ARCHGARCH

ARCHGARCH refers to a family of statistical models used to describe and forecast time-varying volatility in time series, particularly financial returns. The ARCH model, introduced by Engle in 1982, models the conditional variance as a function of past squared innovations. The GARCH model, proposed by Bollerslev in 1986, generalizes ARCH by allowing lagged conditional variances to influence current volatility, creating a more flexible autoregressive structure.

In practice, ARCHGARCH denotes GARCH-family models such as GARCH(p, q) and its common extensions. A typical setup assumes a mean process y_t with a conditional variance h_t, where y_t = mu + epsilon_t, epsilon_t = sqrt(h_t) z_t and z_t are i.i.d. with mean zero and unit variance. The volatility follows h_t = omega + sum_{i=1}^p alpha_i epsilon_{t-i}^2 + sum_{j=1}^q beta_j h_{t-j}. Special cases include integrated, exponential, and threshold variants (IGARCH, EGARCH, TGARCH), each providing different dynamics for volatility persistence and leverage effects.

Estimation is usually performed via maximum likelihood under assumed error distributions (e.g., normal or t-distribution). ARCHGARCH models are widely implemented in statistical software, such as R packages (for example, rugarch) and Python libraries (such as arch), enabling volatility forecasting, risk management, and option pricing.

Applications commonly include forecasting volatility for Value-at-Risk, hedging, and portfolio optimization. Limitations involve model mis-specification, sensitivity to outliers, and the assumption of a particular parametric form for volatility dynamics. Researchers also explore extensions like long-memory GARCH, multivariate GARCH, and stochastic volatility as alternatives.