Bounds on functionals of the distribution 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 if 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 the Makarov bounds on a functional of the c.d.f. of the individual treatment effect distribution are not best possible.

Water for life : the impact of the privatization of water services on child mortality

While most countries are committed to increasing access to safe water and thereby reducing child mortality, there is little consensus on how to actually improve water services. One important proposal under discussion is whether to privatize water provision. In the 1990s Argentina embarked on one of the largest privatization campaigns in the world including the privatization of local water companies covering approximately 30 percent of the country’s municipalities. Using the varia tion in ownership of water provision across time and space generated by the privatization process, we find that child mortality fell 8 percent in the areas that privatized their water services; and that the effect was largest (26 percent) in the poorest areas. We check the robustness of these estimates using cause specific mortality. While privatization is associated with significant reductions in deaths from infectious and parasitic diseases, it is uncorrelated with deaths from causes unrelated to water conditions.