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.