On the Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions

Abstract

We show by example that empirical likelihood and other commonly used tests for moment restrictions are unable to control the (exponential) rate at which the probability of a Type I error tends to zero. It follows that the optimality of empirical likelihood asserted in Kitamura (2001) does not hold without additional assumptions. 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.

Existence and Characterization of Conditional Density Projections

“…Note, however, that our definition of φ-divergence (and thus that of Kitamura and Stutzer (1997) and Kitamura (2001)) slightly differs from that of Ali andSilvey (1966) or Csiszár (1995) who allow the divergence to possibly remain finite even if μ 1 is not dominated by μ 2 . In this case, there is an indeterminacy problem caused by the singular component of μ 1 whenever lim u→∞ φ(u)/u < +∞.…”

“…This formulation is used by Kitamura and Stutzer (1997) and Kitamura (2001), for example. 4 The class of φ-divergences D φ generally includes many distances used in econometrics and statistics.…”

Gel Methods for Nonsmooth Moment Indicators

“…Therefore, the EL version of LR defines a δ-optimal test of (2.1). Imposition of a further regularity condition results in this test being asymptotically efficient in the Hoeffding (1965) sense; see Kitamura (2001, Corollary 1, p.1665.…”

“…1 The assumptions required are precisely those given by Newey and McFadden (1994) for the two-step GMM (2S-GMM) estimator, i.e., the regularity conditions for GEL asymptotic normality are no more stringent than those for Kitamura (2001) is the exception to a general absence of theoretical results available for discrimination between asymptotically equivalent tests in a moment condition setting. Kitamura (2001) proves the large deviation optimality of EL-based tests of overidentifying moment conditions under assumptions also appropriate for the non-smooth moment indicator context studied here. Parametric restrictions test performance together with estimator bias are therefore examined in a set of Monte Carlo experiments.…”

Estimation with overidentifying inequality moment conditions

“…While generalized method of moments (GMM) is currently the most widely used procedure in practice, information-theoretic estimators such as empirical likelihood (EL) estimators have emerged as an attractive alternative to GMM, e.g., Owen (1988), Qin and Lawless (1994), Imbens (1997), Kitamura and Stutzer (1997), and Imbens, Spady, and Johnson (1998). Kitamura (2001) showed that the empirical likelihood ratio test for moment restrictions is asymptotically optimal under the Generalized Neyman-Pearson criterion. Newey and Smith (2004) find that the asymptotic bias of EL estimators does not grow with the number of moment conditions and that bias-corrected EL estimators have higher-order efficiency properties.…”