|
216
Part One
Single-Equation Regression Models
EXERCISES
TABLE 7.5
Y
X
2
X
3
1
1
2
3
2
1
8
3
3
Questions
7.1. Consider the data in Table 7.5.
Based on these data, estimate the following regressions:
Y
i
=
α
1
+
α
2
X
2
i
+
u
1
i
(1)
Y
i
=
λ
1
+
λ
3
X
3
i
+
u
2
i
(2)
Y
i
=
β
1
+
β
2
X
2
i
+
β
3
X
3
i
+
u
i
(3)
Note:
Estimate only the coefficients and not the standard errors.
a.
Is
α
2
=
β
2
? Why or why not?
b.
Is
λ
3
=
β
3
? Why or why not?
What important conclusion do you draw from this exercise?
7.2. From the following data estimate the partial regression coefficients, their standard
errors, and the adjusted and unadjusted
R
2
values:
¯
Y
=
367
.
693
¯
X
2
=
402
.
760
¯
X
3
=
8
.
0
(
Y
i
− ¯
Y
)
2
=
66042
.
269
(
X
2
i
− ¯
X
2
)
2
=
84855
.
096
(
X
3
i
− ¯
X
3
)
2
=
280
.
000
(
Y
i
− ¯
Y
)(
X
2
i
− ¯
X
2
)
=
74778
.
346
(
Y
i
− ¯
Y
)(
X
3
i
− ¯
X
3
)
=
4250
.
900
(
X
2
i
− ¯
X
2
)(
X
3
i
− ¯
X
3
)
=
4796
.
000
n
=
15
7.3. Show that Eq. (7.4.7) can also be expressed as
ˆ
β
2
=
y
i
(
x
2
i
−
b
2 3
x
3
i
)
(
x
2
i
−
b
2 3
x
3
i
)
2
=
net (of
x
3
) covariation between
y
and
x
2
net (of
x
3
) variation in
x
2
where
b
2 3
is the slope coefficient in the regression of
X
2
on
X
3
. (
Hint:
Recall that
b
2 3
=
x
2
i
x
3
i
/
x
2
3
i
.
)
7.4. In a multiple regression model you are told that the error term
u
i
has the following
probability distribution, namely,
u
i
∼
N
(0, 4)
.
How would you set up a
Monte Carlo
experiment to verify that the true variance is in fact 4?
7.5. Show that
r
2
1 2
.
3
=
(
R
2
−
r
2
1 3
)
/
(1
−
r
2
1 3
) and interpret the equation.
7.6. If the relation
α
1
X
1
+
α
2
X
2
+
α
3
X
3
=
0 holds true for all values of
X
1
,
X
2
, and
X
3
,
find the values of the three partial correlation coefficients.
7.7. Is it possible to obtain the following from a set of data?
a. r
2 3
=
0
.
9,
r
1 3
= −
0
.
2,
r
1 2
=
0
.
8
b. r
1 2
=
0
.
6,
r
2 3
= −
0
.
9,
r
3 1
= −
0
.
5
c. r
2 1
=
0
.
01,
r
1 3
=
0
.
66,
r
2 3
= −
0
.
7
guj75772_ch07.qxd 11/08/2008 04:22 PM Page 216
Chapter 7
Multiple Regression Analysis: The Problem of Estimation
217
7.8. Consider the following model:
Y
i
=
β
1
+
β
2
Education
i
+
β
2
Years of experience
+
u
i
Suppose you leave out the years of experience variable. What kinds of problems or
biases would you expect? Explain verbally.
7.9. Show that
β
2
and
β
3
in Eq. (7.9.2) do, in fact, give output elasticities of labor and
capital. (This question can be answered without using calculus; just recall the defin-
ition of the elasticity coefficient and remember that a change in the logarithm of a
variable is a relative change, assuming the changes are rather small.)
7.10. Consider the three-variable linear regression model discussed in this chapter.
a.
Suppose you multiply all the
X
2
values by 2. What will be the effect of this rescal-
ing, if any, on the estimates of the parameters and their standard errors?
b.
Now instead of (
a
), suppose you multiply all the
Y
values by 2. What will be the
effect of this, if any, on the estimated parameters and their standard errors?
7.11. In general
R
2
=
r
2
1 2
+
r
2
1 3
, but it is so only if
r
2 3
=
0
.
Comment and point out the
significance of this finding. (
Hint:
See Eq. [7.11.5].)
7.12. Consider the following models.
*
Model A:
Y
t
=
α
1
+
α
2
X
2
t
+
α
3
X
3
t
+
u
1
t
Model B:
(
Y
t
−
X
2
t
)
=
β
1
+
β
2
X
2
t
+
β
3
X
3
t
+
u
2
t
a.
Will OLS estimates of
α
1
and
β
1
be the same? Why?
b.
Will OLS estimates of
α
3
and
β
3
be the same? Why?
c.
What is the relationship between
α
2
and
β
2
?
d.
Can you compare the
R
2
terms of the two models? Why or why not?
7.13. Suppose you estimate the consumption function
†
Y
i
=
α
1
+
α
2
X
i
+
u
1
i
and the savings function
Z
i
=
β
1
+
β
2
X
i
+
u
2
i
where
Y
=
consumption,
Z
=
savings,
X
=
income, and
X
=
Y
+
Z
, that is,
income is equal to consumption plus savings.
a.
What is the relationship, if any, between
α
2
and
β
2
? Show your calculations.
b.
Will the residual sum of squares, RSS, be the same for the two models? Explain.
c.
Can you compare the
R
2
terms of the two models? Why or why not?
7.14. Suppose you express the Cobb–Douglas model given in Eq. (7.9.1) as follows:
Y
i
=
β
1
X
β
2
2
i
X
β
3
3
i
u
i
If you take the log-transform of this model, you will have ln
u
i
as the disturbance
term on the right-hand side.
a.
What probabilistic assumptions do you have to make about ln
u
i
to be able to
apply the classical normal linear regression model (CNLRM)? How would you
test this with the data given in Table 7.3?
b.
Do the same assumptions apply to
u
i
? Why or why not?
*Adapted from Wojciech W. Charemza and Derek F. Deadman,
Econometric Practice: General to Specific
Modelling, Cointegration and Vector Autogression,
Edward Elgar, Brookfield, Vermont, 1992, p. 18.
†
Adapted from Peter Kennedy,
A Guide to Econometrics,
3d ed., The MIT Press, Cambridge,
Massachusetts, 1992, p. 308, Question #9.
guj75772_ch07.qxd 11/08/2008 04:22 PM Page 217
218
Part One
Single-Equation Regression Models
7.15.
Regression through the origin.
Consider the following regression through the origin:
Y
i
= ˆ
β
2
X
2
i
+ ˆ
β
3
X
3
i
+ ˆ
u
i
a.
How would you go about estimating the unknowns?
b.
Will
ˆ
u
i
be zero for this model? Why or why not?
c.
Will
ˆ
u
i
X
2
i
=
ˆ
u
i
X
3
i
=
0 for this model?
d.
When would you use such a model?
e.
Can you generalize your results to the
k
-variable model?
(
Hint:
Follow the discussion for the two-variable case given in Chapter 6.)
*
I am indebted to Joe Walsh for collecting these data from a major wholesaler in the Detroit
metropolitan area and subsequently processing them.
TABLE 7.6
Quarterly Demand
for Roses in Metro
Detroit Area, from
1971-III to 1975-II
Year and
Quarter
Y
X
2
X
3
X
4
X
5
1971–III
11,484
2.26
3.49
158.11
1
–IV
9,348
2.54
2.85
173.36
2
1972–I
8,429
3.07
4.06
165.26
3
–II
10,079
2.91
3.64
172.92
4
–III
9,240
2.73
3.21
178.46
5
–IV
8,862
2.77
3.66
198.62
6
1973–I
6,216
3.59
3.76
186.28
7
–II
8,253
3.23
3.49
188.98
8
–III
8,038
2.60
3.13
180.49
9
–IV
7,476
2.89
3.20
183.33
10
1974–I
5,911
3.77
3.65
181.87
11
–II
7,950
3.64
3.60
185.00
12
–III
6,134
2.82
2.94
184.00
13
–IV
5,868
2.96
3.12
188.20
14
1975–I
3,160
4.24
3.58
175.67
15
–II
5,872
3.69
3.53
188.00
16
Do'stlaringiz bilan baham: |
