29
CHAPTER
6
Exercise Answers
EXERCISE 6.3
(a)
Let the total variation, unexplained variation and explained variation be denoted by
SST
,
SSE
and
SSR
, respectively. Then, we have
42.8281
SSE
802.0243
SST
759.1962
SSR
(b)
A 95% confidence interval for
2
is
2
(0.975,17)
2
se( )
(0.2343,1.1639)
bt
b
A 95% confidence interval for
3
is
2
(0.975,17)
3
se( )
(1.3704, 2.1834)
b
(c)
To test
H
0
:
2
1 against the alternative
H
1
:
2
< 1, we calculate
1.3658
t
. Since
(0.05,17)
1.3658
1.740
t
, we fail to reject
0
H
. There is insufficient evidence to
conclude
2
1
.
(d)
To test
02 3
:0
H
against the alternative
12
H
and/or
3
0
, we calculate
151
F
. Since
(0.95,2,17)
151
3.59
F
, we reject
H
0
and conclude that the hypothesis
2
=
3
= 0 is not compatible with the data.
(e)
The
t
value for testing
023
:2
H
against the alternative
123
H
is
23
2
0.37862
0.634
se 2
0.59675
bb
t
Since
(0.025,17)
2.11
0.634
2.11
t
, we do not reject
0
H
. There is no evidence to
suggest that
2
.
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Principles of Econometrics, 4e
30
EXERCISE 6.5
(a)
The null and alternative hypotheses are:
02 4
3 5
12 4
:
and
:
or
or both
H
H
(b)
The restricted model assuming the null hypothesis is true is
22
14
5
6
ln(
)
(
)
(
)
WAGE
EDUC
EXPER
EDUC
EXPER
HRSWK
e
(c)
The
F
value is
70.32
F
.The critical value at a 5% significance level is
(0.95,2,994)
3.005
F
. Since the
F
value is greater than the critical value, we reject the null
hypothesis and conclude that education and experience have different effects on
ln(
)
WAGE
.
EXERCISE 6.10
(a)
The restricted and unrestricted least squares estimates and their standard errors appear in
the following table. The two sets of estimates are similar except for the noticeable
difference in sign for ln(
PL
). The positive restricted estimate 0.187 is more in line with our
a priori
views about the crossprice elasticity with respect to liquor than the negative
estimate
0.583. Most standard errors for the restricted estimates are less than their
counterparts for the unrestricted estimates, supporting the theoretical result that restricted
least squares estimates have lower variances.
CONST
ln(
PB
)
ln(
PL
)
ln(
PR
)
ln( )
I
Unrestricted
3.243
1.020
0.583
0.210
0.923
(3.743)
(0.239)
(0.560)
(0.080)
(0.416)
Restricted
4.798
1.299
0.187
0.167
0.946
(3.714)
(0.166)
(0.284)
(0.077)
(0.427)
(b)
The high auxiliary
2
s
R
and sample correlations between the explanatory variables that
appear in the following table suggest that collinearity could be a problem. The relatively
large standard error and the wrong sign for ln(
)
PL
are a likely consequence of this
correlation.
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 Winter '10
 leif
 Statistical hypothesis testing, standard errors, following table

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