Interpretation of nonlinear difference-in-differences: the role of the parallel trends assumption

I argue interpretation of nonlinear difference-in-differences models depends on the form of the parallel trends assumption. When they are assumed in the natural scale of the dependent variable, the treatment effect is the interaction effect (a cross-difference). If they are assumed in the transformed scale, it is a single difference. I further note that assuming parallel trends in one scale implies they do not hold in the other, except in special cases. Finally, I consider log-linear (and related) difference-in-differences models and provide a constant form of the treatment effect that is comparable across applications with different parallel trends assumptions.