Kernel smoothing method is one of the most widely used nonparametric regression meth- ods. The smoothing methods impose few assumptions about the shape of the mean function, and it is a highly flexible, data-driven regression method. Though we do not need to assume any parametric form of the mean function, we need to choose an appropriate bandwidth when we use kernel smoothing, and most of the time, the bandwidth will have a huge impact on the final prediction or estimation.

When the errors in the regression model are independent and identically distributed, cross- validation method is often used to select the bandwidth for kernel smoothing and it will, in general, produce decent results. However, when errors are correlated, the cross-validation method will fail to give good bandwidths in most cases.

Many methods are proposed and studied trying to solve the bandwidth selection problem for correlated data, and most of them, if not all of them, choose to impose assumptions on the correlation structure of the errors. In contrast, in this thesis, we consider to keep the very best of nonparametric regression and choose a way that is able to give us more flexibility in correlation structure. We will discuss a new bandwidth selection method that does not require any parametric assumption on the correlation structure of the errors. First, we will start with a fixed design situation, and then extend it to a more complex partially linear model. Then, we will develop the asymptotic theorems showing that under some conditions, the new method will also work for random design. Finally, we will discuss possible ways of further extending the results to spatial regression.