Introduction to Econometrics
Home assignment # 2
(Suggested solutions)
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 restriction (b),
and both of them together).
Solution:
• First, let us test the restrictions separately.
(a) We can express
β2 = 1 + β3
and plug into the unrestricted model:
y = β1 + (1 + β3)X2 + β3X3 + β4X4 + β5X5 + β6X6 + ε
y − X2 = β1 + β3(X3 + X2) + β4X4 + β5X5 + β6X6 + ε .
We have here J = 1 restriction, n observations and k = 6 parameters.
Hence, to test the restrictions, we should run the unrestricted model
y = β1 + β2X2 + β3X3 + β4X4 + β5X5 + β6X6 + ε
and the restricted model
y − X2 = β1 + β3(X3 + X2) + β4X4 + β5X5 + β6X6 + ε ,
save SSE in both cases and test
F =
(SSER − SSEU )/J
SSEU /(n − k)
=
(SSER − SSEU )/1
SSEU /(n − 6)
∼ F1,n−6 .
Note that since we have only one restriction, we could also use the fact that
√
F ∼ tn−6
and if n is large, also √
F ∼ N(0, 1) .
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(b) When we plug these restrictions into the original model, we obtain
y = β1 + β2X2 + β3X3 + β6(X4 + X6) + ε .
We would apply the same method as in (a) with the only diﬀerence that
now we have J = 2 restrictions and so
F =
(SSER − SSEU )/J
SSEU /(n − k)
=
(SSER − SSEU )/2
SSEU /(n − 6)
∼ F2,n−6 .
• Second, let us test the restrictions together. The restricted model now becomes
y − X2 = β1 + β3(X3 + X2) + β6(X4 + X6) + ε
and J = 3, so we have
F =
(SSER − SSEU )/J
SSEU /(n − k)
=
(SSER − SSEU )/3
SSEU /(n − 6)
∼ F3,n−6 .
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?
(c) Explain how would you alter the speciﬁcation to account for the fact that people
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?
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Solution:
(a) This speciﬁcation allows for diﬀerent intercept for weekends and for working
days (ceteris paribus). This could reﬂect the fact most people go shopping on
weekends, possibly making the revenues larger (so η > 0), which could be a
hypothesis to be tested using a one-sided t-test:
H0 : η ≤ 0, HA : η > 0
t =
η
s.d.(η)
∼ tn−5,0.95
(b) For weekend days the model looks like:
ˆRevt = (ˆα + ˆη) + ˆβInct + ˆδPricet + ˆθPopult ,
whereas during working days it is:
ˆRevt = ˆα + ˆβInct + ˆδPricet + ˆθPopult ,
(c) To recognize the eﬀect of a sales period (January and July), we have to create
a dummy variable indicating if the day in question is in Junuary or July or not:
Salest =
1 if day t is in January or July
0 otherwise
We introduce this dummy variable in our model as an additional independent
variable:
Revt = α + βInct + δPricet + θPopult + ηWeekendt + γSalest + εt ,
(d) If we want to incorporate in our speciﬁcation that people buy more shoes on
weekends if they have higher incomes, we need to allow for diﬀerent coeﬃcients of
Inc for weekends and working days. In other words, we need to allow for both
the diﬀerent intercept and slope coeﬃcient of Inc for weekends and working
days. During weekends the model should look like
Revt = (α + η) + (β + ω)Inct + δPricet + θPopult + εt
and during working days it should look like
Revt = α + βInct + δPricet + θPopult + εt .
We can achieve that by using the dummy variable Weekend in the following
way:
Revt = α + (β + ωWeekendt)Inct + δPricet + θPopult + ηWeekendt + εt ,
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so that ﬁnally we obtain
Revt = α + βInct + ωWeekendtInct + δPricet + θPopult + ηWeekendt + εt ,
If we assume that people buy more shoes on weekends if they have higher incomes,
the coeﬃcient ω should be signiﬁcant and positive, which is a hypothesis
that could be tested using one-sided t-test (H0 : ω ≤ 0, HA : ω > 0).
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.
Solution:
(a) We estimate the model in Gretl and obtain the following results:
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(b) We obtain the result β1 = 0.26. This means that 1% increase in ﬁrm’s sales
increases the CEO’s salary by 0.26%. β2 = 0.016, if ﬁrm’s return on equity
increases by 1, the CEO’s salary increases by 1.6%. β3 = −0.18, this means
that CEOs in ﬁrms with negative return on stock have by 18% lower salary
than those in ﬁrms with positive return.
(c) Testing statistical signiﬁcance of variable sales using a two-sided t-test at 95%
signiﬁcance level:
H0 : β1 = 0, HA : β1 = 0
t =
β1
s.d.(β1)
∼ tn−k(1−α
2
)
t-statistics: t =
0.2633
0.0372
= 7.08
critical value: tn−k(1−α
2
) = t185(0.975) = 1.960
|t| > tn−k(1−α
2
)
Therefore, we reject H0, and conﬁrm that the eﬀect of sales on CEO’s salary is
signiﬁcant. This is also conﬁrmed by the results in the Gretl output, which show
both the calculated t-statistic and the corresponding p-value, which is equal to
2.88∗10−11
. The p-value is thus smaller than the chosen signiﬁcance level (0.05),
we thus reject the null hypothesis and conclude that the eﬀect is signiﬁcant.
(d) i. We can test for a non-linear relationship between roe and salary with an
F-test. Let’s ﬁrst deﬁne the unrestricted model:
ln(salary) = α0 + α1 ln(sales) + α2roe + α3roe2
+ α4roe3
+ α5neg ros + ε ,
The null and alternative hypothesis are:
H0 : α3 = 0 & α4 = 0, HA : α3 = 0 ∨ α4 = 0
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And we get the restricted model by plugging in the restrictions (under the
null hypothesis). The restricted model is thus the same as the original
equation:
ln(salary) = α0 + α1 ln(sales) + α2roe + α5neg ros + ε ,
The F-statistics is as follows:
F =
(SSER − SSEU )/J
SSEU /(n − k)
∼ FJ,n−k ,
To be able to conduct the test, we need to run the restricted and unrestricted
model and save the sum of squared residuals (SSE) from both models. We
already have the results for the restricted model from part (a). Therefore,
we create the variables roe2
and roe3
, and run the unrestricted model. We
obtain the following results:
ii. We can now plug the SSER and SSEU to the test statistic (number of restrictions
J = 2, number of observations n = 189, and number of coeﬃcients
in the unrestricted model k = 6):
F =
(SSER − SSEU )/J
SSEU /(n − k)
=
(43.63565 − 43.62930)/2
43.62930/(189 − 6)
= 0.0133 ,
Next, we compare the calculated statistics to the critical value at 99% conﬁdence
level:
FJ,n−k(1−α) = F2,183(0.99) = 4.61
F < FJ,n−k(1−α)
Therefore, we do not reject the null hypothesis, the second and third order
polynomials in roe are not jointly signiﬁcant, and we thus conclude that
there is no evidence for the presence of a non-linear relationship in roe.
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iii. We obtain the following results of the test of linear restrictions in Gretl:
Results from the F-test in Gretl show both the calculated F-statistic (the
same value as the one calculated by hand) and the corresponding p-value.
The p-value is equal to 0.987, which bigger than the chosen signiﬁcance
level (0.01). Therefore, we do not reject the null hypothesis and conclude
that the variables are not jointly signiﬁcant.
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