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Identification of Temporally Lagged Informational r edundancy
The second necessary condition for causality—as defined in the previous section—is informational
redundancy between a potential cause and its potential effect. As a consequence the time series of the
dependent variable as the output of an unknown causal function associated with the respective rela-
tion must also reflect patterns or variations of independent time series. Very often these patterns are
transformed by the underlying causal function and superposed by the influences of other exogenous or
endogenous variables. Therefore they cannot always be detected a prima facie. If the causal function
which transforms the values of a set of independent variables into the value of a dependent value is of
linear type, the concept of correlation can be used as a measure for informational redundancy. In all
other cases one has to find other techniques to analyze whether two time-series are informationally
redundant or not. For the purpose of this approach it seems to be suitable to restrict to the linear case
because the causality proof per se does not build the model base but is used to select variables for the
following approximation of a nonlinear causal function. Therefore it is not necessary to rule out all
possibilities of α-errors because the ANNs—assigned to perform the causal approximation task—are
supposed to detect the non existing influence of marginally spurious associations. The admissibility of
this theory for different types of causal functions has been shown by Hillbrand (2003, pp. 299ff.).
When considering the third necessary condition for causality of temporal precedence of cause and
effect the inadequacy of the concept of correlation alone to prove cause-and-effect relations becomes
obvious: The correlation of two time series would show that the variations of a independent and a de-
pendent variable are similar and that they take place contemporaneously.
As this is mutually contradictory to the notion of causality as defined in the previous section, the
concept of correlation has to be adopted to measure temporally lagged responses of the variation of an
independent factor within the time series of the dependent variable. Therefore cross correlation
( )
∆
X Y
implies a time lag ∆
t
between a cause
X
and an effect
Y
in the following form:
T
(
)(
)
∑
y Y x
−
−
X
t
t t
−∆
t
=
1
T
( )
∆ =
t
X Y
,
⋅
X
Y
Where
y
t
and
x
t
stand for the values of the variables
Y
and
X
at time
t
,
X
and
Y
stand for the average
values and
σ
x
as well as
σ
y
for the standard deviation of the respective time series. As it is evident from
the above expression, the upper part of the fraction is derived from the construct of covariance.
As a consequence, this approach employs this concept to identify the two causality conditions of
informational redundancy and temporal precedence: By calculating the cross correlations for varying
time lags it is possible to identify a window of impact between an independent and a dependent variable.
This window of impact is characterized by a minimum time lag and a number of subsequent effects
(i.e. the length of the window of impact). For this purpose it is necessary to identify the significance of
a cross correlation at a given time lag. Therefore this approach uses Bartlett's significance test (Bartlett,
1955) following the suggestions of the appropriate literature in this area (Makridakis & Wheelwright,
1978; Levich & Rizzo, 1997): The null hypothesis of this test is formed by the assumption that two
given time series at a certain time lag are independent if their cross correlation shows a value of zero.
This hypothesis has to be accepted if the cross correlation lies within the boundaries which are given
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