Simultaneous sparse model selection and coefficient estimation for heavy-tailed autoregressive processes

TL;DR

This paper introduces a penalized likelihood approach for sparse coefficient estimation and model selection in heavy-tailed autoregressive processes, demonstrating strong theoretical properties and practical effectiveness through simulations and real data application.

Contribution

It develops a novel penalized maximum likelihood method for AR models with heavy-tailed innovations, ensuring consistency and oracle properties under mild conditions.

Findings

The estimator is strongly consistent and asymptotically normal.

LASSO and SCAD penalties are effectively compared.

The method performs well on real US Industrial Production Index data.

Abstract

We propose a sparse coefficient estimation and automated model selection procedure for autoregressive (AR) processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator (PCMLE) satisfies a strong consistency, $O_P(N^{-1/2})$ consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator (LASSO) and smoothly clipped average deviation (SCAD), are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations (see \cite{Resnick}). A simulation study confirms our theoretical results. At the end, we apply our method to a historical price data of the US Industrial Production Index for consumer goods, and obtain very promising results.