Subscribe

Subscribe to our Newsletter and get informed about new publication regulary and special discounts for subscribers!

BMSA > Volume 15 > Chaoticity Properties of Fractionally Integrated...
Chaoticity Properties of Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic Processes
< Back to Volume

Chaoticity Properties of Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic Processes

Full Text PDF

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.

Info:

Periodical:
Bulletin of Mathematical Sciences and Applications (Volume 15)
Pages:
69-82
DOI:
https://doi.org/10.18052/www.scipress.com/BMSA.15.69
Citation:
A. Yilmaz and G. Unal, "Chaoticity Properties of Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic Processes", Bulletin of Mathematical Sciences and Applications, Vol. 15, pp. 69-82, 2016
Online since:
May 2016
Authors:
Adil Yilmaz, Gazanfer Unal
Keywords:
Correlation Dimension, Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity, Lyapunov Exponents, Mutual Information
Export:
RIS, BibTeX, Text
Distribution:
Open Access

Creative Commons License
This work is licensed under a
Creative Commons Attribution 4.0 International License

References:

[1] W.D. Dechert, R. Gençay, Is the largest Lyapunov exponent preserved in embedded dynamics? Physics Letters. A 276 (2000) 59-64.

[2] A. Das, P. Das, Does composite index of NYSE represents chaos in the long time scale? Applied Mathematics and Computation. 174 (2006), 483-489.

[3] A. Das, P. Das, Chaotic analysis of the foreign exchange rates, Applied Mathematics and Computation. 185 (2007), 388-396.

[4] A. Moeini, M. Ahrari, S.S. Madarshahi, Investigating Chaos in Tehran Stock Exchange Index, Iranian economic review. 18 (2007), 103-120.

[5] S. Gunay, Chaotic Structure of the BRIC Countries and Turkey's Stock Market. International Journal of Economics and Financial Issues, 5(2) (2015), 515-522.

[6] H.D. Abarbanel, Analysis of Observed Chaotic data, Springer, New York, (1996).

[7] C. Francq and J.M. Zakoian. GARCH Models Structure, Statistical Inference and Financial Applications, Wiley, (2010).

[8] R. T. Baillie, T. Bollerslev, H. O. Mikkelsen, Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, (1996), 74.

[9] R. T. Baillie, Y. W. Han, R. J. Myers Long Memory and FIGARCH Models for Daily and High Frequency Commodity Prices. Working paper (2007) No. 594.

[10] D. O. Cujaeiro, B. M. Tabak, Testing for long-range dependence in world stock markets. Chaos, Solitons and Fractals, 37 (2008), 918-927.

[11] N. Sviridova, K. Sakai, Human photoplethysmogram: new insight into chaotic characteristics. Chaos, Solitons and Fractals 77 (2015), 53-63.

[12] C. Brooks, Chaos in foreign exchange markets: a sceptical view. Computational Economics, 11 (3) (1998), 265-281.

[13] M. Frezza, Goodness of fit assessment for a fractal model of stock markets. Chaos, Solitons and Fractals. 66(2014), 41-50.

[14] E. J. Kostelich, H. L. Swinney Practical Considerations in Estimating Dimension from Time Series Data. Physica Scripta, Vol. 40 (1989), 436-441.

[15] H.F. Liu, Z.H. Dai, W.F. Li, X. Gong, Z.H. Yu. Noise robust estimates of the largest Lyapunov exponent. Physics Letters A 341 (2005), 119-127.

[16] A Wolf, J.B. Swift, H. Swinney, J. Vastano, Determining Lyapunov exponents from a time series. Physica D 16 (1985), 285-317.

[17] H. Kantz, A robust method to estimate the maximal Lyapunov exponent of a time series. Phys. Lett. A 185 (1994) 77-87.

[18] M. T. Rosenstein, J.J. Collins and J.D.L. Carlo . A Practical Method for Calculating Largest Lyapunov Exponents from small data sets. Physica D 65 (1993).

[19] T. Schreiber, Interdisciplinary application of nonlinear time series methods. Physics Reports 308 (1999), 1-64.

[20] A. M. Fraser and H. L. Swinney Independent coordinates for strange attractors from mutual information Phys. Rev. A 33 (1986) 1134-40.

[21] F. Takens, Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence, Rand D, Young L (eds), Springer-Verlag, Berlin (1981).

[22] S. Kodba, M. Perc, M. Marhl, Detecting Chaos from a Time Series. European Journal of Physics 26 (2005) 205-215.

[23] M.B. Kennel, R. Brown, H.D.I. Abarbanel, Determining embedding dimension for phase-space reconstruction using a geometrical reconstruction. Physical Review A, V. 45 (1992).

[24] Software package on https: /www. kevinsheppard. com/MFE_Toolbox.

[25] Software package on http: /www. oxmetrics. net/index. html.

[26] Software package on http: /www. matjaperc. com/ejp/time. html.

[27] Software package on https: /www. r-project. org.

Show More Hide