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

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  • Yue Xu
  • Fukang Zhu

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.

Suggested Citation

  • Yue Xu & Fukang Zhu, 2022. "A new GJR‐GARCH model for ℤ‐valued time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(3), pages 490-500, May.
    • Handle: RePEc:bla:jtsera:v:43:y:2022:i:3:p:490-500
    DOI: 10.1111/jtsa.12623
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    References listed on IDEAS

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    1. Fokianos, Konstantinos & Rahbek, Anders & Tjøstheim, Dag, 2009. "Poisson Autoregression," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1430-1439.
    2. Aknouche, Abdelhakim & Francq, Christian, 2021. "Count And Duration Time Series With Equal Conditional Stochastic And Mean Orders," Econometric Theory, Cambridge University Press, vol. 37(2), pages 248-280, April.
    3. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
    4. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June.
    5. E. Gonçalves & N. Mendes-Lopes & F. Silva, 2015. "Infinitely Divisible Distributions in Integer-Valued Garch Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(4), pages 503-527, July.
    6. René Ferland & Alain Latour & Driss Oraichi, 2006. "Integer‐Valued GARCH Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(6), pages 923-942, November.
    7. Fukang Zhu, 2011. "A negative binomial integer‐valued GARCH model," Journal of Time Series Analysis, Wiley Blackwell, vol. 32(1), pages 54-67, January.
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