1

THE UNIVERSITY OF EDINBURGH
SCHOOL OF ECONOMICS
Applications of Econometrics
ECNM10056
Exam Date: From and To: Exam Diet:
10th May 2021 13:00 – 16:00 April/May 2021

Please read full instructions before commencing writing
Exam paper information
• Total number of pages: 10 (including this page)
• You have 2 hours to complete the degree exam plus one additional hour to upload your exam
script. The deadline for uploading your script is 16:00
• Please answer ALL questions
• Marked out of 120 overall. Each question is worth 1/3 of overall marks
• Tables can be found on pages 6-10
Special instructions for questions
• This is an open-book exam.
• You must start EACH QUESTION on a separate page. Questions must be clearly numbered in the
left margin.
• You are expected to complete this exam within the 2-hour standard exam duration.
• Immediately following the exam scan and upload your answers to Learn. Please make sure you
have scanned and uploaded every page of your script.
• You must number each answer page.
• Your exam number (e.g. B123456) must be clearly written at the top of each page.
• Your answers must be clearly written in ink on lined paper. It is your responsibility to ensure
your answers are readable.
• Answers may be subject to checks through TurnItIn.
• All work must be completed individually, without collaboration, as a standard exam would be.
Special items
• None.

Examiners: Prof. Tim Worrall (Chair), Prof. Giacomo De Luca (External)
This examination will be marked anonymously
Question 1 (one third of total mark)
Using quarterly data, you estimate the following regression:
ln(yt) = δ0 + δ1xt + δ2xt−1 + ut, where ut = φut−1 + ηt, (1)
where yt is crime rate in quarter t, xt is unemployment rate. ut is an error term and ηt an
i.i.d sequence with zero mean and variance σ2. For some of the questions below you will need
the critical values and statistical tables at the end of the exam.
(a) How would you interpret the δ̂2 coefficient? Under what set of minimal assumptions
are δ̂1 and δ̂2 unbiased estimates of the true causal effect? Explain whether you believe
these assumptions are fulfilled in this case. [8 points]
(b) You are worried that ut in equation (1) might be serially correlated. Why would this be
a concern? Explain two ways how you could test for serial correlation. Given evidence
of weakly serially correlated error terms (suggesting φ ≈ 0.6), give two examples of
techniques that allow you to account for this in your estimation. Provide an intuition
for why they may work. [8 points]
(c) You use the residuals from equation (1) and run the regression:
∆ût = ρ0 + ρ1ût−1 + ρ2∆ût−1 + ϵt (2)
where ϵt is an i.i.d. sequence with zero mean and variance σ2. The estimated coefficient
is ρˆ1 = −0.72 with a standard error of SE (ρˆ1) = 0.23. Describe what kind of test this is
and state the hypothesis. What good does adding ∆uˆt−1? What do the results suggest
about the relationship in equation (1)? [8 points]
(d) You decide to instead run an AR(1) model of the form:
yt = α0 + α1yt−1 + νt (3)
where νt is an i.i.d. sequence with zero mean and variance σ2. You suspect that α1 ≈ 1
and α0 ̸= 0. What kind of test would you suggest to test your suspicion? Given
evidence of α1 ≈ 1 and α0 ̸= 0, what moment condition(s) in the covariance stationarity
assumption are violated? [8 points]
(e) You are told to make a one-step ahead forecast of yt using an AR(1) model. Explain
how you would assess the fit of the forecast based on this model. [8 points]
2
Please read this carefully. It is relevant for both Question 2 and Question 3.
We consider a standard question in labour economics: what is the effect of having children on
women’s labour supply? We make use of “Understanding Society”, a UK panel dataset. In
Question 2 we pool three years of survey data: 2009, 2010, and 2011. We don’t observe all
mothers in all years, so the panel is not balanced. Some mothers are only observed once or
twice. In Question 3 we only use data from 2010. Because we will make use of a particular
instrumental variable, we limit the sample to mothers age 21-40 with at least 2 children in both
questions. We also restrict the sample to mothers who were born in the UK. The following
questions refer to the three tables below. In all tables standard errors are given in parentheses
below the coefficient estimates. You can assume that we have a random sample from the cross
section and there are no collinearity problems. You can find statistical tables at the end of
the exam.
3
Question 2 (one third of total mark)
Say we start by writing down a model for hours of work of mothers as
hoursit =β0 + β1nchildrenit + β2ageit + β3age
2
it + β4agefirstbirthi + . . .
. . .+ β5year10t + β6year11t + ai + uit
where i denotes an individual respondent and t is the time period (calendar year). hoursit
refers to the usual number of hours worked per week (zero if not employed), nchildrenit is the
number of children, ageit is the respondent’s age in t and age2it its square, agefirstbirthi is
the age at which the first child was born, and year10t and year11t control for calendar year
(2009 is omitted). ai denotes an unobserved individual effect and uit denotes the error term.
(a) To get a larger sample and thus more precise estimates we are pooling three years of
survey data. Explain what the potential problems are with this as opposed to using just
one year of data. [8 points]
(b) The result of estimation by pooled OLS (POLS) is shown in column (1) of Table 1 below.
Assume pooling is appropriate. Interpret the coefficient estimate for nchildren (βˆ1) in
column (1) and calculate whether it is statistically significant. [8 points]
(c) Consider again the POLS results shown in column (1) of Table 1 below. Assume pooling
is appropriate. Explain why the coefficient estimate βˆ1 in column (1) is likely not the
causal effect of having more children on hours of work (why it is biased and inconsis-
tent). In your explanation, comment on the likely direction of the bias if nchildren is
endogenous (correlated with ai and/or uit) or suffers from measurement error. [8 points]
(d) Consider the results shown in column (2) of Table 1 below, where we estimate the
parameters by fixed effects. Assume pooling is appropriate. Explain why the coefficient
estimate on nchildren (βˆ1) changes from column (1) to column (2) and what potential
problems remain so that the estimates in column (2) are still not causal effects of the
number of children. [8 points]
(e) Consider the results shown in columns (2) and (3) in Table 1 below. Column (2) uses
fixed effects, column (3) uses first differences. Assume pooling is appropriate. Discuss
whether the estimates in column (2) or column (3) are better in terms of the magnitude
and precision that they give us for the effect of nchildren (things like consistency and
efficiency). Although you don’t necessarily have to be technical in your explanation if
you are able to convey the key points in words, it might help make your point by making
precise statements in terms of assumptions and properties those assumptions imply. [8
points]
4
Question 3 (one third of total mark)
For this question we use Tables 2 and 3 below. We use the same data as in Question 2 above,
we only limit the sample to a cross section of women in 2010 (we drop the years 2009 and
2011). So we no longer have a panel. You can still assume that we have a random sample
(from the population of 21-40 year old women born in the UK with at least 2 children) and
that there are no collinearity problems throughout this question.
(a) To deal with the concern that nchildren is not exogenous we pursue an instrumental
variables strategy. The instrument we will use for nchildren is samesex. That’s a
binary variable that captures whether a woman’s first two children had the same gender
(girl/girl or boy/boy). In column (2) of Table 2 we show the first stage, in column (3) we
show the 2SLS second stage results. Discuss the validity of this instrumental variable.
You can assume that a child’s gender is random. That’s a reasonable assumption for a
country like the UK. [8 points]
(b) Compare the results shown in columns (1) and (3) of Table 2 (OLS vs 2SLS). The 2SLS
coefficient is much more negative (-17.9492 vs -3.8278). Explain how we could make
sense of this assuming the instrument is valid. [8 points]
(c) Explain how we could test whether we actually have to use an instrumental variables
strategy or whether OLS would be consistent as well. You can assume that the instru-
ment is valid. [8 points]
(d) In column (1) of Table 3 we show what happens if we use an LPM model to explain
the probability of employment (in 2010 since we’re using only the data for 2010). So
instead of hours the dependent variable is now employed, which can be zero or one,
depending on whether the respondent had usual hours of work greater than zero or
not. In column (2) we use a Logit model for the same dependent variable employed.
Calculate the probability of employment for a woman age 25 with 4 children who had
her first child at age 18 using the LPM and Logit estimates (show your calculations, the
result is not enough). Explain how you would calculate the partial effect of nchildren
for that woman (you don’t have to actually calculate it). Note that Λ(z) = exp(z)
1+exp(z) and
that ∂Λ(z)/∂z = exp(z)
[1+exp(z)]2 . [8 points]
(e) Throughout these questions we have limited the sample to women age 21-40 born in
the UK with at least two children (in part to be able to use the samesex instrument).
Explain why this means that our estimates are potentially not representative for the
population of all mothers in the UK. Briefly comment on whether the Heckman model
would potentially help us solve this problem. [8 points]
5
Table 1: Panel Estimates of Labour Supply Effect of Children
(1) (2) (3)
POLS FixedEffects
First
Differences
nchildren -3.8807 -1.1740 -1.0188
(0.2224) (0.4229) (0.4378)
age 3.2120
(0.4990)
age2 -0.0377 0.0124 0.0090
(0.0076) (0.0158) (0.0184)
agefirstbirth -0.0763
(0.0421)
year10 -0.2250 -0.6153
(0.4510) (1.1137)
year11 -0.1162 -0.7558 0.5590
(0.4591) (2.1748) (0.3376)
_cons -40.0236 1.9776 -0.4933
(8.0120) (17.7920) (1.2912)
N 6,586 6,586 2,762
R2 0.095 0.009 0.003
Standard errors in parentheses
6
Table 2: OLS and 2SLS Estimates of Labour Supply Effect of Children (Only for Year 2010)
(1) (2) (3)
Dependent variable hours nchildren hours
Estimator OLS 2SLS
(first stage) (second stage)
nchildren -3.8278 -17.9492
(0.3461) (8.7718)
age 3.4950 0.2421 7.8724
(0.7660) (0.0420) (2.3769)
age2 -0.0420 -0.0032 -0.0998
(0.0117) (0.0006) (0.0324)
agefirstbirth -0.1483 -0.0640 -1.3106
(0.0655) (0.0034) (0.5693)
samesex 0.0846
(0.0260)
_cons -43.2097 -0.4749 -50.7753
(12.2403) (0.6761) (17.6950)
N 2,734 2,734 2,734
R2 0.090 0.123
Standard errors in parentheses
7
Table 3: Probability of Employment (Only for Year 2010)
(1) (2)
LPM Logit
nchildren -0.1237 -0.5571
(0.0109) (0.0578)
age 0.1460 0.6774
(0.0248) (0.1268)
age2 -0.0019 -0.0090
(0.0004) (0.0019)
agefirstbirth 0.0029 0.0113
(0.0023) (0.0100)
_cons -1.9296 -11.2025
(0.3923) (2.0472)
N 2734 2734
(pseudo) R2 0.087 0.066
Log likelihood -1,765.25
Standard errors in parentheses.
(robust standard errors for LPM)
8
9
— END OF EXAMINATION PAPER —
10