Introduction

Simulation evidence increasingly indicates that for many models specified by unconditional moment restrictions the generalized method of moments (GMM) estimator (Hansen, 1982) may be substantially biased in finite samples, especially so when there are large numbers of moment conditions. See, for example, Altonji and Segal (1996), Imbens and Spady (2001), Judge and Mittelhammer (2001), Ramalho (2001) and Newey, Ramalho and Smith (2005). Newey and Smith (2004), henceforth NS, provides theoretical underpinning for these findings. Alternative estimators which are first-order asymptotically equivalent to GMM include empirical likelihood (EL) (Owen, 1988; Qin and Lawless, 1994; and Imbens 1997), the continuous updating estimator (CUE) (Hansen, Heaton and Yaron, 1996), and exponential tilting (ET) (Kitamuraand Stutzer, 1997 and Imbens, Spady and Johnson, 1998). See also Owen (2001). NS show that these estimators and those from the Cressie and Read (1984) power divergence family of discrepancies are members of a class of generalized empirical likelihood (GEL) estimators and have a common structure; see Brown and Newey (1992, 2002) and Smith (1997, 2001). Correspondingly NS also demonstrate that GEL and GMM estimators are asymptotically equivalent and thus possess the same first-order asymptotic properties. For the unconditional context, NS describe the higher-order efficiency of bias-corrected EL. Also see Kitamura (2001).

The principal aim of this chapter is adapt the GEL method to the conditional moment context and, thereby, to describe GEL estimators which achieve the semi-parametric efficiency lower bound.