Bayesian bandwidth selection for a nonparametric regession model with mixed types of regressors

We propose a sampling approach to bandwidth estimation for a nonparametric regression model with continuous and discrete types of regressors and unknown error density. The unknown error density is approximated by a location-mixture of Gaussian densities with means being the individual errors, and variance a constant parameter. This error density has a form of a kernel density estimator of errors with its bandwidth being the common standard deviation. We derive an approximate likelihood and posterior for bandwidth parameters, and a sampling algorithm is also developed. Monte Carlo simulation studies show that the proposed Bayesian sampling approach leads to better accuracy of the resulting estimators, especially the error density estimator, than the cross-validation. We apply the proposed sampling method to bandwidth estimation for a nonparametric regression model of the Australian All Ordinaries (Aord) daily returns on the overnight S&P 500 return and an indicator of the FTSE return. With the estimated bandwidths, we obtain the one-day-ahead density forecast of the Aord return and a distribution-free measure of value-at-risk. We also use the proposed sampling method to estimate bandwidths for the kernel estimator of the joint density of GDP growth rate, its year level and OECD status.