Self-excited Threshold Poisson Autoregression

TL;DR

This paper introduces a self-excited threshold Poisson autoregressive model for count time series, allowing negative correlation and providing theoretical properties, estimation methods, and real-world application to earthquake data.

Contribution

It develops a novel two-regime Poisson autoregressive model with theoretical analysis and demonstrates its effectiveness on earthquake count data.

Findings

Model has a unique invariant measure under certain conditions.

Asymptotic properties of maximum likelihood estimates are established.

Applied successfully to real earthquake data.

Abstract

This paper studies theory and inference of an observation-driven model for time series of counts. It is assumed that the observations follow a Poisson distribution conditioned on an accompanying intensity process, which is equipped with a two-regime structure according to the magnitude of the lagged observations. The model remedies one of the drawbacks of the Poisson autoregression model by allowing possibly negative correlation in the observations. Classical Markov chain theory and Lyapunov's method are utilized to derive the conditions under which the process has a unique invariant probability measure and to show a strong law of large numbers of the intensity process. Moreover the asymptotic theory of the maximum likelihood estimates of the parameters is established. A simulation study and a real data application are considered, where the model is applied to the number of major earthquakes in the world.