COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1722 On the Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions Yuichi Kitamura, Andres Santos and Azeem M. Shaikh August 2009 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. |
