Rate-adaptive GMM estimations for linear time series models

In this paper we analyze Generalized Method of Moments (GMM) estimators for time series models as advocated by Hansen and Singleton. It is well known that these estimators achieve efficiency bounds if the number of lagged observations in the instrument set goes to infinity. However, to this date no data dependent way of selecting the number of instruments in a finite sample is available. This paper derives an asymptotic mean squared error (MSE) approximation for the GMM estimator. The optimal number of instruments is selected by minimizing a criterion based on the MSE approximation. It is shown that the fully feasible version of the GMM estimator is higher order adaptive. In addition a new version of the GMM estimator based on kernel weighted moment conditions is proposed. The kernel weights are selected in a data-dependent way. Expressions for the asymptotic bias of kernel weighted and standard GMM estimators are obtained. It is shown that standard GMM procedures have a larger asymptotic bias and MSE than optimal kernel weighted GMM. A bias correction for both standard and kernel weighted GMM estimators is proposed. It is shown that the bias corrected version achieves a faster rate of convergence of the higher order terms of the MSE than the uncorrected estimator. Keywords: Time Series, Feasible GMM, Number of Instruments, Rate-adaptive Kernels, Higher Order Adaptive, Bias Correction. JEL Classification: C13, C32.