A Proof of Orthogonal Double Machine Learning with $Z$-Estimators
This is an incremental contribution aimed at researchers in econometrics or statistics, clarifying existing theoretical results without introducing new methods.
The paper tackles the problem of proving statistical properties for two-stage estimation methods by providing an alternative proof for a theorem from prior work, showing that under certain conditions, second-stage estimates achieve root-n consistency and asymptotic normality. The result is presented as an expository note rather than a new technical contribution.
We consider two stage estimation with a non-parametric first stage and a generalized method of moments second stage, in a simpler setting than (Chernozhukov et al. 2016). We give an alternative proof of the theorem given in (Chernozhukov et al. 2016) that orthogonal second stage moments, sample splitting and $n^{1/4}$-consistency of the first stage, imply $\sqrt{n}$-consistency and asymptotic normality of second stage estimates. Our proof is for a variant of their estimator, which is based on the empirical version of the moment condition (Z-estimator), rather than a minimization of a norm of the empirical vector of moments (M-estimator). This note is meant primarily for expository purposes, rather than as a new technical contribution.