Transcript Chapter 6 exercise 6.7 (EC220)

Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 6)
Slideshow: exercise 6.7
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 6). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/132/
Available in LSE Learning Resources Online: May 2012
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EXERCISE 6.7
6.7* A social scientist thinks that the level of activity in the shadow
economy, Y, depends either positively on the level of the tax
burden, X, or negatively on the level of government
expenditure to discourage shadow economy activity, Z . Y
might also depend on both X and Z. International crosssection data on Y, X, and Z, all measured in US $ million, are
obtained for a sample of 30 industrialized countries and a
second sample of 30 developing countries. The social
scientist and regresses log Y (1) on both log X and log Y, (2) on
log X alone, and (3) on log Z alone, for each sample, with the
following results (standard errors in parentheses):
1
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
log X was positively correlated with log Z in both samples (0.81
and 0.75). Perform the appropriate statistical tests and write a
short report advising how these results should be interpreted.
2
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
We will start with the sample of 30 industrialized countries. In the multiple regression, the
estimates of the elasticities of Y with respect to X and Z are both highly significant and have
the expected signs. We will therefore adopt this as our preferred specification.
3
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
 X i  X Zi  Z 

E(b2 )   2   3
2


X

X
 i
In model (2), the estimate of the elasticity of Y with respect to X will be subject to omitted
variable bias.
4
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
 X i  X Zi  Z 

E(b2 )   2   3
2


X

X
 i
It is reasonable to assume that 3 is negative, particularly since its estimate in the multiple
regression is negative and highly significant.
5
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
 X i  X Zi  Z 

E(b2 )   2   3
2


X

X
 i
We are told that X and Z are positively correlated, so the numerator of the second factor in
the bias term will be positive. The denominator must be positive. Hence the bias will be
negative, accounting for the fall in the estimate of the elasticity.
6
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
 X i  X Zi  Z 

E(b3 )   3   2
2


Z

Z
 i
Similarly, in model (3), the estimate of the elasticity of Y with respect to Z will be subject to
omitted variable bias. It is reasonable to assume that 2 is positive, and we know that the
numerator and the denominator of the second factor are positive, so the bias is positive.
7
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
 X i  X Zi  Z 

E(b3 )   3   2
2


Z

Z
 i
The bias just about offsets the true value of the coefficient, with the consequence that the
estimated coefficient is close to 0. It appears that log Z has very little effect on log Y, and
this in turn accounts for the very low value of R2.
8
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
 X i  X Zi  Z 

E(b3 )   3   2
2


Z

Z
 i
For similar reasons, R2 in model (2) was also low.
9
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
 X i  X Zi  Z 

E(b3 )   3   2
2


Z

Z
 i
However, in the multiple regression, where the effect of omitted variable bias is eliminated,
we see that in fact log X and log Z account for nearly half the variance in log Y.
10
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
Next we come to the sample of 30 developing countries. In the multiple regression, the
estimate of the elasticity of Y with respect to X is again highly significant, but that of the
elasticity with respect to Z is not.
11
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
If we drop log Z, the estimate of the elasticity with respect to X is a little lower. R2 is virtually
unchanged.
12
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
If instead we drop log X, the estimate of the elasticity with respect to Z changes sign,
suggesting that expenditure on enforcement actually encourages the growth of the shadow
economy. Moreover, the effect appears to be highly significant.
13
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
 X i  X Zi  Z 

E(b3 )   3   2
2


Z

Z
 i
Obviously, this absurd result is attributable to omitted variable bias, because it is clear that
log X belongs in the model.
14
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
 X i  X Zi  Z 

E(b3 )   3   2
2


Z

Z
 i
As in the case of the industrialized countries, the bias is positive, and it is so large that it
dominates the estimate of the elasticity. log Z is acting as a proxy for log X.
15
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
So what do you conclude in the case of the developing countries?
16
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
Clearly, we eliminate model (3). But the other two models both remain in contention.
17
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
It is possible that enforcement expenditure has no effect and model (2) is the correct
specification. If this is the case, the estimator of the elasticity of X in model (1) will be less
efficient and this is the reason that the standard error is larger.
18
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
Alternatively, model (1) might be the correct specification. Enforcement expenditure may
have an effect, but bureaucracy and corruption undermine it and account for the low and
insignificant estimate of its elasticity.
19
EXERCISE 6.7
Industrialized Countries
(1)
(2)
log X
0.70
(0.15)
0.20
(0.11)
log Z
–0.65
(0.16)
constant
–1.14
(0.86)
0.44
R2
–
(3)
–
Developing Countries
(1)
(2)
0.81
(0.14)
0.73
(0.09)
–
(3)
–
–0.05
(0.12)
–0.09
(0.12)
–1.07
(1.07)
1.23
(0.90)
–1.12
(0.87)
–1.02
(0.86)
2.82
(0.84)
0.10
0.01
0.71
0.70
0.33
0.43
(0.12)
logY  1   2 log X   3 log Z  u
In practice, in a situation like this, you would present the regression results for both models
(1) and (2) in your report, giving the reader all the available information.
20
Copyright Christopher Dougherty 2000–2006. This slideshow may be freely copied for
personal use.
25.06.06