Transcript ch08.ppt

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  y i  E ( y i )  y i   1   2 x i
y i   1   2 x i  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( e i )  
2
cov( e i , e j )  0
var( y i )  var( e i )  h ( x i )
(8.4)
yˆ i  83.42  10.21 x i
eˆi  y i  83.42  10.21 x i
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
y i   1   2 x i  ei
var( b2 ) 
var( e i )  

2
(8.5)
2
N
 ( xi  x )
(8.6)
2
i 1
y i   1   2 x i  ei
Principles of Econometrics, 3rd Edition
var( e i )   i
2
(8.7)
Slide 8-8
N
N
var( b 2 ) 

wi  i 
2
2
i 1

i 1

2
  ( xi  x ) 
 i 1

N
N
N
var( b 2 ) 

i 1
Principles of Econometrics, 3rd Edition
2 2
w i eˆi 
 ( x i  x ) 2  i2 



i 1
2
(8.8)
 ( x i  x ) 2 eˆi2 



2
  ( xi  x ) 
 i 1

N
2
(8.9)
Slide 8-9
yˆ i  83.42  10.21 x i
(27.46) (1.81)
(W hite se)
(43.41) (2.09)
(incorrect se)
W hite:
b 2  t c se( b 2 )  10.21  2.024  1.81  [6.55, 13.87 ]
Incorrect:
b 2  t c se( b 2 )  10.21  2.024  2.09  [5.97, 14.45]
Principles of Econometrics, 3rd Edition
Slide 8-10
y i   1   2 x i  ei
(8.10)
E ( ei )  0
Principles of Econometrics, 3rd Edition
var( e i )   i
2
cov( e i , e j )  0
Slide 8-11
var  ei    i   x i
2
2
 1 
 x
i
 1 
  2 
 x 
 x
xi
i 
i


yi

i
y 
yi
xi

i1
x 
Principles of Econometrics, 3rd Edition
1
xi
x

i2

xi
xi

(8.11)

ei


xi

xi
(8.12)

i
e 
ei
xi
(8.13)
Slide 8-12




y i   1 x i1   2 x i 2  e i
 e
var( e )  var  i
 x
i


i
Principles of Econometrics, 3rd Edition
 1
1 2
2
var( e i ) 
 xi  

 x
xi
i

(8.14)
(8.15)
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:
N
e
2
i
2
N

i 1
When
ei


xi
i 1
xi
N
 ( xi
 1/ 2
ei )
2
i 1
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
yˆ i  78.68  10.45 x i
(8.16)
(se) (23.79) (1.39)
ˆ 2  t c se(ˆ 2 )  10.451  2.024  1.386  [7.65,13.26]
Principles of Econometrics, 3rd Edition
Slide 8-16

var( e i )   i   x i
2
2
(8.17)
ln (  i )  ln (  )   ln ( x i )
2
2
 i  exp  ln(  )   ln( x i ) 
2
2
(8.18)
 exp(  1   2 z i )
Principles of Econometrics, 3rd Edition
Slide 8-17
 i  ex p (  1   2 z i 2 
2
  s z iS )
ln (  i )   1   2 z i
2
(8.19)
(8.20)
y i  E ( y i )  ei   1   2 x i  ei
Principles of Econometrics, 3rd Edition
Slide 8-18
2
2
ln ( eˆi )  ln (  i )  v i   1   2 z i  v i
(8.21)
ln ( ˆ i )  .9 3 7 8  2 .3 2 9 z i
2
2
ˆ i  ex p ( ˆ 1  ˆ 1 z i )
 yi 
 1 
 x i   ei 

  1 
  2 


 i 
 i 
 i   i 
Principles of Econometrics, 3rd Edition
Slide 8-19
 ei   1 
 1  2
var 
   2  var( e i )   2   i  1
 i   i 
 i 
 yi 
y 

 ˆ i 
 1 
x 

 ˆ i 

i

i1


x

 xi 


 ˆ i 
(8.23)

y i   1 x i1   2 x i 2  e i
Principles of Econometrics, 3rd Edition

i2
(8.22)
(8.24)
Slide 8-20
y i  1   2 xi 2 
  k x iK  e i
var( e i )   i  ex p (  1   2 z i 2 
2
Principles of Econometrics, 3rd Edition
  s z iS )
(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

2. Estimate  1 ,  2 ,
ln eˆi   1   2 z i 2 
2
Principles of Econometrics, 3rd Edition
2
eˆ i .
,S
by applying least squares to the equation
  S z iS  v i
Slide 8-22
3. Compute variance estimates ˆ i2
 ex p ( ˆ 1  ˆ 2 z i 2 
 ˆ S z iS. )
4. Compute the transformed observations defined by (8.23),
including x i3 ,

, x iK
if
K  2.
5. Apply least squares to (8.24), or to an extended version of (8.24)
if
K 2
.
yˆ i  76.05  10.63 x
(se)
Principles of Econometrics, 3rd Edition
(9.71)
(.97 )
(8.27)
Slide 8-23
W A G E   9.914  1.234 E D U C  .133 E X P E R  1.524 M E T R O
(se)
(1.08)
(.070)
(.015)
W A G E Mi   M 1   2 E D U C Mi   3 E X P E RMi  eMi
W A G E Ri   R 1   2 E D U C Ri   3 E X P E R Ri  e Ri
(.431)
i  1, 2,
i  1, 2,
,NM
,NR
(8.28)
(8.29a)
(8.29b)
b M 1   9.914  1.524   8.39
Principles of Econometrics, 3rd Edition
Slide 8-24
var( e M i )   M
2
var( e R i )   R
2
ˆ M  3 1 .8 2 4
2
2
ˆ R  1 5 .2 4 3
b M 1   9.052
b M 2  1.282
b R 1   6.166
b R 2  .956
Principles of Econometrics, 3rd Edition
(8.30)
b M 3  .1346
b R 3  .1260
Slide 8-25
 W AG E Mi 
 1 
 ED U C Mi 
 E X P E RMi   eMi 












M1
2 
3 





M
M
M


 M 



  M 
i  1, 2,
,NM
 W A G E Ri 
 1 
 E D U C Ri 
 E X P E R Ri   eRi 












R1 
2 
3 





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.
2.
 ˆ M w hen M E T R O i  1
ˆ i  
 ˆ R w hen M E T R O i  0
3. Apply least squares to the transformed model
 W AG Ei 
 1 
 EDUCi 
 E X P E Ri 
 M E T R O i   ei 
















R1 
2 
3 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ






i
i
i
i


 i





  i
Principles of Econometrics, 3rd Edition
(8.32)
Slide 8-27
W A G E   9.398  1.196 E D U C  .132 E X P E R  1.539 M E T R O
(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
F 
2
2
ˆ M  M
ˆ

2
R
F( N M  K M , N R  K R )
2
R
(8.34)
H 0 :  M   R against H 0 :  M   R
2
2
F 
Principles of Econometrics, 3rd Edition
2
2
ˆ M
ˆ
2
R

31.824
2
(8.35)
 2.09
15.243
Slide 8-31
8.4.2 The Goldfeld-Quandt Test
2
ˆ 1  3 5 7 4 .8
2
ˆ 2  1 2, 9 2 1 .9
F 
2
ˆ 2
ˆ
2
1

12, 921.9
 3.61
3574.8
Principles of Econometrics, 3rd Edition
Slide 8-32
8.4.3 Testing the Variance Function
y i  E ( y i )  ei   1   2 x i 2 
  K x iK  ei
var( y i )   i  E ( e i )  h (  1   2 z i 2 
2
h ( 1   2 zi 2 
2
(8.36)
  S z iS )
  S z iS )  exp(  1   2 z i 2 
(8.37)
  S z iS )
h (  1   2 z i )  exp  ln(  )   ln( x i ) 
2
Principles of Econometrics, 3rd Edition
Slide 8-33
8.4.3 Testing the Variance Function
h ( 1   2 zi 2 
  S z iS )   1   2 z i 2 
h ( 1   2 zi 2 
H 0 : 2  3 
  S z iS
(8.38)
  S z iS )  h (  1 )
 S  0
(8.39)
H 1 : 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( y i )   i  E ( e i )   1   2 z i 2 
2
2
ei  E ( ei )  v i   1   2 z i 2 
2
2
2
eˆi   1   2 z i 2 
  NR
2
Principles of Econometrics, 3rd Edition
2
  S z iS
(8.40)
  S z iS  v i
(8.41)
  S z iS  v i
(8.42)
 ( S 1)
(8.43)
2
Slide 8-35
8.4.3a The White Test
E ( y i )  1   2 xi 2   3 xi 3
z2  x2
Principles of Econometrics, 3rd Edition
z 3  x3
z4  x2
2
z 5  x3
2
Slide 8-36
8.4.3b Testing the Food Expenditure Example
S S T  4, 6 1 0, 7 4 9, 4 4 1
R 1
2
S S E  3, 7 5 9, 5 5 6,1 6 9
SSE
 .1 8 4 6
SST
  N  R  40  .1846  7.38
2
2
  N  R  40  .18888  7.555
2
2
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
y i   1   2 x i  ei
E ( ei )  0
var ( e i )   i
cov( e i , e j )  0
2
b2   2 
wi 
Principles of Econometrics, 3rd Edition
 w i ei
(i  j )
(8A.1)
xi  x
  xi  x 
2
Slide 8-40
E  b2   E   2   E   w i ei 
 2 
Principles of Econometrics, 3rd Edition
 w i E  ei    2
Slide 8-41
var  b 2   var   w i e i 




w i var  e i  
2
i j
w

  w i w j cov  ei , e j 
2
i

(8A.2)
2
i
 ( x i  x ) 2  i2 


  ( xi  x ) 


2
Principles of Econometrics, 3rd Edition
2
Slide 8-42
var( b 2 ) 
Principles of Econometrics, 3rd Edition

2
  xi  x 
2
(8A.3)
Slide 8-43
2
eˆi   1   2 z i 2 
F 
(8B.2)
SSE / ( N  S )

i 1
Principles of Econometrics, 3rd Edition
(8B.1)
( SST  SSE ) / ( S  1)
N
SST 
  S z iS  v i
2
2
eˆi  eˆ

2
N
and
SSE 

2
vˆi
i 1
Slide 8-44
  ( S  1)  F 
2
SST  SSE
SSE / ( N  S )
var( e )  var( v i ) 
2
i
 
2
SSE
N S
SST  SSE
 ( S 1)
2
(8B.3)
(8B.4)
(8B.5)
2
i
var( e )
Principles of Econometrics, 3rd Edition
Slide 8-45
SST  SSE
 
2
 e i2 
var  2   2
 e 
2 ˆ
1
4
var( e ) 
Principles of Econometrics, 3rd Edition
1
N
var( e i )  2
var( e i )  2  e
2
e
2
i
(8B.6)
4
e
N
 (eˆ
i 1
2
i
 eˆ ) 
2
2
2
SST
4
(8B.7)
N
Slide 8-46
 
2
SST  SSE
SST / N
SSE 

 N  1 

SST 

 NR
Principles of Econometrics, 3rd Edition
(8B.8)
2
Slide 8-47