Chapter 5 异方差
Heteroskedasticity
1,Recall the assumption for the CMLRM:
(Homoskedasticity)
2,Counterexamples
1) rich family and poor family expenditures;
2) large company and small company sales.
There exists heteroskedasticity in lots of
econometric problems.
2v a r ( ) c on st,1,,
i in
3,What happens if there is heteroske-
dasticity in an econometric problem?
1) The OLS estimators are maybe
not Blues (they are not efficient ).
2) The hypothesis tests for the
parameters do not hold good
though they is very important.
and so on.
4,Tests for Heteroskedasticity
1) Goldfeld-Quandt Test:
Given a sample with size n
(1) Sort it by the order of an
independent variable and then
portion it into three parts.
Sub-sample 1 with size n1
Sub-sample 2 with size n2
Sub-sample 3 with size n3
13n n n
(2) Estimate the regression equation
with sub-sample 1 and 3
respectively.
(3) let
(4) Test
2
2 1
1
1
1
e
nk


2
2 3
3
3
1
e
nk


2 2 2 2
0 1 3 1 1 3,v s,HH
testing statistic:
Given a significant level and if the
critical value is,then
reject when
2) Breusch-Pagan test
3) White test
2
3
312
1
( 1,1 )
F F n k n k

F?
0H FF

5,Heteroskedasticity Correction
1) Suppose
(1)
and
Then
0 1 1i i k k i iY = b b X b X +
22
11
v a r ( ) ( ),
(,,),1,,,a n d 0
i i i
i i i k i
f
X X X i n f


X
X
22
11
v a r( ) v a r( )
()()
1
( ),1,,
()
ii
ii
i
i
ff
f i n
f



XX
X
X
Let
(1) becomes
(2)
where
The variance of the error term in (2)
is constant,
0
11,
( ) ( )j i j i iii
Z X Z
ff

XX
*
0 0 1 1i i i k k i iV b Z b Z b Z
*( ),( )
i i i i i iV Y f f XX
2) WLS (weighted least squares) Method
(加权最小二乘法）
If we give a weight to the
residual squared
then the weighted sum of residual squared is
Applying the OLS procedure to it,we
obtain the WLS estimator,For example,given
the variances of the error terms,let
2
ie
22
0 1 1[ ( ) ]i i i i i k k iW e W Y b b X b X
2
1
i
i
W
and we can obtain the parameter estimators
which are BLUEs.
Homework:
P109 ex6.7