I thank Harald Uhlig and three anonymous referees for numerous helpful comments and suggestions. I also benefitted from comments by Jun Yu, Yong Li, Andriy Norets, and Chris Sims, as well as seminar participants at the Greater New York Metropolitan Area Econometrics Colloquium, the North American Winter Meeting of the Econometric Society, the Tsinghua International Conference in Econometrics, Harvard/MIT, LSE, NYU, Oxford, Northwestern, Princeton, Rice, UBC, UCL, and Virginia.
Risk of Bayesian Inference in Misspecified Models, and the Sandwich Covariance Matrix
Article first published online: 18 SEP 2013
© 2013 The Econometric Society
Volume 81, Issue 5, pages 1805–1849, September 2013
How to Cite
Müller, U. K. (2013), Risk of Bayesian Inference in Misspecified Models, and the Sandwich Covariance Matrix. Econometrica, 81: 1805–1849. doi: 10.3982/ECTA9097
- Issue published online: 18 SEP 2013
- Article first published online: 18 SEP 2013
- Manuscript received June, 2009; final revision received August, 2012.
- Posterior variance;
- pseudo-true parameter value;
- interval estimation
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 frequentist 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.