Semi-parametric Bayesian Partially Identified Models based on Support Function*

We provide a comprehensive semi-parametric study of Bayesian partially identified econometric models. While the existing literature on Bayesian partial identification has mostly focused on the structural parameter, our primary focus is on Bayesian credible sets (BCS’s) of the unknown identified set and the posterior distribution of its support function. We construct a (two-sided) BCS based on the support function of the identified set. We prove the Bernstein-von Mises theorem for the posterior distribution of the support function. This powerful result in turn infers that, while the BCS and the frequentist confidence set for the partially identified parameter are asymptotically different, our constructed BCS for the identified set has an asymptotically correct frequentist coverage probability. Importantly, we illustrate that the constructed BCS for the identified set does not require a prior on the structural parameter. It can be computed efficiently for subset inference, especially when the target of interest is a sub-vector of the partially identified parameter, where projecting to a low-dimensional subset is often required. Hence, the proposed methods are useful in many applications. The Bayesian partial identification literature has been assuming a known parametric likelihood function. However, econometric models usually only identify a set of moment The authors are grateful to Federico Bugni, Ivan Canay, Joel Horowitz, Enno Mammen, Francesca Molinari, Andriy Norets, Adam Rosen, Frank Schorfheide, Jörg Stoye and seminar participants at CREST (LS and LMI), Luxembourg, Mannheim, Tsinghua, THEMA, University of Illinois at Urbana-Champaign, 4 French Econometrics Conference, Bayes in Paris workshop, Oberwolfach workshop, 2013 SBIES meeting, CMES 2013, EMS 2013 in Budapest and ESEM 2013 in Gothenburg for useful comments. Anna Simoni gratefully acknowledges financial support from the University of Mannheim through the DFG-SNF Research Group FOR916, ANR-13-BSH1-0004, labex MMEDII (ANR11-LBX-0023-01) and hospitality from University of Mannheim and CREST. Department of Mathematics, University of Maryland at College Park, College Park, MD 20742 (USA). Email: [email protected] CNRS and THEMA, Université de Cergy-Pontoise 33, boulevard du Port, 95011 Cergy-Pontoise (France). Email: [email protected]