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208 Chapter 5 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
The Economic Value of a Year of Education:
Homoskedasticity or Heteroskedasticity?
On average, workers with more education have
higher earnings than workers with less educa-
tion. But if the best-paying jobs mainly go to the col-
lege educated, it might also be that the spread of the
distribution of earnings is greater for workers with
more education. Does the distribution of earnings
spread out as education increases?
This is an empirical question, so answering it
requires analyzing data. Figure 5.3 is a scatterplot of
the hourly earnings and the number of years of edu-
cation for a sample of 2829 full-time workers in the
United States in 2012, ages 29 and 30, with between
6 and 18 years of education. The data come from
the March 2013 Current Population Survey, which
is described in Appendix 3.1.
Figure 5.3 has two striking features. The first is
that the mean of the distribution of earnings increases
with the number of years of education. This increase
is summarized by the OLS regression line,
Earnings = -7.29 + 1.93Years Education,
(1.10) (0.08)
R2 = 0.162, SER = 10.29. (5.23)
This line is plotted in Figure 5.3. The coefficient
of 1.93 in the OLS regression line means that, on
Figure 5.3 Scatterplot of Hourly Earnings and Years of Education
for 29- to 30-Year-Olds in the United States in 2012
Hourly earnings are plotted against years of education for Average hourly earnings 150 ahe Fitted values
2,829 full-time 29- to 30-year-old workers. The spread
around the regression line increases with the years of
education, indicating that the regression errors are
heteroskedastic.
average, hourly earnings increase by $1.93 for each
additional year of education. The 95% confidence
interval for this coefficient is 1.93 { 1.96 * 0.08, or
1.77 to 2.09.
The second striking feature of Figure 5.3 is that
the spread of the distribution of earnings increases
with the years of education. While some workers
with many years of education have low-paying jobs,
very few workers with low levels of education have
high-paying jobs. This can be quantified by looking
at the spread of the residuals around the OLS regres-
sion line. For workers with ten years of education,
the standard deviation of the residuals is $4.32; for
workers with a high school diploma, this standard
deviation is $7.80; and for workers with a college
degree, this standard deviation increases to $12.46.
Because these standard deviations differ for differ-
ent levels of education, the variance of the residuals
in the regression of Equation (5.23) depends on the
value of the regressor (the years of education); in
other words, the regression errors are heteroskedas-
tic. In real-world terms, not all college graduates will
be earning $50 per hour by the time they are 29, but
some will, and workers with only ten years of educa-
tion have no shot at those jobs.
100
50
0
5 10 15 20
Years of education