# A Proof of Orthogonal Double Machine Learning with $Z$-Estimators

## TL;DR

This paper provides an alternative proof for the asymptotic properties of orthogonal Z-estimators in two-stage estimation, simplifying the understanding of their consistency and normality under certain conditions.

## Contribution

It offers a simplified, expository proof of the asymptotic normality of orthogonal Z-estimators in two-stage models, extending prior results to a variant based on empirical moment conditions.

## Key findings

- Orthogonal Z-estimators are $\,\sqrt{n}$-consistent and asymptotically normal.
- Sample splitting and $n^{1/4}$-consistency of the first stage are sufficient.
- The proof simplifies understanding of the estimator's asymptotic behavior.

## 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.