Strong consistent model selection for general causal time series

TL;DR

This paper introduces a new penalized quasi-likelihood criterion for strongly consistent model selection in a wide range of causal time series models, including AR, ARCH, and GARCH processes, with proven theoretical guarantees.

Contribution

It proposes a novel model selection method that achieves strong consistency without requiring dependence between regularization parameters and model structure.

Findings

The proposed criterion is strongly consistent under certain conditions.

The estimator follows the law of the iterated logarithm.

No dependence between regularization parameter and model structure is needed.

Abstract

We consider the strongly consistent question for model selection in a large class of causal time series models, including AR($\infty$), ARCH($\infty$), TARCH($\infty$), ARMA-GARCH and many classical others processes. We propose a penalized criterion based on the quasi likelihood of the model. We provide sufficient conditions that ensure the strong consistency of the proposed procedure. Also, the estimator of the parameter of the selected model obeys the law of iterated logarithm. It appears that, unlike the result of the weak consistency obtained by Bardet {\it et al.} \cite{Bardet2020}, a dependence between the regularization parameter and the model structure is not needed.