Modelling sample selection using Archimedean copulas

TLDR

Copulas provide a way to construct multivariate distributions from marginal distributions, and the copula approach is widely used, though selectivity modelling has traditionally relied on multivariate normal specifications. This paper introduces Archimedean copula families and applies them to construct self‑selection, switching‑regime, and double‑selection models for data with selectivity bias. The authors derive log‑likelihood and score expressions for these models, enabling maximum‑likelihood estimation. Archimedean copulas offer desirable distributional properties, allow non‑normal selection modelling, and, as shown in labour‑supply and hospital‑duration examples, facilitate efficient estimation.

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.