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

Chapitre 5. Model selection of weak VARMA models 158
to the observations, where θm is a km-dimensional parameter. The discrepancy between
the candidate and the true models can be measured by the Kullback-Leibler divergence
(or information)
∆{fm(·,θm)|f0} = Ef0 log
where
d{fm(·,θm)|f0} = −2Ef0logfm(X,θm) = −2
f0(X)
fm(X,θm) = Ef0 logf0(X)+ 1
2 d{fm(·,θm)|f0},
{logfm(x,θm)}f0(x)µ(dx)
is sometimes called the Kullback-Leibler contrast (or the discrepancy between the approximating
and the true models). Using the Jensen inequality, we have
∆{fm(·,θm)|f0} = − log fm(x,θm)
f0(x)µ(dx)
f0(x)
fm(x,θm)
≥ −log f0(x)µ(dx) = 0,
f0(x)
with equality if and only if fm(·,θm) = f0. This is the main property of the Kullback-
Leibler divergence. Minimizing ∆{fm(·,θm)|f0} with respect to fm(·,θm) is equivalent
to minimizing the contrast d{fm(·,θm)|f0}. Let
θ0,m = arginfd{fm(·,θm)|f0}
= arginf−2Elogfm(X,θm)
θm
θm
be an optimal parameter for the model m corresponding to the density fm(·,θm) (assuming
that such a parameter exists). We estimate this optimal parameter by QMLE
ˆθn,m.
5.5 Criteria for VARMA order selection
Let
˜ℓn(θ) = − 2
n log˜ Ln(θ)
= 1
n
dlog(2π)+logdetΣe + ˜e
n
′ t(θ)Σ −1
e ˜et(θ) .
t=1
In Boubacar Mainassara and Francq (2009), it is shown that ℓn(θ) = ˜ ℓn(θ) +o(1) a.s,
where
ℓn(θ) = − 2
n logLn(θ)
= 1
n
dlog(2π)+logdetΣe +e
n
′ t (θ)Σ−1 e et(θ) ,
t=1