Long Memory Properties and Complex Systems
1991 ) and the V / S Test work in order to detect the presence of long memory components . Aiming to explain how the Modified R / S Statistic Test works , it is useful to discuss the original R / S Statistic developed by Hurst ( 1951 ). If one defines the following quantities :
Then , the original R / S Statistic can be defined as :
where
with c 1 being a positive constant number that does not depend on n , and H denoting the Hurst Exponent within the range ] 0.5,1 [, if there is a long memory component , and H = 0.5 if the observations X i come from a short-range dependent model .
Having this classical procedure in mind , Lo ( 1991 ) has identified weaknesses in this standard procedure , where short-memory presence may generate distortions in the detection of the presence of long-range dependency . Thus , in order to consider short memory components , the author proposed to modify the original R / S Statistic Test , in order to accommodate autocovariance components instead of the simple variance . Furthermore , instead of considering multiple lags , the test focuses only on lag n = N , the length of the series .
Thus , defining S 2 ( N ) as the sample variance of the series , Lo ( 1991 ) proposed the following measure for S , which now depends on a parameter q :
where y j is the sample autocovariance at lag j , w j ( q ) is defined according to :
and q is the truncation lag .
Hence , the choice of q plays an important role in the sensitivity of the Modified R / S Test . On the other hand , the V / S Statistic Test , developed by Giraitis et al . ( 2003 ), is less sensitive to the choice of the truncation lag . This test consists of the following statistics :
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