Higher Order Bias Correcting Moment Equation for M-Estimation and its Higher Order Efficiency

This paper studies an alternative bias correction for the M-estimator, which is obtained by correcting the moment equation in the spirit of Firth (1993). In particular, this paper compares the stochastic expansions of the analytically bias-corrected estimator and the alternative estimator and finds that the third-order stochastic expansions of these two estimators are identical. This implies that at least in terms of the third order stochastic expansion, we cannot improve on the simple one-step bias correction by using the bias correction of moment equations. Though the result in this paper is for a fixed number of parameters, our intuition may extend to the analytical bias correction of the panel data models with individual specific effects. Noting the M-estimation can nest many kinds of estimators including IV, 2SLS, MLE, GMM, and GEL, our finding is a rather strong result.