Computational Economics

, Volume 21, Issue 3, pp 257–276 | Cite as

Is it Possible to Study Chaotic and ARCH Behaviour Jointly? Application of a Noisy Mackey–Glass Equation with Heteroskedastic Errors to the Paris Stock Exchange Returns Series

  • Catherine Kyrtsou
  • Michel Terraza
Article

Abstract

Most recent empirical works that apply sophisticated statistical proceduressuch as a correlation-dimension method have shown that stock returns arehighly complex. The estimated correlation dimension is high and there islittle evidence of low-dimensional deterministic chaos. Taking the complexbehaviour in stock markets into account, we think it is more robust than thetraditional stochastic approach to model the observed data by a nonlinearchaotic model disturbed by dynamic noise. In fact, we construct a model havingnegligible or even zero autocorrelations in the conditional mean, but a richstructure in the conditional variance. The model is a noisy Mackey–Glassequation with errors that follow a GARCH(p,q) process. This model permits usto capture volatility-clustering phenomena. Its characteristic is thatvolatility clustering is interpreted as an endogenous phenomenon. The mainobjective of this article is the identification of the underlying process ofthe Paris Stock Exchange returns series CAC40. To this end, we apply severaldifferent tests to detect longmemory components and chaotic structures.Forecasting results for the CAC40 returns series, will conclude this paper.

Mackey–Glass equation noisy chaos volatility clustering correlation dimension Lyapunov exponents GARCH effects forecasting 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bartlett, M.S. (1990).Chance or chaos? tJournal of the Royal Statistical Society B, 153 (Part 3), 321-347.Google Scholar
  2. Bollt, E.M. (2000). Model selection, confidence and scaling in predicting chaotic time-series. International Journal of Bifurcation and Chaos, 10(6), 1407-1422.Google Scholar
  3. Bourbonnais, R. and Terraza, M. (1998). Analyse des séries temporelles en économie. Éditions PUF, Paris.Google Scholar
  4. Brock, W.A, Dechert, W.D. and Scheinkman, J.A. (1987). A test for independence based on the correlation dimension. Working paper, University of Wisconsin.Google Scholar
  5. Brock, W.A. and Hommes, C.H. (1998). Heterogeneous beliefs and routes to chaos in a simple asset pricing model. Journal of Economic Dynamics & Control, 22, 1235-1274.Google Scholar
  6. Brock, W.A., Hsieh, D.A. and LeBaron, B. (1992). Nonlinear Dynamics, Chaos and Instability, second edition. The MIT Press, Cambridge.Google Scholar
  7. Chan, K.S. and Tong, H. (1994). A note on noisy chaos. Journal of the Royal Statistical Society B, 2, 301-311.Google Scholar
  8. Chen, S-H., Lux, T. and Marchesi, M. (1999). Testing for non-linear structure in an artificial financial market. Discussion paper B-447, University of Bonn.Google Scholar
  9. Cheung, Y-W. (1993). Tests for fractional integration: a Monte Carlo investigation. Journal of Time Series Analysis, 14 (4), 331-345.Google Scholar
  10. Chiarella, C., Dieci, R. and Gardini, L. (2000). Speculative behaviour and complex asset price dynamics. Forthcoming in Journal of Economic Behavior and Organization.Google Scholar
  11. Chiarella, C. and He, X-Z. (1999). Heterogeneous beliefs, risk and learning in a simple asset pricing model. Research paper 18, Quantitative Finance Research Group, University of Technology, Syndey.Google Scholar
  12. Cromwell, J.B., Labys, W.L. and Terraza, M. (1994). Univariate Tests for Time Series Models. Sage publications.Google Scholar
  13. Dechert, W.D. and Gençay, R. (1990). Estimating Lyapunov exponents with multilayer feedforward network learning. Working paper, Department of Economics, University of Houston.Google Scholar
  14. Farmer, D.J. and Sidorowitch, J.J. (1987). Predicting chaotic time series. Physical Review Letters, 59, 845-848.Google Scholar
  15. Gallant, R.A. and White, H. (1992). On learning the derivatives of an unknown mapping with multilayer feedforward networks, Artificial Neural Networks, 206-223. Blackwell Publishers, Cambridge.Google Scholar
  16. Gaunersdorfer, A. (2000). Endogenous fluctuations in a simple asset pricing model with heterogeneous agents. Journal of Economic Dynamics & Control, 24, 799-831.Google Scholar
  17. Gençay, R. and Dechert, W.D. (1992). An algorithm for the n Lyapunov exponents of an ndimensional unknown dynamical system. Physica D, 59, 142-157.Google Scholar
  18. Gençay, R. and Liu, T. (1996). Nonlinear modelling and prediction with feedforward and recurrent networks. Working paper, Department of Economics, University of Windsor, Canada.Google Scholar
  19. Geweke, J. (1993). Inference and forecasting for deterministic non-linear time series observed with measurement error. In R.H. Day and P. Chen (eds.), Nonlinear Dynamics and Evolutionary Dynamics. Oxford University Press.Google Scholar
  20. Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models. Journal of Time Series Analysis, 4, 221-238.Google Scholar
  21. Goffe, W.L., Ferrier, G.D. and Rogers, J. (1994). Global optimization of statistical functions with simulated annealing. Journal of Econometrics, 60, 65-99.Google Scholar
  22. Granger, C.W.J. and Joyeux, R. (1980). An introduction to long memory time series and fractional differencing. Journal of Time Series Analysis, 1, 1-15.Google Scholar
  23. Grassberger, P. and Procaccia, I. (1983). Measuring the strangeness of strange attractors, Physica, 9D, 189-208.Google Scholar
  24. Guegan, D. (1994). Stochastic versus deterministic chaos. Working Paper, CREST, No. 9438.Google Scholar
  25. Haugen, R.A. (1999). Beast on Wall Street-How Stock Volatility Devours Our Wealth. Upper Saddle River, N.J., Prentice Hall.Google Scholar
  26. Hsieh, D.A. (1989). Testing nonlinear dependence in daily foreign exchange rates. Journal of Business, 62 (3), 339-368.Google Scholar
  27. Hsieh, D.A. (1991). Chaos and nonlinear dynamics: Application to financial markets. The Journal ofFinance, XLVI(5), 1839-1877.Google Scholar
  28. Iori, G. (1999). A microsimulation of traders activity in the stock market: the role of heterogeneity, agents' interactions and trade frictions. Forthcoming in International Journal of Modern Physics C.Google Scholar
  29. Kuan, C-M. and Liu, T. (1995). Artificial neural networks: an econometric perspective. Journal of Applied Econometrics, 10, 347-364.Google Scholar
  30. Kugiumtzis, D., Lingjaerde, O.C. and Christophersen, N. (1998). Regularized local linear prediction of chaotic time series. Physica D, 112, 344-360.Google Scholar
  31. Kyrtsou, C. and Terraza, M. (1998). Determinism versus stochasticity in emerging financial markets. Forecasting Financial Markets: Proceedings of the Fifth International Conference in London.Google Scholar
  32. Kyrtsou, C., Guiraud, V., Asimakopoulos, G., Siriopoulos, C. and Terraza, M. (1999). Evidence for nonlinearity in small european capital markets: results for financial risk management. In C. Siriopoulos (ed.), Topics in Financial Risk Management, ∏αρατηρητης editions, Thessalonique.Google Scholar
  33. Kyrtsou, C., Labys, W. and Terraza, M. (1998). Testing for nonlinearity in commodity prices: determinism or stochasticity? Proceedings of the Conference ‘Dynamique des Prix et des Marchés de Matières premières’ in Grenoble, France.Google Scholar
  34. Kyrtsou, C. and Terraza, M. (2002c). Stochastic chaos or ARCH effects in stock series? A comparative study. Forthcoming in International Review of Financial Analysis.Google Scholar
  35. Kyrtsou, C. and Terraza, M. (2002a). L'effet du bruit dans les données à haute fréquence: le cas de la série boursière d'Athènes. In Logiques Economiques, éditions Harmattan, Paris.Google Scholar
  36. Kyrtsou, C. and Terraza, M. (2001). High-dimensional chaos, endogenous volatility and interacting agents. Working paper, LAMETA, University of Montpellier I, France.Google Scholar
  37. Kyrtsou, C. and Terraza, M. (2002b). Misleading properties for a noisy chaotic model. Working paper, LAMETA, University of Montpellier I, France.Google Scholar
  38. Lux, T. (1995). Herd behaviour, bubbles and crashes. The Economic Journal, 105, 881-896.Google Scholar
  39. Lux, T. (1998). The socio-economic dynamics of speculative markets: interacting agents, chaos and the fat tails of returns distributions. Journal of Economic Behavior and Organization, 33,143-165.Google Scholar
  40. Mackey, M. and Glass, L. (1977). Oscillation and chaos in physiological control systems. Science, 50, 287-289.Google Scholar
  41. Malliaris, A.G. and Stein, J.L. (1999).Methodological issues in asset pricing: random walk or chaotic dynamics, Journal of Banking & Finance, 23, 1605-1635.Google Scholar
  42. Martin, V. and Sawyer, K. (1994). Statistical techniques for modeling non-linearities. In J. Creedy, V.L. Martin (eds.), Chaos and Non-Linear Models in Economics, Edward Edgar Publishes.Google Scholar
  43. Mizrach, B. (1996). Learning and conditional heteroscedasticity in asset returns. Working paper No. 95-26, Department of Economics, Rutgers University.Google Scholar
  44. Nychka, D., Ellner, S., Gallant, A.R. and McCaffrey, D. (1992). Finding chaos in noisy systems. Journal of Royal Statistical Society B, 54(2), 399-426.Google Scholar
  45. Ogata, S., Iwayama, T. and Terachi, S. (1997). Effect of system noise on chaotic behavior in Roessler type nonlinear system. International Journal of Bifurcation and Chaos, 7(12), 2871-2879.Google Scholar
  46. Sengupta, J.K. and Zheng, Y. (1995). Empirical tests of chaotic dynamics in market volatility. Applied Financial Economics, 5, 291-300.Google Scholar
  47. Scheinkman, J.A. and LeBaron, B. (1989). Nonlinear dynamics and stock returns. Journal of Business, 62(3), 311-337.Google Scholar
  48. Siriopoulos, C., Terraza, M. and Venetis, I. (1997). Fractional cointegration and generalized long memory processes. VI International Conference AEDEM 97, September.Google Scholar
  49. Takens, F. (1981). Detecting strange attractors in turbulence. In D. Rand and L.S. Young (eds.), Dynamical Systems and Turbulence, Lecture Notes in Mathematics 89. Springer, Berlin.Google Scholar
  50. Tong, H. (1990). Non-Linear Time Series: A Dynamical Systems Approach. Oxford University Press, Oxford, UK.Google Scholar
  51. Tong, H. (1992). Discussion on chaos. Journal of the Royal Statistical Society B, 54 (2), 451-474.Google Scholar
  52. Willey, T. (1995). Testing for nonlinear dependence in daily stock indices. In Trippi R.R. (ed.), Chaos and Non-Linear Dynamics in the Financial Markets, 105-135. Irwin.Google Scholar
  53. Wolf, A., Swift, J.B., Swinney, H.L. and Vastano, J.A. (1985). Determining Lyapunov exponents from a time series. Physica, 16D, 285-317.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Catherine Kyrtsou
    • 1
  • Michel Terraza
    • 1
  1. 1.Department of Applied Economics, Lameta, Espace RichterUniversity of Montpellier IMontpellier, Cedex 1France

Personalised recommendations