Bounds on functionals of the distribution treatment effects

Abstract

Bounds on the distribution function of the sum of two random variableswith known marginal distributions obtained by Makarov (1981) canbe used to bound the cumulative distribution function (c.d.f.) of individualtreatment effects. Identification of the distribution of individualtreatment effects is important for policy purposes if we are interested infunctionals of that distribution, such as the proportion of individuals whogain from the treatment and the expected gain from the treatment forthese individuals. Makarov bounds on the c.d.f. of the individual treatmenteffect distribution are pointwise sharp, i.e. they cannot be improvedin any single point of the distribution. We show that the Makarov boundsare not uniformly sharp. Specifically, we show that the Makarov boundson the region that contains the c.d.f. of the treatment effect distributionin two (or more) points can be improved, and we derive the smallest setfor the c.d.f. of the treatment effect distribution in two (or more) points.An implication is that the Makarov bounds on a functional of the c.d.f.of the individual treatment effect distribution are not best possible.