An Estimator for Some Binary-Outcome Selection Models Without Exclusion Restrictions

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Abstract
This article provides a new maximum‐likelihood estimator for selection models with dichotomous dependent variables when identical factors affect the selection equation and the equation of interest. Such situations arise naturally in game‐theoretic models where selection is typically nonrandom and identical explanatory variables influence all decisions under investigation. When identical explanatory variables influence selection and a subsequent outcome of interest, the commonly used Heckman‐type estimators identify from distributional assumptions about the residuals alone. When its own identifying assumption is reasonable, the new estimator allows the researcher to avoid the painful choice between identifying from distributional assumptions alone and adding a theoretically unjustified variable to the selection equation in a mistaken attempt to “boost” identification. The article uses Monte Carlo methods to compare the small‐sample properties of the estimator with those of the Heckman‐type estimator and ordinary probit.

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