Locally Robust Inference with Outcomes Predicted by Machine Learning

Abstract

Many parameters of interest in economics depend on outcomes generated by machine learning (ML).These generated outcomes are subsequently used in standard econometric estimation and typically treated as observed, a practice that threatens the validity of downstream inference. Empirically relevant cases include dynamic treatment effects under sequential exogeneity, long-term effects with surrogates, the partially linear model with an ML-predicted outcome, productivity persistence with an unrestricted production function and constructed instruments, and outcomes residualized of confounding variation. This paper develops a three-step locally robust GMM framework for valid inference when the dependent variable is generated by ML. We construct a debiased moment condition that augments the identifying moment with first- and second-step influence functions, making inference on the parameter of interest locally insensitive to regularization and model selection biases at both steps. We establish consistency and asymptotic normality of the proposed estimator and illustrate its finite-sample performance through Monte Carlo simulations. We also apply the method to U.S. mortgage data to study conditional Black--White disparities in a race-blind ML default-risk score. The score predicts subsequent default well overall, but the conditional disparity embedded in the score is much smaller than the corresponding disparity in realized serious delinquency; accounting for first-step estimation also raises the naive out-of-sample estimate of the score disparity.