Horowitz-Manski-Lee Bounds with Multilayered Sample Selection

This paper studies partial identification of treatment effects in the presence of sample selection, where treatment affects both selection into the sample and sorting across layers with heterogeneous outcomes. We show that canonical Lee bounds identify a total effect that combines the within-layer causal effect of treatment with a sorting effect reflecting outcome differences across layers. We derive sharp bounds on the within-layer causal effect using a two-step approach that extends Horowitz and Manski (1995) to a system of mixture equations with cross-equation dependence. Further, we show that under additional restrictions, these within-layer effects are sufficient for welfare analysis. Two empirical applications to job training experiments illustrate the approach; our estimates show that even when Lee bounds are strictly positive, within-firm bounds can be tight around zero, suggesting that Lee bounds capture a pure sorting effect.