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Hodges-Lehmann Optimality for Testing Moment

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Abstract

This paper studies the Hodges and Lehmann (1956) optimality of tests in a general setup. The tests are compared by the exponential rates of growth to one of the power functions evaluated at a fixed alternative while keeping the asymptotic sizes bounded by some constant. We present two sets of sufficient conditions for a test to be Hodges-Lehmann optimal. These new conditions extend the scope of the Hodges-Lehmann optimality analysis to setups that cannot be covered by other conditions in the literature. The general result is illustrated by our applications of interest: testing for moment conditions and overidentifying restrictions. In particular, we show that (i) the empirical likelihood test does not necessarily satisfy existing conditions for optimality but does satisfy our new conditions; and (ii) the generalized method of moments (GMM) test and the generalized empirical likelihood (GEL) tests are Hodges-Lehmann optimal under mild primitive conditions. These results support the belief that the Hodges-Lehmann optimality is a weak asymptotic requirement.

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  • Ivan Canay & Taisuke Otsu, 2011. "Hodges-Lehmann Optimality for Testing Moment," Cowles Foundation Discussion Papers 1789, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1789
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    File URL: http://cowles.yale.edu/sites/default/files/files/pub/d17/d1789.pdf
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    Keywords

    Asymptotic optimality; Large deviations; Moment condition; Generalized method of moments; Generalized empirical likelihood;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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