Bayesian and classical approaches to instrumental variable regression

We establish relationships between certain Bayesian and classical approaches to instrumental variable regression. We determine the form of priors that lead to posteriors for structural parameters that have similar properties as classical 2SLS and LIML and in doing so provide some new insight, especially in the context of weak instruments, to the small sample behavior of Bayesian and classical procedures. Using a reduced rank restriction on a multivariate linear model, we determine the priors that give rise to posteriors that are identical in functional form to the sampling densities of the classical 2SLS and LIML estimators.