0.1 ARCH Models
Classical linear models (ARMA) assume constant (conditional) variance. Financial returns instead show volatility clustering — large changes tend to be followed by large changes — motivating models where the conditional variance, not just the conditional mean, depends on the past.
Let \(r_t = \mu_t + \varepsilon_t\) with \(\varepsilon_t = \sigma_t Z_t\), \(Z_t \sim IID(0,1)\). The process is ARCH(q) if \[ \sigma_t^2 = \omega + \alpha_1\varepsilon_{t-1}^2 + \dots + \alpha_q\varepsilon_{t-q}^2, \qquad \omega>0,\ \alpha_i \geq 0. \] Squaring shows \(\varepsilon_t^2\) itself follows an \(AR(q)\)-like structure with noise \(\eta_t = \varepsilon_t^2 - \sigma_t^2\): \(\varepsilon_t^2 = \omega + \sum_i \alpha_i \varepsilon_{t-i}^2 + \eta_t\), so ARCH order can be diagnosed by inspecting the ACF/PACF of squared residuals (or via Engle’s LM test for ARCH effects).
Properties. \(\varepsilon_t\) is unconditionally stationary with mean \(0\) and variance \(\omega/(1-\sum\alpha_i)\) when \(\sum_i \alpha_i < 1\), but is unconditionally heavy-tailed (excess kurtosis) even when \(Z_t\) is Gaussian — capturing the fat tails observed in asset returns.
Limitations. Capturing realistic volatility persistence typically requires a large lag order \(q\), which is parameter-inefficient; this motivated the GARCH extension, which adds autoregressive terms in \(\sigma_t^2\) itself (an \(ARMA\)-type generalization of ARCH’s pure-\(AR\) structure).