In this paper, we investigate what can be learned about average counterfactual outcomes when it is assumed that treatment response functions are smooth. The smoothness conditions in this paper amount to assuming that the differences in average counterfactual outcomes are bounded under different...

We analyze identification of nonseparable models under three kinds of exogeneity assumptions weaker than full statistical independence. The first is based on quantile independence. Selection on unobservables drives deviations from full independence. We show that such deviations based on quantile...

A breakdown frontier is the boundary between the set of assumptions which lead to a specific conclusion and those which do not. In a potential outcomes model with a binary treatment, we consider two conclusions: First, that ATE is at least a specific value (e.g., nonnegative) and second that the...

This paper studies measuring various average effects of X on Y in general structural systems with unobserved confounders U, a potential instrument Z, and a proxy W for U. We do not require X or Z to be exogenous given the covariates or W to be a perfect one‐to‐one mapping of U. We study the...

This paper analyzes partial identification of parameters that measure a distribution’s spread, for example, the variance, Gini coefficient, entropy, or interquartile range. The core results are tight, two-dimensional identification regions for the expectation and variance, the median and...