|
11A
11A.1
Proof of Equation (11.2.2)
From Appendix 3A, Section 3A.3, we have
var (
ˆ
β
2
)
=
E
k
2
1
u
2
1
+
k
2
2
u
2
2
+ · · · +
k
2
n
u
2
n
+
2 cross-product terms
=
E
k
2
1
u
2
1
+
k
2
2
u
2
2
+ · · · +
k
2
n
u
2
n
since the expectations of the cross-product terms are zero because of the assumption of no serial
correlation,
var (
ˆ
β
2
)
=
k
2
1
E
u
2
1
+
k
2
2
E
u
2
2
+ · · · +
k
2
n
E
u
2
n
since the
k
i
are known. (Why?)
var (
ˆ
β
2
)
=
k
2
1
σ
2
1
+
k
2
2
σ
2
2
+ · · · +
k
2
n
σ
2
n
since
E
(
u
2
i
)
=
σ
2
i
.
var (
ˆ
β
2
)
=
k
2
i
σ
2
i
=
x
i
x
2
i
2
σ
2
i
since
k
i
=
x
i
x
2
i
(11.2.2)
=
x
2
i
σ
2
i
x
2
i
2
11A.2
The Method of Weighted Least Squares
To illustrate the method, we use the two-variable model
Y
i
=
β
1
+
β
2
X
i
+
u
i
. The unweighted least-
squares method minimizes
ˆ
u
2
i
=
(
Y
i
− ˆ
β
1
− ˆ
β
2
X
i
)
2
(1)
guj75772_ch11.qxd 14/08/2008 11:00 AM Page 409
410
Part Two
Relaxing the Assumptions of the Classical Model
to obtain the estimates, whereas the weighted least-squares method minimizes the weighted residual
sum of squares:
w
i
ˆ
u
2
i
=
w
i
(
Y
i
− ˆ
β
∗
1
− ˆ
β
∗
2
X
i
)
2
(2)
where
β
∗
1
and
β
∗
2
are the weighted least-squares estimators and where the weights
w
i
are such that
w
i
=
1
σ
2
i
(3)
that is, the weights are inversely proportional to the variance of
u
i
or
Y
i
conditional upon the given
X
i
,
it being understood that var (
u
i
|
X
i
)
=
var (
Y
i
|
X
i
)
=
σ
2
i
.
Differentiating Eq. (2) with respect to
ˆ
β
∗
1
and
ˆ
β
∗
2
, we obtain
∂
w
i
ˆ
u
2
i
∂β
∗
1
=
2
w
i
(
Y
i
− ˆ
β
∗
1
− ˆ
β
∗
2
X
i
)(
−
1)
∂
w
i
ˆ
u
2
i
∂β
∗
2
=
2
w
i
(
Y
i
− ˆ
β
∗
1
− ˆ
β
∗
2
X
i
)(
−
X
i
)
Setting the preceding expressions equal to zero, we obtain the following two normal equations:
w
i
Y
i
= ˆ
β
∗
1
w
i
+ ˆ
β
∗
2
w
i
X
i
(4)
w
i
X
i
Y
i
= ˆ
β
∗
1
w
i
X
i
+ ˆ
β
∗
2
w
i
X
2
i
(5)
Notice the similarity between these normal equations and the normal equations of the unweighted
least squares.
Solving these equations simultaneously, we obtain
ˆ
β
∗
1
= ¯
Y
∗
− ˆ
β
∗
2
¯
X
∗
(6)
and
ˆ
β
∗
2
=
w
i
w
i
X
i
Y
i
−
w
i
X
i
w
i
Y
i
w
i
w
i
X
2
i
−
w
i
X
i
2
(11.3.8)
=
(7)
The variance of
ˆ
β
∗
2
shown in Eq. (11.3.9) can be obtained in the manner of the variance of
ˆ
β
2
shown
in Appendix 3A, Section 3A.3.
Note:
¯
Y
∗
=
w
i
Y
i
/
w
i
and
¯
X
∗
=
w
i
X
i
/
w
i
. As can be readily verified, these weighted
means coincide with the usual or unweighted means
¯
Y
and
¯
X
when
w
i
=
w
, a constant, for all
i
.
11A.3
Proof that
E
(ˆ
σ
2
)
σ
2
in the Presence
of Heteroscedasticity
Consider the two-variable model:
Y
i
=
β
1
+
β
2
X
i
+
u
i
(1)
where var (
u
i
)
=
σ
2
i
Now
ˆ
σ
2
=
ˆ
u
2
i
n
−
2
=
(
Y
i
− ˆ
Y
i
)
2
n
−
2
=
[
β
1
+
β
2
X
i
+
u
i
− ˆ
β
1
− ˆ
β
2
X
i
]
2
n
−
2
=
[
−
(
ˆ
β
1
−
β
1
)
−
(
ˆ
β
2
−
β
2
)
X
i
+
u
i
]
2
n
−
2
(2)
guj75772_ch11.qxd 12/08/2008 07:04 PM Page 410
Chapter 11
Heteroscedasticity: What Happens If the Error Variance Is Nonconstant?
411
Noting that (
ˆ
β
1
−
β
1
)
= −
(
ˆ
β
2
−
β
2
)
¯
X
+ ¯
u
, and substituting this into Eq. (2) and taking expecta-
tions on both sides, we get:
E
(
ˆ
σ
2
)
=
1
n
−
2
−
x
2
i
var (
ˆ
β
2
)
+
E
(
u
i
− ¯
u
)
2
=
1
n
−
2
−
x
2
i
σ
2
i
x
2
i
+
(
n
−
1)
σ
2
i
n
(3)
where use is made of Eq. (11.2.2).
As you can see from Eq. (3), if there is homoscedasticity, that is,
σ
2
i
=
σ
2
for each
i
,
E
(
ˆ
σ
2
)
=
σ
2
.
Therefore, the expected value of the conventionally computed
ˆ
σ
2
=
ˆ
u
2
/
(
n
−
2) will not be equal
to the true
σ
2
in the presence of heteroscedasticity.
1
11A.4
White’s Robust Standard Errors
To give you some idea about White’s heteroscedasticity-corrected standard errors, consider the two-
variable regression model:
Y
i
=
β
1
+
β
2
X
i
+
u
i
var (
u
i
)
=
σ
2
i
(1)
As shown in Eq. (11.2.2),
var (
ˆ
β
2
)
=
x
2
i
σ
2
i
x
2
i
2
(2)
Since
σ
2
i
are not directly observable, White suggests using
ˆ
u
2
i
, the squared residual for each
i
, in place
of
σ
2
i
and estimating the var (
ˆ
β
2
) as follows:
var (
ˆ
β
2
)
=
x
2
i
ˆ
u
2
i
x
2
i
2
(3)
White has shown that Eq. (3) is a consistent estimator of Eq. (2), that is, as the sample size increases
indefinitely, Eq. (3) converges to Eq. (2).
2
Incidentally, note that if your software package does not contain White’s robust standard error pro-
cedure, you can do it as shown in Eq. (3) by first running the usual OLS regression, obtaining the
residuals from this regression, and then using formula (3).
White’s procedure can be generalized to the
k
-variable regression model
Y
i
=
β
1
+
β
2
X
2
i
+
β
3
X
3
i
+ · · · +
β
k
X
ki
+
u
i
(4)
The variance of any partial regression coefficient, say
ˆ
β
j
, is obtained as follows:
var (
ˆ
β
j
)
=
ˆ
w
2
ji
ˆ
u
2
i
ˆ
w
2
ji
2
(5)
where
ˆ
u
i
are the residuals obtained from the (original) regression (4) and
ˆ
w
j
are the residuals
obtained from the (auxiliary) regression of the regressor
X
j
on the remaining regressors in Eq. (4).
Obviously, this is a time-consuming procedure, for you will have to estimate Eq. (5) for each
X
variable. Of course, all this labor can be avoided if you have a statistical package that does this rou-
tinely. Packages such as PC-GIVE,
EViews
, MICROFIT, SHAZAM, STATA, and LIMDEP now
obtain White’s heteroscedasticity-robust standard errors very easily.
1
Further details can be obtained from Jan Kmenta,
Elements of Econometrics,
2d. ed., Macmillan, New
York, 1986, pp. 276–278.
2
To be more precise,
n
times Eq. (3) converges in probability to
E
[(
X
i
−
µ
X
)
2
u
2
i
]
/
(
σ
2
X
)
2
, which is the
probability limit of
n
times Eq. (2), where
n
is the sample size,
µ
x
is the expected value of
X
, and
σ
2
X
is
the (population) variance of
X
. For more details, see Jeffrey M. Wooldridge,
Introductory Econometrics:
A Modern Approach,
South-Western Publishing, 2000, p. 250.
guj75772_ch11.qxd 14/08/2008 10:01 AM Page 411
412
Chapter
12
Autocorrelation: What
Happens If the Error
Terms Are Correlated?
The reader may recall that there are generally three types of data that are available for
empirical analysis: (1) cross section, (2) time series, and (3) combination of cross sec-
tion and time series, also known as pooled data. In developing the classical linear regres-
sion model (CLRM) in
Part 1
we made several assumptions, which were discussed
in Section 7.1. However, we noted that not
all
of these assumptions would hold in
every type of data. As a matter of fact, we saw in the previous chapter that the assumption
of homoscedasticity, or equal error variance, may not always be tenable in cross-
sectional data. In other words, cross-sectional data are often plagued by the problem of
heteroscedasticity.
However, in cross-section studies, data are often collected on the basis of a random
sample of cross-sectional units, such as households (in a consumption function analysis) or
firms (in an investment study analysis) so that there is no prior reason to believe that the
error term pertaining to one household or firm is correlated with the error term of another
household or firm. If by chance such a correlation is observed in cross-sectional units, it is
called
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