On the Validity of Edgeworth Expansions and Moment Approximations for Three Indirect Inference Estimators

This paper deals with higher order asymptotic properties for three indirect inference estimators. We provide conditions that ensure the validity of locally uniform Edgeworth approximations. When these are of sufficiently high order they also form integrability conditions that validate locally uniform moment approximations. We derive the relevant 2nd order bias and MSE approximations for the three estimators as functions of the respective approximations for the auxiliary estimator. We allow the possibility of stochastic weighting in any of the steps of the estimation procedure. We confirm that in the special case of deterministic weighting and affinity of the binding function, one of them is second order unbiased. The other two estimators do not have this property under the same conditions. Moreover, in this case, the second order approximate MSEs imply the superiority of the first estimator. We generalize to multistep procedures that provide recursive indirect estimators which are locally uniformly unbiased at any given order. Furthermore, in a particular case, we manage to validate locally uniform Edgeworth expansions for one of the estimators without any differentiability requirements for the estimating equations.