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On the Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions

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  • Yuichi Kitamura
  • Andres Santos
  • Azeem M. Shaikh

Abstract

In this paper we make two contributions. First, we show by example that empirical likelihood and other commonly used tests for parametric moment restrictions, including the GMM-based J-test of Hansen (1982), are unable to control the rate at which the probability of a Type I error tends to zero. From this it follows that, for the optimality claim for empirical likelihood in Kitamura (2001) to hold, additional assumptions and qualifications need to be introduced. The example also reveals that empirical and parametric likelihood may have non-negligible differences for the types of properties we consider, even in models in which they are first-order asymptotically equivalent. Second, under stronger assumptions than those in Kitamura (2001), we establish the following optimality result: (i) empirical likelihood controls the rate at which the probability of a Type I error tends to zero and (ii) among all procedures for which the probability of a Type I error tends to zero at least as fast, empirical likelihood maximizes the rate at which probability of a Type II error tends to zero for "most" alternatives. This result further implies that empirical likelihood maximizes the rate at which probability of a Type II error tends to zero for all alternatives among a class of tests that satisfy a weaker criterion for their Type I error probabilities.
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Suggested Citation

  • Yuichi Kitamura & Andres Santos & Azeem M. Shaikh, 2012. "On the Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions," Econometrica, Econometric Society, vol. 80(1), pages 413-423, January.
  • Handle: RePEc:ecm:emetrp:v:80:y:2012:i:1:p:413-423 DOI: ECTA8773
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    References listed on IDEAS

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    1. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, pages 1029-1054.
    2. Ausubel, Lawrence M & Deneckere, Raymond J, 1993. "A Generalized Theorem of the Maximum," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), pages 99-107.
    3. Esmeralda A. Ramalho & Richard J. Smith, 2013. "Discrete Choice Non-Response," Review of Economic Studies, Oxford University Press, pages 343-364.
    4. Whitney K. Newey & Richard J. Smith, 2004. "Higher Order Properties of Gmm and Generalized Empirical Likelihood Estimators," Econometrica, Econometric Society, pages 219-255.
    5. Guido W. Imbens & Richard H. Spady & Phillip Johnson, 1998. "Information Theoretic Approaches to Inference in Moment Condition Models," Econometrica, Econometric Society, pages 333-358.
    6. Yuichi Kitamura, 2001. "Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions," Econometrica, Econometric Society, pages 1661-1672.
    7. Hansen, Lars Peter & Heaton, John & Yaron, Amir, 1996. "Finite-Sample Properties of Some Alternative GMM Estimators," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(3), pages 262-280, July.
    8. Guido W. Imbens & Phillip Johnson & Richard H. Spady, 1995. "Information Theoretic Approaches to Inference in Moment Condition Models," Harvard Institute of Economic Research Working Papers 1736, Harvard - Institute of Economic Research.
    9. Bent Nielsen, 1995. "Bartlett correction of the unit root test in autoregressive models," Economics Papers 11 & 98., Economics Group, Nuffield College, University of Oxford.
    10. David F. Hendry & Michael P. Clements, 2004. "Pooling of forecasts," Econometrics Journal, Royal Economic Society, vol. 7(1), pages 1-31, June.
    11. Guido W. Imbens & Phillip Johnson & Richard H. Spady, 1995. "Information Theoretic Approaches to Inference in Moment Condition Models," NBER Technical Working Papers 0186, National Bureau of Economic Research, Inc.
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    Cited by:

    1. Marmer, Vadim & Otsu, Taisuke, 2012. "Optimal comparison of misspecified moment restriction models under a chosen measure of fit," Journal of Econometrics, Elsevier, pages 538-550.
    2. Guggenberger, Patrik, 2012. "A note on the (in)consistency of the test of overidentifying restrictions and the concepts of true and pseudo-true parameters," Economics Letters, Elsevier, pages 901-904.
    3. Otsu, Taisuke, 2010. "On Bahadur efficiency of empirical likelihood," Journal of Econometrics, Elsevier, pages 248-256.
    4. Canay, Ivan A., 2010. "EL inference for partially identified models: Large deviations optimality and bootstrap validity," Journal of Econometrics, Elsevier, pages 408-425.
    5. Canay, Ivan A. & Otsu, Taisuke, 2012. "Hodges–Lehmann optimality for testing moment conditions," Journal of Econometrics, Elsevier, pages 45-53.
    6. Mardi Dungey & Vitali Alexeev & Jing Tian & Alastair R. Hall, 2015. "Econometricians Have Their Moments: GMM at 32," The Economic Record, The Economic Society of Australia, vol. 91, pages 1-24, June.

    More about this item

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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