RISK OF BAYESIAN INFERENCE IN MISSPECIFIED MODELS, AND THE SANDWICH COVARIANCE MATRIX

• Published in: Econometrica Vol. 81; no. 5; pp. 1805 - 1849
• Main Author: Müller, Ulrich K.
• Format: Journal Article
• Language: English
• Published: Oxford, UK Econometric Society 01-09-2013; Blackwell Publishing Ltd
• Subjects: Algorithms; Asymptotic methods; Bayesian analysis; Bayesian method; Covariance; Covariance matrices; Distribution; Economic models; Estimators; Incumbents; Inference; interval estimation; Interval estimators; Matrix; Maximum likelihood estimation; Maximum likelihood method; Parametric models; Posterior variance; pseudo‐true parameter value; quasi‐likelihood; Risk assessment; Sandwiches; Statistical variance; Studies
• ISSN: 0012-9682; 1468-0262
• DOI: 10.3982/ECTA9097

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.