Lee Bounds with a Continuous Treatment in Sample Selection

TL;DR

This paper extends Lee bounds to continuous treatments in sample selection models, introducing a new assumption and a double debiased machine learning estimator to improve causal inference with high-dimensional data.

Contribution

It generalizes Lee bounds for multivalued treatments, proposes a novel treatment value assumption, and develops a flexible estimator for better bounds tightening.

Findings

Applied to Job Corps and CCC evaluations, confirming prior results with weaker assumptions.

Demonstrated the estimator's ability to incorporate covariates and heterogeneity.

Showed improved bounds in empirical applications.

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.