Semiparametric Localized Bandwidth Selection in Kernel Density Estimation

Tingting Cheng,Jiti Gao,X Zhang

doi:10.2139/ssrn.2435478

Abstract

Since conventional cross-validation bandwidth selection methods don't work for the case where the data considered are dependent time series, 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 dependent time series as discussed in an empirical application of this paper. Nonetheless, a difficult issue is how we can consistently estimate a localized bandwidth. In this paper, we propose a semiparametric estimation method, for which we establish an asymptotic theory for the proposed semiparametric 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.

Highlights

Kernel density estimation is an important tool for exploring the distributional properties of a random variable in an unknown population (Silverman, 1986)
The main contributions of this paper are: (i) We develop an asymptotic theory for our proposed semiparametric localized bandwidth selection method; (ii) We present a likelihood approach to hyperparameter estimation and show that it works very well in Monte Carlo simulation and empirical studies; (iii) We conduct simulation studies to examine the finite–sample performance of our proposed semiparametric localized bandwidth selection, as well as the performance of likelihood approach to hyperparameter estimation; (iv) We apply the proposed semiparametric localized bandwidth selection method to the kernel density estimates of daily Eurodollar deposit rate and the S&P 500 daily return, respectively
In this paper, we have investigated the asymptotic properties of a semiparametric localized bandwidth (SLB) estimator for kernel density estimation for stationary time series data

Summary

INTRODUCTION

Kernel density estimation is an important tool for exploring the distributional properties of a random variable in an unknown population (Silverman, 1986). We assume the prior of bandwidth is an inverse Gamma density with two hyperparameters We find that these two hyperparameters play an important role in the performance of the resulting bandwidth. Atchadé (2011) developed an adaptive Monte Carlo strategy for sampling from posterior in empirical Bayes analysis These methods are not applicable to bandwidth estimation discussed in this paper. A likelihood function is constructed based on this pseudo sample and is maximized to derive hyperparameter estimates This likelihood approach is semiparametric because the density for constructing the likelihood is approximated by its kernel estimator. (iv) We apply the proposed semiparametric localized bandwidth selection method to the kernel density estimates of daily Eurodollar deposit rate and the S&P 500 daily return, respectively. The proofs of the main theorems are given in a supplemental document

SEMIPARAMETRIC LOCALIZED BANDWIDTH

ASYMPTOTIC THEORY

MONTE CARLO SIMULATION RESULTS

Simulation results

ESTIMATING AND FORECASTING THE DENSITY OF EURODOLLAR DEPOSIT RATE

CONCLUSIONS