Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: exercise 6.4 Original citation: Dougherty, C.
Download ReportTranscript Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: exercise 6.4 Original citation: Dougherty, C.
Slide 1
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 6)
Slideshow: exercise 6.4
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
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
Slide 2
EXERCISE 6.4
6.4* The table gives the results of multiple and simple regressions
of LGFDHO, the logarithm of annual household expenditure on
food eaten at home, on LGEXP, the logarithm of total annual
household expenditure, and LGSIZE, the logarithm of the
number of persons in the household, using a sample of 869
households in the 1995 Consumer Expenditure Survey. The
correlation coefficient for LGEXP and LGSIZE was 0.45.
Explain the variations in the regression coefficients.
1
Slide 3
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
–
(3)
–
0.63
(0.02)
rLGEXP,LGSIZE = 0.45
2
Slide 4
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
The first column of the table gives the result of a multiple regression on LGEXP, the
logarithm of total annual household expenditure, and LGSIZE, the logarithm of the number
of people in the household (standard errors in parentheses).
3
Slide 5
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
We will assume that this is the correct specification. The estimated elasticities are both
significantly different from 0 at a very high significance level.
4
Slide 6
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
The second column gives the result of regressing LGFDHO on LGEXP only. We see that the
coefficient of LGEXP is much larger than before.
5
Slide 7
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b2 ) 2 3
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGEXP )
The reason is that the coefficient is subject to omitted variable bias, and we can
demonstrate that the bias is likely to be positive.
6
Slide 8
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b2 ) 2 3
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGEXP )
As a matter of common sense, 2 is certainly positive. The fact that the estimated
coefficient in the multiple regression is positive and highly significant provides powerful
supporting evidence.
7
Slide 9
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b2 ) 2 3
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGEXP )
The correlation between LGEXP and LGSIZE is also positive, as might be expected, and
hence their covariance must be positive.
8
Slide 10
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b2 ) 2 3
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGEXP )
The variance of LGEXP is positive. Hence all the components of the bias term are positive.
9
Slide 11
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b3 ) 3 2
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGSIZE )
For similar reasons, the coefficient of LGSIZE is biased upwards when LGEXP is omitted.
10
Slide 12
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b3 ) 3 2
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGSIZE )
2 is certainly positive. As with 3, the fact that the estimated coefficient in the multiple
regression is positive and highly significant provides powerful supporting evidence.
11
Slide 13
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b3 ) 3 2
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGSIZE )
We have already seen that Cov(LGEXP, LGSIZE) is positive, and Var(LGSIZE) is
automatically positive, so the bias is positive.
12
Slide 14
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
Finally, note the values of R2. In the simple regression on LGEXP only, R2 is 0.31. However
this exaggerates the explanatory power of LGEXP because it is acting partly as a proxy for
the missing LGSIZE.
13
Slide 15
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
Similarly, in the simple regression on LGSIZE only, R2 is 0.42. This exaggerates the
explanatory power of LGSIZE because it is acting partly as a proxy for the missing LGEXP.
14
Slide 16
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
The simple regressions might seem to suggest that LGEXP and LGSIZE jointly account for
0.31 + 0.42 = 0.73 of the variance of LGFDHO. However the multiple regression reveals that
they account for only 0.52 of the variance.
15
Slide 17
Copyright Christopher Dougherty 2000–2006. This slideshow may be freely copied for
personal use.
30.08.06
Christopher Dougherty
EC220 - Introduction to econometrics
(chapter 6)
Slideshow: exercise 6.4
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
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows
the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user
credits the author and licenses their new creations under the identical terms.
http://creativecommons.org/licenses/by-sa/3.0/
http://learningresources.lse.ac.uk/
Slide 2
EXERCISE 6.4
6.4* The table gives the results of multiple and simple regressions
of LGFDHO, the logarithm of annual household expenditure on
food eaten at home, on LGEXP, the logarithm of total annual
household expenditure, and LGSIZE, the logarithm of the
number of persons in the household, using a sample of 869
households in the 1995 Consumer Expenditure Survey. The
correlation coefficient for LGEXP and LGSIZE was 0.45.
Explain the variations in the regression coefficients.
1
Slide 3
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
–
(3)
–
0.63
(0.02)
rLGEXP,LGSIZE = 0.45
2
Slide 4
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
The first column of the table gives the result of a multiple regression on LGEXP, the
logarithm of total annual household expenditure, and LGSIZE, the logarithm of the number
of people in the household (standard errors in parentheses).
3
Slide 5
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
We will assume that this is the correct specification. The estimated elasticities are both
significantly different from 0 at a very high significance level.
4
Slide 6
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
The second column gives the result of regressing LGFDHO on LGEXP only. We see that the
coefficient of LGEXP is much larger than before.
5
Slide 7
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b2 ) 2 3
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGEXP )
The reason is that the coefficient is subject to omitted variable bias, and we can
demonstrate that the bias is likely to be positive.
6
Slide 8
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b2 ) 2 3
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGEXP )
As a matter of common sense, 2 is certainly positive. The fact that the estimated
coefficient in the multiple regression is positive and highly significant provides powerful
supporting evidence.
7
Slide 9
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b2 ) 2 3
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGEXP )
The correlation between LGEXP and LGSIZE is also positive, as might be expected, and
hence their covariance must be positive.
8
Slide 10
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b2 ) 2 3
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGEXP )
The variance of LGEXP is positive. Hence all the components of the bias term are positive.
9
Slide 11
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b3 ) 3 2
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGSIZE )
For similar reasons, the coefficient of LGSIZE is biased upwards when LGEXP is omitted.
10
Slide 12
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b3 ) 3 2
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGSIZE )
2 is certainly positive. As with 3, the fact that the estimated coefficient in the multiple
regression is positive and highly significant provides powerful supporting evidence.
11
Slide 13
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
(3)
–
–
LGFDHO
0.63
(0.02)
1 2 LGEXP
E (b3 ) 3 2
3 LGSIZE
u
Cov ( LGEXP , LGSIZE )
Var ( LGSIZE )
We have already seen that Cov(LGEXP, LGSIZE) is positive, and Var(LGSIZE) is
automatically positive, so the bias is positive.
12
Slide 14
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
Finally, note the values of R2. In the simple regression on LGEXP only, R2 is 0.31. However
this exaggerates the explanatory power of LGEXP because it is acting partly as a proxy for
the missing LGSIZE.
13
Slide 15
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
Similarly, in the simple regression on LGSIZE only, R2 is 0.42. This exaggerates the
explanatory power of LGSIZE because it is acting partly as a proxy for the missing LGEXP.
14
Slide 16
EXERCISE 6.4
(1)
(2)
LGEXP
0.29
(0.02)
0.48
(0.02)
LGSIZE
0.49
(0.03)
constant
4.72
(0.22)
3.17
(0.24)
7.50
(0.02)
0.52
0.31
0.42
R2
rLGEXP,LGSIZE = 0.45
–
LGFDHO
1 2 LGEXP
(3)
–
0.63
(0.02)
3 LGSIZE
u
The simple regressions might seem to suggest that LGEXP and LGSIZE jointly account for
0.31 + 0.42 = 0.73 of the variance of LGFDHO. However the multiple regression reveals that
they account for only 0.52 of the variance.
15
Slide 17
Copyright Christopher Dougherty 2000–2006. This slideshow may be freely copied for
personal use.
30.08.06