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

Chapitre 5. Model selection of weak VARMA models 161
We have
E∆( ˆ θn) = Enlogdet ˆ
Σe +nETr ˆΣ −1
e S(ˆ
θn) , (5.6)
where ˆ Σe = Σe( ˆ θn) is the estimated error variance matrix under ˆ θn, with Σe(θ) =
n−1n t=1et(θ)e ′ t (θ). Then the
first term on the right-hand side of (5.6) can be estimated
without bias by nlogdet n−1n t=1et( ˆ θn)e ′ t (ˆ
θn) . Hence, only an estimate for the
second term needs to be considered. Moreover, in view of (5.2), a Taylor expansion of
et(θ) around θ (1)
0 yields
where
et(θ) = et(θ0)+ ∂et(θ0)
∂θ (1)′ (θ (1) −θ (1)
0 )+Rt, (5.7)
Rt = 1
2 (θ(1) −θ (1)
0 )′ ∂2et(θ∗ )
∂θ (1) ∂θ (1)′(θ (1) −θ (1) 2
0 ) = OP π ,
with π = θ (1) −θ (1)
and θ∗ is between θ (1)
where
0
0 and θ (1) . We then obtain
∂et(θ0)
S(θ) = S(θ0)+E
∂θ (1)′ (θ (1) −θ (1)
0 )e′ t (θ0)
+E et(θ0)(θ (1) −θ (1)
0 ) ′∂e′ t (θ0)
∂θ (1)
+D θ (1)
+ERt (θ (1) −θ (1)
0 )′∂e′ t(θ0)
∂θ (1)
+Eet(θ0)Rt
∂et(θ0)
+E
∂θ (1)′ (θ (1) −θ (1)
0 )
Rt +ER 2 t ,
D θ (1) = E
+ERte ′ t (θ0)
∂et(θ0)
∂θ (1)′ (θ (1) −θ (1)
0 )(θ (1) −θ (1)
0 ) ′∂e′ t(θ0)
∂θ (1)
.
Using the orthogonality between et(θ0) and any linear combination of the past values
of et(θ0) (in particular ∂et(θ0)/∂θ ′ and ∂ 2 et(θ0)/∂θ∂θ ′ ), and the fact that Eet(θ0) = 0,
we have
S(θ) = S(θ0)+D(θ (1) )+O π 4 = Σe0 +D(θ (1) )+O π 4 ,
where Σe0 = Σe(θ0). Thus, we can write the expected discrepancy quantity in (5.6) as
E∆( ˆ θn) = Enlogdet ˆ
Σe +nETr ˆΣ −1
e Σe0
+nETr ˆΣ −1
e D(ˆ θ (1)
n )
+nE Tr ˆΣ −1 1
e OP
n2
. (5.8)
As in the classical multivariate regression model, an analog of (5.3) is
Σe0 ≈
n
n−d(p+q) E
ˆΣe
=
dn
E
dn−k1
ˆΣe
.