Horowitz-Manski-Lee Bounds with Multilayered Sample Selection

This paper investigates the causal effect of job training on wage rates in the presence of firm heterogeneity. When training affects the sorting of workers to firms, sample selection is no longer binary but is ``multilayered". This paper extends the canonical Heckman (1979) sample selection model -- which assumes selection is binary -- to a setting where it is multilayered. In this setting Lee bounds set identifies a total effect that combines a weighted-average of the causal effect of job training on wage rates across firms with a weighted-average of the contrast in wages between different firms for a fixed level of training. Thus, Lee bounds set identifies a policy-relevant estimand only when firms pay homogeneous wages and/or when job training does not affect worker sorting across firms. We derive analytic expressions for sharp bounds for the causal effect of job training on wage rates at each firm that leverage information on firm-specific wages. We illustrate our partial identification approach with two empirical applications to job training experiments. Our estimates demonstrate that even when conventional Lee bounds are strictly positive, our within-firm bounds can be tight around 0, showing that the canonical Lee bounds may capture only a pure sorting effect of job training.