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Valid Heteroskedasticity Robust Testing

Benedikt M. Pötscher, David Preinerstorfer

TL;DR

This paper tackles the persistent problem that heteroskedasticity-robust tests in linear regression often overreject when conventional asymptotic critical values are used. It proves the existence of smallest size-controlling critical values $C(\alpha)$ for both unrestricted- and restricted-residual test statistics, provides verifiable conditions for their existence, and offers algorithms (with an R package $\texttt{hrt}$) to compute them. The authors show that using these critical values yields valid level-$\alpha$ tests and can improve power relative to standard approaches, while also identifying situations where certain tests become trivial under size control. The framework extends beyond Gaussian errors to elliptical and semiparametric settings and accommodates stochastic regressors, broadening practical applicability. Overall, the work delivers a principled, computable approach to robust hypothesis testing under unknown forms of heteroskedasticity with clear guidance for practitioners.

Abstract

Tests based on heteroskedasticity robust standard errors are an important technique in econometric practice. Choosing the right critical value, however, is not simple at all: conventional critical values based on asymptotics often lead to severe size distortions; and so do existing adjustments including the bootstrap. To avoid these issues, we suggest to use smallest size-controlling critical values, the generic existence of which we prove in this article for the commonly used test statistics. Furthermore, sufficient and often also necessary conditions for their existence are given that are easy to check. Granted their existence, these critical values are the canonical choice: larger critical values result in unnecessary power loss, whereas smaller critical values lead to over-rejections under the null hypothesis, make spurious discoveries more likely, and thus are invalid. We suggest algorithms to numerically determine the proposed critical values and provide implementations in accompanying software. Finally, we numerically study the behavior of the proposed testing procedures, including their power properties.

Valid Heteroskedasticity Robust Testing

TL;DR

This paper tackles the persistent problem that heteroskedasticity-robust tests in linear regression often overreject when conventional asymptotic critical values are used. It proves the existence of smallest size-controlling critical values for both unrestricted- and restricted-residual test statistics, provides verifiable conditions for their existence, and offers algorithms (with an R package ) to compute them. The authors show that using these critical values yields valid level- tests and can improve power relative to standard approaches, while also identifying situations where certain tests become trivial under size control. The framework extends beyond Gaussian errors to elliptical and semiparametric settings and accommodates stochastic regressors, broadening practical applicability. Overall, the work delivers a principled, computable approach to robust hypothesis testing under unknown forms of heteroskedasticity with clear guidance for practitioners.

Abstract

Tests based on heteroskedasticity robust standard errors are an important technique in econometric practice. Choosing the right critical value, however, is not simple at all: conventional critical values based on asymptotics often lead to severe size distortions; and so do existing adjustments including the bootstrap. To avoid these issues, we suggest to use smallest size-controlling critical values, the generic existence of which we prove in this article for the commonly used test statistics. Furthermore, sufficient and often also necessary conditions for their existence are given that are easy to check. Granted their existence, these critical values are the canonical choice: larger critical values result in unnecessary power loss, whereas smaller critical values lead to over-rejections under the null hypothesis, make spurious discoveries more likely, and thus are invalid. We suggest algorithms to numerically determine the proposed critical values and provide implementations in accompanying software. Finally, we numerically study the behavior of the proposed testing procedures, including their power properties.

Paper Structure

This paper contains 46 sections, 17 theorems, 97 equations, 5 figures, 6 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Framework
  3. Heteroskedasticity robust test statistics using unrestricted residuals
  4. Some intuition on why conventional critical values can lead to overrejection
  5. Size control results for $T_{Het}$ and $T_{uc}$ when $\mathfrak{C}=\mathfrak{C}_{Het}$
  6. Some examples
  7. Some variations on BakiSzek2005
  8. Results for heteroskedasticity robust test statistics using restricted residuals
  9. Size control results for $\tilde{T}_{Het}$ and $\tilde{T}_{uc}$ when $\mathfrak{C}=\mathfrak{C}_{Het}$
  10. Tests obtained from $\tilde{T}_{uc}$ or $\tilde{T}_{Het}$ can be trivial
  11. The case of $\tilde{T}_{uc}$
  12. The case of $\tilde{T}_{Het}$
  13. Generalizations
  14. Generalizations beyond Gaussianity
  15. Generalizations to stochastic regressors
  16. ...and 31 more sections

Key Result

Lemma 3.1

(a) $\hat{\Omega}_{Het}\left( y\right)$ is nonnegative definite for every $y\in \mathbb{R}^{n}$. (b) $\hat{\Omega}_{Het}\left( y\right)$ is singular (zero, respectively) if and only if $\limfunc{rank}\left( B(y)\right) <q$ ($B(y)=0$, respectively). (c) The set $\mathsf{B}$ given by $\left\{ y\in \ma

Figures (5)

  • Figure 1: Power functions for $n_{1}=3$. Left column: tests based on unrestricted residuals (cf. legend). Right column: tests based on restricted residuals (cf. legend). The rows corresponds to $\Sigma _{a}$ for $a=1,5,9$ from top to bottom. The abscissa shows $\delta$. In the left panel the HC0-HC4-curves turn out to be barely distinguishable, with the HC1-curve lying on top of the HC0-curve. In the right panel the HC4R-curve lies on top of the HC0R-HC3R-curves. See the text for an explanation.
  • Figure 2: Power functions for $n_{1}=9$. Left column: tests based on unrestricted residuals (cf. legend). Right column: tests based on restricted residuals (cf. legend). The rows corresponds to $\Sigma _{a}$ for $a=1,5,9$ from top to bottom. The abscissa shows $\delta$. In the left panel the HC0-HC4-curves turn out to be barely distinguishable, with the HC1-curve lying on top of the HC0-curve. In the right panel the HC4R-curve lies on top of the HC0R-HC3R-curves. See the text for an explanation.
  • Figure 3: Power functions for the design matrix considered in Section \ref{['sec:powhostX']}. Left column: tests based on unrestricted residuals (cf. legend). Right column: tests based on restricted residuals (cf. legend). The rows from top to bottom correspond to $\Sigma _{a}^{\ast }$ for $a=0,1,2,3,4$, the case $a=0$ corresponding to homoskedasticity. The abscissa shows $\delta$. In the left panel the HC1-curve lies on top of the HC0-curve. In the right panel the HC4R-curve lies on top of the HC0R-HC3R-curves. See the text for an explanation.
  • Figure 4: Power functions for $n_{1}=15$. Left column: tests based on unrestricted residuals (cf. legend). Right column: tests based on restricted residuals (cf. legend). The rows corresponds to $\Sigma _{a}$ for $a=1,5,9$ from top to bottom. The abscissa shows $\delta$. In the left panel the HC4-curve lies on top of the HC0--HC3-curves and the UC-curve. In the right panel the HC4R-curve lies on top of the HC0R-HC3R-curves and the UCR-curve. See the text for an explanation.
  • Figure 5: Regressor used in Example \ref{['ex:G3']}.

Theorems & Definitions (62)

  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 5.1
  • Remark 5.2
  • Remark 5.3
  • Remark 5.4
  • Proposition 5.5
  • Remark 5.6
  • Proposition 5.7
  • ...and 52 more