遗漏变量下多元非参数核估计渐近偏误的一致估计框架  

A Uni ed Consistent Estimation Framework for the Asymptotic Biases of the Nonparametric Kernel Regressions with Omitted Variables

作　　者：郝士铭 罗汉[1] HAO Shiming;LUO Han(School of Mathematics and Econometrics,Hunan University,Changsha 410082)

机构地区：[1]湖南大学数学与计量经济学院,长沙410082

出　　处：《工程数学学报》2023年第2期231-250,共20页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(10101006,70773038);湖南省高校创新平台开放基金(15K026).

摘　　要：在经济学、社会学、医学、生物学、农业等诸领域的研究中,由于数据获取的困难、实验条件的限制、研究经验的不足以及失误等因素,研究者往往会在回归模型的设定中遗漏掉关键的解释变量,使得遗漏变量模型的识别与处理成为一个广泛存在的问题。由此,提出了一种统一的识别、估计与比较框架,使得针对遗漏变量回归模型的任意非参数核估计量的渐近偏误都可以得到识别与估计。应用此框架,考察了遗漏变量下Nadaraya-Watson估计量、Gasser-Müller估计量以及局部线性估计量的精确渐近性质,发现遗漏变量下Gasser-Müller估计量与局部线性估计量的渐近偏误一样大,且都比Nadaraya-Watson估计量的渐近偏误小。此外遗漏变量下线性参数模型估计量的渐近性质也可以通过本文提出的框架与方法推导出来。在此基础上,进一步探讨了局部线性核估计量的一个没被注意到的优良性质。In the research of economics,sociology,medicine,biology,agriculture and other fields,due to the difficulties in data acquisition,the limitations of experimental conditions,the lack of research experience and errors,researchers often omit the key explanatory variables in the setting of regression models,making the identification and processing of omitted variable models a widespread problem.In this paper,a unified identification,estimation and comparison framework is proposed,which enables the identification and estimation of the asymptotic bias of any nonparametric kernel estimator in the regression model with omitted variables.Under this framework,we investigate the exact asymptotic properties of the Nadaraya-Watson estimator,the Gasser-Müller estimator as well as the local linear estimator with omitted variables.It is found that the asymptotic errors of the Gasser-Müller estimator and the local linear estimator with omitted variables are the same,and smaller than that of the Nadaraya-Watson estimator.In addition,the asymptotic properties of the linear parameter estimator with omitted variables can also be derived through the proposed framework and method.On this basis,an unnoticed good property of local linear kernel estimator proposed in some references is further discussed.

关 键 词：遗漏变量 非参数模型 渐近偏误 三角联立方程 

分 类 号：O212.1[理学—概率论与数理统计]