 10% 46.0 42.8 36.9 10.9 10.3 9.2
II : Weak VARMA(1,1) model (4.13)-(4.15) with θ0 = (0.225,−0.313,0.750)
Francq, and Zakoïan, J-M. (2005) Recent results for linear time series models with
non independent innovations. In Statistical Modeling and Analysis for Complex
Data Problems, Chap. 12 (eds P. Duchesne and B. Rémillard). New York :
Springer Verlag, 137–161.
Herrndorf, N. (1984) A Functional Central Limit Theorem for Weakly Dependent
Sequences of Random Variables. The Annals of Probability 12, 141–153.
Hosking, J. R. M. (1980) The multivariate portmanteau statistic, Journal of the
American Statistical Association 75, 602–608.
Hosking, J. R. M. (1981a) Equivalent forms of the multivariate portmanteau statistic,
Journal of the Royal Statistical Society B 43, 261–262.
Hosking, J. R. M. (1981b) Lagrange-tests of multivariate time series models, Journal
of the Royal Statistical Society B 43, 219–230.