  In this paper we propose and study local linear and polynomial based
estimators for implementing Approximate Bayesian Computation (ABC) style
indirect inference and GMM estimators. This method makes use of nonparametric
regression in the computation of GMM and Indirect Inference models. We provide
formal conditions under which frequentist inference is asymptotically valid and
demonstrate the validity of the estimated posterior quantiles for confidence
interval construction. We also show that in this setting, local linear kernel
regression methods have theoretical advantages over local constant kernel
methods that are also reflected in finite sample simulation results. Our
results also apply to both exactly and over identified models. These estimators
do not need to rely on numerical optimization or Markov Chain Monte Carlo
(MCMC) simulations. They provide an effective complement to the classical
M-estimators and to MCMC methods, and can be applied to both likelihood based
models and method of moment based models.
