A Moment-Based Representation for Heteroskedasticity Robust Standard Errors

Ben Gillen

doi:10.1515/jem-2024-0023

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A Moment-Based Representation for Heteroskedasticity Robust Standard Errors

Ben Gillen

Published/Copyright: October 6, 2025

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From the journal Journal of Econometric Methods

Abstract

Heteroskedasticity robust standard errors are often presented as a formula that is not directly related to classical standard errors derived under homoskedasticity. This short paper introduces a moment-based result relating these two estimators through the correlation between squared residuals and squared regressors. Though the result does not rely on normality, it admits a simple approximation when all variables are normally distributed. This representation can be useful both for pedagogical purposes in undergraduate courses that do not use matrix algebra and in highlighting the relative magnitude of robust to non-robust standard errors.

Keywords: robust standard errors; econometric pedagogy; linear regression

JEL Classification: A22; C01; C1; C13; C2

Corresponding author: Ben Gillen, Robert Day School of Economics and Finance, Claremont McKenna College, Claremont, CA, 91711, USA, E-mail: benjamin.gillen@claremontmckenna.edu

Appendix: Proof of Theorem 1

The result is a simple matter of algebraic manipulation. Throughout, sample estimates are defined using method of moments estimators that do not include an adjustment for degrees of freedom. That is:

E ̂ x = 1 n ∑ i = 1 n x i var ̂ x = 1 n ∑ i = 1 n x i − E ̂ x 2 cov ̂ x , y = 1 n ∑ i = 1 n x i − E ̂ x y i − E ̂ y Kurt ̂ x = 1 n ∑ i = 1 n x i − x ̄ 4 var ̂ x 2

corr ̂ x , y = cov ̂ x , y var ̂ x var ̂ y

To proceed, note that the mean of x is projected out by the inclusion of the constant term, so let x* denote the demeaned version of x and x i * ≡ x i − E ̂ x be the ith observation of x minus the mean of the series.

var ̂ Robust β ̂ 1 = 1 n ∑ i = 1 n x i * 2 u ̂ i 2 n 1 n ∑ i = 1 n x i * 2 1 n ∑ i = 1 n x i * 2 = cov ̂ x * 2 , u ̂ 2 + var ̂ ( x ) var ̂ ( u ̂ ) n var ̂ ( x ) var ̂ ( x ) = cov ̂ x * 2 , u ̂ i 2 n var ̂ ( x ) var ̂ ( x ) + var ̂ ( u ̂ ) n var ̂ ( x ) = cov ̂ x * 2 , u ̂ i 2 var ̂ ( x * 2 ) var ̂ ( u ̂ 2 ) var ̂ ( x * 2 ) var ̂ ( u ̂ 2 ) n var ̂ ( x ) var ̂ ( x ) + var ̂ ( u ̂ ) n var ̂ ( x ) = corr ̂ x * 2 , u ̂ 2 var ̂ ( x * 2 ) var ̂ ( u ̂ 2 ) n var ( x ) ̂ var ( x ) ̂ + var ̂ ( u ̂ ) n var ̂ ( x )

Since x* is the demeaned version of x,

var ̂ ( x * 2 ) = E ̂ x * 4 − E ̂ x * 2 2 = Kurt ̂ ( x ) var ̂ ( x ) 2 − var ̂ ( x ) 2 = Kurt ̂ ( x ) − 1 var ̂ ( x ) 2 .

By similar argument, var ̂ ( u ̂ 2 ) = Kurt ̂ ( u ̂ ) − 1 var ̂ ( u ̂ ) 2 . Substituting these into the above formula gives:

var ̂ Robust β ̂ 1 = corr ̂ x * 2 , u ̂ 2 Kurt ̂ ( x ) − 1 var ̂ ( x ) 2 Kurt ̂ ( u ̂ ) − 1 var ̂ ( u ̂ ) 2 n var ̂ ( x ) var ̂ ( x ) + var ̂ ( u ̂ ) n var ̂ ( x ) = corr ̂ x * 2 , u ̂ 2 var ̂ ( u ̂ ) Kurt ̂ ( x ) − 1 Kurt ̂ ( u ̂ ) − 1 n var ̂ ( x ) + var ̂ ( u ̂ ) n var ̂ ( x ) = corr ̂ x * 2 , u ̂ 2 Kurt ̂ ( x ) − 1 Kurt ̂ ( u ̂ ) − 1 + 1 n − 2 n var ̂ Classical β ̂ 1

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Received: 2024-07-03

Accepted: 2025-09-11

Published Online: 2025-10-06

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https://doi.org/10.1515/jem-2024-0023

Keywords for this article

robust standard errors; econometric pedagogy; linear regression