Cross-Fitting-Free Debiased Machine Learning with Multiway Dependence

This paper develops an asymptotic theory for two-step debiased machine learning (DML) estimators in generalised method of moments (GMM) models with general multiway clustered dependence, without relying on cross-fitting. While cross-fitting is commonly employed, it can be statistically inefficient and computationally burdensome when first-stage learners are complex and the effective sample size is governed by the number of independent clusters. We show that valid inference can be achieved without sample splitting by combining Neyman-orthogonal moment conditions with a localisation-based empirical process approach, allowing for an arbitrary number of clustering dimensions. The resulting DML-GMM estimators are shown to be asymptotically linear and asymptotically normal under multiway clustered dependence. A central technical contribution of the paper is the derivation of novel global and local maximal inequalities for general classes of functions of sums of separately exchangeable arrays, which underpin our theoretical arguments and are of independent interest.

💡 Research Summary

This paper develops a comprehensive asymptotic theory for two‑step debiased machine learning (DML) estimators within a generalized method of moments (GMM) framework when the data exhibit multi‑way clustered dependence. Unlike the prevailing DML literature, which relies heavily on cross‑fitting (sample splitting) to break the dependence between first‑stage nuisance estimation and second‑stage moment evaluation, the authors show that valid inference can be achieved without any sample splitting. The motivation is clear: in multi‑way clustered settings the effective sample size is driven by the number of independent clusters rather than the raw number of observations, so splitting the data can dramatically reduce the information available for estimating high‑dimensional or non‑parametric nuisance components, leading to substantial efficiency loss and computational burden.

The core of the methodology rests on two pillars. First, the authors construct Neyman‑orthogonal moment functions ψ(·) that satisfy a zero‑derivative condition with respect to the nuisance parameters at the true values. This orthogonalization guarantees that first‑order perturbations of the nuisance estimators have no impact on the target parameter’s estimating equation, thereby neutralizing the usual bias that cross‑fitting is designed to control. Second, they employ a localisation argument: the full‑sample nuisance estimator η̂ is shown to lie with high probability in a shrinking deterministic neighbourhood Γₙ(η₀) around the true nuisance η₀ (radius of order n⁻¹⁄⁴). Within this neighbourhood the empirical process Gₙ(f(η̂))—where f(η)=ψ(·,η)−ψ(·,η₀)—can be uniformly bounded using newly derived maximal inequalities.

A major technical contribution is the derivation of global and local maximal inequalities for empirical processes indexed by classes of functions of sums of separately exchangeable (SE) arrays. The SE and dissociation assumptions capture the multi‑way clustering structure: observations are exchangeable across each clustering dimension and independent across disjoint index sets. Existing maximal‑inequality results either restrict the number of clustering dimensions, the moment order q, or the size of the function class. By introducing a transversal partition of the index set and applying the Hoeffding–Jørgensen inequality, the authors obtain sharp high‑moment bounds that hold for arbitrary K (the number of clustering dimensions), any q∈