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

Chapitre 5. Model selection of weak VARMA models 165
Remark 5.3. Given a collection of competing families of approximating models, the
one that minimizes E∆( ˆ θn) might be preferred. For model selection, we then choose ˆp
and ˆq as the set which minimizes the information criterion (5.4).
Remark 5.4. Consider the univariate case d = 1. We then have
AICM = nˆσ 2
n
e + n+
n−(p+q)
1
ˆσ 4
+∞
i,j,i ′ =1
ˆγ(i,j) ˆλj ˆ λ −1′
i ′ ⊗ ˆ λi ˆ λ −1′
i ′
,
where ˆσ 2 e is the variance estimate of the univariate process and where ˆγ(i,j) are the
estimators of γ(i,j) = +∞
h=−∞E(etet−iet−het−j−h) and ˆ λ ′ i are the estimators of λ ′ i ∈
Rp+q given in Section 5.2.
Proof of Remark 5.4 : For d = 1, we have
AICM = nˆσ 2 e +
In view of Section 5.2, we obtained
vec ˆ J11 = 2
i ′ ≥1
ˆλi ′ ⊗ ˆ λi ′
n
n−(p+q)
′
n+ 1
vec
2
Î′ ′
11
and vec Î11 = 4
ˆσ 4
+∞
i,j=1
vec ˆ J −1
11
.
ˆγ(i,j) ˆλj ⊗ ˆ ′
λi ,
where ˆγ(i,j) are the estimators of γ(i,j) = +∞
h=−∞E(etet−iet−het−j−h) and ˆ λ ′ i are the
estimators of λ ′ i ∈ Rp+q given in Section 5.2. Using the last expressions of vec ˆ J11 and
vecÎ11, we then have
AICM = nˆσ 2 e +
n
n+
n−(p+q)
1
ˆσ 4
+∞
i,j,i ′ =1
Using (A⊗B)(C ⊗D) = AC ⊗BD, we have
AICM = nˆσ 2
n
e + n+
n−(p+q)
1
ˆσ 4
The proof is complete. ✷
+∞
i,j,i ′ =1
ˆγ(i,j) ˆλj ⊗ ˆ
λi
ˆλi ′ ⊗ ˆ λi ′
−1 ′
.
ˆγ(i,j) ˆλj ˆ λ −1′
i ′ ⊗ ˆ λi ˆ λ −1′
i ′
.