The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

FIGURE 11.8
Hypothetical patterns
of estimated squared
residuals.
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Part Two
Relaxing the Assumptions of the Classical Model
(
a
)
(
b
)
(
c
)
u
2
u
2
u
2
u
2
(
e
)
X
X
X
X
0
0
0
0
0
u
2
(
d
)
X
FIGURE 11.9
Scattergram of
estimated squared
residuals against 
X
.
11
R. E. Park, “Estimation with Heteroscedastic Error Terms,’’ 
Econometrica
, vol. 34, no. 4, October
1966, p. 888. The Park test is a special case of the general test proposed by A. C. Harvey in 
“Estimating Regression Models with Multiplicative Heteroscedasticity,’’ 
Econometrica
, vol. 44, no. 3,
1976, pp. 461–465.
plotting it against 
X
i
, and therefore Figure 11.9 is similar to Figure 11.8. But this is not the
situation when we consider a model involving two or more 
X
variables; in this instance, 
ˆ
u
2
i
may be plotted against any 
X
variable included in the model.)
A pattern such as that shown in Figure 11.9
c
, for instance, suggests that the variance of
the disturbance term is linearly related to the 
X
variable. Thus, if in the regression of sav-
ings on income one finds a pattern such as that shown in Figure 11.9
c
, it suggests that the
heteroscedastic variance may be 
proportional
to the value of the income variable. This
knowledge may help us in transforming our data in such a manner that in the regression on
the transformed data the variance of the disturbance is homoscedastic. We shall return to
this topic in the next section.