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Essentials of Econometrics
ECNM10052
Monday 18 December 2023
14:30:00–16:30:00
Number of questions: 3
Total number of marks: 100
IMPORTANT PLEASE READ CAREFULLY
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2.Complete PART A and PART B above. By completing PART B you are accepting the
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2.This paper has 2 sections. Section A (worth 80%) has two questions. Answer BOTH
questions. Section B (worth 20%) has ten multiple choice questions. Answer all TEN
questions. For each exercise in the multiple choice part, there is only one correct
answer, and please put down one answer only. If you mark no answer, you will be
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answer the multiple choice questions on the multiple choice question sheet provided.
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At the end of the examination
1. This examination script must not be removed from the examination venue.
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script. Write your examination number on the top of each additional sheet.
Examiners:Prof. Jesper Bagger (Chair), Prof. Sarah Jewell(External)
ECNM10052Do not write above this lineDecember 2023
Section A
1. You want to evaluate a retraining programme that was offered to unemployed workers in the UK
in 2021. Participation was completely voluntary. You have assembled data on 2,000 workers that
were unemployed at the start of the programme in 2021. You have data on their programme
participation in 2021 (train, a dummy variable equalling 1 for those who participated), and data
on their total earnings in 2022 (earn, measured in GBP. Total earnings is the sum of wage earnings,
jobseeker’s allowance and any other income the people might have had). You start with the simple
bivariate regression (Model 1):
log(earn) =β
0
+β
1
train+u
(a) What kinds of factors are contained in u? Are these likely to be correlated with training
participation?[4 marks]
If you have used additional space for working then please tick here:
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(b) What would be the consequences for your OLS estimator ofβ
1
ifuand training participation
were correlated?[3 marks]
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You instead augment the model by controlling for a dummy for being female (female), age
(age, measured in years), education (educ, measured in years), and a dummy for whether the
person had been unemployed for longer than 6 months at the beginning of 2021 (longterm).
You assume that for this model, assumptions MLR1-MLR4 hold. The regression model for
Model 2is:
log(earn) =β
0
+β
1
train+β
2
female+β
3
age+β
4
educ+β
5
longterm+u
You get the following results:
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(c) Perform a hypothesis test of the null hypothesis that programme participation had no effect
on earnings against the two-sided alternative at the 5 percent level of significance. Are the
returns to programme participation statistically significantly different from zero?[3 marks]
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(d) Your dataset also contains the variableunem22, which codes the number of months spent
unemployed during the year 2022. A colleague of yours suggests to include this variable as
additional explanatory variable in Model 2. Discuss this proposal.[4 marks]
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If you have used additional space for working then please tick here:
(e) You are worried that assumption MLR 5 (homoskedasticity) does not hold in this context.
Explain how you would use the Breusch-Pagan test to test for heteroskedasticity. Be specific
about which steps you need to do to perform the test.[6 marks]
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(f) You are particularly worried thatV ar(u|train,female,age,educ,longterm) =σ
2
educ. De-
rive a transformed equation of Model 2 that has a homoskedastic error term.[6 marks]
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(g) Besides transforming the model, what else could you do to at least solve some of the prob-
lems caused by heteroskedasticity? What are the advantages and disadvantages of the other
approach compared to transforming the model?[4 marks]
If you have used additional space for working then please tick here:
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You are also worried that men and women have completely different earnings functions, so
you augment your model further and estimateModel 3:
log(earn) =β
0
+β
1
train+β
2
female+β
3
age+β
4
educ+β
5
longterm+β
6
female·train
+β
7
female·age+β
8
female·educ+β
9
female·longerm+u
With the following results:
(h) Based on these results, by how much did programme participation increase earnings for
women? Is the programme effect statistically significantly different for men and women?[5 marks]
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(i) Given your previous results, is there evidence that allowing men and women to have different
coefficients for training, age, education, and long-term unemployment improves the model
fit?Hint: The formula for an F-statistic isF=
(R
2
ur
−R
2
r
)/q
(1−R
2
ur
)/df
ur
[5 marks]
If you have used additional space for working then please tick here:
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2. Questions (a) to (d) concerns generic time series analysis and have a theoretical bend. Questions
(e) to (l) concerns a particular application of time series analysis. Please note that you do not
need the answers to questions (a) to (d) to answer questions (e) to (l).
Let{ε
t
}be an iid process withE(ε
t
) = 0 andVar(ε
t
) =σ
2
, lety
t
=ρy
t−1
+ε
t
with|ρ|<1, and
letz
t
=z
t−1
+ε
t
with initial conditionz
0
= 0. Note that{y
t
}is a stationaryAR(1) process and
{z
t
}is a Random Walk. Recall thatE(y
t
) = 0,Var(y
t
) =
σ
2
1−ρ
2
, andCov(y
t
,y
t+h
) =
ρ
h
σ
2
1−ρ
2
. Also
recall thatE(z
t
) = 0,Var(z
t
) =tσ
2
andCov(z
t
,z
t+h
) =tσ
2
.
Let{x
t
}be a time series. You consider two models for{x
t
}:
Model 1 (deterministic trend):x
t
=α+δt+y
t
,
Model 2 (stochastic trend):x
t
=α+z
t
,
whereαandδare coefficients and{y
t
}and{z
t
}are described above.
(a) ComputeE(x
t
),Var(x
t
) and, forh≥1,Cov(x
t
,x
t+h
) for Model 1 and Model 2.[2 marks]
If you have used additional space for working then please tick here:
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(b) What are the properties of a covariance stationary time series? Is Model 1 covariance sta-
tionary? Is Model 2 covariance stationary? Explain your answer.[4 marks]
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(c) Describe the notion of weak dependence in one sentence. Model 2 is not weakly dependent.
Is Model 1 weakly dependent? Explain your answer.
Hint:You do not need to provide a formal definition of weak dependence.[4 marks]
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(d) Both Model 1 and Model 2 can be used to model a time series that exhibits a trend. Model
1 is a deterministic trend model. Model 2 is a stochastic trend model. Suppose you have
data on two time series that are both trending. You want to regress one time series on the
other, but you cannot determine whether the trends are deterministic or stochastic. Would
you ignore the trend and run your regression using the original series in levels, or would you
include a trend in your regression, or would you run your regression using first-differenced
series? Explain your answer.[2 marks]
If you have used additional space for working then please tick here:
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Suppose you are interested in understanding how the price of gasoline impacts the approval
ratings of the President of the United States (POTUS). You are working with historical
data from George W. Bush’s presidency and you have monthly time series data on Gallup’s
approval rate of POTUS for the period from February 2001 to July 2007 (n= 78). The
following variables are of interest:
•approval: Gallup approval rate, percent
•rgasprice: Real gas price in cents per gallon
•lcpifood: Log of consumer price index (CPI) for food
•unemploy: Unemployment rate, percent
•sep11: = 1 for 09/2001 and two months following; = 0 otherwise
•iraqinvade: = 1 for three months after Iraq invasion; = 0 otherwise
Figure 1 plots{approval
t
;t= 1,2,...,n}and{rgasprice
t
;t= 1,2,...,n}
Figure 1: Approval rates and log real gasoline price
To control for the effect of the food price inflation, uemployment, the impact of 9/11 and
the Iraq invasion as well as trends, you initially consider the following static regression of
{approval
t
}on{rgasprice
t
},{lcpifood
t
},{unemploy
t
},{sep11
t
},{iraqinvade
t
}and a linear
trendt:
approval
t
=α+δ
0
t+δ
1
rgasprice
t
+β
1
lcpifood
t
+β
2
unemploy
t
+β
3
sep11
t
+β
4
iraqinvade
t
+ε
t
,
where{ε
t
}is an error term andα,δ
0
,δ
1
,β
1
,β
2
,β
3
andβ
4
are coefficients.
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You first compute serial correlations of order 1 through to 12 for the detrendedapproval
t
-
series and the detrendedrgasprice
t
-series. For the detrendedapproval
t
-series, these corre-
lations range between 0.968 and 0.894. For the detrendedrgasprice
t
series, the correlations
range between 0.973 and 0.860.
(e) In light of this information and the plots in Figure 1, do you think the proposed specification
is appropriate? Explain your answer.[4 marks]
If you have used additional space for working then please tick here:
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ECNM10052Do not write above this lineDecember 2023
Upon reflection, you decide to work with the differenced time series instead. Let ∆y
t
=
y
t
−y
t−1
fory∈{approval,rgasprice,lcpifood,unemploy,sep11,iraqinvade}. You consider
the following model:
∆approval
t
=δ
0
+δ
1
∆rgasprice
t
+β
1
∆lcpifood
t
+β
2
∆unemploy
t
+β
3
∆sep11
t
+β
4
∆iraqinvade
t
+e
t
,
wheree
t
≡∆ε
t
=ε
t
−ε
t−1
is the error term. You estimate the regression model by OLS and
obtain the following output:
Table 1: OLS estimates, differenced equation
Note: Fudged the Drgasprice numbers a bit!
Source | SS df MS Number of obs = 77
-------------+---------------------------------- F(5, 71) = 5.86
Model | 507.256034 5 101.451207 Prob > F = 0.0001
Residual | 1228.24819 71 17.2992703 R-squared = 0.2923
-------------+---------------------------------- Adj R-squared = 0.2424
Total | 1735.50423 76 22.8355819 Root MSE = 4.1592
------------------------------------------------------------------------------
Dapprove | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
Drgasprice | -.0947359 .0420608 -2.25 0.024 -.1771751 -.0012297
Dlcpifood | -128.727 222.5121 -0.58 0.565 -572.4035 314.9495
Dunemploy | -2.16217 1.511978 -1.43 0.157 -5.17697 .852629
Dsep11 | 14.9459 2.977234 5.02 0.000 9.009462 20.88233
Diraqinvade | 1.195415 3.020669 0.40 0.693 -4.827627 7.218457
_cons | -.001905 .6824474 -0.00 0.998 -1.362667 1.358857
------------------------------------------------------------------------------
In the regression output tableDapproveindicates ∆approve,Drgaspriceindicates ∆rgasprice,
Dlcpifoodindicates ∆lcpifood,Dunemployindicates ∆unemploy,Dsep11indicates ∆sep11
andDiraqinvadeindicates ∆iraqinvade.
(f) Interpret the estimated coefficient on ∆rgasprice
t
. Do you consider the estimated effect of
the real gas price on the approval rate to be small or large?
Hint:The standard deviation of ∆rgasprice
t
is 7.31 cents. The standard deviation of
∆approve
t
is 4.78 percentage points.[4 marks]
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ECNM10052Do not write above this lineDecember 2023
(g) Interpret the estimated coefficient on ∆sep11
t
[4 marks]
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One of your colleagues asserts that it may take some time for gas prices to fully feed through
to approval ratings. You therefore estimate the following finite distributed lag model of order
2:
∆approval
t
=δ
0
+δ
1
∆rgasprice
t
+δ
2
∆rgasprice
t−1
+δ
3
∆rgasprice
t−2
+β
1
∆lcpifood
t
+β
2
∆unemploy
t
+β
3
∆sep11
t
+β
4
∆iraqinvade
t
+e
t
,
wheree
t
is the error term. You obtain the following output:
Table 2: OLS estimates, differenced equation with lag distribution
Note: Fudged the Drgasprice numbers a bit!
Source | SS df MS Number of obs = 77
-------------+---------------------------------- F(5, 71) = 5.86
Model | 507.256034 5 101.451207 Prob > F = 0.0001
Residual | 1228.24819 71 17.2992703 R-squared = 0.2923
-------------+---------------------------------- Adj R-squared = 0.2424
Total | 1735.50423 76 22.8355819 Root MSE = 4.1592
------------------------------------------------------------------------------
Dapprove | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
Drgasprice | -.0947359 .0420608 -2.25 0.024 -.1771751 -.0012297
Dlcpifood | -128.727 222.5121 -0.58 0.565 -572.4035 314.9495
Dunemploy | -2.16217 1.511978 -1.43 0.157 -5.17697 .852629
Dsep11 | 14.9459 2.977234 5.02 0.000 9.009462 20.88233
Diraqinvade | 1.195415 3.020669 0.40 0.693 -4.827627 7.218457
_cons | -.001905 .6824474 -0.00 0.998 -1.362667 1.358857
Source | SS df MS Number of obs = 75
-------------+---------------------------------- F(7, 67) = 4.65
Model | 561.873788 7 80.2676839 Prob > F = 0.0003
Residual | 1156.66742 67 17.2636928 R-squared = 0.3269
-------------+---------------------------------- Adj R-squared = 0.2566
Total | 1718.5412 74 23.2235298 Root MSE = 4.155
------------------------------------------------------------------------------
Dapprove | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
Drgasprice | -.0824628 .0284890 -2.89 0.004 -.1383013 -.0266243
L1Drgasprice | -.0624628 .0290589 -2.15 0.031 -.1194182 -.0055074
L2Drgasprice | -.0280732 .0158180 -1.77 0.076 -.0590765 .0029302
Dlcpifood | -115.8944 223.1176 -0.52 0.605 -561.2391 329.4502
Dunemploy | -2.084555 1.523475 -1.37 0.176 -5.125425 .9563143
Dsep11 | 15.2941 2.982045 5.13 0.000 9.341918 21.24629
Diraqinvade | 1.593799 3.04939 0.52 0.603 -4.492808 7.680406
_cons | .0713774 .6895598 0.10 0.918 -1.30499 1.447745
------------------------------------------------------------------------------
whereL1Drgaspriceindicates ∆rgasprice
t−1
andL2Drgaspriceindicates ∆rgasprice
t−2
.
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ECNM10052Do not write above this lineDecember 2023
(h) Sketch the estimated lag distribution in a graph with lags of ∆rgasprice
t
on the horizontal
axis and coefficient estimates on the vertical axis. Consider a temporary single-month increase
in the real price per gallon of gasoline of 1 cent in montht. What does the estimated lag
distribution tell us about the impact of this shock on the approval rate in montht, month
t+ 1, montht+ 2 and montht+ 3?
Hint:Assume everything else remains constant at the montht−1 levels.[4 marks]
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ECNM10052Do not write above this lineDecember 2023
(i) Does the estimated finite distributed lag regression model support your colleague’s assertion?
Explain your answer.[2 marks]
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(j) Use the estimated finite distributed lag model to compute the long run response of the
approval rating to a permanent 10 cent increase in the real price per gallon of gasoline
holding everything else constant.[4 marks]
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ECNM10052Do not write above this lineDecember 2023
Let
b
e
t
be the residuals from your estimated finite distributed lag regression. You compute
the first order serial correlation in the residuals, i.e.Corr(
b
e
t
,
b
e
t+1
) and find that it is 0.5946
and highly statistically significant.
(k) What concerns does this residual serial correlation raise?[4 marks]
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ECNM10052Do not write above this lineDecember 2023
(l) How could you alleviate these concerns?[2 marks]
If you have used additional space for working then please tick here:
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ECNM10052Do not write above this lineDecember 2023
3. This section contains ten multiple-choice questions. Each question hasONLY ONEcorrect
choice.
Please answer the questions on the multiple choice question sheet provided.
Use only a pencil.
Write your student number and name on the multiple-choice answer sheet.
Each question contains four possible answers, only one of which is correct. Place a firm horizontal
pencil mark in the appropriate place, e.g., if the answer to question 5 was (C)
Q5 [ A ] [ B ] [
C] [ D ]
Faint lines will not be read!
Very thin marks made with a sharp, hard pencil may not be read.
If a soft pencil is used there should be no need to press heavily.
This should make clear rubbing out easy.
Erasures must be completed clean without smudging. An unclean or incomplete erasure may be
read as an answer!
(a) You estimate a model of the formy=β
0
+β
1
x
1
+β
2
x
2
+uby OLS. Assumptions MLR1-4
hold, but assumption MLR5 (homoskedasticity) does not. Which of the following statements
is true?[2 marks]
A) The OLS estimators will be biased
B) The OLS estimators will be inconsistent
C) The OLS estimators will not be efficient
D) Homoskedasticity has no consequences for OLS estimators
(b) You have data from the years 2012, 2013, and 2014 stored in the variableyear. Using Stata,
you want to see the summary statistics for variableXfor the year 2014 only. Which command
do you use?[2 marks]
A) if year==2014: summarize X
B) summarize X if year==2014
C) bysort X: summarize year
D) regress X year if year==2014, summarize
(c) You want to explain a student’s math test score in 10th grade (math10) with a dummy
for whether the student is female (female), a dummy for whether the student went to pri-
vate school (private), the interaction of the two (female·private) and the average years
of schooling of both parents (pareduc). You obtain the following results (rounded to two
decimals):
\
math10 = 49.26 + 2.31female+ 1.04private−0.21female·private+ 0.23pareduc
Based on this, what is the predicted math test score for a male student who went to private
school and whose parents have an average of 10 years of schooling?[2 marks]
A) 51.56
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ECNM10052Do not write above this lineDecember 2023
B) 52.60
C) 53.87
D) 54.70
(d) In which of the following bivariate regression models doesβ
1
give you the elasticity of y with
respect to x?[2 marks]
A)y=β
0
+β
1
x+u
B)log(y) =β
0
+β
1
x+u
C)y=β
0
+β
1
log(x) +u
D)log(y) =β
0
+β
1
log(x) +u
(e) You are estimating a wage regression with education (educ, measured in years of education),
experience (exper, measured in years) and its square (expersq) as explanatory variable. You
obtain the following results (rounded to three decimals)
\
logwage= 12.80 + 0.09educ+ 0.035exper−0.001expersq
At which level of experience does the marginal effect of an additional year of experience turn
negative?[2 marks]
A) 0.035
B) 17.5
C) 35
D) 306.26
(f) Miami experienced a sudden influx of immigration during a very short time in 1980, Los
Angeles (LA) did not. LetEmp
it
be a an employment indicator = 1 if individualiis employed
in yeart, = 0 otherwise. You have pooled cross section data onEmpfornindividuals from
Miami and Los Angeles for 1979 and 1981. You estimate the following regression:
ˆ
Emp= 0.80 + 0.03Miami+ 0.02D81−0.01Miami×D81,
whereMiamiis a dummy variable = 1 if the observation is for a Miami resident and = 0
if the observation is for a Los Angeles resident, andD81 is a dummy variable = 1 if the
observation is from 1981 and = 0 if it is from 1979. According to the estimated regression,
in response to the immigration influx, the employment rate in Miami ...[2 marks]
A) ... increased by 80 percent.
B) ... increased by 3 percent.
C) ... increased by 2 percent.
D) ... declined by 1 percent.
(g) What is the parallel trends assumption in the difference-in-difference estimator?[2 marks]
A) There is no trend in the outcome variable.
B) The trends in the outcome variable are the same for the treatment and the control group.
C) There is only a trend in the outcome variable for the control group.
D) There is only a trend in the outcome variable for the treatment group.
(h) In a panel data setting, the fixed effects estimator ...[2 marks]
A) ... estimates the effects of time-invariant variables on an outcome while controlling for
time-invariant unobserved heterogeneity.
B) ... estimates the effects of time-varying variables on an outcome while controlling for
time-invariant unobserved heterogeneity.
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ECNM10052Do not write above this lineDecember 2023
C) ... estimates the effects of time-invariant variables on an outcome while controlling for
time-varying unobserved heterogeneity.
D) ... estimates the effects of time-varying variables on an outcome while controlling for
time-varying unobserved heterogeneity.
(i) Consider a panel data set.Tis the number of time periods in the panel. SupposeT >2.
Which of the following statements isFALSE?[2 marks]
A) The fixed effects estimator and the first differenced estimator yields exactly identical
estimates.
B) The fixed effects estimator is more efficient than the first differenced estimator if the
classical assumptions hold.
C) The first differenced estimator is the appropropriate choice of estimator if the outcome
variable exhibits severe serial correlation.
D) The fixed effects estimator preserves more data than the first differenced estimator in
unbalanced panels.
(j) The difference-in-difference-in-difference (DDD) estimator allows you to control for some vi-
olations of the parallel trends assumption in a difference-in-difference (DD) estimator. How-
ever, the DDD estimator requires richer data than the DD estimator. What is the additional
data requirement?[2 marks]
A) The DDD estimator requires 1 treatment group and 2 control groups
B) The DDD estimator requires 2 treatment groups and 1 control group
C) The DDD estimator requires 3 treatment groups and 3 control groups
D) The DDD estimator requires experimental data with explicit randomization and is there-
fore not suitable for analyzing natural experiments.
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relates, otherwise your working may not be marked.
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relates, otherwise your working may not be marked.
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relates, otherwise your working may not be marked.
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relates, otherwise your working may not be marked.
Page 25 of 25
Exam Hall Regulations
The following is a copy of a Notice which is displayed in Edinburgh University Examination Halls for the
information of students and staff.
The University of Edinburgh Exam Hall Regulations
1. An examination attendance sheet is laid on the desk for each student to complete upon arrival. These
are collected by an invigilator after thirty minutes have elapsed from the start of the examination.
Students are not normally allowed to enter the examination hall more than thirty minutes after the start
of the examination.
2. Students arriving after the start of the examination are required to complete a “Late arrival form” which
requires them to sign a statement that they understand that they are not entitled to any additional time.
Students are not allowed to leave the examination hall less than thirty minutes after the
commencement of the examination or within the last fifteen minutes of the examination.
3. Books, papers, briefcases and cases must be left at the back or sides of the examination room. It is an
offence against University discipline for a student to have in their possession in the examination any
material relevant to the work being examined unless this has been authorised by the examiners.
4. Students must take their seats within the block of desks allocated to them and must not communicate
with other students either by word or sign, nor let their papers be seen by any other student.
5. Students are prohibited from deliberately doing anything that might distract other students. Students
wishing to attract the attention of an invigilator shall do so without causing a disturbance. Any student
who causes a disturbance in an examination room may be required to leave the room, and shall be
reported to the University Secretary.
6. Personal handbags must be placed on the floor at the student’s feet; they should be opened only in full
view of an invigilator.
7. An announcement will be made to students that they may start the examination. Students must stop
writing immediately when the end of the examination is announced.
8. Answers should be written in the script book provided. Rough work, if any, should be completed within
the script book and subsequently crossed out. Script books must be left in the examination hall.
9. During an examination, students will be permitted to use only such dictionaries, other reference books,
computers, calculators and other electronic technology as have been issued or specifically authorised
by the examiners. Such authorisation must be confirmed by the Registry.
10. The use of mobile telephones is not permitted and mobile telephones must be switched off during an
examination.
11. It is an offence against University discipline for any student knowingly
•to make use of unfair means in any University examination
•to assist a student to make use of such unfair means
•to do anything prejudicial to the good conduct of the examination, or
•to impersonate another student or allow another student to impersonate them
12. Students will be required to display their University Card on the desk throughout all written degree
examinations and certain other examinations. If a card is not produced, the student will be required to
make alternative arrangements to allow their identity to be verified before the examination is marked.
13. Smoking and eating are not allowed inside the examination hall.
14. If an invigilator suspects a student of cheating, they shall impound any prohibited material and shall
inform the Examinations Office as soon as possible.
15. Cheating is an extremely serious offence, and any student found by the Discipline Committee to have
cheated or attempted to cheat in an examination may be deemed to have failed that examination or
the entire diet of examinations, or be subject to such penalty as the Discipline Committee considers
appropriate.
