Introduction to Econometrics
Sample exam
You have 80 minutes to complete this setup. The exam is worth 60 points in total, exact
amount of points is indicated for each exercise.
1. (10 points) Describe the heteroskedasticity problem: explain briefly what it is and
how it affects the estimation. Give the name of at least one of the tests for heteroskedasticity
and state its null hypothesis. Provide at least one solution of the
heteroskedasticity problem.
2. (6 points) Describe the two properties a valid instrumental variable must satisfy.
3. (5 points) Decide if the following claim is true or false (and explain why): “I run a
regression of y on x and I save the residuals e = y − y. If I find that Cov(x, e) = 0,
I have the right to conclude that the variable x was exogenous in my regression.”
4. Suppose a sample of adults is classified into groups 1, 2 and 3 on the basis of whether
their education stopped at the end of elementary school, high school, or university,
respectively. The relationship
y = β1 + β2D2 + β3D3 + ε
is specified, where y is income, Di = 1 for those in group i and zero for all others.
(a) (3 points) Explain why D1 is not included in the regression.
(b) (4 points) In terms of the parameters of the model, what is the expected
income of people whose education stopped at the end of university? What is
the expected income of people whose education stopped at the end of elementary
school?
(c) (6 points) Suppose some respondents who only finished the elementary school
were embarrassed about their lack of education and lied, claiming that they
graduated from high school. What would be the impact of this lie on the
estimates of β2 and β3?
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5. (26 points) You have data for 732 student-athletes from a large university for fall
semester. Your primary question of interest is this: Do athletes perform more poorly
in school during the semester when their sport is in season?
(a) You run a regression with the following variables:
trmgpa . . . the student’s GPA (grade point average) for the semester
season . . . dummy, equal to 1 if the student’s sport is in season
that semester, 0 otherwise
hsrank . . . the student’s performance at high school, measured as
the rank among his/her classmates
crsgpa . . . the course GPA (average GPA over all students taking the course)
for the semester
You obtain the following result:
_cons -.0829782 .327399 -0.25 0.800 -.725737 .5597806
crsgpa .9246755 .1165595 7.93 0.000 .6958426 1.153508
hsrank -.0021305 .0002344 -9.09 0.000 -.0025907 -.0016703
season -.1091708 .0546164 -2.00 0.046 -.2163952 -.0019463
trmgpa Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 420.296956 731 .574961636 Root MSE = .69367
Adj R-squared = 0.1631
Residual 350.300799 728 .481182416 R-squared = 0.1665
Model 69.996157 3 23.3320523 Prob > F = 0.0000
F( 3, 728) = 48.49
Source SS df MS Number of obs = 732
i. Interpret the coefficient on season. Is it significant at 95% confidence level?
ii. Test if the coefficient on crspga is significantly different from 1 at 95%
confidence level1
. Interpret your finding.
iii. Explain what hsrank controls for in the regression. (Hint: the lower
hsrank, the better the high school performance of the student).
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for critical values, see Appendix on p.3
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(b) Most of the athletes who play their sport in the fall are football players. You
suppose the academic performance of football players differ systematically from
those of other athletes.
You include in you regression a new variable football, which is a dummy equal
to one if the student is football player, zero otherwise. You get the following
result:
_cons .1937855 .3151154 0.61 0.539 -.4248594 .8124303
football -.4479337 .0543623 -8.24 0.000 -.5546595 -.3412078
crsgpa .8978952 .1115946 8.05 0.000 .6788091 1.116981
hsrank -.001966 .0002252 -8.73 0.000 -.0024081 -.0015238
season .0018219 .0539756 0.03 0.973 -.1041449 .1077886
trmgpa Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 420.296956 731 .574961636 Root MSE = .66384
Adj R-squared = 0.2335
Residual 320.380658 727 .440688664 R-squared = 0.2377
Model 99.9162975 4 24.9790744 Prob > F = 0.0000
F( 4, 727) = 56.68
Source SS df MS Number of obs = 732
Is the coefficient on season significant at 95% confidence level now? Does this
confirm there was a bias in part (a)? If yes, explain how this bias was created
and what was its sign (intuitive explanation is sufficient).
Appendix:
Extract from statistical table of Student t-distribution (area under right-hand tail)
d.f. 0.05 0.025 0.01
40 1.684 2.021 2.423
60 1.671 2.000 2.390
120 1.658 1.980 2.358
∞ 1.645 1.960 2.326
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