  Exogenous heterogeneity, for example, in the form of instrumental variables
can help us learn a system's underlying causal structure and predict the
outcome of unseen intervention experiments. In this paper, we consider linear
models in which the causal effect from covariates $X$ on a response $Y$ is
sparse. We provide conditions under which the causal coefficient becomes
identifiable from the observed distribution. These conditions can be satisfied
even if the number of instruments is as small as the number of causal parents.
We also develop graphical criteria under which identifiability holds with
probability one if the edge coefficients are sampled randomly from a
distribution that is absolutely continuous with respect to Lebesgue measure and
$Y$ is childless. As an estimator, we propose spaceIV and prove that it
consistently estimates the causal effect if the model is identifiable and
evaluate its performance on simulated data. If identifiability does not hold,
we show that it may still be possible to recover a subset of the causal
parents.
