Threshold negative binomial autoregressive model

ABSTRACT

This article studies an observation-driven model for time series of counts, which allows for overdispersion and negative serial dependence in the observations. The observations are supposed to follow a negative binomial distribution conditioned on past information with the form of thresh old models, which generates a two-regime structure on the basis of the magnitude of the lagged observations. We use the weak dependence approach to establish the stationarity and ergodicity, and the inference for regression parameters are obtained by the quasi-likelihood. Moreover, asymptotic properties of both quasi-maximum likelihood estimators and the threshold estimator are established, respectively. Simulation studies are considered and so are two applications, one of which is the trading volume of a stock and another is the number of major earthquakes.

KEYWORDS:

• INGARCH
• negative binomial
• quasi-likelihood inference
• threshold model
• time series of counts
• weak dependence

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 62M10
• 62F10

Acknowledgments

The authors are very grateful to Editor, Associate Editor and two anonymous referees for providing several exceptionally helpful comments which led to a significant improvement of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The considered NBAR model differs from the negative binomial INGARCH(1,1) in Zhu [5], and it has been studied by Christou and Fokianos [11] and is a special case of TNBAR defined below, i.e., $b_1=b_2$.

2 See the online version of this paper for the colourful Figure 1, where the original observations are black. In the first plot, the series fitted by SETPAR is marked as blue; and that fitted by NBAR and TNBAR are marked as red and green, respectively, in later two plots.

Additional information

Funding

This work is supported by National Natural Science Foundation of China [grant numbers 11871027, 11731015], Science and Technology Developing Plan of Jilin Province [grant number 20170101057JC], Science and Technology Program of Jilin Educational Department during the “13th Five-Year” Plan Period (grant number 2016-399), and Cultivation Plan for Excellent Young Scholar Candidates of Jilin University.