Renormalized maximum likelihood for multivariate autoregressive models

Renormalized maximum likelihood (RNML) is a powerful concept from information theory. We show how it can be used to derive a criterion for selecting the order of vector autoregressive (VAR) processes. We prove that RNML criterion is strongly consistent. We also demonstrate empirically its good performance for examples of VAR which have been considered in recent literature because they possess a particular type of sparsity. In our experiments, we pay a special attention to models for which the inverse spectral density matrix (ISDM) has a specific sparsity pattern. The interest on these models is motivated by the relationship between sparse structure of ISDM and the problem of inferring the conditional independence graph for multivariate time series.