| This dissertation discusses regression models with a long memory heteroscedastic error process with long memory parameter H . When the regression function is formulated nonparametrically and uniformly on the unit interval, the consistency and the finite dimensional weak convergence of the regression function and variance function estimators are established. For the regression function estimators, the asymptotic normality is established for the values of the long memory parameter 1/2 < H < 1; while for the heteroscedastic function estimators, the asymptotic normality is established for 1/2 < H < 3/4, non-normality for 3/4 < H < 1. We also establishes the uniform convergence rate of the regression function estimators for a large class of innovations, including bounded and Gaussian innovations. Additionally, the local Whittle estimator of H based on the standardized nonparametric residuals is shown to be log(n)-consistent and the finite dimensional distributions of the studentized versions of the regression function estimators are shown to be asymptotically normal.; While when the regression function is linear, the design is long memory Gaussian with the long memory parameter h, in some circumstance, the first order asymptotic distribution of the least square estimator of the slope parameter is observed to be degenerate. Under some additional mild conditions, the second order asymptotic distribution of this estimator is shown to be normal whenever h+H < 3/2; non-normal otherwise. The asymptotic distribution of the kernel type estimators of the heteroscedasticity function is found to be normal whenever H < (1 + h)/2, and non-normal otherwise. In addition, an estimator of H based on pseudo residuals in a more general heteroscedastic regression model is shown to be log(n)-consistent. We also discuss the consistency of a cross validation type estimator of the heteroscedasticity function in a more general regression model under the assumed long memory set up.; All of these findings are then used to propose a lack-of-fit test of a parametric regression model. Some simulations and an application to currency exchange rate data sets are included in this study. |
