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

Formal Methods
Park Test
11
Park formalizes the graphical method by suggesting that 
σ
2
i
is some function of the
explanatory variable 
X
i
. The functional form he suggests is
σ
2
i
=
σ
2
X
β
i
e
vi
or
ln
σ
2
i
=
ln
σ
2
+
β
ln
X
i
+
v
i
(11.5.1)
where 
v
i
is the stochastic disturbance term.
guj75772_ch11.qxd 12/08/2008 07:04 PM Page 378

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Chapter 11
Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
379
Since 
σ
2
i
is generally not known, Park suggests using 
ˆ
u
2
i
as a proxy and running the
following regression:
ln
ˆ
u
2
i
=
ln
σ
2
+
β
ln
X
i
+
v
i
=
α
+
β
ln
X
i
+
v
i
(11.5.2)
If 
β
turns out to be statistically significant, it would suggest that heteroscedasticity is
present in the data. If it turns out to be insignificant, we may accept the assumption of
homoscedasticity. The Park test is thus a two-stage procedure. In the first stage we run the
OLS regression disregarding the heteroscedasticity question. We obtain 
ˆ
u
i
from this
regression, and then in the second stage we run the regression (11.5.2).
Although empirically appealing, the Park test has some problems. Goldfeld and Quandt
have argued that the error term
v
i
entering into Eq. (11.5.2) may not satisfy the OLS assump-
tions and may itself be heteroscedastic.
12
Nonetheless, as a strictly exploratory method, one
may use the Park test.
12
Stephen M. Goldfeld and Richard E. Quandt, 
Nonlinear Methods in Econometrics
, North Holland
Publishing Company, Amsterdam, 1972, pp. 93–94.
13
The particular functional form chosen by Park is only suggestive. A different functional form may reveal
significant relationships. For example, one may use 
ˆ
u
2
i
instead of ln
ˆ
u
2
i
as the dependent variable.
14
H. Glejser, “A New Test for Heteroscedasticity,’’ 
Journal of the American Statistical Association
, vol. 64,
1969, pp. 316–323.