Eco311, First Midterm Exam, Spring 2017
Prof. Bill Even
Your Name (Please print) ____KEY – FORM 1
Your row (Please circle)
1
2
3
4
5
Directions
Place your answers to all questions in the space provided. Organize your answer so it is easy for me to
see the logic behind your answer. Clearly circle your answer to each question so it can be easily
distinguished from your work. Round all numerical answers to the nearest 100th (e.g. 1.23) unless
specifically told otherwise. Each question is worth 5 points.
The formula sheet and a table with the standard normal CDF are attached to the last page of the exam.
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Eco311, First Midterm Exam, Spring 2017
Prof. Bill Even
To answer the next 3 questions, consider the following relationship between weekly earnings, age, and
age-squared that was estimated using data from the 2015 Current Population Survey.
Wkearn= -722 + 72*age - .72*age2
1. If a worker is 30 years old, use a derivative to calculate the marginal effect of one more year of
age on weekly earnings.
𝜕𝑤𝑘𝑒𝑎𝑟𝑛
= 72 − (2)(. 72)(30) = $28.80
𝜕𝑎𝑔𝑒
In the CPS data, workers can range between 16 and 85 years of age.
2.
What age maximizes weekly earnings? (Your answer must between 16 and 85 years of age).
𝜕𝑤𝑘𝑒𝑎𝑟𝑛
𝜕𝑎𝑔𝑒
= 72 − 2(. 72) ∗ 𝑎𝑔𝑒 = 0 → 𝑎𝑔𝑒 = 50.0
This is a maximum because
𝜕2 𝑤𝑘𝑒𝑎𝑟𝑛
𝜕𝑎𝑔𝑒 2
= −1.4 < 0 (𝑖. 𝑒. 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑐𝑜𝑛𝑐𝑎𝑣𝑒).
3. What age minimizes weekly earnings? (Your answer must between 16 and 85 years of age).
wkearn at age of 16=$246; at age of 85=$196. Hence, minimum must be at age of 85.
4. As an alternative to the linear model above, consider the following log-linear model of weekly
earnings:
ln(wkearn)=3.9 +.12*age-.001*age2
For a worker who is 30 years old, what is the marginal effect of an additional year of age on
weekly earnings (in percentage terms)? Be sure to include the percent sign in your answer so
that there is no ambiguity.
𝜕 ln(𝑤𝑘𝑒𝑎𝑟𝑛)
= .12 − .002 ∗ 𝑎𝑔𝑒 = .060 𝑎𝑡 𝑎𝑔𝑒 = 30
𝜕𝑎𝑔𝑒
→ 𝐴𝑛 𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑦𝑒𝑎𝑟 𝑜𝑓 𝑎𝑔𝑒 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑤𝑒𝑒𝑘𝑙𝑦 𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑏𝑦 6.0% 𝑎𝑡 𝑎𝑔𝑒 30.
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Eco311, First Midterm Exam, Spring 2017
Prof. Bill Even
To answer the next 2 questions, consider the following hypothetical data on the number of years of
schooling among adults and the corresponding probabilities of each level of schooling.
Years of
schooling
Probability
10
12
14
16
18
20
.1
.4
.1
.2
.1
.1
5. What is the expected value of years of schooling?
∑𝑝𝑖 𝑆𝑐ℎ𝑜𝑜𝑙𝑖 = 𝟏𝟒. 𝟐
6. What is the variance of years of schooling?
∑𝑝𝑖 (𝑠𝑐ℎ𝑜𝑜𝑙𝑖 − 14.2)2 = 𝟗. 𝟏𝟔
7. The “skewness” of the distribution of is measured as 𝐸[(𝑥 − 𝜇)3 ] where 𝜇 = 𝐸(𝑥).
Distributions with positive, negative, and zero skewness are given below.
What is the measure of skewnesss for years of schooling?
∑𝑝𝑖 (𝑠𝑐ℎ𝑜𝑜𝑙𝑖 − 14.2)3 = 𝟏𝟒. 𝟓𝟎
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Eco311, First Midterm Exam, Spring 2017
Prof. Bill Even
To answer the next 3 questions, use the following information. Based on an analysis of birthweight for
120 infants, the mean birth weight (in ounces) is 120. The standard deviation of birth weight is 20
ounces. Assume that birthweight has a normal distribution.
8. Provide an 85% confidence interval for birthweight.
𝟏𝟐𝟎 ±1.44*20=(91.2,148.8)
9. What is the probability that a randomly sampled infant would weigh between 6 and 7 pounds (i.e.
between 96 and 112 ounces)?
Pr(96<weight<112)=Pr[(96-120)/20 < z < (112-120)/20] = .23 [Z has distribution of N(0,1)
10. What is the probability that a randomly sampled infant would weight more than 9 pounds (i.e. 144
ounces)?
Pr[weight>144]=Pr[z>(144-120)/20]=.12
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Eco311, First Midterm Exam, Spring 2017
Prof. Bill Even
To answer the next 9 questions, consider the following data on campus crime obtained from 97
universities in 1992.
lcrimei: log(crimes) where crimesi is number of crimes reported at universityi during 1992.
lenroll: log(enrollment) where enrollmenti is the number of students enrolled at universityi in 1992.
. sum lcrime lenroll
Variable
Obs
Mean
lcrime
lenroll
97
97
5.277149
9.378556
Std. Dev.
1.381121
.8317719
Min
Max
0
7.494986
7.62657
10.93934
Using the above information, answer the following questions. Round answers to the nearest 100th (e.g.
123.45).
To answer the next 3 questions, consider the regression 𝑙𝑐𝑟𝑖𝑚𝑒𝑖 = 𝑏0 + 𝑏1 𝑙𝑒𝑛𝑟𝑜𝑙𝑙𝑖 + 𝑢𝑖 . Using the
information above and assuming that the variance of 𝒖𝒊 is .8, provide estimates of the following:
11.
𝑏̂1 =cov(lcrime,lenroll)/var(lenroll)=.878/.692= 1.27
12.
𝑏̂0 = 𝑙𝑐𝑟𝑖𝑚𝑒 − 𝑏̂1 ∗ 𝑙𝑒𝑛𝑟𝑜𝑙𝑙 = 5.277 − 1.27 ∗ 9.378= -6.63
13.
̂1 ) = 𝜎 2 /(𝑁 ∗ 𝑣𝑎𝑟(𝑙𝑒𝑛𝑟𝑜𝑙𝑙)=.8/*97*.692)= .01
𝑉𝑎𝑟(𝑏
14. Based on the estimates provided above, if campus enrollment increases from 20,000 to 21,000,
what is the predicted percentage change in crime? Be sure to use the percent sign in your answer to
avoid ambiguity.
if enrollment increases from 20,000 to 21,000, this is an increase of 5% (i.e. ln(enrollment increases by
approximately .05)). As a result, lcrime will increase by .05*1.27=.0635. Hence, crime increases by
6.35%.
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Eco311, First Midterm Exam, Spring 2017
Prof. Bill Even
As an alternative to the log-linear model above, consider the following linear model of campus crime
defined as 𝑐𝑟𝑖𝑚𝑒𝑖 = 𝑎0 + 𝑎1 𝑒𝑛𝑟𝑜𝑙𝑙𝑖 + 𝑒𝑖 where crimei is the number of crimes committed on campus i
in 1992 and enrolli is the number of students enrolled at campus I in 1992.
. reg crime enroll
Source
SS
df
MS
Model
Residual
14247032.6
6135857.43
1
95
14247032.6
64587.973
Total
20382890
96
212321.771
crime
Coef.
enroll
_cons
.0313226
-109.0989
Std. Err.
.002109
42.60723
t
14.85
-2.56
Number of obs
F(1, 95)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.012
=
=
=
=
=
=
97
220.58
0.0000
0.6990
0.6958
254.14
[95% Conf. Interval]
.0271357
-193.6849
.0355094
-24.51285
15. Based on the information provided for the linear model of crime, what is the predicted number of
crimes for Miami if its enrollment is 20,000 students?
-109.10+.0313*20,000=516.9 crimes
16. Based on the information provided for the linear model of crime, if actual crime at Miami is 400,
what is the prediction error (i.e. the estimated residual or error term) for Miami’s number of crimes? Be
sure to indicate whether the prediction error is positive or negative.
Prediction error=actual crime-predicted crime=400-516.9= -116.9
17. Based on the above linear regression, if enrollment increases from 20,000 to 21,000, what is the
expected percentage change in crime? Be sure to include a percent sign in your answer so there is no
ambiguity.
% change in crime = 100 ∗
Δ𝑐𝑟𝑖𝑚𝑒
𝑐𝑟𝑖𝑚𝑒
=
.0313∗1000
400
= .0783 = 7.83%
OR
% change in crime = 100 ∗
Δ𝑐𝑟𝑖𝑚𝑒
𝑐𝑟𝑖𝑚𝑒
=
.0313∗1000
516.9
= .0601 = 6.01%
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Eco311, First Midterm Exam, Spring 2017
Prof. Bill Even
18. Notice that R2 in the above regression is 0.699. Also, notice that the standard deviation of the
residual (root mean squared error) is 254.14. Using this information, what is the variance of crime
across campuses?
Defining 𝑦𝑖 as crime:
𝑆𝑆𝑅
̂2
∑𝑢
𝑛𝜎 2
𝜎2
𝑖
𝑅 2 = 1 − 𝑆𝑆𝑇 = 1 − ∑(𝑦 −𝑦)
2 = 1 − 𝑛𝑣𝑎𝑟(𝑦) → 𝑣𝑎𝑟(𝑦) = 1−𝑅 2 =
𝑖
254.142
1−.699
=214,575
19. Indicate whether each of the following would increase (+), decrease (-) , or not affect (0) the
variance of the estimated coefficient on enrollment.
a. an increase in the variance of enrollment b. an increase in the sample size for the regression c. an increase in the variance of the residual (error) in the regression +
20. Suppose that larger campuses are more likely to be in urban areas. Explain in a couple of
sentences how this could lead to a bias in the estimated coefficient on enrollment. Be sure to point out
which of the assumptions necessary for an unbiased estimator might be violated and how you
determined the direction of the bias.
Suppose that, ceteris paribus, urban campus have higher crime due to the larger population surrounding
campus. This causes cov(enrollment, u)>0 because urban campuses have higher crime (i.e. positive u)
and also have higher enrollment. This violates the conditional mean independence assumption and will
cause the estimated effect of enrollment to be biased upward.
Score _________ /100
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Eco311, First Midterm Exam, Spring 2017
Prof. Bill Even
1. Derivatives:
𝑑𝑦
=𝑏
𝑑𝑥
𝑑𝑦
𝐼𝑓 𝑦 = 𝑎 + 𝑏𝑥 𝑛 → 𝑑𝑥 = 𝑏𝑛𝑥 𝑛−1
𝑑𝑦
𝑏
𝐼𝑓 𝑦 = 𝑎 + 𝑏 ∗ ln(𝑥) → 𝑑𝑥 = 𝑥
dln(y)
𝐼𝑓 ln(y) = a + b ∗ ln(x) → dln(x)
a. 𝐼𝑓 𝑦 = 𝑎 + 𝑏𝑥 →
b.
c.
d.
dy
y
dx
x
=
%Δy
= %ΔX = b (i.e. b is an elasticity)
e. 𝐼𝑓 ln(𝑦) = 𝑎 + 𝑏 ∗ 𝑥
𝑑𝑦
𝑑𝑙𝑛(𝑦)
𝑦
→
=
𝑑𝑥
𝑑𝑥
→ %Δ𝑦=100*b∗ Δ𝑥
2. 𝑅𝑢𝑙𝑒𝑠 𝑓𝑜𝑟 𝑙𝑜𝑔 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠
a.
b.
c.
d.
ln(x1*x2)=ln(x1) +ln(x2)
ln(x1/x2)=ln(x1)-ln(x2)
ln(xc) = c*ln(x)
ln(1+x) ≈ x for “small” x.
3. E(aX+bY)=aE(X)+bE(Y)
4. Var(aX + bY) = a2Var(X) + b2Var(Y) +
2ab*cov(X,Y)
5. 𝑐𝑜𝑣(𝑎1 𝑋 + 𝑏1 , 𝑎2 𝑌 + 𝑏2 ) = 𝑎1 𝑎2 𝑐𝑜𝑣(𝑋, 𝑌)
6. Simple linear regression: 𝑦 = 𝛽0 + 𝛽1 𝑥𝑖 + 𝑢𝑖
a. 𝛽̂0 = 𝑦 − 𝛽̂1 𝑥
b. 𝛽̂1 = 𝐶𝑜𝑣(𝑥, 𝑦)/𝑉𝑎𝑟(𝑥)
c. Var(𝛽̂1 ) = 𝜎 2 /(𝑁 ∗ 𝑉𝑎𝑟(𝑥))
̂0 ) =
d. Var(𝛽
2
∑𝑁
𝑖=1(𝑥𝑖 /𝑁)
𝑁∗𝑉𝑎𝑟(𝑥)
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