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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 111
3.8 Verification of expression of I and J on Examples
The following examples illustrate how the matrices I and J depend on θ0 and terms
involving the distribution of the innovations et.
Example 3.1. Consider for instance a univariate ARMA(1,1) of the form Xt =
aXt−1 − bǫt−1 + ǫt, with variance σ2 . Then, with our notations we have θ = (a,b) ′ ,
θ0 = (a0,b0) ′ , A∗ i = ai , B∗ i = bi , λi = A∗ i−1 ,B∗
i−1 i−1 2
i−1 = (a ,b ), Eǫt = σ2 ,
Γ(i,j) = +∞
h=−∞E(ǫtǫt−iǫt−hǫt−j−h)I4 = γ(i,j)I4 and
Thus, we have
vecJ = 2
vecI = 4
σ 4 0
1
1−a2 0
1
1−a0b0
λi ⊗λi = a 2(i−1) ,(ab) i−1 ,(ab) i−1 ,b 2(i−1) .
i≥1
a 2(i−1)
0 ,(a0b0) i−1 ,(a0b0) i−1 ,b 2(i−1)
′
0
+∞
i,j=1
1
1−a0b0
1
1−b2 0
and
γ(i,j) a 2(i−1)
0 ,(a0b0) i−1 ,(a0b0) i−1 ,b 2(i−1)
′
0 ,
where σ0 is the true value of σ. We then deduce that
J = 2
and I = 4
σ4 +∞
γ(i,j)
0
i,j=1
1
1−a2 0
1
1−a0b0
Example 3.2. Now we consider a bivariate VAR(1) of the form
2
α1 0
σ11 0
Xt = Xt−1 +et, Σe =
0 α2
With our notations, we have θ = (α1,α2) ′ , θ0 = (α01,α02) ′ ,
A(L) =
1−α1L 0
0 1−α2L
, Mij(z) =
A ∗
h 1 0 α1 11,h =
0 0
0
A ∗
h 0 1 α1 12,h =
0 0
0
A ∗
h 0 0 α1 21,h =
1 0
0
A ∗
h 0 0 α1 22,h =
0 1
0
0 α h 2
0 α h 2
0 α h 2
0 α h 2
∞
h=0
Eij
0 σ 2 22
α h 1 0
0 α h 2
h α1 0
=
0 0
h 0 α2 =
0 0
0 0
=
α h 1 0
0 0
=
0 α h 2
,
,
,
.
.
z h
1
1−a0b0
1
1−b2 0
.
i,j = 1,2,