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On the use of cross-fitting in causal machine learning with correlated units

Salvador V. Balkus, Hasan Laith, Nima S. Hejazi

TL;DR

In causal machine learning, nuisance-function estimation via ML can introduce empirical process bias; cross-fitting is used to mitigate this bias, but correlated units raise concerns about fold design. The authors formalize an AS-Independent cross-fitting framework and prove that the empirical process term remains negligible under a bound on the number of correlated pairs, preserving valid inference with dependent data. They provide a main theorem (with $r_n(\mathsf{P}_n-\mathsf{P}) f_{\eta_0} \overset{d}{\to} N(0,\sigma^2)$ and $\mathsf{P}(f_{\eta_0}-f_{\hat{\eta}_{S_1}})^2 = o_{\mathsf{P}}(1)$) and illustrate the approach in clustered, network, and time-series settings, showing negligible bias and often favorable finite-sample performance compared to correlation-aware folding schemes. This work suggests that, under mild dependence, one can deploy as-independent cross-fitting broadly, simplifying practical causal inference with ML nuisances while highlighting the need for appropriate variance estimation under correlation.

Abstract

In causal machine learning, the fitting and evaluation of nuisance models are typically performed on separate partitions, or folds, of the observed data. This technique, called cross-fitting, eliminates bias introduced by the use of black-box predictive algorithms. When study units may be correlated, such as in spatial, clustered, or time-series data, investigators often design bespoke forms of cross-fitting to minimize correlation between folds. We prove that, perhaps contrary to popular belief, this is typically unnecessary: performing cross-fitting as if study units were independent usually still eliminates key bias terms even when units may be correlated. In simulation experiments with various correlation structures, we show that causal machine learning estimators typically have the same or improved bias and precision under cross-fitting that ignores correlation compared to techniques striving to eliminate correlation between folds.

On the use of cross-fitting in causal machine learning with correlated units

TL;DR

In causal machine learning, nuisance-function estimation via ML can introduce empirical process bias; cross-fitting is used to mitigate this bias, but correlated units raise concerns about fold design. The authors formalize an AS-Independent cross-fitting framework and prove that the empirical process term remains negligible under a bound on the number of correlated pairs, preserving valid inference with dependent data. They provide a main theorem (with and ) and illustrate the approach in clustered, network, and time-series settings, showing negligible bias and often favorable finite-sample performance compared to correlation-aware folding schemes. This work suggests that, under mild dependence, one can deploy as-independent cross-fitting broadly, simplifying practical causal inference with ML nuisances while highlighting the need for appropriate variance estimation under correlation.

Abstract

In causal machine learning, the fitting and evaluation of nuisance models are typically performed on separate partitions, or folds, of the observed data. This technique, called cross-fitting, eliminates bias introduced by the use of black-box predictive algorithms. When study units may be correlated, such as in spatial, clustered, or time-series data, investigators often design bespoke forms of cross-fitting to minimize correlation between folds. We prove that, perhaps contrary to popular belief, this is typically unnecessary: performing cross-fitting as if study units were independent usually still eliminates key bias terms even when units may be correlated. In simulation experiments with various correlation structures, we show that causal machine learning estimators typically have the same or improved bias and precision under cross-fitting that ignores correlation compared to techniques striving to eliminate correlation between folds.
Paper Structure (10 sections, 6 theorems, 29 equations, 8 figures)

This paper contains 10 sections, 6 theorems, 29 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Cross-fitting theory
  3. Setup
  4. Main Theorem
  5. Applications
  6. Clustered Data
  7. Network and Spatial Data
  8. Time Series Data
  9. Concluding Remarks
  10. Acknowledgments

Key Result

Theorem 1

Suppose that $r_n(\mathsf{P}_n - \mathsf{P})f_{\eta_0} \overset{d}{\rightarrow}N(0, \sigma^2)$ for some $r_n$, and that $\mathsf{P} (f_{\eta_0} - f_{\hat{\eta}_{S_1}} )^2 = o_{\mathsf{P}}(1)$. Assume that the number of correlated pairs in the data grows no faster than $n^2/r_n^2$. Under an as-IID sa

Figures (8)

  • Figure 1: As-independent cross-fitting of three models across three folds.
  • Figure 2: Simulation of one-step ATE estimation using neural networks to learn the propensity score and outcome regression. Without cross-fitting, the bias fails to shrink fast enough for valid inference.
  • Figure 3: Two-way cross-fitting introduced by Chiang2021
  • Figure 4: Operating characteristics of one-step ATE estimators in the cluster setting, comparing as-independent sample-splitting to the two-way method of Chiang2021.
  • Figure 5: Two-fold version of the cross-fitting method from Emmenegger2025 .
  • ...and 3 more figures

Theorems & Definitions (13)

  • Definition 1: As-independent sample splitting
  • Theorem 1: As-independent sample-splitting in correlated data.
  • Lemma 1: Unbiasedness of as-independent sample splitting
  • proof
  • Lemma 2: Variance of as-independent sample splitting
  • proof
  • proof : Proof of Theorem \ref{['thm:as-ind-ss']}
  • Corollary 2: Cross-fitting in clustered data
  • proof
  • Corollary 3: as-IID cross-fitting in network data
  • ...and 3 more