GARCH11

GARCH11, commonly written as GARCH(1,1), is a widely used econometric model for the conditional variance of a return series. It allows volatility to cluster by letting today’s conditional variance depend on past squared innovations and past variances. The model is a standard tool in financial econometrics for modeling time-varying volatility.

The formal specification can be written as follows. The return process is r_t = mu + epsilon_t, with epsilon_t = sigma_t z_t, and z_t are independent and identically distributed with E[z_t] = 0 and Var(z_t) = 1. The conditional variance evolves as sigma_t^2 = alpha0 + alpha1 epsilon_{t-1}^2 + beta1 sigma_{t-1}^2. Practical estimation requires parameter constraints: alpha0 > 0, alpha1 >= 0, beta1 >= 0, and alpha1 + beta1 < 1 for stationarity.

Interpretation focuses on two components: alpha1 captures the impact of past shocks to volatility, while beta1 captures the persistence of volatility over time. A sum of alpha1 and beta1 near one indicates highly persistent volatility. Estimation is typically performed by maximum likelihood, often under normal or t-distributed error assumptions, with quasi-maximum likelihood robust to non-normality. Software implementations are widely available in statistical packages.

Key properties include the unconditional variance of epsilon_t given by alpha0 / (1 - alpha1 - beta1). While GARCH(1,1) is parsimonious and effective at capturing volatility clustering, it assumes a symmetric response to shocks and may be misspecified if leverage effects are present. Extensions such as EGARCH and GJR-GARCH address asymmetry. GARCH(1,1) remains a foundational model for volatility forecasting and risk management in finance.