EFFICIENCY BOUNDS FOR SEMIPARAMETRIC ESTIMATION OF INVERSE CONDITIONAL-DENSITY-WEIGHTED FUNCTIONS
Consider the unconditional moment restriction E[<bold>m</bold>(<bold>y</bold>, <italic>υ</italic>, <bold>w</bold>; <bold>π</bold><sub>0</sub>)/<italic>f</italic><sub>null</sub> (<italic>υ</italic>|<bold>w</bold>) −<bold>s</bold> (<bold>w</bold>; <bold>π</bold><sub>0</sub>)] = 0, where <bold>m</bold>(·) and <bold>s</bold>(·) are known vector-valued functions of data (<bold>y</bold><sup>┬</sup>, <italic>υ</italic>, <bold>w</bold><sup>┬</sup>)<sup>┬</sup>. The smallest asymptotic variance that <inline-graphic>null</inline-graphic>-consistent regular estimators of <bold>null</bold><sub>0</sub> can have is calculated when <italic>f</italic><sub>null</sub>(·) is only known to be a bounded, continuous, nonzero conditional density function. Our results show that “plug-in” kernel-based estimators of <bold>null</bold><sub>0</sub> constructed from this type of moment restriction, such as Lewbel (1998, <italic>Econometrica</italic> 66, 105–121) and Lewbel (2007, <italic>Journal of Econometrics</italic> 141, 777–806), are semiparametric efficient.
