New insights on AR order selection with information theoretic criteria based on localized estimators

• We focus on selecting the order of autoregressions when forgetting factor least-squares algorithms are used.
• We analyze sequentially normalized maximum likelihood and sequentially discounting normalized maximum likelihood.
• Some modifications of the criteria are discussed.
• The implications of the modifications are evaluated.
• Theoretical results are illustrated by numerical examples.

Selecting the order of autoregressions when the parameters of the model are estimated with least-squares algorithms (LSA) is a well researched topic. This type of approach assumes implicitly that the analyzed time series is stationary, which is rarely true in practical applications. It is known since long time that, in the case of nonstationary signals, is recommended to employ forgetting factor least-squares algorithms (FF-LSA) instead of LSA. This makes necessary to modify the selection criteria originally designed for LSA in order to become compatible with FF-LSA. Sequentially normalized maximum likelihood (SNML), which is one of the newest model selection criteria, has been modified independently by two groups of researchers such that to be used in conjunction with FF-LSA. As the proposals coming from the two groups have not been compared in the previous literature, we conduct in this work a theoretical and empirical study for clarifying the relationship between the existing solutions. As part of our study, we also investigate some possibilities to further modify the criteria. Based on our findings, we provide guidance which can potentially be useful for the practitioners.