Introduction to Econometrics
Home assignment # 2
(Deadline: Friday, November 10, 11:00 a.m., by email or a hard-copy in class, absolutely
no late submissions will be accepted)
In this assignment, there is one computer exercise that is to be computed using Gretl.
No other statistical software is allowed. You should present your results as a printout from
the program (e.g. copy the output from Gretl to MS Word and print it out, or print it out
directly from Gretl). When you are asked to comment your results, you can do so in the
printout or on a separate sheet. Do not forget that when you are asked to test a hypothesis,
it is not suﬃcient to present just the result of the test as it is presented in Gretl: you have
to provide a clear conclusion whether you reject or not the null hypothesis, which has to
be formally stated.
1. Your are given the following model
y = β1 + β2X2 + β3X3 + β4X4 + β5X5 + β6X6 + ε
Assume that you want to test the following set of restrictions:
(a) β2 − β3 = 1
(b) β4 = β6 and β5 = 0
Construct models that incorporate restrictions (a) and (b), separately and together.
Describe what test you will use to test the restrictions, including its distribution and
parameters (i.e., describe how would you test: the restriction (a), the restrictions
(b), and all of them together).
2. Imagine you are interested in the determinants of the revenues in shoe stores in
Prague. Suppose you have speciﬁed the following model:
Revt = α + βInct + δPricet + θPopult + ηWeekendt + εt ,
where Revt denotes the amount of revenues in the Prague shoe stores on a particular
day t, Inct is per capita income in Prague, Pricet is a price index for shoes relative to
other goods in Prague, Popult is number of people living in Prague, and Weekendt
is a dummy variable for weekend days.
(a) This speciﬁcation recognizes that people might go shopping for shoes more often
on weekends than on working days. Explain how would you test for such a
hypothesis.
(b) What are the predicted revenues (in terms of the coeﬃcients of the model) for
weekends and for working days?
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(c) Explain how would you alter the speciﬁcation to account for the fact that people
may buy more shoes during the sales period, which is in January and July.
(d) If people have higher income, they buy more shoes on weekends (i.e., the eﬀect
of per capita income on revenues is larger on weekends compared to working
days). Is this incorporated in your speciﬁcation? If not, how would you do it?
How would you test for the hypothesis that if people have higher income, they
buy more shoes on weekends?
3. Use data ceosal2.gdt for this exercise. Consider an equation to explain salaries of
CEOs in terms of annual ﬁrm sales:
ln(salary) = β0 + β1 ln(sales) + β2roe + β3neg ros + ε ,
where
salary . . . CEO’s salary in thousands USD
sales . . . ﬁrm’s sales in millions USD
roe . . . ﬁrm’s return on equity
neg ros . . . dummy, equal to 1 if return on ﬁrm’s stock is negative
(a) Deﬁne the variables you need and estimate the equation.
(b) What is the interpretation of the coeﬃcients β1, β2, and β3?
(c) Test for the presence of a signiﬁcant impact of ﬁrm’s sales on CEO’s salary
by hand (using only the estimated coeﬃcient and the standard error from the
Gretl output) and then compare your results to the results of this test in Gretl.
Deﬁne the null and alternative hypothesis, the test statistic, its distribution,
and interpret the results of the test.
(d) You wonder if the impact of ﬁrm’s return on equity on the CEO’s salary is indeed
linear. You decide to test for the presence of a non-linear relationship, which you
approximate by a third order polynomial of roe (i.e., α1roe + α2roe2
+ α3roe3
).
i. Deﬁne the null and alternative hypothesis, the test statistics and its distribution.
Describe all speciﬁcations you need to be able to conduct the
test, construct the necessary variables, and estimate these speciﬁcations in
Gretl.
ii. Calculate the test statistics by hand, compare to the critical value at 99%
signiﬁcance level, and interpret the results.
iii. Conduct the test in Gretl and compare the results.
4. Suppose that you have a sample of n individuals who apart from their mother tongue
(Czech) can speak English, German, or are trilingual (i.e., all individuals in your
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sample speak in addition to their mother tongue at least one foreign language). You
estimate the following model:
wage = β0 + β1educ + β2IQ + β3exper + β4DM + β5Germ + β6Engl + ε ,
where
educ . . . years of education
IQ . . . IQ level
exper . . . years of on-the-job experience
DM . . . dummy, equal to one for males and zero for females
Germ . . . dummy, equal to one for German speakers and zero otherwise
Engl . . . dummy, equal to one for English speakers and zero otherwise
(a) Explain why a dummy equal to one for trilingual people and zero otherwise is
not included in the model.
(b) Explain how you would test for discrimination against females (in the sense that
ceteris paribus females earn less than males). Be speciﬁc: state the hypothesis,
give the test statistic and its distribution.
(c) Explain how you would measure the payoﬀ (in term of wage) to someone of
becoming trilingual given that he can already speak (i) English, (ii) German.
(d) Explain how you would test if the inﬂuence of on-the-job experience is greater
for males than for females. Be speciﬁc: specify the model, state the hypothesis,
give the test statistic and its distribution.
5. Your aim is to estimate how the number of prenatal examinations and several other
characteristics inﬂuence the birth weight of a baby. Your initial hypothesis is that
more responsible pregnant women visit the doctor more often and this leads to healthier
and thus also bigger babies.
(a) In your ﬁrst speciﬁcation, you run the following model:
bwght = β0 + β1 npvis + β2 npvis2
+ β3 monpre + β4 male + ε ,
where bwght is birth weight of the baby (in grams), npvis is the number of
prenatal doctor’s visits, monpre is the month on pregnancy in which the prenatal
care began and male is a dummy, equal to one if the baby is a boy and zero if
it is a girl. You obtain the following results form Stata:1
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Stata is a statistical software, which can be used to for econometric purposes. The Stata output
is quite similar to the Gretl output you are familiar with. In particular, Coef. denotes the estimated
coeﬃcients, Std.Err. denotes the standard errors of these coeﬃcients, t denotes the t-statistic of the test
of signiﬁcance of the coeﬃcients, P > |t| denotes the corresponding p-value.
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_cons 2853.196 101.3073 28.16 0.000 2654.498 3051.895
male 76.69243 27.76083 2.76 0.006 22.24391 131.141
monpre 30.47033 12.40794 2.46 0.014 6.134091 54.80657
npvissq -1.173175 .3591552 -3.27 0.001 -1.877601 -.4687481
npvis 53.50974 11.41313 4.69 0.000 31.12468 75.8948
bwght Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 582851231 1725 337884.772 Root MSE = 575.5
Adj R-squared = 0.0198
Residual 570003184 1721 331204.639 R-squared = 0.0220
Model 12848047.5 4 3212011.87 Prob > F = 0.0000
F( 4, 1721) = 9.70
Source SS df MS Number of obs = 1726
i. Is there strong evidence that npvissq (stands for npvis2
) should be included
in the model?
ii. How do you interpret the negative coeﬃcient of npvissq?
iii. Holding npvis and monpre ﬁxed, test the hypothesis that newborn boys
weight by 100 grams more than newborn girls (at 95% conﬁdence level).
(b) A friend of yours, student of medicine, reminds you of the fact that the age of
the parents (especially of the mother) might be a decisive factor for the health
and for the weight of the baby. Therefore, in your second speciﬁcation, you
decide to include in your model also the age of the mother (mage) and of the
father (fage). The results of your estimation are now the following:
_cons 2592.813 139.6173 18.57 0.000 2318.974 2866.651
fage 8.697342 3.465973 2.51 0.012 1.899357 15.49533
mage .5285275 4.218069 0.13 0.900 -7.744582 8.801637
male 74.45482 27.75247 2.68 0.007 20.02252 128.8871
monpre 34.35661 12.69477 2.71 0.007 9.457725 59.2555
npvissq -1.138545 .3585648 -3.18 0.002 -1.841816 -.4352743
npvis 52.43859 11.40558 4.60 0.000 30.06826 74.80891
bwght Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 579528396 1719 337131.121 Root MSE = 573.42
Adj R-squared = 0.0247
Residual 563258231 1713 328813.912 R-squared = 0.0281
Model 16270165.8 6 2711694.3 Prob > F = 0.0000
F( 6, 1713) = 8.25
Source SS df MS Number of obs = 1720
i. Comment on the signiﬁcance of the coeﬃcients on mage and fage separately:
are they in line with your friend’s claim?
ii. Test the hypothesis that mage and fage are jointly signiﬁcant (at 95%
conﬁdence level). Is the result in line with your friend’s claim?
iii. How can you reconcile you ﬁndings from the two previous questions?
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(c) In your third speciﬁcation, you decide to drop fage and you get the following
results:
_cons 2648.851 137.2778 19.30 0.000 2379.602 2918.1
mage -6.91257 3.137972 -2.20 0.028 -13.06721 -.757928
male 79.38175 27.75667 2.86 0.004 24.94136 133.8221
monpre 35.25912 12.58328 2.80 0.005 10.57898 59.93927
npvissq -1.142647 .3590214 -3.18 0.001 -1.846811 -.4384821
npvis 52.27885 11.41406 4.58 0.000 29.89196 74.66575
bwght Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 582851231 1725 337884.772 Root MSE = 574.86
Adj R-squared = 0.0220
Residual 568399545 1720 330464.852 R-squared = 0.0248
Model 14451685.6 5 2890337.13 Prob > F = 0.0000
F( 5, 1720) = 8.75
Source SS df MS Number of obs = 1726
Comment on the signiﬁcance of the coeﬃcient on mage, compared to the results
from part (b). Is your ﬁnding in line with your reasoning in part (b)? Does it
conﬁrm your friend’s claim?
(d) Having regained trust in your friend, you consult your results once more with
him. Together, you come up with an interesting question: whether smoking
during pregnancy can aﬀect the weight of the baby. Fortunately, you have at
your disposition the variable cigs, standing for the average number of cigarettes
each woman in your sample smokes per day during the pregnancy, and so you
can include it in your model. However, your friend warns you that women
who smoke during pregnancy are in general less responsible than those who do
not smoke, and that these women also tend to visit the doctor less often. (In
other words, the more the women smokes, the less prenatal doctor’s visits she
has). This is an important fact that you have to take into consideration while
interpreting your ﬁnal results, which are:
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_cons 2748.856 141.868 19.38 0.000 2470.591 3027.12
cigs -10.209 3.398309 -3.00 0.003 -16.87456 -3.54344
mage -6.980738 3.227181 -2.16 0.031 -13.31064 -.6508356
male 82.39438 28.34937 2.91 0.004 26.78897 137.9998
monpre 31.77658 12.78156 2.49 0.013 6.706395 56.84676
npvissq -.8948737 .3624432 -2.47 0.014 -1.605782 -.1839653
npvis 42.43442 11.59582 3.66 0.000 19.68999 65.17885
bwght Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 537842203 1621 331796.547 Root MSE = 569.22
Adj R-squared = 0.0235
Residual 523281374 1615 324013.235 R-squared = 0.0271
Model 14560828.9 6 2426804.81 Prob > F = 0.0000
F( 6, 1615) = 7.49
Source SS df MS Number of obs = 1622
i. Interpret the coeﬃcient on cigs.
ii. What evidence do you ﬁnd that cigs really should be included in the model?
List at least two arguments.
iii. Compare the coeﬃcient on npvis with the one you obtained in part (c). Do
you think there was a bias? If yes, explain where it came from and interpret
its sign.
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