The Decaying Missing-at-Random Framework: Doubly Robust Causal Inference with Partially Labeled Data

In real-world scenarios, data collection limitations often result in partially labeled datasets, leading to difficulties in drawing reliable causal inferences. Traditional approaches in the semi-supervised (SS) and missing data literature may not adequately handle these complexities, leading to biased estimates. To address these challenges, our paper introduces a novel decaying missing-at-random (decaying MAR) framework. This framework tackles missing outcomes in high-dimensional settings and accounts for selection bias arising from the dependence of labeling probability on covariates. Notably, we relax the need for a positivity condition, commonly required in the missing data literature, and allow uniform decay of labeling propensity scores with sample size, accommodating faster growth of unlabeled data. Our decaying MAR framework enables easy rate double-robust (DR) estimation of average treatment effects, succeeding where other methods fail, even with correctly specified nuisance models. Additionally, it facilitates asymptotic normality under model misspecification. To achieve this, we propose adaptive new targeted bias-reducing nuisance estimators and asymmetric cross-fitting, along with a novel semi-parametric approach that fully leverages large volumes of unlabeled data. Our approach requires weak sparsity conditions. Numerical results confirm our estimators' efficacy and versatility, addressing selection bias and model misspecification.