A Simple Estimator of Two‐Dimensional Copulas, with Applications1
| Author | Artem Prokhorov,Eddie Anderson,Yajing Zhu |
| Published date | 01 December 2021 |
| Date | 01 December 2021 |
| DOI | http://doi.org/10.1111/obes.12371 |
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©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 82, 6 (2020) 0305–9049
doi: 10.1111/obes.12371
A Simple Estimator of Two-Dimensional Copulas,
with Applications*
Eddie Anderson†,‡, Artem Prokhorov†,§ and Yajing Zhu††
†University of Sydney Business School, Sydney, Australia.
(e-mail: edward.anderson@sydney.edu.au)
‡Imperial College London, London, UK
§St.Petersburg State University, St.Petersburg, Russia
(e-mail: artem.prokhorov@sydney.edu.au)
††Concordia University, Montreal, Canada(e-mail: yajing.s.zhu@gmail.com)
Abstract
Copulas are distributions with uniform marginals. Non-parametric copula estimates may
violate the uniformity condition in finite samples. Welook 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 convexoptimization 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.
I. Introduction
Copulas are multivariate distributions of marginalprobability integral transfor ms.They are
widely used for modelling dependence between the marginals and they have found many
JEL Classification numbers: C13, C14.
*The authors acknowledge the Sydney Informatics Hub and the University of Sydney’s high-performance com-
puting cluster Artemis for providingthe high-perfor mance computing resources that havecontributed to the research
results reported within this paper. Research for this paper was supported by a grant from the Russian Science Foun-
dation (Project No. 16-18-10432).
1376 Bulletin
applications in economics and finance (see, e.g. Fan and Patton, 2014, for a survey).A key
feature of copulas is that all marginals of a copula are uniform on [0, 1]. This condition
is what distinguishes copula density estimation from estimation of general densities on a
hypercube.
In the context of non-parametric copula estimation, the uniformity condition is often
difficult to impose. It involves integral equations based on the copula estimator, which
translate into restrictions on the parameter space and the kernel or basis functions used.
For example, the Bernstein copula estimator (see, e.g. Sancetta and Satchell, 2004) can be
shown to satisfy these conditions if and only if the matrix of copula parameters is doubly
stochastic. The penalized exponential series copula estimator (see, e.g. Gao, Zhang and
Wu, 2015) satisfies these conditions under a set of nonlinear constraints on parameters and
basis functions. Due to such difficulties, estimators that do impose the uniformity condition
may exhibit computational issues (see, e.g. Qu, Qian and Xie, 2009; Qu andYin, 2012).
In this paper, we propose a class of copula density estimators obtained using B-splines
over a possibly sparse grid.The uniform marginal property imposes certain constraints on
the density surface that are easiest to handle on an equi-spaced grid using a triangular basis
function. We work out the constraints and explore the difficulties arising from a direct
application of triangulation in approximating bivariate copula densities with piecewise
linear surfaces while guaranteeing the uniform marginal property. The main difficulty is
that such copula constructions require a mixed integer optimization which is hard to work
with.
Wethen develop a straightforward spline-based method using a specific basis function,
which reduces this problem to a convex non-parametric maximum likelihood estimation,
subject to linear equality constraints – an easy problem to handle in most availablesoftware
packages. The estimator also has well known statistical properties, being a member of a
constrained MLE family of estimators (see, e.g. Aitchison and Silvey, 1958). We provide
sufficient conditions for the copula property to hold inside grid cells provided it holds at
grid knots. We then generalize the spline estimator to higher degree splines and non-equi-
spaced grids. The latter contribution is important because it provides a natural but often
overlooked way of imposing a finer grid at a corner and sparser elsewhere. That is to say,
our estimator has a localization property. We do not pursue uneven grids in the paper but
we use B-splines of a higher order in simulations to illustrate the additional computational
burden associated with this generalization.
The use of B-splines is not new in statistics literature on copula modelling. Kauerman,
Schellhase and Ruppert (2013) propose a spline-based multivariate copula estimator with
a focus on dimensionality reduction permitted by a sparsity pattern. Shen, Zhu and Song
(2008) consider linear splines as a means of improving over the empirical copula and
Erdely (2016) shows that the linear spline estimators are actually a checkerboard copula
of Li, Mikusinski and Taylor (1997). The paper is also related to the literature on methods
of constructing bivariate copulas using the copula values at some points in the unit square,
such as a copula obtained from a diagonal section (see, e.g. Nelsen et al., 2008), a horizontal
section (see, e.g. Klement et al., 2007) and using rectangles (see, e.g. Durante, Saminger-
Platz and Sarkoci, 2009). Just like these other papers, we offer a copula construction
method but our focus is on developing a spline-based estimator imposing the uniformity
of marginals.
©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
Estimator of two-dimensional copulas 1377
We compare our estimator with a battery of commonly used copula density estimators,
including the empirical beta copula, Bernstein polynomials, exponential series, data-mirror
and naive kernel estimators. The list is by no means complete but representative of what is
used in practice. Our estimator performs very well overall and, in particular, we show in
a simulation study that, for various strengths of dependence and various sample sizes, the
estimator is able to capture key features of the true copula no worse than the competitors.
As applications, we provide new insights into several well-studied econometric data
sets and into a recent independence test. First, we reconsider the dependence between
intergenerational body mass indices and uncover a stronger dependence at the upper end
of the parent–child body mass index (BMI) distribution. Second, we use our estimators
to non-parametrically model a macroeconomic phenomenon known as Gibson’s paradox,
which has up to now been modelled using onlyrestrictive parametric distributions. Finally,
we look at the power of a copula-based test for arbitrary dependence between two random
variables proposed by Belalia et al. (2017). The test power is visibly increased when the
copula property is imposed and our estimator dominates the Bernstein polynomial estimator
initially used by Belalia et al. (2017). The key observation in all our applications is that by
imposing the copula property the new estimators provide confidence in dependence-based
measures and tests.
The paper is organized as follows. Section II discusses the uniform marginals prop-
erty and what it implies for the construction of a piecewise linear surface and defines
the estimator we propose. Section III lists selected copula density estimators which serve
as benchmarks. Section IV discusses simulation results. Section V discusses the applica-
tions to intergenerational body mass index estimation and to Gibson’s paradox. SectionVI
discusses the application to independence testing. Section VII concludes.
II. Copula property and proposed estimator
Let Hdenote an absolutely continuous bivariatedistribution function with one-dimensional
marginals F1,F2. A copula function C: [0, 1]2→[0, 1] can be obtained from the equation
H(x1,x2)=C(F1(x1), F2(x2))
by inversion as follows:
C(u)=H(F−1
1(u1), F−1
2(u2)),
where u=(u1,u2)∈[0, 1]2and F−1
jis the generalized inverse of Fj,j=1,2. The Sklar
(1959) theorem states that Cis unique for continuous distributions. When it exists, the
copula density c(u) is defined as @2
@u1@u2
C(u).
A key property of c(u) is that it has uniform marginals (see, e.g. Ibragimov and
Prokhorov, 2017, Definition 1.1). This property can be equivalently stated in terms of
the distribution function
C(1, u)=C(u,1)=u, for all u∈[0, 1],
©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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