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12
AlicjaGanczarek-Gamrot
1.1.Assessinglongmemoryestimation
Inthispartofthepaperthreelongmemoryestimationmethodsarepresent-
ed:R/Sstatistics,GPHstatisticsandperiodogrammethod.
Rangeoverstandarddeviation(R/S)isthemostknowntestoflong
memory.ItwasproposedbyHurst
7andmodifiedbyMandelbrot8andfinallyby
Lo
9totaketheform:
R/S
=
σ
∧
T
1
(
q
)
⎡
⎢
⎣
max
1
≤
k
≤
T
1
∑
j
k
=
1
(
y
j
−
y
−
)
−
min
1
≤
k
≤
T
1
∑
j
k
=
1
(
y
j
−
y
−
)
⎤
⎥
⎦
,
(7)
−
where
ymeanofyt,sTstandarddeviationofyt,
∧
σ
T
(q
)
isarootoflongtermvariancewithbandwidthq:
σ
∧
T
2
(
q
)
=
s
T
2
+
T
2
∑
j
q
=
1
ω
j
(
q
)
⎡
⎢
⎣
i
=
∑
T
j
+
1
(
y
i
−
y
_
)(
y
i
−
j
−
y
_
)
⎤
⎥
⎦
,
ω
j
(
q
)
=
1
−
q
+
j
1
−Bartlett,s
weights
q
=
⎡
⎢
⎢
⎣
4
⎛
⎜
⎝
100
T
⎞
⎟
⎠
1
4
⎤
⎥
⎥
⎦
,[x]−integralpartofx.
Lo
10showedthatifstationarytimeseriesy
thasshortmemory,thenR/Ssta-
tisticconvergestoaBrownianbridgeatrateT
1/2.Mandelbrot11showedthat,if
stationarytimeseriesythaslongmemory,thenR/SstatisticconvergestoBrown-
ianbridgeatrateT
H.TheR/SstatisticisusedtoestimateHurstcoefficientinthe
followingway:
1)k1observationsfromempiricaltimeseries(largek1)arechosen,
2)valuesofR/Sstatisticsforki=f*ki-1wherei=2,…,sarecalculated,
3)alinefitofalltheseR/Sstatisticsversuski,i=1,…,s,onthelog-logscale
yieldsanestimateoftheHurstcoefficientH.
7H.E.Hurst:Op.cit.
8B.B.Mandelbrot:LimitTheoremsontheSelf-NormalizedRangeforWeaklyandStrongly
DependentProcesses."Zeitschriftf¨urWahrscheinlichkeitstheorieundverwandteGebiete”1975,
No.31,p.271-285.
9A.W.Lo:LongTermMemoryinStockMarketPrices."Econometrica”1951,No.59,
p.1279-1313.
10Ibidem.
11B.B.Mandelbrot:Op.cit.
