<s>
In	O
statistics	O
,	O
a	O
sequence	O
(	O
or	O
a	O
vector	O
)	O
of	O
random	O
variables	O
is	O
homoscedastic	B-General_Concept
(	O
)	O
if	O
all	O
its	O
random	O
variables	O
have	O
the	O
same	O
finite	O
variance	O
;	O
this	O
is	O
also	O
known	O
as	O
homogeneity	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
.	O
</s>
<s>
The	O
complementary	O
notion	O
is	O
called	O
heteroscedasticity	B-General_Concept
,	O
also	O
known	O
as	O
heterogeneity	O
of	O
variance	O
.	O
</s>
<s>
The	O
spellings	O
homoskedasticity	B-General_Concept
and	O
heteroskedasticity	B-General_Concept
are	O
also	O
frequently	O
used	O
.	O
</s>
<s>
Assuming	O
a	O
variable	O
is	O
homoscedastic	B-General_Concept
when	O
in	O
reality	O
it	O
is	O
heteroscedastic	B-General_Concept
(	O
)	O
results	O
in	O
unbiased	O
but	O
inefficient	O
point	O
estimates	O
and	O
in	O
biased	O
estimates	O
of	O
standard	B-General_Concept
errors	I-General_Concept
,	O
and	O
may	O
result	O
in	O
overestimating	O
the	O
goodness	O
of	O
fit	O
as	O
measured	O
by	O
the	O
Pearson	O
coefficient	O
.	O
</s>
<s>
The	O
existence	O
of	O
heteroscedasticity	B-General_Concept
is	O
a	O
major	O
concern	O
in	O
regression	O
analysis	O
and	O
the	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
,	O
as	O
it	O
invalidates	O
statistical	O
tests	O
of	O
significance	B-General_Concept
that	O
assume	O
that	O
the	O
modelling	O
errors	O
all	O
have	O
the	O
same	O
variance	O
.	O
</s>
<s>
While	O
the	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
estimator	O
is	O
still	O
unbiased	O
in	O
the	O
presence	O
of	O
heteroscedasticity	B-General_Concept
,	O
it	O
is	O
inefficient	O
and	O
generalized	O
least	O
squares	O
should	O
be	O
used	O
instead	O
.	O
</s>
<s>
Because	O
heteroscedasticity	B-General_Concept
concerns	O
expectations	O
of	O
the	O
second	O
moment	O
of	O
the	O
errors	O
,	O
its	O
presence	O
is	O
referred	O
to	O
as	O
misspecification	O
of	O
the	O
second	O
order	O
.	O
</s>
<s>
The	O
econometrician	O
Robert	O
Engle	O
was	O
awarded	O
the	O
2003	O
Nobel	O
Memorial	O
Prize	O
for	O
Economics	O
for	O
his	O
studies	O
on	O
regression	O
analysis	O
in	O
the	O
presence	O
of	O
heteroscedasticity	B-General_Concept
,	O
which	O
led	O
to	O
his	O
formulation	O
of	O
the	O
autoregressive	O
conditional	O
heteroscedasticity	B-General_Concept
(	O
ARCH	O
)	O
modeling	O
technique	O
.	O
</s>
<s>
Consider	O
the	O
linear	B-General_Concept
regression	I-General_Concept
equation	O
where	O
the	O
dependent	O
random	O
variable	O
equals	O
the	O
deterministic	O
variable	O
times	O
coefficient	O
plus	O
a	O
random	O
disturbance	O
term	O
that	O
has	O
mean	O
zero	O
.	O
</s>
<s>
The	O
disturbances	O
are	O
homoscedastic	B-General_Concept
if	O
the	O
variance	O
of	O
is	O
a	O
constant	O
;	O
otherwise	O
,	O
they	O
are	O
heteroscedastic	B-General_Concept
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
disturbances	O
are	O
heteroscedastic	B-General_Concept
if	O
the	O
variance	O
of	O
depends	O
on	O
or	O
on	O
the	O
value	O
of	O
.	O
</s>
<s>
One	O
way	O
they	O
might	O
be	O
heteroscedastic	B-General_Concept
is	O
if	O
(	O
an	O
example	O
of	O
a	O
scedastic	O
function	O
)	O
,	O
so	O
the	O
variance	O
is	O
proportional	O
to	O
the	O
value	O
of	O
.	O
</s>
<s>
More	O
generally	O
,	O
if	O
the	O
variance-covariance	O
matrix	B-Architecture
of	O
disturbance	O
across	O
has	O
a	O
nonconstant	O
diagonal	O
,	O
the	O
disturbance	O
is	O
heteroscedastic	B-General_Concept
.	O
</s>
<s>
The	O
disturbance	O
in	O
matrix	B-Architecture
A	O
is	O
homoscedastic	B-General_Concept
;	O
this	O
is	O
the	O
simple	O
case	O
where	O
OLS	B-General_Concept
is	O
the	O
best	O
linear	O
unbiased	O
estimator	O
.	O
</s>
<s>
The	O
disturbances	O
in	O
matrices	O
B	O
and	O
C	O
are	O
heteroscedastic	B-General_Concept
.	O
</s>
<s>
In	O
matrix	B-Architecture
B	O
,	O
the	O
variance	O
is	O
time-varying	O
,	O
increasing	O
steadily	O
across	O
time	O
;	O
in	O
matrix	B-Architecture
C	O
,	O
the	O
variance	O
depends	O
on	O
the	O
value	O
of	O
.	O
</s>
<s>
The	O
disturbance	O
in	O
matrix	B-Architecture
D	O
is	O
homoscedastic	B-General_Concept
because	O
the	O
diagonal	O
variances	O
are	O
constant	O
,	O
even	O
though	O
the	O
off-diagonal	O
covariances	O
are	O
non-zero	O
and	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
is	O
inefficient	O
for	O
a	O
different	O
reason	O
:	O
serial	O
correlation	O
.	O
</s>
<s>
Heteroscedasticity	B-General_Concept
often	O
occurs	O
when	O
there	O
is	O
a	O
large	O
difference	O
among	O
the	O
sizes	O
of	O
the	O
observations	O
.	O
</s>
<s>
A	O
classic	O
example	O
of	O
heteroscedasticity	B-General_Concept
is	O
that	O
of	O
income	O
versus	O
expenditure	O
on	O
meals	O
.	O
</s>
<s>
The	O
data	O
you	O
collect	O
would	O
exhibit	O
heteroscedasticity	B-General_Concept
.	O
</s>
<s>
One	O
of	O
the	O
assumptions	O
of	O
the	O
classical	O
linear	B-General_Concept
regression	I-General_Concept
model	O
is	O
that	O
there	O
is	O
no	O
heteroscedasticity	B-General_Concept
.	O
</s>
<s>
Breaking	O
this	O
assumption	O
means	O
that	O
the	O
Gauss	O
–	O
Markov	O
theorem	O
does	O
not	O
apply	O
,	O
meaning	O
that	O
OLS	B-General_Concept
estimators	O
are	O
not	O
the	O
Best	O
Linear	O
Unbiased	O
Estimators	O
(	O
BLUE	O
)	O
and	O
their	O
variance	O
is	O
not	O
the	O
lowest	O
of	O
all	O
other	O
unbiased	O
estimators	O
.	O
</s>
<s>
Heteroscedasticity	B-General_Concept
does	O
not	O
cause	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
coefficient	O
estimates	O
to	O
be	O
biased	O
,	O
although	O
it	O
can	O
cause	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
estimates	O
of	O
the	O
variance	O
(	O
and	O
,	O
thus	O
,	O
standard	B-General_Concept
errors	I-General_Concept
)	O
of	O
the	O
coefficients	O
to	O
be	O
biased	O
,	O
possibly	O
above	O
or	O
below	O
the	O
true	O
of	O
population	O
variance	O
.	O
</s>
<s>
Thus	O
,	O
regression	O
analysis	O
using	O
heteroscedastic	B-General_Concept
data	O
will	O
still	O
provide	O
an	O
unbiased	O
estimate	O
for	O
the	O
relationship	O
between	O
the	O
predictor	O
variable	O
and	O
the	O
outcome	O
,	O
but	O
standard	B-General_Concept
errors	I-General_Concept
and	O
therefore	O
inferences	O
obtained	O
from	O
data	O
analysis	O
are	O
suspect	O
.	O
</s>
<s>
Biased	O
standard	B-General_Concept
errors	I-General_Concept
lead	O
to	O
biased	O
inference	O
,	O
so	O
results	O
of	O
hypothesis	O
tests	O
are	O
possibly	O
wrong	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
OLS	B-General_Concept
is	O
performed	O
on	O
a	O
heteroscedastic	B-General_Concept
data	O
set	O
,	O
yielding	O
biased	O
standard	B-General_Concept
error	I-General_Concept
estimation	O
,	O
a	O
researcher	O
might	O
fail	O
to	O
reject	O
a	O
null	O
hypothesis	O
at	O
a	O
given	O
significance	B-General_Concept
level	I-General_Concept
,	O
when	O
that	O
null	O
hypothesis	O
was	O
actually	O
uncharacteristic	O
of	O
the	O
actual	O
population	O
(	O
making	O
a	O
type	O
II	O
error	O
)	O
.	O
</s>
<s>
Under	O
certain	O
assumptions	O
,	O
the	O
OLS	B-General_Concept
estimator	O
has	O
a	O
normal	O
asymptotic	O
distribution	O
when	O
properly	O
normalized	O
and	O
centered	O
(	O
even	O
when	O
the	O
data	O
does	O
not	O
come	O
from	O
a	O
normal	O
distribution	O
)	O
.	O
</s>
<s>
This	O
result	O
is	O
used	O
to	O
justify	O
using	O
a	O
normal	O
distribution	O
,	O
or	O
a	O
chi	O
square	O
distribution	O
(	O
depending	O
on	O
how	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
calculated	O
)	O
,	O
when	O
conducting	O
a	O
hypothesis	O
test	O
.	O
</s>
<s>
This	O
holds	O
even	O
under	O
heteroscedasticity	B-General_Concept
.	O
</s>
<s>
More	O
precisely	O
,	O
the	O
OLS	B-General_Concept
estimator	O
in	O
the	O
presence	O
of	O
heteroscedasticity	B-General_Concept
is	O
asymptotically	O
normal	O
,	O
when	O
properly	O
normalized	O
and	O
centered	O
,	O
with	O
a	O
variance-covariance	O
matrix	B-Architecture
that	O
differs	O
from	O
the	O
case	O
of	O
homoscedasticity	B-General_Concept
.	O
</s>
<s>
In	O
1980	O
,	O
White	O
proposed	O
a	O
consistent	O
estimator	O
for	O
the	O
variance-covariance	O
matrix	B-Architecture
of	O
the	O
asymptotic	O
distribution	O
of	O
the	O
OLS	B-General_Concept
estimator	O
.	O
</s>
<s>
This	O
validates	O
the	O
use	O
of	O
hypothesis	O
testing	O
using	O
OLS	B-General_Concept
estimators	O
and	O
White	O
's	O
variance-covariance	O
estimator	O
under	O
heteroscedasticity	B-General_Concept
.	O
</s>
<s>
Heteroscedasticity	B-General_Concept
is	O
also	O
a	O
major	O
practical	O
issue	O
encountered	O
in	O
ANOVA	B-General_Concept
problems	O
.	O
</s>
<s>
The	O
F	B-General_Concept
test	I-General_Concept
can	O
still	O
be	O
used	O
in	O
some	O
circumstances	O
.	O
</s>
<s>
However	O
,	O
it	O
has	O
been	O
said	O
that	O
students	O
in	O
econometrics	O
should	O
not	O
overreact	O
to	O
heteroscedasticity	B-General_Concept
.	O
</s>
<s>
In	O
addition	O
,	O
another	O
word	O
of	O
caution	O
was	O
in	O
the	O
form	O
,	O
"	O
heteroscedasticity	B-General_Concept
has	O
never	O
been	O
a	O
reason	O
to	O
throw	O
out	O
an	O
otherwise	O
good	O
model.	O
"	O
</s>
<s>
With	O
the	O
advent	O
of	O
heteroscedasticity-consistent	O
standard	B-General_Concept
errors	I-General_Concept
allowing	O
for	O
inference	O
without	O
specifying	O
the	O
conditional	O
second	O
moment	O
of	O
error	O
term	O
,	O
testing	O
conditional	O
homoscedasticity	B-General_Concept
is	O
not	O
as	O
important	O
as	O
in	O
the	O
past	O
.	O
</s>
<s>
For	O
any	O
non-linear	O
model	O
(	O
for	O
instance	O
Logit	O
and	O
Probit	O
models	O
)	O
,	O
however	O
,	O
heteroscedasticity	B-General_Concept
has	O
more	O
severe	O
consequences	O
:	O
the	O
maximum	O
likelihood	O
estimates	O
(	O
MLE	O
)	O
of	O
the	O
parameters	O
will	O
be	O
biased	O
,	O
as	O
well	O
as	O
inconsistent	O
(	O
unless	O
the	O
likelihood	O
function	O
is	O
modified	O
to	O
correctly	O
take	O
into	O
account	O
the	O
precise	O
form	O
of	O
heteroscedasticity	B-General_Concept
)	O
.	O
</s>
<s>
Yet	O
,	O
in	O
the	O
context	O
of	O
binary	O
choice	O
models	O
(	O
Logit	O
or	O
Probit	O
)	O
,	O
heteroscedasticity	B-General_Concept
will	O
only	O
result	O
in	O
a	O
positive	O
scaling	O
effect	O
on	O
the	O
asymptotic	O
mean	O
of	O
the	O
misspecified	O
MLE	O
(	O
i.e.	O
</s>
<s>
the	O
model	O
that	O
ignores	O
heteroscedasticity	B-General_Concept
)	O
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
misspecified	O
Probit	O
and	O
Logit	O
MLE	O
will	O
be	O
asymptotically	O
normally	O
distributed	O
which	O
allows	O
performing	O
the	O
usual	O
significance	B-General_Concept
tests	O
(	O
with	O
the	O
appropriate	O
variance-covariance	O
matrix	B-Architecture
)	O
.	O
</s>
<s>
However	O
,	O
regarding	O
the	O
general	O
hypothesis	O
testing	O
,	O
as	O
pointed	O
out	O
by	O
Greene	O
,	O
“	O
simply	O
computing	O
a	O
robust	O
covariance	O
matrix	B-Architecture
for	O
an	O
otherwise	O
inconsistent	O
estimator	O
does	O
not	O
give	O
it	O
redemption	O
.	O
</s>
<s>
There	O
are	O
five	O
common	O
corrections	O
for	O
heteroscedasticity	B-General_Concept
.	O
</s>
<s>
Apply	O
a	O
weighted	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
estimation	O
method	O
,	O
in	O
which	O
OLS	B-General_Concept
is	O
applied	O
to	O
transformed	O
or	O
weighted	O
values	O
of	O
X	O
and	O
Y	O
.	O
</s>
<s>
Heteroscedasticity-consistent	O
standard	B-General_Concept
errors	I-General_Concept
(	O
HCSE	O
)	O
,	O
while	O
still	O
biased	O
,	O
improve	O
upon	O
OLS	B-General_Concept
estimates	O
.	O
</s>
<s>
HCSE	O
is	O
a	O
consistent	O
estimator	O
of	O
standard	B-General_Concept
errors	I-General_Concept
in	O
regression	O
models	O
with	O
heteroscedasticity	B-General_Concept
.	O
</s>
<s>
This	O
method	O
corrects	O
for	O
heteroscedasticity	B-General_Concept
without	O
altering	O
the	O
values	O
of	O
the	O
coefficients	O
.	O
</s>
<s>
This	O
method	O
may	O
be	O
superior	O
to	O
regular	O
OLS	B-General_Concept
because	O
if	O
heteroscedasticity	B-General_Concept
is	O
present	O
it	O
corrects	O
for	O
it	O
,	O
however	O
,	O
if	O
the	O
data	O
is	O
homoscedastic	B-General_Concept
,	O
the	O
standard	B-General_Concept
errors	I-General_Concept
are	O
equivalent	O
to	O
conventional	O
standard	B-General_Concept
errors	I-General_Concept
estimated	O
by	O
OLS	B-General_Concept
.	O
</s>
<s>
Several	O
modifications	O
of	O
the	O
White	O
method	O
of	O
computing	O
heteroscedasticity-consistent	O
standard	B-General_Concept
errors	I-General_Concept
have	O
been	O
proposed	O
as	O
corrections	O
with	O
superior	O
finite	O
sample	O
properties	O
.	O
</s>
<s>
Residuals	O
can	O
be	O
tested	O
for	O
homoscedasticity	B-General_Concept
using	O
the	O
Breusch	B-General_Concept
–	I-General_Concept
Pagan	I-General_Concept
test	I-General_Concept
,	O
which	O
performs	O
an	O
auxiliary	O
regression	O
of	O
the	O
squared	O
residuals	O
on	O
the	O
independent	O
variables	O
.	O
</s>
<s>
From	O
this	O
auxiliary	O
regression	O
,	O
the	O
explained	O
sum	O
of	O
squares	O
is	O
retained	O
,	O
divided	O
by	O
two	O
,	O
and	O
then	O
becomes	O
the	O
test	B-General_Concept
statistic	I-General_Concept
for	O
a	O
chi-squared	O
distribution	O
with	O
the	O
degrees	O
of	O
freedom	O
equal	O
to	O
the	O
number	O
of	O
independent	O
variables	O
.	O
</s>
<s>
The	O
null	O
hypothesis	O
of	O
this	O
chi-squared	O
test	O
is	O
homoscedasticity	B-General_Concept
,	O
and	O
the	O
alternative	O
hypothesis	O
would	O
indicate	O
heteroscedasticity	B-General_Concept
.	O
</s>
<s>
Since	O
the	O
Breusch	B-General_Concept
–	I-General_Concept
Pagan	I-General_Concept
test	I-General_Concept
is	O
sensitive	O
to	O
departures	O
from	O
normality	O
or	O
small	O
sample	O
sizes	O
,	O
the	O
Koenker	O
–	O
Bassett	O
or	O
'	O
generalized	O
Breusch–Pagan	O
 '	O
test	O
is	O
commonly	O
used	O
instead	O
.	O
</s>
<s>
From	O
the	O
auxiliary	O
regression	O
,	O
it	O
retains	O
the	O
R-squared	O
value	O
which	O
is	O
then	O
multiplied	O
by	O
the	O
sample	O
size	O
,	O
and	O
then	O
becomes	O
the	O
test	B-General_Concept
statistic	I-General_Concept
for	O
a	O
chi-squared	O
distribution	O
(	O
and	O
uses	O
the	O
same	O
degrees	O
of	O
freedom	O
)	O
.	O
</s>
<s>
Although	O
it	O
is	O
not	O
necessary	O
for	O
the	O
Koenker	O
–	O
Bassett	O
test	O
,	O
the	O
Breusch	B-General_Concept
–	I-General_Concept
Pagan	I-General_Concept
test	I-General_Concept
requires	O
that	O
the	O
squared	O
residuals	O
also	O
be	O
divided	O
by	O
the	O
residual	O
sum	O
of	O
squares	O
divided	O
by	O
the	O
sample	O
size	O
.	O
</s>
<s>
Testing	O
for	O
groupwise	O
heteroscedasticity	B-General_Concept
can	O
be	O
done	O
with	O
the	O
Goldfeld	B-General_Concept
–	I-General_Concept
Quandt	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Although	O
tests	O
for	O
heteroscedasticity	B-General_Concept
between	O
groups	O
can	O
formally	O
be	O
considered	O
as	O
a	O
special	O
case	O
of	O
testing	O
within	O
regression	O
models	O
,	O
some	O
tests	O
have	O
structures	O
specific	O
to	O
this	O
case	O
.	O
</s>
<s>
Two	O
or	O
more	O
normal	O
distributions	O
,	O
are	O
both	O
homoscedastic	B-General_Concept
and	O
lack	O
Serial	O
correlation	O
if	O
they	O
share	O
the	O
same	O
diagonals	O
in	O
their	O
covariance	O
matrix	B-Architecture
,	O
and	O
their	O
non-diagonal	O
entries	O
are	O
zero	O
.	O
</s>
<s>
Homoscedastic	B-General_Concept
distributions	O
are	O
especially	O
useful	O
to	O
derive	O
statistical	O
pattern	O
recognition	O
and	O
machine	O
learning	O
algorithms	O
.	O
</s>
<s>
One	O
popular	O
example	O
of	O
an	O
algorithm	O
that	O
assumes	O
homoscedasticity	B-General_Concept
is	O
Fisher	B-General_Concept
's	I-General_Concept
linear	I-General_Concept
discriminant	I-General_Concept
analysis	O
.	O
</s>
<s>
The	O
concept	O
of	O
homoscedasticity	B-General_Concept
can	O
be	O
applied	O
to	O
distributions	O
on	O
spheres	O
.	O
</s>
<s>
The	O
study	O
of	O
homescedasticity	O
and	O
heteroscedasticity	B-General_Concept
has	O
been	O
generalized	O
to	O
the	O
multivariate	O
case	O
,	O
which	O
deals	O
with	O
the	O
covariances	O
of	O
vector	O
observations	O
instead	O
of	O
the	O
variance	O
of	O
scalar	O
observations	O
.	O
</s>
<s>
Bartlett	B-General_Concept
's	I-General_Concept
test	I-General_Concept
for	O
heteroscedasticity	B-General_Concept
between	O
grouped	O
data	O
,	O
used	O
most	O
commonly	O
in	O
the	O
univariate	O
case	O
,	O
has	O
also	O
been	O
extended	O
for	O
the	O
multivariate	O
case	O
,	O
but	O
a	O
tractable	O
solution	O
only	O
exists	O
for	O
2	O
groups	O
.	O
</s>
<s>
Approximations	O
exist	O
for	O
more	O
than	O
two	O
groups	O
,	O
and	O
they	O
are	O
both	O
called	O
Box	B-General_Concept
's	I-General_Concept
M	I-General_Concept
test	I-General_Concept
.	O
</s>
