Interpretation of Nonlinear Difference-in-Differences : The Role of the Parallel Trends Assumption

I discuss nonlinear difference-in-differences models, arguing their interpretation depends on the context of their application. When parallel trends are assumed in the natural scale of the dependent variable, I contend the treatment effect is the interaction effect (a cross-difference), while if parallel trends are assumed in the transformed scale, it is a single difference. Thus, interpretation is driven by the form of the parallel trends assumption. I further note that assuming parallel trends in one scale implies they do not hold in the other, except in special cases. Therefore, researchers should be careful about inadvertently making inconsistent assumptions when mixing models that imply different forms of parallel trends. Finally, I consider the special case of the log-linear model, providing a form of the treatment effect in percentage terms that is constant, easily calculated, and valid to compare across applications that might assume parallel trends in different scales