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ECNM10056 Applications of Econometrics
From and To: Exam Diet:
Exam Date:
02 May 2023
Exam: 13:00:00 – 15:00:00 May 2023
Please read full instructions before commencing writing
Exam paper information
•Total number of pages: 9 (including this page).
•This paper has 3 sections.
•Pl
ease answer ALL questions.
Special instructions
•Attach your personalised barcode to EACH script book.
•You should use a total of three booklets, with the personalized barcodes attached,
and your personal information completely and correctly.
•You must complete the exam within the time specified at the top of this page.
•You must start EACH QUESTION on a separate page. Questions must be clearly
numbered in the left margin.
•Your answers must be clearly written.
•Your exam number (e.g., B123456) must be clearly written at the top of each page.
•Answers may be subject to checks through TurnItIn.
•This examination will be marked anonymously.
•Students are allowed to keep the question sheet.
Special items
•Non-programmable calculators are permitted in this exam.
Examiners: Tim Worrall (Chair), Giacomo De Luca (External)
Question 1
Lethy6
t
denote the three-month holding yield (in percent) from buying a six-month T-bill at
timet−1and selling it at timet(three months hence) as a three-month T-bill. Lethy3
t−1
be the three-month holding yield from buying a three-month T-bill at timet−1. The line
graphs ofhy6andhy3are shown in Figure 1 below.
A model that relateshy3
t−1
tohy6
t
could be
hy6
t
=β
0
+β
1
hy3
t−1
+u
t
.(1)
(a) Briefly describe how you would test for seasonality and serial correlation in Eq. (1).
[8 marks]
(b) At timet−1,hy3
t−1
is known, whereashy6
t
is unknown because the price of three-
month T-bills is unknown at timet−1. Theexpectations hypothesissays that these two
different three-month investments should be the same, on average. Mathematically, we
can write this as a conditional expectation:
E(hy6
t
|I
t−1
) =hy3
t−1
whereI
t−1
denotes all observable information up through timet−1. This is what
motivates Eq. (1) and suggests we can test the expectations hypothesis by testing
H
0
:β
1
= 1. (We can also test H
0
:β
0
= 0, but we often allow for aterm premium
for buying assets with different maturities, so thatβ
0
6
= 0.) The regression output
for Eq. (1) is presented in Table 1. Does the expectations hypothesis hold at the 5%
significance level? Why might Figure 1 raise concerns with the test that you performed?
Hint: Consider spurious regression.[8 marks]
(c) After estimating Eq. (1), you obtain the residualŝu
t
and plan on estimating
̂u
t
=α+φ̂u
t−1
+e
t
,(2)
However, your lab partner has estimated
∆̂u
t
=α+γ̂u
t−1
+e
t
,(3)
where∆̂u
t
=̂u
t
−̂u
t−1
.The lab partner reports the regression output under Eq. (3) in
Table 1. Show that testingH
0
:φ= 1in Eq. (2) is equivalent to testingH
0
:γ= 0
in Eq. (3). Then, use the estimated parameters and the appropriate critical value from
Table 2 to address your concerns from part (b).Hint: The concern from part (b) might
be spurious regression.[8 marks]
(d) Inspired by your findings from part (c), you decide to augment Eq. (1) such that
hy6
t
=β
0
+β
1
hy3
t−1
+β
2
spread
t−1
+u
t
,(4)
wherespread
t−1
=r6
t−1
−r3
t−1
, the difference between six-month (r6) and three-month
T-bill rates (r3) at timet−1. The regression output is provided under Eq. (4) in Table 1.
What condition mustspreadsatisfy to justify its inclusion? According to the estimates,
if, at timet−1,r6is abover3, should you invest in six-month or three-month T-bills?
[8 marks]
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(e) In Eq. (4), what Gauss-Markov assumption is violated ifhy3
t−1
is correlated with
u
s
, s= 1,2, . . . , t, . . .? Comment on how this affects the properties of the OLS estimator
and suggest a solution.[8 marks]
0
2
4
6
Holding yield for hy6
0255075100125
t
(a) Holding yield:hy6
t
0
1
2
3
4
Holding yield for hy3
0255075100125
t
(b) Holding yield:hy3
t
Figure 1: Three-month holding yield from buying a six-month T-bill at timet−1and selling
it att(left), and three-month holding yield from buying a three-month T-bill att−1(right).
Table 1: Regression output for equations (1), (3), and (4)
Eq. (1)Eq. (3)Eq. (4)
Dependent variable:hy6
t
∆̂u
t
hy6
t
hy3
t−1
1.10431.0535
(0.0395)(0.0385)
̂u
t−1
-0.9919
(0.0913)
spread
t−1
0.4800
(0.1086)
constant-0.0579-0.0005-0.1229
(0.0700)(0.0299)(0.0668)
N123122123
Notes:Each column represents one regression; the dependent variable is listed as
the column header and the explanatory variables are listed in rows. Note that∆is
the first difference operator. Standard errors are in parentheses.
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Table 2: Critical values for different tests at 5% significance level
Test and specification5%
Dickey-Fuller: constant, no trend-2.86
Dickey-Fuller: constant and trend-3.41
Engle-Granger: constant, no trend-3.34
Engle-Granger: constant and trend-3.78
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Question 2
A researcher wants to know whether it is true that workers who are members of labour unions
earn higher hourly wages. They decide to use two waves of data from the UK Labour Force
Survey (UKLFS) that follow a sample of workers from 2020 to 2021. Consider the following
model for log hourly wages of workeriobserved in yeart
lnwage
it
=β
0
+β
1
union
it
+β
2
y2021
t
+β
3
educ
i
+β
4
age
it
+β
5
agesq
it
+a
i
+u
it
wherelnwage
it
refers to the natural logarithm of gross hourly pay,union
it
is a dummy for
union status of workeri(= 1if in a union int, zero otherwise),y2021
t
is a dummy for
t= 2021,educ
i
is workeri’s years of education,age
it
andagesq
it
measure age and its square,
respectively.a
i
denotes an unobserved individual effect andu
it
denotes the error term. The
following questions refer to Table 3.
(a) Interpret the POLS coefficient estimates onunion
it
,
ˆ
β
1
, and the POLS coefficient es-
timate ony2021
t
,
ˆ
β
2
, in Table 3 column (1).Hint: One sentence per coefficient for
interpretation is enough. You don’t have to explain or discuss anything.[8 marks]
(b) The researcher finds a strong positive effect ofunion
it
on wages using POLS in col-
umn (1) but they are worried that this is biased because of omitting education and age,
and because there might be ana
i
term. Considering the results in Table 3 columns (2)
and (3), which show POLS with additional controls and fixed effects, discuss whether
you think their concern was justified.[10 marks]
(c) The researcher wonders whetheragecontributes significantly to explaining the variation
inlnwagein the POLS regression in column (2). After the POLS estimation they run
the appropriate test and Stata shows the following output:
test age agesq
( 1) age = 0
( 2) agesq = 0
F( 2, 1138) = 43.01
Prob > F =0.0000
Briefly explain the test the researcher ran and what it shows.[4 marks]
(d) Considering the similarity between POLS and fixed effects coefficients, the researcher
thinks random effects might be the right estimator for this problem. The results of this
are shown in Table 3 column (4). Explain which potential problem with the results
in Table 3 column (2) this estimator would solve and whether you think it worked.
[10 marks]
(e) The researcher shows their table to their advisor thinking they now have a good set
of results. The advisor tells them that they should have been using a Tobit model
instead because wages cannot be negative. Explain whether you think the advisor is
right with this advice. In your explanation, briefly discuss what the potential problems
are that OLS faces when estimating a (log) wage equation like the one we estimated
above.[8 marks]
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Table 3: Effect of Union Status on Hourly Wages
(1)(2)(3)(4)
EstimatorPOLSPOLS
Fixed
Effects
Random
Effects
union0.11730.04680.06130.0487
(0.0321)
∗∗∗
(0.0291)(0.0800)(0.0343)
y20210.04010.03740.06040.0396
(0.0301)(0.0269)(0.0679)(0.0165)
∗∗
educ0.07130.0716
(0.0056)
∗∗∗
(0.0072)
∗∗∗
age0.07570.0734
(0.0085)
∗∗∗
(0.0107)
∗∗∗
agesq-0.0008-0.0002-0.0008
(0.0001)
∗∗∗
(0.0007)(0.0001)
∗∗∗
_cons2.6236-0.37393.1167-0.3379
(0.0237)
∗∗∗
(0.1924)
∗
(1.5928)
∗
(0.2426)
R
2
0.0130.2180.012
N1,1441,1441,1441,144
Standard errors in parentheses.
∗
p <0.10,
∗∗
p <0.05,
∗∗∗
p <0.01
Dependent variable islnwagein all four columns.
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Question 3
We are still using the UK Labour Force Survey in this question, but only data from 2020. We
are interested in predicting whether workers are members of a labour union. For this question
we use Table 4 below. Column (1) shows LPM estimates, column (2) shows Logit coefficients
and column (3) shows average partial effects based on the Logit estimates.f emaleis a dummy
for whether the worker is female,degreeis a dummy equal to one if the worker has a degree,
and zero otherwise.ageis the worker’s age in years.
(a) Calculate the predicted probability to be in a labour union for a man age 30 without
a degree in the LPM and Logit. The formula for predicted probabilities in the Logit
model is
ˆ
P
Logit
=
exp
(
ˆ
β
0
+
ˆ
β
1
f emale+
ˆ
β
2
degree+
ˆ
β
3
age
)
1 +exp
(
ˆ
β
0
+
ˆ
β
1
f emale+
ˆ
β
2
degree+
ˆ
β
3
age
)
[4 marks]
(b) Calculate the predicted probability to be in a labour union for a man age 30 with a
degree in the LPM and Logit. Then calculate the discrete probability effect of having a
degree for a man age 30 for both models.[4 marks]
(c) Explain why the predicted probabilities and the discrete probability effect in (a) and
(b) are different in the LPM compared to Logit. If you couldn’t calculate them you can
assume that the predicted probabilities are smaller in the LPM compared to Logit and
that the discrete probability effect is bigger in the LPM compared to Logit.[8 marks]
For the remaining two questions we return to estimating the effect of union status on hourly
wages, but only using data from 2020. Consider the following model for log hourly wages of
workeri
lnwage
i
=β
0
+β
1
union
i
+β
2
abil
i
+e
i
(d) Unfortunately we do not observeabil
i
. It is an unobserved ability variable. This means
we are concerned thatCov(union
i
, u
i
)6
= 0, whereu
i
=β
2
abil
i
+e
i
. Explain why this
means that the OLS estimate
ˆ
β
OLS
1
is inconsistent. You can use algebra if it helps make
your point.Hint: One way to write the omitted variable bias formula is
plim
n→∞
ˆ
β
OLS
1
=β
1
+
Cov(union, u)
V ar(union)
[12 marks]
(e) Assume we know whether personi’s father was in a union and letf un
i
= 1if that’s the
case andf un
i
= 0if workeri’s father was not a union member. Assume that
E(union
i
|f un
i
= 1)−E(union
i
|f un
i
= 0)6
= 0.
Explain whether you thinkf un
i
is a good instrument forunion
i
under this assumption,
and whether/how we could test iff un
i
is a good instrument.[12 marks]
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Table 4: Explaining Union Membership
(1)(2)(3)
LPM
Logit
Coefficients
Logit
APE
female0.08490.46620.0850
(0.0187)
∗∗∗
(0.1038)
∗∗∗
(0.0187)
∗∗∗
degree0.07440.40550.0748
(0.0191)
∗∗∗
(0.1041)
∗∗∗
(0.0192)
∗∗∗
age0.00150.00850.0015
(0.0007)
∗∗
(0.0042)
∗∗
(0.0008)
∗∗
_cons0.0969-1.9757
(0.0373)
∗∗∗
(0.2354)
∗∗∗
(Pseudo-)R
2
0.0168
N2,0822,0822,082
Standard errors in parentheses.
∗
p <0.10,
∗∗
p <0.05,
∗∗∗
p <0.01
Dependent variable isunion(0 or 1) in all columns.
— END OF EXAMINATION PAPER —
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