Bivariate extended skew-elliptical Heckman models: mathematical characterization and an application in economic sciences

Statistical inference methods that correct sample bias are a common challenge in empirical studies, as sample selection restricts observations to a specific subset of the target population. This selection bias can arise from a latent decision process, such as the decision of a person to participate in the labor market, respond to a survey, or undergo medical treatments. Heckman models are widely used to correct this selection bias in the estimation of regression parameters by incorporating an equation that models the latent process causing such a bias. In addition to the selection equation, there is an outcome equation that models the response variable through a linear regression conditioned on the selection. The errors associated with these two models are assumed to follow a bivariate normal distribution. However, this normality assumption may not be suitable for all datasets. To handle data that do not follow bivariate normality, several extensions to Heckman models have been proposed. The present study introduces novel Heckman models based on extended skew-elliptical distributions, which can accommodate asymmetric and heavy-tailed distributions, as well as other types of situations, which are specially helpful in Heckman frameworks and often observed in real-world scenarios. Furthermore, the introduced model allows the integration of explanatory variables into the dispersion and correlation (relationship between selection and outcome) parameters, providing high flexibility and accuracy in modeling. After the mathematical characterization of the new model, a Monte Carlo simulation study is conducted to evaluate the performance of the introduced models. To illustrate the practical applicability of the models, a real dataset on female labor supply in the United States is analyzed. The results demonstrate that Heckman models based on extended skew-elliptical distributions provide an improved fit over traditional models. This fit underscores the robustness and effectiveness of our methodology in accurately analyzing and correcting for sample selection bias in empirical data.