Chapter 11
Heteroscedasticity,What
Happens if the Error
Variance is Nonconstant
11.1 The Nature of Heteroscedasticity
? Homoscedasticity,equal variance.
? Heteroscedasticity,unequal variance.
Heteroscedasticity is usually found
in cross-sectional data.
11.2 Consequences of Heteroscedasticity
? 1,OLS estimators are still linear.
? 2,They are still unbiased.
? 3,But they no longer have minimunm variance.
? 4,The usual formulas to estimate the variances of OLS
estimators are generally biased.
? 5,The bias arises from the fact that,namely,,
is no longer an unbiased estimator of,
? 6,The usual confidence intervals and hypothesis tests
based on t and F distributions are unreliable.
2?
/d, f,e 2i?
2?
? In short,in the presence of heteroscedasticity,
the usual hypothesis-testing routine is not
reliable,raising the possibility of drawing
misleading conclusions.
? Heteroscedasticity is potentially a serious
problem,for it might destroy the whole
edifice of the standard,and so routinely used,
OLS estimation and hypothesis-testing
procedure.
11.3 Detection of Heteroscedasticity,
How Do We Know When There is a
Heteroscedasticity Problem?
? 1,Nature of the Problem
In cross-sectional data involving
heterogeneous units,heteroscedasticity
may be the rule rather than the
exception.
? 2,Graphical Examination of Residuals
? These residuals can be plotted against
the observation to which they belong or
against one or more of the explanatory
variables or against,the estimated
mean value of,
For example,we can plot,the
residuals squared,against sales.
i??
i??
2ie
? 3,Park Test.
? we can regress on one or more of
the X variables.
(11.4)
? Park suggests using ei as proxies for ui,
(11.5)
ii222i vl n XBBln σ ???
ii222i vl n XBBeln ???
2iσ
? ( 1 ) Run the original regression despite the
heteroscedasticity problem
? (2) Obtain the residuals ei,square them,and
take their logs,
? (3)Regress ei2 against each X variable.
Alternatively,run the regression against,
the estimated Y.
? (4 )Null hypothesis,B2=0;
i??
? (5) Serious problem with the Park test,
the error term vi may itself be
heteroscedastic!
4.Glejser Test
(1)Obtaining residuals ei from the original model,
(2)Regressing the absolute values of ei,|ei|,on
the X variable,
|ei| =B1+B2Xi +vi (11.7)
|ei| =B1+B2 +vi (11.8)
|ei| =B1+B2 +vi (11.9)
(3)Null hypotheses:
There is no heteroscedasticity,that is, B2=0.
If this hypothesis is rejected,there is probably
evidence of heteroscedasticity.
iX
1?
?
??
?
?
iX
Note:
? The error term vi in the regressions.
can itself be heteroscedastic.
? Glejser,however,has maintained that
in large samples the preceding models
are fairly good in detecting
heteroscedasticity.
5.White Test
(11.13)
( 1 ) Eestimate regression (11.13) by OLS,obtaining the
residuals,ei.
(2) Auxiliary regression:
(11.14)
Regressed on all the original variables,their squared
values,and their cross-products,Additional powers of
the original X variables can also be added.
i2i22i uXBBY ???
i3i2i623i522i43i32i212i vXXAXAXAXAXAAe ???????
(3) Obtain the R2
Null hypothesis,B2=0
There is no heteroscedasticity,all the slope
coefficients in Eq,(11.14) are zero.
（ 11.15）
Reject H0,there is heteroscedasticity
Do not reject,there is heteroscedasticity
(4) Problem of White Test
—— rapidly consumes degrees of freedom.
Sometimes one can drop the cross-product
terms of the various variables in regression.
d, f,X~Rn 22?
11.4 What to do if Heteroscedasticity is
Observed,Remedial Measures
1,When Is Known,The Method of Weighted Least Squares
(WLS)
Yi = B1+B2Xi +ui
（ 1） Model Transformation,
(2) Estimate the transformed model with OLS,get the
estimators of b1,b2,they are BLUE
2iσ
i
i
i
i
21
i
i
σ
u
σ
XB
σ
1B
σ
Y ?
???
?
???
???
?
??
?
??
i
i
i σ
uv ?
? ?
? ?
? ?
1
( 1 1, 1 )E of b ec au s e,σ
σ
1
k n o w n is σ s i n c e,uE
σ
1
σ
u
EvE
q
2
i
2
i
2
i
2
i
2
i
2
i
2
i2
i
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2i
2i
2i
σ
uv ?
2,When True Is Unknown
(1)Plot the residuals from this regression against
the explanatory variable X and observe a
pattern similar.
(2) Transform the model
A,The error variance is proportional to X∶
(11.23)
(11.24)
? ? i22i XσuE ?
ii2
i
1
i
i
i
i
2
i
1
i
i
vXB
X
1
B
X
u
X
X
B
X
1
B
X
Y
???
???
B.Error variance proportional to,
estimated residuals
(11.27)
(11.28)
(3) Estimate the transformed model with OLS,
get the estimators of b1,b2,they are BLUE
2iX
? ? 2i22i XσuE ?
i2
i
1
i
i
2
i
1
i
i
vB
X
1
B
X
u
B
X
1
B
X
Y
????
?
?
??
?
?
?
??
?
?
??
?
?
????
?
?
??
?
?
?
? 3,Respecification of the Model
(1)If we estimate the model in the log form,it
often reduces heteroscedasticity.
(11.29)
(2)Which model we should use in a given
instance has to be determined by theoretical
and other considerations.
ii21i ul nXBBl nY ???
11.5 White’s Heteroscedasticity-
corrected Standard Errors and t Statistics
? As a result,we can continue to use
the t and F tests,except that they are
now valid asymptotically,that is,in
large samples.
Heteroscedasticity,What
Happens if the Error
Variance is Nonconstant
11.1 The Nature of Heteroscedasticity
? Homoscedasticity,equal variance.
? Heteroscedasticity,unequal variance.
Heteroscedasticity is usually found
in cross-sectional data.
11.2 Consequences of Heteroscedasticity
? 1,OLS estimators are still linear.
? 2,They are still unbiased.
? 3,But they no longer have minimunm variance.
? 4,The usual formulas to estimate the variances of OLS
estimators are generally biased.
? 5,The bias arises from the fact that,namely,,
is no longer an unbiased estimator of,
? 6,The usual confidence intervals and hypothesis tests
based on t and F distributions are unreliable.
2?
/d, f,e 2i?
2?
? In short,in the presence of heteroscedasticity,
the usual hypothesis-testing routine is not
reliable,raising the possibility of drawing
misleading conclusions.
? Heteroscedasticity is potentially a serious
problem,for it might destroy the whole
edifice of the standard,and so routinely used,
OLS estimation and hypothesis-testing
procedure.
11.3 Detection of Heteroscedasticity,
How Do We Know When There is a
Heteroscedasticity Problem?
? 1,Nature of the Problem
In cross-sectional data involving
heterogeneous units,heteroscedasticity
may be the rule rather than the
exception.
? 2,Graphical Examination of Residuals
? These residuals can be plotted against
the observation to which they belong or
against one or more of the explanatory
variables or against,the estimated
mean value of,
For example,we can plot,the
residuals squared,against sales.
i??
i??
2ie
? 3,Park Test.
? we can regress on one or more of
the X variables.
(11.4)
? Park suggests using ei as proxies for ui,
(11.5)
ii222i vl n XBBln σ ???
ii222i vl n XBBeln ???
2iσ
? ( 1 ) Run the original regression despite the
heteroscedasticity problem
? (2) Obtain the residuals ei,square them,and
take their logs,
? (3)Regress ei2 against each X variable.
Alternatively,run the regression against,
the estimated Y.
? (4 )Null hypothesis,B2=0;
i??
? (5) Serious problem with the Park test,
the error term vi may itself be
heteroscedastic!
4.Glejser Test
(1)Obtaining residuals ei from the original model,
(2)Regressing the absolute values of ei,|ei|,on
the X variable,
|ei| =B1+B2Xi +vi (11.7)
|ei| =B1+B2 +vi (11.8)
|ei| =B1+B2 +vi (11.9)
(3)Null hypotheses:
There is no heteroscedasticity,that is, B2=0.
If this hypothesis is rejected,there is probably
evidence of heteroscedasticity.
iX
1?
?
??
?
?
iX
Note:
? The error term vi in the regressions.
can itself be heteroscedastic.
? Glejser,however,has maintained that
in large samples the preceding models
are fairly good in detecting
heteroscedasticity.
5.White Test
(11.13)
( 1 ) Eestimate regression (11.13) by OLS,obtaining the
residuals,ei.
(2) Auxiliary regression:
(11.14)
Regressed on all the original variables,their squared
values,and their cross-products,Additional powers of
the original X variables can also be added.
i2i22i uXBBY ???
i3i2i623i522i43i32i212i vXXAXAXAXAXAAe ???????
(3) Obtain the R2
Null hypothesis,B2=0
There is no heteroscedasticity,all the slope
coefficients in Eq,(11.14) are zero.
（ 11.15）
Reject H0,there is heteroscedasticity
Do not reject,there is heteroscedasticity
(4) Problem of White Test
—— rapidly consumes degrees of freedom.
Sometimes one can drop the cross-product
terms of the various variables in regression.
d, f,X~Rn 22?
11.4 What to do if Heteroscedasticity is
Observed,Remedial Measures
1,When Is Known,The Method of Weighted Least Squares
(WLS)
Yi = B1+B2Xi +ui
（ 1） Model Transformation,
(2) Estimate the transformed model with OLS,get the
estimators of b1,b2,they are BLUE
2iσ
i
i
i
i
21
i
i
σ
u
σ
XB
σ
1B
σ
Y ?
???
?
???
???
?
??
?
??
i
i
i σ
uv ?
? ?
? ?
? ?
1
( 1 1, 1 )E of b ec au s e,σ
σ
1
k n o w n is σ s i n c e,uE
σ
1
σ
u
EvE
q
2
i
2
i
2
i
2
i
2
i
2
i
2
i2
i
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2i
2i
2i
σ
uv ?
2,When True Is Unknown
(1)Plot the residuals from this regression against
the explanatory variable X and observe a
pattern similar.
(2) Transform the model
A,The error variance is proportional to X∶
(11.23)
(11.24)
? ? i22i XσuE ?
ii2
i
1
i
i
i
i
2
i
1
i
i
vXB
X
1
B
X
u
X
X
B
X
1
B
X
Y
???
???
B.Error variance proportional to,
estimated residuals
(11.27)
(11.28)
(3) Estimate the transformed model with OLS,
get the estimators of b1,b2,they are BLUE
2iX
? ? 2i22i XσuE ?
i2
i
1
i
i
2
i
1
i
i
vB
X
1
B
X
u
B
X
1
B
X
Y
????
?
?
??
?
?
?
??
?
?
??
?
?
????
?
?
??
?
?
?
? 3,Respecification of the Model
(1)If we estimate the model in the log form,it
often reduces heteroscedasticity.
(11.29)
(2)Which model we should use in a given
instance has to be determined by theoretical
and other considerations.
ii21i ul nXBBl nY ???
11.5 White’s Heteroscedasticity-
corrected Standard Errors and t Statistics
? As a result,we can continue to use
the t and F tests,except that they are
now valid asymptotically,that is,in
large samples.
