Testing exclusion restrictions and additive separability in sample selection models

Standard sample selection models with non-randomly censored outcomes assume (i) an exclusion restriction (i.e., a variable affecting selection, but not the outcome) and (ii) additive separability of the errors in the selection process. This paper proposes tests for the joint satisfaction of these assumptions by applying the approach of Huber and Mellace (2011) (for testing instrument validity under treatment endogeneity) to the sample selection framework. We show that the exclusion restriction and additive separability imply two testable inequality constraints that come from both point identifying and bounding the outcome distribution of the subpopulation that is always selected/observed. We apply the tests to two variables for which the exclusion restriction is frequently invoked in female wage regressions: non-wife/husband's income and the number of (young) children. Considering eight empirical applications, our results suggest that the identifying assumptions are likely violated for the former variable, but cannot be refuted for the latter on statistical grounds.