Introduction to Econometrics
Home assignment # 1
(To be submitted on the lecture, Friday October 20)
1. Imagine you work in a bulb factory as a supervisory technician, who is asked to check
the lifespan of light bulbs. You know that the lifespan of light bulbs produced in
your factory is distributed N(200, 400), that is, normally distributed with a mean of
200 hours and a variance of 400 hours. What is the probability that you find a light
bulb with a life span longer than 245 hours? (Hint: convert the distribution to a
standard normal one and then refer to statistical tables.)
2. The basic ingredients of beer are water; a starch source, such as malted barley, able to
be fermented (converted into alcohol); a brewer’s yeast to produce the fermentation;
and a flavoring, such as hops, to offset the sweetness of the malt. You receive a
unique dataset that includes information about barley production in South Bohemia
as well as information about agricultural production techniques and local weather.
Obviously, you are very curious about what is the effect of soil fertilization and
weather on barley yields. You run an OLS regression of annual yields on fertilizer
intensity and rainfall and obtain the following results:
Yt = −120 + 0.10 · Ft + 5.33 · Rt
where Yt = the barley yield (bushels/acre) in year t
Ft = fertilizer intensity (pounds/ acre) in year t
Rt = rainfall (inches) in year t
(a) Carefully interpret the meaning of the coefficients of 0.10 and 5.33 in terms of
impact of F and R on Y.
(b) Use the estimates to determine the yield in years with 100, 500 and 1500 inches
of rainfall, given that the fertilizer intensity was always 100 pounds per acre.
(c) Does the constant term of -120 mean that negative amounts of barley are possible?
If not, what is the meaning of that estimate?
(d) Suppose that you were told that the true value of βF is known to be 0.20. Does
this show that the estimate is biased? Why or why not?
3. Let us investigate the results of an experiment in broiler (poultry meat) production.
The average weight of an experimental lot of broilers and their corresponding level of
average feed consumption was tabulated over the time period in which they changed
from baby chickens to mature broilers ready for market. At time t, we denote the
average weight of the experimental group of broilers as the output yt, the total feed
consumed by the input xt. The observations are summarized in Table 1.
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Average Weight Average Cumulative
End of of Broiler Feed Inputs
Time Period in Pounds in Pounds
t yt xt
1 0.57 1.00
2 1.01 2.00
3 1.20 3.00
4 1.27 4.00
5 1.91 5.00
6 2.52 6.00
7 2.55 7.00
8 2.92 8.00
9 3.38 9.00
10 3.43 10.00
11 3.53 11.00
12 4.11 12.00
13 4.26 13.00
14 4.33 14.00
15 4.41 15.00
Table 1: Data on poultry production
Let us suppose that the underlying model is
yt = β0 + β1xt + β2x2
t + εt .
Answer the following questions. Do not use statistical software (Stata etc.) for the
solution!!! You can use Excel for multiplication and inversion of the matrices in
question1
, but otherwise you are asked to do all estimation and computation “by
hand” - meaning according to the formulae we introduced on the lecture. Make sure
you show in your solution all the matrices you construct and compute.
(a) Using the Least Squares formula, find the coefficients β0, β1 and β2.
(b) Find and list the residuals of the model.
(c) What is the sign of β2? How do you interpret this coefficient?
(d) Based on your estimation result, find ∂yt
∂xt
, the marginal productivity of the feed
input. Interpret your your result.
(e) We denote the price of broilers by pb and the price of feed by pf . Explain why
∂yt
∂xt
=
pf
pb
. [Hint: Solve for the firm’s profit maximization problem.]
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For Excel, you may want to check out the commands =MMULT() and =MINVERSE()
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