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.^[1] 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. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition 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 fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test. This gives the entry an operational identity rather than merely a historical label.

A useful analysis keeps three layers separate. The constitutive layer says what must be true: 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 evidential layer asks what observation or proof warrants the claim: 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. The use layer asks what reasoning becomes available once the identity is established: recognizing and comparing instances of Breusch–Godfrey test, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Conflating the layers is the most common source of scope inflation.

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.

Structural Signature¶

Carrier: the typed econometrics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets
Inputs or antecedent state: the exact econometrics carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Breusch–Godfrey test
Constitutive operation: 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.
Invariant: 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
Recognition test: 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
Output or consequence: recognizing and comparing instances of Breusch–Godfrey test, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions
Failure boundary: the carrier is mistyped, the condition 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 fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test

What It Is Not¶

It is not the whole field of econometrics. The field contains many questions and methods that do not instantiate Breusch–Godfrey test.
It is not its most familiar example. A canonical instance directly demonstrates 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. exhibits the structure, but the example is evidence for the abstraction rather than its definition.
It is not the neighboring catalog concept Durbin–Watson statistic. Durbin–Watson is primarily a first-order residual-correlation diagnostic with restrictions around lagged dependent variables; Breusch–Godfrey handles general chosen lag order.
It is not a claim that every boundary case has one uncontested classification. a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Breusch–Godfrey test must control the decision
It is not an unrestricted metaphor for any process that seems similar. Outside econometrics, the vocabulary and validity conditions do not transfer literally.

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. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.^[2]

Definition and recognition. Determine whether a proposed instance satisfies the constitutive conditions rather than merely sharing terminology.
Construction or evolution. Track how the exact econometrics carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Breusch–Godfrey test are converted, constrained, or organized by 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..
Comparison. Compare instances using carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior, without treating convenience measures as the definition.
Boundary analysis. Diagnose cases where a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Breusch–Godfrey test must control the decision and state which convention or theorem controls the decision.
Downstream reasoning. Use the established identity to support recognizing and comparing instances of Breusch–Godfrey test, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions while preserving the assumptions under which the inference is valid.

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. A bare label is insufficient because the name Breusch–Godfrey test can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated. The disciplined statement is: given the exact econometrics carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Breusch–Godfrey test, the structure counts as Breusch–Godfrey test exactly when 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.

This format also separates identity from measurement. Empirical, computational, or documentary proxies support recognition only under declared validity and uncertainty assumptions; formal cases require proof rather than measurement. Measurements can be noisy, implementations can approximate, and proofs can use equivalent characterizations; none of those facts licenses changing the object being measured. When reports disagree, first check scope and convention, then data or proof, and only then interpret the disagreement as substantive.

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.

The compression has a price. A single label can hide canonical, generalized, restricted, approximate, computational, empirical, and historically variant formulations of Breusch–Godfrey test. Good use therefore carries a small declaration of assumptions alongside the name. The abstraction manages complexity when it reduces the state space of the question while keeping the failure boundary visible; it mismanages complexity when the label substitutes for that boundary analysis.

Abstract Reasoning¶

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.
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. This step prevents the canonical example from becoming the definition.
Derive consequences. From 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, infer recognizing and comparing instances of Breusch–Godfrey test, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Record each assumption used so that a later change of setting does not silently preserve an invalid conclusion.
Test adversarial cases. Examine a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Breusch–Godfrey test must control the decision and an object that resembles Breusch–Godfrey test in purpose or vocabulary but does not satisfy its invariant is outside the class. A robust identity explains why the first is convention-sensitive and why the second is outside the class.
Compare and refine. Use carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior to compare legitimate instances, and refine the model when discrepancies reflect hidden variation rather than failure of the abstraction itself.

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. A theorem, diagnostic, or modeling warning can travel when those roles remain literal. For example, the distinction between constitutive identity and a convenient observable transfers from A canonical instance directly demonstrates 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. to An applied instance preserves the same invariant under changed scale, notation, jurisdiction, dataset, or implementation..^[3]

Transfer outside the home domain is weaker. The skeletal pattern—type the carrier, apply the defining mechanism of Breusch–Godfrey test, preserve its invariant, and derive only consequences licensed by the stated boundary—may suggest an analogy, but the domain-specific mechanisms, admissible evidence, and consequences do not come along automatically. The safe transfer procedure maps each role explicitly, checks the invariant again, and refuses the name when only a superficial resemblance remains.

Examples¶

Canonical¶

A canonical instance directly demonstrates 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. The example exposes the carrier and directly tests 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; changing incidental notation preserves the identity, while removing that condition destroys it. This example is canonical because every role can be inspected: the carrier is the typed econometrics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets; the operative rule is 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 invariant is 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; and the result supports recognizing and comparing instances of Breusch–Godfrey test, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions.^[1] Changing incidental notation or scale leaves the structure intact, while removing 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 destroys the classification.

Mapped back: 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. → 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 → recognizing and comparing instances of Breusch–Godfrey test, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions

Applied / In Practice¶

An applied instance preserves the same invariant under changed scale, notation, jurisdiction, dataset, or implementation. The applied case qualifies only because the same invariant and boundary test remain literal under changed parameters or implementation. The applied case is not licensed merely by vocabulary. It qualifies because the same recognition test—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—can be run and because the same failure boundary—the carrier is mistyped, the condition 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 fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test—remains meaningful.^[2] The case also shows why practical outputs should report assumptions, resolution, and uncertainty instead of a naked label.

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Axiomatic identity vs. operational recognition. The defining conditions may be exact while empirical or computational recognition is approximate. Neither pole can be removed without changing the analytical task. Diagnostic: Can the reviewer state both the exact condition and the evidence used to infer it?
T2: Local roles vs. global consequence. The mechanism is enacted through local relations, but the abstraction is usually valued for a global classification or prediction. Neither pole can be removed without changing the analytical task. Diagnostic: Does the claimed global result actually follow from the declared local conditions?
T3: Ideal form vs. finite representation. Theory states a clean invariant while data structures, measurements, or proofs expose only finite representations. Neither pole can be removed without changing the analytical task. Diagnostic: Would increasing resolution converge toward the same classification?
T4: Canonical convention vs. legitimate variants. A standard formulation supports communication, while variants may preserve the same core under changed assumptions. Neither pole can be removed without changing the analytical task. Diagnostic: Which role is invariant across variants, and which convention-specific conclusion changes?
T5: Compression vs. hidden assumptions. The name compresses a complex argument but can conceal prerequisites. Neither pole can be removed without changing the analytical task. Diagnostic: Can each downstream inference be traced to an explicit assumption?
T6: Autonomous residual vs. reduction to catalog neighbors. The candidate uses broader structures but adds an identity-bearing residual. Neither pole can be removed without changing the analytical task. Diagnostic: After subtracting the proposed parent and named neighbors, does the constitutive residual still support independent diagnostics?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is type the carrier, apply the defining mechanism of Breusch–Godfrey test, preserve its invariant, and derive only consequences licensed by the stated boundary. Its identity-bearing terms—Breusch–Godfrey test, carrier, parameter, invariant, boundary, evidence, model, transformation, and application—derive their meaning from econometrics and cannot be replaced by generic systems language without losing the tests that distinguish valid from invalid instances.

This mixed character explains why the abstraction is reusable inside the domain yet does not meet the Prime bar. The structure organizes reasoning, but its claims still depend on domain-specific objects, evidence, and intervention semantics.

Structural Core vs. Domain Accent¶

The structural core consists of a carrier, 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., a recognition invariant, and a consequence. That skeleton may resemble patterns elsewhere, especially type the carrier, apply the defining mechanism of Breusch–Godfrey test, preserve its invariant, and derive only consequences licensed by the stated boundary. The domain accent is not decorative: Breusch–Godfrey test, carrier, parameter, invariant, boundary, evidence, model, transformation, and application determine what counts as an admissible carrier, a valid transition, and successful evidence.

The abstraction therefore remains domain-specific. A cross-domain reuse that preserves only words such as 'balance,' 'cut,' 'sequence,' 'loss,' or 'simulation' is metaphor. Literal transfer requires the original role structure and diagnostics, which in this case remain anchored in econometrics.

This entry is a kind of Statistical Inference.

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

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

Not to Be Confused With¶

Durbin–Watson statistic. Durbin–Watson is primarily a first-order residual-correlation diagnostic with restrictions around lagged dependent variables; Breusch–Godfrey handles general chosen lag order.
One canonical example. An instance demonstrates the structure but does not define the whole abstraction.
Measurement or implementation of Breusch–Godfrey test. A proxy or realization is evidence for the abstraction, not the abstraction itself.
Generalized Breusch–Godfrey test. An extension qualifies only when its changed axioms and retained invariant are stated.

References¶

[1] T. S Breusch, 'Testing for Autocorrelation in Dynamic Linear Models', Australian Economic Papers, 1978, doi:10.1111/j.1467-8454.1978.tb00635.x. registry ↩a ↩b

[2] L. G Godfrey, 'Testing Against General Autoregressive and Moving Average Error Models when the Regressors Include Lagged Dependent Variables', Econometrica, 1978, doi:10.2307/1913829. registry ↩a ↩b

[3] Dimitrios Asteriou, Stephen G Hall, 'Applied Econometrics', Palgrave Macmillan, 2011.not authoritative registry ↩