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A Simple Estimator of Two‐Dimensional Copulas, with Applications

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  • Eddie Anderson
  • Artem Prokhorov
  • Yajing Zhu

Abstract

Copulas are distributions with uniform marginals. Non‐parametric copula estimates may violate the uniformity condition in finite samples. We look at whether it is possible to obtain valid piecewise linear copula densities by triangulation. The copula property imposes strict constraints on design points, making an equi‐spaced grid a natural starting point. However, the mixed‐integer nature of the problem makes a pure triangulation approach impractical on fine grids. As an alternative, we study the ways of approximating copula densities with triangular functions which guarantees that the estimator is a valid copula density. The family of resulting estimators can be viewed as a non‐parametric MLE of B‐spline coefficients on possibly non‐equally spaced grids under simple linear constraints. As such, it can be easily solved using standard convex optimization tools and allows for a degree of localization. A simulation study shows an attractive performance of the estimator in small samples and compares it with some of the leading alternatives. We demonstrate empirical relevance of our approach using three applications. In the first application, we investigate how the body mass index of children depends on that of parents. In the second application, we construct a bivariate copula underlying the Gibson paradox from macroeconomics. In the third application, we show the benefit of using our approach in testing the null of independence against the alternative of an arbitrary dependence pattern.

Suggested Citation

  • Eddie Anderson & Artem Prokhorov & Yajing Zhu, 2020. "A Simple Estimator of Two‐Dimensional Copulas, with Applications," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 82(6), pages 1375-1412, December.
  • Handle: RePEc:bla:obuest:v:82:y:2020:i:6:p:1375-1412
    DOI: 10.1111/obes.12371
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    References listed on IDEAS

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    1. Zhang, Qingzhao & Li, Deyuan & Wang, Hansheng, 2013. "A note on tail dependence regression," Journal of Multivariate Analysis, Elsevier, vol. 120(C), pages 163-172.
    2. Berghaus, Betina & Segers, Johan, 2017. "Weak convergence of the weighted empirical beta copula process," LIDAM Discussion Papers ISBA 2017015, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Valentyn Panchenko & Artem Prokhorov, 2011. "Efficient estimation of parameters in marginals in semiparametric multivariate models," Working Papers 11001, Concordia University, Department of Economics.
    4. 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.
    5. Yichen Gao & Yu Zhang & Ximing Wu, 2015. "Penalized exponential series estimation of copula densities with an application to intergenerational dependence of body mass index," Empirical Economics, Springer, vol. 48(1), pages 61-81, February.
    6. Bouezmarni, Taoufik & Rombouts, Jeroen V.K. & Taamouti, Abderrahim, 2010. "Asymptotic properties of the Bernstein density copula estimator for [alpha]-mixing data," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 1-10, January.
    7. Lan Xue & Hua Liang, 2010. "Polynomial Spline Estimation for a Generalized Additive Coefficient Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(1), pages 26-46, March.
    8. Burda, Martin & Prokhorov, Artem, 2014. "Copula based factorization in Bayesian multivariate infinite mixture models," Journal of Multivariate Analysis, Elsevier, vol. 127(C), pages 200-213.
    9. Segers, Johan & Sibuya, Masaaki & Tsukahara, Hideatsu, 2017. "The empirical beta copula," LIDAM Reprints ISBA 2017005, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    10. Shen, Xiaojing & Zhu, Yunmin & Song, Lixin, 2008. "Linear B-spline copulas with applications to nonparametric estimation of copulas," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3806-3819, March.
    11. Bouezmarni Taoufik & Ghouch El & Taamouti Abderrahim, 2013. "Bernstein estimator for unbounded copula densities," Statistics & Risk Modeling, De Gruyter, vol. 30(4), pages 343-360, December.
    12. Segers, Johan & Sibuya, Masaaki & Tsukahara, Hideatsu, 2017. "The empirical beta copula," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 35-51.
    13. Sancetta, Alessio & Satchell, Stephen, 2004. "The Bernstein Copula And Its Applications To Modeling And Approximations Of Multivariate Distributions," Econometric Theory, Cambridge University Press, vol. 20(3), pages 535-562, June.
    14. Ying Zhang & Lei Hua & Jian Huang, 2010. "A Spline‐Based Semiparametric Maximum Likelihood Estimation Method for the Cox Model with Interval‐Censored Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(2), pages 338-354, June.
    15. Marin Bozic & John Newton & Cameron S. Thraen & Brian W. Gould, 2014. "Tails Curtailed: Accounting for Nonlinear Dependence in Pricing Margin Insurance for Dairy Farmers," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 96(4), pages 1117-1135.
    16. M. Belalia & T. Bouezmarni & F. C. Lemyre & A. Taamouti, 2017. "Testing independence based on Bernstein empirical copula and copula density," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(2), pages 346-380, April.
    17. Qin Shao & Lijian Yang, 2017. "Oracally efficient estimation and consistent model selection for auto-regressive moving average time series with trend," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 507-524, March.
    18. Qu, Leming & Yin, Wotao, 2012. "Copula density estimation by total variation penalized likelihood with linear equality constraints," Computational Statistics & Data Analysis, Elsevier, vol. 56(2), pages 384-398.
    19. Lu, Minggen, 2010. "Spline-based sieve maximum likelihood estimation in the partly linear model under monotonicity constraints," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2528-2542, November.
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