Long Memory Properties and Complex Systems
的有趣见解 , 并讨论可能会对他们造成影响的长记忆性 。 关键词 : 长记忆 ; 复杂系统 ; 多主体模型 ; 蒙特卡罗分析
Introduction
Long Memory Processes , generally
speaking , are stochastic processes that exhibit nonexponential decay in their respective autocorrelation functions , as usually observed in what can be called “ short-memory processes .” Therefore , the main feature of this type of stochastic process is a hyperbolic decay in its respective autocorrelation function , which points out that perturbations occurred far away in time are still able to explain part of the current state of the system .
Consequently , this kind of stochastic process exhibits persistence that is neither consistent with the presence of a unit root nor with its complete absence . Hence , in order to have the necessary flexibility to deal with this apparent dilemma , it is introduced a fractional difference coefficient , which tries to accommodate stochastic processes between those with a unit root process ( d = 1 ) and those with no unit root ( d = 0 ).
Thus , a long memory stochastic process can be defined as :
where is a stochastic disturbance ( white noise ), L the lag operator , x t the contemporaneous observation of the stochastic process and d the fractional difference operator . Furthermore , b can be defined as :
The study of this kind of phenomenon is a relatively old field in mathematics and physics , which started to be investigated right at the beginning of the 1950s , by Hurst ( 1951 ), followed by Mandelbrot and Wallis ( 1969 ), and Mc- Leod and Hipel ( 1978 ), among others . Then , during the 1980s and 1990s the field grew almost exponentially , where the most important works were those written by Hosking ( 1981 ), Geweke and Porter-Hudak ( 1983 ), Robinson
( 1995 ), and Baillie ( 1996 ). The main focus of the pioneer works was the search for empirical evidences of long memory over a different range of problems . The focus of the posterior works was the mathematical / statistical analysis of such properties in terms of its stochastic components and the development of proper mathematical tools and statistical tests aiming at the calculation of the fractional difference coefficient .
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