Non-linear Treatment Effects, Distributions and Interactions

• In Event: Poster Session: Posters in Political Methodology

Abstract

Many relationships studied in political science are conditional in nature. For example, it could be that higher inequality affects redistribution preferences, but that this effect is conditional on income levels (Dimick et al. 2016). These heterogeneous effects may indeed allow researchers to investigate the mechanisms: The share of immigrants in the workplace makes attitudes towards immigration more positive, and finding that the effect is solely due to same-skill relations in small workplaces, we can understand that this could be due to contact between natives and immigrants (Andersson and Dehdari,
2021). However, the question of which is the estimand of interest when studying interactions remains understudied.

This paper addresses critical issues in analyzing interactions with continuous treatment variables. First, a theoretical estimand of interest is proposed. Briefly, it is argued that applied researchers are interested in the average differential effect of the treatment variable at different levels of the moderator, holding all other factors constant, implying that comparisons should be made at similar levels of the treatment variable. This estimand is compared with the common practice in the literature, which is to estimate the average marginal effects (AME) of the treatment at different levels of the moderator. It is shown that the difference in AME may be a biased estimand of the interactive effects when there are nonlinear treatment effects and the distribution of the treatment is not independent of the moderator.

I propose two solutions that account for each of these dimensions (i.e., the nonlinearity of the treatment effects and the different distribution of the treatment variable). First, I propose to use a reweighting scheme so that the distribution of the treatment at different levels of the moderator is equal to the overall distribution of the treatment variable. Second, I estimate a non-linear model and compute the interactive effects according to the definition of the theoretical estimand of interest. These approaches are compared with other approaches. First, it is shown that it is not just a matter of misspecifying the models: regressing a non-linear model but retrieving the average marginal effects from it would lead to biased estimates of the interactive effects. Moreover, the paper takes the opposite position to new accounts that analyze non-linear models such as logit and propose to calculate the AME at different levels of the moderator, taking into account that the distribution of controls may be different (Zhirnov et al., 2023). Finally, it is shown that a simple discretization of the treatment and the moderator does not solve the problem, as the comparisons between groups are not clear when the distribution of the treatment variable is different at different levels of the moderator.

To illustrate the usefulness of these approaches, simulations are conducted in which the relationship between the treatment variable and the dependent variables follows different polynomial relationships and the distribution of the treatment variable is different at different terciles of the moderator. Further, empirical applications are presented.

Author

• Josep Serrano-Serrat, Universidad Carlos III de Madrid