Heteroskedasticity:
Nature and Detection
11.1
Aims and Learning Objectives
By the end of this session students should be able to:
• Explain the nature of heteroskedasticity
• Understand the causes and consequences of
heteroskedasticity
• Perform tests to determine whether a regression
model has heteroskedastic errors
11.2
Nature of Heteroskedasticity
Heteroskedasticity is a systematic pattern in
the errors where the variances of the errors
are not constant.
Ordinary least squares assumes that all
observations are equally reliable.
11.3
Regression Model
Yi = 1 + 2Xi + Ui
Homoskedasticity:
Var(Ui) =  2
Or E(Ui2) =  2
Heteroskedasticity:
Var(Ui) = i 2
Or E(Ui2) = i 2
11.4
Homoskedastic pattern of errors
Consumption
Yi
.
.
.
..
.
.
.
.
.
.
.
.
.
. ..
.
.
.
..
.
.
.
.
.
.
.
.
.
. . . . .
..
. .
.
Income
Xi
11.5
The Homoskedastic Case
f(Yi)
.
.
X1
X2
X3
X4
.
.
Income
Xi
11.6
Heteroskedastic pattern of errors
Consumption
.
Yi
.
.
.
.
. . .
. .
. .
.
.
. .
.
.
.
.
. .. . .
.
.
.
.
.
.
.
.
.
. .
.
.
.
. . . .
.
. .
Income
Xi
11.7
The Heteroskedastic Case
f(Yi)
.
.
.
rich people
poor people
X1
X2
X3
Income
Xi
11.8
Causes of Heteroskedasticity
Common Causes
Direct
Indirect
• Scale Effects
• Omitted Variables
• Structural Shift
• Outliers
•Learning Effects
•Parameter Variation
11.9
Consequences of Heteroskedasticity
1. Ordinary least squares estimators still linear
and unbiased.
2. Ordinary least squares estimators
not efficient.
3. Usual formulas give incorrect standard
errors for least squares.
4. Confidence intervals and hypothesis tests
based on usual standard errors are wrong.
11.10
^
^
Yi = 1 + 2Xi + ei
heteroskedasticity:
Var(ei) = i 2
Formula for ordinary least squares variance
2

(homoskedastic disturbances): Var ( ˆ ) 
2
2
x
 i
Formula for ordinary least squares variance
2 2
(heteroskedastic disturbances):
xi  i

ˆ
Var (  2 ) 
2 2
 x 
i
Therefore when errors are heteroskedastic ordinary
11.11
least squares estimators are inefficient (i.e. not “best”)
Detecting Heteroskedasticity
Yi  ˆ1  ˆ2 X 2i  ˆ3 X 3i  ei
ei2 : squared residuals provide proxies for Ui2
Preliminary Analysis
• Data - Heteroskedasticity often occurs in cross
sectional data (exceptions: ARCH, panel data)
• Graphical examination of residuals - plot ei or
ei2 against each explanatory variable or against
predicted Y
11.12
Residual Plots
Plot residuals against one variable at a time
after sorting the data by that variable to try
to find a heteroskedastic pattern in the data.
.
ei
0
. . . . ..
. .. . .
.
.
.
.. .
.
.
.
.
.
.
.
.
.
Xi
.
.
.
.
11.13
Formal Tests for Heteroskedasticity
The Goldfeld-Quandt Test
1. Sort data according to the size of a potential
proportionality factor d (largest to smallest)
2. Omit the middle r observations
3. Run separate regressions on first n1 observations
and last n2 observations
4. If disturbances are homoskedastic then Var(Ui)
should be the same for both samples.
11.14
The Goldfeld-Quandt Test
5. Specify null and alternative hypothesis
Ho:
1 2 = 2 2
H1:
1 2 >  2 2
6. Test statistic
ˆ
GQ 
~ F (n1  k1 , n2  k 2 )
ˆ
2
1
2
2
Compare test statistic value with critical value from
F-distribution table
11.15
White’s Test
1. Estimate
Yˆi  ˆ1  ˆ2 X 2i
And obtain the residuals
 ˆ3 X 3i
2. Run the following auxiliary regression:
e  A0  A1 X 2i  A2 X 3i  A3 X  A4 X  A5 X 2i X 3i  Vi
2
i
2
2i
2
3i
3. Calculate White test statistic from auxiliary
regression
2
2
nR ~  d . f .
4. Obtain critical value from 2 distribution
(df = no. of explanatory variables in auxiliary regression)
5. Decision rule: if test statistic > critical 2 value
then reject null hypothesis of no heteroskedasticity
11.16
Summary
In this lecture we have:
1. Analysed the theoretical causes and
consequences of heteroskedasticity
2. Outlined a number of tests which can be used to
detect the presence of heteroskedastic errors
11.17