Bounds on Functionals of the Distribution of Treatment Effects

Bounds on the distribution function of the sum of two random variables with known marginal distributions obtained by Makarov (1981) can be used to bound the cumulative distribution function (c.d.f.) of individual treatment effects. Identification of the distribution of individual treatment effects is important for policy purposes when we are interested in functionals of that distribution, such as the proportion of individuals who gain from the treatment and the expected gain from the treatment for these individuals. Makarov bounds on the c.d.f. of the individual treatment effect distribution are pointwise sharp, i.e. they cannot be improved in any single point of the distribution. We show that the Makarov bounds are not uniformly sharp. Specifically, we show that the Makarov bounds on the region that contains the c.d.f. of the treatment effect distribution in two (or more) points can be improved, and we derive the smallest set for the c.d.f. of the treatment effect distribution in two (or more) points. An implication is that if we want to bound a functional of the c.d.f. of the individual treatment effect distribution, such as the expected gain for those who benefit from the treatment, then the bounds of this functional on the set of c.d.f. that are within the Makarov bounds cannot be sharp, because they are smaller and larger, respectively, than the smallest and largest values on the set of c.d.f. within the improved bounds that we derive