Self-exciting threshold -valued autoregressive processes for non-stationary time series of counts

Abstract

To handle non-stationary integer-valued time series of counts with piecewise characteristics and linear trends, this paper introduced a class of self-exciting threshold $\mathbb{Z}$-valued autoregressive processes. The process is defined with a free type distribution for the innovation, which enhances the flexibility of the model. The basic probabilistic and statistical properties of the proposed model are discussed. Conditional least squares (CLS) estimator and modified quasi-likelihood (MQL) estimator, as well as their asymptotic properties are obtained. A searching algorithm for estimating the threshold parameter is also provided. Some simulation studies are conducted to show the performances of the proposed methods. Finally, an application to a real data example is provided.

Keywords:

• Integer-valued time series
• non-stationary time series
• threshold autoregressive model
• signed binomial thinning
• quasi-likelihood estimation

Acknowledgments

We gratefully acknowledge the anonymous reviewers for their careful work and thoughtful suggestions on an early version of this work that have helped improve this article substantially.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Natural Science Foundation of China (No. 11901053, 12101417), Natural Science Foundation of Jilin Province (No. YDZJ202301ZYTS393, 20210101149JC), Scientific Research Project of Jilin Provincial Department of Education (No. JJKH20230665KJ).