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Maximum likelihood estimation of higher‐order integer‐valued autoregressive processes

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  • Ruijun Bu
  • Brendan McCabe
  • Kaddour Hadri

Abstract

. In this article, we extend the earlier work of Freeland and McCabe [Journal of time Series Analysis (2004) Vol. 25, pp. 701–722] and develop a general framework for maximum likelihood (ML) analysis of higher‐order integer‐valued autoregressive processes. Our exposition includes the case where the innovation sequence has a Poisson distribution and the thinning is binomial. A recursive representation of the transition probability of the model is proposed. Based on this transition probability, we derive expressions for the score function and the Fisher information matrix, which form the basis for ML estimation and inference. Similar to the results in Freeland and McCabe (2004), we show that the score function and the Fisher information matrix can be neatly represented as conditional expectations. Using the INAR(2) specification with binomial thinning and Poisson innovations, we examine both the asymptotic efficiency and finite sample properties of the ML estimator in relation to the widely used conditional least squares (CLS) and Yule–Walker (YW) estimators. We conclude that, if the Poisson assumption can be justified, there are substantial gains to be had from using ML especially when the thinning parameters are large.

Suggested Citation

  • Ruijun Bu & Brendan McCabe & Kaddour Hadri, 2008. "Maximum likelihood estimation of higher‐order integer‐valued autoregressive processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(6), pages 973-994, November.
    • Handle: RePEc:bla:jtsera:v:29:y:2008:i:6:p:973-994
    DOI: 10.1111/j.1467-9892.2008.00590.x
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    1. Bu, Ruijun & McCabe, Brendan, 2008. "Model selection, estimation and forecasting in INAR(p) models: A likelihood-based Markov Chain approach," International Journal of Forecasting, Elsevier, vol. 24(1), pages 151-162.
    2. Freeland, R. K. & McCabe, B. P. M., 2004. "Forecasting discrete valued low count time series," International Journal of Forecasting, Elsevier, vol. 20(3), pages 427-434.
    3. Feike C. Drost & Ramon Van Den Akker & Bas J. M. Werker, 2008. "Local asymptotic normality and efficient estimation for INAR(p) models," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(5), pages 783-801, September.
    4. R. K. Freeland & B. P. M. McCabe, 2004. "Analysis of low count time series data by poisson autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(5), pages 701-722, September.
    5. Jung, Robert C. & Tremayne, A.R., 2006. "Coherent forecasting in integer time series models," International Journal of Forecasting, Elsevier, vol. 22(2), pages 223-238.
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    Cited by:

    1. Robert Jung & A. Tremayne, 2011. "Useful models for time series of counts or simply wrong ones?," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 95(1), pages 59-91, March.
    2. Chen, Zezhun Chen & Dassios, Angelos & Tzougas, George, 2023. "A first order binomial mixed poisson integer-valued autoregressive model with serially dependent innovations," LSE Research Online Documents on Economics 112222, London School of Economics and Political Science, LSE Library.
    3. Kai Yang & Han Li & Dehui Wang & Chenhui Zhang, 2021. "Random coefficients integer-valued threshold autoregressive processes driven by logistic regression," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(4), pages 533-557, December.
    4. Yousung Park & Hee-Young Kim, 2012. "Diagnostic checks for integer-valued autoregressive models using expected residuals," Statistical Papers, Springer, vol. 53(4), pages 951-970, November.
    5. Mirko Armillotta & Paolo Gorgi, 2023. "Pseudo-variance quasi-maximum likelihood estimation of semi-parametric time series models," Tinbergen Institute Discussion Papers 23-054/III, Tinbergen Institute.
    6. Yang Lu, 2021. "The predictive distributions of thinning‐based count processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(1), pages 42-67, March.
    7. Jentsch, Carsten & Weiß, Christian, 2017. "Bootstrapping INAR models," Working Papers 17-02, University of Mannheim, Department of Economics.
    8. Mohammadipour, Maryam & Boylan, John E., 2012. "Forecast horizon aggregation in integer autoregressive moving average (INARMA) models," Omega, Elsevier, vol. 40(6), pages 703-712.
    9. Christian Weiß, 2015. "A Poisson INAR(1) model with serially dependent innovations," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(7), pages 829-851, October.
    10. Robert C. Jung & Andrew R. Tremayne, 2020. "Maximum-Likelihood Estimation in a Special Integer Autoregressive Model," Econometrics, MDPI, vol. 8(2), pages 1-15, June.
    11. Luisa Bisaglia & Margherita Gerolimetto, 2019. "Model-based INAR bootstrap for forecasting INAR(p) models," Computational Statistics, Springer, vol. 34(4), pages 1815-1848, December.
    12. Zezhun Chen & Angelos Dassios & George Tzougas, 2023. "INAR approximation of bivariate linear birth and death process," Statistical Inference for Stochastic Processes, Springer, vol. 26(3), pages 459-497, October.
    13. Xanthi Pedeli & Anthony C. Davison & Konstantinos Fokianos, 2015. "Likelihood Estimation for the INAR( p ) Model by Saddlepoint Approximation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(511), pages 1229-1238, September.

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