Inference For Semiparametric Models And Nearly Unit Root Processes

In chapter 1 of this paper, we propose a variable selection procedure for generalized semiparametric varying coefficient model. We propose a variable selection procedure for GVCPLM model with diverging number of parameters based on the basis function approximation and SCAD penalty. In contrast to Li and Liang (2008), our method offers the following improvements. Firstly, we generalize the model in Li and Liang (2008) to high dimensional case. Secondly, our method can select significant variables in the nonparametric components by SCAD penalty, which implies that our method can avoid the heavy computational burden. Thirdly, our method can select significant variables in both parametric and nonparametric components simultaneously. Compared with Zhao and Xue (2009), our method generalized the model to the generalized and high dimensional case.Chapter 2 consider variable selection for semiprametric additive models with measurement errors. With basis function approximations and group lasso penalty function, we propose a bias-corrected variable selection procedure for nonpara-metric components and parametric components in semiparametric addtive model Furthermore, by choosing proper tuing parameters, we show that the estimators are consistent and sparse. Our method extends the group lasso variable selection procedure, proposed by Yuan and Lin (2006) for parametric model, to semi-paramtric model with measurement errors. Specifically, our variable selection procedure offers the following improvements. Firstly, we consider a semipara-metric additive model which is more general than that considered in Huang, Horowitz and Wei (2010). Secondly, our method simultaneously selects signifi-cant variables in the parametric components and the nonparametric components, which is different from that in Li and Lin (2010). And our procedure can save substantial computation. Finally, we consider variable selection under measure-ment errors circumstance.In chapter 3, we consider estimation in a generalized varying coefficient model. We will use nonparametric quasi-likelihood method to draw a robust estimation for the case that the variance function is unknown. Our model is similar to that of Cai, Fan and Li (2000), but their results are established basing on known variance function and independent samples and their method is based on likelihood estimation. Our method and model are also different from that of Zhang (2004) and Qu and Li (2006) in two aspects:(a) their varying-coefficient functions are nonparametric functions with fixed design instead of our random design; (b) they used spline method, but we use local linear regression.In chapter 4, we estimate the nearly unit root model with GARCH errors. We derive the asymptotic theory forφn with GARCH(p,q) errors under only finite second order moment for both errors and innovations, which generalizes the results in the literature for unit root case, such as Wang (2006) and Li and Li (2009). Furthermore, we construct a confidence intervals forφn by empirical likelihood method. Under the nearly unit root circumstance, applying empirical likelihood method to construct confidence intervals forφn is an interesting topic and very useful. It avoids estimating a nuisance parameter, the variance of zt. which is involved in the limiting distribution ofφn.