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

Chapitre 2. Estimating weak structural VARMA models 73
Proof of Lemma 3 : Let ϑ = (ϑ1,...,ϑk0) ′ . In view of (2.15), (2.16) and (2.17),
for all i ∈ {1,...,k0}, we have
∂lt(ϑ)
= Tr Σ
∂ϑi
−1∂Σe
e −Σ
∂ϑi
−1
e et(ϑ)e ′ t (ϑ)Σ−1
∂Σe
e +2
∂ϑi
∂e′ t(ϑ)
Σ
∂ϑi
−1
e et(ϑ). (2.19)
Using the previous relations and (2.18), for all i,j ∈ {1,...,k0}, we have
∂ 2 lt(ϑ)
∂ϑi∂ϑj
= Tr
Σ −1
e
∂ 2 Σe
∂ϑi∂ϑj
−Σ −1
e
∂Σe
∂ϑi
Σ −1∂Σe
e −Σ
∂ϑj
−1
+Σ −1∂Σe
e Σ
∂ϑi
−1
e et(ϑ)e ′ t (ϑ)Σ−1
∂Σe
e +Σ
∂ϑj
−1
−Σ −1∂Σe
e Σ
∂ϑi
−1∂et(ϑ)e
e
′
t(ϑ)
+2
∂ϑj
∂2e ′ t(ϑ)
∂ϑi∂ϑj
+2 ∂e′ t(ϑ)
Σ
∂ϑi
−1
∂et(ϑ)
e −2Tr Σ
∂ϑj
−1
e et(ϑ) ∂e′ t(ϑ)
∂ϑi
e et(ϑ)e ′ t (ϑ)Σ−1 e
e et(ϑ)e ′ t (ϑ)Σ−1 e
Σ −1
e et(ϑ)
Σ −1
e
∂Σe
∂ϑj
∂Σe
∂ϑi
∂ 2 Σe
∂ϑi∂ϑj
Σ −1∂Σe
e
∂ϑj
. (2.20)
Using Eete ′ t = Σe, Eet = 0, the uncorrelatedness between et and the linear past Ht−1,
∂et(ϑ0)/∂ϑi ∈ Ht−1, and ∂ 2 et(ϑ0)/∂ϑi∂ϑj ∈ Ht−1, we have
E ∂2 lt(ϑ0)
∂ϑi∂ϑj
= Tr Σ −1∂Σe(ϑ0)
e0 Σ
∂ϑi
−1
∂Σe(ϑ0)
e0 +2E
∂ϑj
∂e′ t (ϑ0)
Σ
∂ϑi
−1∂et(ϑ0)
e0
∂ϑj
= J(i,j). (2.21)
The ergodic theorem and the next lemma conclude. ✷
Lemma 4. Under the assumptions of Theorem 2.2, the matrix
is invertible.
and
with
J = E ∂2 lt(ϑ0)
∂ϑ∂ϑ ′
Proof of Lemma 4 : In view of (2.21), we have J = J1 +J2, where
J1(i,j) = Tr
Σ −1/2
e0
J2 = 2E ∂e′ t (ϑ0)
∂ϑ Σ−1 e0
∂Σe(ϑ0)
Σ
∂ϑi
−1/2
hi = (Σ −1/2
e0 ⊗Σ −1/2
e0 Σ −1/2
e0
∂et(ϑ0)
∂ϑ ′
∂Σe(ϑ0)
∂ϑj
e0 )di, di = vec ∂Σe(ϑ0)
Σ −1/2
e0 = h ′ ihj, In the previous derivations, we used the well-known relations Tr(A ′ B) = (vecA) ′ vecB
and vec(ABC) = (C ′ ⊗A)vecB. Note that the matrices J, J1 and J2 are semi-definite
∂ϑi
.