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

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 91
A7 : We have Eǫt4+2ν < ∞ and ν
∞
k=0 {αǫ(k)} 2+ν < ∞ for some ν > 0.
The reader is referred to Boubacar Mainassara and Francq (2009) for a discussion
of these assumptions. Note that (ǫt) can be replaced by (Xt) in A4, because
Xt = A −1
θ0 (L)Bθ0(L)ǫt and ǫt = B −1
θ0 (L)Aθ0(L)Xt, where L stands for the backward
operator. Note that from A1 the matrices A0 and B0 are invertible. Introducing the
innovation process et = A −1
00B00ǫt, the structural representation Aθ0(L)Xt = Bθ0(L)ǫt
can be rewritten as the reduced VARMA representation
Xt −
p
i=1
A −1
00A0iXt−i = et −
q
i=1
We thus recursively define ˜et(θ) for t = 1,...,n by
˜et(θ) = Xt −
p
i=1
A −1
0 AiXt−i +
q
i=1
A −1
00B0iB −1
00 A00et−i.
A −1
0 BiB −1
0 A0˜et−i(θ),
with initial values ˜e0(θ) = ··· = ˜e1−q(θ) = X0 = ··· = X1−p = 0. The gaussian
quasi-likelihood is given by
n 1
Ln(θ,Σe) =
(2π) d/2√
exp −
detΣe
1
2 ˜e′ t (θ)Σ−1 e ˜et(θ)
, Σe = A −1
0 B0ΣB ′ 0A−1′ 0 .
t=1
A quasi-maximum likelihood estimator (QMLE) of θ and Σe is a measurable solution
( ˆ θn, ˆ Σe) of
( ˆ θn, ˆ Σe) = arg max Ln(θ,Σe).
θ∈Θ,Σe
We now use the matrix Mθ0 of the coefficients of the reduced form to that made by
Boubacar Mainassara and Francq (2009), where
Mθ0 = [A −1
00A01 : ... : A −1
00A0p : A −1 −1
00B01B00 A00 : ... : A −1 −1
00B0qB00 A00].
Now we need an assumption which specifies how this matrix depends on the parameter
θ0. Let Mθ0 be the matrix ∂vec(Mθ)/∂θ ′ evaluated at θ0.
A8 : The matrix
Mθ0 is of full rank k0.
Under Assumptions A1–A8, Boubacar Mainassara and Francq (2009) have showed the
consistency ( ˆ θn → θ0 a.s. as n → ∞) and the asymptotic normality of the QMLE :
√
n ˆϑn L→ −1 −1
−ϑ0 N(0,Ω := J IJ ), (3.2)
where J = J(θ0,Σe0) and I = I(θ0,Σe0), with
and
2 ∂
J(θ,Σe) = lim
n→∞ n
2
∂θ∂θ ′ logLn(θ,Σe) a.s.
I(θ,Σe) = lim Var
n→∞ 2 ∂
√
n∂θ
logLn(θ,Σe). (3.3)