Bounds on Functionals of the Distribution of Treatment Effects

42 Pages Posted: 20 Oct 2008

See all articles by Sergio Firpo

Sergio Firpo

Insper

Geert Ridder

University of Southern California

Date Written: September 22, 2008

Abstract

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.

Keywords: Treatment effects, Bounds, Social welfare

JEL Classification: C31

Suggested Citation

Firpo, Sergio and Ridder, Geert, Bounds on Functionals of the Distribution of Treatment Effects (September 22, 2008). IEPR Working Paper No. 08-09. Available at SSRN: https://ssrn.com/abstract=1284116 or http://dx.doi.org/10.2139/ssrn.1284116

Sergio Firpo (Contact Author)

Insper ( email )

R Quata 300
Sao Paulo, 04542-030
Brazil

HOME PAGE: http://https://sites.google.com/site/sergiopfirpo/

Geert Ridder

University of Southern California ( email )

Kaprielian Hall
Los Angeles, CA 90089
United States
213-740-2110 (Phone)
213-740-8543 (Fax)

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