The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

Critical values of 
R
2
when true 
R
2
0.
Equation (8.4.11) gave the relationship be-
tween 
F
and 
R
2
under the hypothesis that all partial slope coefficients are simultane-
ously equal to zero (i.e., 
R
2
=
0). Just as we can find the critical 
F
value at the 
α
level
of significance from the 
F
table, we can find the critical 
R
2
value from the following
relation:
R
2
=
(
k
−
1)
F
(
k
−
1)
F
+
(
n
−
k
)
where 
k
is the number of parameters in the regression model including the intercept
and where 
F
is the critical 
F
value at the 
α
level of significance. If the observed 
R
2
exceeds the critical 
R
2
obtained from the preceding formula, we can reject the
hypothesis that the true 
R
2
is zero.
Establish the preceding formula and find out the critical 
R
2
value (at 
α
=
5 per-
cent) for the regression (8.1.4).
8.7. From annual data for the years 1968–1987, the following regression results were 
obtained:
ˆ
Y
t
= −
859
.
92
+
0
.
6470
X
2
t
−
23
.
195
X
3
t
R
2
=
0
.
9776
(1)
ˆ
Y
t
= −
261
.
09
+
0
.
2452
X
2
t
R
2
=
0
.
9388
(2)
where 
Y
=
U.S. expenditure on imported goods, billions of 1982 dollars, 
X
2
=
per-
sonal disposable income, billions of 1982 dollars, and 
X
3
=
trend variable. 
True or
false:
The standard error of 
X
3
in (1) is 4.2750. Show your calculations. (
Hint:
Use
the relationship between 
R
2
, 
F
, and 
t
.)
8.8. Suppose in the regression
ln (
Y
i
/
X
2
i
)
=
α
1
+
α
2
ln
X
2
i
+
α
3
ln
X
3
i
+
u
i
the values of the regression coefficients and their standard errors are known.
*
From
this knowledge, how would you estimate the parameters and standard errors of the
following regression model?
ln
Y
i
=
β
1
+
β
2
ln
X
2
i
+
β
3
ln
X
3
i
+
u
i
8.9. Assume the following:
Y
i
=
β
1
+
β
2
X
2
i
+
β
3
X
3
i
+
β
4
X
2
i
X
3
i
+
u
i
where 
Y
is personal consumption expenditure, 
X
2
is personal income, and 
X
3
is per-
sonal wealth.
†
The term (
X
2
i
X
3
i
) is known as the 
interaction term.
What is meant
by this expression? How would you test the hypothesis that the marginal propensity
to consume (MPC) (i.e., 
β
2
) is independent of the wealth of the consumer?
*
Adapted from Peter Kennedy, 
A Guide to Econometrics,
the MIT Press, 3d ed., Cambridge, Mass.,
1992, p. 310.
†
Ibid., p. 327.
guj75772_ch08.qxd 12/08/2008 10:03 AM Page 263

---

264
Part One
Single-Equation Regression Models
8.10. You are given the following regression results:
ˆ
Y
t
=
16,899
−
2978.5
X
2
t
R
2
=
0.6149
t
=
(8.5152)
(
−
4.7280)
ˆ
Y
t
=
9734.2
−
3782.2
X
2
t
+
2815
X
3
t
R
2
=
0.7706
t
=
(3.3705)
(
−
6.6070)
(2.9712)
Can you find out the sample size underlying these results? (
Hint:
Recall the relation-
ship between 
R
2
, 
F
, and 
t
values.)
8.11. Based on our discussion of 
individual
and 
joint
tests of hypothesis based, respec-
tively, on the 
t
and 
F
tests, which of the following situations are likely?
1. Reject the joint null on the basis of the 
F
statistic, but do not reject each separate
null on the basis of the individual 
t
tests.
2. Reject the joint null on the basis of the 
F
statistic, reject one individual hypothe-
sis on the basis of the 
t
test, and do not reject the other individual hypotheses on
the basis of the 
t
test.
3. Reject the joint null hypothesis on the basis of the 
F
statistic, and reject each sep-
arate null hypothesis on the basis of the individual 
t
tests.
4. Do not reject the joint null on the basis of the 
F
statistic, and do not reject each
separate null on the basis of individual 
t
tests.
5. Do not reject the joint null on the basis of the 
F
statistic, reject one individual hy-
pothesis on the basis of a 
t
test, and do not reject the other individual hypotheses
on the basis of the 
t
test.
6. Do not reject the joint null on the basis of the 
F
statistic, but reject each separate
null on the basis of individual 
t
tests.
*
*
Quoted from Ernst R. Berndt, 
The Practice of Econometrics: Classic and Contemporary,
Addison-Wesley,
Reading, Mass., 1991, p. 79.
†
See Badi H. Baltagi, 
Econometrics,
Springer-Verlag, New York, 1998, p. 111.