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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 92
3.3 Expression for the derivatives of the VARMA residuals
The reduced VARMA representation can be rewritten under the compact form
Aθ(L)Xt = Bθ(L)et(θ),
whereAθ(L) = Id− p
i=1 AiL i andBθ(L) = Id− q
i=1 BiL i , withAi = A −1
0 Ai andBi =
A −1
0 BiB −1
0 A0. For ℓ = 1,...,p and ℓ ′ = 1,...,q, let Aℓ = (aij,ℓ), Bℓ ′ = (bij,ℓ ′), aℓ =
vec[Aℓ] and bℓ ′ = vec[Bℓ ′]. We denote respectively by
a := (a ′ 1 ,...,a′ p )′
and b := (b ′ 1 ,...,b′ q )′ ,
the coefficients of the multivariate AR and MA parts. Thus we can rewrite θ = (a ′ ,b ′ ) ′ ,
where a ∈ R k1 depends on A0,...,Ap, and where b ∈ R k2 depends on B0,...,Bq, with
k1 + k2 = k0. For i,j = 1,...,d, let Mij(L) and Nij(L) the (d×d)−matrix operators
defined by
Mij(L) = B −1 −1
θ (L)EijAθ (L)Bθ(L) and Nij(L) = B −1
θ (L)Eij,
where Eij is the d×d matrix with 1 at position (i,j) and 0 elsewhere. We denote by
A∗ ij,h and B∗ij,h the (d×d) matrices defined by
Mij(z) =
∞
A ∗ ij,hzh , Nij(z) =
h=0
∞
B ∗ ij,hzh , |z| ≤ 1
for h ≥ 0. Take A∗ ij,h = B∗ij,h = 0 when h < 0. Let respectively A∗h =
∗ A11,h : A∗ 21,h : ... : A∗
∗
dd,h and Bh = B∗ 11,h : B∗21,h : ... : B∗
3
dd,h the d × d matrices.
We consider the white noise "empirical" autocovariances
Γe(h) = 1
n
For k,k ′ ,m,m ′ = 1,...,∞, let
and
Γ(k,k ′ ) =
∞
h=−∞
n
t=h+1
h=0
Γm,m ′ = (Γ(k,k′ ))1≤k≤m,1≤k ′ ≤m ′ =
ete ′ t−h , for 0 ≤ h < n.
E {et−k ⊗et}{et−h−k ′ ⊗et−h} ′ ,
∞
h=−∞
where Υt = (e ′ t−1 ,...,e′ t−m )′ ⊗et. Let the d×d 3 (p+q) matrix
Cov(Υt,Υt−h)
λh(θ) = −A ∗ h−1 : ... : −A∗ h−p : B∗ h−1 : ... : B∗ h−q
. (3.4)