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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 97
Let
ˆMn ij,h := 1
n
Consider the matrix
n−|h|
t=1
ˆΓn(i,j) :=
′
ê t−h ⊗ Id2 (p+q) ⊗ê ′
′
t−j−h ⊗ ê t ⊗ Id2 (p+q) ⊗ê ′
t−i .
+Tn
h=−Tn
f(hbn) ˆ Mn ij,h and Tn =
where [x] denotes the integer part of x. In view of Proposition 3, we define an estimator
În of I by
vec În = 4
+∞
i,j=1
ˆΓn(i,j) Id ⊗ ˆ λ ′
i ⊗ Id ⊗ ˆ λ ′
j vec
a
bn
vec ˆ Σ −1
e0
,
vec ˆ Σ −1
e0
′
.
We are now able to state the following theorem, which shows the weak consistency of
În.
Theorem 3.2. Under Assumptions A1–A8, we have
În → I in probability as n → ∞.
Therefore Theorems 3.1 and 3.2 show that
ˆΩn := ˆ J −1
n În ˆ J −1
n
is a weakly estimator of the asymptotic covariance matrix Ω := J −1 IJ −1 .
3.6 Technical proofs
Proof of Proposition 1 : Because θ ′ = (a ′ ,b ′ ), Lemmas 3.3 and 3.4 below show
that
∂et(θ)
∂θ ′ = [Vt−1(θ) : ... : Vt−p(θ) : Ut−1(θ) : ... : Ut−q(θ)]
∞
=
=
=
h=0
−A ∗ h−1 (I d 2 ⊗et−h(θ)) : ... : −A ∗ h−p (I d 2 ⊗et−h(θ)) :
B ∗ h−1(Id2 ⊗et−h(θ)) : ... : B ∗
2
h−q(Id ⊗et−h(θ))
∞ ∗
−Ah−1 : ... : −A ∗ h−p : B ∗ h−1 : ... : B ∗ h−q
h=0
∞
λh(θ) Id2 (p+q) ⊗et−h(θ) .
h=1
Id 2 p ⊗et−h(θ)
I d 2 q ⊗et−h(θ)