第2章 多元回归分析

计量经济学课程课件 赵西亮

第二章 多元回归分析:估计y = β0 + β1x1 + β2x2 + . . . βkxk + u

计量经济学课程课件 赵西亮

Multiple Regression Analysisy = β0 + β1x1 + β2x2 + . . . βkxk + u 1. Estimation

计量经济学课程课件 赵西亮

Parallels with Simple Regressiony = β0 + β1x1 + β2x2 + . . . βkxk + u

β0 is still the intercept β1 to βk all called slope parametersu is still the error term (or disturbance) Still need to make a zero conditional mean assumption, so now assume that E(u|x1,x2, …,xk) = 0 Still minimizing the sum of squared residuals, so have k+1 first order conditions

计量经济学课程课件 赵西亮

Obtaining OLS EstimatesIn the general case with k independent variables, from the first order condition, we can get k + 1 we seek estimates β + βunknowns the equition linear equations in k0 ,11 ,K , β k in β 0 , β1 ,K , β k : n + y = β 0 +β1 x1 L + β k xk ∑ yi β0 minimize theβsum of squared residuals: therefore, β1 xi1 L k xik = 0 i =n 1 2 n β x L β x ∑xi1yi yi β0β0 1βi11xi1 L kβ kikxik = 0 ∑

∑ x ( y βMi =1 n i2 i

i =1 i =1 n

( ((

) ))

0

β1 xi1 L β k xik = 0

)

∑ x ( y βi =1 ik i

β1 xi1 L β k xik = 0. 04

)

计量经济学课程课件 赵西亮

Obtaining OLS Estimates, cont. y = β 0 + β 1 x1 + β 2 x 2 L + β k x kThe above estimated equation is called the OLS regression line or the sample regression function (SRF) the above equation is the estimated equation, is not the really equation. The really equation is population regression line which we don’t know. We only estimate it. So, using a different sample, OLS slope estimates OLS intercept estimate we can get another different estimated equation line. The population regression line is

E ( y | x) = β 0 + β1 x1 + β 2 x2 L + β k xk5

计量经济学课程课件 赵西亮

Interpreting Multiple Regression

y = β 0 + β1 x1 + β 2 x2 + ... + β k xk , so y = β x + β x + ... + β x ,1 1 2 2 k k

so holding x2 ,..., xk fixed implies that y = β x , that is each β has1 1

a ceteris paribus interpretation6

计量经济学课程课件 赵西亮

An Example (Wooldridge, p76)The determination of wage (dollars per hour), wage:Years of education, educ Years of labor market experience, exper Years with the current employer, tenure

The relationship btw. wage and educ, exper, tenure:wage=β0+β1educ+β2exper+β3tenure+u log(wage)=β0+β1educ+β2exper+β3tenure+u

The estimated equation as below:wage=2.873+0.599educ+0.022exper+0.169tenure log(wage)=0.284+0.092educ+0.0041exper+0.022tenure

The STATA commandUse [path]wage1.dta (insheet using [path]wage1.raw/wage1.txt) Reg wage educ exper tenure Reg lwage educ exper tenure7

计量经济学课程课件 赵西亮

A “Partialling Out” Interpretation

Consider the case where k = 2, i.e. y = β + β x + β x , then β1 = (∑ ri1 yi )0 1 1 2 2 2 i1

∑ r

, where ri1 are

the residuals from the estimated regression x1 = γ0 + γ2 x28

计量经济学课程课件 赵西亮

“Partialling Out” continuedPrevious equation implies that regressing y on x1 and x2 gives same effect of x1 as regressing y on residuals from a regression of x1 on x2 This means only the part of xi1 that is uncor

related with xi2 are being related to yi so we’re estimating the effect of x1 on y after x2 has been “partialled out”9

计量经济学课程课件 赵西亮

The wage determinationsThe estimated equation as below:wage=2.873+0.599educ+0.022exper+0.169tenure log(wage)=0.284+0.092educ+0.0041exper+0.022tenure

Now, we first regress educ on exper and tenure to patial out the exper and tenure’s effects. Then we regress wage on the residuals of educ on exper and tenure. Whether we get the same result.?educ=13.575-0.0738exper+0.048tenure wage=5.896+0.599resid log(wage)=1.623+0.092resid

We can see that the coefficient of resid is the same of the coefficien of the variable educ in the first estimated equation. And the same to log(wage) in the second equation.10

计量经济学课程课件 赵西亮

Simple vs Multiple Reg Estimate~ ~ ~=β +β x Compare the simple regression y 0 1 1 with the multiple regression y = β 0 + β1 x1 + β 2 x2 ~ Generally, β1 ≠ β1 unless : β = 0 (i.e. no partial effect of x ) OR2 2

x1 and x2 are uncorrelated in the sample11

计量经济学课程课件 赵西亮

The wage determinations: exempleThe estimated equation as below:wage=2.873+0.599educ+0.022exper+0.169tenure log(wage)=0.284+0.092educ+0.0041exper+0.022tenure

The estimated equations without tenurewage=3.391+0.644educ+0.070exper log(wage)=0.217+0.098educ+0.0103exper wage=0.905+0.541educ log(wage)=0.584+0.083educ12

计量经济学课程课件 赵西亮

Goodness-of-FitWe can think of each observation as being made up of an explained part, and an unexplained part, yi = yi + ui We then define the following :

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