Misspecified regressions with mixed regressors: robust inference and causal interpretation

2026-07-10

AI Summary

This study addresses the challenge that existing statistical methods struggle to effectively estimate average treatment effects in experiments involving both randomly assigned treatments and fixed covariates. The authors develop a unified theoretical framework that, for the first time, establishes a general estimating equation theory for misspecified linear regression models with mixed regressors—combining random treatment indicators and fixed covariates—and extends this framework to clustered data settings. By integrating estimating equations, misspecification-robust analysis, and causal inference techniques, the proposed approach yields valid causal interpretations of regression coefficients and their standard errors even under model misspecification. This methodology is broadly applicable to practical experimental designs, including completely randomized trials.

Abstract

For analytic convenience, existing statistical frameworks either assume random or fixed regressors. However, it is a little awkward that they do not cover the practical case of estimating the average treatment effect in experiments with randomized treatments and non-randomized, fixed pretreatment covariates. We unify the literature by providing the theory for regressions with mixed regressors that contain both random and fixed components. Importantly, our theory allows for misspecification of the regression functions. We first establish general results for estimating equations with both random and fixed components and then use it to analyze misspecified linear regression, with applications to completely randomized experiments. We focus on the causal interpretation of the regression coefficients and standard errors even when the models are wrong. We start with the theory for independent data and then extend the discussion to clustered data.

Problem

• mixed regressors
• misspecified regression
• average treatment effect
• causal inference
• robust inference

Innovation

• mixed regressors
• model misspecification
• causal inference
• randomized experiments
• robust inference