On the Finite Dimensional Laws of Threshold GARCH Processes

Abstract

In this chapter we establish bounds for the finite dimensional laws of a threshold GARCH process, X, with generating process Z. In this class of models the conditional standard deviation has different reactions according to the sign of past values of the process. So, we firstly find lower and upper bounds for the law of $\left ({X}_{1}^{+},-{X}_{1}^{+},\ldots,{X}_{n}^{+},-{X}_{n}^{+}\right )$, in certain regions of ${\mathbb{R}}^{2n}$, and use them to find bounds of the law of $\left ({X}_{1},\ldots,{X}_{n}\right )$. Some of these bounds only depend on the parameters of the model and on the distribution function of the independent generating process, Z. An application of these bounds to control charts for time series is presented.