GARCHtype

GARCH-type refers to a broad class of econometric models used to describe and forecast time-varying volatility in financial return series. The central idea is that the conditional variance of returns depends on past information, including squared shocks and past volatility, allowing volatility to cluster over time.

In a standard GARCH(p,q) specification, let r_t be the return and μ_t its conditional mean, with ε_t = r_t − μ_t. The conditional variance h_t follows h_t = ω + ∑_{i=1}^q α_i ε_{t−i}^2 + ∑_{j=1}^p β_j h_{t−j}, where ω > 0, α_i ≥ 0, β_j ≥ 0. The model implies that large shocks raise expected future volatility and that volatility itself feeds back into future variance. Stationarity typically requires ∑ α_i + ∑ β_j < 1.

GARCH-type models encompass several important variants. EGARCH (exponential GARCH) models the log of the variance, allowing asymmetric responses to shocks and ensuring positivity without explicit parameter constraints. TGARCH or GJR-GARCH introduces leverage effects so negative and positive shocks have different impacts on volatility. IGARCH imposes a unit root in the volatility process, implying persistent volatility. Other extensions adapt distributions of innovations (e.g., t-distributed errors) to capture heavy tails.

Applications of GARCH-type models include volatility forecasting for risk management, portfolio optimization, and option pricing. Estimation is typically done by maximum likelihood, assuming a distribution for the standardized residuals, often Gaussian or Student’s t. Software implementations are widespread in statistical packages, with common use in research and industry for modeling and forecasting financial volatility.