Chaoticity Properties of Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic Processes

Adil Yilmaz,Gazanfer Unal

doi:10.18052/www.scipress.com/bmsa.15.69

Adil Yilmaz, Gazanfer Unal

Open Access

https://doi.org/10.18052/www.scipress.com/bmsa.15.69

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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 computing mutual information, correlation dimensions, FNNs (False Nearest Neighbour), the largest Lyapunov 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 produced 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 later 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 deterministic chaotic process.

Highlights

Detection of chaotic behavior in financial and economic data has been the topic of numerous scientific studies such as (Dechert and Gençay, 2000; Das and Das, 2006-7; Moeni et al, 2007; Günay, 2015)
We focus on the chaoticity properties of Fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) (Fractionally Integrated generalized autoregressive conditional heteroscedasticity) model by considering correlation dimension and Lyapunov exponents
This paper investigates the existence of chaoticity in nonlinear FIGARCH model by using simulated time series and nonlinear difference equation directly

Summary

Introduction

Detection of chaotic behavior in financial and economic (both micro and macro) data has been the topic of numerous scientific studies such as (Dechert and Gençay, 2000; Das and Das, 2006-7; Moeni et al, 2007; Günay, 2015). We focus on the chaoticity properties of FIGARCH (Fractionally Integrated generalized autoregressive conditional heteroscedasticity) model by considering correlation dimension and Lyapunov exponents. Grassberger and Procaccia (1983) was introduced a useful method in order to compute correlation dimension which measure an attractor dependent on a contraction rate of a fractal measure in some phase space in a given set They defined correlation sum which approximates the probability of having pair of points with separation distance less than a given size ε as,. In order to find out embedding dimensions for Figarch simulations, for the application of correlation dimension and the false nearest neighbor method, Kostelich and Swinney suggest that both methods work well when applied to low dimensional (3 or less) chaotic attractors. Equation below provides computation of maximal Lyapunov exponent in a direct way

M tevolv

Conclusion