Title: A Proof of Orthogonal Double Machine Learning with $Z$-Estimators,
Abstract: 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.