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

Abstract

We propose a sparse coefficient estimation and automated model selection procedure for autoregressive 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 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 and smoothly clipped average deviation, 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 [Feigin, Resnick. Pitfalls of fitting autoregressive models for heavy-tailed time series. Extremes. 1999;1:391–422]. 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.

Keywords:

• autoregressive process
• causality
• heavy tails
• penalized maximum likelihood estimation
• oracle properties
• strong consistency

AMS Subject Classifications:

• 62M10
• 60G10
• 60F05

Acknowledgement

The authors thank the editor and the referees for carefully reading the manuscript and for the suggestions that improved the presentation.