Annals of Statistics

Bayesian analysis in moment inequality models

Yuan Liao and Wenxin Jiang

Full-text: Open access


This paper presents a study of the large-sample behavior of the posterior distribution of a structural parameter which is partially identified by moment inequalities. The posterior density is derived based on the limited information likelihood. The posterior distribution converges to zero exponentially fast on any δ-contraction outside the identified region. Inside, it is bounded below by a positive constant if the identified region is assumed to have a nonempty interior. Our simulation evidence indicates that the Bayesian approach has advantages over frequentist methods, in the sense that, with a proper choice of the prior, the posterior provides more information about the true parameter inside the identified region. We also address the problem of moment and model selection. Our optimality criterion is the maximum posterior procedure and we show that, asymptotically, it selects the true moment/model combination with the most moment inequalities and the simplest model.

Article information

Ann. Statist., Volume 38, Number 1 (2010), 275-316.

First available in Project Euclid: 31 December 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference 62N01: Censored data models
Secondary: 62F99: None of the above, but in this section

Identified region limited information likelihood consistent set estimation maximum posterior model and moment selection


Liao, Yuan; Jiang, Wenxin. Bayesian analysis in moment inequality models. Ann. Statist. 38 (2010), no. 1, 275--316. doi:10.1214/09-AOS714.

Export citation


  • Andrews, D. and Soares, G. (2007). Inference for parameters defined by moment inequalities using generalized moment selection. Yale Univ. Manuscript.
  • Andrews, D. and Jia, P. (2008). Inference for parameters defined by moment inequalities: A recommended moment selection procedure. Yale Univ. Manuscript.
  • Beresteanu, A. and Molinari, F. (2008). Asymptotic properties for a class of partially identified models. Econometrica 76 763–814.
  • Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
  • Bugni, F. (2007). Bootstrap inference in partially identified models. Northwestern Univ. Manuscript.
  • Canay, I. (2008). EL Inference for partially identified models: Large deviations optimality and bootstrap validity. Northwestern Univ. Manuscript.
  • Chernozhukov V., Hong H. and Tamer E. (2007). Estimation and confidence regions for parameter sets in econometric models. Econometrica 75 1243–1284.
  • Cover, T. and Thomas, J. (1991). Elements of Information Theory. Wiley, New York.
  • Gelfand, A. and Sahu, S. (1999). Identifiability, improper priors, and Gibbs sampling for generalized liner models. J. Amer. Statist. Assoc. 94 247–253.
  • Gustafson, P. (2005). On model expansion, model contraction, identifiability and prior information: Two illustrative scenarios involving mismeasured variables. Statist. Sci. 20 111–140.
  • Horowitz, J. and Manski, F. (2000). Nonparametric analysis of randomized experiments with missing covariate and outcome data. J. Amer. Statist. Assoc. 95 77–84.
  • Imbens, G. and Manski, C. F. (2004). Confidence intervals for partially identified parameters. Econometrica 72 1845–1857.
  • Kim, J. (2002). Limited information likelihood and Bayesian analysis. J. Econometrics 107 175–193.
  • Liao, Y. and Jiang, W. (2008). Bayesian analysis of moment inequality models: Supplement material. Technical report, Northwestern Univ. Available at
  • Liu, X. and Shao, Y. (2003). Asymptotics for likelihood ratio tests under loss of identifiability. Ann. Statist. 31 807–832.
  • Manski, C. F. and Tamer, E. (2002). Inference on regressions with interval data on a regressor or outcome. Econometrica 70 519–547.
  • Moon, H. and Schorfheide, F. (2009). Bayesian and frequentist inference in partially identified models. Univ. South California and Univ. Pennsylvania. Manuscript.
  • Neath A. and Samaniego, F. (1997). On the efficacy of Bayesian inference for nonidentifiable models. Amer. Statist. 51 225–232.
  • Pakes, A., Porter, J., Ho, K. and Ishii, J. (2006). Moment inequalities and their application. Harvard Univ. Working paper.
  • Poirier, D. (1998). Revising beliefs in nonidentified models. Econometric Theory 14 483–509.
  • Romano, J. and Shaikh, A. (2008). Inference for identifiable parameters in partially identified econometric models. J. Statist. Plann. Inference 138 2786–2807.
  • Rosen, A. (2008). Confidence sets for partially identified parameters that satisfy a finite number of moment inequalities. J. Econometrics. 146 107–117.
  • Zellner, A. (1994). Model, prior information and Bayesian analysis. J. Econometrics 75 51–68.