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Negative Binomial Autoregressive Process


  • Christian Gouriéroux

    (University of Toronto; Toulouse School of Economics; CREST)

  • Yang Lu

    (University of Paris 13; Department of Economics (CEPN))


We introduce Negative Binomial Autoregressive (NBAR) processes for (univariate and bivariate) count time series. The univariate NBAR process is defined jointly with an underlying intensity process, which is autoregressive gamma. The resulting count process is Markov, with negative binomial conditional and marginal distributions. The process is then extended to the bivariate case with a Wishart autoregressive matrix intensity process. The NBAR processes are Compound Autoregressive, which allows for simple stationarity condition and quasi-closed form nonlinear forecasting formulas at any horizon, as well as a computationally tractable generalized method of moment estimator. The model is applied to a pairwise analysis of weekly occurrence counts of a contagious disease between the greater Paris region and other French regions.

Suggested Citation

  • Christian Gouriéroux & Yang Lu, 2018. "Negative Binomial Autoregressive Process," Working Papers 2018-03, Center for Research in Economics and Statistics.
  • Handle: RePEc:crs:wpaper:2018-03

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    References listed on IDEAS

    1. Joann Jasiak & Christian Gourieroux, 2006. "Autoregressive gamma processes," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 25(2), pages 129-152.
    2. Serge Darolles & Christian Gourieroux & Joann Jasiak, 2006. "Structural Laplace Transform and Compound Autoregressive Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(4), pages 477-503, July.
    3. Heinen, Andreas & Rengifo, Erick, 2007. "Multivariate autoregressive modeling of time series count data using copulas," Journal of Empirical Finance, Elsevier, vol. 14(4), pages 564-583, September.
    4. Bockenholt, Ulf, 1998. "Mixed INAR(1) Poisson regression models: Analyzing heterogeneity and serial dependencies in longitudinal count data," Journal of Econometrics, Elsevier, vol. 89(1-2), pages 317-338, November.
    5. Gourieroux, C. & Jasiak, J., 2004. "Heterogeneous INAR(1) model with application to car insurance," Insurance: Mathematics and Economics, Elsevier, vol. 34(2), pages 177-192, April.
    6. Richard Blundell & Rachel Griffith & John van Reenen, 1999. "Market Share, Market Value and Innovation in a Panel of British Manufacturing Firms," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 66(3), pages 529-554.
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    More about this item


    Negative Binomial Process; Autoregressive Gamma; Poisson-Gamma Conjugacy; Intensity; Compound Autoregressive Process; Common Factor; Pairwise Analysis; Health Insurance;
    All these keywords.

    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models


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