Engle ARCH LM Test Calculator
Enter ordered regression residuals, a fixed squared-residual lag and any fitted-parameter correction to calculate Engle’s ARCH LM and companion F references. The page exposes the complete auxiliary regression but does not declare an ARCH state, forecast volatility, validate a strategy or issue a trade signal.
Enter residuals and fixed ARCH settings
Use oldest-to-newest residual order. The chosen lag and fitted-parameter correction are applied exactly with no hidden optimization.
Whole lags from 1 through 12. The selected lag is used exactly.
Use 0 when no correction is maintained; otherwise enter the fitted model parameter count you intend to disclose.
Enter 20 to 500 finite residuals without percent signs, oldest to newest, from one fitted model.
Squared-residual auxiliary regression
Entered Residual Diagnostics 1.0.0.
On smaller screens, scroll horizontally to inspect the complete audit table.
| Row | Residual squared | Fitted square | Auxiliary residual |
|---|
How Engle’s ARCH LM calculation works
LM = (n − q − ddof) × R²
Version 1.0.0 squares every entered residual, removes the first q rows and regresses each retained square on an intercept and its previous q squared residuals. The entered lag is used exactly and is never selected from the data.
The LM statistic multiplies auxiliary R-squared by the regression-row count minus the declared fitted-parameter correction. Its upper-tail reference uses chi-square with q degrees of freedom, matching statsmodels 0.14.6.
The companion F statistic compares the joint contribution of all q lag coefficients with an intercept-only auxiliary model. Coefficients and per-row fitted values remain visible so the result can be reconciled rather than treated as a black-box volatility label.
Worked example from the audited fixture
The governed fixture enters 48 residuals, selects three squared-residual lags and declares a fitted-parameter correction of two.
- The auxiliary regression retains 45 rows and R-squared is 0.10237714, with lag coefficients 0.31662493, −0.19624584 and −0.06137585.
- The LM statistic is 4.40221714 with chi-square probability 0.22117990; the F statistic is 1.55873290 with probability 0.21397414.
Reproduce it: select “Load audited example” above. The governed engine retains full precision and rounds only the visible interface.
How to interpret the result
- Read the declared lag and correction with the statistic. Changing either specification changes the reference and must remain in the audit trail.
- The maintained question concerns linear dependence in squared residuals over the selected lags. It does not describe every nonlinear or asymmetric volatility process.
- The fitted-parameter correction may be useful when residuals come from an estimated ARMA model, but the page cannot determine the correct count from residuals alone.
- Trying several lags and reporting only the smallest p-value creates an unreported search. Version 1.0.0 performs no multiple-testing adjustment.
- Conditional variance evidence does not supply a volatility forecast, position size, stop distance, expected return or proof that a trading rule is profitable.
Assumptions and limits
- Enter 20 to 500 finite residuals in meaningful chronological order and select one to 12 lags.
- The retained regression rows must exceed the intercept-plus-lag parameter count; otherwise the model fails closed.
- The fitted-parameter correction must be a whole number from zero through 20 and smaller than the retained row count.
- The calculator cannot verify residual provenance, equal spacing, original model order, correction choice, stationarity, structural stability, outliers or missing intervals.
- Chi-square and F references are asymptotic and can be unreliable under misspecification, dependence outside the chosen lag structure or selected samples.
- No ARCH/no-ARCH, volatility regime, model-adequacy, predictive-edge, strategy-validation, forecast, grade, signal, position instruction or recommendation verdict is generated.
Which residual diagnostic answers which question?
These four diagnostics do not create one interchangeable model-quality score. They use residual levels or squares, different regressors and different reference systems. Keep the data transformation, maintained null and selected specification visible before interpreting any statistic or probability.
| Diagnostic | Primary question | Output reference | Declared specification | Auxiliary regression |
|---|---|---|---|---|
| Durbin–Watson | Adjacent residual-level movement | Statistic only; design-dependent bounds withheld | Residual order | No |
| Ljung–Box | Joint residual autocorrelation through h | Chi-square upper-tail reference | Maximum lag and fitted orders | No |
| Breusch–Pagan / Koenker | Variance related to entered predictor | Chi-square and F upper-tail references | Predictor and variant | Yes |
| Engle ARCH LM | Squared residuals related to own lags | Chi-square and F upper-tail references | Lag and fitted correction | Yes |
Frequently asked questions
- It fits current squared residuals on an intercept and the selected number of their own lags, then reports conditional asymptotic references.
- It multiplies auxiliary R-squared by the retained regression-row count minus the entered fitted-parameter correction.
- It sets how many prior squared residuals enter the auxiliary regression and also supplies the LM chi-square degrees of freedom.
- It is a manually disclosed adjustment for parameters estimated by the model that produced the residuals; the calculator cannot infer the correct count.
- It is the auxiliary regression’s joint F reference for the selected squared-residual lag coefficients under the same fitted rows.
- No. Version 1.0.0 uses the entered lag exactly and performs no AIC, BIC, p-value search or multiple-testing correction.
- No. A squared-residual dependence reference does not estimate a GARCH model, future variance path, position size, stop distance or expected return.
- No. It generates no ARCH/no-ARCH verdict, volatility regime, model grade, forecast, signal, position instruction or recommendation.
Sources and methodology
- statsmodels — Engle ARCH Test — Official implementation documentation for squared-residual lags, correction and returned LM/F references.
- Engle (1982) — Original Econometrica paper introducing autoregressive conditional heteroskedasticity.
Version 1.0.0 was locked after its statistics, reference probabilities, auxiliary coefficients and worked examples were independently recomputed with statsmodels 0.14.6 and SciPy 1.13.1. The browser calculator performs local arithmetic and does not upload entered observations. Reference probabilities remain conditional on the disclosed model and data assumptions.
Audit the residual and volatility evidence
Verify the source model before testing its residuals
Reconcile the exact symbol, fitted equation, observation timestamps, timezone, frequency, missing rows, spread, commission, financing, currency conversion, rollover adjustments and preprocessing before entering residuals. Correct diagnostic arithmetic cannot repair a misspecified, selected, misaligned or cost-inconsistent source model.
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