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

Chapitre 4. Multivariate portmanteau test for weak structural VARMA models 128
|i−j| 1 ρ
Let, Σij =
ρ |i−j|
the covariance matrix of the vector (ηit,ηj
1
t) ′ . A joint
probability density function of the bivariate random vector (ηit,ηj t) ′ with mean vector
zero is of the form
f(x) = (2π) −1 |Σij| −1/2 exp[(−1/2)x ′ Σ −1
ij x],
with |Σij| = 1−ρ 2|i−j| . If |ρ| < 1 the matrix Σij is invertible and
Σ −1
ij =
1 1 −ρ
×
1−ρ 2|i−j| |i−j|
−ρ |i−j|
.
1
Then we have
f(x1,x2) =
1
2π
−1
exp
1−ρ 2|i−j| 2(1−ρ 2|i−j|
2
x1 +x
)
2 2 −2ρ|i−j|
x1x2
.
Letting β = 1/2π 1−ρ 2|i−j| and α = 1−ρ 2|i−j| , we calculate the integral
Ex 2 1 x2 2
= β
R 2
x 2 1x22 e
x1 −1 −ρ
2
|i−j| 2 x2 +x α
2
2
dx1dx2.
Setting u = x1 −ρ |i−j|
x2 /α and s = x2, the Jacobian of this transformation is |J| =
α. Then we have
Ex 2 1x 2
2 = βα
R2 2 −1
|i−j| 2
αu+ρ s s e 2 (u2 +s2 ) duds,
= βα s 2 e −1
2 s2
2 2 |i−j| 2|i−j| 2
α u +2αρ us+ρ s e −1
2 u2
du ds.
R
Integrating by parts and passing to polar coordinates, we have
Ex 2 1x 2
2 = βα α 2√ 2π +ρ 2|i−j|√ 2πs 2
Thus, we have
then, we have
R
R
= βα 2πα 2 +6πρ 2|i−j|
= 1+2ρ 2|i−j| .
s 2 e −1
2 s2
ds,
Cov ǫ 2 it,ǫ 2 2
j t = Eǫitǫ 2 j t − Eǫ 2 2 2 2
it Eǫj t
= E
2k
η 2 it−mη2 j t−m −1
m=0
= 1+2ρ 2|i−j|2k+1 −1,
Cov ǫ 2 it ,ǫ2
2|i−j|
j t−h = 1+2ρ 2k−h+1
−1.