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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 100
where Eij = ∂Bℓ ′/∂bij,ℓ ′ is the d×d matrix with 1 at position (i,j) and 0 elsewhere.
We then have
∞ ∂et(θ)
= B ∗ ij,het−ℓ ′ −h(θ). (3.10)
∂bij,ℓ ′
Similarly to Lemma 3.4, we have
∂et(θ)
∂b ′ ℓ ′
h=0
= [N11(L)et−ℓ ′(θ) : N21(L)et−ℓ ′(θ) : ... : Ndd(L)et−ℓ ′(θ)]
d×d2 = N(L)(Id 2 ⊗et−ℓ ′(θ)) = N(L)et−ℓ ′(θ) = Ut−ℓ ′(θ),
where N(L) = [N11(L) : N21(L) : ... : Ndd(L)] and Ut−ℓ ′ are respectively the d×d3 and
d×d 2 matrices. Then, we have
∂et(θ)
∂b ′ ℓ ′
=
∞
B ∗ het−ℓ ′ −h(θ) =
h=0
∞
B ∗ k−ℓ ′et−k(θ) = Ut−ℓ ′(θ),
where B ∗ k−ℓ ′ = 0 when k < ℓ′ . With these notations, we obtain
∂et(θ)
∂b ′
=
=
k=0
∞
∗
Bk−1et−k(θ) : B ∗ k−2et−k(θ) : ... : B ∗
k−qet−k(θ)
d×d2q ∞
B ∗ θ,k (Id2q ⊗et−k(θ)),
k=0
k=0
whereB ∗ θ,k = B ∗ k−1 : B∗ k−2 : ... : B∗ k−q
Let
is thed×d 3 q matrix. The conclusion follows. ✷
Proof of Proposition 2 : Let
∂
∂θ ′et(θ0)
∂
= et(θ0),...,
∂θ1
∂
et(θ0)
∂θk0
˜ℓn(θ) = − 2
n log˜ Ln(θ)
= 1
n
dlog(2π)+logdetΣe + ˜e
n
′ t(θ)Σ −1
e ˜et(θ) .
t=1
In Boubacar Mainassara and Francq (2009), it is shown that ℓn(θ) = ˜ ℓn(θ)+o(1) a.s.,
where
ℓn(θ) = 1
n
dlog(2π)+logdetΣe +e
n
′ t (θ)Σ−1 e et(θ) .
t=1
.