| 摘要: |
| 在讨论协变量和响应变量关系时，常会遇到内生变量，已有关于内生变量的研究大多是在最小二乘目标函数的框架下讨论，然而该方法不具有稳健性，鉴于此，本文采用指数平方损失估计方法，构造模型中回归系数的稳健估计。为了克服内生变量对估计产生的偏差，利用工具变量消除协变量的内生性，再构造回归系数的指数平方损失估计；针对指数平方损失目标函数，提出选取有效的调节参数估计过程；在一些正则条件下，研究所提出估计的渐近正态性；模拟研究比较了朴素最小二乘估计、朴素M估计、朴素指数平方损失估计、基于工具变量的最小二乘估计、基于工具变量的M估计、基于工具变量的指数平方损失估计等6种估计方法，模拟结果表明：本文提出的基于工具变量的指数平方损失估计有效地消除了协变量的内生性，且具有较好的稳健性；最后，利用本文提出的方法分析了孪生双胞胎“收入-教育程度”的数据。 |
| 关键词: 内生变量；工具变量 线性模型 指数平方损失 |
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| Exponential Squared Loss Estimation of Linear Regression Models Using Instrumental Variables |
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ZHANG Wei，YANG Yi-ping1,2
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1. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China;2.2. Chongqing Key Laboratory of Social Economy and Applied Statistics, Chongqing Technology and Business University,Chongqing 400067, China
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| Abstract: |
| Endogenous variables are often encountered when discussing the relationship between covariates and response variables. Most of the existing researches on endogenous variables are discussed in the framework of least squares objective function. However,this method is not robust. In this paper, the exponential squared loss estimation method is used to construct the robust estimation of regression coefficient in the model. In order to overcome the bias of endogenous variables on the estimation, instrumental variables are used to eliminate the endogeneity of covariates, and then the exponential squared loss estimation of regression coefficients is constructed. For the exponential squared loss objective function, the estimation process of selecting effective adjustment parameters is proposed. Under some regular conditions, the asymptotic normality of the proposed estimator is studied. In the simulation study, six estimation methods are compared, including the naive least squares estimation, the naive M estimation, the naive exponential squared loss estimation, the least squares estimation using instrumental variables, the M estimation using instrumental variables, and the exponential squared loss estimation using instrumental variables. The simulation results show that the proposed method can effectively eliminate the endogeneity of covariates, and has good robustness. Finally, the wage and schooling data collected from the survey of identical twins is analyzed by our proposed method. |
| Key words: endogenous variable instrumental variables linear model exponential squared loss |
