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2206
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Economics
Date
Apr 3, 2024
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2206 PROBLEM SET 1
LILY EDWARDS BLASIKIEWICZ - Z5418961
TUTORIAL MONDAY 9:00
Note -
I will be rounding all decimals to two decimal places for the final answer, but all computation
will use the unrounded values given by STATA.
Question 1
Given this output from regressing wage on education, we get the following model:
\
Wage
= 1
.
57 + 0
.
36
Educ,
N
= 1260
, R
2
= 0
.
0451
The estimated coefficient
c
α
1
= 0
.
36 implies that an extra year of education is associated with an increase
in that individuals hourly wage by
$
0.36 dollars.
Question 2
Using a log-linear model:
Date
: March 2024.
1
2
LILY EDWARDS BLASIKIEWICZ - Z5418961
TUTORIAL MONDAY 9:00
We predict this model for Wage:
\
ln
Wage
= 0
.
90 + 0
.
06
Educ,
N
= 1260
, R
2
= 0
.
0708
The estimated coefficient
c
β
1
= 0
.
06 implies that an extra year of education is associated with an increase
in that individuals hourly wage of 6%.
Question 3
Linear regression models must be linear in the parameters, hence (a), (c), and (d) are not linear
regressions models, whereas (b) is.
Question 4
The Zero Conditional Mean assumption in mathematical terms essentially says that the expected value
of the error term must have no reliance on any of the independent variables, i.e.
E
(
U
|
X
i
) = 0. In less
mathematical terms, there must be no confounding factors in the error term, else we don’t have an un-
biased estimator.
If we are to look at model (1), the ZCM assumption assumes that the error term
U
has no reliance
on education. It’s quite easy to see why ZCM may not hold in this case due to the fact that the model
explains so little, an example that would violate the ZCM, could be parental income, which has a great
effect on the quality of education an individual might receive, as well as the years of study that individual
is able to complete. If parental income is low, the child may be less capable of for university or being
supported financially, and hence might enter the workforce after completing high school, rather than
getting a university degree.
This would violate ZCM due to parental income being omitted from the
model, and education having some correlation with parental income.
If ZCM doesn’t hold, then
b
(
α
1
) would be influenced by the omitted variable and would hence be biased.
2206 PROBLEM SET 1
3
Question 5
a.
Hence the predicted model is:
\
ln
Wage
= 0
.
37 + 0
.
08
Educ
+ 0
.
22
Noncog,
N
= 1260
, R
2
= 0
.
2022
b.
The estimated coefficient
b
γ
1
= 0
.
08 implies that an extra year of education is associated with an
increase in that individuals hourly wage by 8%.
The estimated coefficient
b
γ
2
= 0
.
22 implies that one
extra non-cognitive skill is associated with an increase in that individuals hourly wage by 22%.
c.
Evidently considering model (2) has
β
1
= 0
.
06 compared to model (3)’s
b
γ
1
= 0
.
08, shows that model
(2) likely had omitted variables, it also shows that too little weight was given to education in the second
model when compared to the third model.
d.
The bias for
β
1
is negative, model (2) essentially tells us that education is less important as a factor of
hourly wage when there are no other independent variables considered. When you consider non-cognitive
skills as well, education matters more. If we average across people with a similar number of non-cognitive
skills, education has a higher weight, but when averaging across all the different possible numbers of
non-cognitive skills (what model (2) does by only considering education) education happens to matters
less, implying a negative bias.
We can also check this using the formula
E
(
ˆ
β
1
)
−
ˆ
γ
1
= ˆ
γ
2
ˆ
δ
1
, we know
E
(
ˆ
β
1
)
−
ˆ
γ
1
= 0
.
0602839
−
0
.
0758434 which is negative.
e.
Education and non-cognitive skills are negatively correlated, which we can see from the previous
question and the relationship between bias and correlation.
E
(
ˆ
β
1
)
−
ˆ
γ
1
=
ˆ
γ
2
ˆ
δ
1
=
⇒
−
0
.
0155595 =
0
.
2199816
·
ˆ
δ
1
and hence
ˆ
δ
1
≈ −
0
.
07.
Question 6
To check whether education is statistically significant at the 1% significance level. We first define the
null and alternate hypotheses:
H
0
:
γ
1
= 0
,
H
1
:
γ
1
̸
= 0
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