Nonparametric Modeling in Financial Time Series

  • Jürgen FrankeEmail author
  • Jens-Peter Kreiss
  • Enno Mammen


In this chapter, we deal with nonparametric methods for discretely observed financial data. The main ideas of nonparametric kernel smoothing are explained in the rather simple situation of density estimation and regression. For financial data, a rather relevant topic is nonparametric estimation of a volatility function in a continuous-time model such as a homogeneous diffusion model. We review results on nonparametric estimation for discretely observed processes, sampled at high or at low frequency. We also discuss application of nonparametric methods to testing, especially model validation and goodness-of-fit testing. In risk measurement for financial time series, conditional quantiles play an important role and nonparametric methods have been successfully applied in this field too. At the end of the chapter we discuss Grenander’s sieve methods and other more recent advanced nonparametric approaches.


Regression Quantile Nonparametric Estimation Stochastic Volatility Nonparametric Regression Nonparametric Estimator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abberger, K. (1996): Nichtparametrische Schaetzung bedingter Quantile in Zeitreihen. Mit Anwendung auf Finanzmarktdaten. Hartung–Gore, Konstanz.Google Scholar
  2. Ai, C. and Chen, X. (2003): Efficient estimators of models with conditional moment restrictions containing unknown functions. Econometrica 71, 1795–1843.zbMATHMathSciNetCrossRefGoogle Scholar
  3. Aït-Sahalia, Y. (1996a): Testing continuous-time models of the spot interest rate. Review of Financial Studies 9, 385–426.CrossRefGoogle Scholar
  4. Aït-Sahalia, Y. (1996b): Nonparametric pricing of interest rate derivative securities. Econometrica 64, 527–560.zbMATHCrossRefGoogle Scholar
  5. Aït-Sahalia, Y., Duarte, J. (2003): Nonparametric option pricing und shape restrictions. Journal of Econometrics 116, 9–47.zbMATHMathSciNetCrossRefGoogle Scholar
  6. Aït-Sahalia, Y., Lo, A.W. (1998): Nonparametric estimation of state-price densities implicit in financial asset prices. Journal of Finance 53, 499–547.CrossRefGoogle Scholar
  7. Aït-Sahalia, Y., Bickel, P.J., Stoker, T.M. (2001): Goodness-of-fit tests for kernel regression with an application to option implied volatilities. Journal of Econometrics 105, 363–412.zbMATHMathSciNetCrossRefGoogle Scholar
  8. Aït-Sahalia, Y., Fan, J. and Peng, H. (2005): Nonparametric transition-based tests for jump-diffusions. Preprint. Available at SSRN:
  9. Ango Nze, P.A., Doukhan, P. (2004): Weak dependence: Models and applications in econometrics. Econometric Theory 20, 995–1045.zbMATHMathSciNetGoogle Scholar
  10. Ango Nze, P.A., Bühlmann, P., Doukhan, P. (2002): Weak dependence beyond mixing and asymptotics for nonparametric regression. Annals of Statistics 30, 397–430.zbMATHMathSciNetCrossRefGoogle Scholar
  11. Arapis, M. and Gao, J. (2006): Empirical comparisons in short-term interest rate models using nonparametric methods. Journal of Financial Econometrics 4, 310–345.CrossRefGoogle Scholar
  12. P. Artzner, P., Delbaen, F., Eber, F.-J. and Heath, D. (1997): Thinking Coherently. Risk Magazine 10, 68–71.Google Scholar
  13. Audrino, F. and Bühlmann, P. (2001): Tree-structured GARCH models. Journal of the Royal Statistical Society Series B 63, 727–744.zbMATHCrossRefGoogle Scholar
  14. Bandi, F. and Nguyen, T. H. (2000): Fully nonparametric estimators for diffusions: Small sample analysis. Working paper Graduate School of Business, The University of Chicago.Google Scholar
  15. Bandi, F. M. and Nguyen, T. H. (2003): On the functional estimation of jump-diffusion models. Journal of Econometrics 116, 293–328.zbMATHMathSciNetCrossRefGoogle Scholar
  16. Bandi, F. and Phillips, P.C.B. (2003): Fully nonparametric estimation of scalar diffusion models. Econometrica 71, 241–283.zbMATHMathSciNetCrossRefGoogle Scholar
  17. Benko, M., Härdle, W. and Kneip, A. (2008): Common functional principal components. Annals of Statistics to appear.Google Scholar
  18. Boente, G. and Fraiman, R. (1995). Asymptotic distribution of smoothers based on local means and local medians under dependence. Journal of Multivariate Analysis 54, 77–90.zbMATHMathSciNetCrossRefGoogle Scholar
  19. Bol, G., Nakhaeizadeh, G. and Vollmer, K.-H. (Eds.) (1996): Finanzmarktanalyse und-prognose mit innovativen quantitativen Verfahren. Physica, Heidelberg.Google Scholar
  20. Borak, S., Härdle, W., Mammen, E. and Park, B. (2008): Time series modelling with semiparametric factor dynamics. Preprint.Google Scholar
  21. Bosq, D. (1996): Bosq, D. (1996): Nonparametric statistics for stochastic processes: estimation and prediction. Lecture Notes in Statistics 110. Springer, New York.Google Scholar
  22. Breiman, L., Friedman, J. H., Olshen, R. A. and Stone, C. J. (1984): Classification and Regression Trees. Wadsworth, Belmont.zbMATHGoogle Scholar
  23. Brüggemann, R., Härdle, W., Mungo, J. and Trenkler, C. (2008): VAR modeling for dynamic semiparametric factors of volatility strings. Journal of Financial Econometrics 6, 361–381.CrossRefGoogle Scholar
  24. Cai, Z. (2002): Regression quantile for time series. Econometric Theory 18, 169–192.zbMATHMathSciNetCrossRefGoogle Scholar
  25. Cai, Z. and Hong, Y. (2003): Nonparametric methods in continuous-time fiance: A selective review. In: Akritas, M. G. and Politis, D. N. (Eds.): Recent Advances and Trends in Nonparametric Statistics, 283–302. Elsevier, Amsterdam.CrossRefGoogle Scholar
  26. Casas, I. and Gao, J. (2008): Econometric estimation in long-range dependent volatility models: Theory and practice. Journal of Econometrics to appear.Google Scholar
  27. Chen, S. X., Gao, J. and Tang, C. (2008): A test for model specification of diffusion processes. Annals of Statistics 36, 167–198.zbMATHMathSciNetCrossRefGoogle Scholar
  28. Chen, X. and Shen, X. (1998): Sieve extremum estimates for weakly dependent data. Econometrica 66, 289–314.zbMATHMathSciNetCrossRefGoogle Scholar
  29. Chen, X. and White, H. (1999): Improved rates and asymptotic normality for nonparametric neural network estimators. IEEE Transactions on Information Theory 45, 682–691.zbMATHMathSciNetCrossRefGoogle Scholar
  30. Christmann, A. (2004): An approach to model complex high-dimensional insurance data. Allgemeines Statistisches Archiv 88, 375–396.zbMATHMathSciNetCrossRefGoogle Scholar
  31. Collomb, G. (1984): Propriétés de convergence presque complète du predicteur à noyau. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 66, 441–460.zbMATHMathSciNetCrossRefGoogle Scholar
  32. Connor, G., Hagmann, M. and Linton, O.B. (2007): Efficient estimation of a semiparametric characteristic-based factor model of security returns. Preprint.Google Scholar
  33. Cont, R. and da Fontseca, J. (2002): The dynamics of implied volatility surfaces. Quantitative Finance 2, 45–60.MathSciNetCrossRefGoogle Scholar
  34. Csörgö, S. and Mielniczuk, J. (1995): Density estimation under long-range dependence Annals of Statistics 23, 990–999.zbMATHMathSciNetCrossRefGoogle Scholar
  35. Donald, S.G. (1997): Inference concerning the number of factors in a multivariate nonparametric relationship. Econometrica 65, 103–131.zbMATHMathSciNetCrossRefGoogle Scholar
  36. Doukhan, P., Louhichi, S. (1999): A new weak dependence condition and applications to moment inequalities. Stochastic Processes and Their Applications 84, 313–342.zbMATHMathSciNetCrossRefGoogle Scholar
  37. Dürkes, A. and Kreiss, J.-P. (2006): Weak dependence of nonparametric GARCH-models. Technical report, TU Braunschweig.Google Scholar
  38. Engle, R.F. and Manganelli, S. (2004): CAViaR: Conditional autoregressive value at risk by regression quantiles. Journal of Business and Economic Statistics 22, 367–381.MathSciNetCrossRefGoogle Scholar
  39. Evans, O. (1997): Short-term currency forecasting using neural networks. ICL Systems Journal 11, 1–17.Google Scholar
  40. Fan, J. (1992): Design-adaptive nonparametric regression. Journal of the American Statistical Association 87, 998–1004.zbMATHMathSciNetCrossRefGoogle Scholar
  41. Fan, J. (1993): Local linear regression smoothers and their minimax efficiencies. Annals of Statistics 21, 196–216.zbMATHMathSciNetCrossRefGoogle Scholar
  42. Fan, J. (2005): A selective overview of nonparametric methods in financial econometrics (with discussion). Statistical Science 20, 317–357.zbMATHMathSciNetCrossRefGoogle Scholar
  43. Fan, J. and Gijbels, I. (1995): Local Polynomial Modelling and Its Applications. Theory and Methodologies. Chapman and Hall, New York.Google Scholar
  44. Fan, J. and Guo, J. (2003): Semiparametric estimation of value at risk. Econometrics Journal 6, 261–290.zbMATHMathSciNetCrossRefGoogle Scholar
  45. Fan, J. and Yao, Q. (1998): Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645–660.zbMATHMathSciNetCrossRefGoogle Scholar
  46. Fan, J. and Yao, Q. (2003): Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.zbMATHCrossRefGoogle Scholar
  47. Fan, J. and Zhang, C.(2003): A re-examination of Stanton's diffusion estimations with applications to financial model validation. Journal of the American Statistical Association 98, 118–134.zbMATHMathSciNetCrossRefGoogle Scholar
  48. Fan, J., Fan, Y. and Lv, J. (2007a): Aggregation of nonparametric estimators for volatility matrix. Journal of Financial Econometrics 5, 321–357.CrossRefGoogle Scholar
  49. Fan, J., Fan, Y. and Lv, J. (2007b): High dimensional covariance matrix estimation using a factor model. Journal of Econometrics to appear.Google Scholar
  50. Fengler, M., Härdle, W. and Mammen, E. (2007): Implied volatility string dynamics. Journal of Financial Econometrics 5, 189–218.CrossRefGoogle Scholar
  51. Franke, J. (1998): Nonlinear and nonparametric methods for analyzing financial time series. In: Kall, P. and Luethi, H.-J. (eds.): Operation Research Proceedings 98. Springer, Heidelberg.Google Scholar
  52. Franke, J. (2000): Portfolio management and market risk quantification using neural networks. In: Chan, W.S., Li, W.K. and Tong, H. (Eds.): Statistics and Finance: An Interface. Imperial College Press, London.Google Scholar
  53. Franke, J. and Diagne, M. (2006): Estimating market risk with neural networks. Statistics and Decisions 24, 233–253.zbMATHMathSciNetCrossRefGoogle Scholar
  54. Franke, J. and Klein, M. (1999): Optimal portfolio management using neural networks—a case study. Report in Wirtschaftsmathematik 49. University of Kaiserslautern.Google Scholar
  55. Franke, J. and Mwita, P. (2003): Nonparametric estimates for conditional quantiles of time series. Report in Wirtschaftsmathematik 87, University of Kaiserslautern.Google Scholar
  56. Franke, J., Kreiss, J.-P., Mammen, E. (2002a): Bootstrap of kernel smoothing in nonlinear time series. Bernoulli 8, 1–37.zbMATHMathSciNetGoogle Scholar
  57. Franke, J., Kreiss, J.-P., Mammen, E., Neumann, M.H. (2002b): Properties of the nonparametric autoregressive bootstrap. Journal of Time Series Analysis 23, 555–585.zbMATHMathSciNetCrossRefGoogle Scholar
  58. Franke, J., Härdle, W. and Kreiss, J.-P. (2003): Nonparametric estimation in a stochastic volatility model. In: Akritas, M. G. and Politis, D. N. (Eds.): Recent Advances and Trends in Nonparametric Statistics, 302–313. Elsevier, Amsterdam.Google Scholar
  59. Franke, J., Neumann, M.H. and Stockis, J.-P. (2004): Bootstrapping nonparametric estimates of the volatility function. Journal of Econometrics 118, 189–218.zbMATHMathSciNetCrossRefGoogle Scholar
  60. Franke, J., Stockis, J.-P. and Tadjuidje, J. (2007): Quantile sieve estimates for time series. Report in Wirtschaftsmathematik 105, University of Kaiserslautern.Google Scholar
  61. Gao, J. (2007): Nonlinear Time Series: semiparametric and nonparametric methods. Monographs on Statistics and Applied Probability 108. Chapman and Hall, London.Google Scholar
  62. Gao, J., King, M. L., Lu, Z. and Tjøstheim, D. (2007): Specification Testing in Nonlinear Time Series. Preprint, School of Economics, The University of Adelaide.Google Scholar
  63. Gobet, E., Hoffmann, M. and Reiss, M. (2004): Nonparametric estimation of scalar diffusions based on low-frequency data. Annals of Statistics 32, 2223–2253.zbMATHMathSciNetCrossRefGoogle Scholar
  64. Gouriéroux, C. (1997): ARCH Models and Financial Applications. Springer, Heidelberg.zbMATHGoogle Scholar
  65. Gouriéroux, C. and Montfort, A. (1992): Qualitative threshold ARCH models. Journal of Econometrics 52, 159–199.zbMATHMathSciNetCrossRefGoogle Scholar
  66. Grenander, U. (1981): Abstract Inference. Wiley, New York.zbMATHGoogle Scholar
  67. Györfy, L., Kohler, M., Krzyzak, A. and Walk, H. (2002): A Distribution-Free Theory of Nonparametric Regression. Springer, Heidelberg.Google Scholar
  68. Hall, P., Lahiri, S.N. and Truong, Y.K. (1995): On bandwidth choice for density estimation with dependent data. Annals of Statistics 23, 2241–2263.zbMATHMathSciNetCrossRefGoogle Scholar
  69. Härdle, W. and Linton, O. (1994): Applied nonparametric methods. In: Engle, R. and McFadden, D. (Eds.): Handbook of Econometrics IV. North-Holland, Amsterdam.Google Scholar
  70. Härdle, W. and Mammen, E. (1993): Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1947.zbMATHMathSciNetCrossRefGoogle Scholar
  71. Härdle, W. and Tsybakov, A.B. (1997): Local polynomial estimators of the volatility function in nonparametric autoregression. Journal of Econometrics 81, 223–242.zbMATHMathSciNetCrossRefGoogle Scholar
  72. Härdle, W., Lütkepohl, H. and Chen, R. (1997): A review of nonparametric time series analysis. International Statistical Review 65, 49–72.zbMATHCrossRefGoogle Scholar
  73. Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov (1998): Wavelets approximation and statistical applications. Springer Lecture Notes in Statistics 129. Springer, New York.Google Scholar
  74. Hart, J.D. (1996): Some automated methods of smoothing time-dependent data. Journal of Nonparametric Statistics 6, 115–142.zbMATHMathSciNetCrossRefGoogle Scholar
  75. Hart, J.D. and Vieu, P. (1990): Data-driven bandwidth choice for density estimation based on dependent data. Annals of Statistics 18, 873–890.zbMATHMathSciNetCrossRefGoogle Scholar
  76. Hastie, T. and Tibshirani, R. (1991): Generalized Additive Models. Chapman and Hall, London.Google Scholar
  77. Heitkamp, D. (1996): Methodische Aspekte bei der Entwicklung von Tradingmodellen auf der Basis Neuronaler Netze. Wirtschaftsinformatik 38, 238–292.Google Scholar
  78. Hjellvik, V. and Tjøstheim, D. (1995): Nonparametric tests of linearity for time series. Biometrika 82, 351–368.zbMATHMathSciNetCrossRefGoogle Scholar
  79. Hoffmann, M. (2001): On estimating the diffusion coefficient: Parametric versus nonparametric. Annales de l’Institut Henri Poincare 37, 339–372.zbMATHGoogle Scholar
  80. Hong, Y. and Li, H. (2002): Nonparametric specification testing for continuous time models with applications to term structure of interest rates. Review of Financial Studies 18, 37–84.CrossRefGoogle Scholar
  81. Jacod J. (2000): Non-parametric kernel estimation of the diffusion coefficient of a diffusion. Scandinavian Journal of Statistics 27, 83–96.zbMATHMathSciNetCrossRefGoogle Scholar
  82. Jiang, G.J. and Knight, J.L. (1997): A nonparametric approach to the estimation of diffusion processes, with application to a short term interest model. Econometric Theory 13, 615–645MathSciNetCrossRefGoogle Scholar
  83. Karlsen, H. and Tjøstheim, D. (2001): Nonparametric estimation in null-recurrent time series. Annals of Statistics 29, 372–416.zbMATHMathSciNetCrossRefGoogle Scholar
  84. Koenker, R. (2005). Quantile Regression Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  85. Koenker, R. and Bassett, G. (1978): Regression quantiles. Econometrica 46, 33–50.zbMATHMathSciNetCrossRefGoogle Scholar
  86. Kreiss, J.-P. (2000): Nonparametric estimation and bootstrap for financial time series. In: Chan, W.S., Li, W.K. and Tong, H. (Eds.): Statistics and Finance: An Interface. Imperial College Press, London.Google Scholar
  87. Kreiss, J.-P., Neumann, M.H. and Yao, Q. (2008): Bootstrap tests for simple structures in nonparametric time series regression. Statistics and Its Interface to appear.Google Scholar
  88. Linton, O. B. (2008): Semiparametric and nonparametric ARCH modelling. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 156–167. Springer, New York.Google Scholar
  89. Linton, O. B. and Mammen, E. (2005): Estimating semiparametric ARCH(∞) models by kernel smoothing methods. Econometrica 73, 771–836.zbMATHMathSciNetCrossRefGoogle Scholar
  90. Linton, O. B. and Mammen, E. (2008): Nonparametric transformation to white noise. Journal of Econometrics 142, 241–264.MathSciNetCrossRefGoogle Scholar
  91. Linton, O. B. and Sancetta, A. (2007): Consistent estimation of a general nonparametric regression function in time series. Preprint, The London School of Economics.Google Scholar
  92. Linton, O. B., Mammen, E., Nielsen, J. P. and Tanggaard, C. (2001): Estimating yield curves by kernel smoothing methods. Journal of Econometrics 105, 185–223.zbMATHMathSciNetCrossRefGoogle Scholar
  93. Linton, O., Nielsen, J. P. and Nielsen, S. F. (2008): Nonparametric regression with a latent time series. Preprint.Google Scholar
  94. Lu, Z. (2001): Asymptotic normality of kernel density estimators under dependence. Annals of the Institute of Statistical Mathematics 53, 447–468.zbMATHMathSciNetCrossRefGoogle Scholar
  95. Lu, Z., Linton, O. (2007): Local linear fitting under near epoch dependence. Econometric Theory 23, 37–70.MathSciNetCrossRefGoogle Scholar
  96. Mammen, E. (1991): Estimating a smooth monotone regression function. Annals of Statistics 19, 724–740.zbMATHMathSciNetCrossRefGoogle Scholar
  97. Mammen, E., Linton, O. B. and Nielsen, J. P. (1999): The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Annals of Statistics 27, 1443–1490.zbMATHMathSciNetGoogle Scholar
  98. Mammen, E., Marron, J.S., Turlach, B.A., Wand, M.P. (2001): A general projection framework for constrained smoothing. Statistical Science 16, 232–248.zbMATHMathSciNetCrossRefGoogle Scholar
  99. Mammen E., Stove, B. and Tjøstheim, D. (2008): Nonparametric additive models for panels of time series. Preprint.Google Scholar
  100. Masry, E. (1996a): Multivariate regression estimation: Local polynomial fitting for time series. Stochastic Processes and Their Applications 65, 81–101 [Correction (1997). 67, 281].zbMATHMathSciNetCrossRefGoogle Scholar
  101. Masry, E. (1996b): Multivariate local polynomial regression for time series: uniform strong consistency and rates. Journal of Time Series Analysis 17, 571–599.zbMATHMathSciNetCrossRefGoogle Scholar
  102. Masry, E. and Fan, J. (1997): Local polynomial estimation of regression functions for mixing processes. Scandinavian Journal of Statistics. 24, 165–179.zbMATHMathSciNetCrossRefGoogle Scholar
  103. Masry, E. and Tjøstheim, D. (1994): Nonparametric estimation and identification of non-linear ARCH time series: Strong convergence and asymptotic normality. Econometric Theory 11, 258–289.CrossRefGoogle Scholar
  104. Meir, R. (2000): Nonparametric time series prediction through adaptive model selection. Machine Learning 39, 5–34.zbMATHCrossRefGoogle Scholar
  105. Modha, D.S. and Masry, E. (1996): Minimum complexity regression estimation with weakly dependent observations. IEEE Transacvtions on Information Theory 42, 2133–2145.zbMATHMathSciNetCrossRefGoogle Scholar
  106. Modha, D.S. and Masry, E. (1998): Memory-universal prediction of stationary random processes. IEEE Transactions on Infomation Theory 44, 117–133.zbMATHMathSciNetCrossRefGoogle Scholar
  107. Moloche, G. (2001): Kernel regression for non-stationary harris-recurrent processes. MIT working paper.Google Scholar
  108. Neumann, M.H. and Kreiss, J.-P. (1998): Regression-type inference in nonparametric autorregression. Annals of Statistics 26, 1570–1613.zbMATHMathSciNetCrossRefGoogle Scholar
  109. Neumann, M.H. and Reiss, M. (2007): Nonparametric estimation for Lévy processes from low-frequency observations. Preprint, arXiv:0709.2007v1 [math.ST].Google Scholar
  110. Newey, W.K. (1997): Convergence rates and asymptotic normality for series estimators. Journal of Econometrics 79, 147–168.zbMATHMathSciNetCrossRefGoogle Scholar
  111. Pagan, A, and Ullah, A. (1999): Nonparametric Econometrics. Cambridge University Press, Cambridge.Google Scholar
  112. Phillips, P.C.B. and Park, J.Y. (1998): Nonstationary density estimation and kernel autoregression. Cowles Foundation Discussion Paper 1181.Google Scholar
  113. Pritsker, M. (1998): Nonparametric density estimation and tests of continuous time interest rate models. Review of Financial Studies 11, 449–487.CrossRefGoogle Scholar
  114. Refenes, A.-P., Burgess, A.N. and Bentz, Y. (1996): Neural networks in financial engineering: A study in methodology. IEEE Transactions on Neural Networks 8, 1222–1267.CrossRefGoogle Scholar
  115. Robinson, P.M. (1983): Nonparametric Estimators for Time Series. Journal of Time Series Analysis 4, 185–207.zbMATHMathSciNetCrossRefGoogle Scholar
  116. Schienle, M. (2007): Nonparametric nonstationary regression. Preprint, Department of Economics, Universität Mannheim.Google Scholar
  117. Stanton, R. (1997): A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance 52, 1973–2002.CrossRefGoogle Scholar
  118. Stone, C.J. (1977): Consistent nonparametric regression. Annals of Statistics 5, 595–620.zbMATHMathSciNetCrossRefGoogle Scholar
  119. Stone, C.J. (1994): The use of polynomial splines and their tensor products in mutlivariate function estimation (with discussion). Annals of Statistics 22, 118–184.zbMATHMathSciNetCrossRefGoogle Scholar
  120. Tjøstheim, D. (1994): Non-linear time series: A selective review. Scandinavian Journal of Statistics 21, 97–130.Google Scholar
  121. Thompson, S. (2008): Identifying term structure volatility from the LIBOR–swap curve. Review of Financial Studies 21, 819–854.CrossRefGoogle Scholar
  122. Tong, H. (1990): Non-linear Time Series. A Dynamic System Approach. Oxford University Press, Oxford.Google Scholar
  123. Tsybakov, A.B. (1986): Robust reconstruction of functions by the local approximation method. Problems of Information Transmission 22, 133–146.zbMATHGoogle Scholar
  124. Truong, Y.K. and Stone, C.J. (1992): Nonparametric function estimation involving time series. Annals of Statistics 20, 77–97.zbMATHMathSciNetCrossRefGoogle Scholar
  125. Van Es, B., Spreij, P. and van Zanten, H. (2003): Nonparametric volatility density estimation. Bernoulli 9, 451–465.zbMATHMathSciNetCrossRefGoogle Scholar
  126. Van Es, B., Spreij, P. and van Zanten, H. (2005): Nonparametric volatility density estimation for discrete time models. Nonparametric Statistics 17, 237–251.zbMATHMathSciNetCrossRefGoogle Scholar
  127. Whaba, G. (1990): Spline models for observational data. CBMS-NSF Regional Conference Series in Apllied Mathematics 59. Society for Industrial and Applied Mathematics, Philadelphia.Google Scholar
  128. White, H. and Wooldridge, W. (1990): Some results for sieve estimation with dependent observations. In: Barnett, W., Powell, J. and Tauchen, G. (Eds.): Nonparametric and Semi-Parametric Methods in Econometrics and Statistics. Cambridge University Press, Cambridge..Google Scholar
  129. Yang, L., Härdle, W. and Nielsen, J. (1999): Nonparametric autoregression with multiplicative volatility and additive mean. Journal of Time Series Analysis 20, 579–604.zbMATHMathSciNetCrossRefGoogle Scholar
  130. Zhao, Z. (2008a): Parametric and nonparametric models and methods in financial econometrics. Statistics Surveys, arXiv:0801.1599v1 [stat.ME].Google Scholar
  131. Zhao, Z. (2008b): Nonparametric model validation for hidden Markov models with application to financial econometrics. Preprint, Department of Statistics, Pennsylvania State University.Google Scholar
  132. Zhao, Z. and Wu, W. B. (2007): Confidence bands in nonparametric time series regression. Annals of Statistics 36, 1854–1878.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jürgen Franke
    • 1
    Email author
  • Jens-Peter Kreiss
    • 2
  • Enno Mammen
    • 3
  1. 1.Department of MathematicsUniversität KaiserslauternErwin–Schrödinger-StrasseGermany
  2. 2.Institut für Mathematische StochastikTechnische Universität BraunschweigBraunschweigGermany
  3. 3.Abteilung VolkswirtschaftslehreUniversität MannheimMannheimGermany

Personalised recommendations