Heteroskedasticityintermediate

Testing for Heteroskedasticity: Breusch-Pagan and White

The Breusch-Pagan and White tests both ask whether the error variance depends on the regressors, and both work by running an auxiliary regression of the squared OLS residuals $\hat{u}^2$ on a set of explanatory terms. Breusch-Pagan regresses $\hat{u}^2$ on the original regressors; the White test adds their squares and cross products, so it also catches variance that depends on nonlinear combinations. The null is homoskedasticity ( $\mathrm{H_0}:$ constant variance); a small p-value rejects it. In Stata, `estat hettest` runs Breusch-Pagan and `estat imtest, white` runs the White test, both issued right after `regress`.

Why it matters

If the spread of the errors really is constant, then nothing about the regressors should help you predict how big a residual is. So square the residuals, which turns spread into a level you can model, and regress them on the $x$ variables. If those variables explain a meaningful chunk of the squared residuals, the spread is moving with $x$ and you have heteroskedasticity. Breusch-Pagan looks for a straight-line link to the regressors; the White test casts a wider net by also including squares and interactions, at the price of using up more degrees of freedom.

Formulas

Breusch-Pagan auxiliary regression

\hat{u}^2 = \delta_0 + \delta_1 x_1 + \delta_2 x_2 + \cdots + \delta_k x_k + \text{error}

Test

\mathrm{H_0}: \delta_1 = \cdots = \delta_k = 0

. Reject (heteroskedasticity) if the joint

F

, or the

LM = n \cdot R_{\hat{u}^2}^2

statistic, is large.

White test LM statistic

LM = n \cdot R^2_{\hat{u}^2} \;\sim\; \chi^2_{q}

R^2_{\hat{u}^2}

comes from regressing

\hat{u}^2

on the regressors, their squares, and their cross products;

q

is the number of those terms. A large

LM

rejects homoskedasticity.

Worked examples

Scenario

After a housing price regression you want to test formally whether the error variance is constant, using both the Breusch-Pagan and White tests.

Solution

Run `regress price lotsize sqrft bdrms`, then `estat hettest` for Breusch-Pagan and `estat imtest, white` for White. Each returns a chi-squared statistic and a p-value testing $\mathrm{H_0}$ of homoskedasticity. If `estat hettest` reports a p-value below 0.05, you reject constant variance and should move to robust SEs (or WLS). The White test, having more terms, may reject when Breusch-Pagan does not if the variance depends on squares or interactions.

Note`estat hettest` defaults to the fitted values version; add the regressors as arguments to test against them specifically. `estat imtest, white` is the full White test.

Common mistakes

✗The null hypothesis is heteroskedasticity, so a small p-value means the errors are fine. It is the reverse. $\mathrm{H_0}$ is homoskedasticity, so a small p-value rejects constant variance and signals heteroskedasticity.
✗The tests regress the residuals $\hat{u}$ themselves on the regressors. They regress the squared residuals $\hat{u}^2$ , because squaring converts the unobserved spread into a level the auxiliary regression can model.
✗The White test is uniformly better than Breusch-Pagan. It is more general because it includes squares and cross products, but it uses many more degrees of freedom, which can cost power in small samples; the two answer slightly different questions.
✗Failing to reject means heteroskedasticity is definitely absent. A non-rejection is only weak evidence; with a small sample the test may simply lack power. Many practitioners report robust SEs regardless of the test outcome.

Revision bullets

•Both tests use an auxiliary regression of $\hat{u}^2$ on explanatory terms
•Breusch-Pagan: $\hat{u}^2$ on the regressors; White: also squares and cross products
• $\mathrm{H_0}$ is homoskedasticity; small p-value rejects it
•Statistics: joint $F$ or $LM = n R^2_{\hat{u}^2}$ from the auxiliary fit
•Stata: `estat hettest` (Breusch-Pagan), `estat imtest, white` (White)

Quick check

In both the Breusch-Pagan and White tests, the null hypothesis is:

What is the dependent variable in the auxiliary regression for these tests?

How does the White test differ from the Breusch-Pagan test?

Connected topics

Fitted / Resid Gauss-Markov F test Heteroskedastic Robust SE WLS / FGLS

Sources

Wooldridge (2019), §8.3
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach, 7th ed., Section 8.3, Testing for Heteroskedasticity. Cengage, 2019.
Presents the Breusch-Pagan and White tests as auxiliary regressions of the squared residuals, with the F and LM forms of the statistic.
Breusch & Pagan (1979)
Breusch, T. S., and A. R. Pagan. A Simple Test for Heteroscedasticity and Random Coefficient Variation. Econometrica 47(5): 1287-1294, 1979.
Original Breusch-Pagan test based on regressing squared residuals on the explanatory variables.

How to cite this page

Dr. Phil's Quant Lab. (2026). Testing for Heteroskedasticity: Breusch-Pagan and White. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-heteroskedasticity-tests

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