We consider the problem of estimating a causal effect in a multi-domain setting. The causal effect of interest is confounded by an unobserved confounder and can change between the different domains. We assume that we have access to a proxy of the hidden confounder and that all variables are discrete or categorical. We propose methodology to estimate the causal effect in the target domain, where we assume to observe only the proxy variable. Under these conditions, we prove identifiability (even when treatment and response variables are continuous). We introduce two estimation techniques, prove consistency, and derive confidence intervals. The theoretical results are supported by simulation studies and a real-world example studying the causal effect of website rankings on consumer choices.

翻译：本文研究多领域设置下的因果效应估计问题。感兴趣的因果效应受到未观测混杂变量的干扰，且可能在不同领域间发生变化。我们假设可获取隐藏混杂变量的代理变量，且所有变量均为离散或分类变量。我们提出了一种在目标领域估计因果效应的方法，其中假设仅能观测到代理变量。在此条件下，我们证明了因果效应的可识别性（即使处理变量和响应变量为连续型）。我们引入了两种估计技术，证明了其一致性，并推导了置信区间。理论结果通过模拟研究和真实案例（研究网站排名对消费者选择的因果效应）得到验证。