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

Chapitre 4. Multivariate portmanteau test for weak structural VARMA models 143
√ n( ˆ Se ⊗ ˆ Se −Se ⊗Se) = OP(1) and, using (4.5) and the ergodic theorem, we obtain
n vec( ˆ S −1
e ˆ Γe(h) ˆ S −1
e )−vec(S−1 e ˆ Γe(h)S −1
e )
= n ( ˆ S −1
e ⊗ ˆ S −1
e )vecˆ Γe(h)−(S −1
e ⊗S−1 e )vecˆ
Γe(h)
= n ( ˆ Se ⊗ ˆ Se) −1 vec ˆ Γe(h)−(Se ⊗Se) −1 vec ˆ
Γe(h)
= ( ˆ Se ⊗ ˆ Se) −1√ n(Se ⊗Se − ˆ Se ⊗ ˆ Se)(Se ⊗Se) −1√ nvec ˆ Γe(h)
= OP(1).
In the previous equalities we also use vec(ABC) = (C ′ ⊗A)vec(B) and (A⊗B) −1 =
A−1 ⊗B−1 when A and B are invertible. It follows that
ˆρm = vecˆ ′
Re(1) ,..., vecˆ
′
′
Re(m)
= ( ˆ Se ⊗ ˆ Se) −1 vecˆ ′
Γe(1) ,..., ( ˆ Se ⊗ ˆ Se) −1 vecˆ
′
′
Γe(m)
= Im ⊗( ˆ Se ⊗ ˆ Se) −1
ˆΓm = Im ⊗(Se ⊗Se) −1 Γm
ˆ +OP(n −1 ).
We now obtain (4.6) from (4.5). Hence, we have
Var( √ nˆρm) = Im ⊗(Se ⊗Se) −1 Σˆ Γm
The proof is complete. ✷
Im ⊗(Se ⊗Se) −1 .
Proof of Theorem 4.4. The proof is similar to that given by Francq, Roy and
Zakoïan (2005) for Theorem 5.1. ✷