伍德里奇计量经济学 (3)

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Motivation: An Example
Consider a model that says family
consumption is a quadratic function of family income:
Cons = b0 + b1 inc+b2 inc2 +u
Now the marginal propensity to consume is approximated by
Introductory Econometrics
3
Motivation : Advantage
Multiple regression models can accommodate many explanatory variables that may be correlated.
Important for drawing inference about causal relations between y and explanatory variables when using non-experimental data. 。
Introductory Econometrics
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Properties
The sample average of the residuals is zero. The sample covariance between each independent variable and the OSL residuals is zero. The point (x1, x2, , xk , y) is always on the OLS regression line.
Introductory Econometrics
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Holding other factors fixed
The power of multiple regression analysis is that it allows us to do in non-experimental environments what natural scientists are able to do in a controlled laboratory setting: keep other factors fixed.
Introductory Econometrics
4
Motivation : Advantage
It can explain more of the variation in the dependent variable.
It can incorporate more general functional form.
the residuals from the estimated
regression xˆ1 ˆ0 ˆ2 xˆ2
Introductory Econometrics
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A “Partialling Out” Interpretation
Regress our first independent variable x1 on our second independent variable x2 ,
pcolGPA: predicted values of college grade point average
pcolGPA:大学绩点预测值
hsGPA : high school GPA hsGPA : 高中绩点
ACT : achievement test score ACT :成绩测验分数
pcolGPA = 1.29 + 0.453hsGPA+0.0094ACT
bbˆˆ11cann
still
rˆ1i
yi
be written as in n rˆ1i2 , but the
rxe1soidnuxa2l…r1
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
and then obtain the residual r1 .
Then, do to obtain
a simple bˆ1 .
regression
of
y
on
r1
Introductory Econometrics
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“Partialling Out” continued
Previous equation implies that regressing y
exper: years of labor market experience
wage b0 b1educ b2exper u
In this example experience is explicitly taken out of the error term.
Introductory Econometrics
Introductory Econometrics
8
Parallels with Simple Regression
b0 is still the intercept b1 to bk all called slope parameters
u is still the error term (or disturbance)
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Example: Determinants of College GPA
One-independent-variable regression
pcolGPA = 2.4 +0.0271ACT
The coefficients on ACT is three times larger.
If these two regressions were both true, they can be considered as the results of two different experiments.
The multiple regression model is the most widely used vehicle for empirical analysis.
Introductory Econometrics
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Motivation: An Example
Consider a simple version of the wage equation for obtaining the effect of education on hourly wage:
3. Multiple Regression
Analysis: Estimation
yi = b0 + b1x1i + b2x2i + . . . bkxki + ui
Introductory Econometrics
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Motivation: Advantage
The primary drawback of the simple regression analysis for empirical work is that it is very difficult to draw ceteris paribus conclusions about how x affects y.
x1 and x2 are uncorrelated in the sample
Introductory Econometrics
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“Partialling Out” continued
In the general model with k explanatory
variables, equation
If other factors that affecting y are not correlated with x, changing x can ensure that u is not changed, and the effect of x on y can be identified.
Multiple regression analysis is more amenable to ceteris paribus analysis because it allows us to explicitly control for many other factors that simultaneously affect the dependent variable.
Introductory Econometrics
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Simple vs Multiple Reg Estimate
Compare thesimple regression ~y b~0 b~1x1 with themultiple regression yˆ bˆ0 bˆ1x1 bˆ2x2 Generally, b~1 bˆ1 unless : bˆ2 0 (i.e. no partial effectof x2 ) OR
MPC= b1 +2b2 inc
Introductory Econometrics
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The Model with k Independent Variables
The general multiple linear regression model can be written as
yi b0 b1x1i b2 x2i bk xki ui
Introductory Econometrics
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Two-independent-variable regression
Introductory Econometrics
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One-independent-variable regression
Introductory Econometrics
yˆ bˆ1x1, that is each b has
a ceteris paribus interpretation
Introductory Econometrics
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Example 3.4: Determinants of College
GPA (GPA1.dta)
Two-independent-variable regression
Introductory Econometrics
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Interpreting Multiple Regressi百度文库n
yˆ bˆ0 bˆ1x1 bˆ2 x2 ... bˆk xk , so yˆ bˆ1x1 bˆ2 x2 ... bˆk xk ,
so holding x2,...,xk fixed implies that
Whether the ceteris paribus effects are reliable or not depends on whether the conditional mean assumption is realistic.
Introductory Econometrics
2
Motivation: Advantage
Introductory Econometrics
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A “Partialling Out” Interpretation
Consider the case where k 2, i.e.
yˆ bˆ0 bˆ1x1 bˆ2 x2 , then
bˆ1 rˆ1i yi
rˆ12i , where rˆ1i are
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 x1 that is uncorrelated with x2 are being related to y so we’re estimating the effect of x1 on y after x2 has been “partialled out”