Heteroskedasticity
Prepared by Vera Tabakova, East Carolina University

8.1 The Nature of Heteroskedasticity

8.2 Using the Least Squares Estimator

8.3 The Generalized Least Squares Estimator

8.4 Detecting Heteroskedasticity
Principles of Econometrics, 3rd Edition
Slide 8-2
E ( y )  1  2 x
ei  yi  E ( yi )  yi  1  2 xi
yi  1  2 xi  ei
Principles of Econometrics, 3rd Edition
(8.1)
(8.2)
(8.3)
Slide 8-3
Figure 8.1 Heteroskedastic Errors
Principles of Econometrics, 3rd Edition
Slide 8-4
E (ei )  0
var(ei )  2
cov(ei , e j )  0
var( yi )  var(ei )  h( xi )
(8.4)
yˆi  83.42  10.21 xi
eˆi  yi  83.42  10.21xi
Principles of Econometrics, 3rd Edition
Slide 8-5
Figure 8.2 Least Squares Estimated Expenditure Function and Observed Data Points
Principles of Econometrics, 3rd Edition
Slide 8-6
The existence of heteroskedasticity implies:

The least squares estimator is still a linear and unbiased estimator, but
it is no longer best. There is another estimator with a smaller
variance.

The standard errors usually computed for the least squares estimator
are incorrect. Confidence intervals and hypothesis tests that use these
standard errors may be misleading.
Principles of Econometrics, 3rd Edition
Slide 8-7
yi  1 2 xi  ei
var(b2 ) 
var(ei )  2
(8.5)
2
N
 ( xi  x )
2
(8.6)
i 1
yi  1 2 xi  ei
Principles of Econometrics, 3rd Edition
var(ei )  i2
(8.7)
Slide 8-8
N
N
var(b2 )   wi2i2 
i 1
2 2

(
x

x
)
i 
 i
i 1

2
( xi  x ) 
 
i 1

N
2
(8.8)
N
N
var(b2 )   wi2eˆi2 
i 1
Principles of Econometrics, 3rd Edition
2 2

(
x

x
)
eˆi 
 i
i 1

2
( xi  x ) 
 
i 1

N
2
(8.9)
Slide 8-9
yˆi  83.42  10.21xi
(27.46) (1.81)
(White se)
(43.41) (2.09)
(incorrect se)
White:
b2  tcse(b2 )  10.21  2.024 1.81  [6.55, 13.87]
Incorrect:
b2  tcse(b2 )  10.21  2.024  2.09  [5.97, 14.45]
Principles of Econometrics, 3rd Edition
Slide 8-10
yi  1  2 xi  ei
(8.10)
E (ei )  0
Principles of Econometrics, 3rd Edition
var(ei )  i2
cov(ei , e j )  0
Slide 8-11
yi
y 
xi

i
var  ei   i2  2 xi
(8.11)
 1 
 xi  ei
yi
 1 
 2 



 x 
 x 
xi
xi
i


 i
(8.12)
1
x 
xi

i1
Principles of Econometrics, 3rd Edition
xi
ei

x 
 xi ei 
xi
xi

i2
(8.13)
Slide 8-12
yi  1xi1  2 xi2  ei
(8.14)
 ei  1
1 2
var(e )  var 
 var(ei )   xi  2

 x  xi
xi
 i
(8.15)

i
Principles of Econometrics, 3rd Edition
Slide 8-13
To obtain the best linear unbiased estimator for a model with
heteroskedasticity of the type specified in equation (8.11):
1.
Calculate the transformed variables given in (8.13).
2.
Use least squares to estimate the transformed model given in (8.14).
Principles of Econometrics, 3rd Edition
Slide 8-14
The generalized least squares estimator is as a weighted least
squares estimator. Minimizing the sum of squared transformed errors
that is given by:
2
N
e
2
i
 ei   x   ( xi 1/2ei )2
i 1
i 1 i
i 1
N
When
N
xi
is small, the data contain more information about the
regression function and the observations are weighted heavily.
When
xi
is large, the data contain less information and the
observations are weighted lightly.
Principles of Econometrics, 3rd Edition
Slide 8-15
yˆi  78.68  10.45 xi
(se) (23.79) (1.39)
(8.16)
ˆ 2  tcse(ˆ 2 )  10.451  2.024 1.386  [7.65,13.26]
Principles of Econometrics, 3rd Edition
Slide 8-16
var(ei )  i2  2 xi
(8.17)
ln(i2 )  ln(2 )   ln( xi )
i2  exp  ln( 2 )   ln( xi ) 
(8.18)
 exp(1   2 zi )
Principles of Econometrics, 3rd Edition
Slide 8-17
i2  exp(1  2 zi 2 
 s ziS )
ln(i2 )  1  2 zi
(8.19)
(8.20)
yi  E ( yi )  ei  1  2 xi  ei
Principles of Econometrics, 3rd Edition
Slide 8-18
ln(eˆi2 )  ln(i2 )  vi  1  2 zi  vi
(8.21)
ˆ i2 )  .9378  2.329 zi
ln(
ˆ1 
ˆ 1zi )
ˆ i2  exp(

 yi 
1
 xi   ei 
   1    2     
 i 
 i 
 i   i 
Principles of Econometrics, 3rd Edition
Slide 8-19
 ei   1 
 1  2
var     2  var(ei )   2  i  1
 i   i 
 i 
 yi 
y  
 ˆ i 

i
1
x  
 ˆ i 

i1
 xi 
x  
 ˆ i 
yi  1xi1  2 xi2  ei
Principles of Econometrics, 3rd Edition

i2
(8.22)
(8.23)
(8.24)
Slide 8-20
yi  1  2 xi 2 
 k xiK  ei
var(ei )  i2  exp(1  2 zi 2 
Principles of Econometrics, 3rd Edition
 s ziS )
(8.25)
(8.26)
Slide 8-21
The steps for obtaining a feasible generalized least squares estimator
for 1 , 2 , ,  K are:

1. Estimate (8.25) by least squares and compute the squares of the
least squares residuals eˆi2 .

2. Estimate 1 ,  2 , ,  S by applying least squares to the equation
ln eˆi2  1  2 zi 2 
Principles of Econometrics, 3rd Edition
 S ziS  vi
Slide 8-22
3. Compute variance estimates ˆ i2  exp(ˆ 1  ˆ 2 zi 2 
ˆ S ziS. )

4. Compute the transformed observations defined by (8.23),
including xi3 , , xiK if K  2.
5. Apply least squares to (8.24), or to an extended version of (8.24)
if K  2 .
yˆi  76.05  10.63x
(se) (9.71)
Principles of Econometrics, 3rd Edition
(.97)
(8.27)
Slide 8-23
WAGE  9.914  1.234EDUC  .133EXPER  1.524METRO
(se)
(1.08)
(.070)
(.015)
WAGEMi  M 1  2 EDUCMi  3 EXPERMi  eMi
WAGERi   R1  2 EDUCRi  3 EXPERRi  eRi
(.431)
i  1, 2,
i  1, 2,
(8.28)
, N M (8.29a)
, NR
(8.29b)
bM 1  9.914  1.524  8.39
Principles of Econometrics, 3rd Edition
Slide 8-24
var(eMi )  2M
ˆ 2M  31.824

var(eRi )  2R
ˆ 2R  15.243

bM 1  9.052
bM 2  1.282
bM 3  .1346
bR1  6.166
bR 2  .956
bR 3  .1260
Principles of Econometrics, 3rd Edition
(8.30)
Slide 8-25
 WAGEMi 
 1

  M 1 

M


 M

 EDUCMi 
 EXPERMi   eMi 
  2 
  3 





M
M




  M
i  1, 2,
, NM
 WAGERi 
 1 
 EDUCRi 
 EXPERRi   eRi 








 2
 3
 
R1 
 R 
 R 
 R 
 R
  R 
i  1, 2,
Principles of Econometrics, 3rd Edition
(8.31a)
(8.31b)
, NR
Slide 8-26
Feasible generalized least squares:
1. Obtain estimated ˆ M and ˆ R by applying least squares separately to
the metropolitan and rural observations.
ˆ

M when METROi  1
2. ˆ i  

ˆ R when METROi  0
3. Apply least squares to the transformed model
 WAGEi 
1
 EDUCi 
 EXPERi   METROi   ei 








 2
 3
  
 
R1 
ˆ
ˆ
ˆ
ˆ
ˆ
i
 i 
 i 
 i 
 i  
  ˆ i 
Principles of Econometrics, 3rd Edition
(8.32)
Slide 8-27
WAGE  9.398  1.196 EDUC  .132 EXPER  1.539METRO
(se)
(1.02) (.069)
Principles of Econometrics, 3rd Edition
(.015)
(.346)
(8.33)
Slide 8-28
Remark: To implement the generalized least squares estimators
described in this Section for three alternative heteroskedastic
specifications, an assumption about the form of the
heteroskedasticity is required. Using least squares with White
standard errors avoids the need to make an assumption about the
form of heteroskedasticity, but does not realize the potential
efficiency gains from generalized least squares.
Principles of Econometrics, 3rd Edition
Slide 8-29
8.4.1 Residual Plots

Estimate the model using least squares and plot the least squares
residuals.

With more than one explanatory variable, plot the least squares
residuals against each explanatory variable, or against yˆ i , to see if
those residuals vary in a systematic way relative to the specified
variable.
Principles of Econometrics, 3rd Edition
Slide 8-30
8.4.2 The Goldfeld-Quandt Test
ˆ 2M 2M
F 2 2
ˆ R  R
F( NM  KM , N R  K R )
H0 : 2M  2R against H0 : 2M  2R
(8.34)
(8.35)
ˆ 2M 31.824
F 2 
 2.09
ˆ R 15.243
Principles of Econometrics, 3rd Edition
Slide 8-31
8.4.2 The Goldfeld-Quandt Test
ˆ 12  3574.8

ˆ 22  12,921.9

ˆ 22 12,921.9
F 2
 3.61
ˆ 1
3574.8
Principles of Econometrics, 3rd Edition
Slide 8-32
8.4.3 Testing the Variance Function
yi  E ( yi )  ei  1  2 xi 2 
  K xiK  ei
var( yi )  i2  E(ei2 )  h(1  2 zi 2 
h(1   2 zi 2 
(8.36)
 S ziS )
  S ziS )  exp(1   2 zi 2 
(8.37)
  S ziS )
h(1  2 zi )  exp  ln(2 )   ln( xi ) 
Principles of Econometrics, 3rd Edition
Slide 8-33
8.4.3 Testing the Variance Function
h(1   2 zi 2 
  S ziS )  1   2 zi 2 
h(1   2 zi 2 
H 0 :  2  3 
  S ziS
(8.38)
  S ziS )  h(1 )
 S  0
(8.39)
H1 : not all the  s in H 0 are zero
Principles of Econometrics, 3rd Edition
Slide 8-34
8.4.3 Testing the Variance Function
var( yi )  i2  E(ei2 )  1  2 zi 2 
ei2  E(ei2 )  vi  1  2 zi 2 
eˆi2  1  2 zi 2 
2  N  R 2
Principles of Econometrics, 3rd Edition
 S ziS
(8.40)
 S ziS  vi
(8.41)
 S ziS  vi
(8.42)
(2S 1)
(8.43)
Slide 8-35
8.4.3a The White Test
E ( yi )  1  2 xi 2  3 xi 3
z2  x2
Principles of Econometrics, 3rd Edition
z3  x3
z4  x22
z5  x32
Slide 8-36
8.4.3b Testing the Food Expenditure Example
SST  4,610,749,441
SSE  3,759,556,169
SSE
R  1
 .1846
SST
2
2  N  R 2  40  .1846  7.38
2  N  R 2  40  .18888  7.555
Principles of Econometrics, 3rd Edition
p-value  .023
Slide 8-37














Breusch-Pagan test
generalized least squares
Goldfeld-Quandt test
heteroskedastic partition
heteroskedasticity
heteroskedasticity-consistent
standard errors
homoskedasticity
Lagrange multiplier test
mean function
residual plot
transformed model
variance function
weighted least squares
White test
Principles of Econometrics, 3rd Edition
Slide 8-38
Principles of Econometrics, 3rd Edition
Slide 8-39
yi  1  2 xi  ei
E (ei )  0
var(ei )  i2
cov(ei , e j )  0 (i  j )
b2  2   we
i i
wi 
Principles of Econometrics, 3rd Edition
(8A.1)
xi  x
  xi  x 
2
Slide 8-40
E  b2   E  2   E   wi ei 
 2   wi E  ei   2
Principles of Econometrics, 3rd Edition
Slide 8-41
var  b2   var   wi ei 
  wi2 var  ei    wi w j cov  ei , e j 
i j
 w 
2
i
2
i
(8A.2)
2 2

(
x

x
)
i 

i


2 2
  ( xi  x ) 
Principles of Econometrics, 3rd Edition
Slide 8-42
var(b2 ) 
Principles of Econometrics, 3rd Edition
2
  xi  x 
2
(8A.3)
Slide 8-43
eˆi2  1  2 zi 2 
 S ziS  vi
(8B.1)
( SST  SSE ) / ( S  1)
F
SSE / ( N  S )
N

SST   eˆi2  eˆ2
i 1
Principles of Econometrics, 3rd Edition

2
(8B.2)
N
and SSE   vˆi2
i 1
Slide 8-44
SST  SSE
  ( S  1)  F 
SSE / ( N  S )
2
SSE
var(e )  var(vi ) 
N S
2
i
 
2
SST  SSE
2
i
 (2S 1)
(8B.3)
(8B.4)
(8B.5)
var(e )
Principles of Econometrics, 3rd Edition
Slide 8-45
SST  SSE
 
2ˆ e4
2
 ei2 
var  2   2
 e 
1
2
var(
e
i )2
4
e
1 N 2 2 2 SST
var(e )   (eˆi  eˆ ) 
N i 1
N
2
i
Principles of Econometrics, 3rd Edition
(8B.6)
var(ei2 )  2e4
(8B.7)
Slide 8-46
SST  SSE
 
SST / N
2
 SSE 
 N  1 

SST


(8B.8)
 N  R2
Principles of Econometrics, 3rd Edition
Slide 8-47