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

Step 4.
Compute the ratio
λ
=
RSS
2
/
df
RSS
1
/
df
(11.5.11)
If we assume u
i
are normally distributed
(which we usually do), and 
if the assumption
of homoscedasticity is valid
, then it can be shown that 
λ
of Eq. (11.5.10) follows the 
F
distribution with numerator and denominator df each of (
n
−
c
−
2
k
)
/
2.
If in an application the computed 
λ
(
=
F
) is greater than the critical 
F
at the chosen
level of significance, we can reject the hypothesis of homoscedasticity, that is, we can say
that heteroscedasticity is very likely.
Before illustrating the test, a word about omitting the 
c
central observations is in order.
These observations are omitted to sharpen or accentuate the difference between the small
variance group (i.e., RSS
1
) and the large variance group (i.e., RSS
2
). But the ability of the
Goldfeld–Quandt test to do this successfully depends on how 
c
is chosen.
19
For the two-
variable model the Monte Carlo experiments done by Goldfeld and Quandt suggest that 
c
is about 8 if the sample size is about 30, and it is about 16 if the sample size is about 60.
But Judge et al. note that 
c
= 
4 if 
n
= 
30 and 
c
= 
10 if 
n
is about 60 have been found sat-
isfactory in practice.
20
Before moving on, it may be noted that in case there is more than one
X
variable in the model,
the ranking of observations, the first step in the test, can be done according to any one of them.
Thus in the model:
Y
i
=
β
1
+
β
2
X
2
i
+
β
3
X
3
i
+
β
4
X
4
i
+
u
i
, we can rank-order the data accord-
ing to any one of these
X
’s. If a priori we are not sure which
X
variable is appropriate, we can
conduct the test on each of the
X
variables, or via a Park test, in turn, on each
X
.
19
Technically, the 
power
of the test depends on how 
c
is chosen. In statistics, the 
power of a test
is mea-
sured by the probability of rejecting the null hypothesis when it is false (i.e., by 1 
−
Prob [type II error]).
Here the null hypothesis is that the variances of the two groups are the same, i.e., homoscedasticity. For
further discussion, see M. M. Ali and C. Giaccotto, “A Study of Several New and Existing Tests for 
Heteroscedasticity in the General Linear Model,’’ 
Journal of Econometrics
, vol. 26, 1984, pp. 355–373.
20
George G. Judge, R. Carter Hill, William E. Griffiths, Helmut Lütkepohl, and Tsoung-Chao Lee, 
Introduction to the Theory and Practice of Econometrics
, John Wiley & Sons, New York, 1982, p. 422.