  We consider the problem of estimating the causal effect of a treatment on an
outcome in linear structural causal models (SCM) with latent confounders when
we have access to a single proxy variable. Several methods (such as
difference-in-difference (DiD) estimator or negative outcome control) have been
proposed in this setting in the literature. However, these approaches require
either restrictive assumptions on the data generating model or having access to
at least two proxy variables. We propose a method to estimate the causal effect
using cross moments between the treatment, the outcome, and the proxy variable.
In particular, we show that the causal effect can be identified with simple
arithmetic operations on the cross moments if the latent confounder in linear
SCM is non-Gaussian. In this setting, DiD estimator provides an unbiased
estimate only in the special case where the latent confounder has exactly the
same direct causal effects on the outcomes in the pre-treatment and
post-treatment phases. This translates to the common trend assumption in DiD,
which we effectively relax. Additionally, we provide an impossibility result
that shows the causal effect cannot be identified if the observational
distribution over the treatment, the outcome, and the proxy is jointly
Gaussian.
  Our experiments on both synthetic and real-world datasets showcase the
effectiveness of the proposed approach in estimating the causal effect.
