GARCHpqmodel

GARCHpqmodel, commonly written as GARCH(p,q), is a class of statistical models used to describe time-varying volatility in financial time series. It extends the ARCH family by allowing the current conditional variance to depend on both past squared innovations and past conditional variances. The model is widely used in risk management, asset pricing, and econometric forecasting.

Let r_t denote the return at time t. Then r_t = mu + e_t, with e_t = sigma_t z_t and z_t following an independent and identically distributed distribution with mean zero and unit variance. The conditional variance is specified as sigma_t^2 = omega + sum_{i=1}^p alpha_i e_{t-i}^2 + sum_{j=1}^q beta_j sigma_{t-j}^2. Parameters satisfy omega > 0, alpha_i >= 0, beta_j >= 0. A common stationarity condition is sum alpha_i + sum beta_j < 1.

Estimation is typically performed by maximum likelihood, assuming a distribution for z_t such as normal or Student's t. Model selection uses information criteria like AIC or BIC. Diagnostics often involve checking squared residuals for remaining ARCH effects and using ARCH LM tests.

Extensions and related models include GARCH-M (volatility feeding into the mean), Exponential GARCH (EGARCH), and GJR-GARCH for asymmetry and leverage effects. Multivariate GARCH models handle volatility spillovers across assets. Practical applications encompass forecasting volatility for value-at-risk, option pricing, and portfolio optimization. Limitations include sensitivity to model mis-specification, nonstationarity, and heavy-tailed or skewed error distributions.

Note on terminology: the term GARCHpqmodel is not standard in the literature; the conventional designation is GARCH(p,q). The article describes the general GARCH(p,q) framework and its common interpretations and uses.