Abstract
The unknown error density of a nonparametric regression model is approximated by a mixture of Gaussian densities with means being the individual error realizations and variance a constant parameter. Such a mixture density has the form of a kernel density estimator of error realizations. An approximate likelihood and posterior for bandwidth parameters in the kernel-form error density and the Nadaraya–Watson regression estimator are derived, and a sampling algorithm is developed. 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. The proposed approach is demonstrated through a nonparametric regression model of the Australian All Ordinaries daily return on the overnight FTSE and S&P 500 returns. With the estimated bandwidths, the one-day-ahead posterior predictive density of the All Ordinaries return is derived, 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.
Highlights
A simple and commonly used estimator of the regression function in a nonparametric regression model is the Nadaraya–Watson (NW) estimator, whose performance is mainly determined by the choice of bandwidths
We apply the proposed sampling algorithm to the estimation of the bandwidths for the nonparametric regression of the Australian All Ordinaries (Aord) daily return on the overnight FTSE and S&P 500 returns with its error density being the kernel–form
Our Bayesian sampling procedure represents a data–driven solution to the problem of simultaneously estimating bandwidths for the kernel estimators of the regression function and error density. Applying it to the nonparametric regression of the All Ordinaries daily return on the overnight FTSE and S&P 500 returns, we have obtained the bandwidth estimates for the kernel estimator of the regression under the three error–density assumptions
Summary
A simple and commonly used estimator of the regression function in a nonparametric regression model is the Nadaraya–Watson (NW) estimator, whose performance is mainly determined by the choice of bandwidths. In a class of nonlinear regression models, Yuan and de Gooijer (2007) constructed an approximate likelihood through the kernel density estimator of prefitted residuals with its bandwidth pre-chosen by the rule–of–thumb They proved that under some regularity conditions, the resulting maximum likelihood estimates of parameters are consistent, asymptotically normal and efficient. Our second application is the one discussed by Zhang et al (2009) who estimated the bandwidths for a nonparametric regression model so as to estimate the state–price density (SPD) of the S&P 500 index at the maturity of its call option In this application, we assume that the unknown error density is approximated by the kernel–form density and find that this robust error density is favored with very strong evidence against the Gaussian error density.
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