Advertisement

Non-Parametric and Flexible Time Series Estimators

  • Jürgen Franke
  • Wolfgang Karl Härdle
  • Christian Matthias Hafner
Chapter
Part of the Universitext book series (UTX)

Abstract

With the analysis of (financial) time series, one of the most important goals is to produce forecasts. Using past data one can argue about the future mean, the future volatility and so on; however, a flexible method of producing such estimates will be introduced in this chapter.

References

  1. Ango Nze, P. (1992). Critères d’ergodicité de quelques modèles à représentation markovienne. Technical Report 315, ser. 1. Paris: C.R. Acad. Sci.Google Scholar
  2. Bossaerts, P., & Hillion, P. (1993). Test of a general equilibrium stock option pricing model. Mathematical Finance, 3, 311–347.CrossRefGoogle Scholar
  3. Carroll, R. J., Härdle, W., & Mammen, E. (2002). Estimation in an additive model when the components are linked parametrically. Econometric Theory, 18, 886–912.MathSciNetCrossRefGoogle Scholar
  4. Chan, K., & Tong, H. (1985). On the use of deterministic Lyapunov functions for the ergodicity of stochastic difference equations. Advanced Applied Probability, 17, 666–678.MathSciNetCrossRefGoogle Scholar
  5. Chen, R., & Tsay, R. S. (1993a). Functional-coefficient autoregressive models. Journal of the American Statistical Association, 88, 298–308.MathSciNetzbMATHGoogle Scholar
  6. Chen, R., & Tsay, R. S. (1993b). Nonlinear additive ARX models. Journal of the American Statistical Association, 88, 955–967.CrossRefGoogle Scholar
  7. Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association, 74, 829–836.MathSciNetCrossRefGoogle Scholar
  8. Collomb, G. (1984). Propriétés de convergence presque complète du prédicteur à noyau. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 66, 441–460.MathSciNetCrossRefGoogle Scholar
  9. Diebolt, J., & Guégan, D. (1990). Probabilistic properties of the general nonlinear autoregressive process of order one. Technical Report 128. Paris: Université de Paris VI.Google Scholar
  10. Doukhan, P., & Ghindès, M. (1980). Estimation dans le processus xn+1 = f(xn) + εn+1. C.R. Acad. Sci. Paris, Sér. A, 297, 61–64.zbMATHGoogle Scholar
  11. Doukhan, P., & Ghindès, M. (1981). Processus autorégressifs non-linéaires. C.R. Acad. Sci. Paris, Sér. A, 290, 921–923.zbMATHGoogle Scholar
  12. Duan, J.-C. (1995). The GARCH option pricing model. Mathematical Finance, 5, 13–32.MathSciNetCrossRefGoogle Scholar
  13. Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica, 50, 987–1008.MathSciNetCrossRefGoogle Scholar
  14. Engle, R. F., & Gonzalez-Rivera, G. (1991). Semiparametric ARCH models. Journal of Business and Economic Statistics, 9, 345–360.Google Scholar
  15. Fan, J., & Gijbels, I. (1996). Local polynomial modeling and its application – Theory and methodologies. Chapman and Hall.zbMATHGoogle Scholar
  16. Fan, J., & Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika, 85, 645–660.MathSciNetCrossRefGoogle Scholar
  17. Fan, J., & Yao, Q. (2003). Nonlinear time series: Nonparametric and parametric methods. New York: Springer-Verlag.CrossRefGoogle Scholar
  18. Föllmer, H., & Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In M. H. A. Davis, & R. J. Elliot (Eds.), Applied stochastic analysis (pp. 389–414). London: Gordon and Breach.Google Scholar
  19. Föllmer, H., & Sondermann, D. (1991). Hedging of non-redundant contingent claims. In W. Hildenbrand, & A. Mas-Colell (Eds.), Contributions to mathematical economics (pp. 205–223). North Holland: Amsterdam.Google Scholar
  20. Franke, J. (1999). Nonlinear and nonparametric methods for analyzing financial time series. In P. Kall, & H.-J. Luethi (Eds.), Operation research proceedings 98, Heidelberg: Springer-Verlag.Google Scholar
  21. Franke, J., Härdle, W., & Kreiss, J. (2003). Nonparametric estimation in a stochastic volatility model. Recent Advances and Trends in Nonparametric Statistics, 303–314.Google Scholar
  22. Franke, J., Kreiss, J., & Mammen, E. (2002). Bootstrap of kernel smoothing in nonlinear time series. Bernoulli, 8(1), 1–37.MathSciNetzbMATHGoogle Scholar
  23. Glosten, L., Jagannathan, R., & Runkle, D. (1993). On the relationship between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48, 1779–1801.CrossRefGoogle Scholar
  24. Gouriéroux, C., & Monfort, A. (1992). Qualitative threshold ARCH models. Journal of Econometrics, 52, 159–199.MathSciNetCrossRefGoogle Scholar
  25. Gregory, A. (1989). A nonparametric test for autoregressive conditional heteroscedasticity: A Markov chain approach. Journal of Business and Economic Statistics, 7, 107–115.Google Scholar
  26. Hafner, C. (1998). Estimating high frequency foreign exchange rate volatility with nonparametric ARCH models. Journal of Statistical Planning and Inference, 68, 247–269.MathSciNetCrossRefGoogle Scholar
  27. Härdle, W. (1990). Applied nonparametric regression. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  28. Härdle, W., & Hafner, C. (2000). Discrete time option pricing with flexible volatility estimation. Finance and Stochastics, 4, 189–207.MathSciNetCrossRefGoogle Scholar
  29. Härdle, W., Lütkepohl, H., & Chen, R. (1997). Nonparametric time series analysis. International Statistical Review, 12, 153–172.zbMATHGoogle Scholar
  30. Härdle, W., Müller, M., Sperlich, S., & Werwatz, A. (2004). Non- and semiparametric modelling. Heidelberg: Springer-Verlag.CrossRefGoogle Scholar
  31. Härdle, W., & Tsybakov, A. (1997). Local polynomial estimation of the volatility function. Journal of Econometrics, 81, 223–242.MathSciNetCrossRefGoogle Scholar
  32. Härdle, W., Tsybakov, A., & Yang, L. (1996). Nonparametric vector autoregression. Journal of Statistical Planning and Inference, 68, 221–245.MathSciNetCrossRefGoogle Scholar
  33. Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatilities. Journal of Finance, 42, 281–300.CrossRefGoogle Scholar
  34. Katkovnik, V. (1979). Linear and nonlinear methods for nonparametric regression analysis (in Russian). Avtomatika i Telemehanika, 35–46.Google Scholar
  35. Katkovnik, V. (1985). Nonparametric identification and data smoothing. Nauka.zbMATHGoogle Scholar
  36. McKeague, I., & Zhang, M. (1994). Identification of nonlinear time series from first order cumulative characteristics. Annals of Statistics, 22, 495–514.MathSciNetCrossRefGoogle Scholar
  37. Melino, A., & Turnbull, S. M. (1990). Pricing foreign currency options with stochastic volatility. Journal of Econometrics, 45, 239–265.CrossRefGoogle Scholar
  38. Mokkadem, A. (1987). Sur un modèle autorégressif nonlinéaire. ergodicité et ergodicité géometrique. Journal of Time Series Analysis, 8, 195–204.MathSciNetCrossRefGoogle Scholar
  39. Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59, 347–370.MathSciNetCrossRefGoogle Scholar
  40. Rabemananjara, R., & Zakoian, J. M. (1993). Threshold ARCH models and asymmetries in volatility. Journal of Applied Econometrics, 8, 31–49.CrossRefGoogle Scholar
  41. Renault, E., & Touzi, N. (1996). Option hedging and implied volatilities in a stochastic volatility model. Mathematical Finance, 6, 277–302.CrossRefGoogle Scholar
  42. Robinson, P. (1983). Non-parametric estimation for time series models. Journal of Time Series Analysis, 4, 185–208.MathSciNetCrossRefGoogle Scholar
  43. Robinson, P. (1984). Robust nonparametric autoregression. In J. Franke, W. Härdle, & Martin (Eds.), Robust and nonlinear time series analysis. Heidelberg: Springer-Verlag.Google Scholar
  44. Stone, C. (1977). Consistent nonparametric regression. Annals of Statistics, 5, 595–645.MathSciNetCrossRefGoogle Scholar
  45. Tong, H. (1983). Threshold models in nonlinear time series analysis, Vol. 21 of Lecture notes in statistics. Heidelberg: Springer-Verlag.CrossRefGoogle Scholar
  46. Tsybakov, A. (1986). Robust reconstruction of functions by the local-approximation method. Problems of Information Transmission, 22, 133–146.MathSciNetzbMATHGoogle Scholar
  47. Vieu, P. (1995). Order choice in nonlinear autoregressive models, Discussion Paper, Laboratoire de Statistique et Probabilités, Université Toulouse.zbMATHGoogle Scholar
  48. Wiggins, J. (1987). Option values under stochastic volatility: Theory and empirical estimates. Journal of Financial Economics, 19.CrossRefGoogle Scholar
  49. Yang, L., Härdle, W., & Nielsen, J. (1999). Nonparametric autoregression with multiplicative volatility and additive mean. Journal of Time Series Analysis, 20(5), 579–604.MathSciNetCrossRefGoogle Scholar
  50. Zakoian, J. (1994). Threshold heteroskedastic functions. Journal of Economic Dynamics and Control, 18, 931–955.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jürgen Franke
    • 1
  • Wolfgang Karl Härdle
    • 2
  • Christian Matthias Hafner
    • 3
  1. 1.Department of MathematicsTechnische Universität KaiserslauternKaiserslauternGermany
  2. 2.Ladislaus von Bortkiewicz Chair of StatisticsHumboldt-Universität BerlinBerlinGermany
  3. 3.Louvain Institute of Data Analysis and Modeling in Economics and StatisticsUCLouvainLouvain-la-NeuveBelgium

Personalised recommendations