COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1789 Hodges-Lehmann Optimality for Testing Moment Conditions Ivan Canay and Taisuke Otsu March 2011 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. |
