  In many observational studies, researchers are often interested in studying
the effects of multiple exposures on a single outcome. Standard approaches for
high-dimensional data such as the lasso assume the associations between the
exposures and the outcome are sparse. These methods, however, do not estimate
the causal effects in the presence of unmeasured confounding. In this paper, we
consider an alternative approach that assumes the causal effects in view are
sparse. We show that with sparse causation, the causal effects are identifiable
even with unmeasured confounding. At the core of our proposal is a novel
device, called the synthetic instrument, that in contrast to standard
instrumental variables, can be constructed using the observed exposures
directly. We show that under linear structural equation models, the problem of
causal effect estimation can be formulated as an $\ell_0$-penalization problem,
and hence can be solved efficiently using off-the-shelf software. Simulations
show that our approach outperforms state-of-art methods in both low-dimensional
and high-dimensional settings. We further illustrate our method using a mouse
obesity dataset.
