Augmented GARCH (p,q) process and its diffusion limit

A family of parametric GARCH models, defined in terms of an auxiliary process and referred to as the augmented GARCH process, is proposed. The strict stationarity of the augmented GARCH process is characterized and this process is shown to contain many existing parametric GARCH models. The augmented GARCH process can serve as a general alternative for Lagrange Multiplier test of many existing GARCH specifications. The diffusion limit of the augmented GARCH process is shown to contain many bivariate diffusion processes that are commonly used for modeling stochastic volatility in the finance literature. This convergence result generalizes that of Nelson (1990a) to cover a substantially larger class of GARCH (1,1) models and also extends to the GARCH (p,q) specification. The augmented GARCH process can be used as a direct approximation to the stochastic volatility models, or as the score generator in the efficient method of moments (Gallant and Tauchen, 1996) estimation of these models.