The Consequences of Model Misspecification for the Estimation of Nonlinear Interaction Effects

Abstract

Recent research has shown that interaction effects may often be nonlinear (Hainmueller, Mummolo, and Xu [2019, Political Analysis 27, 163–192]). As standard interaction effect specifications assume a linear interaction effect, that is, the moderator conditions the effect at a constant rate, this can lead to bias. However, allowing nonlinear interaction effects, without accounting for other nonlinearities and nonlinear interaction effects, can also lead to biased estimates. Specifically, researchers can infer nonlinear interaction effects, even though the true interaction effect is linear, when variables used for covariate adjustment that are correlated with the moderator have a nonlinear effect upon the outcome of interest. We illustrate this bias with simulations and show how diagnostic tools recommended in the literature are unable to uncover the issue. We show how using the adaptive Lasso to identify relevant nonlinearities among variables used for covariate adjustment can avoid this issue. Moreover, the use of regularized estimators, which allow for a fuller set of nonlinearities, both independent and interactive, is more generally shown to avoid this bias and more general forms of omitted interaction bias.

Systematic mapping of climate and environmental framing experiments and re-analysis with computational methods points to omitted interaction bias

“…In this section, we therefore outline descriptive and causal inference in the exploration of sub-group effects and the fundamental challenges to inference through the example of omitted interaction bias. We also examine how incorporating additional interaction effects in a manner suggested by previous research [38,44,45,55] affects the estimates of previously published subgroup effects. Finally, we reflect on these challenges in the pursuit of causal inference for sub-group effects, while also highlighting recent work that allows for more principled descriptive examination of sub-group effects.…”

“…One example of a threat to causal inference from considering sub-group effects in isolation is omitted interaction bias [38,44,45,55]. Omitted interaction bias occurs where differences between the sub-groups on other characteristics, such as age, education, and income, also result in heterogeneous treatment effects that are left unmodelled, which are absorbed by the included interaction effect.…”

“…Recent research points towards the use of machine learning in estimating such heterogeneous framing effects across population sub-group effects and prevent omitted interaction bias [38,44,45,55] (for further details, see Methods). With uncertainty about the true data generation process, the number of relevant heterogeneous treatment effects and conditional effects amongst relevant covariates results in large numbers of parameters to be estimated.…”