Non-Linear Treatment Effects, Distributions and Interactions

This paper examines critical aspects of analyzing interactions with continuous treatment variables. A theoretical estimand is defined, the Average Interactive Effects, which is the mean of the difference in the slopes of the treatment variable at different levels of the moderator. Crucially, this theoretical estimand is distinct from the difference in Conditional Average Marginal Effects (CAME) at different levels of the moderator. The paper proposes a flexible parametric model that can be used to estimate three important pieces of information: i) predicted values at different levels of the moderator; ii) the difference in the slopes of the relationship between the treatment and the dependent variable at different values of the moderator; and iii) the mean of these slope differences. Simulations show that this approach is better than models that only estimatethe differences in the CAMEs. However, since the unbiasedness of the proposed method depends on the correct specification of the functional form, it is proposed to complement the parametric model with results from Generalized Additive Models. Finally, the effectiveness of the proposed approach is illustrated by two examples.