Empirical Economics

, Volume 38, Issue 2, pp 325–345 | Cite as

Seasonal Mackey–Glass–GARCH process and short-term dynamics

  • Catherine Kyrtsou
  • Michel Terraza
Original Paper


The aim of this article is the study of complex structures which are behind the short-term predictability of stock returns series. In this regard, we employ a seasonal version of the Mackey–Glass–GARCH(p,q) model, initially proposed by Kyrtsou and Terraza (Computat Econ 21:257–276, 2003) and generalized by Kyrtsou (Int J Bifurcat Chaos 15(10):3391–3394, 2005). To unveil short or long memory components and non-linear structures in the French Stock Exchange (CAC40) returns series, we apply the test of Geweke and Porter-Hudak (J Time Ser Anal 4:221–238, 1983), the Brock et al. (Econom Rev 15:197–235, 1996) and Dechert (An application of chaos theory to stochastic and deterministic observations. Working paper, University of Houston, 1995) tests, the correlation-dimension method of Grassberger and Procaccia (Phys 9D:189–208, 1983), the Lyapunov exponents method of Gençay and Dechert (Phys D 59:142–157, 1992), and the Recurrence quantification analysis introduced by Webber and Zbilut (J Appl Physiol 76:965–973, 1994). As a confirmation procedure of the dynamics generating future movements in CAC40, we perform forecast with the use of a seasonal Mackey–Glass–GARCH(1,1) model. The interest of the forecasting exercise is found in the inclusion of high-dimensional non-linearities in the mean equation of returns.


Noisy chaos Short-term dynamics Correlation dimension Lyapunov exponents Recurrence quantifications Forecasting 

JEL Classification

C49 C51 C52 C53 D84 G12 G14 


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  1. Ashley R (2008) On the origins of conditional heteroskedasticity in time series. Working paper, Department of Economics, Virginia Tech.
  2. Baumol JW, Quandt RE (1995) Chaos models and their implications for forecasting. In: Trippi RT (ed) Chaos and nonlinear dynamics in the financial markets: theory, evidence and applications. Irwin, USGoogle Scholar
  3. Bask M, Gençay R (1998) Testing chaotic dynamics via Lyapunov exponents. Phys D 114: 1–2CrossRefGoogle Scholar
  4. Beller K, Nofsinger JR (1998) On stock return seasonality and conditional heteroskedasticity. J Financ Res XXI(2): 229–246Google Scholar
  5. Belaire-Franch J (2004) Testing for non-linearity in an artificial financial market: a recurrence quantification approach. J Econ Behav Organ 54: 483–494CrossRefGoogle Scholar
  6. Belaire-Franch J, Contreras D, Tordera-Lledo L (2002) Assessing nonlinear structures in real exchange rates using recurrence plot strategies. Phys D 171: 249–264CrossRefGoogle Scholar
  7. Bollerslev T (1987) A conditionally heteroskedastic time series model for speculative prices and rates of return. Rev Econ Stat 69: 542–547CrossRefGoogle Scholar
  8. Bourbonnais R, Terraza M (1998) Analyse des séries temporelles en économie. PUF, ParisGoogle Scholar
  9. Brock WA, Dechert WD, Scheinkman JA, LeBaron B (1996) A test for independence based on the correlation dimension. Econom Rev 15: 197–235CrossRefGoogle Scholar
  10. Brock WA, Hommes CH (1998) Heterogeneous beliefs and routes to chaos in a simple asset pricing model. J Econ Dyn Control 22: 1235–1274CrossRefGoogle Scholar
  11. Brock WA, Hsieh DA, LeBaron B (1992) Nonlinear dynamics, chaos and instability, 2nd edn. MIT Press, CambridgeGoogle Scholar
  12. Chen S-H, Lux T, Marchesi M (2001) Testing for non-linear structure in an artificial financial market. J Econ Behav Organ 46: 327–342CrossRefGoogle Scholar
  13. Chiarella C, Dieci R, Gardini L (2002) Speculative behaviour and complex asset price dynamics. J Econ Behav Organ 49(1): 173–197CrossRefGoogle Scholar
  14. Dechert WD (1995) An application of chaos theory to stochastic and deterministic observations. Working paper, University of HoustonGoogle Scholar
  15. Dechert WD, Gençay R (1996) The topological invariance of Lyapunov exponents in embedded dynamics. Phys D 90: 40–55CrossRefGoogle Scholar
  16. Dechert WD, Gençay R (2000) Is the largest Lyapunov exponent preserved in embedded dynamics?. Phys Lett A 276: 59–64CrossRefGoogle Scholar
  17. Donoho DL (1995) De-noising by soft-thresholding. IEEE Trans Inform Theory 41(3): 613–627CrossRefGoogle Scholar
  18. Donoho DL, Johnstone IM (1994) Ideal spatial adaptation via wavelet shrinkage. Biometrika 81: 425–455CrossRefGoogle Scholar
  19. Donoho DL, Johnstone IM (1995) Adapting to unknown smoothing via wavelet shrinkage. J Am Stat Assoc 90: 1200–1224CrossRefGoogle Scholar
  20. Eckmann JP, Kamphorst SO, Ruelle D (1987) Recurrence plots of dynamical systems. Europhys Lett 4(9): 973–977CrossRefGoogle Scholar
  21. Engle RF (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50(4): 987–1007CrossRefGoogle Scholar
  22. Fama E (1965) The behaviour of stock market prices. J Bus 38: 34–105CrossRefGoogle Scholar
  23. Farmer DJ, Sidorowith JJ (1987) Predicting chaotic time series. Phys Rev Lett 59: 845–848CrossRefGoogle Scholar
  24. Freedman D, Pisani R, Purves R (1978) Statistics. W.W. Norton, New YorkGoogle Scholar
  25. Gallant RA, White H (1992) On learning the derivatives of an unknown mapping with multilayer feedforward networks. Artificial Neural Networks, Blackwell, Cambridge, pp. 206–223Google Scholar
  26. Gaunersdorfer A (2000) Endogenous fluctuations in a simple asset pricing model with heterogeneous agents. J Econ Dyn Control 24: 799–831CrossRefGoogle Scholar
  27. Gaunersdorfer A (2001) Adaptive beliefs and the volatility of asset prices. Cent Eur J Oper Res 9: 5–30Google Scholar
  28. Gençay R, Dechert WD (1992) An algorithm for the n Lyapunov exponents of an n-dimensional unknown dynamical system. Phys D 59: 142–157CrossRefGoogle Scholar
  29. Gençay R, Liu T (1997) Nonlinear modelling and prediction with feedforward and recurrent networks. Phys D 108: 119–134CrossRefGoogle Scholar
  30. Geweke J (1993) Inference and forecasting for deterministic non-linear time series observed with measurement error. In: Day RH, Chen P(eds) Nonlinear dynamics and evolutionary dynamics. Oxford University Press, OxfordGoogle Scholar
  31. Geweke J, Porter-Hudak S (1983) The estimation and application of long memory time series models. J Time Ser Anal 4: 221–238CrossRefGoogle Scholar
  32. Goffe WL, Ferrier GD, Rogers J (1994) Global optimization of statistical functions with simulated annealing. J Econom 60: 65–99CrossRefGoogle Scholar
  33. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Phys 9D 189–208Google Scholar
  34. Hommes C (2006) Heterogeneous agent models in economics and finance. In: Tesfatsion L, Judd K(eds) Handbook of computational economics. North-Holland, AmsterdamGoogle Scholar
  35. Hommes C, Manzan S (2006) Comments on testing for nonlinear structure and chaos in economic time series. J Macroecon 28(1): 169–174CrossRefGoogle Scholar
  36. Hristu-Varsakelis D, Kyrtsou C (2008) Evidence for nonlinear asymmetric causality in US inflation, metal and stock returns. Dis Dyn Nature Soc, 2008 138547: 7. doi: 10.1155/2008/138547 Google Scholar
  37. Kantz H, Schreiber T (1997) Nonlinear time series analysis, Cambridge nonlinear science series, vol 7. Cambridge University Press, CambridgeGoogle Scholar
  38. Kaplan D, Glass L (1995) Understanding nonlinear dynamics. Springer, New YorkGoogle Scholar
  39. Kyrtsou C (2005) Evidence for neglected linearity in noisy chaotic models. Int J Bifurcat Chaos 15(10): 3391–3394CrossRefGoogle Scholar
  40. Kyrtsou C (2008) Re-examining the sources of heteroskedasticity: the paradigm of noisy chaotic models. Phys A 387: 6785–6789CrossRefGoogle Scholar
  41. Kyrtsou C, Karanasos M (2008) Analyzing the link between stock volatility and volume by a Mackey–Glass GARCH-type model: the case of Korea. Int J Financ Mark Inst (accepted subject to revisions)Google Scholar
  42. Kyrtsou C, Labys W (2006) Evidence for chaotic dependence between US inflation and commodity prices. J Macroecon 28(1): 256–266CrossRefGoogle Scholar
  43. Kyrtsou C, Labys W (2007) Detecting positive feedback in multivariate time series: the case of metal prices and US inflation. Phys A 377(1): 227–229CrossRefGoogle Scholar
  44. Kyrtsou C, Labys W, Terraza M (2004) Noisy chaotic dynamics in commodity markets. Emp Econ 29(3): 489–502CrossRefGoogle Scholar
  45. Kyrtsou C, Malliaris A (2009) The impact of information signals on market prices, when agents have non-linear trading rules. Econ Modell 26(1): 167–176CrossRefGoogle Scholar
  46. Kyrtsou C, Terraza M (2002) Stochastic chaos or ARCH effects in stock series? A comparative study. Int Rev Financ Anal 11: 407–431Google Scholar
  47. Kyrtsou C, Terraza M (2003) 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. Comput Econ 21: 257–276CrossRefGoogle Scholar
  48. Kyrtsou C, Vorlow C (2005) Complex dynamics in macroeconomics: A novel approach. In: Diebolt C, Kyrtsou C(eds) New trends in macroeconomics. Springer, HeidelbergGoogle Scholar
  49. Kyrtsou C, Vorlow C (2009) Modelling nonlinear comovements between time series. J Macroecon 31(1): 200–211CrossRefGoogle Scholar
  50. Lux T (1995) Herd behaviour, bubbles and crashes. Econ J 105: 881–896CrossRefGoogle Scholar
  51. Lux T (1998) The socio-economic dynamics of speculative markets: interacting agents, chaos, and the fat tails of returns distributions. J Econ Behav Organ 33: 143–165CrossRefGoogle Scholar
  52. Mackey M, Glass L (1977) Oscillation and chaos in physiological control systems. Science 50: 287–289CrossRefGoogle Scholar
  53. Malliaris AG, Stein JL (1999) Methodological issues in asset pricing: random walk or chaotic dynamics. J Banking Financ 23: 1605–1635CrossRefGoogle Scholar
  54. Mizrach (2009) Learning and conditional heteroskedasticity in assets returns. In: Kyrtsou C, Vorlow C (eds) Progress in financial markets research. NOVA Publishers, New York (forthcoming)Google Scholar
  55. Strozzi F, Zaldivar J-M, Zbilut JP (2002) Application of nonlinear time series analysis techniques to high-frequency currency exchange data. Phys A 312: 520–538CrossRefGoogle Scholar
  56. Takens F (1981) Detecting strange attractors in turbulence. In: Rand D, Young LS (eds) Dynamical systems and turbulence. Lecture notes in mathematics, vol 89, Springer, BerlinGoogle Scholar
  57. Trulla LL, Giuliani A, Zbilut JP, Webber CL (1996) Recurrence quantification analysis of the logistic equation with transients. Phys Lett A 223: 225–260CrossRefGoogle Scholar
  58. Tsay RS (1986) Nonlinearity tests for time series. Biometrica 73: 461–466CrossRefGoogle Scholar
  59. White H (1989) Some asymptotic results for learning in single hidden layer feedforward networks models. J Am Stat Assoc 84: 1003–1013CrossRefGoogle Scholar
  60. Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Phys 16D 285–317Google Scholar
  61. Webber CL, Zbilut JP (1994) Dynamical assessment of physiological systems and states using recurrence plot strategies. J Appl Physiol 76: 965–973Google Scholar
  62. Zbilut JP, Giuliani A, Webber CL (2000) Recurrence quantification analysis as an empirical test to distinguish relatively short deterministic versus random number series. Phys Lett A 267: 174–178CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of MacedoniaThessalonikiGreece
  2. 2.Department of Economics, LAMETAUniversity of Montpellier IMontpellier Cedex 1France

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