# On the Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions

*Yuichi Kitamura* (),
*Andres Santos* and
*Azeem Shaikh* ()

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Yuichi Kitamura: Cowles Foundation, Yale University, http://economics.yale.edu/people/yuichi-kitamura

No 1722, Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University

**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.

**Keywords:** Empirical likelihood; Large deviations; Hoeffding optimality; Moment restrictions (search for similar items in EconPapers)

**JEL-codes:** C12 C14 (search for similar items in EconPapers)

**New Economics Papers:** this item is included in nep-ecm

**Date:** 2009-08

**Note:** CFP 1378.

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**Published** in Econometrica (March 2013), 81(2): 581-607

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Journal Article: On the Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions (2012)

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