Violation of Distributional Assumptions in Latent Interaction Models

Abstract

Violating the distributional assumptions of latent interaction models can lead to biased estimates and invalid inference. However, no statistical procedure is readily available to verify whether these assumptions hold. We develop several specification tests contrasting consistent latent interaction estimators—which are robust to violations of distributional assumptions (i.e., Extended Unconstrained Indicator Approach and Model-Implied Instrumental Variables method)—to an efficient estimator (i.e., Latent Moderated Structural Equations), which is inconsistent under non-normally distributed linear latent variables. We compare these estimators under a variety of conditions. The robust Hausman test we propose works well in identifying misspecifications due to violations of distributional assumptions of the latent variables. Moreover, our results indicate that the Latent Moderated Structural Equations method is severely bias under non-ideal conditions. Thus, it should not be used as the default approach and its results should always be compared to consistent estimators that are robust to distributional assumptions.