Lee Bounds with a Continuous Treatment in Sample Selection

Abstract

We study causal inference in sample selection models where a continuous or multivalued treatment affects both outcome and their observability (eg., employment or survey response). We generalized the widely used Lee (2009)'s bounds for binary treatment effects. Our key innovation is a sufficient treatment value assumption that imposes weak restrictions on selection heterogeneity and is implicit in separable threshold-crossing models, including monotone effects on selection. Our double debiased machine learning estimator enables nonparametric and high-dimensional methods, using covariates to tighten the bounds and capture heterogeneity. Applications to Job Corps and Civilian Conservation Corps (CCC) program evaluations reinforce prior findings under weaker assumptions.