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Fuzzy differences-in-differences

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  • Clément de Chaisemartin

    () (Institute for Fiscal Studies and Warwick University)

  • Xavier d'Haultfoeuille

    (Institute for Fiscal Studies)

Abstract

In many applications of the differences-in-differences (DID) method, the treatment increases more in the treatment group, but some units are also treated in the control group. In such fuzzy designs, a popular estimator of treatment effects is the DID of the outcome divided by the DID of the treatment, or OLS and 2SLS regressions with time and group fixed effects estimating weighted averages of this ratio across groups. We start by showing that when the treatment also increases in the control group, this ratio estimates a causal effect only if treatment effects are homogenous in the two groups. Even when the distribution of treatment is stable, it requires that treatment effects be constant over time. As this assumption is not always applicable, we propose two alternative estimators. The first estimator relies on a generalization of common trends assumptions to fuzzy designs, while the second extends the changes-in-changes estimator of Athey & Imbens (2006). When the distribution of treatment changes in the control group, treatment effects are partially identified. Finally, we prove that our estimators are asymptotically normal and use them to revisit applied papers using fuzzy designs.

Suggested Citation

  • Clément de Chaisemartin & Xavier d'Haultfoeuille, 2015. "Fuzzy differences-in-differences," CeMMAP working papers CWP69/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    • Handle: RePEc:ifs:cemmap:69/15
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    1. Mirko Draca & Stephen Machin & Robert Witt, 2011. "Panic on the Streets of London: Police, Crime, and the July 2005 Terror Attacks," American Economic Review, American Economic Association, vol. 101(5), pages 2157-2181, August.
    2. Victor Chernozhukov & Iván Fernández‐Val & Blaise Melly, 2013. "Inference on Counterfactual Distributions," Econometrica, Econometric Society, vol. 81(6), pages 2205-2268, November.
    3. Xiaohong Chen & Oliver Linton & Ingrid Van Keilegom, 2003. "Estimation of Semiparametric Models when the Criterion Function Is Not Smooth," Econometrica, Econometric Society, vol. 71(5), pages 1591-1608, September.
    4. Katz, Lawrence F. & Autor, David H., 1999. "Changes in the wage structure and earnings inequality," Handbook of Labor Economics,in: O. Ashenfelter & D. Card (ed.), Handbook of Labor Economics, edition 1, volume 3, chapter 26, pages 1463-1555 Elsevier.
    5. Horowitz, Joel L & Manski, Charles F, 1995. "Identification and Robustness with Contaminated and Corrupted Data," Econometrica, Econometric Society, vol. 63(2), pages 281-302, March.
    6. Victor Chernozhukov & Iv·n Fern·ndez-Val & Alfred Galichon, 2010. "Quantile and Probability Curves Without Crossing," Econometrica, Econometric Society, vol. 78(3), pages 1093-1125, May.
    7. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731, Fall.
    8. David S. Lee, 2009. "Training, Wages, and Sample Selection: Estimating Sharp Bounds on Treatment Effects," Review of Economic Studies, Oxford University Press, vol. 76(3), pages 1071-1102.
    9. Newey, Whitney K., 1997. "Convergence rates and asymptotic normality for series estimators," Journal of Econometrics, Elsevier, vol. 79(1), pages 147-168, July.
    10. Victor Chernozhukov & Sokbae Lee & Adam M. Rosen, 2013. "Intersection Bounds: Estimation and Inference," Econometrica, Econometric Society, vol. 81(2), pages 667-737, March.
    11. Donald W. K. Andrews & Panle Jia Barwick, 2012. "Inference for Parameters Defined by Moment Inequalities: A Recommended Moment Selection Procedure," Econometrica, Econometric Society, vol. 80(6), pages 2805-2826, November.
    12. Joshua Angrist & Victor Chernozhukov & Iván Fernández-Val, 2006. "Quantile Regression under Misspecification, with an Application to the U.S. Wage Structure," Econometrica, Econometric Society, vol. 74(2), pages 539-563, March.
    13. Joshua D. Angrist & Kathryn Graddy & Guido W. Imbens, 2000. "The Interpretation of Instrumental Variables Estimators in Simultaneous Equations Models with an Application to the Demand for Fish," Review of Economic Studies, Oxford University Press, vol. 67(3), pages 499-527.
    14. Newey, Whitney K, 1994. "The Asymptotic Variance of Semiparametric Estimators," Econometrica, Econometric Society, vol. 62(6), pages 1349-1382, November.
    15. Ruben Enikolopov & Maria Petrova & Ekaterina Zhuravskaya, 2011. "Media and Political Persuasion: Evidence from Russia," American Economic Review, American Economic Association, vol. 101(7), pages 3253-3285, December.
    16. Marianne Bertrand & Esther Duflo & Sendhil Mullainathan, 2004. "How Much Should We Trust Differences-In-Differences Estimates?," The Quarterly Journal of Economics, Oxford University Press, vol. 119(1), pages 249-275.
    17. Clément De Chaisemartin, 2012. "Fuzzy differences in differences," PSE Working Papers halshs-00671368, HAL.
    18. Joseph P. Romano & Azeem M. Shaikh & Michael Wolf, 2014. "A Practical Two‐Step Method for Testing Moment Inequalities," Econometrica, Econometric Society, vol. 82, pages 1979-2002, September.
    19. de Chaisemartin, Clement & D'Haultfoeuille, Xavier, 2014. "Fuzzy Changes-in Changes," CAGE Online Working Paper Series 184, Competitive Advantage in the Global Economy (CAGE).
    20. Field, Erica Marie, 2005. "Property Rights and Investment in Urban Slums," Scholarly Articles 3634150, Harvard University Department of Economics.
    21. Richard Blundell & Monica Costa Dias & Costas Meghir & John Van Reenen, 2004. "Evaluating the Employment Impact of a Mandatory Job Search Program," Journal of the European Economic Association, MIT Press, vol. 2(4), pages 569-606, June.
    22. Erica Field, 2007. "Entitled to Work: Urban Property Rights and Labor Supply in Peru," The Quarterly Journal of Economics, Oxford University Press, vol. 122(4), pages 1561-1602.
    23. repec:spo:wpecon:info:hdl:2441/5rkqqmvrn4tl22s9mc4b6ga2g is not listed on IDEAS
    24. Alberto Abadie & Matthew M. Chingos & Martin R. West, 2013. "Endogenous Stratification in Randomized Experiments," NBER Working Papers 19742, National Bureau of Economic Research, Inc.
    25. Erica Field, 2005. "Property Rights and Investment in Urban Slums," Journal of the European Economic Association, MIT Press, vol. 3(2-3), pages 279-290, 04/05.
    26. Heckman, James J. & Robb, Richard Jr., 1985. "Alternative methods for evaluating the impact of interventions : An overview," Journal of Econometrics, Elsevier, vol. 30(1-2), pages 239-267.
    27. Guido W. Imbens & Donald B. Rubin, 1997. "Estimating Outcome Distributions for Compliers in Instrumental Variables Models," Review of Economic Studies, Oxford University Press, vol. 64(4), pages 555-574.
    28. Edward Vytlacil, 2002. "Independence, Monotonicity, and Latent Index Models: An Equivalence Result," Econometrica, Econometric Society, vol. 70(1), pages 331-341, January.
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    More about this item

    JEL classification:

    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Spatio-temporal Models

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