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 outcomes and their observability (e.g., employment or survey responses). We generalized the widely used Lee (2009)'s bounds for binary treatment effects. Our key innovation is a sufficient treatment values 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 CCC program evaluations reinforce prior findings under weaker assumptions.