计量经济第二章2-1.ppt

1、Inroductory Econometrics Lijun Jia,1,The Simple Regression Model (1) 简单二元回归,y = b0 + b1x + u,Inroductory Econometrics Lijun Jia,2,Chapter Outline 本章大纲,Definition of the Simple Regression Model 简单回归模型的定义 Deriving the Ordinary Least Squares Estimates 普通最小二乘法的推导 Mechanics of OLS OLS的操作技巧 Units of Measu

2、rement and Functional Form 测量单位和函数形式 Expected Values and Variances of the OLS estimators OLS估计量的期望值和方差 Regression through the Origin 过原点回归,Inroductory Econometrics Lijun Jia,3,Lecture Outline 讲义大纲,Some Terminology 一些术语的注解 A Simple Assumption 一个简单假定 Zero Conditional Mean Assumption 条件期望零值假定 What is O

3、rdinary Least Squares 何为普通最小二乘法 Deriving OLS Estimates 普通最小二乘法的推导,Inroductory Econometrics Lijun Jia,4,Some Terminology 术语注解,In the simple linear regression model, where y = b0 + b1x + u, we typically refer to y as the Dependent Variable, or Left-Hand Side Variable, or Explained Variable, or respons

4、e variable, or Predicted variable or Regressand 在简单二元回归模型y = b0 + b1x + u中， y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。,Inroductory Econometrics Lijun Jia,5,Some Terminology术语注解,In the simple linear regression of y on x, we typically refer to x as the Independent Variable, or Right-Hand Side Variable, or

5、Explanatory Variable, or Control Variables, or Covariate, or predictor variable Regressor 在y 对 x进行回归的简单二元回归模型中， x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 。,Inroductory Econometrics Lijun Jia,6,Some Terminology术语注解,Equation 2.1 y = b0 + b1x + u has only one nonconstant regressor x, it is called a simple line

6、ar regression model, or two-variables regression model, or bivariate linear regression model. 等式y = b0 + b1x + u只有一个非常数回归元。我们称之为简单回归模型， 两变量回归模型或双变量回归模型.,Inroductory Econometrics Lijun Jia,7,Some Terminology术语注解,The coefficients b0 , b1 are called the regression coefficients or parameter. b0 is also

7、called the constant term or the intercept term, or intercept parameter. b1 represents the marginal effects of the regressor, x. It is also called the slope parameter. b0 , b1被称为回归系数。 b0也被称为常数项或截矩项，或截矩参数。 b1代表了回归元x的边际效果，也被成为斜率参数。,Inroductory Econometrics Lijun Jia,8,Some Terminology术语注解,The variable

8、u is called the error term or disturbance in the relationship. It represents factors other than x that can affect y. u 为误差项或扰动项，它代表了除了x之外可以影响y的因素。,Inroductory Econometrics Lijun Jia,9,Some Terminology术语注解,Meaning of linear: linear means linear in parameters, not necessarily mean that y and x must ha

9、ve a linear relationship. There are many cases that y and x have nonlinear relationship, but after some transformation, they are linear in parameters. For example, y=eb0+b1x+u . 线性的含义： y 和x 之间并不一定存在线性关系，但是，只要通过转换可以使y的转换形式和x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。,Inroductory Econometrics Lijun Jia,10,Examples

10、简单二元回归模型例子,A simple wage equation 2.4 wage= b0 + b1(educ) + u b1 : if education increase by one year, how much more wage will one gain. 上述简单工资函数描述了受教育年限和工资之间的关系, b1 衡量了多接受一年教育工资可以增加多少.,Inroductory Econometrics Lijun Jia,11,A Simple Assumption关于u的假定,The average value of u, the error term, in the popu

11、lation is 0. That is, E(u) = 0(2.5) It it restrictive? 我们假定总体中误差项u的平均值为零. 该假定是否具有很大的限制性呢?,Inroductory Econometrics Lijun Jia,12,A Simple Assumption关于u的假定,If for example, E(u)=5. Then y = (b0 +5)+ b1x + (u-5), therefore, E(u)=E(u-5)=0. This is not a restrictive assumption, since we can always use b0

12、to normalize E(u) to 0. 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, 因此该假定的限制性不大.,Inroductory Econometrics Lijun Jia,13,Zero Conditional Mean Assumption 条件期望零值假定,We need to make a crucial assumption about how u and x are related We want it to be the case that knowing something about x does not give us any informa

13、tion about u, so that they are completely unrelated. That is E(u|x) = E(u)。 我们需要对u和 x之间的关系做一个关键假定。理想状况是对x的了解并不增加对u的任何信息。换句话说，我们需要u和 x完全不相关。,Inroductory Econometrics Lijun Jia,14,Zero Conditional Mean Assumption 条件期望零值假定,Since we have assumed E(u) = 0, therefore, E(u|x) = E(u) = 0. (2.6) What does it

14、 mean? 由于我们已经假定了E(u) = 0，因此有E(u|x) = E(u) = 0。该假定是何含义？,Inroductory Econometrics Lijun Jia,15,Zero Conditional Mean Assumption 条件期望零值假定,In the example of education, suppose u represents innate ability, zero conditional mean assumption means E(ability|edu=6)=E(ability|edu=18)=0. The average level of a

15、bility is the same regardless of years of education. 在教育一例中，假定u 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，该能力的平均值相同。,Inroductory Econometrics Lijun Jia,16,Zero Conditional Mean Assumption 条件期望零值假定,Question: Suppose that a score on a final exam, score, depends on classes attended (attend) and unobserved factors t

16、hat affect exam performance (such as student ability). Then consider model score =b0 + b1attend +u When would you expect it satisfy (2.6)? 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（2.6）何时能够成立？,Inroductory Econometrics Lijun Jia,17,Zero Conditional Mean Assumption 条件期望零值假定,(2.6) implies the popul

17、ation regression function, E(y|x) , satisfies E(y|x) = E(b0 /x)+ E( b1x /x) + E(u /x ) = b0 + b1x. E(y|x) as a linear function of x, where for any x the distribution of y is centered about E(y|x). (2.6)说明总体回归函数应满足E(y|x) = b0 + b1x。该函数是x的线性函数，y的分布以它为中心。,Inroductory Econometrics Lijun Jia,18,.,.,.,.,y

18、4,y1,y2,y3,x1,x2,x3,x4,u1,u2,u3,u4,x,y,Population regression line, sample data points and the associated error terms 总体回归线，样本观察点和相应误差,E(y|x) = b0 + b1x,Inroductory Econometrics Lijun Jia,19,So：,or,Ui is called stochastic disturbance, or stochastic error,两边取X的条件期望值，可推出E(ui/Xi)=0,Inroductory Econometr

19、ics Lijun Jia,20,Deriving the Ordinary Least Squares Estimates 普通最小二乘法的推导,Basic idea of regression is to estimate the population parameters from a sample Let (xi,yi): i=1, ,n denote a random sample of size n from the population For each observation in this sample, it will be the case that yi = b0 +

20、b1xi + ui 回归的基本思想是从样本去估计总体参数。 我们用(xi,yi): i=1, ,n 来 表示一个随机样本，并假定每一观测值满足yi = b0 + b1xi + ui。,Inroductory Econometrics Lijun Jia,21,预备知识,附录A5: 附录A7 附录A8,Inroductory Econometrics Lijun Jia,22,Deriving OLS Estimates普通最小二乘法的推导,To derive the OLS estimator we need to realize that our main assumption of E(u

21、|x) = E(u) = 0 also implies that Cov(x,u) = E(xu) = 0 Why? Remember from basic probability that Cov(X,Y) = E(XY) E(X)E(Y) 由E(u|x) = E(u) = 0 可得Cov(x,u) = E(xu) = 0 。,Inroductory Econometrics Lijun Jia,23,Deriving OLS continued普通最小二乘法的推导,We can write our 2 restrictions just in terms of x, y, b0 and b

22、1 , since u = y b0 b1x E(y b0 b1x) = 0 Ex(y b0 b1x) = 0 These are called moment restrictions 可将u = y b0 b1x代入以得上述两个矩条件。,Inroductory Econometrics Lijun Jia,24,Derivation of OLS普通最小二乘法的推导,The sample versions are as follows:,Inroductory Econometrics Lijun Jia,25,Derivation of OLS普通最小二乘法的推导,Given the de

23、finition of a sample mean, and properties of summation, we can rewrite the first condition as follows 根据样本均值的定义以及加总的性质，可将第一个条件写为,Inroductory Econometrics Lijun Jia,26,Derivation of OLS普通最小二乘法的推导,Inroductory Econometrics Lijun Jia,27,So the OLS estimated slope is因此OLS估计出的斜率为,Inroductory Econometrics

24、Lijun Jia,28,OLS推导的思路,（1）2.102.122.142.162.17 （2）2.112.132.15 （3）plug 2.17 into 2.152.19,Inroductory Econometrics Lijun Jia,29,Summary of OLS slope estimateOLS斜率估计法总结,The slope estimate is the sample covariance between x and y divided by the sample variance of x. If x and y are positively correlated, the slope will be positive. If x and y are negatively correlated, the slope will be negative. Only need x to vary in our samp