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Testing many moment inequalities

Author

Listed:
  • Victor Chernozhukov

    () (Institute for Fiscal Studies and MIT)

  • Denis Chetverikov

    () (Institute for Fiscal Studies and UCLA)

  • Kengo Kato

    (Institute for Fiscal Studies)

Abstract

This paper considers the problem of testing many moment inequalities where the number of moment inequalities, denoted by p, is possibly much larger than the sample size n. There are variety of economic applications where the problem of testing many moment inequalities appears; a notable example is a market structure model of Ciliberto and Tamer (2009) where p = 2m+1 with m being the number of fi rms. We consider the test statistic given by the maximum of p Studentized (or t-type) statistics, and analyze various ways to compute critical values for the test statistic. Speci cally, we consider critical values based upon (i) the union bound combined with a moderate deviation inequality for self-normalized sums, (ii) the multiplier and empirical bootstraps, and (iii) two-step and three-step variants of (i) and (ii) by incorporating selection of uninformative inequalities that are far from being binding and novel selection of weakly informative inequalities that are potentially binding but do not provide fi rst order information. We prove validity of these methods, showing that under mild conditions, they lead to tests with error in size decreasing polynomially in n while allowing for p being much larger than n; indeed p can be of order exp(nc) for some c > 0. Importantly, all these results hold without any restriction on correlation structure between p Studentized statistics, and also hold uniformly with respect to suitably large classes of underlying distributions. Moreover, when p grows with n, we show that all of our tests are (minimax) optimal in the sense that they are uniformly consistent against alternatives whose "distance" from the null is larger than the threshold (2(log p)=n)1/2, while any test can only have trivial power in the worst case when the distance is smaller than the threshold. Finally, we show validity of a test based on block multiplier bootstrap in the case of dependent data under some general mixing conditions.

Suggested Citation

  • Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2016. "Testing many moment inequalities," CeMMAP working papers CWP42/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:42/16
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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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    Citations

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    as


    Cited by:

    1. Donald W.K. Andrews & Xiaoxia Shi, 2015. "Inference Based on Many Conditional Moment Inequalities," Cowles Foundation Discussion Papers 2010, Cowles Foundation for Research in Economics, Yale University.
    2. repec:eee:ecolet:v:162:y:2018:i:c:p:124-126 is not listed on IDEAS
    3. Andrews, Donald W.K. & Shi, Xiaoxia, 2017. "Inference based on many conditional moment inequalities," Journal of Econometrics, Elsevier, vol. 196(2), pages 275-287.

    More about this item

    Keywords

    Many moment inequalities; moderate deviation; multiplier and empirical bootstrap; non-asymptotic bound; self-normalized sum.;

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