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

Chapitre 5
Model selection of weak VARMA
models
Abstract We consider the problem of orders selections of vector autoregressive
moving-average (VARMA) models under the assumption that the errors are uncorrelated
but not necessarily independent. We relax the standard independence assumption
to extend the range of application of the VARMA models, and allow to cover linear representations
of general nonlinear processes. We propose a modified Akaike information
criterion (AIC).
Keywords : AIC, discrepancy, Kullback-Leibler information, QMLE/LSE, order selection,
structural representation, weak VARMA models.
5.1 Introduction
The class of vector autoregressive moving-average (VARMA) models and the subclass
of vector autoregressive (VAR) models are used in time series analysis and econometrics
to describe not only the properties of the individual time series but also
the possible cross-relationships between the time series (see Reinsel, 1997, Lütkepohl,
2005, 1993). In a VARMA(p,q) models, the choice of p and q is particularly important
because the number of parameters, (p + q + 3)d 2 where d is the number of the
series, quickly increases with p and q, which entails statistical difficulties. If orders lower
than the true orders of the VARMA(p,q) models are selected, the estimate of the
parameters will not be consistent and if too high orders are selected, the accuracy of
the estimation parameters is likely to be low. This paper is devoted to the problem of
the choice (by minimizing an information criterion) of the orders of VARMA models
under the assumption that the errors are uncorrelated but not necessarily independent.