Choosing the number of moments in conditional moment restriction models

Properties of GMM estimators are sensitive to the choice of instruments. Usingmany instruments leads to high asymptotic asymptotic efficiency but can causehigh bias and/or variance in small samples. In this paper we develop and implementasymptotic mean square error (MSE) based criteria for instrumental variablesto use for estimation of conditional moment restriction models. The models weconsider include various nonlinear simultaneous equations models with unknownheteroskedasticity. We develop moment selection criteria for the familiar two-stepoptimal GMM estimator (GMM), a bias corrected version, and generalized empiricallikelihood estimators (GEL), that include the continuous updating estimator(CUE) as a special case. We also find that the CUE has lower higher-order variancethan the bias-corrected GMM estimator, and that the higher-order efficiencyof other GEL estimators depends on conditional kurtosis of the moments.