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

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  • Xibin Zhang

    ()

  • Maxwell L. King

    ()

  • Han Lin Shang

    ()

Abstract

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.

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Bibliographic Info

Paper provided by Monash University, Department of Econometrics and Business Statistics in its series Monash Econometrics and Business Statistics Working Papers with number 20/13.

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Date of creation: 2013
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Handle: RePEc:msh:ebswps:2013-20

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Related research

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

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  1. Kim, Sangjoon & Shephard, Neil & Chib, Siddhartha, 1998. "Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models," Review of Economic Studies, Wiley Blackwell, vol. 65(3), pages 361-93, July.
  2. Oliver Linton & Zhijie Xiao, 2001. "A nonparametric regression estimator that adapts to error distribution of unknown form," LSE Research Online Documents on Economics 2120, London School of Economics and Political Science, LSE Library.
  3. Rothe, Christoph, 2009. "Semiparametric estimation of binary response models with endogenous regressors," Journal of Econometrics, Elsevier, vol. 153(1), pages 51-64, November.
  4. 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.
  5. Xibin Zhang & Robert D. Brooks & Maxwell L. King, 2007. "A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation," Monash Econometrics and Business Statistics Working Papers 11/07, Monash University, Department of Econometrics and Business Statistics.
  6. 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.
  7. 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.
  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. Geweke, John & Keane, Michael, 2007. "Smoothly mixing regressions," Journal of Econometrics, Elsevier, vol. 138(1), pages 252-290, May.
  10. Y.K. Tse & Xibin Zhang & Jun Yu, 2002. "Estimation of Hyperbolic Diffusion Using MCMC Method," Monash Econometrics and Business Statistics Working Papers 18/02, Monash University, Department of Econometrics and Business Statistics.
  11. David J. Nott & Robert Kohn, 2005. "Adaptive sampling for Bayesian variable selection," Biometrika, Biometrika Trust, vol. 92(4), pages 747-763, December.
  12. 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.
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