Introduction to Econometrics
Home assignment # 2
(To be submitted on the lecture, Tuesday November 15, you may work alone or in groups
of two, but copying from other groups will lead to zero points)
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.
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