Identification and Inference in Nonlinear Difference-in-Differences Models


  • Susan Athey,

    1. Dept. of Economics, Stanford University, Stanford, CA 94305-6072, U.S.A., and National Bureau of Economic Research;;
      Dept. of Economics and Dept. of Agricultural and Resource Economics, University of California Berkeley, Berkeley, CA 94720-3880, U.S.A., and National Bureau of Economic Research;;
    Search for more papers by this author
  • Guido W. Imbens

    Search for more papers by this author
    • We are grateful to Alberto Abadie, Joseph Altonji, Don Andrews, Joshua Angrist, David Card, Esther Duflo, Austan Goolsbee, Jinyong Hahn, Caroline Hoxby, Rosa Matzkin, Costas Meghir, Jim Poterba, Scott Stern, Petra Todd, Edward Vytlacil, seminar audiences at the University of Arizona, UC Berkeley, the University of Chicago, University of Miami, Monash University, Harvard/MIT, Northwestern University, UCLA, USC, Yale University, Stanford University, the San Francisco Federal Reserve Bank, the Texas Econometrics conference, SITE, NBER, and AEA 2003 winter meetings, the 2003 Joint Statistical Meetings, and, especially, Jack Porter for helpful discussions. We are indebted to Bruce Meyer, who generously provided us with his data. Four anonymous referees and a co-editor provided insightful comments. Richard Crump, Derek Gurney, Lu Han, Khartik Kalyanaram, Peyron Law, Matthew Osborne, Leonardo Rezende, and Paul Riskind provided skillful research assistance. Financial support for this research was generously provided through NSF grants SES-9983820 and SES-0351500 (Athey), SBR-9818644, and SES 0136789 (Imbens).


This paper develops a generalization of the widely used difference-in-differences method for evaluating the effects of policy changes. We propose a model that allows the control and treatment groups to have different average benefits from the treatment. The assumptions of the proposed model are invariant to the scaling of the outcome. We provide conditions under which the model is nonparametrically identified and propose an estimator that can be applied using either repeated cross section or panel data. Our approach provides an estimate of the entire counterfactual distribution of outcomes that would have been experienced by the treatment group in the absence of the treatment and likewise for the untreated group in the presence of the treatment. Thus, it enables the evaluation of policy interventions according to criteria such as a mean–variance trade-off. We also propose methods for inference, showing that our estimator for the average treatment effect is root-N consistent and asymptotically normal. We consider extensions to allow for covariates, discrete dependent variables, and multiple groups and time periods.