Unifying Design-based Inference: On Bounding and Estimating the Variance of any Linear Estimator in any Experimental Design

TL;DR

This paper introduces a unified, design-based framework for variance estimation applicable to any experimental design and linear estimator, enabling systematic comparison and interpretation of various variance bounds and estimators.

Contribution

It provides a general, matrix spectral analysis-based approach that unifies and extends existing variance estimation methods across diverse experimental setups.

Findings

Reproduces Eicker-Huber-White standard errors as a special case

Unifies cluster-robust and heteroskedasticity-consistent standard errors

Offers a systematic basis for comparing variance estimators across designs

Abstract

This paper provides a design-based framework for variance (bound) estimation in experimental analysis. Results are applicable to virtually any combination of experimental design, linear estimator (e.g., difference-in-means, OLS, WLS) and variance bound, allowing for unified treatment and a basis for systematic study and comparison of designs using matrix spectral analysis. A proposed variance estimator reproduces Eicker-Huber-White (aka. "robust", "heteroskedastic consistent", "sandwich", "White", "Huber-White", "HC", etc.) standard errors and "cluster-robust" standard errors as special cases. While past work has shown algebraic equivalences between design-based and the so-called "robust" standard errors under some designs, this paper motivates them for a wide array of design-estimator-bound triplets. In so doing, it provides a clearer and more general motivation for variance estimators.