Exploiting Independent Instruments: Identification and Distribution Generalization

When estimating the causal function between a vector of covariates X and a response Y in the presence of unobserved confounding, standard regression procedures such as ordinary least squares (OLS) are even asymptotically biased. Instrumental variable approaches (Wright, 1928; Imbens and Angrist, 1994; Newey, 2013) exploit the existence of exogenous heterogeneity in the form of an instrumental variable (IV) Z and estimate, under suitable conditions, the causal function consistently. Importantly, the errors in Y and the hidden confounders U should be uncorrelated with the instruments Z. Usually, this has to be argued for with background knowledge. When the data generating process is modeled by a structural causal model (SCM) (Pearl, 2009; Bongers et al., 2021) (so that the distribution is Markov with respect to the induced graph), then the above condition is satisfied if Y and U are d-separated from Z in the graph obtained by removing the edge from X to Y. Furthermore, in this case the errors in Y and U are even independent from Z. Using that the errors and instruments are not only uncorrelated but also independent comes with several benefits. For example, even in settings, where the causal function can be identified by classical approaches based on uncorrelatedness, the independence can be exploited to construct estimators that achieve the semiparametric efficiency bound, at least when the error distribution comes from a known, parametric family (Hansen et al., 2010). Furthermore, the independence constraint is stronger than uncorrelatedness and therefore yields stronger identifiability results, which has been reported in the field of econometrics (e.g., Imbens and Newey, 2009; Chesher, 2003).