Chapter 7 Multiple Regression－－Estimation and Hypothesis Testing.ppt

Chapter 7 Multiple Regression: --Estimation and Hypothesis Testing
Multiple Regression Model: A regression model with more than one explanatory variable, multiple because multiple influences (., variables)affect the dependent variable.
The Three-variable Linear Regression Model
three-variable PRF:
nonstochastic form: E(Yt)=B1+B2X2t+B3X3t ()
stochastic form: Yt=B1+B2X2t+B3X3t+ut ()
= E(Yt)+ut

B2, B3~partial regression coefficients, partial slope coefficient
B2: the change in the mean value of Y, E(Y), per unit change in X2, holding the value of X3 constant.
B3: the change in the mean value of Y per unit change in X3, holding the value of X2 constant.
Assumptions of Multiple Linear Regression Model
. X2 and X3 are uncorrelated with the disturbance term u.
. The error term u has a zero mean value E(ui)=0 ()
. Homoscedasticity, that is , the variance of u, is constant:
var(ui)=σ2 ()
. For hypothesis testing, the error term u follows the
normal distribution with mean zero and (homoscedastic)
variance σ2 . That is , ui ~N(0, σ2 ) ( )
. No autocorrelation exists between the error terms ui and uj :
cov(ui , uj) i≠j ()
. No exact collinearity exists between X2 and X3 ; that is ,
there is no exact linear relationship between the two
explanatory variables.
--no collinearity,or no multicollinearity,
①exact linear relationship
②high or near perfect collinearity
Estimation of Parameters of Multiple Regression
Ordinary Least Squares(OLS)Estimators
SRF: Stochastic form: Yt=b1+b2X2t+b3X3t+et ()
Nonstochastic form: =b1+b2X2t+b3X3t ()
et=Yt-
et=Yt –b1-b2X2t+b3X3t ()
RSS: ()
  OLS Estimators:
b1= ()
b2= ()
b3= ()
Variance and Standard Errors of OLS Estimators
We need the standard errors for two main purposes:
(1) to establish confidence intervals for the true parameter values
(2) to test statistical hypotheses.
1. ui ~N(0, σ2 ) →b1~N(B