Transcript GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL . reg EARNINGS S EXP Source | SS df MS -------------+-----------------------------Model | 22513.6473 2 11256.8237 Residual | 89496.5838 537 166.660305 -------------+-----------------------------Total |

GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. reg EARNINGS S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 22513.6473
2 11256.8237
Residual | 89496.5838
537 166.660305
-------------+-----------------------------Total | 112010.231
539 207.811189
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
67.54
0.0000
0.2010
0.1980
12.91
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
2.678125
.2336497
11.46
0.000
2.219146
3.137105
EXP |
.5624326
.1285136
4.38
0.000
.3099816
.8148837
_cons | -26.48501
4.27251
-6.20
0.000
-34.87789
-18.09213
------------------------------------------------------------------------------
EAR Nˆ INGS   26 . 49  2 . 68 S  0 . 56 EXP
The output above shows the result of regressing EARNINGS, hourly earnings in dollars, on
S, years of schooling, and EXP, years of work experience.
1
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
120
Hourly earnings ($)
100
80
60
40
20
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
-20
Years of schooling (highest grade completed)
Suppose that you were particularly interested in the relationship between EARNINGS and S
and wished to represent it graphically, using the sample data.
2
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
120
Hourly earnings ($)
100
80
60
40
20
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
-20
Years of schooling (highest grade completed)
A simple plot would be misleading.
3
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. cor S EXP
(obs=540)
|
S
ASVABC
--------+-----------------S|
1.0000
EXP| -0.2179
1.0000
120
Hourly earnings ($)
100
80
60
40
20
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
-20
Years of schooling (highest grade completed)
Schooling is negatively correlated with work experience. The plot fails to take account of
this, and as a consequence the regression line underestimates the impact of schooling on
earnings.
4
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. cor S EXP
(obs=540)
|
S
ASVABC
--------+-----------------S|
1.0000
EXP| -0.2179
1.0000
120
Hourly earnings ($)
100
80
60
40
20
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
-20
Years of schooling (highest grade completed)
We will investigate the distortion mathematically when we come to omitted variable bias.
5
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. cor S EXP
(obs=540)
|
S
ASVABC
--------+-----------------S|
1.0000
EXP| -0.2179
1.0000
120
Hourly earnings ($)
100
80
60
40
20
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
-20
Years of schooling (highest grade completed)
To eliminate the distortion, you purge both EARNINGS and S of their components related to
EXP and then draw a scatter diagram using the purged variables.
6
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. reg EARNINGS EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 617.717488
1 617.717488
Residual | 111392.514
538 207.049282
-------------+-----------------------------Total | 112010.231
539 207.811189
Number of obs
F( 1,
538)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
2.98
0.0847
0.0055
0.0037
14.389
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------EXP |
.2414715
.1398002
1.73
0.085
-.0331497
.5160927
_cons |
15.55527
2.442468
6.37
0.000
10.75732
20.35321
-----------------------------------------------------------------------------. predict EEARN, resid
We start by regressing EARNINGS on EXP, as shown above. The residuals are the part of
EARNINGS which is not related to EXP. The ‘predict’ command is the Stata command for
saving the residuals from the most recent regression. We name them EEARN.
7
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. reg S EXP
Source |
SS
df
MS
-------------+-----------------------------Model | 152.160205
1 152.160205
Residual | 3052.82313
538 5.67439243
-------------+-----------------------------Total | 3204.98333
539 5.94616574
Number of obs
F( 1,
538)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
26.82
0.0000
0.0475
0.0457
2.3821
-----------------------------------------------------------------------------S |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------EXP | -.1198454
.0231436
-5.18
0.000
-.1653083
-.0743826
_cons |
15.69765
.4043447
38.82
0.000
14.90337
16.49194
-----------------------------------------------------------------------------. predict ES, resid
We do the same with S. We regress it on EXP and save the residuals as ES.
8
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
80
60
40
20
0
-8
-6
-4
-2
0
2
4
6
-20
-40
Now we plot EEARN on ES and the scatter is a faithful representation of the relationship,
both in terms of the slope of the trend line (the red line) and in terms of the variation about
that line.
9
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
80
60
40
20
0
-8
-6
-4
-2
0
2
4
6
-20
-40
As you would expect, the trend line is steeper that in scatter diagram which did not control
for EXP (reproduced here as the black dashed line).
10
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. reg EEARN ES
Source |
SS
df
MS
Number of obs =
540
-------------+-----------------------------F( 1,
538) = 131.63
Model | 21895.9298
1 21895.9298
Prob > F
= 0.0000
Residual | 89496.5833
538 166.350527
R-squared
= 0.1966
-------------+-----------------------------Adj R-squared = 0.1951
Total | 111392.513
539 206.665145
Root MSE
= 12.898
-----------------------------------------------------------------------------EEARN |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------ES |
2.678125
.2334325
11.47
0.000
2.219574
3.136676
_cons |
8.10e-09
.5550284
0.00
1.000
-1.090288
1.090288
------------------------------------------------------------------------------
Here is the regression of EEARN on ES.
11
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. reg EEARN ES
Source |
SS
df
MS
Number of obs =
540
-------------+-----------------------------F( 1,
538) = 131.63
Model | 21895.9298
1 21895.9298
Prob > F
= 0.0000
Residual | 89496.5833
538 166.350527
R-squared
= 0.1966
-------------+-----------------------------Adj R-squared = 0.1951
Total | 111392.513
539 206.665145
Root MSE
= 12.898
-----------------------------------------------------------------------------EEARN |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------ES |
2.678125
.2334325
11.47
0.000
2.219574
3.136676
_cons |
8.10e-09
.5550284
0.00
1.000
-1.090288
1.090288
-----------------------------------------------------------------------------From multiple regression:
. reg EARNINGS S EXP
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
2.678125
.2336497
11.46
0.000
2.219146
3.137105
EXP |
.5624326
.1285136
4.38
0.000
.3099816
.8148837
_cons | -26.48501
4.27251
-6.20
0.000
-34.87789
-18.09213
------------------------------------------------------------------------------
A mathematical proof that the technique works requires matrix algebra. We will content
ourselves by verifying that the estimate of the slope coefficient is the same as in the
multiple regression.
12
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. reg EEARN ES
Source |
SS
df
MS
Number of obs =
540
-------------+-----------------------------F( 1,
538) = 131.63
Model | 21895.9298
1 21895.9298
Prob > F
= 0.0000
Residual | 89496.5833
538 166.350527
R-squared
= 0.1966
-------------+-----------------------------Adj R-squared = 0.1951
Total | 111392.513
539 206.665145
Root MSE
= 12.898
-----------------------------------------------------------------------------EEARN |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------ES |
2.678125
.2334325
11.47
0.000
2.219574
3.136676
_cons |
8.10e-09
.5550284
0.00
1.000
-1.090288
1.090288
-----------------------------------------------------------------------------From multiple regression:
. reg EARNINGS S EXP
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
2.678125
.2336497
11.46
0.000
2.219146
3.137105
EXP |
.5624326
.1285136
4.38
0.000
.3099816
.8148837
_cons | -26.48501
4.27251
-6.20
0.000
-34.87789
-18.09213
------------------------------------------------------------------------------
Finally, a small and not very important technical point. You may have noticed that the
standard error and t statistic do not quite match. The reason for this is that the number of
degrees of freedom is overstated by 1 in the residuals regression.
13
GRAPHING A RELATIONSHIP IN A MULTIPLE REGRESSION MODEL
. reg EEARN ES
Source |
SS
df
MS
Number of obs =
540
-------------+-----------------------------F( 1,
538) = 131.63
Model | 21895.9298
1 21895.9298
Prob > F
= 0.0000
Residual | 89496.5833
538 166.350527
R-squared
= 0.1966
-------------+-----------------------------Adj R-squared = 0.1951
Total | 111392.513
539 206.665145
Root MSE
= 12.898
-----------------------------------------------------------------------------EEARN |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------ES |
2.678125
.2334325
11.47
0.000
2.219574
3.136676
_cons |
8.10e-09
.5550284
0.00
1.000
-1.090288
1.090288
-----------------------------------------------------------------------------From multiple regression:
. reg EARNINGS S EXP
-----------------------------------------------------------------------------EARNINGS |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------S |
2.678125
.2336497
11.46
0.000
2.219146
3.137105
EXP |
.5624326
.1285136
4.38
0.000
.3099816
.8148837
_cons | -26.48501
4.27251
-6.20
0.000
-34.87789
-18.09213
------------------------------------------------------------------------------
That regression has not made allowance for the fact that we have already used up 1 degree
of freedom in removing EXP from the model.
14
Copyright Christopher Dougherty 2012.
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Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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2012.10.28