Econ 120B
Solutions to Chapters 6 and 7 Practice Problems
1.
There is perfect multicollinearity present since one of the three explanatory variables can always be expressed
linearly in terms of the other two (
). Hence, there are not really three pieces of independent
information contained in the three explanatory variables. Dropping one of the three will solve the problem.
X
2
=
X
1
−
X
3
2.
Note: There are more than one answer to part a., and c.
a.
We add and subtract
β
2
+
3
()
X
1
i
Y
i
=
0
+
1
X
1
i
+
2
+
3
X
1
i
−
2
+
3
(
)
X
1
i
+
2
X
2
i
+
3
X
3
i
+
u
i
Y
i
=
0
+
1
+
2
+
3
X
1
i
+
2
X
2
i
−
X
1
i
+
3
X
3
i
−
X
1
i
+
u
i
Y
i
=
0
+
γ
1
X
1
i
+
2
W
i
+
3
V
i
+
u
i
where
,
W
1
=
1
+
2
+
3
i
=
X
2
i
−
X
1
i
(
)
, and
V
i
=
X
3
i
−
X
1
i
(
)
We regress the last equation and obtain the OLS estimators:
)
β
0
,
)
γ
1
,
)
2
,
)
3
We test the following hypothesis,
H
0
:
1
=
1
H
1
:
1
≠
1
where
t
=
)
1
−
1
SE
(
)
1
)
b.
This is not a linear restriction. Hence, you cannot use the
t
-test to test for its validity.
c.
We add and subtract
3
3
X
2
i
Y
i
=
0
+
1
X
1
i
+
2
X
2
i
−
3
3
X
2
i
+
3
3
X
2
i
+
3
X
3
i
+
u
i
Y
i
=
0
+
1
X
1
i
+
2
−
3
3
X
2
i
+
3
3
X
2
i
+
X
3
i
+
u
i
Y
i
=
0
+
1
X
1
i
+
2
X
2
i
+
3
W
i
+
u
i
where
, and
2
=
2
−
3
3
W
i
=
3
X
2
i
+
X
3
i
(
)
We regress the last equation and obtain the OLS estimators:
)
0
,
)
1
,
)
2
,
)
3
We test the following hypothesis,
H
0
:
2
=
2
H
1
:
2
≠
2
where
t
=
)
2
−
2
SE
(
)
2
)
3.
a.
For every additional inch in height, weight increases on average by 5.58 pounds, holding gender
constant. Female students weigh on average 6.36 pounds less than male students, controlling for
height. The regression explains 50 percent of the weight variation among students. It does not make
sense to interpret the intercept, since there are no observations close to the origin, or, put differently,
there are no male individuals who are zero inches tall.
b.
There are now observations close to the origin and you can therefore interpret the intercept. A male
student who is 5ft. tall will weight roughly 105.6 pounds, on average. The two slope coefficients and
their t-statistics will be unaffected, as will be the regression
R
2
. Since the explanatory power of the
regression is unaffected by rescaling, and the dependent variable and the total sums of squares have
remained unchanged, the sums of squared residuals, and hence the
SER
, must also remain the same.