Partial identification of the treatment effect distribution and its functionals

摘要

In the treatment effect problem, the available information is on the marginal distributions of potential outcomes, but not on their joint distribution. The only point identified functional of the treatment effect distribution is its average, the average treatment effect (ATE). Quantiles and other functionals of the distribution of treatment effect are only partially identified. Bounds on a single quantile and on the cumulative distribution function (c.d.f.) in a single point are sharp (Makarov bounds). We show that bounds on functionals of the quantile process that use Makarov bounds are not sharp, because the Makarov bounds are pointwise, but not uniformly sharp. This allows us to propose improved bounds on functionals of the c.d.f. As an intermediate result, we find that the Makarov bounds on the region that contains the c.d.f. of the treatment effect distribution in a finite number of points can be improved. We provide numerical illustrations throughout the paper permitting a clear visualization of how the method works. (C) 2019 Elsevier B.V. All rights reserved.