Midterm - Solution
1. QUESTION A1
The following estimated regression model provides evidence on the relationship between
age (age) and log of wage (log(wage)) after controlling for number of siblings (sibs). The
sample size is 935 and R2 = 0.048. Standard errors are presented in brackets ().
Regression equation here
“Because of the very low R2 (0.048) there is little evidence of a statistically significant
association between age and log(wage)”. This statement is
(a) True
(b) False X
2. QUESTION A2
The following regression model studies the effect of time spent in various activities on
CEOs’ wages. CEOs are asked how many hours they spend each week in three activities:
working, leisure, and sleeping. Any activity is put into one of the three categories, so
that for each CEO, the sum of hours in the three activities must be 168.
wage = β0 + β1work + β2leisure+ β3sleep+ u
Does this model violate any Gauss-Markov assumption?
(a) No, the model does not violate any Gauss-Markov assumption.
(b) The model violates the “no perfect collinearity” assumption. X
(c) The model violates the “zero conditional mean” assumption.
(d) The model violates the “normal distribution of the error term” assumption.
3. QUESTION A3
The following estimated regression model explains fertility represented by the total num-
ber of children born to a women (kids).
ˆkids = 3.012− 0.236 t educ+ 0.053 t age− 0.016 t agesq,
where t educ refers to education of the women in years minus 13,
t age refers to age of the women in years mius 43,
and t agesq is t age squared.
Interpret the constant term.
Answer:
βˆ0 = 3.012: The predicted mean number of children born to a woman aged 43 years old
with 13 years of education is 3 kids. (1 point)
4. QUESTION A4
1
Suppose you take a random sample of undergraduate students drawn from those enrolled
in the University of New South Wales and you find that students who regularly attend
tutorials get a mark 15% higher in the final exam than those who do not attend regularly.
Which of the following is NOT an appropriate conclusion to be drawn from this finding?
(a) Whether students attend tutorials or not will not be random and thus the finding
will likely to be biased.
(b) The evidence suggests there is a positive correlation between attending tutorials
and final-exam marks.
(c) Because the sample is random, it is likely that the result represents the causal effect
of attending tutorials on final-exam marks. X
(d) Whether students attend tutorial or not is likely to be highly correlated with observ-
able factors such as whether they are a foreign student or not, or whether they are
a hard-working student or not. If these factors were controlled for in the regression,
then the result could possibly be given a causal interpretation.
5. QUESTION A5
You have data on all the high school students in Australia on their math score (math)
(scores range 0 to 10) and their school per-student spending (expend) and you are inter-
ested in estimating the following population regression.
math = β0 + β1expend+ u
Suppose that you present your model to your friend, Federico, and he argues that it
is highly unlikely that the error term u is distributed like a normal distribution and
therefore you are not able to use OLS to test whether β > 0.
Is Federico right or wrong? Explain why.
Answer:
The assumption on normal distribution of the error term is required when doing hypothesis
testing for small samples. As we have a large sample size, there is no need to be concerned
about the distribution of the error term. We can use the asymptotic theory to make
inferences about OLS estimates. Therefore, Federico is wrong. (1 point)
The following regression model studies the effect of experience (exper) on log of wage
(log(wage)).
log(wage) = β0 + β1exper + β2exper
2 + β3educ + u (1)
Using the data from a sample of 82 workers to estimate the model, we get the following
result:
2
Regression equation here
where exper is experience, exper sq is experience squared, and educ is education.
QUESTION B4
What is the test statistic of the following test:
23.000± 0.1000 X
22.820± 0.1000 X
22.834± 0.1000 X
H0 : β1 < −1 vs. H1 : β1 > −1
(The numerical answers must be accurate to three decimal places.)
QUESTION B5
What is the 95% confidence interval of β3.
Write your answer in the following way: a; b, where a is the lower bound of the 95% con-
fidence interval and b is the upper bound of the 95% confidence interval. The numerical
answers must be accurate to three decimal places.
0.037; 0.127 X
0.036; 0.128 X
7. QUESTION B1
Provide one example of an omitted variable that would bias the OLS of β3. Describe in
which direction do you think the bias will go.
Answer:
One example of an omitted variable that would bias the OLS of β3: ability (IQ), family
income, mother’s education, father’s education, etc. (give students 0.5 point for any
reasonable example)
Ability (IQ)/ family income/ mother’s education/ father’s education, etc. is positively
correlated with education and wage. When one of these variables is omitted from the
regression, some of the estimated effect of educ is actually due to the effect of the omitted
variable. Therefore, we are likely to overestimate the effect of educ on wages, i.e. the bias is
positive. (0.5 point)
8. QUESTION B6
For this question assume that the Gauss-Markov assumptions from 1 to 4 are satisfied.
What is the effect of an extra year of experience on wages for an individual that has 10
years of experience and 12 years of education?
Write the answer in percentage term with two decimals places. (For example, if you
believe that the wages increase by 20.1321% write in the answer box 20.13)
• 2.34114 ± 0.1 X
3
• 2.27± 0.1 X
9. Answers for questions B4, B5, B6:
QUESTION B4:
tβˆ1 =
βˆ1 − (−1)
se(βˆ1)
=
−0.0113− (−1)
0.0433
= 22.834
4
QUESTION B5:
Degrees of freedom (df)= n - k - 1 = 82 - 3 - 1 = 78, thus from the Table, the critical
value is 1.987 (or 2.000).
Therefore, the confidence interval is:
[0.0819 - 1.987*0.0228; 0.0819 + 1.987*0.0228 ] = [0.037; 0.127]
(or [0.0819 - 2.000*0.0228; 0.0819 + 2.000*0.0228 ] = [0.036; 0.128] )
QUESTION B6:
The effect of an extra year of experience on log(wage) for an individual with 10 years of
experience is:
−0.0113 + 2 ∗ 0.0017 ∗ 10 = 0.0227. We can interpret this as a predicted increase in wage
of about 2.27%.
QUESTION B2:
One year increase in education leads to 8.19% increase in earnings.
QUESTION B3:
The variation in individual's education and experience explain 16.27% of variation in the natural
logarithm of individual earnings.
The critical value corresponding to n=82 is 1.990. Since the t-statistic is greater than the
critical value, we reject the Null Hypothesis.