Neumann-series corrections for regression adjustment in randomized experiments

TL;DR

This paper introduces a Neumann-series based correction method for regression adjustment in randomized experiments, significantly expanding the number of covariates for which asymptotic normality holds without relying on parametric models.

Contribution

It develops a novel Neumann-series decomposition approach that refines the analysis of regression adjustment, allowing for higher-dimensional covariates in finite-population randomized experiments.

Findings

Neumann-series corrections improve the asymptotic analysis of regression adjustment.

The method allows for a larger number of covariates while maintaining asymptotic normality.

The approach is purely design-based and does not depend on outcome model assumptions.

Abstract

We study average treatment effect (ATE) estimation under complete randomization with many covariates in a design-based, finite-population framework. In randomized experiments, regression adjustment can improve precision of estimators using covariates, without requiring a correctly specified outcome model. However, existing design-based analyses establish asymptotic normality only up to $p = o(n^{1/2})$, extendable to $p = o(n^{2/3})$ with a single de-biasing. We introduce a novel theoretical perspective on the asymptotic properties of regression adjustment through a Neumann-series decomposition, yielding a systematic higher-degree corrections and a refined analysis of regression adjustment. Specifically, for ordinary least squares regression adjustment, the Neumann expansion sharpens analysis of the remainder term, relative to the residual difference-in-means. Under mild leverage regularity, we show that the degree-$d$ Neumann-corrected estimator is asymptotically normal whenever $p^{ d+3}(\log p)^{ d+1}=o(n^{ d+2})$, strictly enlarging the admissible growth of $p$. The analysis is purely randomization-based and does not impose any parametric outcome models or super-population assumptions.