Chaoticity Properties of Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic Processes

Abstract

Fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) arises in modeling of financial time series. FIGARCH is essentially governed by a system of nonlinear stochastic difference equations. In this work, we have studied the chaoticity properties of FIGARCH (p,d,q) processes by com- puting mutual information, correlation dimensions, FNNs (False Nearest Neighbour), the largest Lya- punov exponents (LLE) for both the stochastic difference equation and for the financial time series by applying Wolf's algorithm, Kant'z algorithm and Jacobian algorithm. Although Wolf's algorithm pro- duced positive LLE's, Kantz's algorithm and Jacobian algorithm which are subsequently developed methods due to insufficiency of Wolf's algorithm generated negative LLE's constantly. So, as well as experimenting Wolf's methods' inefficiency formerly pointed out by Rosenstein (1993) and more recently Dechert and Gencay (2000), based on Kantz's and Jacobian algorithm's negative LLE outcomes, we concluded that it can be suggested that FIGARCH (p,d,q) is not deter- ministic chaotic process.