All exercises in appendix C and chapters 1 and 2 of the Wooldridge textbook
are good practice exercises.
Question 1
Suppose that I run an experiment to measure the effect of classroom temperature
on final exam scores. I have a total of 300 students and give them the same final
exam. I randomly assign students to one of the three classrooms, each one with
different levels of temperature: 10 C (50 F), 20 C (68 F) and 30 C (86 F). Define Y
i
the grade on the exam (between 0 and 100) and define X
i
the temperature of the
classroom in which the student had to complete the exam (X
i
=10, or X
i
=20 or
X
i
=30).
Assume the regression model: Y
i
=
β
0
+ β
1
X
i
+
ε
i
(a) What does the term
ε
i
mean? Why will different students have different
values of
ε
i
?
(b) Why E(
ε
i
 X
i
) = 0 in this regression model?
(c) The estimated regression is
Ŷ
i
= 45 + 1.2 X
i
. What is the predicted grade of
students who completed the exam in the cold room (10 C)? What about the other
two classrooms (20 C and 30 C)?
(d) In light of your conclusions in part (c), discuss the current specification. Is it
right and if not, what changes would you make?
Question 2
Assume the regression model Y
i
=
α
+ βX
i
+ ε
i
(a) Derive the least squares (OLS) estimator of
β
and verify that it is unbiased.
State the assumptions that you are making.
(b) Suppose you
know that α=4. Derive a formula
for the least squares estimator.
(c) Suppose that the regression model is such that there is no constant in the
model (α=0).
What is the ordinary least square estimator of
β
in this case? Under
what conditions is this estimator unbiased?
Question 3
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 Fall '09
 Louis
 Least Squares, Regression Analysis, regression model, Yi

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