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A sampling algorithm for bandwidth estimation in a nonparametric regression model with a flexible error density


  • Xibin Zhang


  • Maxwell L. King


  • Han Lin Shang



We propose to approximate the unknown 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. This mixture density has the form of a kernel density estimator of error realizations. We derive an approximate likelihood and posterior for bandwidth parameters in the kernel-form error density and the Nadaraya-Watson regression estimator and develop a sampling algorithm. A simulation study shows that when the true error density is non-Gaussian, the kernel-form error density is often favored against its parametric counterparts including the correct error density assumption. Our approach is demonstrated through a nonparametric regression model of the Australian All Ordinaries daily return on the overnight FTSE and S&P 500 returns. Using the estimated bandwidths, we derive the one-day-ahead density forecast of the All Ordinaries return, and a distribution-free value-at-risk is obtained. The proposed algorithm is also applied to a nonparametric regression model involved in state–price density estimation based on S&P 500 options data.

Suggested Citation

  • Xibin Zhang & Maxwell L. King & Han Lin Shang, 2013. "A sampling algorithm for bandwidth estimation in a nonparametric regression model with a flexible error density," Monash Econometrics and Business Statistics Working Papers 20/13, Monash University, Department of Econometrics and Business Statistics.
  • Handle: RePEc:msh:ebswps:2013-20

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    References listed on IDEAS

    1. Yacine Aït-Sahalia & Andrew W. Lo, 1998. "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," Journal of Finance, American Finance Association, vol. 53(2), pages 499-547, April.
    2. Tse, Y.K. & Zhang, Bill & Yu, Jun, 2002. "Estimation of Hyperbolic Diffusion using MCMC Method," Working Papers 182, Department of Economics, The University of Auckland.
    3. 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.
    4. David J. Nott & Robert Kohn, 2005. "Adaptive sampling for Bayesian variable selection," Biometrika, Biometrika Trust, vol. 92(4), pages 747-763, December.
    5. 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.
    6. Sangjoon Kim & Neil Shephard & Siddhartha Chib, 1998. "Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models," Review of Economic Studies, Oxford University Press, vol. 65(3), pages 361-393.
    7. 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.
    8. 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.
    9. Rothe, Christoph, 2009. "Semiparametric estimation of binary response models with endogenous regressors," Journal of Econometrics, Elsevier, vol. 153(1), pages 51-64, November.
    10. 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.
    11. 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.
    12. Geweke, John & Keane, Michael, 2007. "Smoothly mixing regressions," Journal of Econometrics, Elsevier, vol. 138(1), pages 252-290, May.
    13. Ao Yuan & Jan G. De Gooijer, 2007. "Semiparametric Regression with Kernel Error Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(4), pages 841-869.
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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. Tristan Senga Kiessé & Nabil Zougab & Célestin C. Kokonendji, 2016. "Bayesian estimation of bandwidth in semiparametric kernel estimation of unknown probability mass and regression functions of count data," Computational Statistics, Springer, vol. 31(1), pages 189-206, March.
    3. repec:bla:istatr:v:85:y:2017:i:2:p:228-249 is not listed on IDEAS

    More about this item


    Bayes factors; kernel-form error density; Metropolis-Hastings algorithm; state–price density; value-at-risk;

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