Regression Analysis y=Bo+Bx1+Bx2+...Bxk+u Estimation Econometrics 15-Zhuxi @SJTU 1
Econometrics 15 - Zhuxi @SJTU 1 Regression Analysis y = b0 + b1 x1 + b2 x2 + . . . bk xk + u Estimation
Some Terminology In the linear regression model,wherey=Bo+Bx +Bx2+...Bx+u,we typically call 1% xpx2...xk Dependent Variable Independent Variable Left-Hand Side Variable Right-Hand Side Variable Explained Variable Explanatory Variable Regressand Regressor Covariate,Control Variables Econometrics 15-Zhuxi @SJTU 2
Econometrics 15 - Zhuxi @SJTU 2 Some Terminology In the linear regression model, where y = b0 + b1 x1 + b2 x2 + . . . bk xk + u, we typically call y x1 ,x2 , …,xk Dependent Variable Independent Variable Left-Hand Side Variable Right-Hand Side Variable Explained Variable Explanatory Variable Regressand Regressor Covariate, Control Variables
2.1 Motivation for multiple regression We usually make Zero Conditional Mean Hypothesis: E(ux)=0,which implies E(ylx)=Bo+Bx1+B2x2+...Bxxk E(ux)=0 implies that the regression disturbance u contains no systematic information ofx that can be used to predict the mean of y. In other words,all information ofx that can be used to predict the expectation of y has been completely summarized. Econometrics 15-Zhuxi @SJTU 3
Econometrics 15 - Zhuxi @SJTU 3 2.1 Motivation for multiple regression We usually make Zero Conditional Mean Hypothesis: E(u|x) = 0, which implies E(y|x) = b0 + b1 x1 + b2 x2 + . . . bk xk E(u|x) = 0 implies that the regression disturbance u contains no systematic information of x that can be used to predict the mean of y. In other words, all information of x that can be used to predict the expectation of y has been completely summarized
Zero Conditional Mean Hypothesis (ctd) The condition E(ux)=0 is crucial for the validity of economic interpretation of model parameters-- correct model specification for E(vx) y=B。+Bx+B2x2+ If E(ux)=0,then- (y1x)=B 0x1 IfE(ux)=yx2≠0，then E(y)=B+27x 、 8x Econometrics 15-Zhuxi @SJTU 4
Econometrics 15 - Zhuxi @SJTU 4 Zero Conditional Mean Hypothesis (ctd) The condition E(u|x) = 0 is crucial for the validity of economic interpretation of model parameters -- correct model specification for E(y|x). 0 1 1 2 2 1 1 2 1 1 1 1 | If | 0, then . | If | 0, then 2 . y x x u E y x E u x x E y x E u x x x x b b b b b
Motivation of multiple regression For simple model:y=B+Beduc+u, It's unlikely that the condition E(u educ)=0 holds. Q:Why? A:Many important variables are omitted in u. Thus a multi model:y=Bo+Beduc+Bzexperience+u, where E(uleduc,experience)=0, may be more reasonable. Now,what's the economic meaning of B Econometrics 15-Zhuxi @SJTU 5
Econometrics 15 - Zhuxi @SJTU 5 Motivation of multiple regression Now, what’s the economic meaning of b1 . 0 1 0 1 2 For simple model: , It's unlikely that the condition | 0 holds. Q: Why? A: Many important variables are omitted in . Thus a multi model: , | , y educ u E u educ u y educ experience u where E u educ b b b b b 0, may be more reasonable. experience