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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 103
where
Γ(i,j) =
+∞
h=−∞
The proof is complete. ✷
E e ′ t−h ⊗ Id 2 (p+q) ⊗e ′ t−j−h
Proof of Remark 3.1. For d = 1, we have
I(p+q)
⊗2
M := E ×et = σ 2 I (p+q) 2,
where σ 2 is the variance of the univariate process. We also have
Γ(i,j) =
=
+∞
h=−∞
+∞
h=−∞
In view of Proposition 2, we have
′
⊗ e t ⊗ Id2 (p+q) ⊗e ′
t−i .
E
et−het−j−hI(p+q) ⊗ etet−iI(p+q)
E(etet−iet−het−j−h)I (p+q) 2 = γ(i,j)I (p+q) 2. (3.12)
vecJ = 2
M{λ ′ i ⊗λ′ i }σ−2 .
Replacing M by σ2I (p+q) 2 in vecJ, we have
vecJ = 2
{λi ⊗λi} ′ .
i≥1
i≥1
Using (3.12) and in view of Proposition 3, we have
vecI = 4
σ 4
+∞
i,j=1
The proof is complete. ✷
Γ(i,j) λ ′ j ⊗λ′ 4
i =
σ4 +∞
i,j=1
γ(i,j){λj ⊗λi} ′ .
Proof of Theorem 3.1. For any multiplicative norm, we have
vecJ −vec ˆ
Jn
≤ 2
M−
Mn
ˆ
λ
′
⊗2
−1
vec Σ
i≥1
+ ˆ
Mnλ
′ ⊗2
i − ˆ λ ′
⊗2
i
−1
vecΣ
e0
+ ˆ
Mnˆ
λ ′
⊗2
vec
ˆΣ −1
e0 −Σ−1
e0 .
The proof will thus follow from Lemmas 3.5, 3.6 and 3.8 below. ✷
i
i
e0