In the first essay, we investigate the nonlinear quantile regression with mixed discrete and continuous regressors. A local linear smoothing technique with the mixed continuous and discrete kernel function is proposed to estimate the conditional quantile regression function. Under some mild conditions, the asymptotic distribution is established for the proposed nonparametric estimators, which can be seen as the generalization of some existing theory which only handles the case of purely continuous regressors. We also study the choice of the tuning parameters in the estimation procedure which is crucial in kernelbased smoothing approach. We suggest using the cross-validation approach to choose the optimal bandwidths. A simulation study is provided to examine the finite sample behavior of the proposed method and compare it with the naive local linear quantile estimation without smoothing the discrete regressors and the nonparametric inverse-CDF (cumulative distribution function) method. In the second essay, we propose to estimate a nonparametric regression function subject to a monotonicity restriction using the Knn (k-nearest neighbors) method. We also propose using a new convergence criterion to measure the closeness between an unconstrained and the (monotone) constrained Knn estimated curves. This method is an alternative to the monotone kernel methods. We use a bootstrap procedure for testing the validity of the monotone restriction. We apply our method to the 'Job Market Matching' data and find that the unconstrained/constrained Knn estimators work better than kernel estimators for this type of highly unevenly distributed data. In the third essay, we propose a nonparametric methodology to test heterogeneous risk preference against asymmetric value distribution of bidders. By modeling bidders' asymmetry as unobserved heterogeneity, we first show that bid distributions conditional on the heterogeneity are nonparametrically identified. Next, we find that the two alternative models provide distinct implications on the conditional bid distributions. Based on the estimated conditional bid distributions, we are able to distinguish the two models by formally testing the distinct model implications. The Monte Carlo experiments demonstrate the good performance of our method. In an application using the US Forest Service timber auction data, we find that the data support the model with heterogeneity in risk preference.
