Xibin Zhang

A Bayesian approach to bandwidth selection for multivariate kernel density estimation

Kernel density estimation for multivariate data is an important technique that has a wide range of applications. However, it has received significantly less attention than its univariate counterpart. The lower level of interest in multivariate kernel density estimation is mainly due to the increased difficulty in deriving an optimal data-driven bandwidth as the dimension of the data increases. We provide Markov chain Monte Carlo (MCMC) algorithms for estimating optimal bandwidth matrices for multivariate kernel density estimation. Our approach is based on treating the elements of the bandwidth matrix as parameters whose posterior density can be obtained through the likelihood cross-validation criterion. Numerical studies for bivariate data show that the MCMC algorithm generally performs better than the plug-in algorithm under the Kullback-Leibler information criterion, and is as good as the plug-in algorithm under the mean integrated squared error (MISE) criterion. Numerical studies for five-dimensional data show that our algorithm is superior to the normal reference rule. Our MCMC algorithm is the first data-driven bandwidth selector for multivariate kernel density estimation that is applicable to data of any dimension.

A Monte Carlo investigation of some tests for stochastic dominance

This paper compares the performance of several tests for stochastic dominance up to order three using Monte Carlo methods. The tests considered are the Davidson and Duclos (2000) test, the Anderson test (1996) and the Kaur, Rao and Singh (1994) test. Only unpaired samples of independent observations are considered, as this is a restriction for both the Anderson and Kaur–Rao–Singh tests. We find that the Davidson–Duclos test appears to be the best. The Kaur–Rao–Singh test is overly conservative and does not compare favorably against the Davidson–Duclos and Anderson tests in terms of power.

A class of nonlinear stochastic volatility models and its implications for pricing currency options

A class of stochastic volatility (SV) models is proposed by applying the Box-Cox transformation to the volatility equation. This class of nonlinear SV (N-SV) models encompasses all standard SV models, including the well-known lognormal (LN) SV model. It allows to empirically compare and test all standard specifications in a very convenient way and provides a measure of the degree of departure from the classical models. A likelihood-based technique is developed for analyzing the model. Daily dollar/pound exchange rate data provide some evidence against LN model and strong evidence against all the other classical specifications. An efficient algorithm is proposed to study the economic importance of the proposed model on pricing currency options.

The sizes and powers of some stochastic dominance tests: A Monte Carlo study for correlated and heteroskedastic distributions

Testing for stochastic dominance among distributions is an important issue in the study of asset management, income inequality, and market efficiency. This paper conducts Monte Carlo simulations to examine the sizes and powers of several commonly used stochastic dominance tests when the underlying distributions are correlated or heteroskedastic. Our Monte Carlo study shows that the test developed by Davidson and Duclos [R. Davidson, J.Y. Duclos, Statistical inference for stochastic dominance and for the measurement of poverty and inequality, Econometrica 68 (6) (2000) 1435-1464] has better size and power performances than two alternative tests developed by Kaur et al. [A. Kaur, B.L.S.P. Rao, H. Singh, Testing for second order stochastic dominance of two distributions, Econ. Theory 10 (1994) 849-866] and Anderson [G. Anderson, Nonparametric tests of stochastic dominance in income distributions, Econometrica 64 (1996) 1183-1193]. In addition, we find that when the underlying distributions are heteroskedastic, both the size and power of the test developed by Davidson and Duclos [R. Davidson, J.Y. Duclos, Statistical inference for stochastic dominance and for the measurement of poverty and inequality, Econometrica 68 (6) (2000) 1435-1464] are superior to those of the two alternative tests.

Influence Diagnostics in Generalized Autoregressive Conditional Heteroscedasticity Processes

Influence diagnostics have become important tools for statistical analysis since the seminal work by Cook. In this article we present a curvature-based directional diagnostic, set up based on the slope-based diagnostic to assess the local influence of minor perturbations on influence graph in a regression model. Using both slope- and curvature-based diagnostics, we examine local influence in the generalized autoregressive conditional heteroscedasticity (GARCH) model under two perturbation schemes that involve model perturbation and data perturbation. We present a Monte Carlo study to obtain the approximate benchmark for determining the significance of a directional diagnostic, as well as the threshold for locating influential observations. An empirical study involving GARCH modeling of the continuously compounded daily return of the New York Stock Exchange composite index illustrates the effectiveness of the proposed diagnostics. The empirical study also shows that the curvature-based diagnostic can find a cluster of additive shocks that cannot be discovered by the slope-based diagnostic. Because observations may have different effects on the influence graph under different perturbation schemes, and both the slope-based and the curvature-based diagnostics are useful for assessing local influence (especially in GARCH models), it is advisable to assess local influence under different perturbation schemes through both diagnostics.

Bayesian adaptive bandwidth kernel density estimation of irregular multivariate distributions

In this paper, we propose a new methodology for multivariate kernel density estimation in which data are categorized into low- and high-density regions as an underlying mechanism for assigning adaptive bandwidths. We derive the posterior density of the bandwidth parameters via the Kullback-Leibler divergence criterion and use a Markov chain Monte Carlo (MCMC) sampling algorithm to estimate the adaptive bandwidths. The resulting estimator is referred to as the tail-adaptive density estimator. Monte Carlo simulation results show that the tail-adaptive density estimator outperforms the global-bandwidth density estimators implemented using different global bandwidth selection rules. The inferential potential of the tail-adaptive density estimator is demonstrated by employing the estimator to estimate the bivariate density of daily index returns observed from the USA and Australian stock markets.

Estimation of hyperbolic diffusion using the Markov chain Monte Carlo method

In this paper we propose a Bayesian method to estimate the hyperbolic diffusion model. The approach is based on the Markov chain Monte Carlo (MCMC) method with the likelihood of the discretized process as the approximate posterior likelihood. We demonstrate that the MCMC method Provides a useful tool in analysing hyperbolic diffusions. In particular, quantities of posterior distributions obtained from the MCMC outputs can be used for statistical inference. The MCMC method based on the Milstein scheme is unsatisfactory. Our simulation study shows that the hyperbolic diffusion exhibits many of the stylized facts about asset returns documented in the discrete-time financial econometrics literature, such as the Taylor effect, a slowly declining autocorrelation function of the squared returns, and thick tails.

Structural change and lead-lag relationship between the Nikkei spot index and futures price: a genetic programming approach

Abstarct In this paper we adopt a nonparametric genetic programming (GP) approach to identify the structural changes in the Nikkei spot index and futures price. Due to the dominance of the ‘normal’ period in sample data, the lead-lag relationship identified in the spot-futures system based on conventional methods such as the test for Granger causality pertains to the normal period and may not be applicable in an ‘extreme’ period. Using GP we identify the lead-lag relationship based on the chronological ordering of the structural changes in the spot and futures markets. Our results show that in recent periods, major market changes originated from the spot market and spread over to the futures market.

A sampling algorithm for bandwidth estimation in a nonparametric regression model with a flexible error density

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.

Influence Diagnostics for Multivariate GARCH Processes

This article presents diagnostics for identifying influential observations when estimating multivariate generalized autoregressive conditional heteroscedasticity (GARCH) models. We derive influence diagnostics by introducing minor perturbations to the conditional variances and covariances. The derived diagnostics are applied to a bivariate GARCH model of daily returns of the S&P500 and IBM. We find that univariate diagnostic procedures may be unable to identify the influential observations in a multivariate model. Importantly, the proposed curvature-based diagnostic identified influential observations where the correlation between the two series had a major change. These observations were not identified as influential using the univariate diagnostics for each asset separately. When estimating the bivariate GARCH model allowing for weights at influential observations, we found that the time-varying correlations behaved differently from that implied by the model ignoring influential observations. The application therefore highlights the importance of extending univariate diagnostic procedures to multivariate settings.