Journal on Policy and Complex Systems
Thus , as it is widely known that Power Laws are an important characteristic of self-similar processes , and on the other hand , knowing that longrange dependency can arise from such processes , this is one more interesting finding towards the obtainment of empirical evidences of the presence of such property .
This same procedure was applied to other set of experiments made in the complementary case , as shown in Figure 9 .
In Figure 9 , which synthesizes the distribution of the prices in the second case , it can be seen that the prices are not far away from a normal distribution , despite the fact that the statistical test for normality rejects a normal distribution .
This distribution is far more symmetric than the previously presented and does not exhibit huge fat tails , not suggesting a Power Law like distribution , which is an interesting evidence in terms of the absence of long-range dependency .
In terms of the Modified R / S Statistic Test and V / S Statistic Test , with 5 % of significance , the results show that 50 % and 60 % ( respectively ) of the simulated samples have long-range dependency in the heterogeneous samples , while 95 % and 82 % ( respectively ) of the simulated samples do not have long-range dependency in the homogenous samples .
Finally , as these two statistical tests are very sensitive to the bandwidth choice of q , as discussed in the Methodology section , it was calculated the mean of autocorrelation functions of all simulations , in order to analyze the presence of slow hyperbolical decay ( long memory ) versus fast exponential decay ( short memory ).
In Figure 10 , it is possible to see that there is a slow decay in the red line and a fast decay in the blue line , which represents respectively the heterogeneous agents ’ process and the homogeneous agents ’ process , evidencing the rise of long memory , despite possible doubts according to the Modified R / S and V / S Statistic Tests . It is also important to notice that , despite the hyperbolic shape of the blue line , its values are not greater than the confidence interval ( dashed black line ) in order to be considered statistically significant , with a p-value of 1 % ( they are very close to the 5 %, casting doubts ). In other words , the apparent hyperbolic shape of the blue line is probably a mere spurious statistical effect .
Hence , what can be said in terms of agent-based models is that the introduction of heterogeneity among the agents is an important factor in terms of the existence of long memory properties in any process . When heterogeneity is considerably reduced , it seems these properties vanish , at least in this model .
Furthermore , it is worth mentioning that the implied process in the agents discussed in the present section is rather different from those discussed in the traditional literature , as in Granger ( 1980 ) and Lippi and Zaffaroni ( 1999 ).
26