Identification of causal effects in linear models: beyond instrumental variables

The instrumental variable (IV) formula has become widely used to address the issue of identification of a causal effect in linear systems with an unobserved variable that acts as direct confounder. We here propose two alternative formulations to achieve identification when the assumptions underlying the use of IV are violated. Parallel to the IV, the proposed formulas exploit the conditional independence structure of a directed acyclic graph and can be obtained via a series of univariate regressions, a feature that renders the results particularly attractive and easy to implement. By exploiting the notion of Markov equivalence, the derivations can also be applied to regression graphs, thereby enlarging the class of models to which the results are of use.