Abstract

In this article, we propose a novel model for time series of counts

called the hysteretic Poisson autoregressive model with thresholds (HPART) by

extending the linear Poisson autoregressive model into a nonlinear model. Unlike

other approaches that bear the adjective “hysteretic”, our model incorporates

a scientifically relevant and essential controlling factor that produces genuine

hysteresis. Further, we re-analyse the buffered Poisson autoregressive model with

thresholds (BPART). Although the two models share the convenient piecewise

linear structure, the HPART model probes deeper into the intricate dynamics

that governs regime switching.

We study the maximum likelihood estimation

of the parameters of both models and their asymptotic properties in a unified

manner, establish tests of separate families of hypotheses for the non-nested case

involving a BPART model and a HPART model, and demonstrate the finitesample efficacy of parameter estimation and tests with Monte Carlo simulation

in the Supplementary Material. We showcase advantages of the HPART model

with two real time series, including plausible interpretations and improved outof-sample predictions.

Key words and phrases: Buffered Poisson autoregression, Hysteresis, Hysteretic Poisson autoregression, Non-nested models, Separate family of hypotheses, Thresh- olds

Information

Preprint No.SS-2025-0390
Manuscript IDSS-2025-0390
Complete AuthorsXintong Ma, Dong Li, Howell Tong
Corresponding AuthorsXintong Ma
Emailsmxt22@mails.tsinghua.edu.cn

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Acknowledgments

We are grateful to the editor, associate editor, and two referees for their

insightful and valuable comments and suggestions, which have helped us

substantially improve the presentation and quality of our article. Li’s work

is supported by the Beijing Natural Science Foundation (No.F251002).

Supplementary Materials

The Supplementary Material contains simulation study, additional details

for real data analysis results, and proofs of all theorems in the article with

some useful technical lemmas.

Supplementary materials are available for download.