0.1 GARCH Models
\(\varepsilon_t = \sigma_t Z_t\), \(Z_t \sim IID(0,1)\), with conditional variance \[ \sigma_t^2 = \omega + \sum_{i=1}^q \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^p \beta_j \sigma_{t-j}^2, \qquad \omega>0,\ \alpha_i,\beta_j\geq 0. \] This is the volatility analogue of an \(ARMA\) model: ARCH(q) terms are the “MA” part (shocks) and the \(\beta_j\sigma_{t-j}^2\) terms are the “AR” part (persistence), letting a low-order GARCH(1,1) match the volatility persistence that would need a very large ARCH order.
GARCH(1,1) is the workhorse: \(\sigma_t^2 = \omega + \alpha\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2\), covariance-stationary iff \(\alpha+\beta<1\), with unconditional variance \(\omega/(1-\alpha-\beta)\) and \(\alpha+\beta\) close to \(1\) typically observed empirically (high persistence, slow mean reversion of volatility).
Estimation. Usually by (quasi-)maximum likelihood, conditioning on initial values, since \(\sigma_t^2\) is a deterministic (recursively defined) function of past data given parameters — unlike stochastic volatility models, where \(\sigma_t\) has its own noise source and requires filtering/simulation-based estimation (e.g. stochvol-style MCMC).
Extensions. - IGARCH: \(\alpha+\beta=1\), integrated (unit-root) volatility, connecting to long-memory volatility behavior. - EGARCH / GJR-GARCH: allow asymmetric (“leverage”) response, where negative returns raise future volatility more than positive returns of the same size. - GARCH-M: feeds \(\sigma_t^2\) (or \(\sigma_t\)) into the conditional mean equation, pricing a risk premium for volatility. - Multivariate GARCH (BEKK, DCC): extends to a vector of returns with time-varying conditional covariance/correlation matrices.
GARCH models the conditional variance parametrically from returns; an alternative, model-light approach uses high-frequency data to construct a nonparametric ex-post volatility measure — see Realized Volatility and HAR-RV Model.