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Estimation of Copula Models With Discrete Margins via Bayesian Data Augmentation


  • Michael S. Smith
  • Mohamad A. Khaled


Estimation of copula models with discrete margins can be difficult beyond the bivariate case. We show how this can be achieved by augmenting the likelihood with continuous latent variables, and computing inference using the resulting augmented posterior. To evaluate this, we propose two efficient Markov chain Monte Carlo sampling schemes. One generates the latent variables as a block using a Metropolis--Hastings step with a proposal that is close to its target distribution, the other generates them one at a time. Our method applies to all parametric copulas where the conditional copula functions can be evaluated, not just elliptical copulas as in much previous work. Moreover, the copula parameters can be estimated joint with any marginal parameters, and Bayesian selection ideas can be employed. We establish the effectiveness of the estimation method by modeling consumer behavior in online retail using Archimedean and Gaussian copulas. The example shows that elliptical copulas can be poor at modeling dependence in discrete data, just as they can be in the continuous case. To demonstrate the potential in higher dimensions, we estimate 16-dimensional D-vine copulas for a longitudinal model of usage of a bicycle path in the city of Melbourne, Australia. The estimates reveal an interesting serial dependence structure that can be represented in a parsimonious fashion using Bayesian selection of independence pair-copula components. Finally, we extend our results and method to the case where some margins are discrete and others continuous. Supplemental materials for the article are also available online.

Suggested Citation

  • Michael S. Smith & Mohamad A. Khaled, 2012. "Estimation of Copula Models With Discrete Margins via Bayesian Data Augmentation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(497), pages 290-303, March.
  • Handle: RePEc:taf:jnlasa:v:107:y:2012:i:497:p:290-303
    DOI: 10.1080/01621459.2011.644501

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    Cited by:

    1. Azam, Kazim & Pitt, Michael, 2014. "Bayesian Inference for a Semi-Parametric Copula-based Markov Chain," Economic Research Papers 270232, University of Warwick - Department of Economics.
    2. Li, Feng & Kang, Yanfei, 2018. "Improving forecasting performance using covariate-dependent copula models," International Journal of Forecasting, Elsevier, vol. 34(3), pages 456-476.
    3. Pravin Trivedi & David Zimmer, 2017. "A Note on Identification of Bivariate Copulas for Discrete Count Data," Econometrics, MDPI, Open Access Journal, vol. 5(1), pages 1-11, February.
    4. Kazim Azam & Andre Lucas, 2015. "Mixed Density based Copula Likelihood," Tinbergen Institute Discussion Papers 15-003/IV/DSF084, Tinbergen Institute.
    5. Azam, Kazim & Pitt, Michael, 2014. "Bayesian Inference for a Semi-Parametric Copula-based Markov Chain," The Warwick Economics Research Paper Series (TWERPS) 1051, University of Warwick, Department of Economics.
    6. Stöber, Jakob & Hong, Hyokyoung Grace & Czado, Claudia & Ghosh, Pulak, 2015. "Comorbidity of chronic diseases in the elderly: Patterns identified by a copula design for mixed responses," Computational Statistics & Data Analysis, Elsevier, vol. 88(C), pages 28-39.
    7. Jeffrey Racine, 2015. "Mixed data kernel copulas," Empirical Economics, Springer, vol. 48(1), pages 37-59, February.
    8. Calabrese, Raffaella & Degl’Innocenti, Marta & Osmetti, Silvia Angela, 2017. "The effectiveness of TARP-CPP on the US banking industry: A new copula-based approach," European Journal of Operational Research, Elsevier, vol. 256(3), pages 1029-1037.
    9. Ruben Loaiza-Maya & Michael Stanley Smith, 2017. "Variational Bayes Estimation of Discrete-Margined Copula Models with Application to Time Series," Papers 1712.09150,, revised Jul 2018.
    10. Marchese, Scott & Diao, Guoqing, 2017. "Density ratio model for multivariate outcomes," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 249-261.
    11. Zilko, Aurelius A. & Kurowicka, Dorota, 2016. "Copula in a multivariate mixed discrete–continuous model," Computational Statistics & Data Analysis, Elsevier, vol. 103(C), pages 28-55.
    12. Salaheddine El Adlouni, 2018. "Quantile regression C-vine copula model for spatial extremes," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 94(1), pages 299-317, October.
    13. F. Louzada & P. H. Ferreira, 2016. "Modified inference function for margins for the bivariate clayton copula-based SUN Tobit Model," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(16), pages 2956-2976, December.
    14. Sun, Can & Bie, Zhaohong & Xie, Min & Jiang, Jiangfeng, 2016. "Fuzzy copula model for wind speed correlation and its application in wind curtailment evaluation," Renewable Energy, Elsevier, vol. 93(C), pages 68-76.

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