ECON 4551
Econometrics II
Memorial University of Newfoundland
Heteroskedasticity
Adapted from Vera Tabakova’s notes

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)
Food expenditure example:
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
We can use a robust estimator: GRETL offers several options…check the defaults
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
The existence of heteroskedasticity implies:

Why not use robust estimation all the time?

Well, that is a good idea for large samples but for small samples,
homoskedasticity plus normality guarantees that the t ratios are
distributed as t

But robust estimates do not guarantee that, so our inference could be
misleading!

If you have a small sample, check whether there is homoskedasticity
or not!
Principles of Econometrics, 3rd Edition
Slide 8-11
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 8-12
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 8-13




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 8-14
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 8-15
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 8-16
Food example again, where was the problem coming from?
regress food_exp income [aweight = 1/income]
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 8-17

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 8-18
 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 8-19
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 8-20
 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 8-21
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 8-22
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 8-23
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 8-24
For our food expenditure example (GRETL:
#Estimating the skedasticity function and GLS
ols y const x
genr lnsighat = log($uhat*$uhat)
genr z = log(x)
#Obtain prediction of variance:
ols lnsighat const z
genr predsighat = exp($yhat)
#generate weights;
genr w = 1/predsighat
wls w y const x
Principles of Econometrics, 3rd Edition
Slide 8-25
For our food expenditure example (STATA):
gen z = log(income)
regress food_exp income
predict ehat, residual
gen lnehat2 = log(ehat*ehat)
regress lnehat2 z
* -------------------------------------------* Feasible GLS
* -------------------------------------------predict sig2, xb
gen wt = exp(sig2)
regress food_exp income [aweight = 1/wt]
Principles of Econometrics, 3rd Edition
Slide 8-26
Using our wage data (cps2.dta):
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,
b M 1   9.914  1.524   8.39
Principles of Econometrics, 3rd Edition
(8.28)
,NM
,NR
(8.29a)
(8.29b)
???
Slide 8-27
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 8-28
 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 8-29
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 8-30
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)
(.015)
(8.33)
(.346)
. regress wage educ exper metro [aweight = 1/wt]
(sum of wgt is
3.7986e+01)
Source
SS
df
MS
Model
Residual
9797.0667
26284.1488
3
996
3265.6889
26.3897076
Total
36081.2155
999
36.1173328
wage
Coef.
educ
exper
metro
_cons
1.195721
.1322088
1.538803
-9.398362
Principles of Econometrics, 3rd Edition
Std. Err.
.068508
.0145485
.3462856
1.019673
t
17.45
9.09
4.44
-9.22
Number of obs
F(
3,
996)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.000
0.000
0.000
=
=
=
=
=
=
1000
123.75
0.0000
0.2715
0.2693
5.1371
[95% Conf. Interval]
1.061284
.1036595
.8592702
-11.39931
1.330157
.160758
2.218336
-7.397408
Slide 8-31
STATA
Commands:
Principles of Econometrics, 3rd Edition
* -------------------------------------------* Rural subsample regression
* -------------------------------------------regress wage educ exper if metro == 0
scalar rmse_r = e(rmse)
scalar df_r = e(df_r)
* -------------------------------------------* Urban subsample regression
* -------------------------------------------regress wage educ exper if metro == 1
scalar rmse_m = e(rmse)
scalar df_m = e(df_r)
* -------------------------------------------* Groupwise heteroskedastic regression using FGLS
* -------------------------------------------gen rural = 1 - metro
gen wt=(rmse_r^2*rural) + (rmse_m^2*metro)
regress wage educ exper metro [aweight = 1/wt]
Slide 8-32
GRETL
Commands:
#Wage Example
open "c:\Program Files\gretl\data\poe\cps2.gdt"
ols wage const educ exper metro
# Use only metro observations
smpl metro --dummy
ols wage const educ exper
scalar stdm = $sigma
#Restore the full sample
smpl full
Principles of Econometrics, 3rd Edition
Slide 8-33
#Create a dummy variable for rural
genr rural = 1-metro
GRETL
Commands:
#Restrict sample to rural observations
smpl rural --dummy
ols wage const educ exper
scalar stdr = $sigma
#Restore the full sample
smpl full
Principles of Econometrics, 3rd Edition
Slide 8-34








#Generate standard deviations for each metro and rural obs
genr wm = metro*stdm
genr wr = rural*stdr
#Make the weights (reciprocal)
#Remember, Gretl's wls needs these to be variances so you'll need to
square them
genr w = 1/(wm + wr)^2
#Weighted least squares
wls w wage const educ exper metro
Principles of Econometrics, 3rd Edition
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 8-36
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 8-37
8.4.2 The Goldfeld-Quandt Test
F 
2
2
ˆ M  M
ˆ

2
R
2
R
F( N M  K M , N R  K R )
(8.34)
H 0 :  M   R against H 0 :  M   R
2
F 
2
ˆ M
ˆ
2
R

2
31.824
2
2
(8.35)
 2.09
15.243
Principles of Econometrics, 3rd Edition
Slide 8-38
8.4.2 The Goldfeld-Quandt Test
STATA:
* -------------------------------------------* Goldfeld Quandt test
* --------------------------------------------
GRETL:
#Goldfeld Quandt statistic
scalar fstatistic = stdm^2/stdr^2
scalar GQ = rmse_m^2/rmse_r^2
scalar crit = invFtail(df_m,df_r,.05)
scalar pvalue = Ftail(df_m,df_r,GQ)
scalar list GQ pvalue crit
F 
2
ˆ M
ˆ
Principles of Econometrics, 3rd Edition
2
R

31.824
 2.09
15.243
Slide 8-39
8.4.2 The Goldfeld-Quandt Test
2
ˆ 1  3 5 7 4 .8
More generally, the test can be based
Simply on a continuous variable
Split the sample in halves (usually omitting
some from the middle) after ordering
them according to the suspected variable
(income in our food example)
ˆ  1 2, 9 2 1 .9
2
2
F 
2
ˆ 2
ˆ
2
1

12, 921.9
 3.61
3574.8
Principles of Econometrics, 3rd Edition
Slide 8-40
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
For the food expenditure data
You should now be able to obtain
this test statistic
And check whether it exceeds the critical
value

12, 921.9
 3.61
3574.8
Remember that you can probably use
the one-tail version of this test
Why?
Principles of Econometrics, 3rd Edition
Slide 8-41
8.4.3 Testing the Variance Function
For the mean:
y i  E ( y i )  ei   1   2 x i 2 
  K x iK  ei
For the variance, in general:
var( y i )    E ( e i )  h (  1   2 z i 2 
2
i
2
(8.36)
  S z iS )
(8.37)
For example::
h ( 1   2 zi 2 
  S z iS )  exp(  1   2 z i 2 
  S z iS )
h (  1   2 z i )  exp  ln(  )   ln( x i ) 
2
Principles of Econometrics, 3rd Edition
Slide 8-42
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 8-43
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
S is the number of variables used
Slide 8-44




This is a large sample test
It is a Lagrange Multiplier (LM) test, which are
based on an auxiliary regression
In this case named after Breusch and Pagan
Here (and in the textbook) we saw a test
statistic based on a linear function of the
squared residual, but the good thing about
this test is that this form can be used to test
for any form of heteroskedasticity
Principles of Econometrics, 3rd Edition
8.4.3a The White Test
Since we may not know which variables explain heteroskedasticity…
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 8-46
8.4.3b Testing the Food Expenditure Example
S S T  4, 6 1 0, 7 4 9, 4 4 1
S S E  3, 7 5 9, 5 5 6,1 6 9
STATA:
whitetst
R 1
2
SSE
 .1 8 4 6
SST
Or
Breusch-Pagan test
estat imtest, white
  N  R  40  .1846  7.38
2
2
White test
  N  R  40  .18888  7.555
2
2
Principles of Econometrics, 3rd Edition
p -value  .023
GRETL: ols y const x
modtest --breusch-pagan
modtest –white
Slide 8-47
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












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 8-48
Principles of Econometrics, 3rd Edition
Slide 8-49
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 8-50
E  b2   E   2   E   w i ei 
 2 
Principles of Econometrics, 3rd Edition
 w i E  ei    2
Slide 8-51
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 8-52
var( b 2 ) 
Principles of Econometrics, 3rd Edition

2
  xi  x 
2
(8A.3)
Slide 8-53
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 8-54
  ( 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 8-55
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 8-56
 
2
SST  SSE
SST / N
SSE 

 N  1 

SST 

 NR
Principles of Econometrics, 3rd Edition
(8B.8)
2
Slide 8-57