Self-excited hysteretic negative binomial autoregression
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Self-excited hysteretic negative binomial autoregression Mengya Liu1 · Qi Li2 · Fukang Zhu1 Received: 31 January 2019 / Accepted: 12 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract This paper studies an observation-driven model for time series of counts, in which the observations are supposed to follow a negative binomial distribution conditioned on past information with the form of the hysteretic autoregression. As an extension of the classical two-regime threshold process, the hysteretic autoregression enjoys a more flexible regime-switching mechanism. Stability properties of the model are established by the e-chain and Lyapunov’s method. The estimator for regression parameters is obtained by the quasi-maximum likelihood with Poisson-based score estimating function, and the corresponding asymptotic properties are established. Moreover, a reasonable method for selecting search ranges for thresholds is also proposed and simulation studies are considered. As an application, we bring attention to some features of the daily number of trades of Siparex Croissance which have been overlooked in previous studies. Keywords Hysteretic autoregression · Negative binomial · Threshold model · Time series of counts
1 Introduction The classical two-regime threshold autoregressive model introduced in Tong and Lim (1980) has attracted great attention, especially after the publication of Tong (1990). The piecewise linear structure of the threshold autoregressive model makes many nonlinearity-related properties realizable. Meanwhile, there are many valuable nonlinear phenomena in real life which deserve to be studied. Hence, it is worth assuming that the observations follow a particular distribution conditioned on an accompanying intensity process, which is equipped with a form of two-regime threshold autoregressive model. For example, Wang et al. (2014) proposed a self-excited threshold Poisson
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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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M. Liu et al.
autoregressive (SETPAR) model which assumes a two-regime structure of the conditional mean process according to the magnitude of the lagged observations. This is a promotion of the Poisson autoregression (PAR) proposed by Ferland et al. (2006), which is also known as the Poisson integer-valued GARCH (INGARCH) and receives much attention. For example, Neumann (2011) softened the linear assumption on the Poisson autoregressive model to a general contracting evolution rule. Doukhan et al. (2012) showed the existence of moments under similar conditions with Fokianos et al. (2009). And some methods about the estimation and influence analysis for INGARCH were proposed in Zhu et al. (2015, 2016) and Li et al. (2016). As an extension from another perspective, the advantages of introducing thresholds are reflected through the comparison between effects of SETPAR and PAR when applied in the real
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