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Approaches for this dilemma include the solution of separated autoregressive generating processes
for each time series as well as iterative approximation techniques. The separated generating processes
as mentioned above are characterized by partial autocorrelation coefficients. The combination of partial
autocorrelations of dependent and independent time series as well as the cross correlation between the
two of them result in the specific cross correlation pattern observable in the correlogram (cf. Figure 1).
Since the relevant literature knows solutions for this problem which are far from trivial, this chapter
employs an approach which is based on an iterative approximation of an appropriate length for the
window of impact:
1. Prewhitening of independent variables:
a. Determine orders of significant autocorrelation within the time series of every independent
variable
X
t
.
b. For every order
o
of significant autocorrelation for every independent variable
X
t
determine
the respective autoregression coefficient
β
X,o
.
c. For every value
x
i,t
of an autocorrelated independent variable
X
t
subtract past values of the
same time series weighted by the appropriate autoregression coefficient in the following form:
(
)
′
= − ⋅
∑
.
2. Determine significant cross correlations between every prewhitened independent time series
i t
x
x
x
−
i t
,
i t
,
X
i t of
,
i of
,
∀
o
x
′
and the dependent time series
y
t
for different time lags ∆
t
.
3. Identify the windows of impact - determined by the minimum time lag
b
i
and the following ef-
fects
s
i
- from the the correlograms between every prewhitened independent variable
i
X
′
and the
dependent variable
Y
.
4. Determine the regression coefficients
X
∆
between any independent variable
i
X
′
and the dependent
,
i t
element
Y
for time lags ∆
t
with a significant cross correlation.
5. Eliminate all impacts of independent variables on the dependent variable by subtracting the re-
spective past values of the independent time series
i t t
x
−
′
weighted by the appropriate regression
X
∆
according to the prewhitening procedure as described in step 1c.
6. The isolated dependent time series
coefficient
,
i t
y
resulting from the previous step is tested for significant
autocorrelation.
a. If no significant autocorrelation can be identified within the time series of
y
, the procedure
terminates. Therefore the windows of impact for each independent variable
X
i
are given as
determined in step 3.
b. If the time series
y
is significantly autocorrelated, the the raw time series
y
t
is prewhitened
according to steps 1b and 1c. The procedure resumes with a new iteration at step 2 using the
prewhitened dependent time series
y
.
Having an appropriate algorithm to test the second and third necessary condition of causality (in-
formational redundancy and temporal precedence) non-causal or spurious relations can be identified
by the absence of an empirically significant window of impact.
Controlling for Third Variable effects
The remaining potentially causal relations which comply with the first three causality criteria are subject
to a further analysis of common causation by third variables. For this purpose Pearl and Verma (1991)
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