[Paper Review] Efficient Semiparametric Estimation of Average Treatment Effects Under Covariate Adaptive Randomization

Ahnaf Rafi|arXiv (Cornell University)|May 15, 2023

Advanced Causal Inference Techniques4 citations

TL;DR

This paper establishes the semiparametric efficiency bound for estimating the average treatment effect (ATE) in randomized experiments using covariate adaptive randomization (CAR), where treatment assignment is balanced within strata defined by baseline covariates. It shows that a cross-fitted Nadaraya-Watson kernel estimator achieves this bound under weak regularity conditions, enabling efficient inference even when treatment assignment mechanisms induce dependence in outcomes.

ABSTRACT

Experiments that use covariate adaptive randomization (CAR) are commonplace in applied economics and other fields. In such experiments, the experimenter first stratifies the sample according to observed baseline covariates and then assigns treatment randomly within these strata so as to achieve balance according to pre-specified stratum-specific target assignment proportions. In this paper, we compute the semiparametric efficiency bound for estimating the average treatment effect (ATE) in such experiments with binary treatments allowing for the class of CAR procedures considered in Bugni, Canay, and Shaikh (2018, 2019). This is a broad class of procedures and is motivated by those used in practice. The stratum-specific target proportions play the role of the propensity score conditional on all baseline covariates (and not just the strata) in these experiments. Thus, the efficiency bound is a special case of the bound in Hahn (1998), but conditional on all baseline covariates. Additionally, this efficiency bound is shown to be achievable under the same conditions as those used to derive the bound by using a cross-fitted Nadaraya-Watson kernel estimator to form nonparametric regression adjustments.

Motivation & Objective

• To determine whether a well-defined semiparametric efficiency bound (SPEB) exists for ATE estimation under a broad class of covariate adaptive randomization (CAR) procedures.
• To investigate whether a feasible semiparametric estimator can achieve this efficiency bound asymptotically.
• To characterize the conditions under which efficiency is attainable despite dependence induced by CAR mechanisms.
• To extend existing efficiency theory to CAR designs where target assignment proportions vary by stratum and treatment assignment mechanisms affect variance.

Proposed method

• Derives the semiparametric efficiency bound for ATE under CAR by generalizing the bound in [26], conditioning on all baseline covariates.
• Considers a broad class of CAR procedures, including those used in practice, where stratum-specific target proportions are pre-specified.
• Proposes a cross-fitted Nadaraya-Watson kernel estimator for nonparametric regression adjustment to achieve the efficiency bound.
• Uses cross-fitting to ensure asymptotic normality and root-n consistency of the estimator under weak regularity conditions.
• Establishes that the efficiency bound is achievable under the same weak conditions used to derive it, ensuring practical feasibility.
• Analyzes the impact of treatment assignment mechanisms on variance, showing that while some estimators remain sensitive to mechanism choice, the proposed estimator achieves efficiency regardless.

Experimental results

Research questions

• RQ1 Is there a well-defined semiparametric efficiency bound for ATE estimation under covariate adaptive randomization with stratum-specific target proportions?
• RQ2 Can a feasible estimator achieve this efficiency bound under weak regularity conditions?
• RQ3 How does the choice of treatment assignment mechanism affect the asymptotic variance of ATE estimators in CAR designs?
• RQ4 To what extent can nonparametric regression adjustments improve estimation efficiency beyond stratum-fixed effects in CAR experiments?
• RQ5 Does the proposed estimator remain efficient when target proportions vary across strata and treatment assignment mechanisms induce dependence in outcomes?

Key findings

• The semiparametric efficiency bound for ATE estimation under CAR is a special case of the bound in [26], but conditional on all baseline covariates.
• The bound is achievable using a cross-fitted Nadaraya-Watson kernel estimator under the same weak regularity conditions used to derive the bound.
• The proposed estimator achieves root-n consistency and asymptotic normality, enabling valid inference.
• Unlike standard estimators, the proposed method is robust to the choice of treatment assignment mechanism, even when such choices affect variance in other estimators.
• The efficiency gain from nonparametric adjustment via kernel estimation is formally established, improving precision beyond stratum-fixed effects models.
• The results show that even under complex CAR mechanisms, efficient ATE estimation is possible when using appropriate nonparametric adjustment and cross-fitting.