TY - JOUR

T1 - A Simple Estimator of Two-Dimensional Copulas, with Applications

AU - Anderson, Eddie

AU - Prokhorov, Artem

AU - Zhu, Yajing

N1 - Publisher Copyright: © 2020 The Department of Economics, University of Oxford and John Wiley & Sons Ltd

PY - 2021/12

Y1 - 2021/12

N2 - 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.

AB - 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.

KW - MAXIMUM-LIKELIHOOD-ESTIMATION

KW - DENSITY-ESTIMATION

KW - SPLINE ESTIMATION

KW - DEPENDENCE

KW - MODEL

KW - SERIES

UR - http://www.scopus.com/inward/record.url?scp=85085143858&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/99c4560d-c3f2-3e33-a2a2-0b349a1fe75f/

U2 - 10.1111/obes.12371

DO - 10.1111/obes.12371

M3 - Article

AN - SCOPUS:85085143858

VL - 82

SP - 1375

EP - 1412

JO - Oxford Bulletin of Economics and Statistics

JF - Oxford Bulletin of Economics and Statistics

SN - 0305-9049

IS - 6

ER -