Bayesian sampling for bandwidth selection

This thesis aims to investigate three main topics, which are the density-based multiple variance ratio test, Bayesian estimation of partially linear models, and Bayesian estimation of Tobit models. All three topics involve a kernel density estimator whose performance is mainly determined by the choice of bandwidth. Variance ratio testing methods have been extensively used in testing for random walk hypothesis in financial markets. In Chapter 3 of this thesis, we propose a density-based multiple variance ratio test, and the testing procedure is based on the work of King, Zhang and Akram (2011). The proposed variance ratio test is constructed based on the joint density of a vector of variance ratio statistics over multiple holding periods. The joint density is estimated via a multivariate kernel estimator whose bandwidth matrix (or vector) is sampled through an Markov chain Monte Carlo (MCMC) algorithm. The Monte Carlo simulation results show that our proposed variance ratio test has a better power performance than some other conventional multiple variance ratio tests in small samples with little size distortion. In an application to the S&P500 index and FTSE100 index during the sub-prime crisis period, we find evidence that the random walk hypothesis is clearly rejected in the U.S. and U.K. stock markets. In Chapter 4, a major issue in the estimation of a partially linear model is how to estimate the linear coefficients and smoothing parameter simultaneously. We present an MCMC algorithm for such a purpose in a partially linear model, where the nonlinear component is modeled by the kernel estimator. Using the kernel-form error density approach proposed by Zhang and King (2013), we approximate the unknown error density by a kernel-form error density. The proposed Bayesian approach is nonparametric in the sense that it does not require any parametric assumption on the error distribution. We derive the posterior density of the parameters, from which parameters can be sampled using an MCMC algorithm. The Monte Carlo simulation results show that the proposed Bayesian sampling method outperforms two conventional least squares estimators in situations where the errors are assumed to follow a Laplace distribution or a mixture normal distribution. The convergence diagnostics show that the proposed sampling algorithm achieves a very good mixing performance. In an application to the gasoline consumption data in the U.S., we find that gasoline consumption is negatively related to gasoline price, while a nonlinear positive relationship exists between gasoline consumption and household income. A major issue in the estimation of a standard Tobit model is how to construct the likelihood function when the error distribution is unknown. In Chapter 5, we construct the likelihood by using the ideas of the kernel-form error density Zhang and King (2013) and data augmentation tanner and wong (1987). We present an MCMC algorithm for estimating the parameters and simulating the latent variables in a standard Tobit model. The Monte Carlo simulation results show that the proposed Bayesian sampling approach produces more accurate estimates than the semiparametric maximum likelihood estimation method proposed by cosslett (2004) and the Bayesian sampling method proposed by Chib (1992) when the errors are assumed to follow a normal, Student-t, Laplace or mixture normal distribution. Another issue with a standard Tobit model is that some relationships between variables cannot be measured parametrically. To deal with this issue, we develop a partially linear Tobit model with a kernel-form error density. In such a model, we construct the likelihood and derive the posterior density of the parameters, from which parameters can be sampled by using an MCMC algorithm. The Monte Carlo simulation results show that the proposed Bayesian sampling method produces reasonable estimates of parameters. The convergence diagnostics show that the proposed sampling algorithms achieve a very good mixing performance. With an application to the U.S. women's income data, we find that women's income is positively related to education level, while it is negatively related to age and number of children under age six. We also find a nonlinear positive relationship between women's income and working experience.