Bayesian bandwidth estimation in varying–coefficient time series models

This thesis investigates three main topics, which are bandwidth selection for local linear estimation of time–varying coefficient time series models, nonparametric estimation of functional coefficient time series models with trending regressors and semiparametric localised bandwidth selection in kernel density estimation. First, we propose a Bayesian approach to bandwidth selection for local linear estimation of time–varying coefficient time series models, where the errors are assumed to follow the Gaussian kernel error density. A Markov chain Monte Carlo algorithm is presented to simultaneously estimate the bandwidths for local linear estimators in the regression function and the bandwidth for the Gaussian kernel error–density estimator. A Monte Carlo simulation study shows that: 1) our proposed Bayesian approach achieves better performance in estimating the bandwidths for local linear estimators than normal reference rule and cross–validation; and 2) compared with the parametric assumption of either the Gaussian or a mixture of two Gaussians, Gaussian kernel error–density assumption is a data–driven choice and helps gain robustness in terms of different specifications of the true error density. Moreover, we apply our proposed Bayesian sampling method to the estimation of bandwidth for the time–varying coefficient models that explain Okun’s law and the relationship between consumption growth and income growth in the U.S. For each model, we also provide calibrated parametric forms of its time–varying coefficients. Second, we develop a functional coefficient time series model with trending regressors. We propose a local linear estimation method to estimate the unknown coefficient functions. The asymptotic distributions of the proposed local linear estimator are established under mild conditions. We further propose a Bayesian approach to select bandwidths involved in the proposed local linear estimator. Several numerical examples are provided to illustrate the finite sample behavior of the proposed methods. The results show that the local linear estimator works very well and the proposed Bayesian bandwidth selection method is better than cross–validation method. Furthermore, we employ the functional coefficient model to study the relationship between consumption per capita and income per capita in the U.S. and the results show that this functional coefficient model with our proposed local linear estimator and Bayesian bandwidth selection method performs better than other competing models in terms of both in–sample fitting and out–of–sample forecasting. Third, we propose a semiparametric localised bandwidth estimator for kernel density estimation based on strictly stationary mixing processes. We prove that the semiparametric localised bandwidth estimator is asymptotically normally distributed with root–n rate of convergence. To carry out the computation of the semiparametric localised bandwidth estimator for a given sample of data, we propose a sampling–based likelihood approach to hyperparameter estimation. Monte Carlo simulation studies show that the proposed hyperparameter estimation approach works very well, and that the proposed semiparametric localised 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 localised bandwidth. In addition, we present an easily computable expression for integrated squared error of normal density estimators, mixture of two normals density estimators and Gaussian kernel density estimators under different specifications of the true density. This provides a new way of evaluating the performance of the above three common–used density estimators. The numerical studies show that: 1) closed–form of integrated squared error is more accurate than grid–point approximation; 2) gird–point approximation is not robust, especially when the true density is asymmetric; 3) when the true density is neither normal nor mixture normal densities, Gaussian kernel density estimators can provide us with more accurate estimation.