University of British Columbia
Vancouver School of Economics
Econ 325: Introduction to Empirical Methods
Instructor:
M. Emrul Hasan, CFA, FRM, PhD
Problem Set 1
Due on Monday (10 am PST), January 31st, 2022, on Canvas
Instructions: These questions are based on Chapter 1, 2 and 3 of the textbook Newbold et.
al., Statistics for business and economics, 8th Edition, Pearson, and the class lecture notes. Do your
best to explain and make your arguments as rigorous as possible using any resources you want
by keeping in mind the UBC policy of academic dishonesty and plagiarism. Do not copy from
other groups, any online resources, or the solution manuals. You can ask any general course
and topic related questions at any time on Piazza and to the TA.
This is a group work. All the students are randomly divided into groups of 4 to 5. The groups
may change later. We are calling the first set of groups “First Set”. Canvas will also choose a
group leader randomly. The leader will organize the meetings and works among the group
members. If the TA or I have any question, we will always the group leader first. If you do not
want to be the group leader, ask the TA. The TA can always change the leader.
Work on all the questions independently and then discuss with the group members. There will
be only one group submission. Please do not divide the questions among group members. This
may lead to academic misconduct and the whole group will suffer the consequences. Similar
questions will also show up in your exams.
Assignment questions below may come from back of the chapter, relevant articles, and from
other data sources. We know that some students may have unfair advantage by having access
to the solution manual of the book only meant for the teachers or by taking answers from
previous students. So, we will make sure any answer that is same as the solution manual or
assignments from previous terms and groups will get zero. The group will also be reported to
the academic misconduct officer of VSE.
Do not forget to complete the ipeer review. While completing the problem set or right after
submitting the problem set, go to ipeer and complete the peer review right away. Check Canvas
to find out how to complete peer review using ipeer. If you miss ipeer, there will be no second
chance no matter what’s the reason.
Submission:
• Unless otherwise stated, answers should be handwritten only.
• Submit one scanned softcopy (PDF version) per group on Canvas before the due date and
time.
1
Problem 1
Download the data and the data description titled “CPS96_15”. You can use excel or any other
statistical software to answer the following questions (hint: find Descriptive Statistics)
a) What is the mean, median, mode, and standard deviation of the average hourly earnings
and age of the whole sample in the dataset? What is the sample size?
b) Find the correlation matrix among all the variables except year. What is the correlation
between average hourly earnings and age from the correlation matrix? What does the
matrix tell you about the relationship(s) between:
a. Average hourly earnings and education
b. Average hourly earnings and gender
c. Average hourly earnings and age
Problem 2 (Software Output + handwritten)
Problem 3 (Software Output + handwritten)
2
Problem 4
Problem 5
3
Problem 6
4
Problem 7
5
A harmonic mean of €
76.48.
geometric
€
77.26.
TheBaverage
price mean
is bestofrepresented
as the:
B geometric mean of €
77.26.
C arithmetic
of €
78.00.
A harmonic
meanaverage
of €
76.48.
C arithmetic average of €
78.00.
B geometric mean of €
77.26.
C
arithmetic average of €
78.00.
The following
information relates to Questions
Problem
8
The
following
information relates to Questions
27–28and explain the correct answers for question I and II below.
Choose
27–28
The
following information relates to Questions
The following exhibit shows the annual MSCI World Index total returns for a 10-year
27–28
Theperiod.
following exhibit shows the annual MSCI World Index total returns for a 10-year
period.
The following
annual MSCI World Index
total
a 10-year
Year 1 exhibit shows the
15.25%
Year
6 returns for
30.79%
Year 1
15.25%
Year 6
30.79%
period.
Year 2
10.02%
Year 7
12.34%
Year 2
YearYear
1 3
Year 3
YearYear
2 4
Year 4
Year
Year 3 5
Year 5
Year 4
10.02%
20.65%
15.25%
20.65%
9.57%
10.02%
9.57%
−40.33%
20.65%
−40.33%
9.57%
Year 7
YearYear
6 8
Year 8
YearYear
7 9
Year 9
Year
Year 8 10
Year 10
Year 9
12.34%
−5.02%
30.79%
−5.02%
16.54%
12.34%
16.54%
27.37%
−5.02%
27.37%
16.54%
(I)
27 The fourth quintile return for the MSCI World Index is closest to:
−40.33%
Year 10
27 Year
The 5fourth quintile return
for the MSCI World Index
is closest to: 27.37%
A 20.65%.
A 20.65%.
26.03%.
27 TheBfourth
quintile return for the MSCI World Index is closest to:
B 26.03%.
C 27.37%.
A 20.65%.
C 27.37%.
For Year 6–Year 10, the mean absolute deviation of the MSCI World Index total
B28 26.03%.
28 For returns
Year 6–Year 10, the mean absolute deviation of the MSCI World Index total
C 27.37%. is closest to:
(II)
returns is closest to:
10.20%.
28 For AYear
6–Year 10, the mean absolute deviation of the MSCI World Index total
A 10.20%.
B
12.74%.
returns is closest to:
B 12.74%.
C 16.40%.
A 10.20%.
C 16.40%.
B 12.74%.
C 16.40%.
29 Annual returns and summary statistics for three funds are listed in the follow29 Annual
returns and summary statistics for three funds are listed in the following exhibit:
ing exhibit:
29 Annual returns and summary statistics for threeAnnual
funds Returns
are listed
in the follow(%)
Annual
Returns
(%)
ingYear
exhibit:
Fund ABC
Fund XYZ
Fund PQR
Year
Year 1
Year 1
Year
Year 2
Year 2
YearYear
1 3
Year 3
Year
Year 2 4
Year 4
YearYear
3 5
Year 5
Year 4
Year 5
Fund ABC
Fund XYZ
−20.0 Annual Returns
−33.0 (%)
−20.0
−33.0
Fund
ABC
Fund
XYZ
23.0
−12.0
23.0
−12.0
−14.0
−12.0
−20.0
−33.0
−14.0
−12.0
5.0
23.0
−12.0−8.0
5.0
−8.0
−14.0
−14.0
−12.011.0
−14.0
11.0
5.0
−8.0
−14.0
11.0
Fund PQR
−14.0
−14.0
Fund
PQR
−18.0
−18.0
−14.0 6.0
6.0
−18.0−2.0
−2.0
6.0 3.0
3.0
−2.0
3.0
6
impressive because the probability that a firm is a tech firm (from the whole sample)
is only 0.20. In sum, it can be readily seen from the tree map and the underlying
frequency data (Exhibits 19 and 20, respectively) or from the probabilities in Bayes’
formula that there are 160 firms with R > 10%, and 60 of them are tech firms, so P(tech
| R > 10%) = 60/160 = .375.
Problem
9 of Bayesian statistics do not consider probabilities (or likelihoods) to be
Users
known with certainty but that these should be subject to modification whenever new
Case Study: Application of Bayes’ Theorem
information
becomes available.
orthe
probabilities
are continually updated
Answer
the 4 questions
at the endOur
afterbeliefs
reading
vignette below.
as new information arrives over time.
To further illustrate Bayes’ formula, we work through an investment example that
can be adapted to any actual problem. Suppose you are an investor in the stock of
DriveMed, Inc. Positive earnings surprises relative to consensus EPS estimates often
result in positive stock returns, and negative surprises often have the opposite effect.
DriveMed is preparing to release last quarter’s EPS result, and you are interested in
which of these three events happened: last quarter’s EPS exceeded the consensus EPS
estimate, last quarter’s EPS exactly met the consensus EPS estimate, or last quarter’s
EPS fell short of the consensus EPS estimate. This list of the alternatives is mutually
exclusive and exhaustive.
On the basis of your own research, you write down the following prior probabilities (or priors, for short) concerning these three events:
■
P(EPS exceeded consensus) = 0.45
■
P(EPS met consensus) = 0.30
■
P(EPS fell short of consensus) = 0.25
These probabilities are “prior” in the sense that they reflect only what you know now,
before the arrival of any new information.
The next day, DriveMed announces that it is expanding factory capacity in Singapore
and Ireland to meet increased sales demand. You assess this new information. The
decision to expand capacity relates not only to current demand but probably also to
the prior quarter’s sales demand. You know that sales demand is positively related to
EPS. So now it appears more likely that last quarter’s EPS will exceed the consensus.
The question you have is, “In light of the new information, what is the updated
probability that the prior quarter’s EPS exceeded the consensus estimate?”
Bayes’ formula provides a rational method for accomplishing this updating. We
can abbreviate the new information as DriveMed expands. The first step in applying
Bayes’ formula is to calculate the probability of the new information (here: DriveMed
expands), given a list of events or scenarios that may have generated it. The list of
events should cover all possibilities, as it does here. Formulating these conditional
probabilities is the key step in the updating process. Suppose your view, based on
research of DriveMed and its industry, is
P(DriveMed expands | EPS exceeded consensus) = 0.75
P(DriveMed expands | EPS met consensus) = 0.20
P(DriveMed expands | EPS fell short of consensus) = 0.05
Conditional probabilities of an observation (here: DriveMed expands) are sometimes
referred to as likelihoods. Again, likelihoods are required for updating the probability.
7
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Bayes'
Bayes'Formula
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Next,
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youcombine
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theseconditional
conditionalprobabilities
probabilitiesor
orlikelihoods
likelihoodswith
withyour
yourprior
prior
probabilities
probabilitiestotoget
getthe
theunconditional
unconditionalprobability
probabilityfor
forDriveMed
DriveMedexpanding,
expanding,P(DriveMed
P(DriveMed
expands),
expands),asasfollows:
follows:
PP!DriveMed
expands" "
!DriveMedexpands
## PP!DriveMed
DriveMed expands
expands| |EPS
EPSexceeded
exceededconsens
consensuus"s
!
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%%PP!DriveMed
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DriveMed expands
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onsensus"
!
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"
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%%PP!DriveMed
EPSfefellllshort
DriveMed expands
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!
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by
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question by applying Bayes’ formula:
PP!EPS
exceededconsensus
consensus| |DriveMed
DriveMedexpands
expands" "
!EPSexceeded
P !DriveMed expands | EPS exceeded consensus"
## P !DriveMed expands | EPS exceeded consensus"PP!EPS
consensus" "
exceedeeddconsensus
!EPSexceed
PP!DriveMed
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"
!DriveMed expands"
## !00.75
.7500.41
.41"!00.45
.45" # # 1.829268
1.829268!0.45
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Prior
PriortotoDriveMed’s
DriveMed’sannouncement,
announcement,you
youthought
thoughtthe
theprobability
probabilitythat
thatDriveMed
DriveMedwould
would
beat
consensus
expectations
was
45%.
On
the
basis
of
your
interpretation
of
beat consensus expectations was 45%. On the basis of your interpretation ofthe
the
announcement,
you
update
that
probability
to
82.3%.
This
updated
probability
is
called
announcement, you update that probability to 82.3%. This updated probability is called
your
yourposterior
posteriorprobability
probabilitybecause
becauseititreflects
reflectsor
orcomes
comesafter
afterthe
thenew
newinformation.
information.
The
Bayes’
calculation
takes
the
prior
probability,
which
was
45%,
and
The Bayes’ calculation takes the prior probability, which was 45%, andmultiplies
multiplies
ititby
byaaratio—the
ratio—thefirst
firstterm
termon
onthe
therightright-hand
handside
sideofofthe
theequal
equalsign.
sign.The
Thedenominator
denominator
ofofthe
theratio
ratioisisthe
theprobability
probabilitythat
thatDriveMed
DriveMedexpands,
expands,asasyou
youview
viewititwithout
withoutconsiderconsidering
(conditioning
on)
anything
else.
Therefore,
this
probability
is
unconditional.
ing (conditioning on) anything else. Therefore, this probability is unconditional.The
The
numerator
numeratorisisthe
theprobability
probabilitythat
thatDriveMed
DriveMedexpands,
expands,ififlast
lastquarter’s
quarter’sEPS
EPSactually
actually
exceeded
exceededthe
theconsensus
consensusestimate.
estimate.This
Thislast
lastprobability
probabilityisislarger
largerthan
thanunconditional
unconditional
probability
probabilityininthe
thedenominator,
denominator,so
sothe
theratio
ratio(1.83
(1.83roughly)
roughly)isisgreater
greaterthan
than1.1.As
Asaaresult,
result,
your
updated
or
posterior
probability
is
larger
than
your
prior
probability.
Thus,
your updated or posterior probability is larger than your prior probability. Thus,the
the
ratio
reflects
the
impact
of
the
new
information
on
your
prior
beliefs.
ratio reflects the impact of the new information on your prior beliefs.
EXAMPLE
EXAMPLE15
15
Inferring
InferringWhether
WhetherDriveMed’s
DriveMed’sEPS
EPSMet
MetConsensus
ConsensusEPS
EPS
You
Youare
arestill
stillan
aninvestor
investorininDriveMed
DriveMedstock.
stock.To
Toreview
reviewthe
thegivens,
givens,your
yourprior
prior
probabilities
are
P(EPS
exceeded
consensus)
=
0.45,
P(EPS
met
consensus)
probabilities are P(EPS exceeded consensus) = 0.45, P(EPS met consensus)==0.30,
0.30,
and
P(EPS
fell
short
of
consensus)
=
0.25.
You
also
have
the
following
conditional
and P(EPS fell short of consensus) = 0.25. You also have the following conditional
probabilities:
probabilities:
P(DriveMed
P(DriveMedexpands
expands| |EPS
EPSexceeded
exceededconsensus)
consensus)==0.75
0.75
P(DriveMed
P(DriveMedexpands
expands| |EPS
EPSmet
metconsensus)
consensus)==0.20
0.20
P(DriveMed
P(DriveMedexpands
expands| |EPS
EPSfell
fellshort
shortofofconsensus)
consensus)==0.05
0.05
Recall
Recallthat
thatyou
youupdated
updatedyour
yourprobability
probabilitythat
thatlast
lastquarter’s
quarter’sEPS
EPSexceeded
exceededthe
the
8
your updated or posterior probability is larger than your prior probability. Thus, the
ratio reflects the impact of the new information on your prior beliefs.
EXAMPLE 15
Inferring Whether DriveMed’s EPS Met Consensus EPS
You are still an investor in DriveMed stock. To review the givens, your prior
probabilities are P(EPS exceeded consensus) = 0.45, P(EPS met consensus) = 0.30,
and P(EPS fell short of consensus) = 0.25. You also have the following conditional
probabilities:
P(DriveMed expands | EPS exceeded consensus) = 0.75
P(DriveMed expands | EPS met consensus) = 0.20
P(DriveMed expands | EPS fell short of consensus) = 0.05
Recall that you updated your probability that last quarter’s EPS exceeded the
consensus estimate from 45% to 82.3% after DriveMed announced it would
© CFA
Institute.
For candidate
use your
only.other
Not for
distribution.
expand.
Now you
want to update
priors.
Reading 3 ■ Probability Concepts
1
Update your prior probability that DriveMed’s EPS met consensus.
2
Update your prior probability that DriveMed’s EPS fell short of consensus.
3
Show that the three updated probabilities sum to 1. (Carry each probability to four decimal places.)
4
Suppose, because of lack of prior beliefs about whether DriveMed would
meet consensus, you updated on the basis of prior probabilities that all
three possibilities were equally likely: P(EPS exceeded consensus) = P(EPS
met consensus) = P(EPS fell short of consensus) = 1/3.
What is your estimate of the probability P(EPS exceeded consensus | DriveMed
expands)?
Solution to 1:
The probability is P(EPS met consensus | DriveMed expands) =
P !DriveMed expands | EPS met consensus"
P !DriveMed expands"
P !EPS met consensus"
The probability P(DriveMed expands) is found by taking each of the three conditional probabilities in the statement of the problem, such as P(DriveMed expands
| EPS exceeded consensus); multiplying each one by the prior probability of the
conditioning event, such as P(EPS exceeded consensus); then adding the three
products. The calculation is unchanged from the problem in the text above:
P(DriveMed expands) = 0.75(0.45) + 0.20(0.30) + 0.05(0.25) = 0.41, or 41%. The
other probabilities needed, P(DriveMed expands | EPS met consensus) = 0.20
and P(EPS met consensus) = 0.30, are givens. So
P(EPS met consensus | DriveMed expands)
= [P(DriveMed expands | EPS met consensus)/P(DriveMed expands)]P(EPS
met consensus)
= (0.20/0.41)(0.30) = 0.487805(0.30) = 0.146341
After taking account of the announcement on expansion, your updated probability that last quarter’s EPS for DriveMed just met consensus is 14.6% compared
with your prior probability of 30%.
Solution to 2:
P(DriveMed expands) was already calculated as 41%. Recall that P(DriveMed
expands | EPS fell short of consensus) = 0.05 and P(EPS fell short of consensus)
= 0.25 are givens.
9