Transcript EXERCISE 6.13 6.13* The first regression shows the result of regressing LGFDHO, the logarithm of annual household expenditure on food eaten at home,

EXERCISE 6.13
6.13* The first regression shows the result of regressing 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 868
households in the 1995 Consumer Expenditure Survey.
In the second regression, LGFDHOPC, the logarithm of food
expenditure per capita (FDHO/SIZE), is regressed on
LGEXPPC, the logarithm of total expenditure per capita
(EXP/SIZE).
In the third regression LGFDHOPC is regressed on
LGEXPPC and LGSIZE.
1
EXERCISE 6.13
. reg LGFDHO LGEXP LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 138.776549
2 69.3882747
Residual | 130.219231
865 .150542464
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
460.92
0.0000
0.5159
0.5148
.388
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.2866813
.0226824
12.639
0.000
.2421622
.3312003
LGSIZE |
.4854698
.0255476
19.003
0.000
.4353272
.5356124
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
------------------------------------------------------------------------------
2
EXERCISE 6.13
. reg LGFDHOPC LGEXPPC
Source |
SS
df
MS
---------+-----------------------------Model | 51.4364364
1 51.4364364
Residual | 142.293973
866 .164311747
---------+-----------------------------Total |
193.73041
867 .223449146
Number of obs
F( 1,
866)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
313.04
0.0000
0.2655
0.2647
.40535
-----------------------------------------------------------------------------LGFDHOPC |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXPPC |
.376283
.0212674
17.693
0.000
.3345414
.4180246
_cons |
3.700667
.1978925
18.700
0.000
3.312262
4.089072
------------------------------------------------------------------------------
3
EXERCISE 6.13
. reg LGFDHOPC LGEXPPC LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 63.5111811
2 31.7555905
Residual | 130.219229
865 .150542461
---------+-----------------------------Total |
193.73041
867 .223449146
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
210.94
0.0000
0.3278
0.3263
.388
-----------------------------------------------------------------------------LGFDHOPC |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXPPC |
.2866813
.0226824
12.639
0.000
.2421622
.3312004
LGSIZE | -.2278489
.0254412
-8.956
0.000
-.2777826
-.1779152
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
------------------------------------------------------------------------------
1. Explain why the second model is a restricted version of
the first, stating the restriction.
2. Perform an F test of the restriction.
3. Perform a t test of the restriction.
4. Summarize your conclusions from the analysis of the
regression results.
4
EXERCISE 6.13
LGFDHO   1   2 LGEXP   3 LGSIZE  u
The first regression is a straightforward logarithmic regression of expenditure on food
consumed at home on total household expenditure and size of household.
5
EXERCISE 6.13
LGFDHO   1   2 LGEXP   3 LGSIZE  u
LGFDHOPC  1   2 LGEXPPC  u
log FDHO/ SIZE  1   2 log EXP / SIZE  u
The second regression is a simple regression of LGFDHOPC, defined as log FDHO/SIZE, on
LGEXPPC, defined as log EXP/SIZE.
6
EXERCISE 6.13
LGFDHO   1   2 LGEXP   3 LGSIZE  u
LGFDHOPC  1   2 LGEXPPC  u
log FDHO/ SIZE  1   2 log EXP / SIZE  u
LGFDHO LGSIZE  1   2 ( LGEXP  LGSIZE)  u
The logarithmic ratios have been split.
7
EXERCISE 6.13
LGFDHO   1   2 LGEXP   3 LGSIZE  u
LGFDHOPC  1   2 LGEXPPC  u
log FDHO/ SIZE  1   2 log EXP / SIZE  u
LGFDHO LGSIZE  1   2 ( LGEXP  LGSIZE)  u
LGFDHO  1   2 LGEXP  (1   2 ) LGSIZE  u
The LGSIZE terms have been brought together.
8
EXERCISE 6.13
LGFDHO   1   2 LGEXP   3 LGSIZE  u
LGFDHOPC  1   2 LGEXPPC  u
log FDHO/ SIZE  1   2 log EXP / SIZE  u
LGFDHO LGSIZE  1   2 ( LGEXP  LGSIZE)  u
LGFDHO  1   2 LGEXP  (1   2 ) LGSIZE  u
3  1  2
Comparing this equation with that for the first regression, we see that the second
specification is a restricted version of the first with the restriction 3 = 1 – 2.
9
EXERCISE 6.13
. reg LGFDHO LGEXP LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 138.776549
2 69.3882747
Residual | 130.219231
865 .150542464
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
460.92
0.0000
0.5159
0.5148
.388
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.2866813
.0226824
12.639
0.000
.2421622
.3312003
LGSIZE |
.4854698
.0255476
19.003
0.000
.4353272
.5356124
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
------------------------------------------------------------------------------
LGFDHO   1   2 LGEXP   3 LGSIZE  u
3  1  2
Before performing a test of the restriction, we should check whether the estimates of the
elasticities in the unrestricted version appear to satisfy it.
10
EXERCISE 6.13
. reg LGFDHO LGEXP LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 138.776549
2 69.3882747
Residual | 130.219231
865 .150542464
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
460.92
0.0000
0.5159
0.5148
.388
-----------------------------------------------------------------------------LGFDHO |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXP |
.2866813
.0226824
12.639
0.000
.2421622
.3312003
LGSIZE |
.4854698
.0255476
19.003
0.000
.4353272
.5356124
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
------------------------------------------------------------------------------
LGFDHO   1   2 LGEXP   3 LGSIZE  u
3  1  2
b3 is 0.49. 1 – b2 is 0.71. The discrepancy is rather large, compared with the standard errors
of the estimates. We should expect the restriction to be rejected.
11
EXERCISE 6.13
. reg LGFDHO LGEXP LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 138.776549
2 69.3882747
Residual | 130.219231
865 .150542464
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
460.92
0.0000
0.5159
0.5148
.388
Number of obs
F( 1,
866)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
313.04
0.0000
0.2655
0.2647
.40535
. reg LGFDHOPC LGEXPPC
Source |
SS
df
MS
---------+-----------------------------Model | 51.4364364
1 51.4364364
Residual | 142.293973
866 .164311747
---------+-----------------------------Total |
193.73041
867 .223449146
We see that the residual sum of squares increases from 130.2 to 142.3 when we impose the
restriction.
12
EXERCISE 6.13
. reg LGFDHO LGEXP LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 138.776549
2 69.3882747
Residual | 130.219231
865 .150542464
---------+-----------------------------Total | 268.995781
867 .310260416
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
460.92
0.0000
0.5159
0.5148
.388
Number of obs
F( 1,
866)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
313.04
0.0000
0.2655
0.2647
.40535
. reg LGFDHOPC LGEXPPC
Source |
SS
df
MS
---------+-----------------------------Model | 51.4364364
1 51.4364364
Residual | 142.293973
866 .164311747
---------+-----------------------------Total |
193.73041
867 .223449146
H0 : 3  1  2
H1 :  3  1   2
(142.3  130.2) / 1
F (1,865) 
 80.4
130.2 / 865
Fcrit , 0.1% (1,750)  10.9
The F statistic is far above the critical value of F(1,750) at the 0.1% level. The critical value
of F(1,865) must be lower than that for F(1,750). Therefore we reject the null hypothesis and
conclude that the restriction is invalid.
13
EXERCISE 6.13
log FDHO   1   2 log EXP   3 log SIZE  u
3  1  2
2  3  1  0
We will also use the t test approach to test the restriction. First we rewrite the restriction so
that the right side of the definition is zero.
14
EXERCISE 6.13
log FDHO   1   2 log EXP   3 log SIZE  u
3  1  2
2  3  1  0
q  2  3  1
We define q to be equal to the left side. The restriction is now q = 0.
15
EXERCISE 6.13
log FDHO   1   2 log EXP   3 log SIZE  u
3  1  2
2  3  1  0
q  2  3  1
3  q  2  1
We express one of the  coefficients in terms of q and the other  coefficient.
16
EXERCISE 6.13
log FDHO   1   2 log EXP   3 log SIZE  u
3  1  2
2  3  1  0
q  2  3  1
3  q  2  1
log FDHO  1   2 log EXP  q   2  1 log SIZE  u
We substitute for this  in the regression model.
17
EXERCISE 6.13
log FDHO   1   2 log EXP   3 log SIZE  u
3  1  2
2  3  1  0
q  2  3  1
3  q  2  1
log FDHO  1   2 log EXP  q   2  1 log SIZE  u
log FDHO  log SIZE  1   2 log EXP  log SIZE  q log SIZE  u
We bring the 2 components together and take the (+1)log SIZE to the left side of the
equation.
18
EXERCISE 6.13
log FDHO   1   2 log EXP   3 log SIZE  u
3  1  2
2  3  1  0
q  2  3  1
3  q  2  1
log FDHO  1   2 log EXP  q   2  1 log SIZE  u
log FDHO  log SIZE  1   2 log EXP  log SIZE  q log SIZE  u
LGFDHOPC  1   2 LGEXPPC  qLGSIZE  u
This allows us to rewrite the model with the dependent variable the logarithm of expenditure
on food per capita and the explanatory variables the logarithms of total household
expenditure per capita and household size.
19
EXERCISE 6.13
log FDHO   1   2 log EXP   3 log SIZE  u
3  1  2
2  3  1  0
q  2  3  1
3  q  2  1
log FDHO  1   2 log EXP  q   2  1 log SIZE  u
log FDHO  log SIZE  1   2 log EXP  log SIZE  q log SIZE  u
LGFDHOPC  1   2 LGEXPPC  qLGSIZE  u
Having reparameterized the model in this way, we can test the restriction with a simple t test
on the coefficient of LGSIZE.
20
EXERCISE 6.13
log FDHO   1   2 log EXP   3 log SIZE  u
3  1  2
2  3  1  0
q  2  3  1
3  q  2  1
log FDHO  1   2 log EXP  q   2  1 log SIZE  u
log FDHO  log SIZE  1   2 log EXP  log SIZE  q log SIZE  u
LGFDHOPC  1   2 LGEXPPC  qLGSIZE  u
If the coefficient of LGSIZE is significantly different from zero, we need the term and should
stay with the unrestricted model. If it is not, the term could be dropped, giving us the
restricted model as the appropriate specification.
21
EXERCISE 6.13
log FDHO   1   2 log EXP   3 log SIZE  u
3  1  2
2  3  1  0
q  2  3  1
3  q  2  1
log FDHO  1   2 log EXP  q   2  1 log SIZE  u
log FDHO  log SIZE  1   2 log EXP  log SIZE  q log SIZE  u
LGFDHOPC  1   2 LGEXPPC  qLGSIZE  u
H0 : q  2  3  1  0
H1 : q   2   3  1  0
Note that the null hypothesis for the t test is that the restriction is valid. This ties in with the
reasoning in the previous slide. If the restriction is valid, we do not need the LGSIZE term
and the restricted version is the appropriate specification.
22
EXERCISE 6.13
. reg LGFDHOPC LGEXPPC LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 63.5111811
2 31.7555905
Residual | 130.219229
865 .150542461
---------+-----------------------------Total |
193.73041
867 .223449146
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
210.94
0.0000
0.3278
0.3263
.388
-----------------------------------------------------------------------------LGFDHOPC |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXPPC |
.2866813
.0226824
12.639
0.000
.2421622
.3312004
LGSIZE | -.2278489
.0254412
-8.956
0.000
-.2777826
-.1779152
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
------------------------------------------------------------------------------
H0 : q  2  3  1  0
H1 : q   2   3  1  0
Here is the corresponding regression result. We find that the coefficient has a very high
(negative) t statistic. The null hypothesis is rejected and again we conclude that the
restriction is invalid.
23
EXERCISE 6.13
. reg LGFDHOPC LGEXPPC LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 63.5111811
2 31.7555905
Residual | 130.219229
865 .150542461
---------+-----------------------------Total |
193.73041
867 .223449146
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
210.94
0.0000
0.3278
0.3263
.388
-----------------------------------------------------------------------------LGFDHOPC |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXPPC |
.2866813
.0226824
12.639
0.000
.2421622
.3312004
LGSIZE | -.2278489
.0254412
-8.956
0.000
-.2777826
-.1779152
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
------------------------------------------------------------------------------
H0 : q  2  3  1  0
H1 : q   2   3  1  0
The F test and the t test approaches are of course equivalent. The F statistic, 80.4, is the
square of the t statistic and the critical value of F is the square of the critical value of t.
24
EXERCISE 6.13
. reg LGFDHOPC LGEXPPC LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 63.5111811
2 31.7555905
Residual | 130.219229
865 .150542461
---------+-----------------------------Total |
193.73041
867 .223449146
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
210.94
0.0000
0.3278
0.3263
.388
-----------------------------------------------------------------------------LGFDHOPC |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXPPC |
.2866813
.0226824
12.639
0.000
.2421622
.3312004
LGSIZE | -.2278489
.0254412
-8.956
0.000
-.2777826
-.1779152
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
------------------------------------------------------------------------------
H0 : q  2  3  1  0
H1 : q   2   3  1  0
Should we have anticipated this outcome? The restricted version effectively controls for
the size of the household. Why should the size variable have a separate effect?
25
EXERCISE 6.13
. reg LGFDHOPC LGEXPPC LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 63.5111811
2 31.7555905
Residual | 130.219229
865 .150542461
---------+-----------------------------Total |
193.73041
867 .223449146
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
210.94
0.0000
0.3278
0.3263
.388
-----------------------------------------------------------------------------LGFDHOPC |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXPPC |
.2866813
.0226824
12.639
0.000
.2421622
.3312004
LGSIZE | -.2278489
.0254412
-8.956
0.000
-.2777826
-.1779152
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
------------------------------------------------------------------------------
H0 : q  2  3  1  0
H1 : q   2   3  1  0
One possibility is that there are economies of scale in feeding a larger household, or
perhaps less wastage.
26
EXERCISE 6.13
. reg LGFDHOPC LGEXPPC LGSIZE
Source |
SS
df
MS
---------+-----------------------------Model | 63.5111811
2 31.7555905
Residual | 130.219229
865 .150542461
---------+-----------------------------Total |
193.73041
867 .223449146
Number of obs
F( 2,
865)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
868
210.94
0.0000
0.3278
0.3263
.388
-----------------------------------------------------------------------------LGFDHOPC |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------+-------------------------------------------------------------------LGEXPPC |
.2866813
.0226824
12.639
0.000
.2421622
.3312004
LGSIZE | -.2278489
.0254412
-8.956
0.000
-.2777826
-.1779152
_cons |
4.720269
.2209996
21.359
0.000
4.286511
5.154027
------------------------------------------------------------------------------
H0 : q  2  3  1  0
H1 : q   2   3  1  0
Another is that there may be a compositional effect, large households tending to have more
children, who eat less. Perhaps we should be controlling by some notion of the number of
equivalent adults, rather than the unadjusted number of people in the household.
27
Copyright Christopher Dougherty 2000–2007. This slideshow may be freely copied for
personal use.
07.12.07