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.