Interaction and marginal effects in nonlinear models : case of ordered logit and probit models

Interaction and marginal effects are often an important concern, especially when variables are allowed to interact in a nonlinear model. In a linear model, the interaction term, representing the interaction effect, is the impact of a variable on the marginal effect of another variable. In a nonlinear model, however, the marginal effect of the interaction term is different from the interaction effect. This report provides a general derivation of both effects in a nonlinear model and a linear model to clearly illustrate the difference. These differences are then demonstrated with empirical data. The empirical study shows that the corrected interaction effect in an ordered logit or probit model is substantially different from the incorrect interaction effect produced by the margins command in Stata. Based on the correct formulas, this report verifies that the interaction effect is not the same as the marginal effect of the interaction term. Moreover, we must be careful when interpreting the nonlinear models with interaction terms in Stata or any other statistical software package.