Skip to main content

Forecasting in nonlinear univariate time series using penalized splines

Abstract

In this article we discuss penalized splines for fitting and forecasting univariate nonlinear time series models. While penalized splines have been excessively used in smooth regression, their use in nonlinear time series models is less far developed. This paper focuses on univariate autoregressive processes and discuss different nonlinear (functional) time series models including parsimonious estimation and model selection ideas. Furthermore, in simulations and an application we show how this approach compares to common parametric nonlinear models.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Petrov BN, Csaki F (eds) Second international symposium on information theory. Akademiai Kiado, Budapest, pp 267–81

  2. Balke N, Fomby T (1997) Thresholds cointegration. Int Econ Rev 38:627–645

    MathSciNet  Article  MATH  Google Scholar 

  3. Brown B, Mariano R (1989) Measures of deterministic prediction bias in nonlinear models. Int Econ Rev 30:667–684

    MathSciNet  Article  MATH  Google Scholar 

  4. Brumback B, Ruppert D, Wand M (1999) Comment on “variable selection and function estimation in additive nonparametric regression using a data-based prior,” by Shively T., Kohn R. and Wood S. J Am Stat Assoc 94:794–797

    Google Scholar 

  5. Burnham K, Anderson DR (2002) Model selection and multimodel inference. Springer, Berlin

    MATH  Google Scholar 

  6. Cao Y, Lin H, Wuc T, Yu Y (2010) Penalized spline estimation for functional coefficient regression models. Comput Stat Data Anal 54:891–905

    MathSciNet  Article  MATH  Google Scholar 

  7. Carroll R, Fan J, Gijbels I, Wand M (1997) Generalized partially linear single-index models. J Am Stat Assoc 92:477–489

    MathSciNet  Article  MATH  Google Scholar 

  8. Cerrato M, Kim H, MacDonald R (2010) Three-regime asymmetric star modeling and exchange rate reversion. J Money Credit Bank 42(7):1447–1467

    Article  Google Scholar 

  9. Chatfield C (2003) The analysis of time series: an introduction, 6th edn. Chapman and Hall, New York

    MATH  Google Scholar 

  10. de Boor C (2001) A practical guide to splines. Springer, Berlin

    MATH  Google Scholar 

  11. Di Narzo A, Aznarte J, Stigler M (2015) tsDyn: Nonlinear time series models with regime switching. R package version 0.9

  12. Dufrénot G, Mignon V (2002) Recent developments in nonlinear cointegration with applications to macroeconomics and finance. Kluwer Academic Press, Dordrecht

    Book  MATH  Google Scholar 

  13. Eilers PHC, Marx BD (1996) Flexible smoothing with B-splines and penalties. Stat Sci 11(2):89–121

    MathSciNet  Article  MATH  Google Scholar 

  14. Fahrmeir L, Tutz G (1994) Multivariate statistical modelling based on generalized linear models. Springer, New York

    Book  MATH  Google Scholar 

  15. Fan J, Gijbels I (1996) Local polynomial modelling and its applications. Chapman and Hall, London

    MATH  Google Scholar 

  16. Fan J, Yao Q (2003) Nonlinear time series: nonparametric and parametric methods. Springer, New York

    Book  MATH  Google Scholar 

  17. Gao J (2007) Nonlinear time series : semiparametric and nonparametric methods. Chapman and Hall/CRC, London

    Book  MATH  Google Scholar 

  18. Gonzalo J, Pitarakis J (2006) Threshold effects in multivariate error correction models. In: Mills TC, Patterson K (eds) Palgrave handbook of econometrics, vol 1. Palgrave MacMillan, Basingstoke

    Google Scholar 

  19. Green DJ, Silverman BW (1994) Nonparametric regression and generalized linear models. Chapman and Hall, London

    Book  MATH  Google Scholar 

  20. Greven S, Kneib T (2010) On the behaviour of marginal and conditional AIC in linear mixed models. Biometrika 97(4):773–789

    MathSciNet  Article  MATH  Google Scholar 

  21. Hamilton J (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57:357–384

    MathSciNet  Article  MATH  Google Scholar 

  22. Hamilton J (1990) Analysis of time series subject to changes in regime. J Econ 45:39–70

    MathSciNet  Article  MATH  Google Scholar 

  23. Härdle W, Hall P, Ichimura H (1993) Optimal smoothing in single-index models. Ann Stat 21:157–178

    MathSciNet  Article  MATH  Google Scholar 

  24. Härdle W, Lütkepohl H, Chen R (1997) A review of nonparametric time series analysis. Int Stat Rev 65:49–72

    Article  MATH  Google Scholar 

  25. Hartmann P, Manna M, Manzanares A (2001) The microstructure of the Euro money market. J Int Money Financ 20:895–948

    Article  Google Scholar 

  26. Hastie T, Tibshirani R (1990) Generalized additive models. Chapman and Hall, London

    MATH  Google Scholar 

  27. Hastie T, Tibshirani R (1993) Varying-coefficient models. J R Stat Soc Ser B 55:757–796

    MathSciNet  MATH  Google Scholar 

  28. Huang J, Shen H (2004) Functional coefficient regression models for nonlinear time series: a polynomial spline approach. Scand J Stat 31:515–534

    Article  MATH  Google Scholar 

  29. Josep A, Espigares S, Lopez-Moreno A (2012) MSwM: univariate autoregressive markov switching models for linear and generalized models by using the EM algorithm. R package version 1

  30. Kapetanios GYS, Snell A (2006) Testing for cointegration in nonlinear smooth transition error correction models. Econ Theory 22:279–303

    MathSciNet  Article  MATH  Google Scholar 

  31. Kauermann G (2004) A note on smoothing parameter selection for penalised spline smoothing. J Stat Planing Inference 127:53–69

    Article  MATH  Google Scholar 

  32. Kauermann G, Krivobokova T, Fahrmeir L (2009) Some asymptotic results on generalized penalized spline smoothing. J R Stat Soc Ser B 71:487–503

    MathSciNet  Article  MATH  Google Scholar 

  33. Kauermann G, Opsomer J (2011) Data-driven selection of the spline dimension in penalized spline regression. Biometrika 98(1):225–230

    MathSciNet  Article  MATH  Google Scholar 

  34. Koop G, Potter SM (2000) Nonlinearity, structural breaks, or outliers in economic time series. In: Barnett WA, Hendry DF, Hylleberg S, Tersvirta T, Tjøstheim D, Würtz A (eds) Nonlinear econometric modeling in time series analysis. University Press, Cambridge

    Google Scholar 

  35. Krivobokova T, Kauermann G (2007) A note on penalized spline smoothing with correlated errors. J Am Stat Assoc 102:1328–1337

    MathSciNet  Article  MATH  Google Scholar 

  36. Kuan CM (2002) Lecture on the Markov switching model. Institute of Economics Academia Sinica, Taipei

    Google Scholar 

  37. Li Y, Genton MG (2009) Single-index additive vector autoregressive time series models. Scand J Stat 36:369–388

    MathSciNet  Article  MATH  Google Scholar 

  38. Manna M, Pill H, Quirós G (2002) The Eurosystem’s operational framework in the context of the ECB’s monetary policy strategy. Int Financ 20:64–95

    Google Scholar 

  39. Marcellino M, Stock JH, Watson M (2006) A comparison of direct and iterated multistep ar methods for forecasting macroeconomic time series. J Econ 135:499–526

    MathSciNet  Article  MATH  Google Scholar 

  40. Nautz D, Offermanns CJ (2007) The dynamic relationship between the Euro overnight rate, the ECB’s policy rate and the term spread. Int J Financ Econ 12:287–300

    Article  Google Scholar 

  41. Ngo L, Wand M (2004) Smoothing with mixed model software. J Stat Softw 9:1–54

    Article  Google Scholar 

  42. Nychka D, Cummins D (1996) Comment on: Eilers, p., marx, b., flexible smoothing with b-splines and penalties. Stat Sci 11:89–121

    Article  Google Scholar 

  43. O’Sullivan F (1986) A statistical perspective on ill-posed inverse problems (c/r: P519–527). Stat Sci 1:502–518

    Article  MATH  Google Scholar 

  44. Pinheiro JC, Bates DM (2000) Mixed-effect models in S and S-plus. Springer, New York

  45. Ruppert D (2004) Statistics and finance. Springer, New York

    Book  MATH  Google Scholar 

  46. Ruppert D, Wand M, Carroll R (2003) Semiparametric regression. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  47. Ruppert D, Wand M, Carroll R (2009) Semiparametric regression during 2003–2007. Electron J Stati 3:1193–1256

    MathSciNet  Article  MATH  Google Scholar 

  48. Stock J, Watson M (1999) A comparison of linear and nonlinear univariate models for forecasting macroeconomic time series. In: Engle R, White H (eds) Cointegration, causality, and forecasting: a festschrift in honor of Clive W.J. Granger. Oxford, Oxford University Press

    Google Scholar 

  49. Stoker T (1986) Consistent estimation of scaled coefficients. Econometrica 54:1461–1481

    MathSciNet  Article  MATH  Google Scholar 

  50. Teräsvirta T, Tjøstheim D, Granger CWJ (2010) Modelling nonlinear economic time series. Oxford University Press, Oxford

    Book  MATH  Google Scholar 

  51. Tong H (2011) Threshold models in time series analysis-30 years on. Stat Interface 4:107–118

    MathSciNet  Article  MATH  Google Scholar 

  52. Tschernig R (2004) Nonparametric time series modeling. In: Lütkepohl H, Krätzig M (eds) Applied time series econometrics. Cambridge University Press, Cambridge

    Google Scholar 

  53. van Dijk D, Terasvirta T, Franses P (2002) Smooth transition autoregressive models—a survey of recent developments. Econ Rev 21:1–47

    MathSciNet  Article  MATH  Google Scholar 

  54. Wager C, Vaida F, Kauermann G (2007) Model selection for p-spline smoothing using akaike information criteria. Aust N Z J Stat 49:173–190

    MathSciNet  Article  MATH  Google Scholar 

  55. Wand M (2003) Smoothing and mixed models. Comput Stat 18:223–249

    Article  MATH  Google Scholar 

  56. Wand M, Ormerod J (2008) On semiparametric regression with O’Sullivan penalized splines. Aust N Z J Stat 50:179–198

    MathSciNet  Article  MATH  Google Scholar 

  57. Wang L, Yang L (2009) Spline estimation of single index model. Stat Sin 19(2):765–783

    MathSciNet  MATH  Google Scholar 

  58. Wood SN (2006) Generalized additive models: an introduction with R. Chapman and Hall, London

    MATH  Google Scholar 

  59. Xia Y, Härdle W (2006) Semi-parametric estimation of partially linear single-index models. J Multivar Anal 97(5):1162–1184

  60. Yu Y, Ruppert D (2002a) Penalized spline estimation for partially linear single index models. J Am Stat Assoc 97:1042–1054

    MathSciNet  Article  MATH  Google Scholar 

  61. Yu Y, Ruppert D (2002b) Penalized spline estimation for partially linear single index models. J Am Stat Assoc 97:1042–1054

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Göran Kauermann.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wegener, M., Kauermann, G. Forecasting in nonlinear univariate time series using penalized splines. Stat Papers 58, 557–576 (2017). https://doi.org/10.1007/s00362-015-0711-1

Download citation

Keywords

  • Time series
  • Penalized splines
  • Model selection
  • EONIA-rate

Mathematics Subject Classification

  • 62G08
  • 62M10