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

10 Pages Posted: 23 May 2008

See all articles by Patrick A. Puhani

Patrick A. Puhani

Leibniz Universität Hannover; University of St. Gallen - Swiss Institute for International Economics and Applied Economic Research; IZA Institute of Labor Economics; Université Paris II - Panthéon-Assas

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.

Keywords: identification, nonlinear models, limited dependent variable, probit, logit, tobit, difference-in-differences, interaction effect

JEL Classification: C21, C25, H0, I0, J0

Suggested Citation

Puhani, Patrick A., The Treatment Effect, the Cross Difference, and the Interaction Term in Nonlinear 'Difference-in-Differences' Models. IZA Working Paper No. 3478, Available at SSRN: https://ssrn.com/abstract=1136279 or http://dx.doi.org/10.2139/ssrn.1136279

Patrick A. Puhani (Contact Author)

Leibniz Universität Hannover ( email )

Institut für Arbeitsökonomik
Koenigsworther Platz 1
30167 Hannover, DE 30167
Germany

University of St. Gallen - Swiss Institute for International Economics and Applied Economic Research ( email )

Dufourstr. 48
St. Gallen, 9000
Switzerland
+41 71 224 2341 (Phone)
+41 71 224 2298 (Fax)

IZA Institute of Labor Economics

P.O. Box 7240
Bonn, D-53072
Germany

Université Paris II - Panthéon-Assas

12 place du Pantheon
Paris cedex 06, 75231
France

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