Breusch–Pagan test

A Lagrange-multiplier regression diagnostic testing whether disturbance variance in a fitted linear model depends systematically on specified covariates rather than remaining constant.

Version

v1 · 2026-09-28 · History

Domain-specific #

8269

Domain group

Social Sciences

Origin domain

Economics & Finance

Subdomains

Econometrics, Regression Diagnostics → Economics & Finance

Core Idea

The Breusch–Pagan test begins after a linear mean model has been fitted. It uses squared residuals as proxies for latent disturbance variance and asks whether specified covariates explain their magnitude under a null of homoskedasticity.

Its LM statistic and chi-squared calibration are asymptotic and assumption-dependent. Rejection is evidence against constant variance under the chosen specification; it neither identifies a unique variance law nor proves that the mean model is correctly specified.

How would you explain it like I'm…

Are My Misses the Same Size?

Imagine a rule that guesses how tall kids are from their age. Some guesses are off by a little, some by a lot. The Breusch–Pagan test checks whether the misses get bigger for some kinds of kids, like older ones, instead of being about the same size for everyone.

The Uneven Mistakes Check

After economists make a straight-line prediction rule, they look at how far off each prediction was. Ideally, the size of the misses is about the same everywhere, which is called 'constant variance.' The Breusch–Pagan test checks whether some measured things, like income or age, help explain how big the misses are. If they do, that's evidence the misses aren't the same size everywhere, but it doesn't tell you exactly how they change or prove the rule itself is right.

Constant-Variance Test on Residuals

The Breusch–Pagan test is used after fitting a linear regression to check for heteroskedasticity, meaning the variance (spread) of the errors is not constant. The true errors can't be seen, so the test uses squared residuals as a stand-in for how big each error's variance is. It then checks whether chosen variables explain those squared residuals. The null hypothesis is homoskedasticity (constant variance). The test statistic is compared with a chi-squared distribution, but that's an approximation that relies on large samples and other assumptions. If the test rejects, that's evidence against constant variance for the chosen setup, not a description of exactly how the variance behaves and not proof that the regression's main equation is correct.

 

The Breusch–Pagan test is applied after a linear mean model has been estimated, to test the null hypothesis of homoskedasticity, that the disturbance variance is constant. Because the true error variances are latent, the squared residuals serve as proxies for them. The test asks whether a specified set of covariates explains the magnitude of these squared residuals, typically through an auxiliary regression. The resulting Lagrange multiplier statistic is compared with a chi-squared distribution, but that calibration is asymptotic and depends on assumptions. A rejection is evidence against constant variance under the chosen specification of covariates. It does not identify a unique law for how the variance varies, and it does not certify that the mean model is correctly specified.

Scope of Application

• Regression diagnostics. Tests one form of variance misspecification.
• Econometrics. Evaluates homoskedasticity assumptions behind conventional inference.
• Model comparison. Contrasts alternative choices of variance covariates.
• Sensitivity analysis. Compares classical and heteroskedasticity-robust conclusions.
• Teaching. Separates mean specification from conditional-variance specification.

Clarity

Report the primary regression, sample and missing-data handling, residual definition, variance covariates, auxiliary regression, LM statistic, degrees of freedom, p-value or critical rule, assumptions or robust variant, and diagnostic follow-up. Do not interpret non-rejection as proof of constant variance. Inclusion test: Require a fitted linear mean model, a constant-variance null, declared variance covariates, an auxiliary relation using squared residuals, and an LM decision calibrated under explicit assumptions. Exclusion test: Exclude visual residual inspection, the White test's broader generic expansion, tests for serial correlation, robust standard errors treated as a test, and variance patterns caused solely by an incorrectly specified mean model. Nearest boundary: The White test allows a more expansive auxiliary specification, including nonlinear terms and interactions; the original Breusch–Pagan test targets variance dependence on stated regressors or covariates. Exit condition: It stops being this test when the auxiliary statistic, null, or calibration is replaced by another heteroskedasticity diagnostic. Common misclassifications: It is not a test for serial correlation. It is not a residual plot. It is not identical to the White test. It is not a replacement for checking mean-model misspecification. Nearest named distinctions: White Test: White's broader auxiliary expansion targets more general variance dependence. Durbin–Watson Test: That diagnostic concerns serial correlation, not conditional variance. Robust Standard Errors: They alter inference under heteroskedasticity but do not themselves test for it. Residual Plot: A graph is exploratory evidence rather than the named LM procedure.

Manages Complexity

The test makes a hidden second-moment assumption inspectable through a second regression. It reduces the broad question of unequal variance to a declared alternative and a calibrated diagnostic while preserving the distinction between evidence and model repair.

Abstract Reasoning

1. Fit and assess the primary linear mean model.
2. State the homoskedastic null and variance covariates.
3. Compute residuals using a consistent convention.
4. Fit the required auxiliary regression to squared residuals.
5. Construct the LM statistic and correct degrees of freedom.
6. Interpret the result alongside assumptions, sample size, and alternative diagnostics.

Knowledge Transfer

The transferable cargo is an auxiliary-model test of whether unexplained variability depends on observed covariates. It transfers to other variance diagnostics only with their own statistic and reference law; the Breusch–Pagan name stops at its linear-regression LM construction and variants.

Relationships to Other Abstractions

Current abstraction Breusch–Pagan test Domain-specific

Parents (1) — more general patterns this builds on

• Breusch–Pagan test is a kind of Evaluation Prime

  Breusch–Pagan test is a strict kind of Evaluation: its frozen identity entails the parent's defining structure while adding domain-specific restrictions.

Hierarchy path (1) — routes to 1 parentless root

• Breusch–Pagan test → Evaluation → Comparison → Self Checking

Neighborhood in Abstraction Space

Breusch–Pagan test sits in a sparse region of the domain-specific corpus (69^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Statistical Hypothesis Tests & Diagnostics (9 abstractions)

Nearest neighbors

• Augmented Dickey–Fuller Test — 0.87
• Estimation of Covariance Matrices — 0.85
• Factor Regression Model — 0.84
• Ljung–Box Test — 0.84
• Ecosystem Model — 0.83

Computed from structural-signature embeddings · 2026-10-08