Abstract

Proximal causal inference provides a framework for estimating the av

erage treatment effect (ATE) in the presence of unmeasured confounding by

leveraging outcome and treatment proxies. Identification in this framework relies

on the existence of a so-called bridge function. Standard approaches typically

postulate a parametric specification for the bridge function, which is estimated

in a first step and then plugged into an ATE estimator. However, this sequential

procedure suffers from two potential sources of efficiency loss: (i) the difficulty

of efficiently estimating a bridge function defined by an integral equation, and

(ii) the failure to account for the correlation between the estimation steps. To

overcome these limitations, we propose a novel approach that approximates the

integral equation with increasing moment restrictions and jointly estimates the

bridge function and the ATE. We show that, under suitable conditions, our estimator is efficient. Additionally, we provide a data-driven procedure for selecting

the tuning parameter (i.e., the number of moment restrictions). Simulation studies reveal that the proposed method performs well in finite samples, and an

application to the right heart catheterization dataset from the SUPPORT study

demonstrates its practical value.

Key words and phrases: Data-driven method; generalized method of moments; proximal causal inference; semiparametric efficiency

Information

Preprint No.SS-2025-0104
Manuscript IDSS-2025-0104
Complete AuthorsChunrong Ai, Jiawei Shan
Corresponding AuthorsJiawei Shan
Emailsjiawei.shan@wisc.edu

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Acknowledgments

We sincerely thank the editor, associate editor, and two reviewers for their

valuable comments, which led to a significant improvement in our paper.

Chunrong Ai gratefully acknowledges funding support from the National

Natural Science Foundation of China (Project No. 72133005).

Supplementary Materials

The online Supplementary Material contains additional simulation studies,

lemmas, and all technical proofs.

Supplementary materials are available for download.