ARCH-LM Test for Volatility Clustering
Also known as: ARCH-LM Testi ve Volatilite Kümelenmesi Analizi, ARCH LM test, Engle's ARCH test, test for autoregressive conditional heteroscedasticity, volatility clustering test
The ARCH-LM test is Robert Engle's (1982) Lagrange multiplier diagnostic for autoregressive conditional heteroscedasticity in the residuals of a fitted time-series model. It checks whether the error variance changes over time and clusters into calm and turbulent periods, and it is the standard pre-test run before fitting a GARCH-family volatility model.
Key highlights
- Directly targets conditional (time-varying) heteroscedasticity rather than the static heteroscedasticity that Breusch-Pagan or White tests address.
- Simple and fast: a single auxiliary regression of squared residuals on their lags yields the n·R² statistic.
- Provides the standard, literature-grounded justification for moving to a GARCH-family volatility model when the null is rejected.
Intuition
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How it works
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When to use it
Use the ARCH-LM test on the residuals of a fitted time-series model when you suspect time-varying volatility, especially with financial returns or other series that show calm-then-turbulent patterns. It assumes the residuals of a linear mean model have already been computed and that you have a reasonable time-series sample (at least about 30 observations). It is the routine check before committing to GARCH, EGARCH, or GJR-GARCH modelling.
Strengths & limitations
- Directly targets conditional (time-varying) heteroscedasticity rather than the static heteroscedasticity that Breusch-Pagan or White tests address.
- Simple and fast: a single auxiliary regression of squared residuals on their lags yields the n·R² statistic.
- Provides the standard, literature-grounded justification for moving to a GARCH-family volatility model when the null is rejected.
- It is a diagnostic only — it detects the presence of ARCH effects but does not estimate or forecast the volatility process itself.
- Results depend on the chosen number of lags q, and the residuals from the mean model must be correctly specified beforehand.
- It does not distinguish symmetric ARCH effects from asymmetric leverage effects, which require models such as EGARCH or GJR-GARCH.
Common pitfalls
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Applications
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Frequently asked
What does it mean if the ARCH-LM test rejects the null?
Rejection (a small p-value) means the squared residuals are serially related, so the error variance is conditional on the recent past — volatility clustering is present. This indicates that a GARCH-family model is more appropriate than assuming constant variance.
How is the ARCH-LM test different from the Breusch-Pagan or White tests?
Breusch-Pagan and White detect static heteroscedasticity, where the variance depends on the regressors. The ARCH-LM test targets conditional heteroscedasticity in time, where the current variance depends on past squared shocks — the time-series notion of volatility clustering.
How many lags should I use?
The lag count q sets how far back past squared residuals are allowed to influence the current variance. There is no universal rule; a moderate value tied to the data frequency is common, and sensitivity to the choice should be checked because too few or too many lags can distort the result.
What should I do after the test rejects?
Fit a conditional-variance model. A symmetric GARCH captures basic clustering, while EGARCH or GJR-GARCH add the asymmetric leverage effect where negative shocks raise volatility more than positive ones.
Sources
- 1.Engle, R. F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, 50(4), 987-1007.
- 2.Lee, J. H. H. (1991). A Lagrange Multiplier Test for GARCH Models. Economics Letters, 37(3), 265-271.
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ScholarGate. (2026, June 1). ARCH-LM Test. ScholarGate. https://scholargate.app/econometrics/arch-lm-test