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Confidence Intervals for Projections of Partially Identified Parameters

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Listed:
  • Hiroaki Kaido
  • Francesca Molinari
  • Jörg Stoye

Abstract

We propose a bootstrap‐based calibrated projection procedure to build confidence intervals for single components and for smooth functions of a partially identified 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 confidence interval are obtained by extremizing the value of the 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 function of θ, not θ itself, is uniformly asymptotically covered with prespecified probability. This calibration is based on repeatedly checking feasibility of linear programming problems, rendering it computationally attractive. Nonetheless, the program defining an extreme point of the confidence 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, and we establish its rate of convergence. The algorithm is of independent interest for optimization problems with simple objectives and complicated constraints. An empirical application estimating an entry game illustrates the usefulness of the method. Monte Carlo simulations confirm the accuracy of the solution algorithm, the good statistical as well as computational performance of calibrated projection (including in comparison to other methods), and the algorithm's potential to greatly accelerate computation of other confidence intervals.

Suggested Citation

  • Hiroaki Kaido & Francesca Molinari & Jörg Stoye, 2019. "Confidence Intervals for Projections of Partially Identified Parameters," Econometrica, Econometric Society, vol. 87(4), pages 1397-1432, July.
  • Handle: RePEc:wly:emetrp:v:87:y:2019:i:4:p:1397-1432
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    File URL: https://doi.org/10.3982/ECTA14075
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    References listed on IDEAS

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    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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    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.
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    13. Jorg Stoye, 2009. "More on Confidence Intervals for Partially Identified Parameters," Econometrica, Econometric Society, vol. 77(4), pages 1299-1315, July.
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    Citations

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    Cited by:

    1. 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.
    2. repec:eee:pubeco:v:166:y:2018:i:c:p:98-114 is not listed on IDEAS
    3. Xiaohong Chen & Timothy Christensen & Keith O'Hara & Elie Tamer, 2016. "MCMC Confidence sets for Identified Sets," Cowles Foundation Discussion Papers 2037R, Cowles Foundation for Research in Economics, Yale University, revised Jul 2016.
    4. 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.
    5. Yuichi Kitamura & Jörg Stoye, 2018. "Nonparametric Analysis of Random Utility Models," Econometrica, Econometric Society, vol. 86(6), pages 1883-1909, November.
    6. 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.
    7. repec:eee:indorg:v:53:y:2017:i:c:p:241-266 is not listed on IDEAS
    8. 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.
    9. Vishal Kamat, 2017. "Identification with Latent Choice Sets," Papers 1711.02048, arXiv.org, revised Aug 2019.
    10. Christian Bontemps & Rohit Kumar, 2019. "A Geometric Approach to Inference in Set-Identified Entry Games," Working Papers hal-02137356, HAL.
    11. Hiroaki Kaido & Francesca Molinari & Jorg Stoye & Matthew Thirkettle, 2017. "Calibrated Projection in MATLAB: Users' Manual," Papers 1710.09707, arXiv.org.
    12. repec:eee:econom:v:204:y:2018:i:1:p:119-130 is not listed on IDEAS
    13. 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.
    14. repec:anr:reveco:v:9:y:2017:p:103-129 is not listed on IDEAS
    15. Christian Bontemps & Thierry Magnac, 2017. "Set Identification, Moment Restrictions, and Inference," Annual Review of Economics, Annual Reviews, vol. 9(1), pages 103-129, September.
    16. Yu-Wei Hsieh & Xiaoxia Shi & Matthew Shum, 2017. "Inference on Estimators defined by Mathematical Programming," Papers 1709.09115, arXiv.org.
    17. 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.
    18. Levon Barseghyan & Maura Coughlin & Francesca Molinari & Joshua C. Teitelbaum, 2019. "Heterogeneous Choice Sets and Preferences," Papers 1907.02337, arXiv.org.
    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.
    21. 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.

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    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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