Bounding the Distribution of Treatment Effects∗

We use bounds on the distribution function of the sum of two random variables with known marginal distributions obtained by Makarov (1981) to bound the distribution function of the individual treatment effect. Although the Makarov bounds are pointwise sharp if we only consider the marginals, they can be improved by using the fact that we know the mean (and only the mean) of the treatment effect distribution, i.e. the Average Treatment Effect. We propose a procedure that uses this additional information to obtain more informative bounds. We also show that if the treatment effect varies with observable covariates, averaging over these covariates improves the Makarov bounds. We can combine the two types of additional information by using the known conditional treatment effect means to improve the conditional Makarov bounds and by subsequently averaging these conditional bounds. The (improved) Makarov bounds on the cdf of the treatment effect distribution yield bounds on the quantiles of that distribution. Without covariates we can use the ATE to directly obtain bounds on these quantiles, that turn out to be identical to the bounds obtained by using the ATE to improve the cdf bounds. By this equivalence we can use the average improved conditional Makarov bounds that use information on the conditional ATE to improve the bounds on the unconditional quantiles. Bounds on the conditional quantiles cannot be averaged to obtain bounds on the unconditional quantiles. We illustrate the qualitative features of the bounds for normal and dichotomous outcome distributions.