Semiparametric Localized Bandwidth Selection in Kernel Density Estimation

Since conventional cross-validation bandwidth selection methods do not work for the case where the data considered are serially dependent, alternative bandwidth selection methods are needed. In recent years, Bayesian based global bandwidth selection methods have been proposed. Our experience shows that the use of a global bandwidth is however less suitable than using a localized bandwidth in kernel density estimation in the case where the data are serially dependent. Nonetheless, a difficult issue is how we can consistently estimate a localized bandwidth. In this paper, we propose a semiparametric estimation method and establish an asymptotic theory for the proposed estimator. A by-product of this bandwidth estimate is a new sampling-based likelihood approach to hyperparameter estimation. Monte Carlo simulation studies show that the proposed hyperparameter estimation method works very well, and that the proposed bandwidth estimator outperforms its competitors. Applications of the new bandwidth estimator to the kernel density estimation of Eurodollar deposit rate, as well as the S&P 500 daily return under conditional heteroscedasticity, demonstrate the effectiveness and competitiveness of the proposed semiparametric localized bandwidth.