Advanced Time Series: GARCH and Volatility Models

ARCH and GARCH Models

Financial and insurance time series often exhibit volatility clustering: large changes tend to follow large changes. ARCH(q) models conditional variance as a function of past squared residuals: sigma_t^2 = alpha_0 + alpha_1*epsilon_{t-1}^2 + ... + alpha_q*epsilon_{t-q}^2. GARCH(p,q) adds lagged conditional variances: sigma_t^2 = alpha_0 + sum of alpha_i*epsilon_{t-i}^2 + sum of beta_j*sigma_{t-j}^2. GARCH(1,1) with sigma_t^2 = alpha_0 + alpha_1*epsilon_{t-1}^2 + beta_1*sigma_{t-1}^2 captures most volatility dynamics. Stationarity requires alpha_1 + beta_1 < 1.

Extensions and Applications

EGARCH models asymmetric volatility (negative returns increase volatility more than positive returns): ln(sigma_t^2) depends on both epsilon_{t-1}/sigma_{t-1} and its absolute value. GJR-GARCH adds an indicator for negative residuals. IGARCH (alpha + beta = 1) models persistent volatility. In actuarial applications, GARCH models capture time-varying risk in investment returns for variable annuity pricing, interest rate volatility for reserve valuation, and loss ratio volatility for dynamic financial analysis. Exam MAS-II tests GARCH model specification, parameter interpretation, and volatility forecasting.