Causal inference with some invalid instrumental variables: A quasi-Bayesian approach*

In observational studies, instrumental variables estimation is often used to identify causal effects. We propose a quasi-Bayesian approach to make consistent inferences about the causal effect when there are some invalid instruments that violate the exclusion restriction condition. Asymptotic properties of the proposed Bayes estimator, including model selection consistency, are established. A simulation study demonstrates that the proposed Bayesian method produces consistent point estimators and valid credible intervals with correct coverage rates for Gaussian and non-Gaussian data with some invalid instruments. We also demonstrate the proposed method in an application to real data.