In this paper, we propose nonparametric estimators of sharp bounds on the distribution of treatment effects of a binary treatment and establish their asymptotic distributions. We note the possible failure of the standard bootstrap with the same sample size and apply the fewer-than-nbootstrap to making inferences on these bounds. The finite sample performances of the confidence intervals for the bounds based on normal critical values, the standard bootstrap, and the fewer-than-nbootstrap are investigated via a simulation study. Finally we establish sharp bounds on the treatment effect distribution when covariates are available.

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In this paper, we establish sharp bounds on the joint distribution of potential outcomes and the distribution of treatment effects in parametric switching regimes models with generalized hyperbolic errors and in the semiparametric switching regimes models of Heckman (1990). Our results for parametric switching regimes models with generalized hyperbolic errors extend some existing results for Gaussian switching regimes models and our results for semiparametric switching regimes models supplement the point identification results of Heckman (1990). Compared with the corresponding sharp bounds when selection is random, we observe that self selection tightens the bounds on the joint distribution of the potential outcomes and the distribution of treatment effects. These bounds depend on the identified model parameters only and can be easily estimated once the identified model parameters are estimated. We demonstrate the feasibility of inference on the distribution of treatment effects by constructing an asymptotically uniformly valid and non-conservative confidence set in a semiparametric switching regimes model.

We obtain criteria for cubical hyponormality, which provide a tool for finding examples being distinct the classes of 2-hyponormal, cubically hyponormal and quadratically hyponormal operators. ᮊ