Information criteria for multivariate CARMA processes

Abstract

Multivariate continuous-time ARMA$(p,q)$ ($\operatorname{MCARMA} (p,q)$) processes are the continuous-time analog of the well-known vector ARMA$(p,q)$ processes. They have attracted interest over the last years. Methods to estimate the parameters of an MCARMA process require an identifiable parametrization such as the Echelon form with a fixed Kronecker index, which is in the one-dimensional case the degree $p$ of the autoregressive polynomial. Thus, the Kronecker index has to be known in advance before parameter estimation can be done. When this is not the case, information criteria can be used to estimate the Kronecker index and the degrees $(p,q)$, respectively. In this paper, we investigate information criteria for MCARMA processes based on quasi maximum likelihood estimation. Therefore, we first derive the asymptotic properties of quasi maximum likelihood estimators for MCARMA processes in a misspecified parameter space. Then, we present necessary and sufficient conditions for information criteria to be strongly and weakly consistent, respectively. In particular, we study the well-known Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) as special cases.