IDEAS home Printed from https://ideas.repec.org/p/pra/mprapa/76574.html
   My bibliography  Save this paper

Negative binomial quasi-likelihood inference for general integer-valued time series models

Author

Listed:
  • Aknouche, Abdelhakim
  • Bendjeddou, Sara

Abstract

Two negative binomial quasi-maximum likelihood estimates (NB-QMLE's) for a general class of count time series models are proposed. The first one is the profile NB-QMLE calculated while arbitrarily fixing the dispersion parameter of the negative binomial likelihood. The second one, termed two-stage NB-QMLE, consists of four stages estimating both conditional mean and dispersion parameters. It is shown that the two estimates are consistent and asymptotically Gaussian under mild conditions. Moreover, the two-stage NB-QMLE enjoys a certain asymptotic efficiency property provided that a negative binomial link function relating the conditional mean and conditional variance is specified. The proposed NB-QMLE's are compared with the Poisson QMLE asymptotically and in finite samples for various well-known particular classes of count time series models such as the (Poisson and negative binomial) Integer GARCH model and the INAR(1) model. Applications to two real datasets are given.

Suggested Citation

  • Aknouche, Abdelhakim & Bendjeddou, Sara, 2016. "Negative binomial quasi-likelihood inference for general integer-valued time series models," MPRA Paper 76574, University Library of Munich, Germany, revised 03 Feb 2017.
    • Handle: RePEc:pra:mprapa:76574
    as

    Download full text from publisher

    File URL: https://mpra.ub.uni-muenchen.de/76574/1/MPRA_paper_76574.pdf
    File Function: original version
    Download Restriction: no
    File URL: https://mpra.ub.uni-muenchen.de/83082/8/MPRA_paper_83082.pdf
    File Function: revised version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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. Cameron,A. Colin & Trivedi,Pravin K., 2013. "Regression Analysis of Count Data," Cambridge Books, Cambridge University Press, number 9781107667273.
    3. William Kengne & Paul Doukhan, 2015. "Inference and testing for structural change in general Poisson autoregressive models," Post-Print hal-02979913, HAL.
    4. Cameron, A Colin & Trivedi, Pravin K, 1986. "Econometric Models Based on Count Data: Comparisons and Applications of Some Estimators and Tests," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 1(1), pages 29-53, January.
    5. Vasiliki Christou & Konstantinos Fokianos, 2014. "Quasi-Likelihood Inference For Negative Binomial Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(1), pages 55-78, January.
    6. Richard A. Davis & Rongning Wu, 2009. "A negative binomial model for time series of counts," Biometrika, Biometrika Trust, vol. 96(3), pages 735-749.
    7. Ali Ahmad & Christian Francq, 2016. "Poisson QMLE of Count Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(3), pages 291-314, May.
    8. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
    9. Gourieroux, Christian & Monfort, Alain & Trognon, Alain, 1984. "Pseudo Maximum Likelihood Methods: Theory," Econometrica, Econometric Society, vol. 52(3), pages 681-700, May.
    10. Doukhan, Paul & Wintenberger, Olivier, 2008. "Weakly dependent chains with infinite memory," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 1997-2013, November.
    11. Douc, R. & Doukhan, P. & Moulines, E., 2013. "Ergodicity of observation-driven time series models and consistency of the maximum likelihood estimator," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2620-2647.
    12. Fokianos, Konstantinos & Tjøstheim, Dag, 2011. "Log-linear Poisson autoregression," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 563-578, March.
    13. Paul Doukhan & William Kengne, 2015. "Inference and testing for structural change in general Poisson autoregressive models," Post-Print hal-02979929, HAL.
    14. Benjamin M.A. & Rigby R.A. & Stasinopoulos D.M., 2003. "Generalized Autoregressive Moving Average Models," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 214-223, January.
    15. Doukhan, Paul & Fokianos, Konstantinos & Tjøstheim, Dag, 2012. "On weak dependence conditions for Poisson autoregressions," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 942-948.
    16. 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.
    17. Heinen, Andreas, 2003. "Modelling Time Series Count Data: An Autoregressive Conditional Poisson Model," MPRA Paper 8113, University Library of Munich, Germany.
    18. Gourieroux, Christian & Monfort, Alain & Trognon, Alain, 1984. "Pseudo Maximum Likelihood Methods: Applications to Poisson Models," Econometrica, Econometric Society, vol. 52(3), pages 701-720, May.
    19. 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.
    20. HEINEN, Andreas & RENGIFO, Erick, 2003. "Multivariate modelling of time series count data: an autoregressive conditional Poisson model," LIDAM Discussion Papers CORE 2003025, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    21. Fukang Zhu, 2011. "A negative binomial integer‐valued GARCH model," Journal of Time Series Analysis, Wiley Blackwell, vol. 32(1), pages 54-67, January.
    22. Marcelo Bourguignon, 2016. "Poisson–geometric INAR(1) process for modeling count time series with overdispersion," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 70(3), pages 176-192, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ali Ahmad & Christian Francq, 2016. "Poisson QMLE of Count Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(3), pages 291-314, May.
    2. Cui, Yunwei & Zheng, Qi, 2017. "Conditional maximum likelihood estimation for a class of observation-driven time series models for count data," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 193-201.
    3. Vasiliki Christou & Konstantinos Fokianos, 2014. "Quasi-Likelihood Inference For Negative Binomial Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(1), pages 55-78, January.
    4. Aknouche, Abdelhakim & Scotto, Manuel, 2022. "A multiplicative thinning-based integer-valued GARCH model," MPRA Paper 112475, University Library of Munich, Germany.
    5. Aknouche, Abdelhakim & Bentarzi, Wissam & Demouche, Nacer, 2017. "On periodic ergodicity of a general periodic mixed Poisson autoregression," MPRA Paper 79650, University Library of Munich, Germany.
    6. Mamadou Lamine Diop & William Kengne, 2017. "Testing Parameter Change in General Integer-Valued Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(6), pages 880-894, November.
    7. Aknouche, Abdelhakim & Dimitrakopoulos, Stefanos & Touche, Nassim, 2019. "Integer-valued stochastic volatility," MPRA Paper 91962, University Library of Munich, Germany, revised 04 Feb 2019.
    8. Aknouche, Abdelhakim & Almohaimeed, Bader & Dimitrakopoulos, Stefanos, 2020. "Forecasting transaction counts with integer-valued GARCH models," MPRA Paper 101779, University Library of Munich, Germany, revised 11 Jul 2020.
    9. Aknouche, Abdelhakim & Bentarzi, Wissam & Demouche, Nacer, 2018. "On periodic ergodicity of a general periodic mixed Poisson autoregression," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 15-21.
    10. Aknouche, Abdelhakim & Dimitrakopoulos, Stefanos, 2020. "On an integer-valued stochastic intensity model for time series of counts," MPRA Paper 105406, University Library of Munich, Germany.
    11. Aknouche, Abdelhakim & Francq, Christian, 2023. "Two-stage weighted least squares estimator of the conditional mean of observation-driven time series models," Journal of Econometrics, Elsevier, vol. 237(2).
    12. Yunwei Cui & Rongning Wu & Qi Zheng, 2021. "Estimation of change‐point for a class of count time series models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(4), pages 1277-1313, December.
    13. Youngmi Lee & Sangyeol Lee, 2019. "CUSUM test for general nonlinear integer-valued GARCH models: comparison study," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(5), pages 1033-1057, October.
    14. Yan Cui & Fukang Zhu, 2018. "A new bivariate integer-valued GARCH model allowing for negative cross-correlation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(2), pages 428-452, June.
    15. William Kengne & Isidore S. Ngongo, 2022. "Inference for nonstationary time series of counts with application to change-point problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(4), pages 801-835, August.
    16. 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.
    17. Byungsoo Kim & Sangyeol Lee, 2020. "Robust estimation for general integer-valued time series models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(6), pages 1371-1396, December.
    18. Aknouche, Abdelhakim & Gouveia, Sonia & Scotto, Manuel, 2023. "Random multiplication versus random sum: auto-regressive-like models with integer-valued random inputs," MPRA Paper 119518, University Library of Munich, Germany, revised 18 Dec 2023.
    19. Mengya Liu & Qi Li & Fukang Zhu, 2020. "Self-excited hysteretic negative binomial autoregression," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(3), pages 385-415, September.
    20. Abdelhakim Aknouche & Christian Francq, 2022. "Stationarity and ergodicity of Markov switching positive conditional mean models," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(3), pages 436-459, May.

    More about this item

    Keywords

    Integer-valued time series models; Integer GARCH; Integer AR; Generalized Linear Models; Quasi-likelihood; Geometric QMLE; Negative Binomial QMLE; Poisson QMLE; consistency and asymptotic normality.;
    All these keywords.

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:pra:mprapa:76574. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Joachim Winter (email available below). General contact details of provider: https://edirc.repec.org/data/vfmunde.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.