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Confidence intervals for projections of partially identified parameters

Author

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  • 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)

Abstract

We propose a bootstrap-based calibrated projection procedure to build con fidence intervals for single components and for smooth functions of a partially identi fied parameter vector in moment (in)equality models. The method controls asymptotic coverage uniformly over a large class of data generating processes. The extreme points of the calibrated projection confi dence interval are obtained by extremizing the value of the component (or function) of interest subject to a proper relaxation of studentized sample analogs of the moment (in)equality conditions. The degree of relaxation, or critical level, is calibrated so that the component (or function) of , not itself, is uniformly asymptotically covered with prespeci ed probability. This calibration is based on repeatedly checking feasibility of linear programming problems, rendering it computationally attractive. Nonetheless, the program defi ning an extreme point of the confi dence interval is generally nonlinear and potentially intricate. We provide an algorithm, based on the response surface method for global optimization, that approximates the solution rapidly and accurately. The algorithm is of independent interest for inference on optimal values of stochastic nonlinear programs. We establish its convergence under conditions satisfi ed by canonical examples in the moment (in)equalities literature. Our assumptions and those used in the leading alternative approach (a profi ling based method) are not nested. An extensive Monte Carlo analysis con rms the accuracy of the solution algorithm and the good statistical as well as computational performance of calibrated projection, including in comparison to other methods.

Suggested Citation

  • Hiroaki Kaido & Francesca Molinari & Jorg Stoye, 2017. "Confidence intervals for projections of partially identified parameters," CeMMAP working papers CWP49/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:49/17
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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(03), pages 669-709, June.
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    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.
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    Citations

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    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, arXiv.org.
    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, arXiv.org.
    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, arXiv.org, revised Aug 2019.
    15. Levon Barseghyan & Maura Coughlin & Francesca Molinari & Joshua C. Teitelbaum, 2019. "Heterogeneous Choice Sets and Preferences," Papers 1907.02337, arXiv.org.
    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, arXiv.org, 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, arXiv.org, revised Sep 2019.

    More about this item

    Keywords

    Partial identi fication; 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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