Skip to main content
SpringerLink
Log in

Forecasting in nonlinear univariate time series using penalized splines

Statistical Papers volume 58, pages 557–576 (2017)Cite this article

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.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

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

References

  • 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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  • 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 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • 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 

  • 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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

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

    Article  Google Scholar 

  • 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 

  • 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 

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

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

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Quantitative Products, DEKA Investment GmbH, Mainzer Landstrasse 16, Frankfurt (Main), Germany

    Michael Wegener

  2. Department of Statistics, Ludwigs-Maximilians-University Munich, Munich, Germany

    Göran Kauermann

Authors
  1. Michael Wegener
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Göran Kauermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Göran Kauermann.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. 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

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-015-0711-1

Keywords

Mathematics Subject Classification

search

Navigation