A new GJR‐GARCH model for ℤ‐valued time series

Yue Xu and Fukang Zhu

Journal of Time Series Analysis, 2022, vol. 43, issue 3, 490-500

Abstract: The Glosten–Jagannathan–Runkle GARCH (GJR‐GARCH) model is popular in accounting for asymmetric responses in the volatility in the analysis of continuous‐valued financial time series, but asymmetric responses in the volatility are also observed in time series of counts or ℤ‐valued time series, such as the daily number of stock transactions or the daily stock returns divided by tick price (1 cent). Two different integer‐valued GARCH models based on Poisson distribution have been proposed for these two types of discrete data respectively. Shifted geometric distribution is more flexible than Poisson distribution, whose variance is greater than its mean. In this article, we propose a GJR‐GARCH model based on shifted geometric distribution for ℤ‐valued time series exhibiting asymmetric volatility. Basic probabilistic properties of the new model are given, and the maximum likelihood method is used to estimate unknown parameters and the asymptotic normality of corresponding estimators is established. A simulation study is presented to illustrate the estimation method. An empirical application to a real data concerning the daily stock returns divided by tick price is considered to show the proposed model's superiority compared with existing models.

Date: 2022
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