In this dissertation I investigate several topics in the field of nonparametric econometrics. In chapter II, we consider the problem of estimating a nonparametric regression model with only categorical regressors. We investigate the theoretical properties of least squares cross-validated smoothing parameter selection, establish the rate of convergence (to zero) of the smoothing parameters for relevant regressors, and show that there is a high probability that the smoothing parameters for irrelevant regressors converge to their upper bound values thereby smoothing out the irrelevant regressors. In chapter III, we consider the problem of estimating a joint distribution defined over a set of discrete variables. We use a smoothing kernel estimator to estimate the joint distribution, allowing for the case in which some of the discrete variables are uniformly distributed, and explicitly address the vector-valued smoothing parameter case due to its practical relevance. We show that the cross-validated smoothing parameters differ in their asymptotic behavior depending on whether a variable is uniformly distributed or not. In chapter IV, we consider a k-n-n estimation of regression function with k selected by a cross validation method. We consider both the local constant and local linear cases. In both cases, the convergence rate of of the cross validated k is established. In chapter V, we consider nonparametric estimation of regression functions with mixed categorical and continuous data. The smoothing parameters in the model are selected by a cross-validation method. The uniform convergence rate of the kernel regression function estimator function with weakly dependent data is derived.
