Locally robust semiparametric estimation

This paper shows how to construct locally robust semiparametric GMM estimators, meaning equivalently moment conditions have zero derivative with respect to the first step and the first step does not affect the asymptotic variance. They are constructed by adding to the moment functions the adjustment term for first step estimation. Locally robust estimators have several advantages. They are vital for valid inference with machine learning in the first step, see Belloni et. al. (2012, 2014), and are less sensitive to the specification of the first step. They are doubly robust for affine moment functions, where moment conditions continue to hold when one first step component is incorrect. Locally robust moment conditions also have smaller bias that is flatter as a function of first step smoothing leading to improved small sample properties. Series first step estimators confer local robustness on any moment conditions and are doubly robust for affine moments, in the direction of the series approximation. Many new locally and doubly robust estimators are given here, including for economic structural models. We give simple asymptotic theory for estimators that use cross-fitting in the first step, including machine learning.