Patterns › Time series

GARCH / ARCH volatility

A GARCH(p, q) model lets conditional variance depend on q past squared shocks and p past variances, with an ARCH-LM test for volatility clustering.

What is GARCH / ARCH volatility?

Financial returns show volatility clustering — calm and turbulent periods bunch together. GARCH models the conditional variance hₜ = ω + Σα·ε²ₜ₋ᵢ + Σβ·hₜ₋ⱼ, capturing that persistence. GARCH(1,1) is the field's default.

Persistence α + β near 1 means shocks to volatility decay slowly (near-integrated variance). The ARCH-LM test (LM = n·R² from regressing squared values on their lags) checks whether any ARCH effect is present before/after fitting.

When should I use GARCH / ARCH volatility?

Modeling and forecasting the volatility of returns or any series with clustered variance.
Risk work (value-at-risk) needing a conditional-variance model.

What data does it need?

One numeric series (typically returns) + GARCH order p and ARCH order q + optional demeaning.

What does it report?

ω / α / β estimates with SEs, persistence, log-likelihood, AIC, and the ARCH-LM test.

What does it assume?

Mean-zero (or demeaned) series; GARCH models the variance, not the mean.
the fit uses Gaussian QML — heavy tails may warrant a t-distribution (out of scope here).

How do I interpret the result?

Significant α and β confirm ARCH/GARCH effects. α + β ≥ 1 warns of non-stationary variance.

References

Bollerslev (1986). Generalized autoregressive conditional heteroskedasticity. J Econometrics 31.