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

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.