A generalized mixture integer-valued GARCH model
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A generalized mixture integer-valued GARCH model Huiyu Mao1,2 · Fukang Zhu1 · Yan Cui1 Accepted: 15 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We propose a generalized mixture integer-valued generalized autoregressive conditional heteroscedastic model to provide a more flexible modeling framework. This model includes many mixture integer-valued models with different distributions already studied in the literature. The conditional and unconditional moments are discussed and the necessary and sufficient first- and second-order stationary conditions are derived. We also investigate the theoretical properties such as strict stationarity and ergodicity for the mixture process. The conditional maximum likelihood estimators via the EM algorithm are derived and the performances of the estimators are studied via simulation. The model can be selected in terms of both the number of mixture regimes and the number of orders in each regime by several different criteria. A real-life data example is also given to assess the performance of the model. Keywords Ergodicity · Integer-valued time series · Mixture model · Stationarity
1 Introduction Integer-valued time series data are fairly common in various fields of applications such as social science, industry, finance, economy, medicine and ecology, etc. Over the past few years, there has been a growing interest in discrete-valued time series models. An integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) model has been used for dealing with overdispersion. This model was introduced by Ferland et al. (2006) and Fokianos et al. (2009), its properties were then studied by Weiß (2009, 2010), Zhu et al. (2015, 2016), Li et al. (2016), among others.
This work is supported by National Natural Science Foundation of China (Nos. 11871027, 11731015), Science and Technology Developing Plan of Jilin Province (No. 20170101057JC), and Cultivation Plan for Excellent Young Scholar Candidates of Jilin University.
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Yan Cui [email protected]
1
School of Mathematics, Jilin University, 2699 Qianjin Street, Changchun 130012, China
2
Aviation University Air Force, 7855 Renmin Street, Changchun 130022, China
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H. Mao et al.
In count time series analysis, Poisson distribution is frequently used and provides a standard framework [see e.g., Neumann (2011), Fokianos and Tjøtheim (2012) and Doukhan et al. (2012)]. However, due to the fact that Poisson distribution has equal values of the mean and variance, which is not always found in the real data. Thus, there is a need to introduce the integer-valued time series models with other distributions. In the literature, negative binomial (NB) distribution is considered to be a prototype for handling overdispersion in the integer-valued time series. Zhu (2011, 2012b) introduced an NB-INGARCH model and its zero-inflated version to deal with overdispersion and zero-inflation phenomenon, respectively. Later, Christou and Fokianos (2014) considered the NB processes for count time series an
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