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Model selection, estimation and forecasting in INAR(p) models: A likelihood-based Markov Chain approach

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  • Bu, Ruijun
  • McCabe, Brendan

Abstract

This paper considers model selection, estimation and forecasting for a class of integer autoregressive models suitable for use when analysing time series count data. Any number of lags may be entertained, and estimation may be performed by likelihood methods. Model selection is enhanced by the use of new residual processes that are defined for each of the p + 1 unobserved components of the model. Forecasts are produced by treating the model as a Markov Chain, and estimation error is accounted for by providing confidence intervals for the probabilities of each member of the support of the count data variable. Confidence intervals are also available for more complicated event forecasts such as functions of the cumulative distribution function, e.g., for probabilities that the future count will exceed a given threshold. A data set of Australian counts on medical injuries is analysed in detail.

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  • 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.
    • Handle: RePEc:eee:intfor:v:24:y:2008:i:1:p:151-162
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    1. Ruijun Bu & Kaddour Hadri & Brendan McCabe, 2006. "Conditional Maximum Likelihood Estimation of Higher-Order Integer-Valued Autoregressive Processes," Working Papers 200619, University of Liverpool, Department of Economics.
    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. 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.
    4. Corradi, Valentina & Swanson, Norman R., 2006. "Predictive density and conditional confidence interval accuracy tests," Journal of Econometrics, Elsevier, vol. 135(1-2), pages 187-228.
    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.
    6. Ralph D. Snyder & Gael M. Martin & Phillip Gould & Paul D. Feigin, 2007. "An Assessment of Alternative State Space Models for Count Time Series," Monash Econometrics and Business Statistics Working Papers 4/07, Monash University, Department of Econometrics and Business Statistics.
    7. McCabe, B.P.M. & Martin, G.M., 2005. "Bayesian predictions of low count time series," International Journal of Forecasting, Elsevier, vol. 21(2), pages 315-330.
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    Cited by:

    1. Yang, Kai & Yu, Xinyang & Zhang, Qingqing & Dong, Xiaogang, 2022. "On MCMC sampling in self-exciting integer-valued threshold time series models," Computational Statistics & Data Analysis, Elsevier, vol. 169(C).
    2. Muhammed Rasheed Irshad & Christophe Chesneau & Veena D’cruz & Naushad Mamode Khan & Radhakumari Maya, 2022. "Bivariate Poisson 2Sum-Lindley Distributions and the Associated BINAR(1) Processes," Mathematics, MDPI, vol. 10(20), pages 1-24, October.
    3. Petropoulos, Fotios & Apiletti, Daniele & Assimakopoulos, Vassilios & Babai, Mohamed Zied & Barrow, Devon K. & Ben Taieb, Souhaib & Bergmeir, Christoph & Bessa, Ricardo J. & Bijak, Jakub & Boylan, Joh, 2022. "Forecasting: theory and practice," International Journal of Forecasting, Elsevier, vol. 38(3), pages 705-871.
      • Fotios Petropoulos & Daniele Apiletti & Vassilios Assimakopoulos & Mohamed Zied Babai & Devon K. Barrow & Souhaib Ben Taieb & Christoph Bergmeir & Ricardo J. Bessa & Jakub Bijak & John E. Boylan & Jet, 2020. "Forecasting: theory and practice," Papers 2012.03854, arXiv.org, revised Jan 2022.
    4. Serge Darolles & Gaëlle Le Fol & Yang Lu & Ran Sun, 2018. "Bivariate integer-autoregressive process with an application to mutual fund flows," Post-Print hal-04590149, HAL.
    5. 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.
    6. Catania, Leopoldo & Di Mari, Roberto, 2021. "Hierarchical Markov-switching models for multivariate integer-valued time-series," Journal of Econometrics, Elsevier, vol. 221(1), pages 118-137.
    7. Haque, M. Ohidul & Haque, Tariq H., 2018. "Evaluating the effects of the road safety system approach in Brunei," Transportation Research Part A: Policy and Practice, Elsevier, vol. 118(C), pages 594-607.
    8. 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.
    9. Kharin, Yuriy & Voloshko, Valeriy, 2021. "Robust estimation for Binomial conditionally nonlinear autoregressive time series based on multivariate conditional frequencies," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    10. Ng, Jason & Forbes, Catherine S. & Martin, Gael M. & McCabe, Brendan P.M., 2013. "Non-parametric estimation of forecast distributions in non-Gaussian, non-linear state space models," International Journal of Forecasting, Elsevier, vol. 29(3), pages 411-430.
    11. 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.
    12. Raju Maiti & Atanu Biswas, 2015. "Coherent forecasting for stationary time series of discrete data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 99(3), pages 337-365, July.
    13. Masoomeh Forughi & Zohreh Shishebor & Atefeh Zamani, 2022. "Portmanteau tests for generalized integer-valued autoregressive time series models," Statistical Papers, Springer, vol. 63(4), pages 1163-1185, August.
    14. Darolles, Serge & Fol, Gaëlle Le & Lu, Yang & Sun, Ran, 2019. "Bivariate integer-autoregressive process with an application to mutual fund flows," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 181-203.
    15. Kai Yang & Yiwei Zhao & Han Li & Dehui Wang, 2023. "On bivariate threshold Poisson integer-valued autoregressive processes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(8), pages 931-963, November.
    16. Luisa Bisaglia & Margherita Gerolimetto, 2015. "Forecasting integer autoregressive processes of order 1: are simple AR competitive?," Economics Bulletin, AccessEcon, vol. 35(3), pages 1652-1660.
    17. Luisa Bisaglia & Margherita Gerolimetto, 2019. "Model-based INAR bootstrap for forecasting INAR(p) models," Computational Statistics, Springer, vol. 34(4), pages 1815-1848, December.

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