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

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).