Before revision:  I apply this result to estimate sharp nonparametric bounds on the average treatment effect in Lee (2008)'s model of endogenous selection and substantially tighten the bounds on this parameter in Angrist et al. (2006)'s empirical setting.I also apply this result to estimate sharp identified sets for two other parameters - a new parameter, called a partially linear predictor, and the average partial derivative when the outcome variable is recorded in intervals .
After revision:  I apply this result to estimate sharp nonparametric bounds on the average treatment effect in Lee (2008)'s endogenous selection model, I provide the asymptotic theory for the sharp (i.e., the tightest possible) bounds on the Average Treatment Effect in the presence of high-dimensional covariates. Furthermore, I relax the conventional monotonicity assumption and allow the sign of the treatment effect on the selection (e.g., employment) to be determined by covariates. Using JobCorps data set with very rich baseline characteristics, I substantially tighten the bounds on the JobCorps effect on wages under weakened monotonicity assumption .