A two-stage information criterion for stochastic systems revisited

Two-stage information criteria for model selection are constructed by properly penalizing the maximized likelihood. A well known criterion is due to Hannan and Quinn (HQ). The applicability of HQ to multivariable, non-Gaussian, linear stochastic systems has been established by Hannan and Deistler deriving an asymptotic result for the maximized quasi-likelihood function. The objective of the paper is to provide a new, transparent and structured proof of the latter result, based on explicitly stated known techniques, under conditions only slightly stronger than those used by Hannan and Deistler. The advantage of our approach is that it naturally lends itself to the analysis of other models, such as Markov or Hidden Markov Models.