  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 challenging synthetic benchmarks.
