Breusch–Godfrey test

A regression diagnostic testing residual serial correlation through an auxiliary regression that permits higher-order autocorrelation and lagged dependent regressors.

Version

v2 · 2026-09-19 · History

Domain-specific #

3542

Domain group

Social Sciences

Origin domain

Economics & Finance

Subdomain

Econometrics → Economics & Finance

Core Idea

The LM statistic is asymptotically chi-square under regularity conditions; lag order, deterministic terms, heteroskedasticity and small-sample F variants affect interpretation. Residuals from the original model are regressed on its regressors and selected residual lags, and the auxiliary explained variation measures whether lagged errors add systematic predictive power. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of econometrics. It is the domain-specific identity determined by the fitted model and time ordering, null correlation orders, lag choice, auxiliary regression and included regressors, LM or F statistic, reference distribution, sample adjustment, significance and residual assumptions are explicit.

How would you explain it like I'm…

Do Mistakes Follow Each Other?

Scientists sometimes make a guessing rule to predict numbers day after day, and the guesses are a little off each day. The Breusch–Godfrey test checks whether yesterday's mistake gives a clue about today's mistake. If it does, the guessing rule is missing a pattern.

The Linked Mistakes Check

Scientists often make a rule to predict things over time, like sales each month, and the rule is never perfect. The gaps between the guesses and what really happened are called errors. The Breusch–Godfrey test asks whether past errors help predict the next error. To do this, it takes the errors and tries to predict them using the original information plus earlier errors. If the earlier errors help a lot, that is evidence the errors are connected over time, which the original rule did not expect.

Serial Correlation Test on Residuals

The Breusch–Godfrey test is a statistics test used in econometrics to check a time-series regression for serial correlation (also called autocorrelation), meaning the model's errors are correlated with earlier errors. You fit your original model, collect its residuals, and then run a second 'auxiliary' regression of those residuals on the original explanatory variables plus a chosen number of lagged residuals. If the lagged residuals add real predictive power, that's evidence of serial correlation up to that lag. The test statistic (an LM statistic, or an F-version for small samples) is compared with a chi-square distribution, but that comparison is only approximately right in large samples and under certain assumptions. Choices like how many lags to include affect what the result means.

 

The Breusch–Godfrey test is a Lagrange multiplier (LM) test for serial correlation in the errors of a regression model with time-ordered data. After estimating the original model, its residuals are regressed on the original regressors together with a chosen number of lagged residuals. The explained variation of this auxiliary regression measures whether lagged errors carry systematic predictive power, which would be absent under the null hypothesis of no serial correlation up to the chosen order. The LM statistic is asymptotically chi-square under regularity conditions, and small-sample F versions exist. Interpretation depends on the lag order selected, the deterministic terms included, heteroskedasticity, and sample-size adjustments. A proper report states the fitted model and time ordering, the correlation orders under the null, the lag choice, the auxiliary regression and its regressors, whether an LM or F statistic was used, the reference distribution, the significance level and the residual assumptions.

Scope of Application

Breusch–Godfrey test belongs to econometrics and is useful where the analyst can specify the typed econometrics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the fitted model and time ordering, null correlation orders, lag choice, auxiliary regression and included regressors, LM or F statistic, reference distribution, sample adjustment, significance and residual assumptions are explicit. The scope is broad within that domain but bounded by the need for the fitted model and time ordering, null correlation orders, lag choice, auxiliary regression and included regressors, LM or F statistic, reference distribution, sample adjustment, significance and residual assumptions are explicit.

Clarity

The abstraction clarifies a crowded vocabulary by making the fitted model and time ordering, null correlation orders, lag choice, auxiliary regression and included regressors, LM or F statistic, reference distribution, sample adjustment, significance and residual assumptions are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Breusch–Godfrey test. Breusch–Godfrey test compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed econometrics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
2. Lock the constitutive rule. Express the fitted model and time ordering, null correlation orders, lag choice, auxiliary regression and included regressors, LM or F statistic, reference distribution, sample adjustment, significance and residual assumptions are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of econometrics because they reuse the typed econometrics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Residuals from the original model are regressed on its regressors and selected residual lags, and the auxiliary explained variation measures whether lagged errors add systematic predictive power., and type the carrier, state every parameter and convention in the definition, test that the fitted model and time ordering, null correlation orders, lag choice, auxiliary regression and included regressors, LM or F statistic, reference distribution, sample adjustment, significance and residual assumptions are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Breusch–Godfrey test Domain-specific

Parents (1) — outgoing typed relations

• Breusch–Godfrey test is a kind of Statistical Inference Prime

  It remains its own entry because its identity is fixed by the fitted model and time ordering, null correlation orders, lag choice, auxiliary regression and included regressors, LM or F statistic, reference distribution, sample adjustment, significance and residual assumptions.

Hierarchy paths (3) — routes to 3 parentless roots

• Breusch–Godfrey test → is a kind of Statistical Inference → is a kind of Inductive Reasoning

Show alternative paths (2)

Neighborhood in Abstraction Space

Breusch–Godfrey test sits in a crowded region of the domain-specific corpus (26^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Economic & Financial Indicators (42 abstractions)

Nearest neighbors

• KPSS test — 0.91
• Pareto index — 0.90
• Polytomous choice — 0.89
• Full employment — 0.89
• Ecological regression — 0.89

Computed from structural-signature embeddings · 2026-10-25