In this paper, we establish sharp bounds on the joint distribution of potential outcomes and the distribution of treatment effects in parametric switching regime models with normal mean-variance mixture errors and in the semi-parametric switching regime models of Heckman (1990). Our results for parametric switching regime models with normal mean-variance mixture errors extend some existing results for the Gaussian switching regime model and our results for semi-parametric switching regime 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. The important issue of inference is briefly discussed.

By using nonparametric kernel estimation method and a central limit theorem for degenerate U-statistics of order higher than two, the authors develop several consistent model specification tests in the context of a nonparametric regression model. These include tests for omitted variables, tests for a partially linear model, and tests for a semiparametric single index model. The asymptotic normality of the test statistics are established under the respective null hypotheses and consistent estimators of the asymptotic variances are provided. Copyright 1996 by The Econometric Society.

We consider the identification of counterfactual distributions and treatment effects when the outcome variables and conditioning covariates are observed in separate datasets. Under the standard selection on observables assumption, the counterfactual distributions and treatment effect parameters are no longer point identified. However, applying the classical monotone re-arrangement inequality, we derive sharp bounds on the counterfactual distributions and policy parameters of interest.

The Lorenz curve and Gini index are standard tools for the evaluation of inequality in one dimension. However, inequality is inherently multi-dimensional. Extending the Lorenz curve and Gini index to a multidimensional context has proved controversial. This paper proposes a new multivariate extension based on multivariate rearrangements of optimal transport theory, which shares many of the desirable properties of their univariate counterparts. In particular, the corresponding multivariate inequality ordering is equivalent to preference by any social planner with inequality averse multivariate rank dependent social evaluation functional.