Inference on Counterfactual Distributions


  • Victor Chernozhukov,

    1. Dept. of Economics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.;;
    Search for more papers by this author
  • Iván Fernández-Val,

    1. Dept. of Economics, Boston University, Boston, MA 02215, U.S.A.;;
    Search for more papers by this author
  • Blaise Melly

    1. Dept. of Economics, Bern University, 3001 Bern, Switzerland;;
    Search for more papers by this author
    • This paper replaces the earlier independent projects started in 2005, “Inference on Counterfactual Distributions Using Conditional Quantile Models” by Chernozhukov and Fernández-Val, and “Estimation of Counterfactual Distributions Using Quantile Regression” by Melly. We would like to thank the co-editors, five anonymous referees, Isaiah Andrews, Josh Angrist, Manuel Arellano, David Autor, Alexandre Belloni, Moshe Buchinsky, Arun Chandrasekhar, Mingli Chen, Denis Chetverikov, Flavio Cunha, Brigham Frandsen, Jerry Hausman, James Heckman, Michael Jansson, Kengo Kato, Roger Koenker, Joonhwan Lee, Ye Luo, Pierre-Andre Maugis, Justin McCrary, Miikka Rokkanen, and seminar participants at Banff International Research Station Conference on Semiparametric and Nonparametric Methods in Econometrics, Berkeley, Boston University, CEMFI, Columbia, Harvard/MIT, Michigan, MIT, Ohio State, St. Gallen, and the 2008 Winter Econometric Society Meetings for very useful comments that helped improve the paper. Companion software developed by the authors (counterfactual packages for Stata and R) is available from the authors' web sites. We gratefully acknowledge research support from the National Science Foundation.


Counterfactual distributions are important ingredients for policy analysis and decomposition analysis in empirical economics. In this article, we develop modeling and inference tools for counterfactual distributions based on regression methods. The counterfactual scenarios that we consider consist of ceteris paribus changes in either the distribution of covariates related to the outcome of interest or the conditional distribution of the outcome given covariates. For either of these scenarios, we derive joint functional central limit theorems and bootstrap validity results for regression-based estimators of the status quo and counterfactual outcome distributions. These results allow us to construct simultaneous confidence sets for function-valued effects of the counterfactual changes, including the effects on the entire distribution and quantile functions of the outcome as well as on related functionals. These confidence sets can be used to test functional hypotheses such as no-effect, positive effect, or stochastic dominance. Our theory applies to general counterfactual changes and covers the main regression methods including classical, quantile, duration, and distribution regressions. We illustrate the results with an empirical application to wage decompositions using data for the United States.

As a part of developing the main results, we introduce distribution regression as a comprehensive and flexible tool for modeling and estimating the entire conditional distribution. We show that distribution regression encompasses the Cox duration regression and represents a useful alternative to quantile regression. We establish functional central limit theorems and bootstrap validity results for the empirical distribution regression process and various related functionals.