Bayesian Instrumental Variables: Priors and Likelihoods

Instrumental variable (IV) regression provides a number of statistical challenges due to the shape of the likelihood. We review the main Bayesian literature on instrumental variables and highlight these pathologies. We discuss Jeffreys priors, the connection to the errors-in-the-variables problems and more general error distributions. We propose, as an alternative to the inverted Wishart prior, a new Cholesky-based prior for the covariance matrix of the errors in IV regressions. We argue that this prior is more flexible and more robust thanthe inverted Wishart prior since it is not based on only one tightness parameter and therefore can be more informative about certain components of the covariance matrix and less informative about others. We show how prior-posterior inference can be formulated in a Gibbs sampler and compare its performance in the weak instruments case for synthetic as well as two illustrations based on well-known real data.