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

Chapitre 2. Estimating weak structural VARMA models 82
✷
The proof of Theorem 2.1 is completed by the arguments of Wald (1949). More
precisely, the compact set Θ is covered by a neighborhood V(ϑ0) of ϑ0 and a finite
number of neighborhoods V(ϑ1),...,V(ϑk) satisfying (2.31) with ϑ replaced by ϑi,
i = 1,...,k. In view of (2.31) and (2.32), we have almost surely
inf ˜ℓn(ϑ) = min
ϑ∈Θ
inf
i=0,1,...,k ϑ∈V(ϑi)∩Θ
˜ℓn(ϑ) = inf ˜ℓn(ϑ)
ϑ∈V(ϑ0)∩Θ
for n large enough. Since the neighborhood V(ϑ0) can be chosen arbitrarily small, the
conclusion follows.
2.9.3 Details on the proof of Theorem 2.2
Lemma 9. Under the assumptions of Theorem 2.2, we have
√
∂
nsup
˜
ℓn(ϑ) ∂ℓn(ϑ)
− = oP(1).
∂ϑ ∂ϑ
ϑ∈Θ
Proof of Lemma 9 : Similar to (2.30), Assumption A1 entails that, for k =
1,...,k0,
∞ ∂et(ϑ)
= C
∂ϑk i=1
(k) ∂˜et(ϑ) t−1
i (ϑ)Xt−i, = C
∂ϑk i=1
(k)
i (ϑ)Xt−i, (2.33)
C (k)
i (ϑ) = ∂Ci(ϑ)
, supC
∂ϑk ϑ∈Θ
(k)
i (ϑ) ≤ Kρ i . (2.34)
Similar to Lemma 6, we then have
sup
∂˜et(ϑ) ∂et(ϑ)
−
∂ϑ ′ ∂ϑ ′
≤ Kρt . (2.35)
Using (2.19), we have
with
a1 = 2
√ n
a2 = Tr
√ n
ϑ∈Θ
n
′ ∂e t(ϑ)
t=1
Σ −1
e
∂ϑk
∂ ˜ ℓn(ϑ)
∂ϑk
− ∂ℓn(ϑ)
= a1 +a2,
∂ϑk
− ∂˜e′
t(ϑ)
Σ
∂ϑk
−1
e et(ϑ)+ ∂˜e′ t(ϑ)
Σ
∂ϑk
−1
e (et(ϑ)− ˜et(ϑ))
∂Σe
.
{˜et(ϑ)−et(ϑ)}˜e ′ t (ϑ)+et(ϑ){˜et(ϑ)−et(ϑ)} ′ Σ −1
e
∂ϑk