Page 205 -

204 Chapter 5 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals

What Are Heteroskedasticity and Homoskedasticity?

Definitions of heteroskedasticity and homoskedasticity. The error term ui is
homoskedastic if the variance of the conditional distribution of ui given Xi is con-
stant for i = 1, c, n and in particular does not depend on Xi. Otherwise, the
error term is heteroskedastic.

As an illustration, return to Figure 4.4. The distribution of the errors ui is
shown for various values of x. Because this distribution applies specifically for
the indicated value of x, this is the conditional distribution of ui given Xi = x.
As drawn in that figure, all these conditional distributions have the same
spread; more precisely, the variance of these distributions is the same for the
various values of x. That is, in Figure 4.4, the conditional variance of ui given
Xi = x does not depend on x, so the errors illustrated in Figure 4.4 are homo-
skedastic.

In contrast, Figure 5.2 illustrates a case in which the conditional distribution
of ui spreads out as x increases. For small values of x, this distribution is tight, but
for larger values of x, it has a greater spread. Thus in Figure 5.2 the variance of ui
given Xi = x increases with x, so that the errors in Figure 5.2 are heteroskedastic.

The definitions of heteroskedasticity and homoskedasticity are summarized
in Key Concept 5.4.

Figure 5.2 An Example of Heteroskedasticity

Like Figure 4.4, this Test score
shows the conditional 720

distribution of test 700 Distribution of Y when X = 15
scores for three differ- 680 Distribution of Y when X = 20
ent class sizes. Unlike Distribution of Y when X = 25
Figure 4.4, these

distributions become
more spread out (have 660

a larger variance) 640
for larger class sizes.

Because the variance 620 b0 +b1X
of the distribution of

u given X, var(u ͉ X ), 60010 15 20 25 30
depends on X, u is

heteroskedastic. Student–teacher ratio