III.I.1 Heteroskedasticity in
Linear Regression
(III.I.11)
we can
use a two step procedure to solve the problem of heteroskedasticity
as follows:
 divide
each observation by the S.D. of the error term for that
observation  
 apply
LS to the transformed observations  
This
procedure is called the Weighted
Least Squares (WLS).
Of
course there exist a lot of alternative transformations. One of the
most popular transformations is the Neperian logarithm, since it
gives more weight to smallvalued observations and less weight to
large ones. The transformation of time series according to the
logarithmic and related transformations is (in econometrics) mostly
assumed to be theoryrelated.
Another
method is specially designed to solve the problem of multiplicative
heteroskedasticity.
Suppose
(III.I.12)
From
(III.I.12) we find
(III.I.13)
(III.I.14)
The
only question remaining is "How to estimate alpha?".
We
may rewrite (III.I.12) by taking logarithms
(III.I.15)
and
since
(III.I.16)
it
is obvious that
(III.I.17)
Now
we put all t elements from (III.I.17) in matrices and obtain
estimates of alpha for the model
(III.I.18)
by
solving
(III.I.19)
Once
the alpha parameter vector has been computed this information can be
used in the following Estimated
Generalized Least Squares estimator (EGLS)
(III.I.110)
Consider
the following multiple regression equation (to be used in subsequent
illustrations):
Estimation
with OLS:
Endogenous
variable = ship.dba
Variable
Parameter
S.E. tstat
const(0),1.,0,0
+294.5647227 247.2103565
+1.19
employ(0),1.,0,0
+34.36683831 5.160885579
+6.66
expend(0),1.,0,0
+9.572870953 2.108664727
+4.54
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared of stationary series = 0.9193525203
DurbinWatson = 2.472087712
Variance of regression = 1051039.479
Standard Error of regression = 1025.202165
Sum of Squared Residuals = 37837421.25
Degrees of freedom = 36
Correlation
matrix of parameters:
+1.00 0.42 +0.03
0.42 +1.00 0.85
+0.03 0.85 +1.00
Detection
of heteroskedasticity can be achieved by many different tests.
If we assume a linear statistical model of the form
(III.I.111)
then
a test for heteroskedasticity, according to Glejser,
can be obtained by testing
in
one of the following models
(III.I.112)
(and
many others...).
Warning:
this test should only be used if the endogenous variable is NOT used
as lagged exogenous variable. Furthermore the Gleisjer tests assume
ADDITIVE heteroskedasticity. All OLS assumptions should be
satisfied.
Below
you ‘ll find an example of how Gleisjer tests can be applied to
test for heteroskedasticity (this test is applied to our
exampleequation):
Gleisjer
tests:
Estimation
with OLS:
Endogenous
variable = abs(e)
Variable
Parameter
S.E. tstat
const(0),1.,0,0
+186.2884948 148.8382962
+1.25
employ(0),1.,0,0
+7.009636535 1.65355995
+4.24
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared
of stationary series = 0.3392877362
DurbinWatson = 2.107710811
Degrees of freedom
= 37
Variance
of regression = 381232.1935
Standard Error of regression = 617.4400323
Sum of Squared Residuals
= 14105591.16
Correlation
matrix of parameters:
+1.00 0.75
0.75 +1.00
TSTAT
of b in abs(e) =
a + b X = 4.239118477
Estimation
with OLS:
Endogenous
variable = abs(e)
Variable
Parameter
S.E. tstat
const(0),1.,0,0
+247.1919656 131.7411655
+1.88
expend(0),1.,0,0
+3.009214899 0.658345735
+4.57
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared
of stationary series = 0.3613379003
DurbinWatson = 1.709541158
Degrees of freedom
= 37
Variance
of regression = 361985.459
Standard Error of regression = 601.6522741
Sum of Squared Residuals
= 13393461.98
Correlation
matrix of parameters:
+1.00 0.68
0.68 +1.00
TSTAT
of b in abs(e) =
a + b X = 4.570873234
Estimation
with OLS:
Endogenous
variable = abs(e)
Variable
Parameter
S.E. tstat
const(0),1.,0,0
+1453.542646 204.7255451
+7.1
employ(0),1.,0,0
34269.28262 7753.841264
4.42
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared
of stationary series = 0.3458859921
DurbinWatson = 1.762971931
Degrees of freedom
= 37
Variance
of regression = 370690.754
Standard Error of regression = 608.8437845
Sum of Squared Residuals
= 13715557.9
Correlation
matrix of parameters:
+1.00 0.88
0.88 +1.00
TSTAT
of b in abs(e) =
a + b 1/X = 4.419652331
Estimation
with OLS:
Endogenous
variable = abs(e)
Variable
Parameter
S.E. tstat
const(0),1.,0,0
+1296.342082 182.4520168
+7.11
expend(0),1.,0,0
42579.02335 10204.88361
4.17
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared
of stationary series = 0.3203453806
DurbinWatson = 1.553508102
Degrees of freedom
= 37
Variance
of regression = 385163.4666
Standard Error of regression = 620.6153935
Sum of Squared Residuals
= 14251048.26
Correlation
matrix of parameters:
+1.00 0.84
0.84 +1.00
TSTAT
of b in abs(e) =
a + b 1/X = 4.172416362
Estimation
with OLS:
Endogenous
variable = abs(e)
Variable
Parameter
S.E. tstat
const(0),1.,0,0
566.1796621 258.3961199
2.19
employ(0),1.,0,0
+159.8350094 31.50187299
+5.07
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared
of stationary series = 0.4106281997
DurbinWatson = 2.169617364
Degrees of freedom
= 37
Variance
of regression = 333999.7394
Standard Error of regression = 577.9271056
Sum of Squared Residuals
= 12357990.36
Correlation
matrix of parameters:
+1.00 0.93
0.93 +1.00
TSTAT
of b in abs(e) =
a + b sqrt(X) = 5.073825594
Estimation
with OLS:
Endogenous
variable = abs(e)
Variable
Parameter
S.E. tstat
const(0),1.,0,0
318.6307381 205.1846763
1.55
expend(0),1.,0,0
+93.21309421 17.56301183
+5.31
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared
of stationary series = 0.4325531446
DurbinWatson = 1.866343044
Degrees of freedom
= 37
Variance
of regression = 321574.7705
Standard Error of regression = 567.0756303
Sum of Squared Residuals
= 11898266.51
Correlation
matrix of parameters:
+1.00 0.90
0.90 +1.00
TSTAT
of b in abs(e) =
a + b sqrt(X) = 5.30735247
Estimation
with OLS:
Endogenous
variable = abs(e)
Variable
Parameter
S.E. tstat
const(0),1.,0,0
+186.2884948 148.8382962
+1.25
employ(0),1.,0,0
+7.009636535 1.65355995
+4.24
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared
of stationary series = 0.3272823951
DurbinWatson = 2.107710811
Degrees of freedom
= 37
Variance
of regression = 381232.1935
Standard Error of regression = 617.4400323
Sum of Squared Residuals
= 14105591.16
Correlation
matrix of parameters:
+1.00 0.75
0.75 +1.00
TSTAT
of b in abs(e) =
a + b abs(X) = 4.239118477
Estimation
with OLS:
Endogenous
variable = abs(e)
Variable
Parameter
S.E. tstat
const(0),1.,0,0
+247.1919656 131.7411655
+1.88
expend(0),1.,0,0
+3.009214899 0.658345735
+4.57
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared
of stationary series = 0.3612449444
DurbinWatson = 1.709541158
Degrees of freedom
= 37
Variance
of regression = 361985.459
Standard Error of regression = 601.6522741
Sum of Squared Residuals
= 13393461.98
Correlation
matrix of parameters:
+1.00 0.68
0.68 +1.00
TSTAT
of b in abs(e) =
a + b abs(X) = 4.570873234
Estimation
with OLS:
Endogenous
variable = abs(e)
Variable
Parameter
S.E.
tstat
const(0),1.,0,0
+534.6889586
121.5421391
+4.4
employ(0),1.,0,0
+1.52089658e002 6.025380217e003
+2.52
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared
of stationary series = 0.1473776172
DurbinWatson = 1.732573146
Degrees of freedom
= 37
Variance
of regression = 483185.0673
Standard Error of regression = 695.1151468
Sum of Squared Residuals
= 17877847.49
Correlation
matrix of parameters:
+1.00 0.40
0.40 +1.00
TSTAT
of b in abs(e) =
a + b X*X = 2.524150387
Estimation
with OLS:
Endogenous
variable = abs(e)
Variable
Parameter
S.E. tstat
const(0),1.,0,0
+509.6383822
120.1542444 +4.24
expend(0),1.,0,0
+3.702769252e003 1.275457967e003
+2.9
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared
of stationary series = 0.1859772184
DurbinWatson = 1.408225521
Degrees of freedom
= 37
Variance
of regression = 461310.494
Standard Error of regression = 679.1984202
Sum of Squared Residuals
= 17068488.28
Correlation
matrix of parameters:
+1.00 0.43
0.43 +1.00
TSTAT
of b in abs(e) =
a + b X*X = 2.903089987
Another
popular test is the socalled likelihood ratio test for heteroskedasticity
(III.I.113)
(III.I.114)
which
can be used for testing statistical significance.
The
GoldfeldQuandt test for
heteroskedasticity uses testequations for each exogenous variable
(except the constant term). This test is widely applicable, and
fairly unproblematic w.r.t. it’s properties. The GoldfeldQuandt
test uses two regressions of the endogenous variable on each
variable separately; the first regression is based on LOW values of
the exogenous variable, the second regression is based on HIGH
values of the exogenous variable. Note that a prespecified number
of values inbetween LOW and HIGH values are NOT used in these
regressions.
Below
you ‘ll find an example of how the GoldfeldQuandt test can be
applied to test for heteroskedasticity (this test is applied to our
exampleequation):
GoldfeldQuandt
Test
The
GoldfeldQuandt test will be based on two regressions of (T  13)/2 observations.
The first regression on low values of Xi, the second on high values
of Xi.
Estimation
with OLS:
Endogenous
variable = ship.dba
Variable
Parameter
S.E. tstat
const(0),1.,0,0
+1795.714286 130.620347
+13.7
2tailt
at 95 percent = 2.16
1tailt at 95 percent = 1.771
Rsquared
of stationary series = 0.2605089242
DurbinWatson = 2.718602812
Degrees of freedom
= 13
Variance
of regression = 238863.4505
Standard Error of regression = 488.7365861
Sum of Squared Residuals
= 3105224.857
Estimation
with OLS:
Endogenous
variable = ship.dba
Variable
Parameter
S.E.
tstat
const(0),1.,0,0
+6985.071429 1153.776244
+6.05
2tailt
at 95 percent = 2.16
1tailt at 95 percent = 1.771
Rsquared
of stationary series = 1.268073674e003
DurbinWatson = 1.01313106
Degrees of freedom
= 13
Variance
of regression = 18636794.69
Standard Error of regression = 4317.035405
Sum of Squared Residuals
= 242278330.9
GoldfeldQuandt
test for exogenous variable nr. 1 = 78.02279773
DF of numerator = DF of denominator.
Approximate F critical value (95%) (df = {13,13}) = 2.4
Estimation
with OLS:
Endogenous
variable = ship.dba
Variable
Parameter
S.E. tstat
employ(0),1.,0,0
+62.53440159 2.987781852
+20.9
2tailt
at 95 percent = 2.16
1tailt at 95 percent = 1.771
Rsquared
of stationary series = 0.9195408482
DurbinWatson = 1.804461765
Degrees of freedom
= 13
Variance
of regression = 98605.87903
Standard Error of regression = 314.0157305
Sum of Squared Residuals
= 1281876.427
Estimation
with OLS:
Endogenous
variable = ship.dba
Variable
Parameter
S.E. tstat
employ(0),1.,0,0
+56.49942486 3.939457673
+14.3
2tailt
at 95 percent = 2.16
1tailt at 95 percent = 1.771
Rsquared
of stationary series = 1.003004769
DurbinWatson = 1.178141189
Degrees of freedom
= 13
Variance
of regression = 4357873.512
Standard Error of regression = 2087.552038
Sum of Squared Residuals
= 56652355.66
GoldfeldQuandt
test for exogenous variable nr. 2 = 44.194865
DF of numerator = DF of denominator.
Approximate F critical value (95%) (df = {13,13}) = 2.4
Estimation
with OLS:
Endogenous
variable = ship.dba
Variable
Parameter
S.E. tstat
expend(0),1.,0,0
+43.94765568
3.306817926
+13.3
2tailt
at 95 percent = 2.16
1tailt at 95 percent = 1.771
Rsquared
of stationary series = 0.8503625979
DurbinWatson = 2.958320856
Degrees of freedom
= 13
Variance
of regression = 254447.5573
Standard Error of regression = 504.4279505
Sum of Squared Residuals
= 3307818.245
Estimation
with OLS:
Endogenous
variable = ship.dba
Variable
Parameter
S.E.
tstat
expend(0),1.,0,0
+24.00368738
1.992895783
+12.
2tailt
at 95 percent = 2.16
1tailt at 95 percent = 1.771
Rsquared
of stationary series = 0.8703061351
DurbinWatson = 2.720031672
Degrees of freedom
= 13
Variance
of regression = 5853973.458
Standard Error of regression = 2419.498596
Sum of Squared Residuals
= 76101654.96
GoldfeldQuandt
test for exogenous variable nr. 3 = 23.00660113
DF of numerator = DF of denominator.
Approximate F critical value (95%) (df = {13,13}) = 2.4
The
Park tests for
heteroskedasticity uses a testequation for each exogenous variable:
the logarithms of squared residuals are explained by the logarithm
of the absolute values of the exogenous variable.
Warning:
this test should only be used if the endogenous variable is NOT used
as lagged exogenous variable. Furthermore the Park tests assume
MULTIPLICATIVE heteroskedasticity. All OLS assumptions should be
satisfied.
Below
you will find an example of how the Park test can be applied to test
for heteroskedasticity (this test is applied to our exampleequation):
TSTAT
values of b in Simple Regression:
Estimation
with OLS:
Endogenous
variable = ln(e*e)
Variable
Parameter
S.E. tstat
const(0),1.,0,0
+2.673616498 2.32005065
+1.15
employ(0),1.,0,0
+2.255741073 0.5790504391
+3.9
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
TSTAT of b in
ln(e*e) = a + b ln abs(X) = 3.8955865
Estimation
with OLS:
Endogenous
variable = ln(e*e)
Variable
Parameter
S.E. tstat
const(0),1.,0,0
+3.990749162
2.084773124
+1.91
expend(0),1.,0,0
+1.688936824 0.4553619992
+3.71
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
TSTAT of b in
ln(e*e) = a + b ln abs(X) = 3.708998175
The
BreuschPagan test for
heteroskedasticity uses a testequation: the squared residuals
divided by the residual variance are explained by all exogenous
variables. The test statistic is computed as half the difference
between the Total Sum of Squares and the Sum of Squared Residuals,
which has a Chisquare distribution.
Warning:
this test should only be used if the endogenous variable is NOT used
as lagged exogenous variable AND if the number of observations is
VERY LARGE. All OLS assumptions should be satisfied, including
normality of the error term.
Below
you ‘ll find an example of how the BreuschPagan test can be
applied to test for heteroskedasticity (this test is applied to our
exampleequation):
BreuschPagan
test:
Estimation
with OLS:
Endogenous
variable = ê²/(var(ê))
Variable
Parameter
S.E. tstat
const(0),1.,0,0
5.59940532e002 0.3553661
0.158 employ(0),1.,0,0
+9.716468615e004 7.418798335e003 +0.131
expend(0),1.,0,0
+6.694376613e003 3.031215889e003
+2.21
2tailt
at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared
of stationary series = 0.9998009501
DurbinWatson = 2.041789465
Degrees of freedom
= 36
Variance
of regression = 2.171889371
Standard Error of regression = 1.473733141
Sum of Squared Residuals
= 78.18801734
regression:
e_hat**2/(var(e_hat)) = a + X b + v
residual
variance = 2.171889371
(TSSSSR)/2 = 39.35189667
Chisquare (95 percent) critical value = 5.99
The
Squared Residuals versus
Squared Fit test for heteroskedasticity uses a testequation:
the squared residuals are explained by the squared interpolation
forecast of the original regression. This test is fairly
unproblematic and can be used in almost all cases. The tstatistic
of the SquaredFitparameter indicates whether heteroskedasticity is
present or not.
Below
you ‘ll find an example of how the Squared Residuals versus
Squared Fit test can be
applied to test for heteroskedasticity (this test is applied to our
exampleequation):
Squared Residuals versus
Squared Fit:
Estimation with OLS:
Endogenous variable =
SquaredResiduals
Variable
Parameter
S.E.
tstat
constant(0),1.,0,0
+592305.4721
312179.7592
+1.9
SquaredFit(1),1.,0,0 +1.432760191e002
5.425552377e003 +2.64
2tailt at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared of stationary series
= 0.1585866958 DurbinWatson
= 2.020505492
Degrees of freedom = 37
Variance
of regression = 3.002200902e+012
Standard Error of regression = 1732686.037
Sum of Squared Residuals = 1.110814334e+014
Correlation matrix of
parameters:
+1.00 0.46
0.46 +1.00
regression:
e_hat*e_hat = a + y_hat*y_hat*b + v
The
ARCH(p) test is used to
test for Autoregressive Conditional Heteroskedasticity: the squared
residuals are explained by it’s lagged values (p is the number of
lags included in the testequation). The presence of Conditional
Heteroskedasticity is tested by the use of an Fstatistic.
Below
you ‘ll find an example of how the ARCH(p)
test can be applied to test for Conditional Heteroskedasticity
(this test is applied to our exampleequation):
Arch(p) test by Least Squares:
Estimation with OLS:
Endogenous variable =
SquaredResiduals
Variable
Parameter
S.E.
tstat
constant(0),1.,0,0
+362168.8873
318289.0843
+1.14
SqResid(1),1.,0,0
+5.744398869e002 0.1786811315
+0.321
SqResid(2),1.,0,0
+0.4790710887
0.1723985422
+2.78
SqResid(3),1.,0,0
+0.2166609567
0.1908881265
+1.14
2tailt at 95 percent = 2.042
1tailt at 95 percent = 1.697
Rsquared of stationary series
= 0.3602748235 DurbinWatson
= 1.823445676
Degrees of freedom = 32
Variance
of regression = 2.594323444e+012
Standard Error of regression = 1610690.363
Sum of Squared Residuals = 8.301835022e+013
Correlation matrix of
parameters:
+1.00 0.21 0.22 0.11
0.21 +1.00 0.25 0.48
0.22 0.25 +1.00 0.24
0.11 0.48 0.24 +1.00
Fstat = 6.007159936
Critical F value (95%) = 2.84
