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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 102
where
We have
In view of (3.5), we have
I =
=
= 4
Lt(θ) = logdetΣe +e ′ t (θ)Σ−1
e et(θ).
Υt = 2 ∂e′ t (θ0)
∂θ Σ−1 e0 et(θ0)
= 2 e ′ t (θ0)⊗ ∂e′ t (θ0)
vecΣ
∂θ
−1
e0 .
+∞
h=−∞
+∞
h=−∞
+∞
h=−∞
∂
Cov 2
∂θ e′ t (θ0)
Σ −1
e0 et,2
∂
∂θ e′ t−h (θ0)
Σ −1
e0 et−h
Cov 2 e ′ t ⊗ ∂e′
t
vecΣ
∂θ
−1
e0 ,2 e ′ t−h ⊗ ∂e′
t−h
vecΣ
∂θ
−1
e0
E e ′ t ⊗ ∂e′
t
vecΣ
∂θ
−1
−1 ′
e0 vecΣe0 e ′ t−h ⊗ ∂e′ ′
t−h
.
∂θ
Using the elementary relation vec(ABC) = (C ′ ⊗A)vecB, we have
vecI = 4
+∞
h=−∞
E
By Proposition 1, we obtain
vecI = 4
e ′ t−h ⊗ ∂e′
t−h
⊗ e
∂θ
′ t ⊗ ∂e′
t
vec
∂θ
+∞
+∞
h=−∞i,j=1
vecΣ −1
e0
E e ′ t−h ⊗Id2 (p+q) ⊗e ′
′
t−j−h λ j
⊗ e ′ t ⊗ I d 2 (p+q) ⊗e ′ t−i
Using AC ⊗BD = (A⊗B)(C ⊗D), we have
vecI = 4
+∞
+∞
h=−∞i,j=1
′
λ i vec
vecΣ −1
e0
−1 ′
vecΣe0 .
E e ′ t−h ⊗Id2 (p+q) ⊗e ′
t−j−h Id ⊗λ ′
j
⊗ e ′ t ⊗ I d 2 (p+q) ⊗e ′ t−i
{Id ⊗λ ′ i }
vec
Using also AC ⊗BD = (A⊗B)(C ⊗D), we have
vecI = 4
= 4
+∞
+∞
h=−∞i,j=1
E e ′ t−h ⊗ I d 2 (p+q) ⊗e ′ t−j−h
Id ⊗λ ′
j ⊗{Id ⊗λ ′ i} vec
+∞
i,j=1
Γ(i,j) Id ⊗λ ′ j
vecΣ −1
e0
vecΣ −1
e0
−1 ′
vecΣe0 .
−1 ′
vecΣe0 .
′
⊗ e t ⊗ Id2 (p+q) ⊗e ′
t−i
−1 ′
vecΣe0
⊗{Id ⊗λ ′ i }
vec vecΣ −1
e0
−1 ′
vecΣe0 ,