<s>
Clustered	B-General_Concept
standard	I-General_Concept
errors	I-General_Concept
(	O
or	O
Liang-Zeger	O
standard	B-General_Concept
errors	I-General_Concept
)	O
are	O
measurements	O
that	O
estimate	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
a	O
regression	O
parameter	O
in	O
settings	O
where	O
observations	O
may	O
be	O
subdivided	O
into	O
smaller-sized	O
groups	O
(	O
"	O
clusters	O
"	O
)	O
and	O
where	O
the	O
sampling	O
and/or	O
treatment	O
assignment	O
is	O
correlated	O
within	O
each	O
group	O
.	O
</s>
<s>
Clustered	B-General_Concept
standard	I-General_Concept
errors	I-General_Concept
are	O
widely	O
used	O
in	O
a	O
variety	O
of	O
applied	O
econometric	O
settings	O
,	O
including	O
difference-in-differences	O
or	O
experiments	O
.	O
</s>
<s>
Analogous	O
to	O
how	O
Huber-White	O
standard	B-General_Concept
errors	I-General_Concept
are	O
consistent	O
in	O
the	O
presence	O
of	O
heteroscedasticity	B-General_Concept
and	O
Newey	O
–	O
West	O
standard	B-General_Concept
errors	I-General_Concept
are	O
consistent	O
in	O
the	O
presence	O
of	O
accurately-modeled	O
autocorrelation	O
,	O
clustered	B-General_Concept
standard	I-General_Concept
errors	I-General_Concept
are	O
consistent	O
in	O
the	O
presence	O
of	O
cluster-based	O
sampling	O
or	O
treatment	O
assignment	O
.	O
</s>
<s>
Clustered	B-General_Concept
standard	I-General_Concept
errors	I-General_Concept
are	O
often	O
justified	O
by	O
possible	O
correlation	O
in	O
modeling	O
residuals	O
within	O
each	O
cluster	O
;	O
while	O
recent	O
work	O
suggests	O
that	O
this	O
is	O
not	O
the	O
precise	O
justification	O
behind	O
clustering	O
,	O
it	O
may	O
be	O
pedagogically	O
useful	O
.	O
</s>
<s>
Clustered	B-General_Concept
standard	I-General_Concept
errors	I-General_Concept
are	O
often	O
useful	O
when	O
treatment	O
is	O
assigned	O
at	O
the	O
level	O
of	O
a	O
cluster	O
instead	O
of	O
at	O
the	O
individual	O
level	O
.	O
</s>
<s>
However	O
,	O
when	O
estimating	O
the	O
standard	B-General_Concept
error	I-General_Concept
or	O
confidence	O
interval	O
of	O
her	O
statistical	O
model	O
,	O
she	O
realizes	O
that	O
classical	O
or	O
even	O
heteroscedasticity-robust	O
standard	B-General_Concept
errors	I-General_Concept
are	O
inappropriate	O
because	O
student	O
test	O
scores	O
within	O
each	O
class	O
are	O
not	O
independently	O
distributed	O
.	O
</s>
<s>
The	O
researcher	O
can	O
cluster	O
her	O
standard	B-General_Concept
errors	I-General_Concept
at	O
the	O
level	O
of	O
a	O
classroom	O
to	O
account	O
for	O
this	O
aspect	O
of	O
her	O
experiment	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
clustered	B-General_Concept
standard	I-General_Concept
errors	I-General_Concept
account	O
for	O
the	O
uncertainty	O
driven	O
by	O
the	O
fact	O
that	O
the	O
researcher	O
does	O
not	O
observe	O
large	O
parts	O
of	O
the	O
population	O
of	O
interest	O
.	O
</s>
<s>
A	O
useful	O
mathematical	O
illustration	O
comes	O
from	O
the	O
case	O
of	O
one-way	O
clustering	O
in	O
an	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
(	O
OLS	O
)	O
model	O
.	O
</s>
<s>
Given	O
the	O
assumptions	O
that	O
a	O
practitioner	O
deems	O
as	O
reasonable	O
,	O
different	O
types	O
of	O
standard	B-General_Concept
errors	I-General_Concept
solve	O
this	O
problem	O
in	O
different	O
ways	O
.	O
</s>
<s>
For	O
example	O
,	O
classic	O
homoskedastic	B-General_Concept
standard	B-General_Concept
errors	I-General_Concept
assume	O
that	O
is	O
diagonal	O
with	O
identical	O
elements	O
,	O
which	O
simplifies	O
the	O
expression	O
for	O
.	O
</s>
<s>
Huber-White	O
standard	B-General_Concept
errors	I-General_Concept
assume	O
is	O
diagonal	O
but	O
that	O
the	O
diagonal	O
value	O
varies	O
,	O
while	O
other	O
types	O
of	O
standard	B-General_Concept
errors	I-General_Concept
(	O
e.g.	O
</s>
<s>
Clustered	B-General_Concept
standard	I-General_Concept
errors	I-General_Concept
assume	O
that	O
is	O
block-diagonal	O
according	O
to	O
the	O
clusters	O
in	O
the	O
sample	O
,	O
with	O
unrestricted	O
values	O
in	O
each	O
block	O
but	O
zeros	O
elsewhere	O
.	O
</s>
<s>
While	O
no	O
specific	O
number	O
of	O
clusters	O
is	O
statistically	O
proven	O
to	O
be	O
sufficient	O
,	O
practitioners	O
often	O
cite	O
a	O
number	O
in	O
the	O
range	O
of	O
30-50	O
and	O
are	O
comfortable	O
using	O
clustered	B-General_Concept
standard	I-General_Concept
errors	I-General_Concept
when	O
the	O
number	O
of	O
clusters	O
exceeds	O
that	O
threshold	O
.	O
</s>
