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

EXAMPLE 11.2
Relationship
between
Compensation
and Productivity:
The Glejser Test
Continuing with Example 11.1, the absolute value of the residuals obtained from regres-
sion (11.5.3) were regressed on average productivity (
X
), giving the following results:
| ˆ
u
i
| =
407.2783
−
0.0203
X
i
se 
=
(633.1621)
(0.0675)
r
2
=
0.0127
(11.5.5)
t
=
(0.6432) (
−
0.3012)
As you can see from this regression, there is no relationship between the absolute value of
the residuals and the regressor, average productivity. This reinforces the conclusion based
on the Park test.
Spearman’s Rank Correlation Test
In Exercise 3.8 we defined the Spearman’s rank correlation coefficient as
(11.5.6)
r
s
=
1
−
6
d
2
i
n
(
n
2
−
1)
guj75772_ch11.qxd 12/08/2008 07:04 PM Page 380

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Chapter 11
Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
381
where 
d
i
= 
difference in the ranks assigned to two different characteristics of the 
i
th indi-
vidual or phenomenon and 
n
=
number of individuals or phenomena ranked. The preced-
ing rank correlation coefficient can be used to detect heteroscedasticity as follows: Assume
Y
i
=
β
0
+
β
1
X
i
+
u
i
.
Step 1.
Fit the regression to the data on 
Y
and 
X
and obtain the residuals 
ˆ
u
i
.
Step 2.
Ignoring the sign of 
ˆ
u
i
, that is, taking their absolute value 
| ˆ
u
i
|
, rank both 
| ˆ
u
i
|
and 
X
i
(or 
ˆ
Y
i
) according to an ascending or descending order and compute the Spear-
man’s rank correlation coefficient given previously.
Step 3.
Assuming that the population rank correlation coefficient 
ρ
s
is zero and
n
>
8, the significance of the sample 
r
s
can be tested by the 
t
test as follows:
16