A class of indirect inference estimators: higher‐order asymptotics and approximate bias correction

In this paper, we define a set of indirect inference estimators based on moment approximations of the auxiliary estimators. 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 might 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 square error, up to the same order. Our theoretical results are also validated by three Monte Carlo experiments.