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

Chapitre 2. Estimating weak structural VARMA models 75
fixed, the process Y = (Yt,m)t is strongly mixing, with mixing coefficients αY(h) ≤
αǫ(max{0,h−m}). Applying the central limit theorem (CLT) for mixing processes
(see Herrndorf, 1984) we directly obtain
1
√ n
n
t=1
Yt,m
L
→ N(0,Im), Im =
∞
h=−∞
Cov(Yt,m,Yt−h,m).
As in FZ Lemma 3, one can show that I = limm→∞Im exists. Since Zt,m2 → 0 at an
exponential rate when m → ∞, using the arguments given in FZ Lemma 4, one can
show that
lim
m→∞ limsup
P n −1/2
n
Zt,m
> ε = 0 (2.25)
n→∞
for every ε > 0. From a standard result (see e.g. Brockwell and Davis, 1991, Proposition
6.3.9), we deduce that
1
√ n
n
t=1
∂lt(ϑ0)
∂ϑ
which completes the proof. ✷
= 1
√ n
n
t=1
t=1
Yt,m + 1
√ n
n
t=1
Zt,m
L
→ N(0,I),
Proof of Theorem 2.3 : Note that
˜ℓn(ϑ) = 1
n
˜lt(ϑ), ˜lt(ϑ) = dlog(2π)+logdetΣe + ˜e
n
′ t(ϑ)Σ −1
e ˜et(ϑ).
t=1
Under the assumption of the theorem, ∂˜e ′ t(ϑ)/∂ϑ (2) = 0, and (2.19) yields
∂˜lt( ˆ
ϑn)
= Tr ˆΣ
∂ϑi
−1
e Id − ˜et( ˆ ϑ (1)
n )˜e′ t (ˆ ϑ (1)
n )ˆ Σ −1
∂Σe(
e
ˆ
ϑn)
∂ϑi
for i = k1 + 1,...k0, with ˆ Σe such that ˆ ϑ (2)
n = Dvec ˆ Σe. Assumption A6 entails that
the first order condition ∂ ˜ ℓn( ˆ ϑn)/∂ϑ (2) = 0 is satisfied for n large enough. We then have
and
because
1
n
n
t=1
The conclusion follows. ✷
ˆΣe = n −1
n
t=1
˜et( ˆ ϑ (1)
n )˜e′ t (ˆ ϑ (1)
n )
˜ℓn( ˆ ϑn) = dlog(2π)+logdet ˆ Σe +d,
˜e ′ t( ˆ ϑ (1)
n ) ˆ Σ −1
e ˜et( ˆ ϑ (1)
n ) = Tr
1
n
n
t=1
˜et( ˆ ϑ (1)
n )˜e ′ t( ˆ ϑ (1)
n ) ˆ Σ −1
e
= d.