Empirical likelihood inference with applications to some econometric models

In this paper we analyse the higher order asymptotic properties of the empirical likelihood ratio test, by means of the dual likelihood theory. It is shown that when the econometric model is just identified, these tests are accurate to an order o(1/n), and this accuracy can always be improved to an order O(1/n^2) by means of a scale correction, as in standard parametric theory. To show this, we first develop a valid Edgeworth expansion for the empirical likelihood ratio under a local alternative in terms of an "induced" local alternative. As a by-product of the expansion, we find an explicit expression for the Bartlett correction in terms of cumulants of dual likelihood derivatives which is slightly different from the standard adjustment reported in the literature on Bartlett corrections of the empirical likelihood ratio. We then highlight the connection between the empirical likelihood method and the bootstrap by obtaining a valid Edgeworth expansion for a bootstrap based empirical likelihood ratio test. The theory is then applied to some standard econometric models and illustrated by means of some Monte Carlo simulations.