Flexible bivariate Poisson integer-valued GARCH model
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Flexible bivariate Poisson integer-valued GARCH model Yan Cui1 · Qi Li2 · Fukang Zhu1 Received: 17 April 2019 / Revised: 3 August 2019 © The Institute of Statistical Mathematics, Tokyo 2019
Abstract Integer-valued time series models have been widely used, especially integer-valued autoregressive models and integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models. Recently, there has been a growing interest in multivariate count time series. However, existing models restrict the dependence structures imposed by the way they constructed. In this paper, we consider a class of flexible bivariate Poisson INGARCH(1,1) model whose dependence is established by a special multiplicative factor. Stationarity and ergodicity of the process are discussed. The maximization by parts algorithm and its modified version together with the alternative method by using R package Template Model Builder are employed to estimate the parameters of interest. The consistency and asymptotic normality for estimates are obtained, and the finite sample performance of estimators is given via simulations. A real data example is also provided to illustrate the model. Keywords Bivariate · INGARCH model · Multiplicative factor · Poisson distribution · Time series of counts
1 Introduction Integer-valued time series are commonly encountered in many practical situations, such as the number of goods sold in a shopping mall, the monthly number of insurance claim, the daily number of transaction in stock market and so on. Recently, there have
Zhu’s 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. Li’s work is supported by Natural Science Foundation of Changchun Normal University (No. 2018-004).
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Fukang Zhu [email protected]
1
School of Mathematics, Jilin University, 2699 Qianjin Street, Changchun 130012, China
2
College of Mathematics, Changchun Normal University, Changchun 130032, China
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Y. Cui et al.
been plenty of attempts to deal with them, see Weiß (2008) and Scotto et al. (2015) for some excellent reviews on INAR models. As an alternative, the INGARCH model proposed by Ferland et al. (2006) and Fokianos et al. (2009) is also very popular, which is defined as follows:
P(λt ), X t |Ft−1 ∼ q p λt = α0 + i=1 αi λt−i + j=1 β j X t− j ,
∀t ∈ Z,
(1)
where α0 > 0, αi ≥ 0, β j ≥ 0, i = 1, . . . , p, j = 1, . . . , q, p ≥ 0, q ≥ 1, and Ft−1 is the σ -field generated by {X t−1 , X t−2 , . . .}. Zhu (2011, 2012a, b) and Davis and Liu (2016) generalized the Poisson assumption to negative binomial, generalized Poisson, zero-inflated Poisson, negative binomial and exponential family distributions, respectively. Inferential aspects of model (1) and its generalized forms have been well established, including ergodicity, estimating methods and goodness-of-fit tests, see Neumann (2011), Doukhan et al. (2012), Fokianos a
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