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

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
,