Locally robust semiparametric estimation

We give a general construction of debiased/locally robust/orthogonal (LR) moment functions for GMM, where the derivative with respect to first step nonparametric estimation is zero and equivalently first step estimation has no effect on the influence function. This construction consists of adding an estimator of the influence function adjustment term for first step nonparametric estimation to identifying or original moment conditions. We also give numerical methods for estimating LR moment functions that do not require an explicit formula for the adjustment term. LR moment conditions have reduced bias and so are important when the first step is machine learning. We derive LR moment conditions for dynamic discrete choice based on first step machine learning estimators of conditional choice probabilities. We provide simple and general asymptotic theory for LR estimators based on sample splitting. This theory uses the additive decomposition of LR moment conditions into an identifying condition and a first step influence adjustment. Our conditions require only mean square consistency and a few (generally either one or two) readily interpretable rate conditions. LR moment functions have the advantage of being less sensitive to first step estimation. Some LR moment functions are also doubly robust meaning they hold if one first step is incorrect. We give novel classes of doubly robust moment functions and characterize double robustness. For doubly robust estimators our asymptotic theory only requires one rate condition.