Advertisement

Computational Statistics

, Volume 27, Issue 2, pp 319–341 | Cite as

Multi–regime models for nonlinear nonstationary time series

  • Francesco Battaglia
  • Mattheos K. Protopapas
Original Paper

Abstract

Nonlinear nonstationary models for time series are considered, where the series is generated from an autoregressive equation whose coefficients change both according to time and the delayed values of the series itself, switching between several regimes. The transition from one regime to the next one may be discontinuous (self-exciting threshold model), smooth (smooth transition model) or continuous linear (piecewise linear threshold model). A genetic algorithm for identifying and estimating such models is proposed, and its behavior is evaluated through a simulation study and application to temperature data and a financial index.

Keywords

Smooth transition autoregression Threshold model Genetic algorithm 

Mathematics Subject Classification (2000)

62M10 90C59 91B84 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alander JT (1992) On optimal population size of genetic algorithms. In: Proceedings of CompEuro92. IEEE Computer Society Press, pp 65–70Google Scholar
  2. Auer I, Böhm R, Jurkovic A, Lipa W, Orlik A, Potzmann R, Schöner W, Ungersböck M, Matulla C, Briffa K, Jones PD, Efthymiadis D, Brunetti M, Nanni T, Maugeri M, Mercalli L, Mestre O, Moisselin J-M, Begert M, Müller-Westermeier G, Kveton V, Bochnicek O, Stastny P, Lapin M, Szalai S, Szentimrey T, Cegnar T, Dolinar M, Gajic-Capka M, Zaninovic K, Majstorovic Z, Nieplova E (2007) HISTALP—historical instrumental climatological surface time series of the greater alpine region. Int J Climatol 27: 17–46CrossRefGoogle Scholar
  3. Bai J, Perron P (1998) Estimating and testing linear models with multiple structural changes. Econometrica 66: 47–78MathSciNetzbMATHCrossRefGoogle Scholar
  4. Baragona R, Battaglia F, Cucina D (2004) Fitting piecewise linear threshold autoregressive models by means of genetic algorithms. Comput Stat Data Anal 47: 277–295MathSciNetzbMATHCrossRefGoogle Scholar
  5. Baragona R, Battaglia F (2006) Genetic algorithms for building double threshold generalized autoregressive conditional heteroscedastic models of time series. In: Rizzi A, Vichi M (eds) COMPSTAT 2006 Proceedings in Computational Statistics. Physica-Verlag, Heidelberg, pp 441–452Google Scholar
  6. Baragona R, Battaglia F, Poli I (2011) Evolutionary statistical procedures. Springer, BerlinzbMATHCrossRefGoogle Scholar
  7. Battaglia F, Protopapas MK (2011) Time-varying multi-regime models fitting by genetic algorithms. J Time Ser Anal 32: 237–252MathSciNetCrossRefGoogle Scholar
  8. Bhansali RJ, Downham DY (1977) Some properties of the order of an autoregressive model selected by a generalization of Akaike’s EPF criterion. Biometrika 64: 547–551MathSciNetzbMATHGoogle Scholar
  9. Böhm R, Jones PD, Hiebl J, Frank D, Brunetti M, Maugeri M (2010) The early instrumental warm-bias: a solution for long Central European temperature series 1760–2007. Climatic Change 101: 41–67CrossRefGoogle Scholar
  10. Carrasco M (2002) Misspecified structural change, thresholds and Markov-Switching models. J Econom 109: 239–273MathSciNetzbMATHCrossRefGoogle Scholar
  11. Chatterjee S, Laudato M, Lynch LA (1996) Genetic algorithms and their statistical applications: an introduction. Comput Stat Data Anal 22: 633–651zbMATHCrossRefGoogle Scholar
  12. Cai Z, Fan J, Yao Q (2000) Functional coefficient regression models for nonlinear time series. J Am Stat Assoc 95: 941–955MathSciNetzbMATHGoogle Scholar
  13. Chen R, Tsay R (1993) Functional Coefficient Autoregressive Models. J Am Stat Assoc 88: 298–308MathSciNetzbMATHCrossRefGoogle Scholar
  14. Clark TE, West KD (2006) Using out-of-sample mean squared prediction errors to test the martingale difference hypothesis. J Econom 135: 155–186MathSciNetCrossRefGoogle Scholar
  15. Crawford KD, Wainwright RL (1995) Applying genetic algorithms to outlier detection. In: Eshelman LJ (ed) Proceedings of the sixth international conference on genetic algorithms. Morgan Kaufmann, San Mateo, CA, pp 546–550Google Scholar
  16. Davis R, Lee T, Rodriguez-Yam G (2006) Structural break estimation for nonstationary time series models. J Am Stat Assoc 101: 223–239MathSciNetzbMATHCrossRefGoogle Scholar
  17. Dueker MJ, Sola M, Spagnolo F (2007) Contemporaneous threshold autoregressive models: estimation, testing and forecasting. J Econom 141: 517–547MathSciNetCrossRefGoogle Scholar
  18. Dupleich Ulloa MR (2006) Testing for breaks and threshold effects: a non-nested approach. Technical Report, University of Cambridge. http://www.eea-esem.com/files/papers/EEA-ESEM/2006/1197/VerJuly05.pdf. Accessed 16 February 2011
  19. Gaetan C (2000) Subset ARMA model identification using genetic algorithms. J Time Ser Anal 21: 559–570MathSciNetzbMATHCrossRefGoogle Scholar
  20. Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley, ReadingzbMATHGoogle Scholar
  21. Guerrero VM (1993) Time series analysis supported by power transformations. J Forecast 12: 37–48CrossRefGoogle Scholar
  22. Hamilton J (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57: 357–384MathSciNetzbMATHCrossRefGoogle Scholar
  23. Hannan EJ, Quinn BG (1979) The determination of the order of an autoregression. J Roy Stat Soc Ser B 41: 190–195MathSciNetzbMATHGoogle Scholar
  24. Hartmann D, Kempa B, Pierdzioch C (2008) Economic and financial crises and the predictability of U.S. stock returns. J Empir Finance 15: 468–480CrossRefGoogle Scholar
  25. Holland JH (1975) Adaptation in natural and artificial systems. University of Michigan Press, Ann ArborGoogle Scholar
  26. Kamgaing JT, Ombao H, Davis RA (2009) Autoregressive processes with data-driven regime switching. J Time Ser Anal 30: 505–533zbMATHCrossRefGoogle Scholar
  27. Keenan DM (1985) A Tukey nonadditivity-type test for time series nonlinearity. Biometrika 72: 39–44MathSciNetzbMATHCrossRefGoogle Scholar
  28. Kennedy J, Eberhart R (2001) Swarm intelligence. Morgan Kaufman, San MateoGoogle Scholar
  29. Kim D, Kon SJ (1999) Structural change and time dependence in models of stock returns. J Empir Finance 6: 283–308CrossRefGoogle Scholar
  30. Koop G, Potter S (2001) Are apparent finding of nonlinearity due to structural instability in economic time series?. Econom J 4: 37–55zbMATHCrossRefGoogle Scholar
  31. Lin C, Teräsvirta T (1994) Testing the constancy of regression parameters against continuous structural change. J Econom 62: 211–228zbMATHCrossRefGoogle Scholar
  32. Liu L, Hudak GB (1992) Forecasting and time series analysis using the SCA statistical system. Scientific Computing Associates, Oak BrookGoogle Scholar
  33. Lundberg S, Teräsvirta T, van Dijk D (2003) Time-varying smooth transition autoregressive models. J Bus Econ Stat 21: 104–121CrossRefGoogle Scholar
  34. Lu YK, Perron P (2010) Modeling and forecasting stock return volatility using a random level shift model. J Empir Finance 17: 138–156CrossRefGoogle Scholar
  35. Perron P (2006) Dealing with structural breaks. In: Patterson K, Mills TC (eds) Palgrave handbook of econometrics, vol 1: econometric theory. Palgrave Macmillan, Basingstoke, pp 278–352Google Scholar
  36. Pesaran MH, Timmermann A (2002) Market timing and return prediction under model instability. J Empir Finance 9: 495–510CrossRefGoogle Scholar
  37. Price KV, Storn R, Lampinen J (2005) Differential evolution, a practical approach to global optimization. Springer, BerlinzbMATHGoogle Scholar
  38. Priestley MB (1988) Non-linear and non-stationary time series analysis. Academic Press, LondonGoogle Scholar
  39. Reeves CR (1993) Modern heuristic techniques for combinatorial problems. Wiley, New YorkzbMATHGoogle Scholar
  40. Rissanen J (2007) Information and complexity in statistical models. Springer, BerlinGoogle Scholar
  41. Rudolph G (1997) Convergence properties of evolutionary algorithms. Verlag Dr. Kovač, HamburgGoogle Scholar
  42. Teräsvirta T (1994) Specification, estimation and evaluation of smooth transition autoregressive models. J. Am Stat Assoc 89: 208–218Google Scholar
  43. Teräsvirta T (1998) Modeling economic relationships with smooth transition regression. In: Ullah A, Giles DEA (eds) Handbook of applied economic statistics. Marcel Dekker, New York, pp 507–552Google Scholar
  44. Tiao GC, Tsay RS (1994) Some advances in non-linear and adaptive modelling in time series. J Forecast 13: 109–131CrossRefGoogle Scholar
  45. Tong H (1990) Non linear time series: a dynamical system approach. Oxford University Press, OxfordzbMATHGoogle Scholar
  46. Tong H, Lim K (1980) Threshold autoregression, limit cycles and ciclical data. J R Stat Soc Ser B 42: 245–292zbMATHGoogle Scholar
  47. Tsay RS (1986) Nonlinearity tests for time series. Biometrika 73: 461–466MathSciNetzbMATHCrossRefGoogle Scholar
  48. Wu B, Chang CL (2002) Using genetic algorithms to parameters (d, r) estimation for threshold autoregressive models. Comput Stat Data Anal 38: 315–330MathSciNetzbMATHCrossRefGoogle Scholar
  49. Wu S, Chen R (2007) Threshold variable determination and threshold variable driven switching autoregressive models. Statistica Sinica 17: 241–264zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of StatisticsSapienza University of RomeRomeItaly

Personalised recommendations