Econ 120B
Solutions to Chapters 4 and 5 Practice Problems
1.
a.
For every one inch increase in the average height of their parents, the student’s height increases, on
average, by 0.73 of an inch. There is no reasonable interpretation for the intercept (the average height
of a student whose parents have an average height of zero!).
b.
45 percent of the variation in the height of students can be explained by the model, i.e., can be
explained by the average height of the parents. The
SER
is a measure of the spread of the
observations around the regression line. The magnitude of the typical deviation from the regression
line or the typical regression error here is two inches.
c.
19.6 + 0.73
×
70.06 = 70.74.
d.
Tall parents will have, on average, tall children, but they will not be as tall as their parents. Short
parents will have short children, although on average, they will be somewhat taller than their parents.
e.
0
:
1
0
=
β
H
,
t
=7.30, for
0
:
1
1
>
β
H
, the critical value for a two-sided alternative is 1.645. Hence we
reject the null hypothesis.
f.
For the simple linear regression model,
0
:
1
0
=
β
H
implies that
= 0.
Hence it is the same test as
in (e).
2
R
g.
0
:
0
0
=
β
H
,
t
=2.72, for
0
:
0
1
≠
β
H
, the critical value for a two-sided alternative is 2.58. Hence we
reject the null hypothesis in (i). For the slope we have
1
:
1
0
=
β
H
,
t
=-2.70, for
1
:
1
1
≠
β
H
, the
critical value for a two-sided alternative is 1.96. Hence we reject the null hypothesis in (ii).
h.
(0.73x5 – 1.96
×
0.10x5, 0.73x5 + 1.96
×
0.10x5) = (0.53x5, 0.93x5) = (2.65, 4.65)
i.
This is an example of mean reversion. Since the aristocracy was, on average, taller, he was concerned
that their children would be shorter and resemble more the rest of the population. If this conclusion
were true, then eventually everyone would be of the same height. However, we have not observed a
decrease in the variance in height over time.
2.
a.
Interpretation of slope coefficient: A relative increase in the population rate of one percentage point,
from 0.01 to 0.02, say, lowers relative per-capita income by almost 20 percentage points (0.188), on
average. This is a quantitatively important and large effect.
Interpretation of the intercept: Nations which have the same population growth rate as the United
States have, on average, roughly half as much per capita income.
Around 52% of the variation in relative per capita income can be explained by the variations in the
relative population growth rate. The magnitude of a typical deviation from the regression line is about
0.2 (20 percentage points).
The
t
-statistic is 5.93, making the relationship statistically significant, i.e., we can reject the null
hypothesis that the slope is different from zero.
b.
The interpretation of the partial derivative is unaffected, in that the slope still indicates the effect of a
one percentage point increase in the population growth rate. The regression
R
2
will remain the same
since only a constant was removed from the explanatory variable. The intercept will change as a
result of the change in
X
. It does not have any effect in the estimate of the slope and on the standard
errors of the slope, which means that it has no effect on the t-stat of the slope.