Statistics and Its Interface

Volume 16 (2023)

Number 4

Markov-switching Poisson generalized autoregressive conditional heteroscedastic models

Pages: 531 – 544

DOI: https://dx.doi.org/10.4310/22-SII741

Authors

Jichun Liu (School of Mathematical Science, Xiamen University, Xiamen, China)

Yue Pan (Department of Mathematics and Statistics, University of Strathclyde, Glasgow, United Kingdom)

Jiazhu Pan (Department of Mathematics and Statistics, University of Strathclyde, Glasgow, United Kingdom; and School of Mathematics and Statistics, Yangtze Normal University, Chongqing, China)

Abdullah Almarashi (Department of Mathematics and Statistics, University of Strathclyde, Glasgow, United Kingdom)

Abstract

We consider a kind of regime-switching autoregressive models for nonnegative integer-valued time series when the conditional distribution given historical information is Poisson distribution. In this type of models the link between the conditional variance (i.e. the conditional mean for Poisson distribution) and its past values as well as the observed values of the Poisson process may be different when an unobservable (hidden) variable, which is a Markovian Chain, takes different states. We study the stationarity and ergodicity of Markov-switching Poisson generalized autoregressive heteroscedastic (MS-PGARCH) models, and give a condition on parameters under which a MS-PGARCH process can be approximated by a geometrically ergodic process. Under this condition we discuss maximum likelihood estimation for MS-PGARCH models. Simulation studies and application to modelling financial count time series are presented to support our methodology.

Keywords

count time series, Markov regime switching, generalized conditional heteroscedasticity, geometric ergodicity, Poisson GARCH, gmoothing

2010 Mathematics Subject Classification

Primary 62M10. Secondary 37M10, 91B84.

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Received 23 July 2021

Accepted 12 May 2022

Published 14 April 2023