Skip to content

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
Aliases
Breusch–Pagan–Godfrey test, Breusch-Pagan test, BP test

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.

Structural Signature

Sig role-phrases:

  • Fitted primary regression — Supplies residuals and defines the disturbance whose variance is under examination. It is input. Counterfactual: Raw response variation without a mean model is not the target of this test.
  • Variance covariates — Specify the variables along which conditional variance may change. It is hypothesis. Counterfactual: An unspecified form of heteroskedasticity cannot be targeted by the original auxiliary model.
  • Squared residuals — Act as observable proxies for otherwise unobserved disturbance magnitudes. It is proxy. Counterfactual: Residual squares are noisy and inherit errors from the fitted mean model.
  • Auxiliary regression — Measures systematic association between squared residuals and variance covariates. It is operation. Counterfactual: A scatterplot alone does not produce the LM statistic.
  • LM statistic — Aggregates auxiliary explanatory power into a test quantity. It is decision. Counterfactual: A statistic without its degrees of freedom and reference law is uninterpretable.
  • Assumption set — Controls the validity and robustness of the reference distribution. It is validity. Counterfactual: Non-normality, dependence, misspecification, or small samples can alter calibration.

What It Is Not

  • 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.
  • Closest near-miss. 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.

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.

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.

Examples

Canonical

After fitting an income model, an analyst regresses squared residuals on income predictors and compares the LM statistic with the appropriate chi-squared reference.

Mapped back: mean model → linear; null → constant variance; auxiliary → specified.

Applied / In Practice

A significant result prompts examination of both variance structure and omitted nonlinear mean terms before the analyst changes inference.

Mapped back: result → reject; interpretation → diagnostic.

Applied / In Practice

An analyst merely plots residual spread against fitted values; this may reveal a pattern but is not a completed Breusch–Pagan test.

Mapped back: visual → True; LM statistic → False.

Structural Tensions

T1 — Targeted Power versus Broad Sensitivity. A focused variance model can be interpretable and powerful while missing forms not represented by its covariates.

Diagnostic: Which heteroskedastic alternatives are encoded in the auxiliary regression?

T2 — Asymptotic Convenience versus Finite-Sample Reliability. The chi-squared approximation is simple, but calibration depends on sample size and disturbance assumptions.

Diagnostic: Are robust or finite-sample variants required?

Structural–Framed Character

Breusch–Pagan Test is framed: structurally a null-versus-auxiliary variance diagnostic, and statistically framed by regression residuals, chosen covariates, asymptotic calibration, and disturbance assumptions.

Structural Core vs. Domain Accent

The core is a calibrated test of conditional variance dependence. Econometrics supplies fitted residuals, LM theory, chi-squared limits, heteroskedasticity, degrees of freedom, robust variants, and implications for standard errors.

This entry is a kind of Evaluation.

  • Approved root. General hypothesis-testing and regression-diagnostic nodes do not entail this named LM auxiliary construction.

  • Related — heteroskedasticity, Lagrange multiplier test, regression residual, White test, Goldfeld–Quandt test, and robust standard error. These are target, family, inputs, alternatives, or responses.

Relationships to Other Abstractions

Local relationship map for Breusch–Pagan testParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Breusch–Pagan testDOMAINPrime abstraction: Evaluation — is a kind ofEvaluationPRIME

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

Neighborhood in Abstraction Space

Breusch–Pagan test sits in a sparse region of the domain-specific corpus (69th 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

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

Not to Be Confused With

  • White Test. Tell: White's broader auxiliary expansion targets more general variance dependence.
  • Durbin–Watson Test. Tell: That diagnostic concerns serial correlation, not conditional variance.
  • Robust Standard Errors. Tell: They alter inference under heteroskedasticity but do not themselves test for it.
  • Residual Plot. Tell: A graph is exploratory evidence rather than the named LM procedure.

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Breusch%E2%80%93Pagan_test (revision 1358673491).
  • Preserved source candidate: https://bookdown.org/mike/data_analysis/heteroskedasticity-tests.html
  • Preserved source candidate: https://statistics.arabpsychology.com/the-breusch-pagan-test-definition-example/
  • Preserved source candidate: https://mran.microsoft.com/web/packages/car/car.pdf#page=86
  • Preserved source candidate: https://www.rdocumentation.org/packages/lmtest/versions/0.9-36/topics/bptest
  • Preserved source candidate: https://books.google.com/books?id=86rWI7WzFScC&pg=PA101
  • Preserved source candidate: https://mran.microsoft.com/web/packages/plm/plm.pdf#page.71
  • Preserved source candidate: https://cran.r-project.org/web/packages/skedastic/index.html
  • Preserved source candidate: https://www.stata.com/manuals15/rregresspostestimation.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.