Heteroskedasticity-robust tests in linear regression: A review and evaluation of small-sample corrections

James E. Pustejovsky and Gleb Furman


Linear regression

Let’s talk about a basic regression model:

\[ \begin{aligned} y_i &= \beta_0 + \beta_1 x_{1i} + \cdots + \beta_{p-1} x_{p-1,i} + e_i \\ \mathbf{y} &= \mathbf{X} \boldsymbol\beta + \mathbf{e} \end{aligned} \] estimated by ordinary least squares:

\[ \boldsymbol{\hat\beta} = \left(\mathbf{X}'\mathbf{X}\right)^{-1} \mathbf{X}'\mathbf{y}. \]


Potential small-sample improvements

Other potential small-sample improvements

Approximations for the reference distribution of test statistic:


Size of HC* variants, \(\alpha = .05\)

Size of HC* variants, \(\alpha = .01\)

Size of HC* variants, \(\alpha = .005\)

Size of selected tests, \(\alpha = .05\)

Size of selected tests, \(\alpha = .01\)

Size of selected tests, \(\alpha = .005\)


  1. Currently recommended test HC3 does not adequately control type-I error rate.
  2. At the \(\alpha = .05\) level, HC4 maintains most accurate rejection rates of all tests considered.
  3. At smaller \(\alpha\) levels, Satterthwaite and Edgeworth approximations out-perform HC3 and HC4.



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Degree of heteroskedasticity