Andrew M. Jones 607Prieger’s approach builds on Lee’s (1983) selection model that uses a bivariate nor-
mal copula. In both cases the use of copulas to model the joint distribution allows
for the marginal distributions to be non-normal. As the outcome is a measure of
duration, exponential, gamma, log-logistic, log-normal and Weibull specifications
are considered and the gamma is selected as the preferred model. This reflects an
attractive feature of the copula approach, that model selection can focus on find-
ing an acceptable specification of the marginal distributions before turning to the
joint distribution. The specifications of the full selection model are compared using
the AIC, BIC, and the consistent Akaike information criterion (CAIC) as well as the
Voung test for non-nested models. These tests favor the FGM over the bivariate nor-
mal copula in terms of goodness-of-fit. Smith (2003) uses the same data as Prieger
(2002), but a different copula, the Frank copula from the Archimedean class, that
improves the fit of the model and changes the estimates of average length of stay.
Copulas are used by Zimmer and Trivedi (2006) to model dependence in a system
of nonlinear reduced form equations. Their application focuses on couples’ deci-
sions about insurance coverage and health care use and consists of three equations:
one for the husband’s utilization, one for the wife’s and one for whether or not the
couple take out separate health insurance policies. The marginal distributions for
utilization are assumed to be Negbin (NB2) and a probit is used for the insurance
equation. The equations are assumed to be linked by common unobservable fac-
tors and the joint distribution is modeled using a trivariate Frank copula, which
is derived using the mixture-of-powers approach. The use of copulas avoids hav-
ing to select a parametric specification for the unobservable heterogeneity and is
computationally tractable. To emphasize these points, Zimmer and Trivedi com-
pare their results to those derived from an MSL approach, based on multivariate
normality. The model is applied to data from four waves of the US MEPS. An
apparent limitation of the copula approach is that it works by specifying marginal
distributions and then modeling dependence, so that the emphasis is on a sys-
tem of reduced form equations rather than on conditional distributions. However,
Zimmer and Trivedi show how the estimates can be used to derive the condi-
tional distribution and hence compute the ATE of insurance coverage on health
care use.
12.7 Evaluation of treatment effects
12.7.1 Matching
Matching provides a general approach to deal with selection on observables. It
addresses the problem that, in the observed data, confounding factors may be
non-randomly distributed over the treated and controls. Rosenbaum and Rubin
(1983) showed that, rather than matching on an entire set of observable charac-
teristics (x), the dimensions of the problem can be reduced by matching on the
basis of their probability of receiving treatment,P
(
d= 1 |x)
, known as the propen-
sity score. In practice, propensity score matching (PSM) estimators do not rely on