Kernel Single Proxy Control for Deterministic Confounding

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.

@article{xu2025_2308.04585,
  title={ Kernel Single Proxy Control for Deterministic Confounding },
  author={ Liyuan Xu and Arthur Gretton },
  journal={arXiv preprint arXiv:2308.04585},
  year={ 2025 }
}