The principal purpose of this paper is to adapt to the conditional moment context the GEL unconditional moment methods described in Smith(1997, 2001) and Newey and Smith(2004). In particular we develop GEL estimators which achieve the semiparametric efficiency lower bound. The requisite GEL criteria are constructed by local smoothing and parallel the local semiparametric efficient EL method formulated by Kitamura, Tripathi and Ahn (2004) for conditional moment restrictions. A particular advantageof these efficient local methods is the avoidance of the necessity of providing explicit estimators for the Jacobian and conditional variance matrices. The class of local GEL estimators admits a number of alternative first order equivalent estimators such as local EL, local ET and local CUE as in the unconditional moment restrictions case. The paper also provides a local GEL criterion function test statistic for parametric restrictions.

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