ene gies
A icle
Mo ing A e age Ma ke Timing in Eu opean Ene gy
Ma ke s: P oduc ion Ve sus Emissions
Chia-Lin Chang 1, Jukka Ilomäki 2, Hannu Lau ila 2and Michael McAlee 3,4,5,6,7,*
1Depa men o Applied Economics, Depa men o Finance, Na ional Chung Hsing Uni e si y,
Taichung Ci y 402, Taiwan; [email p o ec ed]
2Facul y o Managemen , Uni e si y o Tampe e, 33014 Tampe e, Finland; [email p o ec ed] (J.I.);
[email p o ec ed] (H.L.)
3Depa men o Finance, Asia Uni e si y, Taichung Ci y 41354, Taiwan
4Discipline o Business Analy ics, Uni e si y o Sydney Business School, Sydney 2006, Aus alia
5Econome ic Ins i u e, E asmus School o Economics, E asmus Uni e si y Ro e dam, Ro e dam 3000,
The Ne he lands
6Depa men o Economic Analysis and ICAE, Complu ense Uni e si y o Mad id, 28040 Mad id, Spain
7Ins i u e o Ad anced Sciences, Yokohama Na ional Uni e si y, Yokohama 240-8501, Japan
*Co espondence: [email p o ec ed]; Tel.: +886-4-2332-3456 (ex . 1837)
Recei ed: 12 No embe 2018; Accep ed: 22 No embe 2018; Published: 25 No embe 2018


Abs ac :
This pape sea ches o s ochas ic ends and e u ns p edic abili y in key ene gy asse
ma ke s in Eu ope o e he las decade. The inancial asse s include In e con inen al Exchange
Fu u es Eu ope (ICE-ECX) ca bon emission allowances ( he main d i e o in e es ), Eu opean
Ene gy Exchange (EEX) Coal ARA u u es and ICE B en oil u u es ( e lec ing he wo la ges
ene gy sou ces in Eu ope), S oxx600 Eu ope Oil and Gas Index ( he main ene gy s ock index in
Eu ope), EEX Powe Fu u es ( ep esen ing elec ici y), and S oxx600 Eu ope Renewable Ene gy index
( ep esen ing he sun ise ene gy indus y). This pape inds ha he Mo ing A e age (MA) echnique
bea s andom iming o ca bon emission allowances, coal, and enewable ene gy. In hese asse
ma ke s, he e seems o be signi ican e u ns p edic abili y o s ochas ic ends in p ices. The esul s
a e mixed o B en oil, and he e a e no p edic able ends o he Oil and Gas index. S ochas ic
ends a e also missing in he elec ici y ma ke as he e is an ARFIMA-FIGARCH p ocess in he
day-ahead powe p ices. The empi ical esul s a e in e es ing o se e al easons. We iden i ied he
da a gene a ing p ocess in EU elec ici y p ices as ac ionally in eg a ed (0.5), wi h a ac ionally
in eg a ed Gene alized Au oReg essi e Condi ional He e oscedas ici y (GARCH) p ocess in he
esidual. This is a no el inding. The o de o in eg a ion o o de 0.5 implies ha he p ocess is no
s a iona y bu less non-s a iona y han he non-s a iona y I(1) p ocess, and ha he p ocess has long
memo y. This is p obably because elec ici y canno be s o ed. Re u ns p edic abili y wi h MA ules
equi es s ochas ic ends in p ice se ies, indica ing ha he asse p ices should obey he I(1) p ocess,
ha is, o acili a e long un e u ns p edic abili y. Howe e , all he o he p ice se ies es ed in he
pape a e I(1)-p ocesses, so ha hei e u ns se ies a e s a iona y. The empi ical esul s a e impo an
because hey gi e a simple answe o he ollowing ques ion: When a e MA ules use ul? The answe
is ha , i signi ican s ochas ic ends de elop in p ices, long un e u ns a e p edic able, and ma ke
iming pe o ms be e han does andom iming.
Keywo ds:
s ochas ic ends; e u ns p edic abili y; mo ing a e age; ma ke iming; ene gy ma ke s
1. In oduc ion
Clima e change is possibly he mos se e e h ea o human ci iliza ion, and g eenhouse gas
(GHG) emissions a e conside ed o be a majo ma ke ailu e. The e o e, i is impo an o know how
Ene gies 2018,11, 3281; doi:10.3390/en11123281 www.mdpi.com/jou nal/ene gies
Ene gies 2018,11, 3281 2 o 24
he p ice mechanism in inancial ma ke s accommoda es he issue. This pape ackles he ques ion by
in es iga ing whe he he e a e di e ences in asse p ice p edic abili y (and hence ma ke e iciency)
be ween di e en modes o ene gy sou ces. The ene gy sec o is he main p oduce o GHG emissions
(a 57% in 2016) in Eu ope [
1
]. The aim o he pape is o in es iga e p ice p edic abili y o a main
policy ins umen , namely adable emission allowances.
Eu ope c ea ed he wo ld’s i s ca bon emission allowances ading sys em in 2005 [
2
]. The basic
idea is simple, namely companies ha p oduce ca bon dioxide (CO
2
) emissions mus buy and show
emission allowances enough o co e hei yea ly emissions, o hea y ines a e sen enced. Holding
one emissions allowance gi es a pe mi o p oduce one on o CO
2
emissions. The allowances a e
adable, and ma ke p icing o he emissions is supposed o guide companies o educe hei emissions
e icien ly [3]. The e o e, ca bon emission allowances a e inhe en ly inancial asse s.
The EU Emissions T ading Sys em (EU ETS) cu en ly co e s abou 45% o all GHG emissions in
he EU. The sys em ope a es in all EU coun ies, oge he wi h No way, Iceland, and Liech ens ein,
and has a co-ope a ion ag eemen wi h he ETS o Swi ze land [
3
]. In he sys em, a p ice cap is se by
es ima ing o al g eenhouse gases p oduced by Eu opean companies du ing a mul i-yea phase, and is
g adually diminished o educe o al emissions o e ime.
To da e, he Commission has execu ed wo emission allowance phases, namely phase 1 in
2005–2007, and phase 2 in 2008–2013. The ongoing phase 3 commenced in 2014, and is due o
end in 2020, a e which phase 4 will s a . In phase 1, he i ms ecei ed hei allowance alloca ions
o ee, and he ine was 40
€
pe on o CO
2
emissions. In phases 2 and 3, he ine was li ed o 100
€
pe on. In phase 3, auc ioning became he p ac ice o alloca ing new allowances, and abou 40% o
new allowances we e auc ioned in 2017. The main auc ions a e held a he Eu opean Ene gy Exchange
(EEX) in Ge many, and In e con inen al Exchange Fu u es Eu ope (ICE-ECX). The la e eme ged when
he Ame ican ICE bough he Eu opean ECX in 2010. EEX also se es as he main ading place o
coal, na u al gas, and elec ici y in Con inen al Eu ope.
The pape is inspi ed by Ilomäki e al. [
4
], which uses DJIA s ocks da a om 1988 o 2017.
The pape inds ha ma ke iming wi h MA ules ou pe o ms buy and hold, when he equency
o used obse a ions is educed, and he whole olling window size o 200 ading days is used.
This pape in es iga es e u ns p edic abili y wi h mo e speci ic inancial asse s. The aim is o examine
whe he ma ke iming wi h mo ing a e ages (MA) [
5
] has been success ul o he EU CO
2
emission
allowances, EU coal u u e p ices, B en oil u u es, Dow Jones S oxx 600 Eu ope enewable ene gy
index, Dow Jones S oxx 600 Eu ope oil and gas index, and Eu opean powe p ices ( he da a a e om
Thomson Reu e s Da as eam).
The wo Eu opean s ock indices a e dis inc i e o hea y and low GHG p oduce s in he ene gy
sec o . The EU coal u u e p ices a e ele an as coal is he p ima y ene gy sou ce, pa icula ly in
Ge many. Cu en ly, coal is anked second a e oil as an ene gy sou ce in he wo ld [
6
]. The ading in
u u e con ac s s a ed on 26 Feb ua y 2008 o phase 2. Following he li e a u e, ou da a s a s om
ha da e, and ends on 28 Sep embe 2018.
The pape p oceeds as ollows. Sec ion 2p o ides a li e a u e e iew, and Sec ion 3p esen s
he model speci ica ions. Sec ion 4 epo s he empi ical analysis, and Sec ion 5p o ides some
concluding ema ks
2. Li e a u e Re iew
The economics li e a u e on GHG emissions has g own apidly a e he launch o EU ETS in
2005. Sei e e al. [
7
] a gue ha emission allowance p ices should ollow a ma ingale p ocess wi h
condi ional he e oskedas ici y. Ye , using daily ading da a o e he i s wo yea s, Paolella and
Taschini [
8
] epo ha CO
2
emission allowance e u ns ollow a i s o de au o eg essi e model wi h
Gene alized Au o eg essi e Condi ional He e oscedas ici y (GARCH) dis u bances [
9
,
10
], indica ing an
AR(1)-GARCH(1,1) p ocess o log e u ns ( o de ails on ARMA-GARCH modelling, see Re e ence [
11
].
This sugges s sho - e m e u ns p edic abili y, which iola es he ma ingale hypo hesis.
Ene gies 2018,11, 3281 3 o 24
Mo eo e , Benz and T uck [
12
] ound ha he Ma ko swi ching model [
13
] o e u ns i s he EU
CO
2
da a, such ha he e is a s a iona y non-linea p ocess in e u ns. Mon agnoli and de V ies [
14
]
no e ha he e is signi ican e u ns p edic abili y in phase 1, bu a ma ingale p ocess in phase 2. Thei
da a on phase 2 co e s he pe iod 26 Feb ua y 2008 o 4 Ap il 2009, which is sou ced om BlueNex
ading scene (which was closed in 2012). A ou i e al. [
15
] a gue ha CO
2
spo and u u e p ices
ha e a non-linea ela ionship as he e a e asymme ic and non-linea ela ions be ween spo and
u u e e u ns.
Daskalakis and Ma kellos [
16
] no e ha he e is a ca bon componen in he ma ke p ice o
Eu opean elec ici y as he e is a CO
2
emission allowance isk p emium added o he p ice o elec ic
powe . Bubli z e al. [
17
] a gue ha he decline in ca bon and coal p ices has been he main d i e o he
all in elec ici y p ices in Ge many and Con inen al Eu ope om 2011 o 2015. Vei h e al. [
18
] epo a
posi i e co ela ion be ween CO2emission allowance e u ns and Eu opean ene gy sec o e u ns.
Oes eich and Tsiakas [
19
] ind ha , in Ge man s ock ma ke s, he ee alloca ion o emission
allowances c ea ed a ca bon p emium o hose high ca bon p oduce s ha p oduced signi ican
amoun s o GHG, bu ha he e ec disappea ed in 2009. Bushnell e al. [
20
] examined how he
e ec o he sha p decline in EU CO
2
emissions allowance p ices in 2006 a ec ed ma ke alua ions
in he ene gy sec o , and ound ha low ca bon p oduce s had su e ed mo e han high ca bon
p oduce s. B ouwe s e al. [
21
] in es iga ed he impac s o CO
2
emissions e i ica ion e en s on he
s ock e u ns o Eu opean companies, and no e ha he e en s ha e had a signi ican impac only on
ca bon-in ensi e companies.
C e i e al. [
22
] ind ha , in he ea ly yea s o phase 2, CO
2
emissions allowance p ices ha e been
coin eg a ed [
23
] wi h basic undamen als, namely wi h B en oil, Eu oS oxx 50 index, and wi h a
swi ching p ice be ween coal and gas. Howe e , Koch e al. [
24
] epo ha coin eg a ion does no exis
when da a o e he whole phase 2 a e examined. Tian e al. [
25
] no e ha EU CO
2
e u ns co ela e
posi i ely wi h Eu opean ene gy s ocks e u ns, bu nega i ely wi h ca bon-in ensi e s ocks e u ns.
In addi ion, Cha les e al. [
26
] ind coin eg a ion be ween EU CO
2
emissions spo and u u e p ices,
and in e es a es, du ing phase 2.
Mo e impo an ly, Daskalakis and Ma kellos [
27
] es ma ke e iciency in EU CO
2
p icing wi h
MA ules, namely wi h daily obse a ions, MA50, MA30 and MA15. The conclusion is ha e u ns
we e p edic able in phase 1, possibly because o he lack o ma u i y o he ma ke (A he end o
phase 1, he ma ke was in lux. Acco ding o he EU ETS, he inancial c isis o 2007 c ea ed a majo
su plus o CO2 emissions allowances. A he beginning o Decembe 2007, he equilib ium p ice in he
seconda y ma ke plumme ed o an all- ime low o 0.01 o 0.03
€
pe CO2 on, bo h in he EEX and
ECX. The su plus led o an acu e ise in impo s o in e na ional emissions allowance c edi s ha a e
also adable in he EEX and ECX.). Daskalakis [
28
] e- es ed ma ke e iciency in emission ( u u es)
p ices in phase 2 wi h daily obse a ions, using he MA15 ule. The inding is ha he MA ule bea s
he buy and hold s a egy in e ms o he Sha pe a io in 2008 and 2009, bu no in 2010. This sugges s
a g adually imp o ing ma ke e iciency. Howe e , Medina e al. [
29
] a gue ha imp o emen in
e iciency ( om phase 1 o phase 2) canno be obse ed om in a-day da a. Ibikunle e al. [
30
] used
in a-day da a o 2008–2011, and es ed he co ela ion o liquidi y and ma ke e iciency in EU ca bon
emissions e u ns. The conclusion is ha inc easing liquidi y has imp o ed p icing e iciency, and ha
andom walk p icing beha iou has eme ged in 2010–2011.
C ossland e al. [
31
] ind signi ican ime se ies momen um up o 12 mon hs wi h EU ETS e u ns
be ween ea ly 2008 and mid-2011. Moskowi z e al. [
32
] ind signi ican ime se ies momen um wi h
c ude oil u u e p ices in mon hly da a om Janua y 1985 o Decembe 2009. Yin and Yang [
33
] used
mon hly da a on c ude oil u u e p ices om 1984 o 2013, and ound ha MA ules and ime se ies
momen um ules ou pe o m p edic ions based on mac oeconomic a iables. Liu e al. [34] d ew he
same conclusion using 10 MA ules wi h mon hly da a o 1986–2015.
S a ing om Ma kowi z [
35
], a ional in es o s a e assumed o be isk a e se, indica ing ha hey
moni o bo h e u ns and i s a iabili y. Sha pe [
36
] and Lin ne [
37
] s a e ha e u ns o e iskless
Ene gies 2018,11, 3281 4 o 24
e u ns depend linea ly and posi i ely on ma ke isk adjus ed e u ns. Howe e , LeRoy [
38
] and
Me on [
39
] show ha , i in es o s a e isk a e se, ime- a ying isk adjus ed e u ns exis , and can be
pa ly p edic able because o he non-linea ela ionship be ween e u ns and a iabili y.
Bea ing his in mind, Malkiel [
40
] a gues ha e icien ma ke s do no allow in es o s o gain
abo e a e age e u ns wi hou accep ing abo e a e age a iabili y in e u ns. Me on [
41
] a gues ha
he decision when o buy s ocks and when o sell hem is insigni ican in he long un, and andom
decisions canno be bea en when ma ke s a e e icien . This is based on he idea ha u u e in o ma ion
shocks a e unp edic able, and non-s a iona i y o compe i i e p ices ends o de elop s ochas ic ends
ha could change unexpec edly due o new in o ma ion.
Zhu and Zhou [
42
] showed analy ically ha MA ules add alue o a isk a e se in es o i e u ns
a e p edic able. The empi ical li e a u e on p edic abili y wi h MA ules is ex ensi e. Fo example,
Neely e al. [
43
], Ni e al. [
44
], and Ma shall e al. [
45
] ound ha in es o s bene i om he use o
MA ules, whe eas Hudson e al. [
46
] claim ha MA ules a e useless in in a-day ading. The main
inspi a ion o his pape comes om he indings in Chang e al. [
47
] ha he DJIA index e u ns a e
p edic able in he long un acco ding o MA ules, which p o ides empi ical suppo o he analy ical
esul in Re e ence [42].
Ne o e al. [
48
] p opose a me hodology o isk analysis and po olio op imiza ion o powe
gene a ion asse s in B azil wi h hyd o, wind, and sola powe . Inno a i e s ochas ic models a e
used o gene a e syn he ic ime se ies o wa e in low, wind speed, sola i adiance, empe a u e o
he pho o ol aic panel, and a e age gene a ion capaci y o ene gy. The simula ion is implemen ed
using Mon e simula ions wi h a Cholesky decomposi ion. An economic app oach is p esen ed, aking
in o accoun axa ion and inancing, as well as Ma kowi z po olio heo y. The esul s show ha
he co ela ion be ween ene gy esou ces is al e ed by cash lows and deb . In he di e si ica ion
p ocess, he complemen a i y be ween sou ces educes economic isk. The inc ease in deb inc eases
he co ela ion, dec eases e u n and isk, and a ec s he di e si ica ion esul s. This signi ican ly
educes he hyd oelec ic economic isk and inc ease he inancial e u n, which di ec ly bene i s he
o ma ion o po olios.
Rezec and Schol ens [
49
] no e ha he suppo o inancial ma ke s o he ans o ma ion o
he ene gy sys em o a low ca bon socie y seems c i ical o i s success, and may depend on ma ke
incen i es. The au ho s analyze how equi y indices ha cap u e enewable ene gy in es men s pe o m
compa ed o con en ional benchma k indices, especially inancial ma ke in es o s, such as pension
unds, insu ance companies, and mu ual unds ha use hese o assess and guide hei enewable
ene gy in es men s. The au ho s conside inancial ma ke pa icipan s ha in es indi ec ly in
enewable ene gy as an example o ma ke en i onmen alism wi h espec o commodi ica ion and
ame-shi ing. I is ound ha ha he enewable ene gy indices’ isk-adjus ed e u n is poo , so ha
enewable ene gy does no seem o be a inancially a ac i e po olio in es men .
Fe ei a e al. [
50
] conside an in es men whe eby agen s a e in e es ed in ob aining e u ns bu
p e e less isky asse s. The e a e o he ha migh in luence his selec ion. Re ie ing in o ma ion o
26 sha es o i ms ha p oduce enewable ene gy, he au ho s use he DCCA co ela ion coe icien o
see how hose sha es beha e wi h he main indices. I sha es and indices a e no co ela ed, di e en
ac o s could be seen as isk di e si ica ion. The esul s show ha in he sho un co ela ion exis s
(in scales unde 100 days), bu in he long un mos sha es a e unco ela ed wi h he indices o
hei coun ies (in scales g ea e han 200). This means ha sha es associa ed wi h mo e sus ainable
p inciples could be used o manage po olio isks.
3. Model Speci ica ion
The heo e ical model closely ollows Ilomäki e al. [
4
], and Chang e al. [
47
]. We conside all
he ene gy s ocks as inancial asse s, and assume isk a e se in es o s li e o wo pe iods in an
o e lapping gene a ions economy wi h a con inuum o young and old in es o s
[0, 1]
. An in es o
opens a posi ion in pe iod one, and closes he posi ion in pe iod wo, when all weal h is consumed.
Ene gies 2018,11, 3281 5 o 24
A young in es o jalloca es ini ial weal h
wj
be ween in ini ely-li ed isky asse s,
i=
1, 2, 3,
. . . I
, and
isk- ee asse s ha p oduce he isk- ee a e e u n,
. A isky asse
i
may pay di idend
Di
, and
has
xs
i
ou s anding. A young in es o
j
maximizes u ili y om old age consump ion h ough op imal
alloca ion o ini ial esou ces, wj
, be ween isky and isk- ee asse s.
Igno ing he co ela ion be ween isky asse s, and concen a ing on each isky asse sepa a ely,
a young in es o jmaximizes u ili y wi h espec o each isky asse i:
max xij
E (Pi
+1+Di
+1)
Pi
−(1+ )−νj
2xij2σ2
i
s. .
xij
Pi
≤wj
whe e
E
is he expec a ions ope a o ,
Pi
is he p ice o one sha e o asse
i
,
νj
> 0 is a cons an
isk-a e sion pa ame e o in es o
j
,
σ2
i
> 0 is he a iance o e u ns o he isky asse
i
, and
xij
is he
in es o ’s demand o isky asse i. The i s -o de condi ion yields:
xij
=E (Pi
+1+Di
+1)/Pi
−(1+ )
νjσ2
i
(1)
o in es o j’s op imal demand o he isky asse i. Suppose ha he in es o uses MA ules o ma ke
iming, and alloca es ini ial weal h,
wj
, be ween each isky asse and he isk- ee asse acco ding o
he MA ule o ecas abou he e u n o he asse . The in es o in es s in he isky asse
i
only i
he nume a o on he igh -hand side o Equa ion (1) is posi i e, ha is, i
E (Pi
+1+Di
+1)/Pi
>
(1+ )holds.
4. Empi ical Analysis
The sample da a consis s o daily p ices om 26 Feb ua y 2008 o 28 Sep embe 2018. In choosing
he da a pe iod, we ollowed he common p ac ice in he ene gy economics li e a u e [
14
,
27
], and
excluded phase 1 because o e iden issues du ing ha pe iod. This educes he chances o s uc u al
change du ing he sample pe iod. T ading wi h EU CO
2
emissions allowances in ICE-ECX s a ed a
he beginning o phase 2, ha is, on 26 Feb ua y 2008. The basic olling window is he i s 200 ading
days, which means ha he da a analysis s a s om 2 Decembe 2008 and ends on 28 Sep embe 2018,
yielding 2564 daily obse a ions. We epo he a iables in Eu os, excep o Coal ARA and B en
C ude Oil u u es, which a e epo ed in USD.
The Augmen ed Dickey–Fulle (ADF) es shows ha CO
2
emissions u u es, Coal u u es, B en
Oil u u es, S oxx 600 Eu ope Renewable Ene gy Index, and S oxx 600 Eu ope Oil and Gas Index
a e in eg a ed o o de one, I(1). This indica es ha he na u al loga i hm e u ns o hese se ies a e
s a iona y. The EEX Powe Fu u es is he elec ici y ma ke place o Ge many, Aus ia, F ance, I aly,
Spain, Holland, Czech Republic, Belgium, Swi ze land, Poland, No dic coun ies, Hunga y, Romania,
Slo ak Republic, and he UK.
Figu e 1shows ha he Phelix-DE Powe u u e p ices (base 1–24 h) a e s a iona y, ha is, I(0),
wi h an ADF es alue o
−
12.79. The eigh nega i e Phelix p ices a e concen a ed du ing he holiday
seasons, namely 12/25/2012, 12/26/2012, 12/24/2013, 12/26/2016, 5/1/2017, 12/26/2017, 1/1/2018
and 5/1/2018. Acco ding o he Eu opean Ene gy Exchange, nega i e powe p ices occu when a high
and in lexible (nuclea /ligni e) powe gene a ion appea s simul aneously wi h low elec ici y demand.
This is o en he case on public holidays such as Ch is mas. In hou s o enewable powe supply (wind
and sun), powe p oduce s o e hei elec ici y o nega i e p ices on he exchange. This is o en done
by ma ke e s o enewable powe , bu also by con en ional powe s a ions like nuclea and ligni e
plan s. In his e en , he ma ke clea ing p ice can be se below ze o. In Ge many, enewables ha e
unlimi ed p io i y o e ossil gene a ion.

Ene gies 2018,11, 3281 6 o 24
Ene gies 2018, 11, x FOR PEER REVIEW 6 o 24
Figu e 1. Daily Eu opean Ene gy Exchange (EEX) powe u u e p ices in Eu o/Mwh), 2 Decembe
2008 o 28 Sep embe 2018.
The equilib ium p ice o elec ical powe may p oduce s able equilib ia as elec ici y canno be
s o ed. The a e age p ice is 41.55 € pe Mwh, wi h s anda d de ia ion 11.88 €, and −56.87 € is he
minimum and 98.98 € is he maximum p ice du ing he pe iod. No e ha he e a e some nega i e
obse a ions, usually du ing holiday seasons. This is simply because he gene a ion o enewable
ene gy exceeds demand in he g id, and he enewables ha e unlimi ed p io i y o e ossil ene gy
sou ces in Ge many. The skewness o he s a iona y dis ibu ion is 0.270, and he ku osis is 7.201. As
elec ici y p ices ollow a s a iona y p ocess, he e canno be any long- e m s ochas ic ends.
The e o e, andom iming should be supe io in he long un.
Howe e , as epo ed in Table 1 below, he powe p ice se ies is ac ionally in eg a ed. Thus,
we can iden i y a ac ionally in eg a ed au o eg essi e mo ing a e age p ocess, ARFIMA ( o de ails
o ARFIMA modelling, see Re e ence [51]). The empi ical esul s indica e ha he se ies has bo h long
memo y ( ha c ea es high co ela ion be ween e en s ha a e emo e in ime), and sho memo y
(which las s only one day, as a shock om he p eceding day a ec s he p esen p ice). The ARFIMA
(1,d,1) model can be speci ied as
   
1 1 2 1
1 1
d d
L z L z
   
 
     , whe e
L
is he lag
ope a o , indica ing ha
 
1d
d
L
 
,
2
(0, )
NID

 

,
z y

 
, ha is, he mean

is
sub ac ed om he o iginal se ies,
y
,
1

and
2

a e he es ima ed pa ame e s, and d is he o de
o ac ional in eg a ion.
I
0 0 .5
d
 
, he p ocess is co a iance s a iona y wi h long memo y, assuming ha he
o iginal ARMA p ocess is co a iance s a iona y [51,52] de ines ha , o
0 .5 1
d
 
he se ies is no
s a iona y, bu less non-s a iona y han i d = 1. The a iance also con ains a ac ionally in eg a ed
p ocess, he eby sugges ing an ARFIMA(0,d,1)-FIGARCH(1,d,1) model ha yields non-
au oco ela ed and homoscedas ic esiduals. The FIGARCH p ocess in he esidual,

, is
1 1 2
0 1 1
(1 (1)) 1 (1 ) (1 )(1 )d
h L L L
    
 
 
      
 
, whe e
h
is he condi ional a iance o

, and 0
, ,
  
a e es ima ed pa ame e s. The FIGARCH p ocess implies ha he in luence o
lagged squa ed esiduals slows a a hype bolic a e [53].
Table 1 shows ha , i we ake in o accoun he long memo y (a 0.5) and sho memo y
pa ame e s (a 0.1), he a e age powe p ice has been 50 € pe Mwh wi hin he pe iod.
Figu e 1.
Daily Eu opean Ene gy Exchange (EEX) powe u u e p ices in Eu o/Mwh, 2 Decembe 2008
o 28 Sep embe 2018.
The equilib ium p ice o elec ical powe may p oduce s able equilib ia as elec ici y canno be
s o ed. The a e age p ice is 41.55
€
pe Mwh, wi h s anda d de ia ion 11.88
€
, and
−
56.87
€
is he
minimum and 98.98
€
is he maximum p ice du ing he pe iod. No e ha he e a e some nega i e
obse a ions, usually du ing holiday seasons. This is simply because he gene a ion o enewable
ene gy exceeds demand in he g id, and he enewables ha e unlimi ed p io i y o e ossil ene gy
sou ces in Ge many. The skewness o he s a iona y dis ibu ion is 0.270, and he ku osis is 7.201.
As elec ici y p ices ollow a s a iona y p ocess, he e canno be any long- e m s ochas ic ends.
The e o e, andom iming should be supe io in he long un.
Howe e , as epo ed in Table 1below, he powe p ice se ies is ac ionally in eg a ed. Thus, we
can iden i y a ac ionally in eg a ed au o eg essi e mo ing a e age p ocess, ARFIMA ( o de ails o
ARFIMA modelling, see Re e ence [
51
]). The empi ical esul s indica e ha he se ies has bo h long
memo y ( ha c ea es high co ela ion be ween e en s ha a e emo e in ime), and sho memo y
(which las s only one day, as a shock om he p eceding day a ec s he p esen p ice). The ARFIMA
(1,d,1) model can be speci ied as
(1−L)dz =β1(1−L)dz −1+ε +β2ε −1
, whe e
L
is he lag ope a o ,
indica ing ha
(1−L)d=∆d
,
ε ∼NID(
0,
σ2
ε)
,
z =y −µ
, ha is, he mean
µ
is sub ac ed om he
o iginal se ies, y ,β1and β2a e he es ima ed pa ame e s, and dis he o de o ac ional in eg a ion.
I 0
<d<
0.5, he p ocess is co a iance s a iona y wi h long memo y, assuming ha he
o iginal ARMA p ocess is co a iance s a iona y [
51
,
52
] de ines ha , o 0.5
≤d<
1 he se ies is no
s a iona y, bu less non-s a iona y han i d = 1. The a iance also con ains a ac ionally in eg a ed
p ocess, he eby sugges ing an ARFIMA(0,d,1)-FIGARCH(1,d,1) model ha yields non-au oco ela ed
and homoscedas ic esiduals. The FIGARCH p ocess in he esidual,
ε
, is
h =α0(1−β(1))−1+
h1−(1−β1L)−1(1−φ1L)(1−L)diε2
, whe e
h
is he condi ional a iance o
ε
, and
α0
,
φ
,
β
a e
es ima ed pa ame e s. The FIGARCH p ocess implies ha he in luence o lagged squa ed esiduals
slows a a hype bolic a e [53].
Table 1shows ha , i we ake in o accoun he long memo y (a 0.5) and sho memo y pa ame e s
(a 0.1), he a e age powe p ice has been 50 €pe Mwh wi hin he pe iod.
Ene gies 2018,11, 3281 7 o 24
Table 1.
Elec ici y u u es p ices (Phelix-DE, base 1–24 h) es ima ed by Baillie-Bolle sle -Mikkelsen
me hod [53], 2 Decembe 2008 o 28 Sep embe 2018.
Pa ame e s Coe icien S d.E o - alue -p ob
d ARFIMA 0.52 0.03 17 0.000
MA-1 0.08 0.04 2 0.021
Cons an (Mean) 49.63 11.58 4 0.001
Cons an (Va ) 2.61 0.94 3 0.006
d-FIGARCH 0.58 0.11 5 0.000
phi 0.32 0.10 3 0.001
be a 0.59 0.10 6 0.000
No e ha he MA o Ga ley [
5
] is di e en om he MA p ocess o Box and Jenkins [
54
] ha is
p esen in he ARFIMA es ima ion. The MA o Ga ley de ec s s ochas ic ends in he non-s a iona y
p ice se ies, he eby p oducing a non-s a iona y p ocess, whe eas he MA o Box and Jenkins is always
a s a iona y p ocess wi h sho memo y.
Figu e 2illus a es he log e u ns o ICE-ECX CO
2
EU emission allowances, EEX coal u u es,
ICE-ECX B en oil u u es, Dow Jones S oxx 600 Eu ope Renewable Ene gy, and Oil and Gas Indices
o en yea s. The ADF es s e eal ha all p ocesses a e s a iona y, ha I,I(0).
Ene gies 2018, 11, x FOR PEER REVIEW 7 o 24
Table 1. Elec ici y u u es p ices (Phelix-DE, base 1–24 hou s) es ima ed by Baillie-Bolle sle -
Mikkelsen me hod [53], 2 Decembe 2008 o 28 Sep embe 2018.
Pa ame e s Coe icien
S d.E o
-
alue
-p ob
d ARFIMA 0.52 0.03 17 0.000
MA-1 0.08 0.04 2 0.021
Cons an (Mean) 49.63 11.58 4 0.001
Cons an (Va ) 2.61 0.94 3 0.006
d-FIGARCH 0.58 0.11 5 0.000
phi 0.32 0.10 3 0.001
be a 0.59 0.10 6 0.000
No e ha he MA o Ga ley [5] is di e en om he MA p ocess o Box and Jenkins [54] ha is
p esen in he ARFIMA es ima ion. The MA o Ga ley de ec s s ochas ic ends in he non-s a iona y
p ice se ies, he eby p oducing a non-s a iona y p ocess, whe eas he MA o Box and Jenkins is
always a s a iona y p ocess wi h sho memo y.
Figu e 2 illus a es he log e u ns o ICE-ECX CO2 EU emission allowances, EEX coal u u es,
ICE-ECX B en oil u u es, Dow Jones S oxx 600 Eu ope Renewable Ene gy, and Oil and Gas Indices
o en yea s. The ADF es s e eal ha all p ocesses a e s a iona y, ha I, I(0).
Figu e 2. Re u ns, 2 Decembe 2008 o 28 Sep embe 2018.
Table 2 p o ides desc ip i e s a is ics o he se ies in Figu e 2. We used he h ee mon h Eu ibo
( om Thomson Reu e s Da as eam) as he isk- ee a e, yielding +0.004 o a e age annualized
e u ns du ing he pe iod. The Sha pe a io is calcula ed as [ 0.004] /
i i i
SR

  , whe e
i
is he
a e age annualized e u ns wi h di idends, and
i

is he annualized daily ola ili y o e u ns o
asse
i
.
Table 2 shows ha he a e age annualized e u ns (wi h di idends) o he Oil and Gas s ocks
index is 0.070, wi h annualized ola ili y 0.218. I yields he bes annualized Sha pe Ra io o 0.30 o
he pe iod. CO2 emissions allowances yield a Sha pe a io o 0.05, and coal 0.10, B en oil 0.15, while
ha o he Renewable Ene gy S ock Index is −0.08. These ou comes show ha enewable ene gy
Figu e 2. Re u ns, 2 Decembe 2008 o 28 Sep embe 2018.
Table 2p o ides desc ip i e s a is ics o he se ies in Figu e 2. We used he h ee mon h Eu ibo
( om Thomson Reu e s Da as eam) as he isk- ee a e, yielding +0.004 o a e age annualized
e u ns du ing he pe iod. The Sha pe a io is calcula ed as
[ i−
0.004
]/σi=SRi
, whe e
i
is he
a e age annualized e u ns wi h di idends, and
σi
is he annualized daily ola ili y o e u ns o
asse i.
Table 2shows ha he a e age annualized e u ns (wi h di idends) o he Oil and Gas s ocks
index is 0.070, wi h annualized ola ili y 0.218. I yields he bes annualized Sha pe Ra io o 0.30 o he
pe iod. CO
2
emissions allowances yield a Sha pe a io o 0.05, and coal 0.10, B en oil 0.15, while ha
o he Renewable Ene gy S ock Index is
−
0.08. These ou comes show ha enewable ene gy s ocks
Ene gies 2018,11, 3281 8 o 24
ha e ac ually p oduced a nega i e e u n o isk alue o e he las decade in Eu ope, while ossil
ene gy s ocks ha e pe o med well.
Table 2.
Desc ip i e s a is ics o log e u ns on ca bon dioxide emission allowances, coal, B en oil,
S oxx 600 Eu ope enewable ene gy index wi h di idends, S oxx600 Eu ope Oil and Gas index wi h
di idends, 2 Decembe 2008 o 28 Sep embe 2018.
Va iable Mean S anda d
De ia ion Skewness Ku osis Obs. Ja que-Be a Sha pe
Ra io
CO2e. a. 0.031 0.514 −0.906 21.075 2564 35255 0.05
Coal 0.025 0.214 −0.607 18.459 2564 25690 0.10
B en Oil 0.051 0.323 −0.154 6.538 2564 1347 0.15
Renew Ene gy Index −0.025 0.386 −0.371 7.332 2564 2064 −0.08
Oil and Gas index 0.070 0.218 −0.166 6.431 2564 1269 0.30
Chang e al. [
47
] and Ilomäki e al. [
51
] ind ha ma ke iming wi h MA ules adds alue o a
isk a e se in es o , as compa ed wi h andom iming o Dow Jones Indus ial A e age s ocks, and
wi h he DJIA index i sel . We ollowed he same p ocedu e by choosing a 200 ading days olling
window so ha he sample size o each asse sums o 2564 daily obse a ions. We calcula ed he
empi ical esul s wi h se en equencies o he MA ule. When he MA u ns lowe (highe ) han he
cu en daily closing p ice, we in es in he isky asse ( h ee-mon h Eu ibo ) a he closing p ice o
he nex ading day. The e o e, he ma ke iming s a egy is o in es all weal h, ei he in isky o
isk- ee asse s, while he MA ule ad ises on he iming.
The i s equency ule is o calcula e MA o e e y ading day; he second equency ule akes
in o accoun e e y i h ading day (p oxy o weekly ule); he hi d equency ule is o e e y 22nd
ading day (p oxy o mon hly ule); he ou h equency ule is o e e y 44 h ading day (p oxy
o e e y second mon h); he i h equency ule is o e e y 66 h ading day (p oxy o e e y hi d
mon h); he six h equency ule is o e e y 88 h ading day (p oxy o e e y ou h mon h); and he
se en h equency ule akes in o accoun e e y 110 h ading day (p oxy o e e y i h mon h).
Fo each asse , he MA ules p oduce 2564
×
9 = 23,076 daily e u ns o he i s h ee equencies,
2564
×
4 = 10,256 daily e u ns o he ou h ule, 2564
×
3 = 7692 daily e u ns o he i h ule,
2564
×
2 = 5128 daily e u ns o he six h ule, and 2564 daily e u ns o he se en h ule. A he i s
equency (e e y ading day), we calcula e daily e u ns o MA200, MA180, MA160, MA140, MA120,
MA100, MA80, MA60, and MA40.
These a e calcula ed as:
P −1+P −2+. . . +P −200
200 =X −1;
P −1+P −2+. . . +P −180
180 =X −1;
P −1+P −2+. . . +P −160
160 =X −1;
P −1+P −2+. . . +P −140
140 =X −1;
P −1+P −2+. . . +P −120
120 =X −1;
P −1+P −2+. . . +P −100
100 =X −1;
P −1+P −2+. . . +P −800
80 =X −1
Ene gies 2018,11, 3281 9 o 24
P −1+P −2+. . . +P −60
60 =X −1;
and P −1+P −2+. . . +P −40
40 =X −1
I
X −1<P −1
, we buy he s ock a he closing p ice,
P
, gi ing
R +1=lnP +1
P 
o daily e u ns.
The ansac ion cos is 0.01% pe ansac ion o all MA ule calcula ions.
Figu e 3shows he Eu opean Ca bon Emission Allowance p ices, s a ing om 200 ading days
a e he beginning o phase 2, ha is, om 2 Decembe 2008.
Ene gies 2018, 11, x FOR PEER REVIEW 9 o 24
1 2 800
1
...
80
P P P
X
  

  
  
 
 
1 2 60
1
...
60
P P P
X
  

  
  
 
  ;
and
1 2 40
1
...
40
P P P
X
  

  
  
 
  .
I 11 

PX , we buy he s ock a he closing p ice,
P, gi ing 









P
P
R1
1ln o daily
e u ns. The ansac ion cos is 0.01% pe ansac ion o all MA ule calcula ions.
Figu e 3 shows he Eu opean Ca bon Emission Allowance p ices, s a ing om 200 ading days
a e he beginning o phase 2, ha is, om 2 Decembe 2008.
Figu e 3. Daily u u e p ices o (In e con inen al Exchange Fu u es Eu ope (ICE-ECX)) ca bon
emission allowances in Eu o/ on, 2 Decembe 2008 o 28 Sep embe 2018.
Figu e 3 illus a es a clea nega i e s ochas ic end om sp ing 2011 o sp ing 2013, and a clea
posi i e s ochas ic end om sp ing 2017 o all 2018. Figu e 4 (panels a and b) shows he esul s o
a isk a e se in es o in es ing in CO2 emissions allowances om 2 Decembe 2008 o 28 Sep embe
2018. In he igu e, he s aigh line depic s he heo e ical e icien andom iming e u n o he
ola ili y on ie .
The black squa es on he on ie line ep esen he e u ns o ola ili y pe o mance o an
in es o who chooses andomly he da es o in es in EU ca bon emissions and when o keep he
asse s in h ee-mon h Eu ibo . The squa e close o he e ical axis means in es ing in emissions
allowances o 1026 days (40% om 2564 days), and he squa e a he a end o he line means
in es ing o all 2564 days. Thus, he on ie line is upwa d sloping in bo h panels.
The MA obse a ions in Figu e 4 a e epo ed in Appendix A (Tables A1–A14) (Ca bon Emission
Allowances) as a e ages o e e y equency. Panel a o he igu e e eals ha ma ke iming wi h
MA ules gene ally ou pe o m andom ma ke iming: Daily, weekly, mon hly, e e y o he mon h,
e e y hi d mon h and e e y ou h mon h ules bea andom ma ke iming, on a e age. Howe e ,
he lowes equency, e e y i h mon h, ails o o e come he andom iming on ie .
Figu e 3.
Daily u u e p ices o (In e con inen al Exchange Fu u es Eu ope (ICE-ECX)) ca bon emission
allowances in Eu o/ on, 2 Decembe 2008 o 28 Sep embe 2018.
Figu e 3illus a es a clea nega i e s ochas ic end om sp ing 2011 o sp ing 2013, and a clea
posi i e s ochas ic end om sp ing 2017 o all 2018. Figu e 4(panels a and b) shows he esul s o a
isk a e se in es o in es ing in CO
2
emissions allowances om 2 Decembe 2008 o 28 Sep embe
2018. In he igu e, he s aigh line depic s he heo e ical e icien andom iming e u n o he
ola ili y on ie .
The black squa es on he on ie line ep esen he e u ns o ola ili y pe o mance o an
in es o who chooses andomly he da es o in es in EU ca bon emissions and when o keep he asse s
in h ee-mon h Eu ibo . The squa e close o he e ical axis means in es ing in emissions allowances
o 1026 days (40% om 2564 days), and he squa e a he a end o he line means in es ing o all
2564 days. Thus, he on ie line is upwa d sloping in bo h panels.
The MA obse a ions in Figu e 4a e epo ed in Appendix A(Tables A1–A14) (Ca bon Emission
Allowances) as a e ages o e e y equency. Panel a o he igu e e eals ha ma ke iming wi h
MA ules gene ally ou pe o m andom ma ke iming: Daily, weekly, mon hly, e e y o he mon h,
e e y hi d mon h and e e y ou h mon h ules bea andom ma ke iming, on a e age. Howe e ,
he lowes equency, e e y i h mon h, ails o o e come he andom iming on ie .
Ene gies 2018,11, 3281 16 o 24
Ene gies 2018, 11, x FOR PEER REVIEW 16 o 24
(panel b)
Figu e 10. Re u ns o ola ili y a ios in S oxx 600 Eu ope Renewable Ene gy Index, 2 Decembe 2008
o 28 Sep embe 2018, compa ed wi h andom iming. (panel a) Calcula ed daily, weekly, mon hly,
e e y second mon h, e e y hi d mon h, e e y ou h mon h, and e e y i h mon h; (panel b)
Calcula ed wi h maximal olling window, daily (MA200), weekly (MAW40), mon hly (MA10), e e y
second mon h (MAD5), e e y hi d mon h (MAT4), e e y ou h mon h (MAQ3), and e e y i h
mon h (MAC2).
Figu e 11 shows he de elopmen o Eu opean Oil and Gas S ock Index om 2 Decembe 2008
o 28 Sep embe 2018.
Figu e 11. Daily p ices in S oxx600 Eu ope Oil and Gas Index be o e di idends in Eu o/index sha e,
2 Decembe 2008 o 28 Sep embe 2018.
In Figu e 11, on a e age, no clea las ing s ochas ic ends seem o exis in he de elopmen o
he Eu opean Oil and Gas indus y o e he pe iod. In addi ion, solid gains in s ock p ices and
signi ican a e age annualized di idends ha e p oduced annualized e u ns o +0.070 wi h a e age
Figu e 10.
Re u ns o ola ili y a ios in S oxx 600 Eu ope Renewable Ene gy Index, 2 Decembe 2008 o
28 Sep embe 2018, compa ed wi h andom iming. (
panel a
) Calcula ed daily, weekly, mon hly, e e y
second mon h, e e y hi d mon h, e e y ou h mon h, and e e y i h mon h; (
panel b
) Calcula ed
wi h maximal olling window, daily (MA200), weekly (MAW40), mon hly (MA10), e e y second mon h
(MAD5), e e y hi d mon h (MAT4), e e y ou h mon h (MAQ3), and e e y i h mon h (MAC2).
Figu e 11 shows he de elopmen o Eu opean Oil and Gas S ock Index om 2 Decembe 2008 o
28 Sep embe 2018.
Ene gies 2018, 11, x FOR PEER REVIEW 16 o 24
(panel b)
Figu e 10. Re u ns o ola ili y a ios in S oxx 600 Eu ope Renewable Ene gy Index, 2 Decembe 2008
o 28 Sep embe 2018, compa ed wi h andom iming. (panel a) Calcula ed daily, weekly, mon hly,
e e y second mon h, e e y hi d mon h, e e y ou h mon h, and e e y i h mon h; (panel b)
Calcula ed wi h maximal olling window, daily (MA200), weekly (MAW40), mon hly (MA10), e e y
second mon h (MAD5), e e y hi d mon h (MAT4), e e y ou h mon h (MAQ3), and e e y i h
mon h (MAC2).
Figu e 11 shows he de elopmen o Eu opean Oil and Gas S ock Index om 2 Decembe 2008
o 28 Sep embe 2018.
Figu e 11. Daily p ices in S oxx600 Eu ope Oil and Gas Index be o e di idends in Eu o/index sha e,
2 Decembe 2008 o 28 Sep embe 2018.
In Figu e 11, on a e age, no clea las ing s ochas ic ends seem o exis in he de elopmen o
he Eu opean Oil and Gas indus y o e he pe iod. In addi ion, solid gains in s ock p ices and
signi ican a e age annualized di idends ha e p oduced annualized e u ns o +0.070 wi h a e age
Figu e 11.
Daily p ices in S oxx600 Eu ope Oil and Gas Index be o e di idends in Eu o/index sha e,
2 Decembe 2008 o 28 Sep embe 2018.
In Figu e 11, on a e age, no clea las ing s ochas ic ends seem o exis in he de elopmen
o he Eu opean Oil and Gas indus y o e he pe iod. In addi ion, solid gains in s ock p ices and
signi ican a e age annualized di idends ha e p oduced annualized e u ns o +0.070 wi h a e age

Ene gies 2018,11, 3281 17 o 24
ola ili y o 0.22 o he pas 10 yea s. Figu e 12, Panels a and b, e eals ha , in he absence o s ochas ic
ends in he index wi h solid o al e u ns, andom iming bea s he MA ules in e ms o e u ns o
ola ili y pe o mance.
Ene gies 2018, 11, x FOR PEER REVIEW 17 o 24
ola ili y o 0.22 o he pas 10 yea s. Figu e 12, Panels a and b, e eals ha , in he absence o
s ochas ic ends in he index wi h solid o al e u ns, andom iming bea s he MA ules in e ms o
e u ns o ola ili y pe o mance.
The plo s o Figu e 12 a e om he i h ows in Appendix A (Tables A1–A14) (Oil and Gas
Index). They show ha all MA ma ke iming ules signi ican ly unde pe o m andom ma ke
iming. No e ha andom iming 41% o he ime (1051 ading days) in Oil and Gas index p oduces
+0.030, wi h ola ili y 0.14. In he igu e, he ela ionship s ays he same in bo h Panels. The e o e, he
e u ns o ola ili y esul s wi h MA ma ke iming ules con i m ha he e ha e been no p edic able
e u ns in he Eu opean oil and gas s ocks o e he las 10 yea s.
(panel a)
(panel b)
Figu e 12. Re u ns o ola ili y a ios in S oxx 600 Eu ope Oil and Gas Index, 2 Decembe 2008 o 28
Sep embe 2018. (panel a) Calcula ed daily, weekly, mon hly, e e y second mon h, e e y hi d
mon h, e e y ou h mon h, and e e y i h mon h; (panel b) Calcula ed wi h maximal olling
window, daily (MA200), weekly (MAW40), mon hly (MA10), e e y second mon h (MAD5), e e y
hi d mon h (MAT4), e e y ou h mon h (MAD3), and e e y i h mon h (MAC2).
Figu e 12.
Re u ns o ola ili y a ios in S oxx 600 Eu ope Oil and Gas Index, 2 Decembe 2008 o
28 Sep embe 2018. (
panel a
) Calcula ed daily, weekly, mon hly, e e y second mon h, e e y hi d
mon h, e e y ou h mon h, and e e y i h mon h; (
panel b
) Calcula ed wi h maximal olling window,
daily (MA200), weekly (MAW40), mon hly (MA10), e e y second mon h (MAD5), e e y hi d mon h
(MAT4), e e y ou h mon h (MAD3), and e e y i h mon h (MAC2).
The plo s o Figu e 12 a e om he i h ows in Appendix A(Tables A1–A14) (Oil and Gas Index).
They show ha all MA ma ke iming ules signi ican ly unde pe o m andom ma ke iming. No e
ha andom iming 41% o he ime (1051 ading days) in Oil and Gas index p oduces +0.030, wi h
Ene gies 2018,11, 3281 18 o 24
ola ili y 0.14. In he igu e, he ela ionship s ays he same in bo h Panels. The e o e, he e u ns o
ola ili y esul s wi h MA ma ke iming ules con i m ha he e ha e been no p edic able e u ns in
he Eu opean oil and gas s ocks o e he las 10 yea s.
5. Concluding Rema ks
We ha e es ed e u ns p edic abili y o EU CO
2
emission allowance p ices o e he las en
yea s, as well as he linked asse s, namely EU Coal p ices, B en Oil p ices, DJ S oxx 600 Eu ope
Renewable Ene gy Index, DJ S oxx 600 Eu ope Oil and Gas Index, and EU Elec ici y one-day ahead
u u e p ices. We ollowed he li e a u e by s a ing he sample da a om he beginning o phase
2 o he EU ETS emission allowance alloca ions. The opic is impo an in ligh o he ques ionable
capaci y o he ma ke s o con ol clima e change, and because all he abo e subjec s can be conside ed
as inancial asse s aded in supposedly e icien ma ke s. In e icien inancial ma ke s, ma ke iming
wi h Mo ing A e age (MA) ules should no add alue o isk a e se in es o s, as compa ed wi h
andom ma ke iming.
The empi ical esul s a e in e es ing o se e al easons. Fi s , we iden i ied he da a gene a ing
p ocess in EU elec ici y p ices as ac ionally in eg a ed (0.5), wi h a ac ionally in eg a ed GARCH
p ocess in he esidual. This is a no el inding. The o de o in eg a ion o o de 0.5 implies ha he
p ocess is no s a iona y bu less non-s a iona y han he I(1) p ocess, and ha he p ocess has long
memo y. This is p obably because elec ici y canno be s o ed. Re u ns p edic abili y wi h MA ules
equi es s ochas ic ends in p ice se ies, indica ing ha he asse p ices should obey he I(1) p ocess,
ha is, o acili a e long un e u ns p edic abili y. Howe e , all he o he p ice se ies es ed in he
pape a e I(1)-p ocesses, so ha hei e u ns se ies a e s a iona y.
Second, we ound ha he EU Coal u u e p ice se ies c ea es signi ican e u ns p edic abili y
wi h MA ules, he eby adding alue o a isk a e se in es o . The a e age annualized Sha pe a io
is 0.4, while 100% o ime in es ed in coal p oduces a Sha pe a io o 0.05. The esul s wi h EU CO
2
emission allowances a e simila o hose wi h coal, wi h he lowes equency being an anomaly. This is
an impo an inding since he li e a u e has so a epo ed almos andom walk beha io in daily da a,
indica ing ha ma ke e iciency has imp o ed as he sample size has g own. Howe e , he esul s
wi h B en Oil p ices a e no as clea in his espec .
The Eu opean Renewable Ene gy s ocks (DJ S oxx 600 index) e u ns a e signi ican ly p edic able
wi h MA ules as all equencies p oduce ema kably highe Sha pe a ios han does andom iming.
Mo eo e , he DJ S oxx 600 Oil and Gas index does no p oduce e u ns p edic abili y, and ma ke
iming wi h MA ules p e o ms poo ly. The index has no de eloped long-las ing s ochas ic ends
wi hin he es ing pe iod. The a e age andom iming Sha pe a io is nega i e, bu andom iming
p oduces posi i e Sha pe a ios, and 100% o he ime in he index p oduces a Sha pe a io o 0.30.
The empi ical esul s a e impo an because hey co obo a e he a gumen o Zhu and Zhou [
42
],
and gi e a simple answe o he ollowing ques ion: When a e MA ules use ul? The answe is ha ,
i signi ican s ochas ic ends de elop in p ices, long un e u ns a e p edic able, and ma ke iming
pe o ms be e han does andom iming.
Au ho Con ibu ions: Concep ualiza ion, J.I. and H.L.; Me hodology, C.-L.C., J.I., H.L. and M.M.; So wa e, J.I.;
Valida ion, C.-L.C., H.L. and M.M.; Fo mal Analysis, J.I. and M.M.; In es iga ion, J.I. and H.L.; Resou ces, H.L.
and M.M.; Da a Cu a ion, C.-L.C. and J.I.; W i ing-O iginal D a P epa a ion, J.I. and H.L.; W i ing-Re iew &
Edi ing, M.M.; Visualiza ion, C.-L.C. and J.I.; Supe ision, H.L. and M.M.; P ojec Adminis a ion, H.L. and M.M.;
Funding Acquisi ion, H.L.
Funding: This esea ch ecei ed no ex e nal unding.
Acknowledgmen s:
The au ho s a e mos g a e ul o h ee e iewe s o e y help ul commen s and sugges ions.
Fo inancial suppo , he i s au ho wishes o acknowledge he Minis y o Science and Technology (MOST),
Taiwan, and he ou h au ho is g a e ul o he Aus alian Resea ch Council and Minis y o Science and
Technology (MOST), Taiwan.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Ene gies 2018,11, 3281 19 o 24
Appendix A
Table A1. Annualized daily e u ns o MA40–MA200, a e age e u ns and Sha pe a ios.
S ock B&H MA200 MA180 MA160 MA140 MA120 MA100 MA80 MA60 MA40 A e age
Re u ns
Sha pe
Ra io
Ca bon Emission Allowances 0.031 0.000 0.028 0.032 0.054 0.008 0.014 −0.012 0.014 0.039 0.020 0.07
Coal ARA 0.025 0.074 0.076 0.093 0.087 0.076 0.108 0.120 0.094 0.056 0.087 0.56
B en Oil 0.051 0.025 0.041 0.022 0.017 0.008 0.011 0.030 0.029 −0.020 0.018 0.07
STOXX600 Renewable Ene gy −0.028 0.081 0.105 0.090 0.063 0.066 0.079 0.064 0.058 0.045 0.072 0.30
STOXX600 Oil and Gas 0.030 −0.049 −0.043 −0.026 −0.011 −0.039 −0.043 −0.032 −0.032 −0.071 −0.039 −0.20
No e:
All e u ns a e be o e di idends. S oxx 600 Eu ope Renewable Ene gy Index has he a e age annualized di idend o 0.003 and S oxx 600 Eu ope Oil and Gas Index has he a e age
annualized di idend o 0.039. In addi ion, he a e age annualized isk- ee yield has been +0.004 in he es ed pe iod. The Sha pe a ios a e calcula ed wi h di idends. This p ocedu e is
iden ical om Tables A1–A7.
Table A2. Annualized daily (e e y i h ading day) e u ns o MAW8–MAW40 (W = numbe o weeks), a e age e u ns and Sha pe a ios.
S ock B&H MAW40 MAW36 MAW32 MAW28 MAW24 MAW20 MAW16 MAW12 MAW8 A e age
Re u ns
Sha pe
Ra io
Ca bon Emission Allowances 0.031 0.038 0.056 0.088 0.081 0.062 0.036 0.048 0.030 0.053 0.055 0.16
Coal ARA 0.025 0.082 0.078 0.082 0.074 0.083 0.086 0.106 0.086 0.018 0.077 0.49
B en Oil 0.051 0.045 0.040 0.036 0.011 0.014 0.023 0.050 0.059 0.047 0.036 0.15
STOXX600 Renewable Ene gy −0.028 0.093 0.111 0.130 0.108 0.114 0.129 0.110 0.135 0.116 0.116 0.48
STOXX600 Oil and Gas 0.030 −0.046 −0.049 −0.032 −0.015 −0.016 −0.026 −0.035 −0.021 −0.040 −0.031 −0.13
Table A3. Annualized daily (e e y 22nd ading day) e u ns o MA2–MA10, a e age e u ns and Sha pe a ios.
S ock B&H MA10 MA9 MA8 MA7 MA6 MA5 MA4 MA3 MA2 A e age
Re u ns
Sha pe
Ra io
Ca bon Emission Allowances 0.031 0.020 0.040 0.055 0.037 0.047 0.033 0.092 0.114 0.107 0.061 0.17
Coal ARA 0.025 0.090 0.081 0.085 0.087 0.087 0.092 0.106 0.070 0.014 0.079 0.50
B en Oil 0.051 0.040 0.027 0.046 0.053 0.050 0.061 0.067 0.096 0.085 0.058 0.26
STOXX600 Renewable Ene gy −0.028 0.102 0.102 0.115 0.066 0.136 0.132 0.089 0.095 0.043 0.098 0.39
STOXX600 Oil and Gas 0.030 −0.045 −0.039 −0.011 −0.011 −0.021 0.017 −0.003 −0.011 −0.019 −0.016 −0.03
Ene gies 2018,11, 3281 20 o 24
Table A4.
Annualized daily (e e y o he mon h) e u ns o MAD2–MAD5 (D = e e y o he mon h, 5, 4, 3, 2, a e he numbe s o obse a ions in he olling window),
a e age e u ns and Sha pe a ios.
S ock B&H MAD5 MAD4 MAD3 MAD2 A e age Re u ns Sha pe Ra io
Ca bon Emission Allowances 0.031 0.061 0.063 0.118 0.095 0.084 0.25
Coal ARA 0.025 0.090 0.081 0.085 0.087 0.087 0.55
B en Oil 0.051 0.011 −0.017 −0.015 0.007 −0.003 −0.03
STOXX600 Renewable Ene gy −0.028 0.079 0.079 0.093 0.085 0.084 0.36
STOXX600 Oil and Gas 0.030 −0.045 −0.040 −0.017 −0.042 −0.036 −0.18
Table A5.
Annualized daily (e e y hi d mon h) e u ns o MAT2–MAT4 (T = e e y hi d mon h, and 4, 3, 2, a e he numbe s o obse a ions in he olling window),
a e age e u ns and Sha pe a ios.
S ock B&H MAT4 MAT3 MAT2 A e age Re u ns Sha pe Ra io
Ca bon Emission Allowances 0.031 0.079 0.083 0.150 0.104 0.32
Coal ARA 0.025 0.070 0.055 −0.033 0.030 0.17
B en Oil 0.051 0.013 −0.012 0.078 0.027 0.11
STOXX600 Renewable Ene gy −0.028 0.100 0.066 0.073 0.079 0.34
STOXX600 Oil and Gas 0.030 −0.009 −0.008 0.008 −0.003 0.07
Table A6.
Annualized daily (e e y ou h mon h) e u ns o MAQ2–MAQ3 (Q = e e y ou h mon h, 3, 2, a e he numbe s o obse a ions in he olling window),
a e age e u ns and Sha pe a ios.
S ock B&H MAQ3 MAQ2 A e age Re u ns Sha pe Ra io
Ca bon Emission Allowances 0.031 0.099 0.103 0.101 0.30
Coal ARA 0.025 0.053 0.037 0.045 0.29
B en Oil 0.051 0.036 0.067 0.051 0.22
STOXX600 Renewable Ene gy −0.028 0.056 0.066 0.061 0.24
STOXX600 Oil and Gas 0.030 −0.032 −0.027 −0.029 −0.11
Table A7.
Annualized daily (e e y i h mon h) e u ns o MAC2 (C = e e y i h mon h, 2 is he numbe s o obse a ions in he olling window), a e age e u ns and
Sha pe a ios.
S ock B&H MAC2 A e age Re u ns Sha pe Ra io
Ca bon Emission Allowances 0.031 −0.001 −0.001 −0.02
Coal ARA 0.025 0.048 0.048 0.31
B en Oil 0.051 0.090 0.090 0.39
STOXX600 Renewable Ene gy
−0.028 −0.018 −0.018 −0.08
STOXX600 Oil and Gas 0.030 −0.037 −0.037 −0.18
Ene gies 2018,11, 3281 21 o 24
Table A8. Annualized daily ola ili y o MA40–MA200, a e age annualized ola ili y.
S ock B&H MA200 MA180 MA160 MA140 MA120 MA100 MA80 MA60 MA40 A e age
Ca bon Emission Allowances 0.514 0.293 0.301 0.301 0.304 0.311 0.312 0.315 0.337 0.348 0.314
Coal ARA 0.214 0.147 0.148 0.147 0.147 0.147 0.147 0.151 0.153 0.157 0.149
B en Oil 0.323 0.198 0.201 0.199 0.204 0.206 0.207 0.205 0.212 0.221 0.206
STOXX600 Renewable Ene gy
0.386 0.227 0.233 0.228 0.233 0.237 0.239 0.244 0.258 0.257 0.239
STOXX600 Oil and Gas 0.218 0.130 0.132 0.130 0.132 0.132 0.131 0.134 0.133 0.136 0.132
Table A9. Annualized daily (e e y i h ading day) ola ili y o MAW8–MAW40 (W = numbe o weeks), a e age annualized ola ili y.
S ock B&H MAW40 MAW36 MAW32 MAW28 MAW24 MAW20 MAW16 MAW12 MAW8 A e age
Ca bon Emission Allowances 0.514 0.296 0.301 0.306 0.307 0.311 0.317 0.315 0.338 0.342 0.315
Coal ARA 0.214 0.147 0.146 0.148 0.148 0.147 0.146 0.146 0.150 0.155 0.148
B en Oil 0.323 0.200 0.200 0.202 0.204 0.208 0.210 0.212 0.220 0.212 0.207
STOXX600 Renewable Ene gy
0.386 0.227 0.227 0.227 0.234 0.241 0.244 0.244 0.248 0.254 0.239
STOXX600 Oil and Gas 0.218 0.131 0.130 0.129 0.132 0.132 0.132 0.137 0.135 0.138 0.133
Table A10. Annualized daily (e e y 22nd ading day) ola ili y o MA2–MA10, a e age annualized ola ili y.
S ock B&H MA10 MA9 MA8 MA7 MA6 MA5 MA4 MA3 MA2 A e age
Ca bon Emission Allowances 0.514 0.310 0.316 0.329 0.327 0.324 0.332 0.355 0.354 0.353 0.333
Coal ARA 0.214 0.145 0.146 0.148 0.148 0.150 0.151 0.149 0.155 0.148 0.149
B en Oil 0.323 0.200 0.203 0.206 0.210 0.214 0.212 0.220 0.208 0.219 0.210
STOXX600 Renewable Ene gy
0.386 0.234 0.234 0.223 0.230 0.242 0.243 0.241 0.259 0.253 0.240
STOXX600 Oil and Gas 0.218 0.138 0.140 0.132 0.132 0.133 0.137 0.138 0.136 0.140 0.136
Table A11. Annualized daily (e e y o he mon h) ola ili y o MAD2–MAD5 (D = e e y o he mon h, 5, 4, 3, 2, a e he numbe s o obse a ions in olling window),
a e age annualized ola ili y.
S ock B&H MAD5 MAD4 MAD3 MAD2 A e age
Ca bon Emission Allowances 0.514 0.332 0.336 0.316 0.314 0.324
Coal ARA 0.214 0.146 0.149 0.153 0.160 0.152
B en Oil 0.323 0.194 0.209 0.195 0.209 0.202
STOXX600 Renewable Ene gy 0.386 0.234 0.234 0.245 0.257 0.243
STOXX600 Oil and Gas 0.218 0.142 0.148 0.135 0.129 0.138

Ene gies 2018,11, 3281 22 o 24
Table A12.
Annualized daily (e e y hi d mon h) ola ili y o MAT2–MAT4 (T = e e y hi d mon h, and 4, 3, 2, a e he numbe s o obse a ions in he olling window),
a e age annualized ola ili y.
S ock B&H MAT4 MAT3 MAT2 A e age
Ca bon Emission Allowances 0.514 0.323 0.348 0.283 0.318
Coal ARA 0.214 0.145 0.156 0.145 0.149
B en Oil 0.323 0.198 0.213 0.227 0.213
STOXX600 Renewable Ene gy 0.386 0.234 0.248 0.234 0.239
STOXX600 Oil and Gas 0.218 0.136 0.140 0.125 0.134
Table A13.
Annualized daily (e e y ou h mon h) ola ili y o MAQ2–MAQ3 (Q = e e y ou h mon h, and 3, 2, a e he numbe s o obse a ions in he olling
window), a e age annualized ola ili y.
S ock B&H MAQ3 MAQ2 A e age
Ca bon Emission Allowances 0.514 0.318 0.328 0.323
Coal ARA 0.214 0.136 0.142 0.139
B en Oil 0.323 0.196 0.226 0.211
STOXX600 Renewable Ene gy 0.386 0.233 0.253 0.243
STOXX600 Oil and Gas 0.218 0.153 0.167 0.160
Table A14.
Annualized daily (e e y i h mon h) ola ili y o MAC2 (C = e e y i h mon h, and 2 is he numbe o obse a ions in he olling window), a e age
annualized ola ili y.
S ock B&H MAC2 A e age
Ca bon Emission Allowances 0.514 0.332 0.332
Coal ARA 0.214 0.141 0.141
B en Oil 0.323 0.221 0.221
STOXX600 Renewable Ene gy 0.386 0.242 0.242
STOXX600 Oil and Gas 0.218 0.139 0.139
Ene gies 2018,11, 3281 23 o 24
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