Causal Interaction in High-Dimension ∗

Estimating causal interaction effects is essential for the exploration of heterogeneous treatment effects. In the presence of multiple treatment variables with each having several levels, researchers are often interested in identifying the combinations of treatments that induce large additional causal effects beyond the sum of separate effects attributable to each treatment. We show, however, the standard definition of causal interaction effect, typically estimated with the standard linear regression or ANOVA, suffers from the lack of invariance to the choice of baseline condition and the difficulty of interpretation beyond two-way interaction. We propose an alternative definition of causal interaction effect, called the marginal treatment interaction effect, whose relative magnitude does not depend on the choice of baseline condition while maintaining an intuitive interpretation even for higher-order interaction. The proposed approach enables researchers to effectively summarize the structure of causal interaction in high-dimension by decomposing the total effect of any treatment combination into the marginal effects and the interaction effects. We also establish the identification condition and develop an estimation strategy for the proposed marginal treatment interaction effects. Our motivating example is conjoint analysis where the existing literature largely assumes the absence of causal interaction. Given a large number of interaction effects, we apply a variable selection method to identify significant causal interaction. Our exploratory analysis of a survey experiment on immigration preferences reveals substantive insights the standard conjoint analysis fails to discover.