<s>
The	O
Granger	B-General_Concept
causality	I-General_Concept
test	O
is	O
a	O
statistical	O
hypothesis	O
test	O
for	O
determining	O
whether	O
one	O
time	O
series	O
is	O
useful	O
in	O
forecasting	O
another	O
,	O
first	O
proposed	O
in	O
1969	O
.	O
</s>
<s>
Ordinarily	O
,	O
regressions	O
reflect	O
"	O
mere	O
"	O
correlations	O
,	O
but	O
Clive	O
Granger	O
argued	O
that	O
causality	B-Application
in	O
economics	O
could	O
be	O
tested	O
for	O
by	O
measuring	O
the	O
ability	O
to	O
predict	O
the	O
future	O
values	O
of	O
a	O
time	O
series	O
using	O
prior	O
values	O
of	O
another	O
time	O
series	O
.	O
</s>
<s>
Since	O
the	O
question	O
of	O
"	O
true	O
causality	B-Application
"	O
is	O
deeply	O
philosophical	O
,	O
and	O
because	O
of	O
the	O
post	O
hoc	O
ergo	O
propter	O
hoc	O
fallacy	O
of	O
assuming	O
that	O
one	O
thing	O
preceding	O
another	O
can	O
be	O
used	O
as	O
a	O
proof	O
of	O
causation	O
,	O
econometricians	O
assert	O
that	O
the	O
Granger	B-General_Concept
test	I-General_Concept
finds	O
only	O
"	O
predictive	O
causality	B-Application
"	O
.	O
</s>
<s>
Using	O
the	O
term	O
"	O
causality	B-Application
"	O
alone	O
is	O
a	O
misnomer	O
,	O
as	O
Granger-causality	O
is	O
better	O
described	O
as	O
"	O
precedence	O
"	O
,	O
or	O
,	O
as	O
Granger	O
himself	O
later	O
claimed	O
in	O
1977	O
,	O
"	O
temporally	O
related	O
"	O
.	O
</s>
<s>
Rather	O
than	O
testing	O
whether	O
X	O
causes	O
Y	O
,	O
the	O
Granger	B-General_Concept
causality	I-General_Concept
tests	I-General_Concept
whether	O
X	O
forecasts	O
Y	O
.	O
</s>
<s>
A	O
time	O
series	O
X	O
is	O
said	O
to	O
Granger-cause	O
Y	O
if	O
it	O
can	O
be	O
shown	O
,	O
usually	O
through	O
a	O
series	O
of	O
t-tests	B-General_Concept
and	O
F-tests	B-General_Concept
on	O
lagged	O
values	O
of	O
X	O
(	O
and	O
with	O
lagged	O
values	O
of	O
Y	O
also	O
included	O
)	O
,	O
that	O
those	O
X	O
values	O
provide	O
statistically	B-General_Concept
significant	I-General_Concept
information	O
about	O
future	O
values	O
ofY	O
.	O
</s>
<s>
Granger	O
also	O
stressed	O
that	O
some	O
studies	O
using	O
"	O
Granger	B-General_Concept
causality	I-General_Concept
"	O
testing	O
in	O
areas	O
outside	O
economics	O
reached	O
"	O
ridiculous	O
"	O
conclusions	O
.	O
</s>
<s>
However	O
,	O
it	O
remains	O
a	O
popular	O
method	O
for	O
causality	B-Application
analysis	O
in	O
time	O
series	O
due	O
to	O
its	O
computational	O
simplicity	O
.	O
</s>
<s>
The	O
original	O
definition	O
of	O
Granger	B-General_Concept
causality	I-General_Concept
does	O
not	O
account	O
for	O
latent	O
confounding	O
effects	O
and	O
does	O
not	O
capture	O
instantaneous	O
and	O
non-linear	O
causal	O
relationships	O
,	O
though	O
several	O
extensions	O
have	O
been	O
proposed	O
to	O
address	O
these	O
issues	O
.	O
</s>
<s>
Granger	O
defined	O
the	O
causality	B-Application
relationship	O
based	O
on	O
two	O
principles	O
:	O
</s>
<s>
Given	O
these	O
two	O
assumptions	O
about	O
causality	B-Application
,	O
Granger	O
proposed	O
to	O
test	O
the	O
following	O
hypothesis	O
for	O
identification	O
of	O
a	O
causal	O
effect	O
of	O
on	O
:	O
</s>
<s>
If	O
a	O
time	O
series	O
is	O
a	O
stationary	B-Algorithm
process	I-Algorithm
,	O
the	O
test	O
is	O
performed	O
using	O
the	O
level	O
values	O
of	O
two	O
(	O
or	O
more	O
)	O
variables	O
.	O
</s>
<s>
If	O
the	O
variables	O
are	O
non-stationary	B-Algorithm
,	O
then	O
the	O
test	O
is	O
done	O
using	O
first	O
(	O
or	O
higher	O
)	O
differences	O
.	O
</s>
<s>
The	O
number	O
of	O
lags	O
to	O
be	O
included	O
is	O
usually	O
chosen	O
using	O
an	O
information	O
criterion	O
,	O
such	O
as	O
the	O
Akaike	O
information	O
criterion	O
or	O
the	O
Schwarz	B-General_Concept
information	I-General_Concept
criterion	I-General_Concept
.	O
</s>
<s>
Any	O
particular	O
lagged	O
value	O
of	O
one	O
of	O
the	O
variables	O
is	O
retained	O
in	O
the	O
regression	O
if	O
(	O
1	O
)	O
it	O
is	O
significant	O
according	O
to	O
a	O
t-test	B-General_Concept
,	O
and	O
(	O
2	O
)	O
it	O
and	O
the	O
other	O
lagged	O
values	O
of	O
the	O
variable	O
jointly	O
add	O
explanatory	O
power	O
to	O
the	O
model	O
according	O
to	O
an	O
F-test	B-General_Concept
.	O
</s>
<s>
Then	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
of	O
no	O
Granger	B-General_Concept
causality	I-General_Concept
is	O
not	O
rejected	O
if	O
and	O
only	O
if	O
no	O
lagged	O
values	O
of	O
an	O
explanatory	O
variable	O
have	O
been	O
retained	O
in	O
the	O
regression	O
.	O
</s>
<s>
To	O
test	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
x	O
does	O
not	O
Granger-cause	O
y	O
,	O
one	O
first	O
finds	O
the	O
proper	O
lagged	O
values	O
of	O
y	O
to	O
include	O
in	O
an	O
univariate	O
autoregression	B-Algorithm
of	O
y	O
:	O
</s>
<s>
Next	O
,	O
the	O
autoregression	B-Algorithm
is	O
augmented	O
by	O
including	O
lagged	O
values	O
of	O
x	O
:	O
</s>
<s>
One	O
retains	O
in	O
this	O
regression	O
all	O
lagged	O
values	O
of	O
x	O
that	O
are	O
individually	O
significant	O
according	O
to	O
their	O
t-statistics	O
,	O
provided	O
that	O
collectively	O
they	O
add	O
explanatory	O
power	O
to	O
the	O
regression	O
according	O
to	O
an	O
F-test	B-General_Concept
(	O
whose	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
no	O
explanatory	O
power	O
jointly	O
added	O
by	O
the	O
x	O
's	O
)	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
x	O
does	O
not	O
Granger-cause	O
y	O
is	O
accepted	O
if	O
and	O
only	O
if	O
no	O
lagged	O
values	O
of	O
x	O
are	O
retained	O
in	O
the	O
regression	O
.	O
</s>
<s>
Multivariate	O
Granger	B-General_Concept
causality	I-General_Concept
analysis	O
is	O
usually	O
performed	O
by	O
fitting	O
a	O
vector	O
autoregressive	B-Algorithm
model	I-Algorithm
(	O
VAR	O
)	O
to	O
the	O
time	O
series	O
.	O
</s>
<s>
Granger	B-General_Concept
causality	I-General_Concept
is	O
performed	O
by	O
fitting	O
a	O
VAR	O
model	O
with	O
time	O
lags	O
as	O
follows	O
:	O
</s>
<s>
A	O
time	O
series	O
is	O
called	O
a	O
Granger	B-General_Concept
cause	I-General_Concept
of	O
another	O
time	O
series	O
,	O
if	O
at	O
least	O
one	O
of	O
the	O
elements	O
for	O
is	O
significantly	O
larger	O
than	O
zero	O
(	O
in	O
absolute	O
value	O
)	O
.	O
</s>
<s>
The	O
above	O
linear	O
methods	O
are	O
appropriate	O
for	O
testing	O
Granger	B-General_Concept
causality	I-General_Concept
in	O
the	O
mean	O
.	O
</s>
<s>
However	O
they	O
are	O
not	O
able	O
to	O
detect	O
Granger	B-General_Concept
causality	I-General_Concept
in	O
higher	O
moments	O
,	O
e.g.	O
,	O
in	O
the	O
variance	O
.	O
</s>
<s>
Non-parametric	O
tests	O
for	O
Granger	B-General_Concept
causality	I-General_Concept
are	O
designed	O
to	O
address	O
this	O
problem	O
.	O
</s>
<s>
The	O
definition	O
of	O
Granger	B-General_Concept
causality	I-General_Concept
in	O
these	O
tests	O
is	O
general	O
and	O
does	O
not	O
involve	O
any	O
modelling	O
assumptions	O
,	O
such	O
as	O
a	O
linear	O
autoregressive	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
The	O
non-parametric	O
tests	O
for	O
Granger	B-General_Concept
causality	I-General_Concept
can	O
be	O
used	O
as	O
diagnostic	O
tools	O
to	O
build	O
better	O
parametric	B-General_Concept
models	I-General_Concept
including	O
higher	O
order	O
moments	O
and/or	O
non-linearity	O
.	O
</s>
<s>
As	O
its	O
name	O
implies	O
,	O
Granger	B-General_Concept
causality	I-General_Concept
is	O
not	O
necessarily	O
true	O
causality	B-Application
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
Granger-causality	O
tests	O
fulfill	O
only	O
the	O
Humean	O
definition	O
of	O
causality	B-Application
that	O
identifies	O
the	O
cause-effect	O
relations	O
with	O
constant	O
conjunctions	O
.	O
</s>
<s>
If	O
both	O
X	O
and	O
Y	O
are	O
driven	O
by	O
a	O
common	O
third	O
process	O
with	O
different	O
lags	O
,	O
one	O
might	O
still	O
fail	O
to	O
reject	O
the	O
alternative	B-General_Concept
hypothesis	I-General_Concept
of	O
Granger	B-General_Concept
causality	I-General_Concept
.	O
</s>
<s>
Indeed	O
,	O
the	O
Granger-causality	O
tests	O
are	O
designed	O
to	O
handle	O
pairs	O
of	O
variables	O
,	O
and	O
may	O
produce	O
misleading	O
results	O
when	O
the	O
true	O
relationship	O
involves	O
three	O
or	O
more	O
variables	O
.	O
</s>
<s>
Having	O
said	O
this	O
,	O
it	O
has	O
been	O
argued	O
that	O
given	O
a	O
probabilistic	O
view	O
of	O
causation	O
,	O
Granger	B-General_Concept
causality	I-General_Concept
can	O
be	O
considered	O
true	O
causality	B-Application
in	O
that	O
sense	O
,	O
especially	O
when	O
Reichenbach	O
's	O
"	O
screening	O
off	O
"	O
notion	O
of	O
probabilistic	O
causation	O
is	O
taken	O
into	O
account	O
.	O
</s>
<s>
A	O
similar	O
test	O
involving	O
more	O
variables	O
can	O
be	O
applied	O
with	O
vector	O
autoregression	B-Algorithm
.	O
</s>
<s>
A	O
method	O
for	O
Granger	B-General_Concept
causality	I-General_Concept
has	O
been	O
developed	O
that	O
is	O
not	O
sensitive	O
to	O
deviations	O
from	O
the	O
assumption	O
that	O
the	O
error	O
term	O
is	O
normally	O
distributed	O
.	O
</s>
<s>
Recently	O
,	O
asymmetric	O
causality	B-Application
testing	O
has	O
been	O
suggested	O
in	O
the	O
literature	O
in	O
order	O
to	O
separate	O
the	O
causal	O
impact	O
of	O
positive	O
changes	O
from	O
the	O
negative	O
ones	O
.	O
</s>
<s>
An	O
extension	O
of	O
Granger	O
(	O
non	O
-	O
)	O
causality	B-Application
testing	O
to	O
panel	O
data	O
is	O
also	O
available	O
.	O
</s>
<s>
A	O
modified	O
Granger	B-General_Concept
causality	I-General_Concept
test	O
based	O
on	O
the	O
GARCH	O
(	O
generalized	O
auto-regressive	O
conditional	O
heteroscedasticity	O
)	O
type	O
of	O
integer-valued	O
time	O
series	O
models	O
is	O
available	O
in	O
many	O
areas	O
.	O
</s>
<s>
Recently	O
Granger	B-General_Concept
causality	I-General_Concept
has	O
been	O
applied	O
to	O
address	O
some	O
of	O
these	O
issues	O
with	O
great	O
success	O
.	O
</s>
<s>
Previous	O
Granger-causality	O
methods	O
could	O
only	O
operate	O
on	O
continuous-valued	O
data	O
so	O
the	O
analysis	O
of	O
neural	O
spike	B-Algorithm
train	I-Algorithm
recordings	O
involved	O
transformations	O
that	O
ultimately	O
altered	O
the	O
stochastic	O
properties	O
of	O
the	O
data	O
,	O
indirectly	O
altering	O
the	O
validity	O
of	O
the	O
conclusions	O
that	O
could	O
be	O
drawn	O
from	O
it	O
.	O
</s>
<s>
In	O
2011	O
,	O
however	O
,	O
a	O
new	O
general-purpose	O
Granger-causality	O
framework	O
was	O
proposed	O
that	O
could	O
directly	O
operate	O
on	O
any	O
modality	O
,	O
including	O
neural-spike	O
trains	O
.	O
</s>
<s>
Neural	O
spike	B-Algorithm
train	I-Algorithm
data	O
can	O
be	O
modeled	O
as	O
a	O
point-process	O
.	O
</s>
<s>
This	O
type	O
of	O
binary-valued	O
representation	O
of	O
information	O
suits	O
the	O
activity	O
of	O
neural	O
populations	O
because	O
a	O
single	O
neuron	O
's	O
action	B-Algorithm
potential	I-Algorithm
has	O
a	O
typical	O
waveform	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
what	O
carries	O
the	O
actual	O
information	O
being	O
output	O
from	O
a	O
neuron	O
is	O
the	O
occurrence	O
of	O
a	O
“	O
spike	B-Algorithm
”	O
,	O
as	O
well	O
as	O
the	O
time	O
between	O
successive	O
spikes	B-Algorithm
.	O
</s>
<s>
A	O
point-process	O
can	O
be	O
represented	O
either	O
by	O
the	O
timing	O
of	O
the	O
spikes	B-Algorithm
themselves	O
,	O
the	O
waiting	O
times	O
between	O
spikes	B-Algorithm
,	O
using	O
a	O
counting	O
process	O
,	O
or	O
,	O
if	O
time	O
is	O
discretized	O
enough	O
to	O
ensure	O
that	O
in	O
each	O
window	O
only	O
one	O
event	O
has	O
the	O
possibility	O
of	O
occurring	O
,	O
that	O
is	O
to	O
say	O
one	O
time	O
bin	O
can	O
only	O
contain	O
one	O
event	O
,	O
as	O
a	O
set	O
of	O
1s	O
and	O
0s	O
,	O
very	O
similar	O
to	O
binary	O
.	O
</s>
<s>
So	O
if	O
this	O
unit	O
time	O
is	O
taken	O
small	O
enough	O
to	O
ensure	O
that	O
only	O
one	O
spike	B-Algorithm
could	O
occur	O
in	O
that	O
time	O
window	O
,	O
then	O
our	O
conditional	O
intensity	O
function	O
completely	O
specifies	O
the	O
probability	O
that	O
a	O
given	O
neuron	O
will	O
fire	O
in	O
a	O
certain	O
time	O
.	O
</s>
