592 Panel Data Methods
12.4.5 Copulas
The presence of common unobservables leads to multiple equation models and
the need to specify multivariate distributions. But the menu of parametric forms
available for bivariate and, more generally, multivariate distributions is limited. In
many applications multivariate normality may be unappealing: for example, with
heavily skewed and long-tailed data on costs of care or on quality adjusted life-
years (QALYs) (Quinn, 2005) or for rare events. Copulas provide an alternative and
are a method of constructing multivariate distributions from univariate marginal
distributions (see Trivedi and Zimmer, 2005). A copula is a function that can be
interpreted as a joint probability whose arguments are the univariate cumulative
distribution functions (CDFs) of the marginal distributions. The fact that the CDF
is used means that the marginal distributions are fixed and invariant to transforma-
tions of the random variable. The functional form selected for the copula uniquely
determines the form of the dependence, independently of the functional forms of
the marginal distributions. The attractions of copulas are that they are flexible –
they can mix together marginal distributions of different types, whether they be
continuous, integer valued counts or categorical; they allow for richer concepts of
dependence than the standard linear measure, including measures of tail depen-
dence; and they are computationally tractable and avoid the need for numerical
integration or simulation.
A key result in the theory of copulas is Sklar’s theorem, which shows that all
multivariate distributions can be represented by a copula. So in the bivariate case,
if two random variables have a joint distributionF(x 1 ,x 2 )and marginal distribu-
tionsF 1 (x 1 )andF 2 (x 2 )then Sklar’s theorem establishes that there exists a copula
functionC(·)such that:
F(x 1 ,x 2 )=C(F 1 (x 1 ),F 2 (x 2 )). (12.29)In practice the unique copula that characterizes the true joint distribution is
unknown. So particular functional forms have to be selected and compared in
terms of their goodness of fit. There is a long list of copulas to choose from. Com-
mon choices include the Frank copula and the Farlie–Gumbel–Morgenstern (FGM)
copula, which is a first-order approximation of the Frank copula and is tractable to
use in applied work (Prieger, 2002; Smith, 2003; Zimmer and Trivedi, 2006). The
bivariate form of the Frank copula is:
Cθ(u,v)=−θ− (^1) log
(
1 +
(e−θu− 1 )(e−θv− 1 )
e−θ− 1
)
, (12.30)
where the parameterθcaptures dependence. The bivariate form of the FGM copula
is:
Cθ(u,v)=uv(^1 +θ(^1 −u)(^1 −v)). (12.31)
Other common choices include elliptical copulas, such as the Gaussian and Stu-
dent’s t, and the Clayton copulas. The Gaussian copula is an example of a copula
derived by the method of inversion and takes the form:
Cθ(u,v)=# 2 (#
− 1
1 (u),#
− 1
1 (v)), (12.32)