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 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.

[1]  P. Moran,et al.  Rank correlation and product-moment correlation. , 1948, Biometrika.

[2]  J. Fabius.  Asymptotic behavior of bayes' estimates , 1963 .

[3]  R. Berk,et al.  Limiting Behavior of Posterior Distributions when the Model is Incorrect , 1966 .

[4]  P. J. Huber.  The behavior of maximum likelihood estimates under nonstandard conditions , 1967 .

[5]  R. Berk,et al.  CONSISTENCY A POSTERIORI , 1970 .

[6]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[7]  George E. P. Box,et al.  Sampling and Bayes' inference in scientific modelling and robustness , 1980 .

[8]  H. White.  A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity , 1980 .

[9]  H. White,et al.  Misspecified models with dependent observations , 1982 .

[10]  H. White.  Maximum Likelihood Estimation of Misspecified Models , 1982 .

[11]  C. Gouriéroux,et al.  PSEUDO MAXIMUM LIKELIHOOD METHODS: THEORY , 1984 .

[12]  Bayesian Implications.  On Asymptotic Normality of Limiting Density Functions with , 1985 .

[13]  Chan‐Fu Chen.  On Asymptotic Normality of Limiting Density Functions with Bayesian Implications , 1985 .

[14]  John F. Monahan,et al.  Bootstrap methods using prior information , 1986 .

[15]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[16]  Peter C. B. Phillips,et al.  Towards a Unified Asymptotic Theory for Autoregression , 1987 .

[17]  D. Andrews.  CONSISTENCY IN NONLINEAR ECONOMETRIC MODELS: A GENERIC UNIFORM LAW OF LARGE NUMBERS , 1987 .

[18]  J. Berger.  Statistical Decision Theory and Bayesian Analysis , 1988 .

[19]  R. M. Dudley,et al.  Real Analysis and Probability , 1989 .

[20]  Albert Y. Lo,et al.  Consistent and Robust Bayes Procedures for Location Based on Partial Information , 1990 .

[21]  Harald Uhlig,et al.  Understanding unit rooters: a helicopter tour , 1991 .

[22]  Allen S. Mandel.  Comment … , 1978, British heart journal.

[23]  L. Fahrmeir,et al.  Multivariate statistical modelling based on generalized linear models , 1994 .

[24]  M. Schervish.  Theory of Statistics , 1995 .

[25]  Xiao-Li Meng,et al.  Posterior Predictive Assessment of Model Fitnessvia Realized , 1995 .

[26]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[27]  P. Jeganathan.  Some Aspects of Asymptotic Theory with Applications to Time Series Models , 1995, Econometric Theory.

[28]  H. Raiffa,et al.  Introduction to Statistical Decision Theory , 1996 .

[29]  James E. Stafford.  A robust adjustment of the profile likelihood , 1996 .

[30]  Xiao-Li Meng,et al.  POSTERIOR PREDICTIVE ASSESSMENT OF MODEL FITNESS VIA REALIZED DISCREPANCIES , 1996 .

[31]  Arnold Zellner,et al.  THE BAYESIAN METHOD OF MOMENTS (BMOM) , 1997 .

[32]  O. Bunke,et al.  Asymptotic behavior of Bayes estimates under possibly incorrect models , 1998 .

[33]  Yum K. Kwan.  Asymptotic Bayesian analysis based on a limited information estimator , 1999 .

[34]  C. Whiteman,et al.  International Business Cycles: World, Region, and Country-Specific Factors , 2003 .

[35]  Jae-Young Kim,et al.  Limited information likelihood and Bayesian analysis , 2002 .

[36]  Frank Schorfheide,et al.  Priors from General Equilibrium Models for Vars , 2002 .

[37]  Dudley,et al.  Real Analysis and Probability: Measurability: Borel Isomorphism and Analytic Sets , 2002 .

[38]  Frank Schorfheide,et al.  Testing for Indeterminacy: An Application to U. S. Monetary Policy , 2002 .

[39]  Eric R. Ziegel,et al.  Multivariate Statistical Modelling Based on Generalized Linear Models , 2002, Technometrics.

[40]  T. Lancaster.  A Note on Bootstraps and Robustness , 2003 .

[41]  R. Royall,et al.  Interpreting statistical evidence by using imperfect models: robust adjusted likelihood functions , 2003 .

[42]  S. Sheather.  Density Estimation , 2004 .

[43]  J. Geweke,et al.  Getting It Right , 2004 .

[44]  Marion Kee,et al.  Analysis , 2004, Machine Translation.

[45]  J. Geweke,et al.  Contemporary Bayesian Econometrics and Statistics , 2005 .

[46]  Susanne M. Schennach,et al.  Bayesian exponentially tilted empirical likelihood , 2005 .

[47]  D. Freedman,et al.  On The So-Called “Huber Sandwich Estimator” and “Robust Standard Errors” , 2006 .

[48]  T. Lumley,et al.  Model-Robust Bayesian Regression and the Sandwich Estimator , 2007 .

[49]  David S. Leslie,et al.  A general approach to heteroscedastic linear regression , 2007, Stat. Comput..

[50]  Frank Schorfheide,et al.  DSGE Model-Based Estimation of the New Keynesian Phillips Curve , 2008 .

[51]  C. Shalizi.  Dynamics of Bayesian Updating with Dependent Data and Misspecified Models , 2009, 0901.1342.

[52]  Thomas Lumley,et al.  Model-Robust Regression and a Bayesian `Sandwich' Estimator , 2010, 1101.1402.

[53]  D. Poirier.  Bayesian Interpretations of Heteroskedastic Consistent Covariance Estimators Using the Informed Bayesian Bootstrap , 2011 .

[54]  Roger,et al.  Testing for Indeterminacy : An Application to U . S . Monetary Policy : Comment , 2011 .

[55]  Van Der Vaart,et al.  The Bernstein-Von-Mises theorem under misspecification , 2012 .

[56]  Justinas Pelenis.  Bayesian Semiparametric Regression , 2012 .

[57]  Justinas Pelenis.  Bayesian regression with heteroscedastic error density and parametric mean function , 2014 .