Journal on Policy and Complex Systems
For the purpose of this paper , this model was implemented using the software Insight Maker and 100 simulations were carried out , where each one generated price time series that encompassed 2,000 ticks . An example of a resulting series is shown in Figure 2 .
Figure 2 . Simulated prices over time .
Thus , it was calculated the Künsch ( 1987 ) and GPH ( Geweke & Porter-Hudak , 1983 ) estimates of the fractional difference coefficients over these price series , in order to test the presence of long memory components .
The average Künsch ( 1987 ) estimate for the fractional difference coefficient for the 100 simulations was 0.4869814 , while the average GPH ( 1983 ) estimate was 0.1457286 . If taken into account the fact that the past state is Boolean and the autoregressive part of the function is still weak ( less than 0.9 ), both results provide strong evidences towards the presence of long memory components in this kind of process . The distribution of the fractional difference estimates ( GPH ) is described in Figure 3 .
According to Figure 3 , it is clear that this process exhibits long memory properties , avoiding any spurious result from a single simulation , as it relies on a Bernoulli random variable to generate part of the stochastic fluctuations .
In Figure 4 it is shown the distribution of the fractional difference estimates , according to Künsch ( 1987 ).
According to Figure 4 , it is important to notice that the shape of the distribution is completely different from the GPH ( 1983 ) estimates . Still , according to these results , it suggests the presence of long-range dependency .
Nonetheless , in order to reduce the heterogeneity between the simulated individuals , this experiment was repeated using z ~ N ( 0.5 , 0.05 ) , fixed at the first simulation step .
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