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starting from zero occurs at a time lag of ∆ t = 2 which corresponds exactly to the generating function
of y t as stated above. Therefore the minimum time lag for this potentially causal relation is b = 2 time
periods. This means that the variation of the independent variable X is first repeated within the time
series of the dependent variable Y after two time periods.
The first attempt to determine the appropriate length of the window of impact using the correlogram
in Figure 1 yields time lags between ∆ t = 2 and ∆ t = 4 because the last significant cross correlation occurs
with a time lag of four time periods. As this is not correct with respect to the generating function above,
one has to take the autocorrelation of the two variables into account. It can be shown that significant
autocorrelation of the independent time series leads to the so-called echo effect (Hillbrand 2004, p.
181) which describes the indirect effects of independent values prior to the window of impact through
an autocorrelated dependent time series. In the example of Figure 1 this effect becomes evident when
scrutinizing the generating function of the time series y t . This function includes the term “ … 0.2y t-1
which means that the actual value of the variable Y reflects also a fraction the preceding value y t-1 . As this
in turn is influenced by the independent values x t-3 and x t-4 it becomes evident that the latter is reflected
in y t with the weight 0.2·0.2 = 0.04 (cf. Figure 2). As the autocorrelation of the dependent variable can
be regarded as an infinite series, it can lead to putative random patterns for the echo effect.
Therefore it is crucial for a correct identification of the impact window to eliminate any autocorrelation
within the time series to be analyzed. This preprocessing of the time series is known as prewhitening in
the appropriate literature (Makridakis & Wheelwright, 1978; pp. 382f.). As a prerequisite for this task,
the order of autocorrelation for independent and dependent variables has to be determined by the use of
the autocorrelation coefficient3. 3 . Thus this approach analyzes the autocorrelation coefficients for ascend-
ing orders which exceed the critical value of the Bartlett test (for details see above) and consequently are
denoted as significant. This approach comes to reliable results for the analysis of independent time series.
However, it does not perform well for result variables as a further disturbance effect can be observed
in this context: Influences with a significant number of subsequent effects (i.e. the window of impact is
larger than one time period) tend to induce an autocorrelation-like pattern in the dependent time series
although there is no significant autocorrelation from the generating process of y t . As a consequence it
seems to be necessary to identify the appropriate window of impact before analyzing the autocorrela-
tion of the dependent time series. Therefore a circular dilemma occurs because the autocorrelation of
the dependent variable is a prerequisite for the prewhitening process which is in turn necessary to avoid
echo effects and to identify the appropriate size of the window of impact consequently.
Figure 2. Echo effect
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