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Assessinglongmemorycharacteristicsofenergyprices…
11
Usingproperties(1)and(3)GrangerandJoyeux
4andHosking5haveshown
thatlongmemoryprocesscanbedescribedparametricallybyfractionallyinte-
gratedprocess:
(1−L)
d(y
t−
μ
)=
μ
t,
(5)
whereL
ky
t=yt
−
kisalagoperator,disafractionaldifferenceparameter(d>−1),
μ
istheexpectedvalueofyt,utisastationaryprocesswithshortmemoryand
E(ut)=0,
(
1
−
L
)
d
=
k
∑
∞
=
0
⎛
⎜
⎜
⎝
d
k
⎞
⎟
⎟
⎠
(
−
1
)
k
L
k
isafractionaldifferencefilter,
⎛
⎜
⎜
⎝
d
k
⎞
⎟
⎟
⎠
=
k
!
(
d
d
−
!
k
)
!
=
Γ
(
k
+
Γ
1
)
(
Γ
d
(
+
d
1
−
)
k
+
1
)
.
If|d|>1/2thenytisnon-stationary.If0<d<1/2thenytisstationaryand
haslongmemory.If−1/2<d<0thenytisstationaryandhasshortmemory.Of-
tentheHurstcoefficientisusedinsteadofdinequation(5)followingtherela-
tion:
d=H−1/2.
(6)
Henceif0<HorH>1thenytisnonstationary.If1/2<H<1thenytissta-
tionaryandhaslongmemorysothetimeseriesischaracterizedbypositiveauto-
correlationandsmallrisk,becausethereisahighprobability,thattrenddirection
willnotbechanged.If0<H<1/2thenytisstationaryandhasshortmemory,
timeseriesischaracterizedbynonpositiveautocorrelation,andsmallrisk,be-
causeytwillreactonnewinformationfromthemarket.IfH=0.5thenytis
arandomprocess(forexampleWiener,sprocessorrandomwalk)
6.
4C.W.J.Granger,R.Joyeux:AnIntroductiontoLong-MemoryTimeSeriesModelsand
FractionalDifferencing."JournalofTimeSeriesAnalysis”1980,No.1,p.15-29.
5J.R.M.Hosking:FractionalDifferencing."Biometrika”1981,No.68,p.165-176.
6J.Stawicki,I.Frączek-Miller:Różnicowaniefraktalneszeregówczasowych.W:Dynamiczne
modeleekonometryczne.MateriałynaVOgólnopolskieSeminariumNaukowe.TNOiK„DomOr-
ganizatora”,Toruń1997.
