Transcript EXERCISE 4.5 4.5* Download from the website the OECD data set on employment growth rates and GDP growth rates tabulated in Exercise 1.1, plot.

EXERCISE 4.5
4.5*
Download from the website the OECD data set on
employment growth rates and GDP growth rates tabulated
in Exercise 1.1, plot a scatter diagram and investigate
whether a nonlinear specification might be superior to a
linear one.
1
EXERCISE 4.5
Employment growth rate (%)
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
-1
-2
GDP growth rate (%)
The figure shows the employment growth rates and GDP growth rates for the sample of 25
OECD countries.
2
EXERCISE 4.5
. reg e g
Source |
SS
df
MS
---------+-----------------------------Model | 14.2762167
1 14.2762167
Residual | 9.88359869
23 .429721682
---------+-----------------------------Total | 24.1598154
24 1.00665898
Number of obs
F( 1,
23)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
25
33.22
0.0000
0.5909
0.5731
.65553
-----------------------------------------------------------------------------e |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------g |
.4846863
.0840907
5.764
0.000
.3107315
.6586411
_cons | -.5208643
.2707298
-1.924
0.067
-1.080912
.039183
------------------------------------------------------------------------------
A linear regression produces quite plausible results. The slope coefficient indicates that e
increases by 0.48 percent for each percent increase in g. This makes sense because part of
the variation in g is due to variations in the growth of efficiency.
3
EXERCISE 4.5
. reg e g
Source |
SS
df
MS
---------+-----------------------------Model | 14.2762167
1 14.2762167
Residual | 9.88359869
23 .429721682
---------+-----------------------------Total | 24.1598154
24 1.00665898
Number of obs
F( 1,
23)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
25
33.22
0.0000
0.5909
0.5731
.65553
-----------------------------------------------------------------------------e |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------g |
.4846863
.0840907
5.764
0.000
.3107315
.6586411
_cons | -.5208643
.2707298
-1.924
0.067
-1.080912
.039183
------------------------------------------------------------------------------
The intercept indicates that if there were no output growth, employment would shrink at the
rate of 0.52 percent per year, again as a consequence of increasing productivity.
4
EXERCISE 4.5
. reg e g
Source |
SS
df
MS
---------+-----------------------------Model | 14.2762167
1 14.2762167
Residual | 9.88359869
23 .429721682
---------+-----------------------------Total | 24.1598154
24 1.00665898
Number of obs
F( 1,
23)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
25
33.22
0.0000
0.5909
0.5731
.65553
-----------------------------------------------------------------------------e |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------g |
.4846863
.0840907
5.764
0.000
.3107315
.6586411
_cons | -.5208643
.2707298
-1.924
0.067
-1.080912
.039183
------------------------------------------------------------------------------
R2 is quite high for a simple regression using cross-section data.
5
EXERCISE 4.5
4
eˆ   0 . 52  0 . 48 g
Employment growth rate (%)
3
2
1
0
0
1
2
3
4
5
6
7
8
9
-1
-2
-3
GDP growth rate (%)
However, it is evident that the relationship is nonlinear. A double-logarithmic specification
is ruled out by the negative values of e in the sample and the fact that the relationship
clearly does not pass through the origin.
6
EXERCISE 4.5
. g gsq = g*g
. reg e g gsq
Source |
SS
df
MS
---------+-----------------------------Model |
15.58633
2
7.793165
Residual | 8.57348541
22 .389703882
---------+-----------------------------Total | 24.1598154
24 1.00665898
Number of obs
F( 2,
22)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
25
20.00
0.0000
0.6451
0.6129
.62426
-----------------------------------------------------------------------------e |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------g |
1.171194
.3828871
3.059
0.006
.3771345
1.965253
gsq | -.0810133
.0441844
-1.834
0.080
-.1726461
.0106196
_cons | -1.614902
.6500016
-2.484
0.021
-2.962922
-.2668809
------------------------------------------------------------------------------
The quadratic form is an alternative that sometimes fits simple curves well. R2 rises from
0.59 to 0.65.
7
EXERCISE 4.5
4
Yˆ   1 . 61  1 . 17 X  0 . 081 X
Employment growth rate (%)
3
2
2
1
0
0
1
2
3
4
5
6
7
8
9
-1
-2
-3
GDP growth rate (%)
Visually the quadratic specification is a large improvement on the linear one. The most
obvious defect is that it becomes downward-sloping for high values of g.
8
EXERCISE 4.5
. g gsq = g*g
. reg e g gsq
Source |
SS
df
MS
---------+-----------------------------Model |
15.58633
2
7.793165
Residual | 8.57348541
22 .389703882
---------+-----------------------------Total | 24.1598154
24 1.00665898
Number of obs
F( 2,
22)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
25
20.00
0.0000
0.6451
0.6129
.62426
-----------------------------------------------------------------------------e |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------g |
1.171194
.3828871
3.059
0.006
.3771345
1.965253
gsq | -.0810133
.0441844
-1.834
0.080
-.1726461
.0106196
_cons | -1.614902
.6500016
-2.484
0.021
-2.962922
-.2668809
------------------------------------------------------------------------------
Note that the t statistic for the squared term is rather low. It is only just significant at the 5
percent level, using a one-sided test. (A one-sided test is justified here because we would
not expect the sensitivity of e to g to increase with g.)
9
EXERCISE 4.5
. g gsq = g*g
. reg e g gsq
Source |
SS
df
MS
---------+-----------------------------Model |
15.58633
2
7.793165
Residual | 8.57348541
22 .389703882
---------+-----------------------------Total | 24.1598154
24 1.00665898
. cor g gsq
(obs=26)
Number of obs =
25
F( 2,
22) =
20.00
|
gdp
gdpsq
Prob > F
= 0.0000
--------+-----------------R-squared
= 0.6451
g|
1.0000
Adj R-squared = 0.6129
gsq|
0.9775 = 1.0000
Root MSE
.62426
-----------------------------------------------------------------------------e |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------g |
1.171194
.3828871
3.059
0.006
.3771345
1.965253
gsq | -.0810133
.0441844
-1.834
0.080
-.1726461
.0106196
_cons | -1.614902
.6500016
-2.484
0.021
-2.962922
-.2668809
------------------------------------------------------------------------------
However, a low t statistic is not surprising, given that g and gsq are highly correlated.
10
EXERCISE 4.5
. g lgg=ln(g)
. reg e lgg
Source |
SS
df
MS
---------+-----------------------------Model |
15.116284
1
15.116284
Residual | 9.04353146
23
.39319702
---------+-----------------------------Total | 24.1598154
24 1.00665898
Number of obs
F( 1,
23)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
25
38.44
0.0000
0.6257
0.6094
.62705
-----------------------------------------------------------------------------e |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------lgg |
1.709096
.2756444
6.200
0.000
1.138883
2.27931
_cons |
-.728669
.2830097
-2.575
0.017
-1.314119
-.1432188
------------------------------------------------------------------------------
The semilogarithmic specification, with the explanatory variable logarithmic, is another
possible specification.
11
EXERCISE 4.5
4
eˆ   0 . 73  1 . 70 log g
Employment growth rate (%)
3
2
1
0
0
1
2
3
4
5
6
7
8
9
-1
-2
-3
GDP growth rate (%)
Comparing it with the quadratic specification, it does have the advantage of not being
downward-sloping for high values of g, but it exhibits excessive sensitivity to g for low
values. R2 is about the same as for the quadratic specification.
12
EXERCISE 4.5
. g Z=1/g
e  1 
. reg e Z
Source |
SS
df
MS
---------+-----------------------------Model | 12.7770171
1 12.7770171
Residual | 11.3827983
23 .494904274
---------+-----------------------------Total | 24.1598154
24 1.00665898
2
u
g
Number of obs
F( 1,
23)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
25
25.82
0.0000
0.5289
0.5084
.70349
-----------------------------------------------------------------------------e |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------Z | -3.997462
.7867382
-5.081
0.000
-5.624954
-2.36997
_cons |
2.592225
.3716506
6.975
0.000
1.823407
3.361043
------------------------------------------------------------------------------
A further possibility is a hyperbolic function, with e being regressed on the reciprocal of g.
13
EXERCISE 4.5
4
eˆ  2 . 59 
Employment growth rate (%)
3
4 . 00
g
2
1
0
0
1
2
3
4
5
6
7
8
9
-1
-2
-3
GDP growth rate (%)
However R2 is lower than in the previous specifications. e is systematically underestimated
for high values of g and the specification exhibits excessive sensitivity to g for low values.
14
EXERCISE 4.5
4
eˆ  2 . 59 
Employment growth rate (%)
3
4 . 00
g
2
1
0
0
1
2
3
4
5
6
7
8
9
-1
-2
-3
GDP growth rate (%)
One reason for the poor performance of the hyperbolic function is that it is constrained to
be asymptotically tangential to the vertical axis.
15
EXERCISE 4.5
============================================================
Dependent Variable: e
Method: Least Squares
2
Sample: 1 25
e  1 
Included observations: 25
g  3
Convergence achieved after 18 iterations
e=C(1)+C(2)/(g+C(3))
============================================================
CoefficientStd. Errort-Statistic Prob.
============================================================
C(1)
5.467457
2.826812
1.934142
0.0661
C(2)
-31.07502
41.79895 -0.743440
0.4651
C(3)
4.148426
4.871805
0.851517
0.4037
============================================================
R-squared
0.635231
Mean dependent var 0.833600
Adjusted R-squared
0.602070
S.D. dependent var 1.014519
S.E. of regression
0.639975
Akaike info criteri2.057393
Sum squared resid
9.010510
Schwarz criterion 2.203658
Log likelihood
-22.71741
F-statistic
19.15609
Durbin-Watson stat
0.770997
Prob(F-statistic) 0.000015
============================================================
u
If we relax that assumption, we obtain a much better fit. Note that there is no way of
linearizing this nonlinear specification. We have to use nonlinear least squares.
16
EXERCISE 4.5
============================================================
Dependent Variable: e
Method: Least Squares
2
Sample: 1 25
e  1 
Included observations: 25
g  3
Convergence achieved after 18 iterations
e=C(1)+C(2)/(g+C(3))
============================================================
CoefficientStd. Errort-Statistic Prob.
============================================================
C(1)
5.467457
2.826812
1.934142
0.0661
C(2)
-31.07502
41.79895 -0.743440
0.4651
C(3)
4.148426
4.871805
0.851517
0.4037
============================================================
R-squared
0.635231
Mean dependent var 0.833600
Adjusted R-squared
0.602070
S.D. dependent var 1.014519
S.E. of regression
0.639975
Akaike info criteri2.057393
Sum squared resid
9.010510
Schwarz criterion 2.203658
Log likelihood
-22.71741
F-statistic
19.15609
Durbin-Watson stat
0.770997
Prob(F-statistic) 0.000015
============================================================
u
EViews was used instead of Stata for this purpose because its nonlinear least squares
facility is much easier to use. To fit a nonlinear least squares specification in EViews, you
write it as an explicit mathematical equation, with C(1), C(2), etc being the parameters.
17
EXERCISE 4.5
4
eˆ  5 . 47 
Employment growth rate (%)
3
31 . 08
g  4 . 15
2
1
0
0
1
2
3
4
5
6
7
8
9
-1
-2
-3
GDP growth rate (%)
This specification is compared with the semilogarithmic one. There is very little to choose
between them, and indeed the quadratic specification is virtually as good, at least for the
data range in the sample.
18
Copyright Christopher Dougherty 2001–2006. This slideshow may be freely copied for
personal use.
22.06.06