The treatment effect, the cross difference, and the interaction term in nonlinear “difference-in-differences” models
Highlights
► I consider any (strictly monotonic) nonlinear “difference-in-differences” model. ► I show that the treatment effect is the difference between two cross differences (cd). ► These are the cd of the observed and the cd of the potential non-treatment outcome. ► This equals the incremental effect of the interaction term coefficient.
Introduction
Difference-in-differences estimation is one of the most important identification strategies in applied economics (Angrist and Krueger, 1999, Athey and Imbens, 2006, Bertrand et al., 2004, Blundell and Costa Dias, 2009, Heckman et al., 1999, Lechner, 2011, Meyer, 1995). Especially when using micro data, the dependent variable may be binary or censored. One possibility to address the resulting nonlinearity in the conditional expectation of the outcome is to transfer the difference-in-differences identification strategy to the latent variable in nonlinear models like probit, logit or tobit. The latent linear index then contains an interaction term that is the product of the group and time indicators, the difference-in-differences usually referring to a difference in the differences between groups across time. Ai and Norton (2003) derive the cross difference (or derivative) in a nonlinear model such as probit, and demonstrate that this cross difference (or derivative) need not have the same sign as the coefficient of the interaction term in the nonlinear model.
In this note, I show that in the case of a nonlinear “difference-in-differences” model, the treatment effect, i.e. the parameters of interest, is not a simple cross difference, but a difference between cross differences: it 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. This difference in cross differences simplifies to the incremental effect of the coefficient of the interaction term, so that the treatment effect has the same sign as the coefficient of the interaction effect.
There are many cases in empirical economics where the simple cross difference/derivative in nonlinear models with interaction term, as provided in Ai and Norton (2003), is directly the parameter of interest. Examples include medical or economic applications where all three of the following variables indicate a form of treatment (and are possibly randomly allocated to be interpreted as causal): dose of drug one, dose of drug two and the interaction of the two doses.
In a difference-in-differences model, however, only a single variable indicates treatment: the interaction between the treatment group and the treatment period dummy variables. This is the key to understanding why a simple cross difference is not equal to the treatment effect in a nonlinear “difference-in-differences” model.
Section snippets
Difference-in-differences in the linear model
I start by presenting the commonly known linear difference-in-differences model for a continuous and uncensored outcome. Using a similar potential outcome setup as in Athey and Imbens (2006), define the treatment effect, for simplicity assumed to be constant across the population and across time, by where and are potential outcomes with and without treatment, respectively. and are binary time period and group indicators, which are coded as either 0
Conclusions
Difference-in-differences models constitute a common identification strategy in empirical economics and are often implemented using an interaction term between time and group indicators whose coefficient describes the difference over time in the outcome variable between the differences between the two groups (in the simplest model with two groups and two time periods). In the nonlinear case, one common way to implement this “difference-in-differences” approach is to apply the assumptions made
Acknowledgments
This project was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft) under the project Labor Market Effects of Social Policy (Arbeitsmarkteffekte sozialpolitischer Maßnahmen). Part of this work was done during my visit to CLE, University of California at Berkeley, whose hospitality is gratefully acknowledged. I thank two anonymous referees for helpful comments.
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