Proximal causal inference provides a framework for estimating the average 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.

翻译：近端因果推断通过利用结果与处理的代理变量，为存在未测量混杂时估计平均处理效应（ATE）提供了一个框架。该框架的识别依赖于所谓桥函数的存在。标准方法通常假设桥函数具有参数化形式，先对其进行估计，再将其代入ATE估计量中。然而，这种序贯过程存在两个潜在的效率损失来源：(i) 由积分方程定义的桥函数难以高效估计；(ii) 未能考虑估计步骤之间的相关性。为克服这些限制，我们提出了一种新方法，通过递增矩条件逼近积分方程，并联合估计桥函数与ATE。我们证明，在适当条件下，我们的估计量是高效的。此外，我们提供了一种数据驱动的调参（即矩条件数量）选择方法。模拟研究表明，所提方法在有限样本中表现良好，并且对SUPPORT研究中右心导管术数据集的应用证明了其实用价值。