|
(11.5.7)
with df
=
n
−
2.
If the computed
t
value exceeds the critical
t
value, we may accept the hypothesis of
heteroscedasticity; otherwise we may reject it. If the regression model involves more than
one
X
variable,
r
s
can be computed between
| ˆ
u
i
|
and each of the
X
variables separately and
can be tested for statistical significance by the
t
test given in Eq. (11.5.7).
t
=
r
s
√
n
−
2
1
−
r
2
s
16
See G. Udny Yule and M. G. Kendall,
An Introduction to the Theory of Statistics
, Charles Griffin &
Company, London, 1953, p. 455.
E
i
,
σ
i
,
d,
Average
Standard
Difference
Annual
Deviation
|
u
ˆ
i
|
‡
between
Name of
Return,
of Annual
Residuals,
Rank
Rank
Two
Mutual Fund
%
Return, %
E
ˆ
i
†
|
(
E
i
−
E
ˆ
i
)
|
of
|
u
ˆ
i
|
of
σ
i
Rankings
d
2
Boston Fund
12.4
12.1
11.37
1.03
9
4
5
25
Delaware Fund
14.4
21.4
15.64
1.24
10
9
1
1
Equity Fund
14.6
18.7
14.40
0.20
4
7
−
3
9
Fundamental Investors
16.0
21.7
15.78
0.22
5
10
−
5
25
Investors Mutual
11.3
12.5
11.56
0.26
6
5
1
1
Loomis-Sales Mutual Fund
10.0
10.4
10.59
0.59
7
2
5
25
Massachusetts Investors Trust
16.2
20.8
15.37
0.83
8
8
0
0
New England Fund
10.4
10.2
10.50
0.10
3
1
2
4
Putnam Fund of Boston
13.1
16.0
13.16
0.06
2
6
−
4
16
Wellington Fund
11.3
12.0
11.33
0.03
1
3
−
2
4
Total
0
110
†
Obtained from the regression:
Ê
i
=
5.8194
+
0.4590
σ
i
.
‡
Absolute value of the residuals.
Note:
The ranking is in ascending order of values.
TABLE 11.2
Rank Correlation Test of Heteroscedasticity
EXAMPLE 11.3
Illustration of the
Rank Correlation
Test
To illustrate the rank correlation test, consider the data given in Table 11.2. The data
pertain to the average annual return (
E
, %) and the standard deviation of annual return
(
σ
i
, %) of 10 mutual funds.
(
Continued
)
guj75772_ch11.qxd 14/08/2008 03:50 PM Page 381
382
Part Two
Relaxing the Assumptions of the Classical Model
Goldfeld–Quandt Test
17
This popular method is applicable if one assumes that the heteroscedastic variance,
σ
2
i
, is
positively related to
one
of the explanatory variables in the regression model. For simplic-
ity, consider the usual two-variable model:
Y
i
=
β
1
+
β
2
X
i
+
u
i
Suppose
σ
2
i
is positively related to
X
i
as
σ
2
i
=
σ
2
X
2
i
(11.5.10)
where
σ
2
is a constant.
18
Assumption (11.5.10) postulates that
σ
2
i
is proportional to the square of the
X
variable.
Such an assumption has been found quite useful by Prais and Houthakker in their study of
family budgets. (See Section 11.5, informal methods.)
If Eq. (11.5.10) is appropriate, it would mean
σ
2
i
would be larger, the larger the values
of
X
i
. If that turns out to be the case, heteroscedasticity is most likely to be present in the
model. To test this explicitly, Goldfeld and Quandt suggest the following steps:
Step 1.
Order or rank the observations according to the values of
X
i
, beginning with
the lowest
X
value.
Step 2.
Omit
c
central observations, where
c
is specified a priori, and divide the
remaining (
n
−
c
) observations into two groups each of (
n
−
c
)
2 observations.
Step 3.
Fit separate OLS regressions to the first (
n
−
c
)
2 observations and the last
(
n
−
c
)
2 observations, and obtain the respective residual sums of squares RSS
1
and
The capital market line (CML) of portfolio theory postulates a linear relationship
between expected return (
E
i
) and risk (as measured by the standard deviation,
σ
) of a
portfolio as follows:
E
i
=
β
i
+
β
2
σ
i
Using the data in Table 11.2, the preceding model was estimated and the residuals from
this model were computed. Since the data relate to 10 mutual funds of differing sizes and
investment goals, a priori one might expect heteroscedasticity. To test this hypothesis, we
apply the rank correlation test. The necessary calculations are given in Table 11.2.
Applying formula (11.5.6), we obtain
r
s
=
1
−
6
110
10(100
−
1)
=
0.3333
(11.5.8)
Applying the
t
test given in Eq. (11.5.7), we obtain
t
=
(0
.
3333)(
√
8)
√
1
−
0
.
1110
=
0.9998
(11.5.9)
For 8 df this
t
value is not significant even at the 10 percent level of significance; the
p
value is 0.17. Thus, there is no evidence of a systematic relationship between the ex-
planatory variable and the absolute values of the residuals, which might suggest that there
is no heteroscedasticity.
17
Goldfeld and Quandt, op. cit., Chapter 3.
18
This is only one plausible assumption. Actually, what is required is that
σ
2
i
be monotonically
related to
X
i
.
EXAMPLE 11.3
(
Continued
)
guj75772_ch11.qxd 27/08/2008 12:12 PM Page 382
Chapter 11
Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
383
RSS
2
, RSS
1
representing the RSS from the regression corresponding to the smaller
X
i
values (the small variance group) and RSS
2
that from the larger
X
i
values (the large
variance group). These RSS each have
(
n
−
c
)
2
−
k
or
n
−
c
−
2
k
2
df
where
k
is the number of parameters to be estimated, including the intercept. (Why?)
For the two-variable case
k
is of course 2.
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