计量经济第二章精选PPT.ppt

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1、计量经济第二章Inroductory Econometrics Lijun Jia1第1页，此课件共32页哦Inroductory Econometrics Lijun Jia2Chapter Outline 本章大纲vDefinition of the Simple Regression Model v简单回归模型的定义简单回归模型的定义vDeriving the Ordinary Least Squares Estimates 普通普通最小二乘法的推导最小二乘法的推导vMechanics of OLS OLS的操作技巧的操作技巧vUnits of Measurement and Funct

2、ional Form测量单位和函数形式测量单位和函数形式vExpected Values and Variances of the OLS estimators OLS估估计量的期望值和方差计量的期望值和方差vRegression through the Origin 过原点回归过原点回归第2页，此课件共32页哦Inroductory Econometrics Lijun Jia3Lecture Outline 讲义大纲vSome Terminology 一些术语的注解一些术语的注解vA Simple Assumption 一个简单假定一个简单假定vZero Conditional Mean

3、Assumption 条件期望零值假条件期望零值假定定 vWhat is Ordinary Least Squares 何为普通最小二乘法何为普通最小二乘法vDeriving OLS Estimates 普通最小二乘法的推导普通最小二乘法的推导第3页，此课件共32页哦Inroductory Econometrics Lijun Jia4Some Terminology 术语注解v In the simple linear regression model,where y=b b0+b b1x+u,we typically refer to y as theDependent Variable,

4、or Left-Hand Side Variable,orExplained Variable,or response variable,orPredicted variable or Regressand在简单二元回归模型在简单二元回归模型y=b b0+b b1x+u中，中，y通常被称为因变量，左通常被称为因变量，左边变量，响应变量，被预测变量，被解释变量，或回归子。边变量，响应变量，被预测变量，被解释变量，或回归子。第4页，此课件共32页哦Inroductory Econometrics Lijun Jia5Some Terminology术语注解v In the simple linea

5、r regression of y on x,we typically refer to x as theIndependent Variable,orRight-Hand Side Variable,orExplanatory Variable,orControl Variables,orCovariate,or predictor variableRegressor在在y 对对 x进行回归的简单二元回归模型中，进行回归的简单二元回归模型中，x通常被称为自变量，通常被称为自变量，右边变量，解释变量，控制变量，协变量，或回归元右边变量，解释变量，控制变量，协变量，或回归元。第5页，此课件共32

6、页哦Inroductory Econometrics Lijun Jia6Some Terminology术语注解vEquation 2.1 y=b b0+b b1x+u has only one nonconstant regressor x,it is called a simple linear regression model,or two-variables regression model,or bivariate linear regression model.等式等式y=b b0+b b1x+u只有一个非常数回归元。我们称之为简只有一个非常数回归元。我们称之为简单回归模型，单回

7、归模型，两变量回归模型或双变量回归模型两变量回归模型或双变量回归模型.第6页，此课件共32页哦Inroductory Econometrics Lijun Jia7Some Terminology术语注解vThe coefficients b b0,b b1 are called the regression coefficients or parameter.vb b0 is also called the constant term or the intercept term,or intercept parameter.vb b1 represents the marginal effe

8、cts of the regressor,x.It is also called the slope parameter.b b0,b b1被称为回归系数。被称为回归系数。b b0也被称为常数项或截矩项，或也被称为常数项或截矩项，或截矩参数。截矩参数。b b1代表了回归元代表了回归元x的边际效果，也被成为斜的边际效果，也被成为斜率参数。率参数。第7页，此课件共32页哦Inroductory Econometrics Lijun Jia8Some Terminology术语注解v The variable u is called the error term or disturbance in

9、the relationship.vIt represents factors other than x that can affect y.u 为误差项或扰动项，它代表了除了为误差项或扰动项，它代表了除了x之外可以之外可以影响影响y的因素。的因素。第8页，此课件共32页哦Inroductory Econometrics Lijun Jia9Some Terminology术语注解vMeaning of linear:linear means linear in parameters,not necessarily mean that y and x must have a linear re

10、lationship.vThere are many cases that y and x have nonlinear relationship,but after some transformation,they are linear in parameters.vFor example,y=eb b0+b b1x+u.v线性的含义：线性的含义：y 和和x 之间并不一定存在线性关系，但之间并不一定存在线性关系，但是，只要通过转换可以使是，只要通过转换可以使y的转换形式和的转换形式和x的转换形式的转换形式存在相对于参数的线性关系，该模型即称为线性模型。存在相对于参数的线性关系，该模型即称为线

11、性模型。第9页，此课件共32页哦Inroductory Econometrics Lijun Jia10Examples 简单二元回归模型例子vA simple wage equation 2.4wage=b b0+b b1(educ)+uvb b1:if education increase by one year,how much more wage will one gain.v上述简单工资函数描述了受教育年限和工资之间的关系上述简单工资函数描述了受教育年限和工资之间的关系,b b1 衡量了多接受一年教育工资可以增加多少衡量了多接受一年教育工资可以增加多少.第10页，此课件共32页哦In

12、roductory Econometrics Lijun Jia11A Simple Assumption关于u的假定v The average value of u,the error term,in the population is 0.That is,E(u)=0(2.5)v It it restrictive?v我们假定总体中误差项我们假定总体中误差项u的平均值为零的平均值为零.该假定是否具该假定是否具有很大的限制性呢有很大的限制性呢?第11页，此课件共32页哦Inroductory Econometrics Lijun Jia12A Simple Assumption关于u的假定v

13、If for example,E(u)=5.Then y=(b b0+5)+b b1x+(u-5),therefore,E(u)=E(u-5)=0.vThis is not a restrictive assumption,since we can always use b b0 to normalize E(u)to 0.v上述推导说明我们总可以通过调整常数项来实现误差上述推导说明我们总可以通过调整常数项来实现误差项的均值为零项的均值为零,因此该假定的限制性不大因此该假定的限制性不大.第12页，此课件共32页哦Inroductory Econometrics Lijun Jia13Zero

14、Conditional Mean Assumption 条件期望零值假定 v We need to make a crucial assumption about how u and x are related v We want it to be the case that knowing something about x does not give us any information about u,so that they are completely unrelated.That isE(u|x)=E(u)。我们需要对我们需要对u和和 x之间的关系做一个关键假定。理想状况是之间的关

15、系做一个关键假定。理想状况是对对x的了解并不增加对的了解并不增加对u的任何信息。换句话说，我们需的任何信息。换句话说，我们需要要u和和 x完全不相关。完全不相关。第13页，此课件共32页哦Inroductory Econometrics Lijun Jia14Zero Conditional Mean Assumption 条件期望零值假定 vSince we have assumed E(u)=0,therefore,E(u|x)=E(u)=0.(2.6)vWhat does it mean?由于我们已经假定了由于我们已经假定了E(u)=0，因此有，因此有E(u|x)=E(u)=0。该假定是

16、何含义？。该假定是何含义？第14页，此课件共32页哦Inroductory Econometrics Lijun Jia15Zero Conditional Mean Assumption 条件期望零值假定 vIn the example of education,suppose u represents innate ability,zero conditional mean assumption meansE(ability|edu=6)=E(ability|edu=18)=0.vThe average level of ability is the same regardless of

17、years of education.v在教育一例中，假定在教育一例中，假定u 代表内在能力，条件期望零值假代表内在能力，条件期望零值假定说明不管解释教育的年限如何，该能力的平均值相定说明不管解释教育的年限如何，该能力的平均值相同。同。第15页，此课件共32页哦Inroductory Econometrics Lijun Jia16Zero Conditional Mean Assumption 条件期望零值假定 vQuestion:Suppose that a score on a final exam,score,depends on classes attended(attend)and

18、 unobserved factors that affect exam performance(such as student ability).Then consider model score=b b0+b b1attend+uvWhen would you expect it satisfy(2.6)?v假设期末成绩分数取决于出勤次数和影响学生现场发挥的假设期末成绩分数取决于出勤次数和影响学生现场发挥的因素，如学生个人素质。那么上述模型中假设（因素，如学生个人素质。那么上述模型中假设（2.6）何）何时能够成立？时能够成立？第16页，此课件共32页哦Inroductory Econome

19、trics Lijun Jia17Zero Conditional Mean Assumption 条件期望零值假定 v(2.6)implies the population regression function,E(y|x),satisfies E(y|x)=E(b b0/x)+E(b b1x/x)+E(u/x)=b b0+b b1x.vE(y|x)as a linear function of x,where for any x the distribution of y is centered about E(y|x).v(2.6)说明总体回归函数应满足说明总体回归函数应满足E(y|x

20、)=b b0+b b1x。该函数该函数是是x的线性函数，的线性函数，y的分布以它为中心。的分布以它为中心。第17页，此课件共32页哦Inroductory Econometrics Lijun Jia18.y4y1y2y3x1x2x3x4u1u2u3u4xyPopulation regression line,sample data pointsand the associated error terms总体回归线，样本观察点和相应误差E(y|x)=b b0+b b1x第18页，此课件共32页哦Inroductory Econometrics Lijun Jia19So：orUi is cal

21、led stochastic disturbance,or stochastic error两边取两边取X的条件期望值，可推出的条件期望值，可推出E(ui/Xi)=0第19页，此课件共32页哦Inroductory Econometrics Lijun Jia20Deriving the Ordinary Least Squares Estimates 普通最小二乘法的推导v Basic idea of regression is to estimate the population parameters from a samplev Let(xi,yi):i=1,n denote a ran

22、dom sample of size n from the populationv For each observation in this sample,it will be the case that yi=b0+b1xi+ui回归的基本思想是从样本去估计总体参数。回归的基本思想是从样本去估计总体参数。我们用我们用(xi,yi):i=1,n 来来表示一个随机样本，并假定每一观测值满足表示一个随机样本，并假定每一观测值满足yi=b b0+b b1xi+ui。第20页，此课件共32页哦Inroductory Econometrics Lijun Jia21预备知识v附录A5:v附录A7v附录A

23、8第21页，此课件共32页哦Inroductory Econometrics Lijun Jia22Deriving OLS Estimates普通最小二乘法的推导v To derive the OLS estimator we need to realize that our main assumption of E(u|x)=E(u)=0 also implies thatv Cov(x,u)=E(xu)=0 vWhy?Remember from basic probability that Cov(X,Y)=E(XY)E(X)E(Y)由由E(u|x)=E(u)=0 可得可得Cov(x,u

24、)=E(xu)=0。第22页，此课件共32页哦Inroductory Econometrics Lijun Jia23Deriving OLS continued普通最小二乘法的推导v We can write our 2 restrictions just in terms of x,y,b b0 and b b1 1,since u=y b b0 b b1xv E(y b b0 b b1x)=0v Ex(y b b0 b b1x)=0vThese are called moment restrictionsv可将可将u=y b b0 b b1x代入以得上述两个矩条代入以得上述两个矩条件件。

25、第23页，此课件共32页哦Inroductory Econometrics Lijun Jia24Derivation of OLS普通最小二乘法的推导普通最小二乘法的推导v The sample versions are as follows:第24页，此课件共32页哦Inroductory Econometrics Lijun Jia25Derivation of OLS普通最小二乘法的推导普通最小二乘法的推导vGiven the definition of a sample mean,and properties of summation,we can rewrite the first

26、 condition as follows根据样本均值的定义以及加总的性质，可将第一个条根据样本均值的定义以及加总的性质，可将第一个条件写为件写为第25页，此课件共32页哦Inroductory Econometrics Lijun Jia26Derivation of OLS普通最小二乘法的推导普通最小二乘法的推导第26页，此课件共32页哦Inroductory Econometrics Lijun Jia27So the OLS estimated slope is因此因此OLS估计出的斜率为估计出的斜率为第27页，此课件共32页哦Inroductory Econometrics Liju

27、n Jia28OLS推导的思路v（1）2.102.122.142.162.17v（2）2.112.132.15v（3）plug 2.17 into 2.152.19第28页，此课件共32页哦Inroductory Econometrics Lijun Jia29Summary of OLS slope estimateOLS斜率估计法总结v The slope estimate is the sample covariance between x and y divided by the sample variance of x.v If x and y are positively corr

28、elated,the slope will be positive.v If x and y are negatively correlated,the slope will be negative.v Only need x to vary in our sample.v斜率估计量等于样本中斜率估计量等于样本中x 和和 y 的协方差除以的协方差除以x的方差。若的方差。若x 和和 y 正相关则斜率为正，反之为负正相关则斜率为正，反之为负。第29页，此课件共32页哦Inroductory Econometrics Lijun Jia30More OLS 关于OLS的更多信息v Intuitive

29、ly,OLS is fitting（拟合）（拟合）a line through the sample points such that the sum of squared residuals is as small as possible,hence the term least squares。v The residual,is an estimate of the error term,u,and is the difference between the fitted line(sample regression function)and the sample point。vOLS法是

30、要找到一条直线，使残差平方和最小。法是要找到一条直线，使残差平方和最小。v残差是对误差项的估计，因此，它是直线（样本回归函数）和样本点之残差是对误差项的估计，因此，它是直线（样本回归函数）和样本点之间的距离。间的距离。第30页，此课件共32页哦Inroductory Econometrics Lijun Jia31.y4y1y2y3x1x2x3x41234xySample regression line,sample data pointsand the associated estimated error terms 样本回归线，样本数据点和相关的误差估计项第31页，此课件共32页哦Inroductory Econometrics Lijun Jia32本节重要知识点及考试范围vThe Terminology In Regression Model vTwo AssumptionvDeriving the Ordinary Least Squares Estimates 第32页，此课件共32页哦