ECON 4550
Econometrics
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  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)
Food expenditure example:
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
(
x

x
)
i
 

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
(
x

x
)
i
 

i 1
N
2
(8.9)
Slide 8-9
We can use a robust estimator:
regress food_exp income, robust
yˆi  83.42  10.21xi
(27.46) (1.81)
(43.41) (2.09)
(White se)
(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
In SHAZAM the HETCOV option on the OLS command reports
the White's heteroskedasticity-consistent standard errors
OLS FOOD INCOME / HETCOV
Confidence interval estimate using the White standard errors
CONFID INCOME / TCRIT=2.024
But the White standard errors reported by SHAZAM have numeric
differences compared to our textbook results.
There is a correction we could apply, but we will not worry about it
for now
Principles of Econometrics, 3rd Edition
Slide 8-11
yi  1  2 xi  ei
(8.10)
E (ei )  0
Principles of Econometrics, 3rd Edition
var(ei )  i2
cov(ei , e j )  0
Slide 8-12
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 8-13
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 8-14
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:
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 8-16
Food example again, where was the problem coming from?
regress food_exp income [aweight = 1/income]
yˆi  78.68  10.45 xi
(8.16)
(se) (23.79) (1.39)
ˆ 2  tcse(ˆ 2 )  10.451  2.024 1.386  [7.65,13.26]
Principles of Econometrics, 3rd Edition
Slide 8-17
Food example again, where was the problem coming from?
Specify a weight variable (SHAZAM works with the inverse)
GENR W=1/INCOME
OLS FOOD INCOME / WEIGHT=W
95% confidence interval, p. 205 CONFID INCOME / TCRIT=2.024
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 8-18

The residual statistics reported in a WLS regression (SIGMA**2,
STANDARD ERROR OF THE ESTIMATE-SIGMA, and SUM OF SQUARED
ERRORS-SSE) are all based on the transformed (weighted) residuals

You should remember when making model comparisons, that the highvariance observations are systematically underweighted by this
procedure

This may be a good thing, if you want to avoid having these observations
dominate the model selection comparisons.

But if you want model selection to be based on how well the alternative
models fit the original (untransformed) data, you must base the model
selection tests on the untransformed residuals.
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 8-20
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 8-21
ln(eˆi2 )  ln(i2 )  vi  1  2 zi  vi
(8.21)
ln(ˆ i2 )  .9378  2.329 zi
ˆ1 
ˆ 1zi )
ˆ i2  exp(

 yi 
1
 xi   ei 
   1    2     
 i 
 i 
 i   i 
Principles of Econometrics, 3rd Edition
Slide 8-22
 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 8-23
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 8-24
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 8-25
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 8-26
For our food expenditure example:
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 8-27
The HET command can be used for Maximum Likelihood
Estimation of the model given in Equations (8.25) and
(8.26), p. 207.
This method is an alternative estimation method to the
GLS method discussed in the text (so the results will also
be different):
HET FOOD INCOME (INCOME) / MODEL=MULT
Principles of Econometrics, 3rd Edition
Slide 8-28
Using our wage data (cps2.dta):
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,
bM 1  9.914  1.524  8.39
Principles of Econometrics, 3rd Edition
, NM
, NR
(8.28)
(8.29a)
(8.29b)
???
Slide 8-29
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 8-30
 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 8-31
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 8-32
WAGE  9.398  1.196 EDUC  .132 EXPER  1.539METRO
(se)
(1.02) (.069)
(.015)
(.346)
(8.33)
. 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 8-33
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 8-34
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 8-35
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 8-36
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 8-37
8.4.2 The Goldfeld-Quandt Test
* -------------------------------------------* Goldfeld Quandt test
* -------------------------------------------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
ˆ 2M 31.824
F 2 
 2.09
ˆ R 15.243
Principles of Econometrics, 3rd Edition
Slide 8-38
8.4.2 The Goldfeld-Quandt Test
ˆ 12  3574.8

ˆ 22  12,921.9

For the food expenditure data
You should now be able to obtain
this test statistic
And check whether it exceeds the critical
value
ˆ 22 12,921.9
F 2
 3.61
ˆ 1
3574.8
Principles of Econometrics, 3rd Edition
Slide 8-39
8.4.2 The Goldfeld-Quandt Test
Sort the data by income:
SORT INCOME FOOD / DESC
OLS FOOD INCOME
On the DIAGNOS command the CHOWONE= option reports the Goldfeld-Quandt
test for heteroskedasticity (bottom of page 212) with a p-value for a one-sided test.
The HET option reports the tests for heteroskedasticity reported on page 215.
DIAGNOS / CHOWONE=20 HET
Principles of Econometrics, 3rd Edition
Slide 8-40
8.4.2 The Goldfeld-Quandt Test
SORT INCOME FOOD / DESC
OLS FOOD INCOME
On the DIAGNOS command the CHOWONE= option reports the Goldfeld-Quandt
test for heteroskedasticity (bottom of page 212) with a p-value for a one-sided test.
The HET option reports the tests for heteroskedasticity reported on page 215.
DIAGNOS / CHOWONE=20 HET
Of course, this option also computes the Chow test statistic for structural change
(that is, tests the null of parameter stability in the two subsamples).
Principles of Econometrics, 3rd Edition
Slide 8-41
8.4.2 The Goldfeld-Quandt Test
SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS
N1 N2 SSE1
SSE2
CHOW PVALUE G-Q
DF1 DF2 PVALUE
20 20 0.23259E+06 64346. 0.45855 0.636 3.615
18 18 0.005
CHOW TEST - F DISTRIBUTION WITH DF1= 2 AND DF2= 36
Principles of Econometrics, 3rd Edition
Slide 8-42
8.4.2 The Goldfeld-Quandt Test
SHAZAM considers that the alternative hypothesis is smaller error variance in the
second subset relative to the first subset.
Some authors present the alternative as larger variance in the second subset.
Goldfeld and Quandt recommend ordering the observations by the values of one of
the explanatory variables.
This can be done with the SORT command in SHAZAM. The DESC option on the
SORT command should be used if it is assumed that the variance is positively
related to the value of the sort variable.
Principles of Econometrics, 3rd Edition
Slide 8-43
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 8-44
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 8-45
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 8-46
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 8-47
8.4.3b Testing the Food Expenditure Example
SST  4,610,749,441
SSE
R  1
 .1846
SST
whitetst
2
Or
estat imtest, white
SSE  3,759,556,169
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 8-48


DIAGNOS / HET
Will yield a battery of heteroskedasticity tests
using different specifications
DIAGNOS\HET
REQUIRED MEMORY IS PAR=
7 CURRENT PAR=
22480
DEPENDENT VARIABLE = FOOD
40 OBSERVATIONS
REGRESSION COEFFICIENTS
Same in SLR
10.2096426868
83.4160065402
HETEROSKEDASTICITY TESTS
CHI-SQUARE
D.F.
P-VALUE
TEST STATISTIC
E**2 ON YHAT:
7.384
1
0.00658
E**2 ON YHAT**2:
7.549
1
0.00600
E**2 ON LOG(YHAT**2):
6.516
1
0.01069
E**2 ON LAG(E**2) ARCH TEST:
0.089
1
0.76544
LOG(E**2) ON X (HARVEY) TEST:
10.654
1
0.00110
ABS(E) ON X (GLEJSER) TEST:
11.466
1
0.00071
E**2 ON X
TEST:
KOENKER(R2):
7.384
1
0.00658
B-P-G (SSR) :
7.344
1
0.00673
E**2 ON X X**2
(WHITE) TEST:
KOENKER(R2):
7.555
2
0.02288
B-P-G (SSR) :
7.514
2
0.02336














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 8-51
Principles of Econometrics, 3rd Edition
Slide 8-52
yi  1  2 xi  ei
E (ei )  0
var(ei )  i2
cov(ei , e j )  0 (i  j )
b2  2   wi ei
wi 
Principles of Econometrics, 3rd Edition
(8A.1)
xi  x
  xi  x 
2
Slide 8-53
E  b2   E  2   E   wi ei 
 2   wi E  ei   2
Principles of Econometrics, 3rd Edition
Slide 8-54
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 8-55
var(b2 ) 
Principles of Econometrics, 3rd Edition
2
  xi  x 
2
(8A.3)
Slide 8-56
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 8-57
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 8-58
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 8-59
SST  SSE
 
SST / N
2
 SSE 
 N  1 

SST


(8B.8)
 N  R2
Principles of Econometrics, 3rd Edition
Slide 8-60