Structured Semiparametric Estimation of Average Treatment Effects with Treatment-Specific Non-Gaussian Error Distributions

Abstract

This paper studies average treatment effect (ATE) estimation for continuous outcomes when the treatment-covariate mean is structured but the error distributions are unknown and may differ across treatment arms in scale, skewness, or tail behavior. We introduce a semiparametric model with a finite-dimensional mean, or a prespecified basis expansion, and separate unrestricted additive error laws for treated and control outcomes. The central theoretical contribution is the ATE-efficient influence function under two treatment-specific error nuisance spaces: it combines arm-specific regression-efficient scores with variation in the marginal covariate law and is distinct from both the unrestricted AIPW gradient and a pooled location-shift score. We establish the exact nested ordering of the efficiency bounds under pooled common-error, treatment-specific-error, and unrestricted causal models, including equality conditions. A local-misspecification result further shows that the variance reduction obtained by valid pooling equals the maximal squared first-order bias incurred along unit treatment-specific directions excluded by the pooled model. We develop efficient cross-fitted plug-in and ATE-targeted implementations requiring only two one-dimensional density estimates; the targeted refinement adds a scalar update. Simulations show substantial precision gains from valid pooling under common non-Gaussian errors and protection against invalid pooling when treatment changes error shape, including under weak positivity. Applications to ACTG175 and National Supported Work data illustrate the resulting precision--robustness tradeoff for irregular continuous outcomes.