Introductory Econometrics
Lecture 4: Hypothesis Testing
Suggested Solution
by Hieu Nguyen
Fall 2024
1.
Consider an equation to explain salaries of CEOs in terms of annual firm sales
return on equity (roe in percentage form) and return on the firm’s stock (ros
in percentage form):
log(salary) = β0 + β1 log(sales) + β2roe + β3ros + u
• In terms of the model parameters, state the null hypothesis that after
controlling for sales and roe ros has no effect on CEO salary. State the
alternative that better stock market performance increases a CEO’s salary.
Solution:
H0 : β3 = 0 vs H1 : β3 > 0.
• Using the data in CEOSAL1.RAW the following equation was obtained by
OLS:
log(salary) = 4.32 + 0.280 log(sales) + 0.0174roe + 0.00024ros
(0.32) (0.035) (0.0041) (0.00054)
n = 209 R2
= 0.283
By what percentage is salary predicted to increase if ros increases by 50
points? Does ros have a practically large effect on salary?
Solution: The proportionate effect on salary is 0.00024 × 50 = 0.012.
To obtain the percentage effect we multiply this by 100: 1.2
• Test the null hypothesis that ros has no effect on salary against the alternative
that ros has a positive effect. Carry out the test at the 10%
significance level.
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Solution: The 10% critical value for a one-tailed test using df =
∞ is obtained from T-test table as 1.282. The t statistic on ros is
0.00024/0.00054 ≈ 0.44 which is well below the critical value. Therefore,
we fail to reject H0 at the 10
• Would you include ros in a final model explaining CEO compensation in
terms of firm performance? Explain.
Solution: Based on this sample, the estimated ros coefficient appears
to be different from zero only because of sampling variation. On the other
hand, including ros may not be causing any harm; it depends on how
correlated it is with the other independent variables (although these are
very significant even with ros in the equation).
2.
The variable rdintens is expenditures on research and development (RD) as
a percentage of sales. Sales are measured in millions of dollars. The variable
profmarg is profits as a percentage of sales.
Using the data in RDCHEM.RAW for 32 firms in the chemical industry the
following equation is estimated:
rdintens = 0.472 + 0.321 log(sales) + 0.050profmarg
(1.369) (0.216) (0.046)
n = 32 R2
= 0.099
• Interpret the coefficient on log(sales). In particular, if sales increases by
10% what is the estimated percentage point change in rdintens? Is this
an economically large effect?
Solution: Holding profmarg fixed:
∆rdintens = 0.321∆ log(sales) =
0.321
100
[100·∆ log(sales)] ≈ 0.00321(%∆sales).
Therefore, if ∆ log(sales) = 10, ∆rdintens ≈ 0.032 or only about 3/100
of a percentage point. For such a large percentage increase in sales, this
seems like a practically small effect.
• Test the hypothesis that R&D intensity does not change with sales against
the alternative that it does increase with sales. Do the test at the 5% and
10% levels.
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Solution:
H0 : β1 = 0 vs H1 : β1 > 0.
Where β1 is the population slope on log(sales). The t statistic is 0.321/0.216 ≈
1.486. The 5% critical value for a one-tailed test with df = 32 − 3 = 29 is
obtained from the Table as 1.699; so we cannot reject H0 at the 5% level.
But the 10% critical value is 1.311; since the t statistic is above this value,
we reject H0 in favor of H1 at the 10% level.
• Does profmarg have a statistically significant effect on rdintens?
Solution: Not really. Its t statistic is only 1.087 which is well below
even the 10% critical value for a one-tailed test.
3.
Are rent rates influenced by the student population in a college town? Let
rent be the average monthly rent paid on rental units in a college town in the
United States. Let pop denote the total city population, avginc the average
city income, and pctstu the student population as a percentage of the total
population. One model to test for a relationship is
log(rent) = β0 + β1 log(pop) + β2 log(avginc) + β3pctstu + u
• State the null hypothesis that size of the student body relative to the population
has no ceteris paribus effect on monthly rents. State the alternative
that there is an effect.
Solution:
H0 : β3 = 0 vs H1 : β3 ̸= 0.
• What signs do you expect for β1 and β2?
Solution: Other things equal, a larger population increases the demand
for rental housing, which should increase rents. The demand for overall
housing is higher when average income is higher, pushing up the cost of
housing including rental rates.
• The equation estimated using 1990 data from RENTAL.RAW for 64 college
towns is:
log(rent) = 0.043 + 0.066 log(pop) + 0.507 log(avginc) + 0.0056pctstu
(0.844) (0.039) (0.081) (0.0017)
n = 64 R2
= 0.458
What is wrong with the statement: “A 10% increase in population is
associated with about a 6.6% increase in rent”?
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Solution: The coefficient on log(pop) is an elasticity. A correct statement
is that “a 10% increase in population increases rent by 0.066 × 10 =
0.66%.”
• Test the hypothesis stated in part (a.) at the 1% level.
Solution: With df = 64−4 = 60, the 1% critical value for a two-tailed
test is 2.660. The t statistic is about 3.29, which is well above the critical
value. So β3 is statistically different from zero at the 1% level.
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