GARCH Models

Conditional heteroscedasticity models: volatility clustering in financial and other series

Introduction

GARCH (generalised autoregressive conditional heteroscedasticity) models series with time-varying conditional variance: quiet and volatile periods alternate. They are the workhorse of financial time series and also used in physiological signals.

Prerequisites

ARMA models, conditional distributions.

Theory

GARCH(1, 1):

\[r_t = \mu + \varepsilon_t, \qquad \varepsilon_t = \sigma_t z_t, \; z_t \sim \mathcal{N}(0, 1),\]

\[\sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2.\]

Mean structure can be combined with ARMA; GARCH models the innovations’ variance.

Extensions: EGARCH (asymmetric), TGARCH (threshold), GARCH-M (mean-spillover).

Assumptions

Conditional normality (or t, skewed-t); stationarity of variance requires \(\alpha + \beta < 1\).

R Implementation

library(rugarch)

# Simulate GARCH(1,1) returns
spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
                   mean.model = list(armaOrder = c(0, 0), include.mean = TRUE),
                   distribution.model = "norm")

set.seed(2026)
sim <- ugarchpath(spec = ugarchspec(
  variance.model = list(garchOrder = c(1, 1)),
  mean.model = list(armaOrder = c(0, 0))),
  n.sim = 500)
r <- as.numeric(fitted(sim))

# Fit
fit <- ugarchfit(spec, data = r)
show(fit)
plot(fit, which = 3)   # conditional sd

Output & Results

Parameter estimates, likelihood, conditional-variance plot.

Interpretation

“The fitted GARCH(1, 1) captured volatility clustering, with alpha + beta = 0.95 indicating highly persistent volatility.”

Practical Tips

• Check residuals (standardised): should be approximately iid, often t-distributed.
• Heavier-tailed returns require t or skewed-t innovation distributions.
• GARCH fits in rugarch are the industry standard.
• VaR forecasts use conditional-variance forecasts plus a quantile of the innovation distribution.
• Multivariate GARCH (DCC, BEKK) models time-varying correlations.