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January 2019
EFIM20011
ECONOMETRICS
Time allowed: TWO hours and THIRTY minutes
Answer ALL questions
30 marks are allocated to Section A. 35 marks are allocated to Section B. 35 marks are
allocated to Section C.
Justify all your answers. Non programmable calculators may be used, but candidates
must show the basis of all calculations.
TURN OVER
Section A. [30 marks]
Answer all of the following 6 questions. Each question is worth 5 marks. Give concise
justifications for all your answers.
1. We throw a cubic die n times and compute the average of the n results. If n = 5 the
average is 4.6. If n = 10 the average is 3.1. If n = 50 the average is 3.3 and if n = 100
it is equal to 3.4. Explain this result pattern. What average would we observe if we
threw the die 10,000 times?
2. Let W denote the wage and M be a dummy variable equal to 1 if the individual is
male and equal to 0 if the individual is female. We regressW onM (and a constant)
by OLS using a large iid sample and denote the estimated coefficient of M as bb.
What is the interpretation of bb?
3. Consider the linear model Y = a+ b ·X1 + c ·X2 + d ·X3 + U where X1, X2, X3 are
observed exogenous regressors, a, b, c, d are parameters and U is an unobserved
error term. We estimate the parameters using an OLS regression on an iid sample
with 1,000 observations and we want to test whether the three regressors jointly
have no effect on the outcome Y . Which test should we use and what would be the
distribution of the test statistic under the null assumption?
4. Consider the linear model Y = a + b · X + c · W + U where W is an exogenous
regressor and U is an unobserved error term. Let Z be an observed variable. Write
down the conditions required for Z to be a valid instrument for X in this linear model.
5. We want to know the effect of a training program (for unemployed workers) on unem-
ployment duration. Write down a potential outcome model for this evaluation problem.
6. What are the conditions under which an Instrumental Variable estimator only cap-
tures a Local Average Treatment Effect (LATE)?
Section B. [35 marks]
We study the gender wage gap in a population. We have one large iid sample of male and
female workers. We consider the following variables:
• W is the (monthly) wage of an individual (in US dollars),
• M is a male gender dummy, equal to 1 for men and to 0 for women,
• A is the individual’s age (in years),
• S is the individual’s education, measured as years spent in education.
We define the linear model: W = a+ b ·M + c ·A+ d · S + U , where U is an error term.
1. What is the interpretation of the parameters b, c and d? [3 marks]
We regress W on M , A and S (and a constant) by OLS using our data sample. The
estimates of a, b, c and d (and their estimated variances) are:ba = 550, bb = 950, bc = 42, bd = 29.5bV (ba) = 184, bV ⇣bb⌘ = 1.2, bV (bc) = 0.4, bV ⇣bd⌘ = 0.1
2. Write down the first-order conditions characterising these OLS estimates. You are
not asked to solve this system of equations. [4 marks]
3. What is the interpretation of bb? What is the interpretation of bd? [3 marks]
4. Write down the exogeneity assumption in the linear model and interpret this assump-
tion. If exogeneity holds, how can you interpret the estimates bb, bc and bd? [4 marks]
From now on, we assume that exogeneity holds.
5. Test whether there is no wage difference between men and women at the 5% signif-
icance level, even when controlling for age and education. [3 marks]
6. Is there evidence of a positive wage difference between men and women at the 5%
level, even when controlling for age and education? [3 marks]
7. How would you test for the equality of wage returns to age and wage returns to
education? You are not asked to conduct this test. [3 marks]
8. Suggest a model specification that would allow for the effect of education and the
effect of age to be different across genders. How would you then test whether wage
returns to age are different between men and women? You are not asked to conduct
this test. [7 marks]
9. Explain how an omitted variable could lead to the exogeneity assumption being vio-
lated in the linear model. How could selection also affect the credibility of the exo-
geneity assumption? [5 marks]
TURN OVER
Section C. [35 marks]
This exercise is based on an article published in 2004 in the Journal of Labor Economics.
We study the effect of firm-sponsored training programs on workers’ wage in the Nether-
lands.
We have a large iid sample of Dutch workers and observe their wage W , their age A,
genderM (1 for men, 0 for women), education S (years of schooling) and a dummy variable
T equal to 1 if the worker received training in the past year and equal to 0 otherwise.
We consider the following linear model: (referred to as model 1)
W = ↵+ ·M + · S + ·A+ · T + U,
where U is an unobserved error term and ↵, , , and are parameters.
1. Explain why not controlling for the worker’s skill level may lead an OLS estimation of
model (1) to produce a biased estimate of the effect of training on wages. [4 marks]
2. If firms offer more training opportunities to low-skill workers, would the OLS estimate
of be biased upwards or downwards? [4 marks]
To overcome identification issues, we can use the following tax policy: in the Netherlands,
employers get an extra tax deduction for training their employers who are at least 40 years
old (this does not apply to workers below 40). We then create a dummy variable D equal
to 1 if the worker is at least 40 years old and equal to 0 if the worker is younger than 40.
3. Explain why this policy motivates a fuzzy regression discontinuity design. [5 marks]
4. Present the two conditions for D to be a valid instrument for T in model (1). Discuss
these assumptions. [6 marks]
5. Present the two stages of the 2SLS estimation of model (1) usingD as an instrument
for T . You need to mention precisely which regressors and outcomes are used in
each stage. [5 marks]
6. Which statistic should we use to assess the strength/weakness of the instrument? If
this statistic is equal to 5.14 in the data, what should we conclude? [4 marks]
7. The 2SLS estimator of is equal to -1150 with an estimated standard error of 960.
Give a 95% confidence interval for the effect of training on wages. [4 marks]
8. Can we conclude that is significantly different from 0 at the 1% level? [3 marks]
END OF EXAM