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Modeling Multivariate Distributions Using Copulas: Applications in Marketing

Author

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  • Peter J. Danaher

    () (Melbourne Business School, University of Melbourne, Carlton, Victoria 3053, Australia)

  • Michael S. Smith

    () (Melbourne Business School, University of Melbourne, Carlton, Victoria 3053, Australia)

Abstract

In this research we introduce a new class of multivariate probability models to the marketing literature. Known as "copula models," they have a number of attractive features. First, they permit the combination of any univariate marginal distributions that need not come from the same distributional family. Second, a particular class of copula models, called "elliptical copula," has the property that they increase in complexity at a much slower rate than existing multivariate probability models as the number of dimensions increase. Third, they are very general, encompassing a number of existing multivariate models and providing a framework for generating many more. These advantages give copula models a greater potential for use in empirical analysis than existing probability models used in marketing. We exploit and extend recent developments in Bayesian estimation to propose an approach that allows reliable estimation of elliptical copula models in high dimensions. Rather than focusing on a single marketing problem, we demonstrate the versatility and accuracy of copula models with four examples to show the flexibility of the method. In every case, the copula model either handles a situation that could not be modeled previously or gives improved accuracy compared with prior models.

Suggested Citation

  • Peter J. Danaher & Michael S. Smith, 2011. "Modeling Multivariate Distributions Using Copulas: Applications in Marketing," Marketing Science, INFORMS, vol. 30(1), pages 4-21, 01-02.
  • Handle: RePEc:inm:ormksc:v:30:y:2011:i:1:p:4-21
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    File URL: http://dx.doi.org/10.1287/mksc.1090.0491
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    References listed on IDEAS

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    Cited by:

    1. Göran Kauermann & Christian Schellhase & David Ruppert, 2013. "Flexible Copula Density Estimation with Penalized Hierarchical B-splines," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 685-705, December.
    2. Peter J. Danaher & Michael S. Smith, 2011. "Rejoinder--Estimation Issues for Copulas Applied to Marketing Data," Marketing Science, INFORMS, vol. 30(1), pages 25-28, 01-02.
    3. Sungho Park & Sachin Gupta, 2012. "Handling Endogenous Regressors by Joint Estimation Using Copulas," Marketing Science, INFORMS, vol. 31(4), pages 567-586, July.
    4. Ceballos, Francisco, 2016. "Estimating spatial basis risk in rainfall index insurance: Methodology and application to excess rainfall insurance in Uruguay," IFPRI discussion papers 1595, International Food Policy Research Institute (IFPRI).
    5. Yang, Yang & Ignatavičiūtė, Eglė & Šiaulys, Jonas, 2015. "Conditional tail expectation of randomly weighted sums with heavy-tailed distributions," Statistics & Probability Letters, Elsevier, vol. 105(C), pages 20-28.
    6. Marta Disegna & Fabrizio Durante & Enrico Foscolo, 2013. "A multivariate nonlinear analysis of tourism expenditures," BEMPS - Bozen Economics & Management Paper Series BEMPS10, Faculty of Economics and Management at the Free University of Bozen.
    7. Ebrahimi, Nader & Jalali, Nima Y. & Soofi, Ehsan S., 2014. "Comparison, utility, and partition of dependence under absolutely continuous and singular distributions," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 32-50.
    8. Banciu, M. & Ødegaard, F., 2016. "Optimal product bundling with dependent valuations: The price of independence," European Journal of Operational Research, Elsevier, vol. 255(2), pages 481-495.
    9. repec:eee:ijrema:v:32:y:2015:i:1:p:78-93 is not listed on IDEAS
    10. repec:tiu:tiutis:52e91e47-4a2d-4e7b-bb23-3926b842ae30 is not listed on IDEAS
    11. Shaun P Vahey & Elizabeth C Wakerly, 2013. "Moving towards probability forecasting," BIS Papers chapters,in: Bank for International Settlements (ed.), Globalisation and inflation dynamics in Asia and the Pacific, volume 70, pages 3-8 Bank for International Settlements.
    12. Edward I. George & Shane T. Jensen, 2011. "Commentary--A Latent Variable Perspective of Copula Modeling," Marketing Science, INFORMS, vol. 30(1), pages 22-24, 01-02.
    13. Zhang, Shulin & Okhrin, Ostap & Zhou, Qian M. & Song, Peter X.-K., 2016. "Goodness-of-fit test for specification of semiparametric copula dependence models," Journal of Econometrics, Elsevier, vol. 193(1), pages 215-233.
    14. Friberg, Richard & Huse, Cristian, 2012. "How to use demand systems to evaluate risky projects, with an application to automobile production," CEPR Discussion Papers 9266, C.E.P.R. Discussion Papers.
    15. Smith, Michael S. & Kauermann, Göran, 2011. "Bicycle commuting in Melbourne during the 2000s energy crisis: A semiparametric analysis of intraday volumes," Transportation Research Part B: Methodological, Elsevier, vol. 45(10), pages 1846-1862.
    16. repec:eee:joinma:v:37:y:2017:i:c:p:16-31 is not listed on IDEAS
    17. Tran, Kien C. & Tsionas, Efthymios G., 2015. "Endogeneity in stochastic frontier models: Copula approach without external instruments," Economics Letters, Elsevier, vol. 133(C), pages 85-88.
    18. Emilio Gómez-Déniz & Jorge Pérez-Rodríguez, 2015. "Closed-form solution for a bivariate distribution in stochastic frontier models with dependent errors," Journal of Productivity Analysis, Springer, vol. 43(2), pages 215-223, April.
    19. Abdallah, Anas & Boucher, Jean-Philippe & Cossette, Hélène, 2016. "Sarmanov family of multivariate distributions for bivariate dynamic claim counts model," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 120-133.
    20. repec:eee:insuma:v:79:y:2018:i:c:p:184-193 is not listed on IDEAS
    21. 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.
    22. Shulin Zhang, & Ostap Okhrin, & Qian M. Zhou & Peter X.-K. Song, 2013. "Goodness-of-fit Test for Specification of Semiparametric Copula Dependence Models," SFB 649 Discussion Papers SFB649DP2013-041, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    23. Oliver J. Rutz & Michael Trusov, 2011. "Zooming In on Paid Search Ads--A Consumer-Level Model Calibrated on Aggregated Data," Marketing Science, INFORMS, vol. 30(5), pages 789-800, September.

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