<s>
The	O
Ljung	B-General_Concept
–	I-General_Concept
Box	I-General_Concept
test	I-General_Concept
(	O
named	O
for	O
Greta	O
M	O
.	O
Ljung	O
and	O
George	O
E	O
.	O
P	O
.	O
Box	O
)	O
is	O
a	O
type	O
of	O
statistical	O
test	O
of	O
whether	O
any	O
of	O
a	O
group	O
of	O
autocorrelations	O
of	O
a	O
time	O
series	O
are	O
different	O
from	O
zero	O
.	O
</s>
<s>
This	O
test	O
is	O
sometimes	O
known	O
as	O
the	O
Ljung	O
–	O
Box	O
Q	O
test	O
,	O
and	O
it	O
is	O
closely	O
connected	O
to	O
the	O
Box	B-General_Concept
–	I-General_Concept
Pierce	I-General_Concept
test	I-General_Concept
(	O
which	O
is	O
named	O
after	O
George	O
E	O
.	O
P	O
.	O
Box	O
and	O
David	O
A	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
Ljung	B-General_Concept
–	I-General_Concept
Box	I-General_Concept
test	I-General_Concept
statistic	O
was	O
described	O
explicitly	O
in	O
the	O
paper	O
that	O
led	O
to	O
the	O
use	O
of	O
the	O
Box	O
–	O
Pierce	O
statistic	O
,	O
and	O
from	O
which	O
that	O
statistic	O
takes	O
its	O
name	O
.	O
</s>
<s>
The	O
Box	B-General_Concept
–	I-General_Concept
Pierce	I-General_Concept
test	I-General_Concept
statistic	O
is	O
a	O
simplified	O
version	O
of	O
the	O
Ljung	O
–	O
Box	O
statistic	O
for	O
which	O
subsequent	O
simulation	O
studies	O
have	O
shown	O
poor	O
performance	O
.	O
</s>
<s>
The	O
Ljung	B-General_Concept
–	I-General_Concept
Box	I-General_Concept
test	I-General_Concept
is	O
widely	O
applied	O
in	O
econometrics	O
and	O
other	O
applications	O
of	O
time	O
series	O
analysis	O
.	O
</s>
<s>
A	O
similar	O
assessment	O
can	O
be	O
also	O
carried	O
out	O
with	O
the	O
Breusch	O
–	O
Godfrey	O
test	O
and	O
the	O
Durbin	B-General_Concept
–	I-General_Concept
Watson	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
The	O
Ljung	B-General_Concept
–	I-General_Concept
Box	I-General_Concept
test	I-General_Concept
may	O
be	O
defined	O
as	O
:	O
</s>
<s>
For	O
significance	B-General_Concept
level	I-General_Concept
α	O
,	O
the	O
critical	O
region	O
for	O
rejection	O
of	O
the	O
hypothesis	O
of	O
randomness	O
is	O
:	O
</s>
<s>
The	O
Ljung	B-General_Concept
–	I-General_Concept
Box	I-General_Concept
test	I-General_Concept
is	O
commonly	O
used	O
in	O
autoregressive	O
integrated	O
moving	O
average	O
(	O
ARIMA	O
)	O
modeling	O
.	O
</s>
