Robustify and Tighten the Lee Bounds: A Sample Selection Model under Stochastic Monotonicity and Symmetry Assumptions

In the presence of sample selection, Lee's (2009) nonparametric bounds are a popular tool for estimating a treatment effect. However, the Lee bounds rely on the monotonicity assumption, whose empirical validity is sometimes unclear. Furthermore, the bounds are often regarded to be wide and less informative even under monotonicity. To address these issues, this study introduces a stochastic version of the monotonicity assumption alongside a nonparametric distributional shape constraint. The former enhances the robustness of the Lee bounds with respect to monotonicity, while the latter helps tighten these bounds. The obtained bounds do not rely on the exclusion restriction and can be root-$n$ consistently estimable, making them practically viable. The potential usefulness of the proposed methods is illustrated by their application on experimental data from the after-school instruction programme studied by Muralidharan, Singh, and Ganimian (2019).