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Confi dence Intervals for Projections of Partially Identi fied Parameters

Author

Listed:
  • Hiroaki Kaido

    (Boston University)

  • Francesca Molinari

    (Cornell University)

  • Jorg Stoye

    (Cornell University)

Abstract

This paper proposes a bootstrap-based procedure to build con dence intervals for single components of a partially identi ed parameter vector, and for smooth functions of such components, in moment (in)equality models. The extreme points of our con 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 cov- ered with prespeci ed probability. Calibration of the critical level is based on repeatedly checking feasibility of linear programming problems, rendering it computationally attrac- tive. Computation of the extreme points of the con dence 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 nite sample) than the one used if projecting con dence regions designed to cover the entire parameter vector. Hence, our con dence interval is weakly shorter than the projection of established con dence 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 pro 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 pro ling based methods are not.

Suggested Citation

  • Hiroaki Kaido & Francesca Molinari & Jorg Stoye, 2016. "Confi dence Intervals for Projections of Partially Identi fied Parameters," Boston University - Department of Economics - Working Papers Series wp2016-001, Boston University - Department of Economics.
  • Handle: RePEc:bos:wpaper:wp2016-001
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    File URL: http://people.bu.edu/hkaido/pdf/Projection.pdf
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    References listed on IDEAS

    as
    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(3), 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. Hiroaki Kaido & Francesca Molinari & Jorg Stoye, 2019. "Constraint Qualifications in Partial Identification," Papers 1908.09103, arXiv.org, revised Apr 2021.
    6. 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.
    7. 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.
    8. Jorg Stoye, 2009. "More on Confidence Intervals for Partially Identified Parameters," Econometrica, Econometric Society, vol. 77(4), pages 1299-1315, July.
    9. Joseph P. Romano & Azeem M. Shaikh & Michael Wolf, 2014. "A Practical Two‐Step Method for Testing Moment Inequalities," Econometrica, Econometric Society, vol. 82, pages 1979-2002, September.
    10. Arie Beresteanu & Francesca Molinari, 2008. "Asymptotic Properties for a Class of Partially Identified Models," Econometrica, Econometric Society, vol. 76(4), pages 763-814, July.
    11. 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.
    12. Kaido, Hiroaki, 2016. "A dual approach to inference for partially identified econometric models," Journal of Econometrics, Elsevier, vol. 192(1), pages 269-290.
    13. Xiaohong Chen & Elie Tamer & Alexander Torgovitsky, 2011. "Sensitivity Analysis in Semiparametric Likelihood Models," Cowles Foundation Discussion Papers 1836, Cowles Foundation for Research in Economics, Yale University.
    14. 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.
    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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    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Partial identi cation; Inference on projections; Moment inequalities; Uniform inference;
    All these keywords.

    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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