The synthetic instrument: from sparse association to sparse causation

In many observational studies, researchers are often interested in the effects of multiple exposures on a single outcome. Standard approaches for high-dimensional data, such as the Lasso, assume that the associations between the exposures and the outcome are sparse. However, these methods do not estimate causal effects in the presence of unmeasured confounding. In this paper, we consider an alternative approach that assumes the causal effects under consideration are sparse. We show that under sparse causation, causal effects are identifiable even with unmeasured confounding. Our proposal is built around a novel device called the synthetic instrument, which, in contrast to standard instrumental variables, can be constructed directly from the observed exposures. We demonstrate that, under the assumption of sparse causation, the problem of causal effect estimation can be formulated as an ℓ0-penalization problem and solved efficiently using off-the-shelf software. Simulations show that our approach outperforms state-of-the-art methods in both low- and high-dimensional settings. We further illustrate our method using a mouse obesity dataset.