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3. Multiple Regression Analysis
The general linear regression with
k
explana-tory variables is just an extension of the sim-ple regression as follows(1)
y
i
=
β
0
+
β
1
x
i
1
+
···
+
β
k
x
ik
+
u
i
.
Because(2)
∂y
i
∂x
ij
=
β
j
j
= 1
,...,k
, coeﬃcient
β
j
indicates the marginaleﬀect of variable
x
j
, and indicates the amount
y
is expected to change as
x
j
changes byone unit and other variables are kept con-stant (ceteris paribus).The multiple regression opens up several ad-ditional options to enrich analysis and makemodeling more realistic compared to the sim-ple regression.
1
Example 3.1: Consider the hourly wage example. En-hance the model as(3) log(
w
) =
β
0
+
β
1
x
1
+
β
2
x
2
+
β
3
x
3
,
where
w
= average hourly earnings,
x
1
= years of ed-ucation (educ),
x
2
= years of labor market experience(exper), and
x
3
= years with the current employer(tenure).
Dependent Variable: LOG(WAGE)Method: Least SquaresDate: 08/21/12 Time: 09:16Sample: 1 526Included observations: 526VariableCoefficientStd. Errort-StatisticProb. C0.2843600.1041902.7292300.0066EDUC0.0920290.00733012.555250.0000EXPER0.0041210.0017232.3914370.0171TENURE0.0220670.0030947.1330700.0000R-squared0.316013 Mean dependent var1.623268 Adjusted R-squared0.312082 S.D. dependent var0.531538S.E. of regression0.440862 Akaike info criterion1.207406Sum squared resid101.4556 Schwarz criterion1.239842Log likelihood-313.5478 Hannan-Quinn criter.1.220106F-statistic80.39092 Durbin-Watson stat1.768805Prob(F-statistic)0.000000
For example the coeﬃcient 0.092 means that, hold-ing exper and tenure ﬁxed, another year of educationis predicted to increase wage by approximately 9.2%.Staying another year at the same ﬁrm (educ ﬁxed,
∆
exper=
∆
tenure=1) is expected to result in a salaryincrease by approximately 0
.
4%+2
.
2% = 2
.
6%.
2
Example 3.2: Consider the consumption function
C
=
f
(
Y
), where
Y
is income. Suppose the assump-tion is that as incomes grow the marginal propensityto consume decreases.In simple regression we could try to ﬁt a level-logmodel or log-log model.One possibility also could be
β
1
=
β
1
l
+
β
1
q
Y ,
where according to our hypothesis
β
1
q
<
0. Thus theconsumption function becomes
C
=
β
0
+(
β
1
l
+
β
1
q
Y
)
Y
+
u
=
β
0
+
β
1
l
Y
+
β
1
q
Y
2
+
u
This is a multiple regression model with
x
1
=
Y
and
x
2
=
Y
2
.This simple example demonstrates that we can mean-ingfully enrich simple regression analysis (even thoughwe have essentially only two variables,
C
and
Y
) andat the same time get a meaningful interpretation tothe above polynomial model.The response of
C
to a one unit change in
Y
is now
∂ C ∂ Y
=
β
1
l
+2
β
1
q
Y .
3
EstimationIn order to estimate the model we replace theclassical assumption 3 as3. None of the independent variables is con-stant, and no observation vector of any in-dependent variable can be written as a lin-ear combination of the observation vectorsof any other independent variables.The estimation method again is the OLS,which produces estimates ˆ
β
0
,
ˆ
β
1
,...,
ˆ
β
k
by min-imizing(4)
n
i
=1
(
y
i
−
β
0
−
β
1
x
i
1
−···−
β
k
x
ik
)
2
with respect to the parameters.Again the ﬁrst order solution is to set the(
k
+1) partial derivatives equal to zero.The solution is straightforward although theexplicit form of the estimators become com-plicated.
4

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