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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 106
where θ ∗ n is between ˆ θn and θ0. Using the strong consistency of ˆ θn and (3.18), it is easily
seen that
vec ˆ Mn = vecMn( ˆ θn) → vecM(θ0) = vecM, a.s.
The proof is complete. ✷
Lemma 3.7. Under Assumptions A1–A8, we have
ˆΣe0 → Σe0 a.s, as n → ∞.
Proof of Lemma 3.7 : We have ˆ Σe0 = Σn( ˆ θn) with Σn(θ) = n −1 n
t=1 et(θ)e ′ t(θ). By
the ergodic theorem
Σn(θ) → Σe(θ) := Eet(θ)e ′ t(θ), a.s.
Using the elementary relation vec(aa ′ ) = a ⊗ a, where a is a vector, we have
vec ˆ Σe0 = n−1n t=1et( ˆ θn) ⊗ et( ˆ θn) and vecΣn(θ0) = n−1n t=1et(θ0) ⊗ et(θ0). Using
a Taylor expansion of vec ˆ Σe0 around θ0 and (3.2), we obtain
vec ˆ Σe0 = vecΣn(θ0)+ 1
n
Using the strong consistency of ˆ θn,
it is easily seen that
The proof is complete. ✷
n
et ⊗ ∂et ∂et
+
∂θ ′ ∂θ
t=1
Esup et(θ)
θ∈Θ
2 < ∞ and
′ ⊗et
sup ∂et(θ)
θ∈Θ ∂θ ′
vec ˆ Σe0 → vecΣe(θ0) = vecΣe0, a.s.
Lemma 3.8. Under Assumptions A1–A8, we have
ˆΣ −1
e0 → Σ −1
e0 a.s as n → ∞.
Proof of Lemma 3.8 : For any multiplicative norm, we have
ˆ Σ −1
e0 −Σ−1
e0 = −ˆ Σ −1
ˆΣe0
e0 −Σe0 Σ −1
e0 ≤ ˆ Σ −1
e0
( ˆ θn −θ0)+OP
2
< ∞,
ˆ Σe0 −Σe0
1
.
n
−1
Σ .
In view of Lemma 3.7 and
−1
Σ
e0 < ∞ (because the matrix Σe0 is nonsingular), we
have
ˆ Σ −1
e0 −Σ−1 e0
→ 0 a.s.
e0