A Class of Indirect Inference Estimators: Higher Order Asymptotics and Approximate Bias Correction (Revised)

In this paper we define a set of Indirect Inference estimators based on moment approximations of the auxiliary ones. Their introduction is motivated by reasons of analytical and computational facilitation. Their definition provides an indirect inference framework for some "classical" bias correction procedures. We derive higher order asymptotic properties of these estimators. We demonstrate that under our assumption framework and in the special case of deterministic weighting and affinity of the binding function these are second order unbiased. Moreover their second order approximate Mean Square Errors do not depend on the cardinality of the Monte Carlo or Bootstrap samples that our definition may involve. Consequently, the second order Mean Square Error of the auxiliary estimator is not altered. We extend this to a class of multistep Indirect Inference estimators that have zero higher order bias without increasing the approximate Mean Squared Error, up to the same order. Our theoretical results are also validated by three Monte Carlo experiments.