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The Treatment Effect, the Cross Difference, and the Interaction Term in Nonlinear “Difference-in-Differences” Models

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  • Puhani, Patrick A.

    () (Leibniz University of Hannover)

Abstract

I demonstrate that Ai and Norton’s (2003) point about cross differences is not relevant for the estimation of the treatment effect in nonlinear “difference-in-differences” models such as probit, logit or tobit, because the cross difference is not equal to the treatment effect, which is the parameter of interest. In a nonlinear “difference-in-differences” model, the treatment effect is the cross difference of the conditional expectation of the observed outcome minus the cross difference of the conditional expectation of the potential outcome without treatment. Unlike in the linear model, the latter cross difference is not zero in the nonlinear model. It follows that the sign of the treatment effect in a nonlinear “difference-in-differences” model with a strictly monotonic transformation function is equal to the sign of the coefficient of the interaction term of the time and treatment group indicators. The treatment effect is simply the incremental effect of the coefficient of the interaction term.

Suggested Citation

  • Puhani, Patrick A., 2008. "The Treatment Effect, the Cross Difference, and the Interaction Term in Nonlinear “Difference-in-Differences” Models," IZA Discussion Papers 3478, Institute for the Study of Labor (IZA).
  • Handle: RePEc:iza:izadps:dp3478
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    References listed on IDEAS

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    1. Angrist, Joshua D. & Krueger, Alan B., 1999. "Empirical strategies in labor economics," Handbook of Labor Economics,in: O. Ashenfelter & D. Card (ed.), Handbook of Labor Economics, edition 1, volume 3, chapter 23, pages 1277-1366 Elsevier.
    2. repec:spr:portec:v:1:y:2002:i:2:d:10.1007_s10258-002-0010-3 is not listed on IDEAS
    3. Richard Blundell & Monica Costa Dias, 2009. "Alternative Approaches to Evaluation in Empirical Microeconomics," Journal of Human Resources, University of Wisconsin Press, vol. 44(3).
    4. Susan Athey & Guido W. Imbens, 2006. "Identification and Inference in Nonlinear Difference-in-Differences Models," Econometrica, Econometric Society, vol. 74(2), pages 431-497, March.
    5. Heckman, James J. & Lalonde, Robert J. & Smith, Jeffrey A., 1999. "The economics and econometrics of active labor market programs," Handbook of Labor Economics,in: O. Ashenfelter & D. Card (ed.), Handbook of Labor Economics, edition 1, volume 3, chapter 31, pages 1865-2097 Elsevier.
    6. 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.
    7. Meyer, Bruce D, 1995. "Natural and Quasi-experiments in Economics," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(2), pages 151-161, April.
    8. Lechner, Michael, 2011. "The Estimation of Causal Effects by Difference-in-Difference Methods," Foundations and Trends(R) in Econometrics, now publishers, vol. 4(3), pages 165-224, November.
    9. Ai, Chunrong & Norton, Edward C., 2003. "Interaction terms in logit and probit models," Economics Letters, Elsevier, vol. 80(1), pages 123-129, July.
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    More about this item

    Keywords

    interaction effect; probit; limited dependent variable; nonlinear models; identification; tobit; logit; difference-in-differences;

    JEL classification:

    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C25 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions; Probabilities
    • H0 - Public Economics - - General
    • I0 - Health, Education, and Welfare - - General
    • J0 - Labor and Demographic Economics - - General

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