Causal inference with instruments and other supplementary variables.

Instrumental variables have been used for a long time in the econometrics literature for the identification of the causal effect of one random variable, B, on another, C, in the presence of unobserved confounders. In the classical continuous linear model, the causal effect can be point identified by studying the regression of C on A and B on A, where A is the instrument. An instrument is an instance of a supplementary variable which is not of interest in itself but aids identification of causal effects. The method of instrumental variables is extended here to generalised linear models, for which only bounds on the causal effect can be computed. For the discrete instrumental variable model, bounds have been derived in the literature for the causal effect of B on C in terms of the joint distribution of (A,B,C). Using an approach based on convex polytopes, bounds are computed here in terms of the pairwise (A,B) and (A,C) distributions, in direct analogy to the classic use but without the linearity assumption. The bounding technique is also adapted to instrumental models with stronger and weaker assumptions. The computation produces constraints which can be used to invalidate the model. In the literature, constraints of this type are usually tested by checking whether the relative frequencies satisfy them. This is unsatisfactory from a statistical point of view as it ignores the sampling uncertainty of the data. Given the constraints for a model, a proper likelihood analysis is conducted to develop a significance test for the validity of the instrumental model and a bootstrap algorithm for computing confidence intervals for the causal effect. Applications are presented to illustrate the methods and the advantage of a rigorous statistical approach. The use of covariates and intermediate variables for improving the efficiency of causal estimators is also discussed