GARCH Models

Conditional heteroscedasticity models: volatility clustering in financial and other series

Introduction

GARCH (Generalised Autoregressive Conditional Heteroscedasticity) models capture time-varying conditional variance — the volatility clustering pattern in which quiet periods alternate with volatile ones. Financial returns are the prototypical application: stock prices show stretches of low daily variability punctuated by high-volatility crisis episodes, a structure that conditional-Normal ARMA models cannot accommodate. GARCH extends ARMA by adding an autoregressive equation for the conditional variance itself, making volatility a forecastable quantity.

Prerequisites

A working understanding of ARMA models, conditional distributions, and the distinction between unconditional and conditional variance.

Theory

The GARCH(1, 1) model decomposes returns as

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

with the conditional variance evolving as

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

The mean structure can be combined with an ARMA component; GARCH specifically models the variance of the innovations. Stationary unconditional variance requires \(\alpha + \beta < 1\); values close to 1 indicate highly persistent volatility (typical for daily financial returns).

Extensions handle asymmetric responses to positive vs negative shocks (EGARCH, TGARCH, APARCH), variance feedback into the mean (GARCH-M), and multivariate volatility (DCC, BEKK). For non-Normal returns, \(t\)- or skewed-\(t\)-distributed innovations are standard.

Assumptions

Conditional Normality (or \(t\), skewed-\(t\)) of innovations \(z_t\); stationarity of the conditional variance (\(\alpha + \beta < 1\)); the chosen GARCH order is appropriate for the data.

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

ugarchfit() returns parameter estimates (\(\omega, \alpha, \beta\), mean parameters), log-likelihood, AIC/BIC, and Ljung-Box-style tests on standardised and squared standardised residuals. The conditional-standard-deviation plot shows the time-varying volatility extracted by the model.

Interpretation

A reporting sentence: “A GARCH(1, 1) fit to daily returns estimated \(\hat\omega = 0.0001\), \(\hat\alpha = 0.08\), \(\hat\beta = 0.91\) (\(\hat\alpha + \hat\beta = 0.99\), very persistent volatility); standardised residuals showed no remaining ARCH effects (Ljung-Box on squared residuals \(p = 0.42\)). Conditional-volatility forecasts widen modestly over the 30-day horizon.” Always report the persistence sum and standardised-residual diagnostics.

Practical Tips

• Check standardised residuals (residuals(fit, standardize = TRUE)); they should be approximately iid and often heavier-tailed than Normal, motivating \(t\) or skewed-\(t\) innovations.
• For asymmetric responses to positive vs negative shocks (the leverage effect in equity returns), use EGARCH or TGARCH; standard GARCH is symmetric and misses this.
• rugarch is the industry standard for univariate GARCH; for multivariate volatility (covariance matrices), rmgarch extends to DCC, BEKK, and copula-GARCH.
• Value-at-Risk forecasts come from the conditional-variance forecast plus a quantile of the innovation distribution; ugarchforecast() and ugarchroll() support rolling VaR estimation.
• GARCH on returns is conventional, but the framework applies to any series with volatility clustering — physiological signals, atmospheric data, traffic counts.
• For very long series, the parameter estimates can be stable but the conditional-volatility series may need recursive re-estimation; use ugarchroll() for genuine out-of-sample evaluation.

R Packages Used

rugarch for univariate GARCH and its many extensions; rmgarch for multivariate GARCH (DCC, BEKK); fGarch as an alternative implementation; bayesGARCH for Bayesian GARCH; tsgarch for the modern tidyverse-flavoured interface.