Department of Economics
Fall 2007
University of California
Prof. Woroch
Economics 140:
PROBLEM SET 3 ANSWER SHEET
1
In Problem Set 2, you estimated the returns to education by running a regression of ln(wage) on
years of education. When responding to the questions below, keep in mind that when the dependent
variable is expressed in natural logs, its derivative with respect to an explanatory variable can be
interpreted as the percentage change in the dependent variable associated with an increase of 1 unit
in the explanatory variable.
Formally,
in x.
increase
unit

one
a
from
resulting
y
in
change
percentage
1
)
ln(
=
=
dx
dy
y
dx
y
d
a)
You suspect that the condition E[
u
i

X
] = 0 in the model
i
i
i
u
edu
wage
+
+
=
1
0
)
ln(
β
may not be
satisfied. Think of reasons why this might be the case (you don’t have to write them). What
would be the consequence of violating this assumption on your OLS estimate of
1
?
Answer
: The consequence of violating E[
u
i

X
] = 0 will be a biased OLS estimate of
1
in both small
and large samples.
The direction of the bias depends on the sign of the correlation between X and u.
b)
Someone suggests to you that job tenure might be an important reason why E[
u
i

X
] = 0
is
violated.
People with more job experience tend to have higher wages. You, of course, had
already thought of it in part (a).
i)
What do you think is the sign of the correlation between years of schooling and tenure?
ii)
If the effect of tenure on wages is an important part of
u
i
,
what do you think will be the sign
of cov(
u,edu
)?
iii)
Write an expression for the
]
ˆ
[
1
E
that allows you to predict whether there is any bias
induced by tenure on your OLS estimate of
1
which is denoted by
1
ˆ
.
What is the direction
of the bias in
1
induced by tenure (if any)?
Answer:
i)
Generally, remaining in school for one year more delays job market entry by one year.
Hence, job tenure and schooling for a given age cohort should be negatively correlated.
ii)
Since longer job tenure is associated with higher wages and since education is negatively
correlated with tenure, the sign of cov(
u, edu
) should be negative.
iii)
1
1
1
1
)
var(
)
,
cov(
)
var(
)
,
cov(
)
,
cov(
)
var(
)
,
cov(ln
]
ˆ
[
<
+
=
+
=
=
edu
edu
u
edu
edu
u
edu
edu
edu
edu
wage
E
Since Cov(
u, edu
) < 0,
1
ˆ
is biased downwards from the true slope value.
c)
A way to confirm your point in (b), is to estimate a regression that controls for years of
schooling and job tenure:
i
i
i
i
e
tenure
edu
wage
+
+
+
=
2
1
0
)
ln(
γ
. How do you think
1
ˆ
will
compare to
1
ˆ
if your point in (b) is correct?
Answer
: It should be the case that
1
ˆ
>
1
ˆ
.
More precisely,
edu
u
u
edu
σ
ρ
,
1
1
ˆ
+
→
from Equation
(6.1).
Above we reasoned that
0
,
<
u
edu
and so the OLSE underestimates the true value.
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d)
When looking again at the data on education and wages from PS2 (i.e.,
PS2 Excel file
(ch8_cps_500
).xls) you notice that your strategy in (c) is hard to implement since you don’t
have a
tenure
variable. However, you do have the variable
age.
If you make the strong
assumption that everyone starts working when they finish school, then you can construct an
approximation to
tenure
using this
age
and
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 Fall '08
 DUNCAN
 Economics, Regression Analysis, Standard Error, Statistical hypothesis testing, ACRA, Regression Statistics RSquared

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