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Section A
The questions in this section are based on the work presented in The impact of undergraduate
degrees on early-career earnings by Chris Belfield et al. (IFS Research Report, Nov 2018).
You were asked to read this paper before this examination so that you could answer these
questions. Relevant tables are reproduced in the Appendix.
1. In what ways does this report aim to add to the existing literature on the graduate earnings
premium in the UK. In your answer you should refer to the claim that this is a relatively
new and unique dataset for investigating graduate earnings.
[5 MARKS]
Answer: While there is strong empirical evidence of the existence of a positive graduate
earnings premium, this evidence has mainly been assembled on the basis of survey data.
Here, the authors use linked administrative data which contain high-school achievements,
higher education records and tax office data. This increases the set of variables which can
be used to proxy prior academic attainment and socio-economic background. This allows
a strengthening of the causal argument. The data means that it is possible to estimate
of course-specific graduate earnings premia.
2. Describe how the authors of the study construct a control group against which they
compare earnings outcomes of graduates.
[5 MARKS]
Answer: The dataset contains individuals who did not go to university but, given their
school-leavers grades, would have qualified to go to University. School leavers who did not
have the required grades to go to University are not included in the analysis. Importantly,
and different to previous work, HE dropouts are included in the treatment group.
3. Table 3 reports some summary statistics. What can you learn from this table when
comparing the “no higher education (no HE)” control group with the HE treatment
group? Note that the SES variable is a measure of social deprivation.
[5 MARKS]
Answer: There are some differences between the overall treatment group and the control
group. In particular, the control group fairs worse in academic achievement. For instance,
in the non-HE group only 9% of women have the highest Maths qualification while 31%
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of women in the HE group have the same.
The other area with notable differences is the measurement of social deprivation (SES).
Of course these differences will be more pronounced when comparing the non-HE group
to more specific HE subgroups, e.g. those who do an engineering degree.
4. In Figure 8, the raw differential between someone studying Economics and someone study-
ing Physics is about £10,000. Are there other factors one needs to consider before making
such comparisons?
[10 MARKS]
Answer: What we can see here are raw differentials which are not conditioned in any
way. Only after controlling for any relevant differences between the group of economics
and physics students in terms of prior achievement and social background could one start
making such a statement. Even then one ought to be aware of the potential impact of
unobserved differences.
5. In Model (6) the authors present their main specification, which is rewritten as follows:
Yit = X
′
iγ + ω1t+ ω2t
2 + α1di + α2(dit) + α3(dit
2) + it.
Yit is annual earnings, di is the treatment dummy (treated by HE) and t is age minus 24.
Also, X ′i contains cohort and age controls (cohort of gradution, age at which university
was started), background characteristics (whether attended sixth form, A-level subject
mix, school type, SES background, ethnicity, region) and prior attainment (as measured
by GCSE and A-level points scores).
(a) From this main specification, derive expressions for E(Y |X, t, d = 1) and E(Y |X, t, d =
0). Hence show that the difference, the conditional HE differential, can be written
as α1 + α2t+ α3t
2. Evaluate this expression when aged 29.
[5 MARKS]
Answer:
E(Y |X, t, d = 1) = X ′γ + ω1t+ ω2t2 + α1 + α2t+ α3t2
E(Y |X, t, d = 0) = X ′γ + ω1t+ ω2t2
and so expression follows. When aged 29, t = 5 and so the conditional diff becomes
α1 + 5α2 + 25α3.
(b) The authors label this conditional HE differential as “Overall returns to HE at age
29”. These are reported in Table 7 separately for men and women. Interpret these
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estimates for women in the columns (1) to (4) and explain how and why they change
as additional controls are included.
[15 MARKS]
Mainly taken from the report’s discussion prior to Table 7:
Women who attend HE earn considerably more at age 29 on average: women who
attend HE earn 0.44 log points (55%) more than women who do not. In Column 2,
we control for the age students start university and report the earnings differential
between those who start university at age 18 and those who do not go. This
increases the estimated earnings difference because those who start university later
typically earn less at age 29 due to having less post-HE work experience. Some of
this gap in earnings is a result of HE and non-HE students differing in background
characteristics that affect their earnings potential. We know, for example, that
HE students typically have higher prior attainment and are more likely to come from
better-off families. Columns 3 and 4 add controls for background characteristics and
prior attainment respectively. Controlling for different background characteristics
such as socio-economic status, ethnicity and region does reduce the estimated return
to HE, but only slightly. Controlling for prior attainment, however, has a dramatic
effect on the estimated returns for both men and women.
6. An international student complains that the IFS Report does not allow him to decide
whether studying an Economics degree at The University of Manchester is a worthwhile
investment. Comment.
[5 MARKS]
Indeed international students were not included in the analysis and hence it is not imme-
diately valid to extend the analysis to international students. The reason for this is that
these would not be covered in the National Pupil Database which served as the population
for this study. Having said that it is likely that similar earnings increasing effects (if not
stronger ones) apply to international students especially if the student was to decide to
work in the UK.
Also one point for mentioning that causal interpretation is not guaranteed (even without
mentioning the international issue).
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Section B
7. A dataset records average earnings, gender, and age for a large sample of individuals over
3 successive years. Not all individuals are observed 3 times. Is this dataset (a) a pooled
cross-section or (b) a micro-econometric panel or (c) macro-econometric panel or (d) a
single cross-section? Give 1 or 2 sentences explaining your answer.
[3 MARKS]
Answer: It is (b) a micro-econometric panel. This is because individuals are observed
repeatedly. Micro-econometric panels have lots of units and very few time-periods, some-
times only 2.
One of these types of dataset is fundamentally unsuitable for constructing a difference-
in-difference estimator. Which? Give a one sentence answer.
[3 MARKS]
Answer: (d). A difference-in-difference estimator needs a time dimension, which by defi-
nition, a single cross-section does not have.
8. In the simple regression model
y = α + βx+ u where Cov(u, x) 6= 0,
consider the following statements:
S1 x is said to be endogenous.
S2 The OLS estimator β̂ is unbiased.
S3 The effect of x on y (given by the OLS estimate β̂) is said to be causal.
Write down whether these statements are true or false. Do not explain why.
[6 MARKS]
The incorrect statements are S2 and S3. [2 marks for each correct statement.]
9. Consider the following statement:
S1 In the Potential Outcomes model, under non-random assignment, the ATE and ATT
coincide, and the raw differential can be interpreted as a causal effect.
If this statememt is false, write it out again underlining the corrected word(s). Otherwise,
write “S1 is true”.
[4 MARKS]
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Answer: In the Potential Outcomes model, under random assignment, the ATE and ATT
coincide, and the raw differential can be interpreted as a causal effect.
10. Consider the following regression
y = β0 + δd+ β1T + τdT + u,
where d is a time dummy for after the treatment occurs, and T is a dummy for whether
treated. After 4 appropriate exogeneity assumptions have been imposed:
E(y|d = 0, T = 0) = β0
E(y|d = 0, T = 1) = β0 + β1
E(y|d = 1, T = 0) = β0 + δ
E(y|d = 1, T = 1) = β0 + δ + β1 + τ.
Which of the following statements are true?
S1 τ is the DiD estimator in the population.
S2 β1 is what would have been the effect of being treated, except it happens in the
“before” period d = 0.
S3 δ is a macro effect, for both treated and control groups.
If a statement is false, write it out again underlining the corrected word(s). Otherwise,
write “Sx is true”.
[8 MARKS]
S3 is false. It should read “δ is a macro effect, but only for the control group.” [2 marks
for each statement and 2 marks for correcting S3.]
11. The simplest returns-to-education model is a regression of log earnings y on years-of-
schooling x1:
y = β0 + β1x1 + u.
When estimating the regression using a sample of data from the BHPS/US, the inves-
tigator’s OLS estimate of β1 is 0.08. However, it is well known that age x2 should also
be included in the regression (to proxy experience). Before he re-estimates his amended
earnings regression, he decides to estimate
x2 = δ0 + δ1x1 + ε,
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just to see what is happening with the two RHS variables. His OLS estimate of δ1 is very
small and statistically insignificant. What do you think will happen to the OLS estimate
of β1 in
y = β0 + β1x1 + β2x2 + u?
S1 The new estimate of β1 will go up by a significant amount.
S2 The new estimate of β1 will go down by a significant amount.
S3 The new estimate of β1 will hardly change.
S4 There is not enough information to say what will happen to new estimate of β1.
Which one of these statements is correct? Justify your answer using the OVB formula.
[7 MARKS]
Answer: We are comparing estimates from Long and Short regressions
y = β0 + β1x1 + β2x2 + e
y = β0 + β1x1 + e
Estimating Short, the OVB formula says that
Bias(β̂1) = E(β̂1)− β1 ≡ β2 Cov(x2, x1)
Var(x1)
= β2δ1,
where δ1 is the “slope” parameter in population model that relates x1 and x2:
x2 = δ0 + δ1x1 + error.
In other words:
β˜1 − β̂1 = β̂2δ̂1.
His second regression establishes clearly that δ1 = 0. In words, x1 and x2 are uncorrelated.
It doesnt matter what sign β2 takes. The correct answer is S3.
12. Consider the following model
y = α + βx∗ + β < 0. (1)
The RHS covariate x∗ is measured with error, so that x is the variable actually observed
in the data at hand:
x = x∗ + e,
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where e is the measurement error. The following assumptions are true:
Cov(, x∗) = 0
Cov(e, x∗) = 0
Cov(e, ) = 0
E() = E(e) = 0.
The model that is estimated is:
y = α + βx+ u.
We know that the bias in the OLS estimator is given by:
Bias(β̂) ≡ E(β̂)− β = Cov(u, x)
Var(x)
.
a) Derive a formula for E(β̂) in terms of the true β, Var(x∗) and Var(x).
[8 MARKS]
Answer: Substituting out x∗ into (1):
y = α + β(x− e) + = α + βx+ (− βe).
Comparing with the model that is estimated, the term in brackets is the regression
error:
u = − βe.
Working through this covariance:
Cov(u, x) = Cov(− βe, x∗ + e) =
Cov(, x∗) + Cov(, e)− βCov(e, x∗)− βCov(e, e) =
0 + 0− 0− βVar(e) = −βVar(e) < 0.
The 3 zero covariances are assumed in the question. (Note that Cov(u, x) can be
signed, ie is unambiguously positive as β < 0.) Substituting into the formula given:
E(β̂) = β
(
1− Var(e)
Var(x)
)
= β
Var(x∗)
Var(x)
,
as Var(x) = Var(x∗) + Var(e).
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b) Suppose that 30% of the variation in x is due to variation in the measurement error
e and that the true value of β is –0.4. On average, what will the OLS estimator β̂
turn out to be?
[3 MARKS]
Answer: The question states that Var(e)
Var(x)
= 0.3 and so E(β̂) = (1− 0.3) ∗ (−0.4) =
−0.28.
13. Briefly explain what a spurious regression is and why it is important to understand the
stationarity property of variables included into a regression model.
[10 MARKS]
There are a number of angles one could attack this question, but most likely an answer
should include:
• spurious regresisons indicate statistically significant relationships when there is no
underlying link
• variables are said to be stationary if they have constant means and variances (and
covariances - but not expected).
• if variables that are non-stationary are included in a regression it is much more likely
to find spurious results
• the reason for that being that statistical tests do not have their usual distributions.
• evaluate the stat prop of a series to a) know whether a spurrious relationship is a
possibility and b) to establish whether differencing turned a nonstationarity into a
stationary series.
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1 Appendix
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END OF EXAMINATION
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