We consider the problem of causal effect estimation with an unobserved confounder, where we observe a proxy variable that is associated with the confounder. Although Proxy Causal Learning (PCL) uses two proxy variables to recover the true causal effect, we show that a single proxy variable is sufficient for causal estimation if the outcome is generated deterministically, generalizing Control Outcome Calibration Approach (COCA). We propose two kernel-based methods for this setting: the first based on the two-stage regression approach, and the second based on a maximum moment restriction approach. We prove that both approaches can consistently estimate the causal effect, and we empirically demonstrate that we can successfully recover the causal effect on a synthetic dataset.

翻译：我们考虑存在未观测混杂变量时的因果效应估计问题，其中我们观测到一个与混杂变量相关的代理变量。尽管代理因果学习（PCL）使用两个代理变量来恢复真实的因果效应，但我们证明当结果变量是确定性生成时，单个代理变量足以进行因果估计，这推广了控制结果校准方法（COCA）。针对该设定，我们提出两种基于核的方法：第一种基于两阶段回归方法，第二种基于最大矩约束方法。我们证明这两种方法都能一致估计因果效应，并通过合成数据集的经验验证成功恢复了因果效应。