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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 101
It is also shown uniformly in θ ∈ Θ that
∂ℓn(θ)
∂θ = ∂˜ ℓn(θ)
+o(1) a.s.
∂θ
The same equality holds for the second-order derivatives of ˜ ℓn. We thus have
J = lim
∂θ∂θ ′
n→∞ Jn a.s, where Jn = ∂2 ℓn(θ0)
Using well-known results on matrix derivatives, we have
n
∂ℓn(θ0) 2 ∂
=
∂θ n ∂θ e′ t (θ0)
t=1
.
Σ −1
e0 et(θ0). (3.11)
In view of (3.11), we have
Jn = 2
n
′ ∂e t (θ0)
n ∂θ
t=1
Σ−1
∂et(θ0)
e0
∂θ ′ + ∂2e ′ t (θ0)
Σ−1
∂θ∂θ ′ e0 et(θ0)
2 ′ ∂ e t (θ0)
→ 2E
∂θ∂θ ′
Σ −1
e0 et
∂
+2E
∂θ e′ t (θ0)
Σ −1
∂
e0
∂θ ′et(θ0)
, a.s
by the ergodic theorem. Using the orthogonality between et and any linear combination
of the past values of et, we have 2E{∂ 2 e ′ t (θ0)/∂θ∂θ ′ }Σ −1
e0 et = 0. Thus we have
J = 2E
∂
∂θ e′ t
(θ0)Σ −1
e0
∂
∂θ ′et(θ0)
.
Using vec(ABC) = (C ′ ⊗A)vec(B) with C = Id2 (p+q) and in view of Proposition 1,
we obtain
′ ∂e
vecJ = 2E
t(θ0)
∂θ ⊗ ∂e′
t(θ0)
vecΣ
∂θ
−1
e0
= 2
E Id2 (p+q) ⊗e ′ ′
t−i λ i ⊗ Id2 (p+q) ⊗e ′ ′ −1
t−i λ i vecΣe0 .
i≥1
Using also AC ⊗BD = (A⊗B)(C ⊗D), we have
vecJ = 2
E Id2 (p+q) ⊗e ′ t
i≥1
= 2
E
i≥1
The proof is complete. ✷
⊗ Id 2 (p+q) ⊗e ′ t
Id 2 (p+q) ⊗e ′
⊗2
t λ ′ ⊗2
i vecΣ −1
e0
{λ ′ i ⊗λ ′ i }vecΣ−1
e0
= 2
Proof of Proposition 3 : In view of (3.6), let
Υt = ∂
∂θ Lt(θ0)
∂
= Lt(θ),...,
∂θ1
∂
Lt(θ)
∂θk0
i≥1
′
Mλ ′ ⊗2
i vecΣ −1
e0 .
θ=θ0