A General Approach to Incorporating Selectivity in a Model
William H. Greene · Faculty Digital Archive (New York University Florence) · 2006
Another, similar application to the Poisson regression model is Greene (1994).This approach is inappropriate for several reasons• The impact on the conditional mean of the model of interest will not take the form of an inverse Mills ratio.That is specific to the linear model.(See Terza (1995) for a development in the context of the Poisson regression.)• The bivariate normality assumption needed to justify the inclusion of the inverse Mills ratio in the second model generally does not appear anywhere in the model.• The dependent variable, conditioned on the sample selection, is unlikely to have the distribution described by the model in the absence of selection.That would be needed to use this approach.Note that this even appears in the canonical linear case.The normally distributed disturbance in the absence of sample selection has a nonnormal distribution in the presence of selection.That is the salient feature of Heckman's development.Counterparts to these three problems will show up in any nonlinear model.One cannot generally 'correct for selectivity' by dropping the inverse Mills ratio into the model at a convenient point.We describe an internally consistent method of incorporating 'sample selection' in a model.The method is based on the premise that motivated Heckman's canonical studies on the subject, that the force of 'sample selectivity' is exerted through the behavior of the unobservables in the model.As such, the key to modeling the effect is to introduce the unobservables that might be affected into the model in a reasonable way that maintains the internal consistency of the model itself.For example, in the Poisson model, the standard approach to introducing unobserved heterogeneity is through the conditional mean, specifically, (1)