The Consequences of Model Misspecification for the Estimation of Non-Linear Interaction Effects

Recent research has shown that interaction effects may often be non-linear (Hainmueller, Mummolo and Xu, 2019). As standard interaction effect specifications assume a linear interaction effect, i.e. the moderator conditions the effect at a constant rate, this can lead to bias. However, allowing non-linear interaction effects, without accounting for other non-linearities and non-linear interaction effects, can also lead to biased estimates. Specifically, researchers can infer non-linear interaction effects, even though the true interaction effect is linear, when variables used for covariate adjustment that are correlated with the moderator have a non-linear 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 non-linearities amongst variables used for covariate adjustment can avoid this issue. Moreover, the use of regularised estimators more generally, which allow for a fuller set of non-linearities, both independent and interactive, are shown to avoid this bias and more general forms of omitted interaction bias.