Modelling sample selection using Archimedean copulas

Abstract

By a theorem due to Sklar, a multivariate distribution can be represented in terms of its underlying margins by binding them together using a copula function. By exploiting this representation, the "copula approach" to modelling proceeds by specifying distributions for each margin and a copula function. In this paper, a number of families of copula functions are given, with attention focusing on those that fall within the Archimedean class. Members of this class of copulas are shown to be rich in various distributional attributes that are desired when modelling. The paper then proceeds by applying the copula approach to construct models for data that may suffer from selectivity bias. The models examined are the self-selection model, the switching regime model and the double-selection model. It is shown that when models are constructed using copulas from the Archimedean class, the resulting expressions for the log-likelihood and score facilitate maximum likelihood estimation. The literature on selectivity modelling is almost exclusively based on multivariate normal specifications. The copula approach permits selection modelling based on multivariate non-normality. Examples of self-selection models for labour supply and for duration of hospitalization illustrate the application of the copula approach to modelling. Copyright Royal Economic Society, 2003

Testing the hypothesis of absence of unobserved confounding in semiparametric bivariate probit models

“…Examples are copula or count data endogenous and non-random sample selection models (Bratti and Miranda 2011;Winkelmann 2011;Zimmer and Trivedi 2006;Smith 2003). More generally, this test could be extended to any other context where there is a system of equations and testing their independence is important (Kiefer 1982;Yee and Wild 1996).…”

The Use of Official Statistics in Self-Selection Bias Modeling

“…Indeed, the marginals are very often not normally distributed, especially financial variables. Smith (2003) provides a general copula-based framework for Heckman's model by demonstrating that copulas can be used to extend the standard analysis to any bivariate distribution with given marginals (see also Smith 2005). The use of normal marginals and normal copula leads us to the traditional Heckman's method, as is shown by comparing Equations (10) and (8) (see Bhat and Eluru 2009).…”

“…This concordance measure is generally preferred to the copula's dependence parameter u, since t is invariant with respect to the marginals and to strictly increasing transformations of the variables. For more details about transforming the copula parameter u into Kendall's t, please see Smith (2003). Figure 1 shows the bivariate contour plots of the different types of copulas illustrated in this section, all with standard normal margins and t ¼ 0.5.…”

Quantile selection models: with an application to understanding changes in wage inequality

“…is the Plackett copula family (e.g., Smith, 2003). Lower and upper Fréchet bounds correspond to θ → 0 and θ → +∞, respectively.…”