Robust Inference in Sample Selection Models

Abstract

Summary The problem of non‐random sample selectivity often occurs in practice in many fields. The classical estimators introduced by Heckman are the backbone of the standard statistical analysis of these models. However, these estimators are very sensitive to small deviations from the distributional assumptions which are often not satisfied in practice. We develop a general framework to study the robustness properties of estimators and tests in sample selection models. We derive the influence function and the change‐of‐variance function of Heckman's two‐stage estimator, and we demonstrate the non‐robustness of this estimator and its estimated variance to small deviations from the model assumed. We propose a procedure for robustifying the estimator, prove its asymptotic normality and give its asymptotic variance. Both cases with and without an exclusion restriction are covered. This allows us to construct a simple robust alternative to the sample selection bias test. We illustrate the use of our new methodology in an analysis of ambulatory expenditures and we compare the performance of the classical and robust methods in a Monte Carlo simulation study.

Copula selection models for non‐Gaussian outcomes that are missing not at random

“…This paper discusses a copula‐based selection model, which can help make more plausible assumptions about the distribution of the data and proposes a flexible imputation procedure that generates plausible imputed values from the copula selection model. Our paper expands on recent work on developing selection models that can accommodate departures from bivariate normality, by advocating a flexible approach for addressing MNAR in a wide range of nonnormal (continuous) outcomes (as well as discrete and count endpoints). The paper directly extends previous applications of the copula approach to address MNAR in settings with binary outcomes (see the work of Marra et al) and continuous outcomes assuming a particular copula function such as the Gaussian or Archimedean copulae .…”

“…Recent studies have considered various generalizations to address possible deviations from normality. For example, the Heckman selection model has been extended to accommodate outcomes with heavier tails by considering a bivariate t ‐distribution, whereas Zhelonkin and others introduced a procedure for robustifying the Heckman's two‐step estimator by using M‐estimators of Mallows' type for both steps. However, most of these approaches are restricted to a specific joint distribution for the selection and outcome processes, and extension to other non‐Gaussian outcomes may not be straightforward.…”

Copula based generalized additive models for location, scale and shape with non-random sample selection

“…Frequentist counterparts of these Bayesian methods are given in Marra and Radice (2013b) in the context of binary responses and Marra and Radice (2013a) for continuous Gaussian outcomes. Zhelonkin et al (2016) introduced a procedure for robustifying the Heckman's two stage estimator by using M-estimators of Mallows' type for both stages. Marchenko and Genton (2012) and Ding (2014) considered a bivariate Student-t distribution for the model's errors as a way of tackling heavy-tailed data.…”

The Effect of Hearing Impairments on Wage Earnings: Evidence from a Copula‐Based Spline Selection Model

“…2 Regarding other software for selection models, the R package ssmrob estimates sample selection models for which the assumption of bivariate normality is relaxed (Zhelonkin et al, 2013). The Stata package heckt estimates a Heckman-style model with a bivariate t distribution.…”