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

Chapitre 4. Multivariate portmanteau test for weak structural VARMA models 130
By induction, we have
and
Cov ǫ 2 it,ǫ 2
2k−h+1
it−h = ˜ρ −1 =
Similarly, for i = j we have
2ρ 2 if h = 2k
0 if h ≥ 2k +1
Cov ǫ 2 dt,ǫ 2
2|d−1|
dt−h = 1+2ρ 2k−h+1
−1
2|d−1| 2ρ if h = 2k
=
0 if h ≥ 2k +1
Cov ǫ 2 it ,ǫ2
2
j t−1 = Eηit k
m=1
= 1+2ρ 2|i−j−1|2k −1.
(4.9)
. (4.10)
η 2 j t−1 η2 i+1t−2m+1 η2 j+1t−2m η2 it−2m η2 j t−2m−1 −1
By induction, we have
Cov ǫ 2 it ,ǫ2 2|i−j−1|
j t−h = 1+2ρ 2k−h+1
−1
2|i−j−1| 2ρ if h = 2k
=
. (4.11)
0 if h ≥ 2k +1
From (4.9), (4.10) and (4.11) the ǫt’s are not independent.
Example 4.4. The GARCH models constitute important examples of weak white
noises in the univariate case. These models have numerous extensions to the multivariate
framework (see Bauwens, Laurent and Rombouts (2006) for a review). Jeantheau
(1998) has proposed the simplest extension of the multivariate GARCH with conditional
constant correlation. In this model, the process (ǫt) verifies the following relation
ǫt = Htηt where {ηt = (η1t,...,ηdt) ′ } t is an iid centered process with Var{ηit} = 1 and
Ht is a diagonal matrix whose elements hiit verify
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎜
⎝
h 2 11t
.
h 2 ddt
⎟ ⎜
⎠ = ⎝
c1
.
cd
⎟
⎠+
q
i=1
Ai
⎜
⎝
ǫ 2 1 t−i
.
ǫ 2 dt−i
⎟
⎠+
p
j=1
Bj
⎜
⎝
h 2 11t−j
.
h 2 ddt−j
The elements of the matrices Ai and Bj, as well as the vector ci, are supposed to
be positive. In addition suppose that the stationarity conditions hold. For simplicity,
consider the ARCH(1) case with A1 such that add = 0 and a11 = ··· = a1d−1 = 0.
Then it is easy to see that
Cov ǫ 2 dt ,ǫ2
dt−1 = Cov cd +addǫ 2
2
dt−1 ηdt ,ǫ 2
dt−1 = addVar ǫ 2
dt = 0,
which shows that the ǫt’s are not independent. Thus, the solution of the GARCH equation
satisfies the assumption A1’, but does not satisfy A1 in general.
⎟
⎠.