Download Economics 405

Economics 405
Due—September 23 at 6:30 pm.
Problem Set #1
Instructions: The questions and problems below are like the questions and
problems you will see on my exams. (Indeed, I don’t rule out the possibility that
some will re-surface verbatim on the exams.) In answering, show all of your work
and give complete explanations. As noted on the course syllabus, problem sets will
be graded on a credit/no-credit basis. You must give a substantive answer to each
question/problem. Non-attempts and weak attempts will be docked accordingly. I
do not encourage you to work in groups. After all, you will have to think for
yourself on the exams. To the extent that you do work with a colleague, be
absolutely sure to give answers in your own words. Duplicate answers will
automatically be assigned a 0.0.
1.
Wooldridge states “After data on the relevant variables have been collected,
econometric methods are used to estimate the parameters in the econometric
model…” What does he mean by parameters in this context?
2.
a.
b.
3.
1
Determine the first derivative of the function y  10 x 2  .3( ) .
x
Determine the partial derivative of y with respect to z of the function
y  (2 x  15  3z ) 2 .
Suppose that the random variables X and Y have the marginal and joint
probability density functions given in the tables below:
X
2
4
6
Y
5
10
15
fx(X)
.5
.3
.2
fy(Y)
.2
.5
.3
f x , y ( x, y )
X
2
4
6
5
.15
.03
.02
Y
10
.2
.2
.1
15
.15
.07
.08
a.
Are X and Y independent? Why or why not?
b.
Calculate the unconditional expected value of X.
2
4.
c.
Calculate the relevant conditional expected values for X (i.e.,
E ( X | Y  10), E ( X | Y  15), E ( X | Y  20)) . Compare and comment on
the conditional expected values obtained here versus the unconditional
expected value obtained in b.
d.
Does E ( XY )  E ( X ) E (Y ) ? (Hint: Determine E(XY) using the joint pdf
and determine E(X) and E(Y) using the marginal pdfs.)
Suppose that the random variable X and the random variable Y have the
following probability density functions:
X
-5
5
10
5.
fx(X)
.4
.3
.3
Y
10
15
20
fy(Y)
.2
.5
.3
a.
Assume that X and Y are independent. Construct the joint probability
density function table for X and Y. Briefly explain what you’ve done
here.
b.
Determine the unconditional expected value of Y.
c.
Determine the expected value of Y conditional on X = 10. Compare your
answer in c with that in b and comment.
d.
Show mathematically whether E ( XY )  E ( X ) E (Y ) .
A professor decides to run an experiment to measure the effect of time pressure
on final exam scores. He gives each of the 400 students in his course the same
final exam, but some students get 90 minutes to complete the exam while others
have 120 minutes. Each student is randomly assigned one of the examination
times based on the flip of a coin. Let Yi denote the number of points scored on the
final exam and let Xi denote the amount of time that the student has to complete
the exam. Consider the regression model: Yi   0  1 X i  ui .
a.
Explain what the term ui represents. Why will different students have
different values of ui?
b.
Will the zero conditional mean assumption, E (u | X )  0 , be valid for this
model? Why or why not?
c.
The estimated regression is Yˆi  40  .33 X i . Compute the regression’s
prediction for the average score of students given 90 minutes to complete
the exam. Compute the regression’s prediction for the average score of
3
students given 120 minutes to complete the exam. What is the estimated
gain in score for a student who is given an additional 10 minutes on the
exam?
n
6.
a.
Show algebraically that
 x (x
i 1
b.
7.
i
n
i
 x )   ( xi  x ) 2 .
i 1
Convince me that the result is true for the ACT scores given in problem
2.3 on p. 61 of the text.
A researcher wished to investigate the effect of the federally funded school lunch
program on 10th grade student test scores. The researcher hypothesized that
students participating in the lunch program would have better scores than those
not participating. (That is, full stomachs make for better performance and empty
stomachs lead to worse performance.) Seems like common sense but when the
researcher ran the regression for a randomly selected sample of 408 10th grade
classes from the State of Michigan, the computer spit out the following result:
math10 =32.14 – 0.319lnchprg
n = 408,
where math10 is the school’s average score on a standardized test; and
lnchprg is the percentage of students at the school who are eligible
for the free lunch program (eligibility is determined on the
basis of family income).
a.
What does the fitted model predict will be the effect on test scores of a 10
percentage point increase in eligibility for the free lunch program?
b.
What is the predicted average test score at a school for which 50% of the
students are eligible for the free lunch program? What is the predicted
average test score at a school for which 10% of the students are eligible
for the free lunch program?
c.
Contrary to common sense, the negative sign on the slope coefficient in
the fitted model suggests that better nutrition has an adverse effect on
exam performance! Discuss why the estimated coefficient is negative.
E ( y | x)
E (u | x)
 1 
[Hint: Base your discussion on the relation
,
x
x
which we derived from the population regression model.]
4
n
8.
The formula for the least squares slope coefficient is ̂1 
 (x
i 1
i
 x )( y i  y )
.
n
 (x
i 1
i
 x)
2
Show step-by-step how this formula is obtained using the method of moments
approach.