  Many proposals have emerged as alternatives to the Heckman selection model,
mainly to address the non-robustness of its normal assumption. The 2001 Medical
Expenditure Panel Survey data is often used to illustrate this non-robustness
of the Heckman model. In this paper, we propose a generalization of the Heckman
sample selection model by allowing the sample selection bias and dispersion
parameters to depend on covariates. We show that the non-robustness of the
Heckman model may be due to the assumption of the constant sample selection
bias parameter rather than the normality assumption. Our proposed methodology
allows us to understand which covariates are important to explain the sample
selection bias phenomenon rather than to only form conclusions about its
presence. We explore the inferential aspects of the maximum likelihood
estimators (MLEs) for our proposed generalized Heckman model. More
specifically, we show that this model satisfies some regularity conditions such
that it ensures consistency and asymptotic normality of the MLEs. Proper score
residuals for sample selection models are provided, and model adequacy is
addressed. Simulated results are presented to check the finite-sample behavior
of the estimators and to verify the consequences of not considering varying
sample selection bias and dispersion parameters. We show that the normal
assumption for analyzing medical expenditure data is suitable and that the
conclusions drawn using our approach are coherent with findings from prior
literature. Moreover, we identify which covariates are relevant to explain the
presence of sample selection bias in this important dataset.
