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Conditional Maximum Likelihood Estimation of Higher-Order Integer-Valued Autoregressive Processes


  • Ruijun Bu


  • Kaddour Hadri
  • Brendan McCabe

    () (Management School, University of Liverpool, UK)


In this paper, we extend earlier work of Freeland and McCabe (2004) and develop a general framework for maximum likelihood (ML) estimation of higher-order integer-valued autoregressive (INAR(p)) 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 for the INAR(p) model is proposed. Based on this representation, we derive expressions for the score function and the Fisher information matrix of the INAR(p) model, which form the basis for maximum likelihood 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. These new expressions enhance the interpretation of these quantities and lead naturally to new de?nitions for residuals for the INAR(p) model. Using the INAR(2) speci?cation with Poisson innovations, we examine the asymptotic effciency gain of implementing the ML technique over the widely used conditional least squares (CLS) method. 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.

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  • Ruijun Bu & Kaddour Hadri & Brendan McCabe, 2006. "Conditional Maximum Likelihood Estimation of Higher-Order Integer-Valued Autoregressive Processes," Research Papers 200619, University of Liverpool Management School.
  • Handle: RePEc:liv:livedp:200619

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    References listed on IDEAS

    1. 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.
    2. 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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    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.

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