GARCHmodel

GARCHmodel refers to the generalized autoregressive conditional heteroskedasticity framework used to model time-varying volatility in time series, particularly financial returns. It extends the ARCH approach by allowing past conditional variances to influence current volatility, capturing volatility clustering commonly observed in markets.

In a standard GARCH(p,q) specification, let r_t be the return, r_t = μ + ε_t, with ε_t = σ_t z_t, where z_t are iid with E[z_t] = 0 and Var(z_t) = 1. The conditional variance evolves as σ_t^2 = ω + Σ_{i=1}^q α_i ε_{t-i}^2 + Σ_{j=1}^p β_j σ_{t-j}^2, with ω > 0, α_i ≥ 0, β_j ≥ 0, and the stationarity condition Σ α_i + Σ β_j < 1. The case GARCH(1,1) is the most commonly used form.

Extensions and variants include EGARCH, which models the logarithm of variance to allow asymmetric responses to shocks; TGARCH or GJR-GARCH, which incorporate leverage effects; and integrated or fractionally integrated versions like IGARCH and FIGARCH for persistent volatility. These models aim to better capture features such as volatility asymmetry and long memory.

Estimation is typically done via maximum likelihood, often assuming conditional normal or Student-t errors, with diagnostic checks on standardized residuals and squared residuals. Applications span volatility forecasting, risk management (e.g., Value at Risk), and option pricing. Limitations include sensitivity to model specification, non-stationarity, structural breaks, and potential misspecification of the error distribution.