Identifiability of Sparse Causal Effects using Instrumental Variables

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 prove that the causal coefficient becomes identifiable under weak conditions and may even be identified in models, where the number of instruments is as small as the number of causal parents. We also develop graphical criteria under which the identifiability holds with probability one if the edge coefficients are sampled randomly from a distribution that is absolutely continuous with respect to Lebesgue measure. 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.