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Confidence intervals for projections of partially identified parameters


  • Hiroaki Kaido

    () (Institute for Fiscal Studies and Boston University)

  • Francesca Molinari

    () (Institute for Fiscal Studies and Cornell University)

  • Jorg Stoye

    (Institute for Fiscal Studies and Cornell University)


This paper proposes a bootstrap-based procedure to build confi dence intervals for single components of a partially identifi ed parameter vector, and for smooth functions of such components, in moment (in)equality models. The extreme points of our confi dence interval are obtained by maximizing/minimizing the value of the component (or function) of interest subject to the sample analog of the moment (in)equality conditions properly relaxed. The novelty is that the amount of relaxation, or critical level, is computed so that the component (or function) of ?, instead of ? itself, is uniformly asymptotically covered with prespeci ed probability. Calibration of the critical level is based on repeatedly checking feasibility of linear programming problems, rendering it computationally attractive. Computation of the extreme points of the con fidence interval is based on a novel application of the response surface method for global optimization, which may prove of independent interest also for applications of other methods of inference in the moment (in)equalities literature. The critical level is by construction smaller (in fi nite sample) than the one used if projecting con fience regions designed to cover the entire parameter vector. Hence, our con fidence interval is weakly shorter than the projection of established con fidence sets (Andrews and Soares, 2010), if one holds the choice of tuning parameters constant. We provide simple conditions under which the comparison is strict. Our inference method controls asymptotic coverage uniformly over a large class of data-generating processes. Our assumptions and those used in the leading alternative approach (a profi ling-based method) are not nested. We explain why we employ some restrictions that are not required by other methods and provide examples of models for which our method is uniformly valid but profi ling-based methods are not.

Suggested Citation

  • Hiroaki Kaido & Francesca Molinari & Jorg Stoye, 2016. "Confidence intervals for projections of partially identified parameters," CeMMAP working papers CWP02/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:02/16

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    References listed on IDEAS

    1. Andrews, Donald W.K. & Guggenberger, Patrik, 2009. "Validity Of Subsampling And “Plug-In Asymptotic” Inference For Parameters Defined By Moment Inequalities," Econometric Theory, Cambridge University Press, vol. 25(03), pages 669-709, June.
    2. Federico Ciliberto & Elie Tamer, 2009. "Market Structure and Multiple Equilibria in Airline Markets," Econometrica, Econometric Society, vol. 77(6), pages 1791-1828, November.
    3. Charles F. Manski & Elie Tamer, 2002. "Inference on Regressions with Interval Data on a Regressor or Outcome," Econometrica, Econometric Society, vol. 70(2), pages 519-546, March.
    4. Christian Bontemps & Thierry Magnac & Eric Maurin, 2012. "Set Identified Linear Models," Econometrica, Econometric Society, vol. 80(3), pages 1129-1155, May.
    5. repec:wly:emetrp:v:82:y:2014:i:5:p:1979-2002 is not listed on IDEAS
    6. Joseph P. Romano & Azeem M. Shaikh & Michael Wolf, 2014. "A Practical Two‐Step Method for Testing Moment Inequalities," Econometrica, Econometric Society, vol. 82(5), pages 1979-2002, September.
    7. Hiroaki Kaido & Andres Santos, 2014. "Asymptotically Efficient Estimation of Models Defined by Convex Moment Inequalities," Econometrica, Econometric Society, vol. 82(1), pages 387-413, January.
    8. Arie Beresteanu & Francesca Molinari, 2008. "Asymptotic Properties for a Class of Partially Identified Models," Econometrica, Econometric Society, vol. 76(4), pages 763-814, July.
    9. Donald W. K. Andrews & Gustavo Soares, 2010. "Inference for Parameters Defined by Moment Inequalities Using Generalized Moment Selection," Econometrica, Econometric Society, vol. 78(1), pages 119-157, January.
    10. Canay, Ivan A., 2010. "EL inference for partially identified models: Large deviations optimality and bootstrap validity," Journal of Econometrics, Elsevier, vol. 156(2), pages 408-425, June.
    11. Federico A. Bugni & Ivan A. Canay & Xiaoxia Shi, 2014. "Inference for functions of partially identified parameters in moment inequality models," CeMMAP working papers CWP05/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    12. Donald W. K. Andrews & Panle Jia Barwick, 2012. "Inference for Parameters Defined by Moment Inequalities: A Recommended Moment Selection Procedure," Econometrica, Econometric Society, vol. 80(6), pages 2805-2826, November.
    13. Jorg Stoye, 2009. "More on Confidence Intervals for Partially Identified Parameters," Econometrica, Econometric Society, vol. 77(4), pages 1299-1315, July.
    14. Kaido, Hiroaki, 2016. "A dual approach to inference for partially identified econometric models," Journal of Econometrics, Elsevier, vol. 192(1), pages 269-290.
    15. Brendan Kline & Elie Tamer, 2016. "Bayesian inference in a class of partially identified models," Quantitative Economics, Econometric Society, vol. 7(2), pages 329-366, July.
    16. Federico A. Bugni, 2010. "Bootstrap Inference in Partially Identified Models Defined by Moment Inequalities: Coverage of the Identified Set," Econometrica, Econometric Society, vol. 78(2), pages 735-753, March.
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    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Xiaohong Chen & Timothy Christensen & Elie Tamer, 2016. "MCMC Confidence sets for Identified Sets," Cowles Foundation Discussion Papers 2037, Cowles Foundation for Research in Economics, Yale University.
    2. Xiaohong Chen & Timothy M. Christensen & Elie Tamer, 2018. "Monte Carlo Confidence Sets for Identified Sets," Econometrica, Econometric Society, vol. 86(6), pages 1965-2018, November.
    3. repec:eee:pubeco:v:166:y:2018:i:c:p:98-114 is not listed on IDEAS
    4. Christian Bontemps & Thierry Magnac, 2017. "Set Identification, Moment Restrictions, and Inference," Annual Review of Economics, Annual Reviews, vol. 9(1), pages 103-129, September.
    5. Gualdani, Cristina, 2018. "An Econometric Model of Network Formation with an Application to Board Interlocks between Firms," TSE Working Papers 17-898, Toulouse School of Economics (TSE), revised Jul 2019.
    6. Cherchye, Laurens & Cosaert, Sam & De Rock, Bram & Kerstens, Pieter Jan & Vermeulen, Frederic, 2018. "Individual welfare analysis for collective households," Journal of Public Economics, Elsevier, vol. 166(C), pages 98-114.
    7. Yuichi Kitamura & Jörg Stoye, 2018. "Nonparametric Analysis of Random Utility Models," Econometrica, Econometric Society, vol. 86(6), pages 1883-1909, November.
    8. Yu-Wei Hsieh & Xiaoxia Shi & Matthew Shum, 2017. "Inference on Estimators defined by Mathematical Programming," Papers 1709.09115,
    9. Bontemps, Christian & Kumar, Rohit, 2018. "A Geometric Approach to Inference in Set-Identified Entry Games," TSE Working Papers 18-943, Toulouse School of Economics (TSE), revised Mar 2019.
    10. repec:eee:indorg:v:53:y:2017:i:c:p:241-266 is not listed on IDEAS
    11. Hiroaki Kaido & Jiaxuan Li & Marc Rysman, 2018. "Moment inequalities in the context of simulated and predicted variables," CeMMAP working papers CWP26/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    12. Hiroaki Kaido & Francesca Molinari & Jorg Stoye & Matthew Thirkettle, 2017. "Calibrated Projection in MATLAB: Users' Manual," Papers 1710.09707,
    13. Pakes, Ariel, 2017. "Empirical tools and competition analysis: Past progress and current problems," International Journal of Industrial Organization, Elsevier, vol. 53(C), pages 241-266.
    14. Vishal Kamat, 2017. "Identification with Latent Choice Sets," Papers 1711.02048,, revised Aug 2019.
    15. Levon Barseghyan & Maura Coughlin & Francesca Molinari & Joshua C. Teitelbaum, 2019. "Heterogeneous Choice Sets and Preferences," Papers 1907.02337,
    16. Christian Bontemps & Rohit Kumar, 2019. "A Geometric Approach to Inference in Set-Identified Entry Games," Working Papers hal-02137356, HAL.
    17. repec:eee:econom:v:204:y:2018:i:1:p:119-130 is not listed on IDEAS
    18. Francis J. DiTraglia & Camilo García-Jimeno, 2017. "Mis-classified, Binary, Endogenous Regressors: Identification and Inference," NBER Working Papers 23814, National Bureau of Economic Research, Inc.
    19. Matias D. Cattaneo & Xinwei Ma & Yusufcan Masatlioglu & Elchin Suleymanov, 2017. "A Random Attention Model," Papers 1712.03448,, revised Aug 2019.
    20. JoonHwan Cho & Thomas M. Russell, 2018. "Simple Inference on Functionals of Set-Identified Parameters Defined by Linear Moments," Papers 1810.03180,, revised Sep 2019.

    More about this item


    Partial identification; inference on projections; moment inequalities; uniform inference.;

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General


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