# Partially linear models

Hardle, Wolfgang and LIang, Hua and Gao, Jiti (2000): Partially linear models. Published in: Physica-Verlag (1 September 2000): pp. 1-202.

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## Abstract

In the last ten years, there has been increasing interest and activity in the general area of partially linear regression smoothing in statistics. Many methods and techniques have been proposed and studied. This monograph hopes to bring an up-to-date presentation of the state of the art of partially linear regression techniques. The emphasis of this monograph is on methodologies rather than on the theory, with a particular focus on applications of partially linear regression techniques to various statistical problems. These problems include least squares regression, asymptotically efficient estimation, bootstrap resampling, censored data analysis, linear measurement error models, nonlinear measurement models, nonlinear and nonparametric time series models. We hope that this monograph will serve as a useful reference for theoretical and applied statisticians and to graduate students and others who are interested in the area of partially linear regression. While advanced mathematical ideas have been valuable in some of the theoretical development, the methodological power of partially linear regression can be demonstrated and discussed without advanced mathematics. This monograph can be divided into three parts: part one–Chapter 1 through Chapter 4; part two–Chapter 5; and part three–Chapter 6. In the first part, we discuss various estimators for partially linear regression models, establish theo- retical results for the estimators, propose estimation procedures, and implement the proposed estimation procedures through real and simulated examples. The second part is of more theoretical interest. In this part, we construct several adaptive and efficient estimates for the parametric component. We show that the LS estimator of the parametric component can be modified to have both Bahadur asymptotic efficiency and second order asymptotic efficiency. In the third part, we consider partially linear time series models. First, we propose a test procedure to determine whether a partially linear model can be used to fit a given set of data. Asymptotic test criteria and power investigations are presented. Second, we propose a Cross-Validation (CV) based criterion to select the optimum linear subset from a partially linear regression and estab- lish a CV selection criterion for the bandwidth involved in the nonparametric kernel estimation. The CV selection criterion can be applied to the case where the observations fitted by the partially linear model (1.1.1) are independent and identically distributed (i.i.d.). Due to this reason, we have not provided a sepa- rate chapter to discuss the selection problem for the i.i.d. case. Third, we provide recent developments in nonparametric and semiparametric time series regression. This work of the authors was supported partially by the Sonderforschungs- bereich373“QuantifikationundSimulationO ̈konomischerProzesse”.Thesecond author was also supported by the National Natural Science Foundation of China and an Alexander von Humboldt Fellowship at the Humboldt University, while the third author was also supported by the Australian Research Council. The second and third authors would like to thank their teachers: Professors Raymond Car- roll, Guijing Chen, Xiru Chen, Ping Cheng and Lincheng Zhao for their valuable inspiration on the two authors’ research efforts. We would like to express our sin- cere thanks to our colleagues and collaborators for many helpful discussions and stimulating collaborations, in particular, Vo Anh, Shengyan Hong, Enno Mam- men, Howell Tong, Axel Werwatz and Rodney Wolff. For various ways in which they helped us, we would like to thank Adrian Baddeley, Rong Chen, Anthony Pettitt, Maxwell King, Michael Schimek, George Seber, Alastair Scott, Naisyin Wang, Qiwei Yao, Lijian Yang and Lixing Zhu. The authors are grateful to everyone who has encouraged and supported us to finish this undertaking. Any remaining errors are ours.