Semiparametric regression models with serially correlated and/or heteroscedastic error

To have the best of two worlds, the parametric and nonparametric, statisticians considered the so called semiparametric regression models. These models involve unknown functions as well as unknown finite-dimensional parameters, and embody a compromise between a general nonparametric specification and a fully parametric specification. As a result, semiparametric regression models can reduce the high risk of misspecification relative to a fully parametric model and avoid some serious drawbacks of fully nonparametric methods. So far most of the research about the semiparametric regression models is focused on the case of random design and i.i.d. errors. However, in some applications the regressors are fixed. Moreover, sequentially and/or crossly-collected data often exhibit evident dependence and/or heteroscedasticity. Neglecting dependence and heteroscedasticity will result in inefficient, or even invalid statistical inference. Therefore, it is important to check the serial dependence and heteroscedasticity whenever it is considered a possibility and take them into account when they do exist.;In this dissertation we mainly study the statistical inference for fixed design semiparametric regression models with serially correlated and/or heteroscedastic errors. These semiparametric regression models include the partially linear regression model, the generalized partially linear regression model, the partially nonlinear regression model and the single-index model. We develop test procedures for serial dependence and heteroscedasticity, and the more efficient weighted estimation methods when serial dependence and/or heteroscedasticity are present. The asymptotic properties, including convergence rates and asymptotic distributions of the test statistics and estimators, are established. Our methods are also illustrated by simulation and real data sets.