Robust Estimation and Inference in Categorical Data

Abstract

In empirical science, many variables of interest are categorical. Like any model, models for categorical responses can be misspecified, leading to possibly large biases in estimation. One particularly troublesome source of misspecification is inattentive responding in questionnaires, which is well-known to jeopardize the validity of structural equation models (SEMs) and other survey-based analyses. I propose a general estimator that is designed to be robust to misspecification of models for categorical responses. Unlike hitherto approaches, the estimator makes no assumption whatsoever on the degree, magnitude, or type of misspecification. The proposed estimator generalizes maximum likelihood estimation, is strongly consistent, asymptotically Gaussian, has the same time complexity as maximum likelihood, and can be applied to any model for categorical responses. In addition, I develop a novel test that tests whether a given response can be fitted well by the assumed model, which allows one to trace back possible sources of misspecification. I verify the attractive theoretical properties of the proposed methodology in Monte Carlo experiments, and demonstrate its practical usefulness in an empirical application on a SEM of personality traits, where I find compelling evidence for the presence of inattentive responding whose adverse effects the proposed estimator can withstand, unlike maximum likelihood.