Risk of Bayesian Inference in Misspecified Models, and the Sandwich Covariance Matrix∗

It is well known that in misspecified parametric models, the maximum likelihood estimator (MLE) is consistent for the pseudo-true value and has an asymptotically normal sampling distribution with "sandwich" covariance matrix. Also, posteriors are asymptotically centered at the MLE, normal and of asymptotic variance that is in general different than the sandwich matrix. It is shown that due to this discrepancy, Bayesian inference about the pseudo-true parameter value is in general of lower asymptotic frquentist risk when the original posterior is substituted by an artificial normal posterior centered at the MLE with sandwich covariance matrix. An algorithm is suggested that allows the implementation of this artificial posterior also in models with high dimensional nuisance parameters which cannot reasonably be estimated by maximizing the likelihood. JEL classification: C44, C11