Consistent model selection criteria and goodness-of-fit test for affine causal processes

TL;DR

This paper develops consistent model selection criteria and a goodness-of-fit test for a broad class of causal time series models, including ARMA, GARCH, and their variants, with theoretical guarantees and practical diagnostics.

Contribution

It introduces a penalized quasi-likelihood approach ensuring model selection consistency and proposes a goodness-of-fit test, highlighting limitations of BIC in certain models.

Findings

BIC does not always guarantee consistency in model selection.

Proposed criteria ensure consistency and asymptotic normality of estimators.

Numerical simulations confirm theoretical results and illustrate model diagnostics.

Abstract

This paper studies the model selection problem in a large class of causal time series models, which includes both the ARMA or AR($\infty$) processes, as well as the GARCH or ARCH($\infty$), APARCH, ARMA-GARCH and many others processes. To tackle this issue, we consider a penalized contrast based on the quasi-likelihood of the model. We provide sufficient conditions for the penalty term to ensure the consistency of the proposed procedure as well as the consistency and the asymptotic normality of the quasi-maximum likelihood estimator of the chosen model. It appears from these conditions that the Bayesian Information Criterion (BIC) does not always guarantee the consistency. We also propose a tool for diagnosing the goodness-of-fit of the chosen model based on the portmanteau Test. Numerical simulations and an illustrative example on the FTSE index are performed to highlight the obtained asymptotic results, including a numerical evidence of the non consistency of the usual BIC penalty for order selection of an AR(p) models with ARCH($\infty$) errors.