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

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