Higher order asymptotics and the bootstrap for empirical likelihood J tests

In this paper we obtain a second order Edgeworth approximation to the density of a likelihood ratio type J test for overidentifying restrictions by embedding the moment conditions into the empirical likelihood framework. The resulting asymptotic expansion can be used to correct to an order o n^-1 the critical values of the empirical likelihood ratio J test and to justify the second order correctness of an ``hybrid'' bootstrap procedure which we propose to bypass the difficult calculation of the cumulants appearing in the Edgeworth density of the empirical likelihood ratio J test. The resulting bootstrap calibrated empirical likelihood ratio test seems to perform well, as shown in a small Monte Carlo study, and suggest that the combination of the empirical likelihood method together with a suitable bootstrap procedure is an extremely useful method for estimation/inference in moment based econometric models.