Softplus Beta Negative Binomial Integer-valued GARCH Model

Le Le QI, Fu Kang ZHU

Acta Mathematica Sinica, Chinese Series ›› 2023, Vol. 66 ›› Issue (2) : 293-308.

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中国科学院数学与系统科学研究院期刊网
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Acta Mathematica Sinica, Chinese Series ›› 2023, Vol. 66 ›› Issue (2) : 293-308. DOI: 10.12386/A20210063

Softplus Beta Negative Binomial Integer-valued GARCH Model

  • Le Le QI, Fu Kang ZHU
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Abstract

INGARCH models are often constructed based on Poisson distribution, negative binomial distribution and so on. Beta negative binomial (BNB) distribution is a flexible distribution. Recently, the related BNB-INGARCH model was proposed, whose conditional mean is linear, the parameters are restricted to non-negative and negative autocorrelation cannot be modeled. In this paper, we first propose the log-linear BNB-INGARCH model to solve the above problems, but the simple form of linear mean and ARMA-like structures are lost. So we further construct softplus BNB-INGARCH(p,q) model by using the softplus function, which is the main research object. When p and q are equal to 1, the stationarity and ergodicity of the model are proved and the conditions for the existence of the second moment are given. In addition, the strong consistency and the asymptotic normality of the maximum likehood estimator are shown. Finally, the analysis of real-data examples show the usefulness of the proposed model.

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BNB distribution / INGARCH model / maximum likelihood estimation / asymptotic moment / softplus function

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Le Le QI, Fu Kang ZHU. Softplus Beta Negative Binomial Integer-valued GARCH Model. Acta Mathematica Sinica, Chinese Series, 2023, 66(2): 293-308 https://doi.org/10.12386/A20210063

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