Journal on Policy and Complex Systems
On the other hand , none of these models was developed for this work . Their respective authors separately made them for specific purposes and that is why the present authors have decided for such strategy ( of picking models made by third parties ). Instead of building models ( which usually takes a considerable amount of time to make them work properly ) that might contain biases in terms of finding such long memory properties — as a consequence of the present paper idea — they were chosen , simulated ( in their respective platforms ) and analyzed using the R Statistical Package .
In the following section , it is presented a brief discussion about the capability of the current fractional difference coefficient estimators to distinguish apparent long memory processes from simple autoregressive processes — because autoregressive processes can be seen as a truncated version of long memory processes , and three other long-range dependency methods are presented . Then , in the other three next sections , each one of these three models are presented , simulated and discussed in terms of the stochastic properties found over the results obtained , while pointing out possible reasons for such results as a consequence of agents ’ heterogeneity , local interactions and spatial complexity . After that , there is presented a final section containing a brief conclusion of evidences towards the emergence of long-range dependency because of other kind of interactions beyond explicit long memory behavior of individuals .
Methodology
In order to test the possibility of having a long-range dependency over time , along this paper the usage of four statistical methods is proposed : Geweke and Porter-Hudak ( 1983 )— from now on GPH , Local Whittle Estimator as in Robinson ( 1995 ), where these first two are fractional difference coefficient estimators — Modified R / S Statistic Test as in Lo ( 1991 ) and V / S Statistic Test following Giraitis , Kokoszka , Leipus , and Teyssière ( 2003 ). Moreover , in addition to these four statistical methods , a direct graphical comparison will be carried out between the partial and standard autocorrelation functions in order to detect the presence of long memory components .
As can be seen in Kumar ( 2014 ), the GPH estimator is based on the slope of the spectral density function of the fractionally integrated time series around through a simple linear regression based on the periodogram . The periodogram is defined as :
where is the kth periodogram point and it can be defined as the squared absolute values of the Fourier Transform of the series .
Having calculated the periodogram , the final step is to estimate the fractional difference coefficient by estimating the following regression using Ordinary Least Squares :
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