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Bayesian estimation of bandwidths for a nonparametric regression model with a flexible error density

Author

Listed:
  • Xibin Zhang

    ()

  • Maxwell L. King

    ()

  • Han Lin Shang

    ()

Abstract

We approximate the error density of a nonparametric regression model by a mixture of Gaussian densities with means being the individual error realizations and variance a constant parameter. We investigate the construction of a likelihood and posterior for bandwidth parameters under this Gaussian-component mixture density of errors in a nonparametric regression. A Markov chain Monte Carlo algorithm is presented to sample bandwidths for the kernel estimators of the regression function and error density. A simulation study shows that the proposed Gaussian-component mixture density of errors is clearly favored against wrong assumptions of the error density. We apply our sampling algorithm to a nonparametric regression model of the All Ordinaries daily return on the overnight FTSE and S&P 500 returns, where the error density is approximated by the proposed mixture density. With the estimated bandwidths, we estimate the density of the one-step-ahead point forecast of the All Ordinaries return, and therefore, a distribution-free value-at-risk is obtained. The proposed Gaussian component mixture density of regression errors is also validated through the nonparametric regression involved in the state-price density estimation proposed by Aït-Sahalia and Lo (1998).

Suggested Citation

  • Xibin Zhang & Maxwell L. King & Han Lin Shang, 2011. "Bayesian estimation of bandwidths for a nonparametric regression model with a flexible error density," Monash Econometrics and Business Statistics Working Papers 10/11, Monash University, Department of Econometrics and Business Statistics.
  • Handle: RePEc:msh:ebswps:2011-10
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    File URL: http://business.monash.edu/econometrics-and-business-statistics/research/publications/ebs/wp10-11.pdf
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    References listed on IDEAS

    as
    1. Linton, Oliver & Xiao, Zhijie, 2007. "A Nonparametric Regression Estimator That Adapts To Error Distribution Of Unknown Form," Econometric Theory, Cambridge University Press, vol. 23(03), pages 371-413, June.
    2. Y. K. Tse & Xibin Zhang & Jun Yu, 2004. "Estimation of hyperbolic diffusion using the Markov chain Monte Carlo method," Quantitative Finance, Taylor & Francis Journals, vol. 4(2), pages 158-169.
    3. Xibin Zhang & Maxwell L. King, 2011. "Bayesian semiparametric GARCH models," Monash Econometrics and Business Statistics Working Papers 24/11, Monash University, Department of Econometrics and Business Statistics.
    4. G. M. Martin & C. S. Forbes, 1999. "Using simulation methods for bayesian econometric models: inference, development and communication: some comments," Econometric Reviews, Taylor & Francis Journals, vol. 18(1), pages 113-118.
    5. repec:taf:gnstxx:v:23:y:2011:i:1:p:137-148 is not listed on IDEAS
    6. Rothe, Christoph, 2009. "Semiparametric estimation of binary response models with endogenous regressors," Journal of Econometrics, Elsevier, vol. 153(1), pages 51-64, November.
    7. Zhang, Xibin & King, Maxwell L. & Hyndman, Rob J., 2006. "A Bayesian approach to bandwidth selection for multivariate kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3009-3031, July.
    8. Huynh, Kim & Kervella, Pierre & Zheng, Jun, 2002. "Estimating state-price densities with nonparametric regression," SFB 373 Discussion Papers 2002,40, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    9. John Geweke, 1999. "Using simulation methods for bayesian econometric models: inference, development,and communication," Econometric Reviews, Taylor & Francis Journals, vol. 18(1), pages 1-73.
    10. Zhang, Xibin & Brooks, Robert D. & King, Maxwell L., 2009. "A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation," Journal of Econometrics, Elsevier, vol. 153(1), pages 21-32, November.
    11. W. E. Griffiths, 1999. "Estimating consumer surplus comments on "using simulation methods for bayesian econometric models: inference development and communication"," Econometric Reviews, Taylor & Francis Journals, vol. 18(1), pages 75-87.
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    Citations

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    Cited by:

    1. Xibin Zhang & Maxwell L. King & Han Lin Shang, 2016. "Bayesian Bandwidth Selection for a Nonparametric Regression Model with Mixed Types of Regressors," Econometrics, MDPI, Open Access Journal, vol. 4(2), pages 1-27, April.
    2. Shang, Han Lin, 2013. "Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 185-198.

    More about this item

    Keywords

    Bayes factors; Gaussian-component mixture density; Markov chain Monte Carlo; state-price density; value-at-risk.;

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • G15 - Financial Economics - - General Financial Markets - - - International Financial Markets

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