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5.4 Heteroskedasticity and Homoskedasticity 205

Heteroskedasticity and Homoskedasticity Key Concept

The error term ui is homoskedastic if the variance of the conditional distribution 5.4
of ui given Xi, var(ui ͉ Xi = x), is constant for i = 1, c, n and in particular does
not depend on x. Otherwise, the error term is heteroskedastic.

Example. These terms are a mouthful, and the definitions might seem abstract.
To help clarify them with an example, we digress from the student–teacher ratio/
test score problem and instead return to the example of earnings of male versus
female college graduates considered in the box in Chapter 3 “The Gender Gap in
Earnings of College Graduates in the United States.” Let MALEi be a binary
variable that equals 1 for male college graduates and equals 0 for female gradu-
ates. The binary variable regression model relating a college graduate’s earnings
to his or her gender is

Earningsi = b0 + b1MALEi + ui (5.19)

for i = 1, c, n. Because the regressor is binary, b1 is the difference in the popu-
lation means of the two groups—in this case, the difference in mean earnings
between men and women who graduated from college.

The definition of homoskedasticity states that the variance of ui does not
depend on the regressor. Here the regressor is MALEi, so at issue is whether the
variance of the error term depends on MALEi. In other words, is the variance of
the error term the same for men and for women? If so, the error is homoskedastic;
if not, it is heteroskedastic.

Deciding whether the variance of ui depends on MALEi requires thinking
hard about what the error term actually is. In this regard, it is useful to write
Equation (5.19) as two separate equations, one for men and one for women:

Earningsi = b0 + ui (women) and (5.20)

Earningsi = b0 + b1 + ui (men). (5.21)

Thus, for women, ui is the deviation of the ith woman’s earnings from the popula-
tion mean earnings for women (b0), and for men, ui is the deviation of the ith man’s
earnings from the population mean earnings for men (b0 + b1). It follows that the
statement “the variance of ui does not depend on MALE” is equivalent to the