Problem Set

Please write the names of any students you worked with on the first page of your problem set. Attach your
R output to your problem set. No credit will be given if you report only the final answers without showing
formulas and calculations where appropriate, and no credit will be given if the R output is missing.
For the empirical exercise, you must use the dataset posted on Blackboard and R. Please ask your TA if
you have any questions about coding. You must submit both R output and solutions to the empirical exercise.
1. Suppose we sample 197 college-educated individuals and 201 individuals with less than college
education to gather information on income and education. Based on our sample, we calculate that
individuals with a college education earn $25.20 an hour, while individuals with less than a college
education earn $21.90 an hour. The standard deviation of hourly earnings for college-educated individuals
is 15.4; the standard deviation of hourly earnings of individuals with less than college education is 12.0.
a. Define College as a binary variable equal to one if the individual has at least a college education and zero
if the individual has less than a college education. Write, in regression form, the regression results you
would obtain from an OLS regression of HourlyWage on College. (Write the numerical coefficients you
would get for 𝛽̂
0 and 𝛽̂
1; do not write the standard errors.)
b. Using a t-statistic, test the hypothesis that the difference in average hourly wages between college- and
non-college educated individuals is zero. Clearly state your alternative hypothesis.
c. Calculate the p-value for the same hypothesis test you conducted in part (b). Explain what this p-value
means in words.
d. Repeat part (a) but instead write the regression results you would obtain if you regressed HourlyWage
on LessThanCollege, where LessThanCollege = 1 if an individual has less than a college education and =0
if an individual has at least a college education.
2. (S&W Ch. 6: 6.12) A school district undertakes an experiment to estimate the effect of class size on test
scores in second-grade classes. The district assigns 50% of its previous year’s first graders to small secondgrade classes (18 students per classroom) and 50% to regular-size classes (21 students per classroom).
Students new to the district are handled differently: 20% are randomly assigned to small classes and 80%
to regular-size classes. At the end of the second-grade school year, each student is given a standardized
exam. Let 𝑌𝑖 denote the exam score for the ith student, 𝑋𝑖 denote a binary variable that equals 1 if the student
is assigned to a small class, and 𝑊𝑖 denote a binary variable that equals 1 if the student is newly enrolled.
Let 𝛽1 denote the causal effect on test scores of reducing class size from regular to small.
ECO 321
Spring 2022
2
a. Consider the regression 𝑌𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖
. Do you think that 𝐸(𝑢𝑖
|𝑋𝑖
) = 0? Is the OLS estimator of
𝛽1 unbiased and consistent? Explain.
b. Consider the regression 𝑌𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝛽2𝑊𝑖 + 𝑢𝑖
. Do you think that 𝐸(𝑢𝑖
|𝑋𝑖𝑊𝑖) depends on 𝑋𝑖? Is
the OLS estimator of 𝛽1 unbiased and consistent? Explain. Do you think that 𝐸(𝑢𝑖
|𝑋𝑖𝑊𝑖) depends on 𝑊𝑖?
Will the OLS estimator of 𝛽2 provide an unbiased and consistent estimate of the causal effect of transferring
to a new school (that is, being a newly enrolled student)? Explain.
3. (S&W Ch. 7: 7.8 part (a) and (b) only)
a. Calculate the 𝑅
2
for each of the regressions.
b. Calculate the homoskedasticity-only F-statistic for testing 𝛽3 = 𝛽4 = 0 in the regression shown in
column (5). Is the statistic significant at the 5% level?
4. S&W Empirical Exercise E5.3, parts (a), (b), and (c), and E6.1 part (b) questions (i) and (ii) only. For
this exercise, use the Birthweight_Smoking.csv data set that is posted on Blackboard. (The scanned
exercise from the 4th edition is posted on Blackboard.)
5. S&W Empirical Exercise E7.1, parts (a), (c), and (d) only. For this exercise, use the
Birthweight_Smoking.csv data set that is posted on Blackboard. (The scanned exercise from the 4th edition
is posted on Blackboard.)