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Assessinglongmemorycharacteristicsofenergyprices…
13
Theweaknessoftheaboveprocedureisthechoiceofk1.Ononehandlarge
k1isdesired,ontheotherhandforlargek1onlyfewvaluesoftheR/Sstatistic
canbecalculated.TomitigatethisproblemtheLeastAbsoluteDeviation(LAD)
isusedtoestimateparametersoftheline.
GewekeandPorter-Hudak
12proposedsemi-parametrictest(GPH)basedon
spectraldensityfunction.Thespectraldensityfunctionforfractaldifferenced
processisgivenby:
f
(
ω
)
=
[
4
sin
2
(
ω
2
)]
−
df
u
t
(
ω
)
,
(8)
where
ω
isaFourierfrequency,
f
u
t
(
ω
)
isaspectraldensityfunctionofthepro-
cess(5).
Thelongmemoryparameterd(8)isestimatedbasedonregressionfunction:
ln
f
(
ω
j
)
=
B
−
d
[
4
sin
2
(
ω
2
j
)]
+
ε
j
(9)
forj=1,2,…nj(T);nj(T)=T
α
.
GewekeandPorter-Hudak
13showed,thatif0<α<1,thendhasnormal
distributionforlongtimeseries:
d
∧
~
N
⎛
⎜
⎜
⎜
⎜
⎜
⎝
d
,
6
n
j
∑
=
f
1
(
U
π
j
2
−
U
−
)
2
⎞
⎟
⎟
⎟
⎟
⎟
⎠
;
U
j
=
ln[
4
sin
2
(
ω
2
j
)]
.
Byassumingthatd=0(processhasnolongmemory),GPHteststatisticis
givenby:
t
d
=
0
=
d
∧
⋅
⎛
⎜
⎜
⎜
⎜
⎝
6
n
∑
j
=
f
1
(
U
π
j
2
−
U
−
)
2
⎞
⎟
⎟
⎟
⎟
⎠
−
1
2
.
(10)
12J.Geweke,S.Porter-Hudak:TheEstimationandApplicationofLongMemoryTimeSeries
Models."JournalofTimeSeriesAnalysis”1983,No.4,p.221-237.
13Ibidem.
