Elec ic Powe Sys ems Resea ch 212 (2022) 108144
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Elec ic Powe Sys ems Resea ch
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Do jumps and cojumps ma e o elec ici y p ice o ecas ing? E idence om
he Ge man-Aus ian day-ahead ma ke
Ai o Cia e a a,∗, Pe u Muniain b, Ainhoa Za aga c
aDepa men o Economic Analysis, Uni e si y o he Basque Coun y, UPV/EHU. A da. Lehendaka i Agui e, 83. 48015 Bilbao, Spain
bDepa men o Applied Ma hema ics, Uni e si y o he Basque Coun y, UPV/EHU. To es Que edo Ingenia ia Plaza, 1. 48013 Bilbao, Spain
cDepa men o Quan i a i e Me hods, Uni e si y o he Basque Coun y, UPV/EHU. A da. Lehendaka i Agui e, 83. 48015 Bilbao, Spain
ARTICLE INFO
Keywo ds:
P ice o ecas ing
Jumps
Cojumps
Elas ic ne
Va iance s abilizing ans o ma ions
ABSTRACT
This pape analyzes he po en ial o including jumps and cojumps in elec ici y p ice o ecas ing models.
The s udy is ca ied ou on he Ge man–Aus ian day-ahead elec ici y ma ke wi h a mul i a ia e amewo k
in which each hou o he day is ea ed as an indi idual ime se ies. Th ee models a e speci ied: The ARX
model, he ARX-J model (which includes jumps), and he ARX-J-CJ model (which also includes cojumps).
P ices a e ans o med using se e al a iance s abilizing ans o ma ions. The o ecas ing pe o mance o he
h ee models wi h o iginal and ans o med p ices is compa ed using se e al o ecas ho izons unning om
one day-ahead o one week-ahead. Resul s show ha he o ecas ho izon is c ucial in de e mining whe he
jumps and cojumps should be included in elec ici y p ice o ecas ing. Jumps and cojumps add impo an
in o ma ion o o ecas p ices o ho izons longe han 4 days, bu he e is no gain in o ecas accu acy o
sho e ho izons. The esul s a e o in e es o ma ke pa icipan s o aking op imal decisions and p icing
base week u u es con ac s.
1. In oduc ion
Wi h he libe aliza ion o elec ici y ma ke s, a need o unde s and
elec ici y p ice o ma ion has a isen. Once p ices a e modeled app o-
p ia ely, hey a e p edic ed. Ma ke pa icipan s need di e en p ice
o ecas ho izons depending on he economic decisions ha hey ha e
o make a any gi en ime. A be e unde s anding o elec ici y p ice
dynamics is he e o e c ucial. Howe e , elec ici y p ices ha e unique
cha ac e is ics no ound in o he commodi ies ha make o ecas ing
hem di icul . Elec ici y canno be s o ed, a leas on a la ge scale, and
i mus be a ailable and managed upon demand. This makes elec ici y
p ices e y ola ile and leads o equen spikes. Recen la ge scale
deploymen o sma g id communica ion ne wo ks enables supplie s
and cus ome s o manage hei powe based on me e ed da a, hus
educing ola ili y and sha ing spikes [2].
Taking in o accoun hese ea u es, many esea ch pape s ha e
ackled he modeling and o ecas ing o elec ici y p ices in di e en
ma ke s. Modeling app oaches can be di ided in o ou ca ego ies:
Fundamen al models, a i icial in elligence-based models, hyb id mod-
els, and s a is ical models [3]. Gi en he na u e o elec ici y p ice
o ma ion, inco po a ing jumps in o hese ca ego ies is a leap o wa d
owa ds imp o ing he o ecas ing abili y o he models. [4] analyze
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (A. Cia e a), [email p o ec ed] (P. Muniain), [email p o ec ed] (A. Za aga).
1P ope ies o di e en jump es s a e analyzed in [1].
he ole o jumps in elec ici y p ice modeling. Since hei pape , jump
es s ha e been widely used o de ec spikes in elec ici y p ice se ies
da a.1Fundamen al models equi e an ex ensi e ep esen a ion o he
elec ici y sys em o simula e he ma ke clea ing mechanism be o e
es ima ion and o ecas ing ake place [5]. A i icial in elligence models
include machine-lea ning models and deep-lea ning models (examples
a e [6], and [7]). Hyb id models combine di e en models wi h he
goal o achie ing highe accu acy in o ecas ing (examples a e [8,9],
and [10]). Finally, s a is ical models use his o ical p ice da a and some
ex e nal p ice- ela ed in o ma ion (mainly wea he and load o ecas s)
o p edic p ices o di e en ime ames. Examples include [3]. Ou
wo k belongs o his las ca ego y. S a is ical models ha e he ad an-
age o enabling an in e p e a ion o hei componen s o be ob ained.
They he e o e help ma ke pa icipan s o unde s and he ela ionship
be ween elec ici y p ices and hei de e minan s, and o ake hei
decisions acco dingly. In pa icula , we es ima e au o eg essi e models
wi h exogenous a iables (ARX), whe e p ices depend on hei pas and
o he exogenous ac o s.
The ole o cojumps, de ined as jumps occu ing a he same ime in
di e en ime se ies, has also been s udied in he li e a u e, especially
in inancial ma ke s. Fo ins ance, [11] ind e idence o cojumps in
h ps://doi.o g/10.1016/j.eps .2022.108144
Recei ed 12 Janua y 2022; Recei ed in e ised o m 4 May 2022; Accep ed 26 May 2022
Elec ic Powe Sys ems Resea ch 212 (2022) 108144
2
A. Cia e a e al.
s ock p ices using di e en app oaches, and [12] show ha he o ecas
accu acy o asse e u ns a iance imp o es when jumps and cojumps
a e conside ed. Howe e , o he bes o ou knowledge, he ele ance
o cojumps in elec ici y ma ke s has no been analyzed.
This pape analyzes whe he jumps and cojumps add ele an in-
o ma ion o elec ici y p ice o ecas ing. To ha end, uni a ia e and
mul i a ia e amewo ks can be adop ed. The o me conside s one
ime se ies o all p ices, while he la e di ides he whole ime se ies
in o se e al se ies, one o each load pe iod. As an example, [13] ocus
on ARX models o compa e uni a ia e and mul i a ia e amewo ks
and conclude ha he e is a sligh gain in o ecas accu acy using he
la e . We use Ge man–Aus ian elec ici y day-ahead auc ion p ices
in a mul i a ia e amewo k, so 24 p ice se ies a e cons uc ed, one
o each hou . Following [14], we ans o m p ices applying di e en
a iance s abilizing ans o ma ions, which a e in ended o smoo h
se ies and imp o e o ecas s. We speci y h ee ARX models: Fi s , he
ARX model (which includes no jumps o cojumps); second, he ARX-J
model (which includes jumps); and hi d, he ARX-J-CJ model (which
includes jumps and cojumps).
Jumps and cojumps a e de ec ed in he esiduals o he ARX model
because hey a e seasonally adjus ed. This p e en s spikes ha a e
pu ely seasonal e ec s om being lagged as jumps o cojumps. As
a as we know, he e a e no o he analyses ha use his app oach
o de ec jumps and cojumps. Among he se e al widely-used jump
es s applied in he li e a u e, we use he one p oposed by [15], he e-
ina e LM, which is applicable o daily da a. Cojumps a e cons uc ed
ollowing [11].
We assess he ole o jumps and cojumps in elec ici y p ice o ecas -
ing. To ha end, jumps and cojumps a e embedded in ARX s a is ical
models. To he bes o ou knowledge, in elec ici y ma ke s jumps ha e
mos ly been used o o ecas elec ici y p ice ola ili y and he e is
li le esea ch in o hei use o p edic ing elec ici y p ices di ec ly.
Fu he mo e, cojumps ha e no been analyzed in elec ici y ma ke s o
da e.
ARX models a e usually o e -pa ame e ized, which makes hem
ha d o es ima e using OLS. Es ima ion me hods wi h a sh inkage
p ope y ha e he e o e been applied in he li e a u e. [16] p oposes
he lasso es ima ion me hod, which allows a iable selec ion, and [17]
p opose he elas ic ne es ima ion me hod, which imposes he so-called
lasso and idge penal ies on he OLS es ima ion o educe he numbe o
a iables. [18] apply di e en es ima ion me hods wi h he sh inkage
p ope y in elec ici y p ice o ecas ing, and conclude ha he elas ic
ne is he bes -pe o ming me hod.
The e o e, we es ima e he h ee ARX models using he elas ic
ne o he o iginal and ans o med p ice se ies using a olling win-
dow. Fo ecas ing is hen ca ied ou wi h he ollowing se en days
o each window being o ecas o all 24 h o each day. Finally, he
mean absolu e e o (MAE) and he oo mean squa ed e o (RMSE)
ou -o -sample c i e ia a e used o assess he o ecas ing pe o mance.
The di e ence in o ecas ing pe o mance be ween models in pai s is
compa ed using a mul i a ia e app oach ia he [19] es . In e es ing
esul s a e ob ained ega ding he ole o jumps and cojumps and
p ice ans o ma ions in elec ici y p ice o ecas ing depending on he
o ecas ho izon.
The day-ahead ma ke is he one wi h he highes liquidi y in he
Ge man–Aus ian zone o he Eu opean Powe Exchange (EPEX). The
pa icipa ing agen s need signpos s o decide hei bidding s a egies
op imally. Those signpos s a e he o ecas s made o he ollowing
days’ p ices. This highligh s he impo ance o o ecas ing as accu a ely
as possible. Mo eo e , in he Eu opean Ene gy Exchange (EEX) he e is
ading o base, peak and o -peak p oduc s o elec ici y wi h cash
se lemen in he Ge man–Aus ian zone known as Phelix (Physical
Elec ici y Index).2The unde lying p ices o hese u u e p oduc s a e
based on he Ge man–Aus ian EPEX day-ahead hou ly p ices.3The
2See h ps://www.eex.com/en/ma ke s/ ading- essou ces/indices.
3In EPEX, hou ly, hal -hou ly and qua e -hou ly p ices a e se in he
in aday con inuous ma ke .
Phelix base p ice p oduc is calcula ed as he mean o all hou ly p ices
in he deli e y pe iod. Fo weekly p oduc s i is calcula ed as he
mean o he 168 hou ly p ices. The e o e, we easonably concen a e
on o ecas ing he co esponding unde lying p ices o di e en ime
ho izons. No e ha o o ecas weekly p oduc s i is necessa y o
o ecas p ices o in e media e ho izons. I is in his con ex ha we
explo e he ole o jumps and cojumps in p ice o ecas ing.
In summa y, ou con ibu ion o he li e a u e is he ollowing:
•We build an ARX model ha embeds jumps o /and cojumps.
•We de ec jumps and cojumps in he seasonally adjus ed esiduals
o he ARX model.
•We measu e he accu acy o he models in o ecas ing p ices om
one day o one week ahead.
•We analyze he o ecas o Phelix base week u u es con ac s
including jumps and cojumps in di e en o ecas ing ho izons.
The es o he pape is o ganized as ollows. Sec ion 2explains he
me hodology used. Sec ion 3desc ibes he da a used in he analysis.
Sec ion 4shows he es ima ion and o ecas ing esul s o all models
and ans o ma ions. Sec ion 5summa izes and concludes.
2. Me hodology
P ice o ecas ing ollows se e al s eps. In he i s , he p ice se ies
is di ided in o 24 se ies, one o each hou , which a e ans o med
using se e al a iance s abilizing ans o ma ions (VST). These ans-
o ma ions make he ime se ies smoo he , hus imp o ing o ecas
pe o mance. In he second s ep, an ARX model is speci ied and es-
ima ed using he elas ic ne me hod. Nex , jumps a e de ec ed in
he esiduals o he es ima ed model by applying he LM es . Once
jumps a e de ec ed in each o he 24 ime se ies o he o iginal and
ans o med p ices, cojumps a e de ec ed as pe [11]. The ARX model
is hen expanded including only jumps and bo h jumps and cojumps,
esul ing in he ARX-J and he ARC-J-CJ models, espec i ely, also
es ima ed ia he elas ic ne . Finally, p ice o ecas ing accu acy is
assessed in each model and o o iginal and ans o med p ices using
RMSE and MAE c i e ia. The o ecas ing pe o mance o he di e en
pai s o models is compa ed using he mul i a ia e app oach in he [19]
es , he eina e DM.
The subsec ions below p o ide a de ailed explana ion o each s ep.
2.1. Va iance s abilizing ans o ma ions
Based on [14], di e en VSTs a e applied. All he ans o ma ions
used a e applicable wi h nega i e p ices. The objec i e o hese ans-
o ma ions is o ob ain ans o med p ice se ies which a e easie o
o ecas and hen o apply he in e se o he ans o ma ion o e-
co e he o ecas p ices. In o al 6 di e en ans o ma ions a e used:
3𝜎, logis ic, a ea hype bolic sine, mi o -loga i hmic, and p obabili y
in eg al ans o ma ion using bo h no mal and S uden - cumula i e
dis ibu ions. In he i s ou ans o ma ions s anda dized p ices a e
ob ained be o e he ans o ma ion is applied. In addi ion o he com-
mon s anda diza ion ha uses he s anda d de ia ion o p ices, a second
s anda diza ion ha uses he median absolu e de ia ion o p ices and
is mo e obus o ou lie s, is also applied.4Once p ices a e o ecas , he
s anda diza ion p ocess is undone.
4Fo he es o he pape we use subsc ip s 1 and 2 a e he name o
he VST o indica e ha p ices ha e been s anda dized using he s anda d
de ia ion and he median absolu e de ia ion, espec i ely. Fo he sake o
simpli ying he no a ion, we deno e as 𝑝bo h he o iginal and s anda dized
p ices.
Elec ic Powe Sys ems Resea ch 212 (2022) 108144
3
A. Cia e a e al.
The 3𝜎 ans o ma ion smoo hs he se ies, hus dec easing he e ec
o ou lie s in p ice o ecas ing. Following [14], he ans o ma ion is
made as ollows:
𝑦𝑑,ℎ =⎧
⎪
⎨
⎪
⎩
3sign(𝑝𝑑,ℎ)i |𝑝𝑑,ℎ|>3
𝑝𝑑,ℎ i |𝑝𝑑,ℎ|≤3
whe e 𝑝𝑑,ℎ deno es he p ice a day 𝑑o he ime se ies co esponding
o hou ℎ. By cons uc ion, he 3𝜎 ans o ma ion does no ha e an
in e se.
The logis ic ans o ma ion has o en been applied in da a analy ics,
bu as a as we know, i has only been applied as a VST in elec ici y
p ice o ecas ing by [14]. The ans o ma ion is:
𝑦𝑑,ℎ =(1 + 𝑒−𝑝𝑑,ℎ )−1
A e o ecas ing, he in e se ans o ma ion is used o eco e he
o ecas o he o iginal p ice as:
𝑝𝑑,ℎ = log (𝑦𝑑,ℎ
1 − 𝑦𝑑,ℎ )
The a ea hype bolic sine (asinh) has been used as a VST in elec-
ici y da a when modeling nega i e p ices (see [13,20], and [14]). I
p ese es he beha io o he loga i hmic ans o ma ion o posi i e
p ices bu is also de ined o nega i e p ices. The ans o med p ices
a e calcula ed as:
𝑦𝑑,ℎ =asinh (𝑝𝑑,ℎ)=𝑙𝑜𝑔 (𝑝𝑑,ℎ +√𝑝2
𝑑,ℎ + 1)
wi h he co esponding in e se ans o ma ion:
𝑝𝑑,ℎ =sinh (𝑦𝑑,ℎ)
The mi o -loga i hmic (mlog) ans o ma ion is a gene aliza ion o
he loga i hmic ans o ma ion o make i applicable o nega i e p ices
(see [14]). The ans o ma ion is cons uc ed as:
𝑦𝑑,ℎ =sign (𝑝𝑑,ℎ)[log (|𝑝𝑑,ℎ|+1
𝑐)+ log (𝑐)]
The mlog ans o ma ion depends on he cons an 𝑐, which is se o
𝑐=1
3 ollowing [14]. Consequen ly, he in e se ans o ma ion is:
𝑝𝑑,ℎ =sign (𝑦𝑑,ℎ)[𝑒|𝑦𝑑,ℎ|−log (𝑐)−1
𝑐]
The las ans o ma ion conside ed is based on he so-called p ob-
abili y in eg al ans o ma ion (PIT), cons uc ed using he empi ical
cumula i e dis ibu ion as an app oxima ion o he unknown ue
dis ibu ion o he ime se ies (see [14]):
𝑦𝑑,ℎ =𝛷−1(
𝐹𝑝(𝑝𝑑,ℎ))
whe e 𝛷−1 is he in e se cumula i e dis ibu ion and
𝐹𝑝is he empi ical
cumula i e dis ibu ion o he p ice se ies 𝑝. Bo h he no mal (N-PIT)
and he S uden - wi h eigh deg ees o eedom (T-PIT) cumula i e
dis ibu ions a e conside ed. The in e se o he ans o ma ion is:
𝑝𝑑,ℎ =
𝐹𝑝
−1(𝛷(𝑦𝑑,ℎ))
2.2. Models
Th ee di e en ARX- ype models a e es ima ed. The i s is he ARX
model, based on he ARX model p oposed by [13]:
𝑝𝑑,ℎ =𝛽
⏟⏟⏟
Cons an
+
24
∑
ℎ=1
7
∑
𝑖=1
𝛽𝑖,ℎ𝑝𝑑−𝑖,ℎ
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟
Au o eg essi e e ec s
+
6
∑
𝑗=1
𝛾0,𝑗W𝑗
𝑑
⏟⏞⏞⏞⏟⏞⏞⏞⏟
Day-o - he-week e ec s
+𝜖𝑑,ℎ (1)
whe e 𝑝 e e s o he o iginal p ice o he ans o med p ice (𝑦), W𝑗
𝑑is
a dummy a iable o day 𝑗o he week, and 𝜖𝑑,ℎ is he e o e m wi h
mean 0 by cons uc ion. The second e m accoun s o up o se en h
o de au o eg essi e and c oss-pe iod e ec s (e ec s o each hou om
up o 7 days ago). The hi d e m accoun s o seasonali y.
Jumps a e included in model (1), esul ing in he ARX-J model. The
sign o he jumps migh di e depending on he hou o he day, so his
model conside s bo h posi i e and nega i e jumps:
𝑝𝑑,ℎ =𝛽
⏟⏟⏟
Cons an
+
24
∑
ℎ=1
7
∑
𝑖=1
𝛽𝑖,ℎ𝑝𝑑−𝑖,ℎ
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟
Au o eg essi e e ec s
+
6
∑
𝑗=1
𝛾0,𝑗W𝑗
𝑑
⏟⏞⏞⏞⏟⏞⏞⏞⏟
Day-o - he-week e ec s
+
7
∑
𝑖=1
𝜃𝑝
𝑖𝑃𝐽𝑑−𝑖,ℎ
⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟
Posi i e jumps
+
7
∑
𝑖=1
𝜃𝑛
𝑖𝑁𝐽𝑑−𝑖,ℎ
⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟
Nega i e jumps
+𝜖𝑑,ℎ (2)
whe e 𝑃𝐽𝑑,ℎ is a dummy a iable ha akes a alue o one i he e
is a posi i e jump on day 𝑑in ime se ies ℎ, and ze o o he wise.
Analogously, 𝑁𝐽𝑑,ℎ is a dummy a iable ha akes a alue o one
i he e is a nega i e jump on day 𝑑a hou ℎ, and ze o o he wise.
The ARX-J model is expec ed o cap u e he beha io o p ices mo e
accu a ely a he ails o he dis ibu ion.
The hi d model p oposed, ARX-J-CJ, includes jumps and cojumps:
𝑝𝑑,ℎ =𝛽
⏟⏟⏟
Cons an
+
24
∑
ℎ=1
7
∑
𝑖=1
𝛽𝑖,ℎ𝑝𝑑−𝑖,ℎ
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟
Au o eg essi e e ec s
+
6
∑
𝑗=1
𝛾0,𝑗W𝑗
𝑑
⏟⏞⏞⏞⏟⏞⏞⏞⏟
Day-o - he-week e ec s
+
7
∑
𝑖=1
𝜃𝑝
𝑖𝑃𝐽𝑑−𝑖,ℎ
⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟
Posi i e jumps
+
7
∑
𝑖=1
𝜃𝑛
𝑖𝑁𝐽𝑑−𝑖,ℎ
⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟
Nega i e jumps
+𝜃𝑐𝐶𝐽𝑑−1
⏟⏞⏟⏞⏟
Cojumps
+𝜖𝑑,ℎ (3)
whe e 𝐶𝐽𝑑is a dummy a iable ha akes a alue o one i a
cojump is de ec ed on day 𝑑and 0 o he wise. No e ha i is equal o
all ime se ies. The ARX-J-CJ model accoun s o co ela ion be ween
jumps by conside ing cojumps, which a e jumps ha occu on he same
day ac oss di e en hou s. The ARX-J-CJ model no only accoun s o
co ela ion by c oss-pe iod e ec s; i also akes in o accoun co ela ion
in he ails, h ough he cojump a iable.
Finally, we also conside he ollowing nai e model as a benchma k:
𝑝𝑑,ℎ =𝑝𝑑−1,ℎ +𝜖𝑑,ℎ, whe e p ice a a gi en hou is de e mined by he
p ice a he same hou o he p e ious day.
I should be no ed ha o he ac o s such as load and wea he o e-
cas s migh also a ec p ices. Howe e , hese ac o s a e no included
in he models because he ocus o he pape is o o ecas p ices up o
se en days ahead and da a a e a ailable day-ahead, so hey canno be
used o p edic p ices beyond ho izon one.
2.3. Jump and cojump de ec ion
The LM jump es is applied o he esiduals o he ARX model
es ima ed (Eq. (1)), hus a oiding spikes ha could be explained by
seasonal e ec s. The es compa es he size o a s anda dized obse a-
ion o a h eshold so ha i can be assessed whe he a signi ican jump
has occu ed o no .
Fi s , a window size mus be selec ed. Acco ding o LM, he op imal
choice o he window size is 𝐾= 20.5Thus, he local a ia ion a day
𝑑and o hou ℎis es ima ed as:
𝜎𝑑,ℎ2=1
𝐾− 2
𝑑−1
∑
𝑗=𝑑−𝐾+2 |𝜖𝑗,ℎ||𝜖𝑗−1,ℎ|,
whe e 𝜖𝑗,ℎ is he esidual om he es ima ed ARX model o day 𝑗and
hou ℎ.
5Each hou o he day is analyzed sepa a ely and he da a equency is
daily, so 𝐾=⌈√365⌉.
Elec ic Powe Sys ems Resea ch 212 (2022) 108144
4
A. Cia e a e al.
The s anda dized esidual is 𝑧𝑑,ℎ =𝜖𝑑,ℎ
𝜎𝑑,ℎ . The asymp o ic dis ibu ion
o he maximums o he es s a is ic in he absence o jumps con e ges
o a Gumbel a iable.6The LM es iden i ies signi ican jumps bu does
no indica e hei sign. Hence, he sign o he co esponding p ice is
checked o de e mine he jump sign.
The jump de ec ion p ocedu e is usually applied in a single i e -
a ion. Howe e in his pape an i e a i e jump de ec ion p ocedu e
is ollowed, because i jumps a e close oge he he de ec ion o he
second jump may be a ec ed. Jumps de ec ed in he i e a ion a e
he e o e se o he mean o he p e ious 𝐾obse a ions and he jump
es is e un un il no mo e jumps a e de ec ed o a maximum o i e
i e a ions is eached.
Cojumps o e di e en hou s o he same day a e de ec ed ollowing
he app oach p oposed by [11].7Speci ically, day 𝑑is classi ied as a
cojump day, i.e. 𝐶𝐽𝑑= 1, i a jump is de ec ed in a leas wo hou s o
day 𝑑, i.e. i
24
∑
ℎ=1
𝐽𝑑,ℎ ≥2,
whe e 𝐽𝑑,ℎ = 1 i a jump is de ec ed on day 𝑑and a hou ℎ, and 0
o he wise.
2.4. Es ima ion
Models (1),(2) and (3) a e es ima ed using he elas ic ne me hod
in oduced by [17], hus sol ing he poo es ima ion o OLS when he
numbe o pa ame e s o be es ima ed is la ge.
Es ima ion is ca ied ou using a olling window o size 𝐷. The
size o he window has o be la ge enough o p ope ly es ima e he
model bu no oo la ge, as he e ec o he a iables migh change o e
ime. The window is hen mo ed one day o wa d and he es ima ion
p ocedu e is epea ed. In o al, he e a e 𝑁di e en windows o size
𝐷.
The elas ic ne es ima o is ob ained by sol ing he ollowing op i-
miza ion p oblem (see [17]):
𝜷ℎ=a gmin
𝜷∈R𝐿
⎡⎢⎢⎢⎢⎢⎢⎣
𝐷
∑
𝑑=1 (𝑝𝑑,ℎ −
𝑿𝑑,ℎ𝜷)2
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
OLS es ima ion e m
+𝜆(1 − 𝛼
2
𝐿
∑
𝑖=1
𝛽2
𝑖,ℎ +𝛼
𝐿
∑
𝑖=1 |𝛽𝑖,ℎ|)
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Penal y e m
⎤⎥⎥⎥⎥⎥⎥⎦
,
whe e 𝑝𝑑,ℎ and
𝑿𝑑,ℎ a e he scaled p ice and he scaled eg ession
ma ix on day 𝑑and a hou ℎ, espec i ely, so ha 𝑝𝑑,ℎ and each
column o
𝑿𝑑,ℎ ha e ze o mean and s anda d de ia ion one. 𝐿is he
numbe o pa ame e s o be es ima ed and 𝜆and 𝛼a e he uning
pa ame e s, which ake alues be ween ze o and one, and cha ac e ize
he penal y e m o including a iables.
When 𝛼= 1 he elas ic ne es ima ion me hod is iden ical o he
lasso penal y p oposed by [16], while i 𝛼= 0 he elas ic ne esul s in
he idge penal y i s in oduced by [21]. Following [18], 𝛼is se a
0.5.8
The op imum alue o 𝜆is selec ed by 10- old block c oss- alida ion
(see [22]). [23] show ha he numbe o obse a ions, he num-
be o pa ame e s, he a iance and he co ela ion a e aken in o
conside a ion when selec ing he uning pa ame e .
6Using a 10% signi icance le el, he h eshold o he es s a is ic is
− log(− log(0.9)) = 2.25. See [15] o mo e de ail.
7The au ho s p opose wo di e en app oaches o de ec cojumps. How-
e e , one o hem conside s in aday p ices and canno he e o e be applied
o daily obse a ions. Thus, only one me hod is included in his pape .
8We also use lasso, 𝛼= 1, bu he o ecas esul s do no change
signi ican ly. Resul s a e a ailable upon eques .
Once he pa ame e s a e es ima ed by sol ing he op imiza ion
p oblem, he unscaled elas ic ne es ima ions
𝜷ℎa e ob ained by escal-
ing
𝜷ℎ.
Finally, we compa e he goodness o i o he models using he
adjus ed R-squa ed o each olling window in he es ima ion o he
h ee models (using o iginal and ans o med p ices).
2.5. Fo ecas
Once models (1),(2) and (3) a e es ima ed, p ices o each hou a e
p edic ed o he ollowing 7 days. The MAE and RMSE ou -o -sample
c i e ia a e used o assess o ecas ing pe o mance o e he 𝑁 olling
windows and he 7 ho izons. Bo h c i e ia a e widely used in he li e a-
u e on o ecas ing in elec ici y ma ke s, o ins ance in [13,14,18,24],
and [25]. By cons uc ion, he MAE c i e ion is op imal o median
o ecas s while he RMSE is op imal o mean o ecas s.
The MAE c i e ion o ho izon 𝑘and hou ℎis calcula ed as ollows9:
𝑀𝐴𝐸ℎ,𝑘 =1
𝑁
𝑁
∑
𝑑=1 |𝑝𝑑,ℎ,𝑘 −𝑝𝑑,ℎ,𝑘|,
whe e 𝑝𝑑,ℎ,𝑘 and 𝑝𝑑,ℎ,𝑘 a e he obse ed and p edic ed p ice on day 𝑑, a
hou ℎand ho izon 𝑘, espec i ely. The mean e o ac oss all he hou s
o he day is calcula ed as in [14]:
𝑀𝐴𝐸𝑘=1
24𝑁
24
∑
ℎ=1
𝑁
∑
𝑑=1 |𝑝𝑑,ℎ,𝑘 −𝑝𝑑,ℎ,𝑘|(4)
Analogously, he RMSE measu e o he ho izon 𝑘and hou ℎis
calcula ed using he squa e e o ins ead o he absolu e e o as10:
𝑅𝑀𝑆𝐸ℎ,𝑘 =√
√
√
√1
𝑁
𝑁
∑
𝑑=1
(𝑝𝑑,ℎ,𝑘 −𝑝𝑑,ℎ,𝑘)2,
and he co esponding e o ac oss all he hou s o he day (see [14])
is:
𝑅𝑀𝑆𝐸𝑘=√
√
√
√1
24𝑁
24
∑
ℎ=1
𝑁
∑
𝑑=1
(𝑝𝑑,ℎ,𝑘 −𝑝𝑑,ℎ,𝑘)2(5)
To de e mine whe he he di e ences in o ecas ing pe o mance
o he models is signi ican he mul i a ia e e sion o he DM es is
applied, using he absolu e e o s o he MAE c i e ion and he squa ed
e o s o he RMSE c i e ion as loss unc ions.
3. Da a desc ip ion
The da a used in his pape a e day-ahead p ices om he o me
Ge man–Aus ian elec ici y ma ke .11 This is a ully in eg a ed ma ke
which se s a single p ice o bo h coun ies. On day 𝑑−1 he ma ke se s
p ices o he 24 h o day 𝑑acco ding o he ollowing mechanism: Fi s ,
ma ke agen s submi elec ici y sale and pu chase bids up o 12 pm on
day 𝑑− 1. Then he sys em agg ega es he bids o demand and supply
unc ions, and inally he in e sec ion be ween he supply and demand
cu es de e mines he quan i y aded and he ma ke p ice o each
hou o day 𝑑. This was he ma ke wi h he highes le el o liquidi y in
he EPEX powe exchange ma ke . The da a un om 1s Janua y 2014
o 30 h Sep embe 2018, which was he las day on which he Ge man–
Aus ian day-ahead ma ke ope a ed.12 In o al, he e a e 41,616 hou ly
p ices and 1734 days in he sample pe iod.
9See o example [26].
10 See o example [26].
11 Da a a ailable on he ENTSOE anspa ency pla o m.
12 A e his da e, he Ge man and Aus ian ene gy egula o s ag eed o spli
hei combined day-ahead ma ke zone. This came as a esul o equen
ansmission conges ion be ween he wo g ids and he esul ing cos ly e-
dispa ching o deal wi h i . The Luxembou g elec ici y ma ke subsequen ly
joined he Ge man ma ke o o m a single zone.
Elec ic Powe Sys ems Resea ch 212 (2022) 108144
5
A. Cia e a e al.
Table 1
Desc ip i e s a is ics o o iginal and ans o med p ices.
T ans o ma ion Mean Median Minimum Maximum S d. De . Skewness Ex. Ku osis
O iginal 33.50 32.34 −130.09 163.52 15.04 −0.12 6.42
3𝜎10.10 0.00 −3.00 3.00 1.17 0.15 0.30
Logis ic10.52 0.50 0.00 1.00 0.23 0.05 −0.70
Asinh10.07 −0.00 −3.35 3.13 0.89 0.05 −0.37
Mlog10.02 0.00 −1.75 1.57 0.32 0.06 0.74
N-PIT 0.00 0.00 −4.06 4.06 1.00 0.00 −0.00
T-PIT 0.00 0.00 −7.89 7.89 1.15 0.01 1.39
Desc ip i e s a is ics o o iginal and ans o med p ice se ies using he s anda d de ia ion o s anda diza ion. S d. De . and Ex. Ku osis s and
o S anda d De ia ion and Excess Ku osis, espec i ely.
Fig. 1. Numbe o cojumps de ec ed in each olling window.
Table 1 epo s he main desc ip i e s a is ics o he o iginal and
ans o med p ice se ies.13 As expec ed, he a iabili y o p ices de-
c eases signi ican ly when ans o med da a a e used. Speci ically, o ig-
inal p ices ange om e−130.09 o e163.52/MWh wi h a s anda d
de ia ion o e15.04/MWh. By con as , he la ges sp ead in ans-
o med p ices is 15.78 and occu s o he T-PIT ans o ma ion. The
o iginal p ices show nega i e skewness and excess ku osis. The hea y
ails o he dis ibu ion migh indica e he p esence o jumps. The
dis ibu ion o he ans o med p ices is close- o-no mal excep o he
N-PIT, whose dis ibu ion is no mal.
The o iginal and ans o med p ice se ies a e di ided in o 24 ime
se ies, one o each hou o he day. All he se ies a e s a iona y
acco ding o he ADF uni oo es . To es ima e models (1),(2), and
(3), a olling window o size 𝐷= 730 ( wo yea s) is used. The ini ial
olling window s a s on 1s Janua y 2014 and ends on 31s Decembe
2015. 7 ho izons a e o ecas in each window, and 𝑁= 998 di e en
olling windows a e conside ed. No e ha he i s 730 obse a ions o
he sample a e used o o ecas he i s p ice.
Acco ding o he LM es , he e a e signi ican jumps in bo h o iginal
and ans o med p ices in mos o he 24 h and each olling window.
Fig. 1 shows he numbe o cojumps de ec ed in he esiduals o he
ARX model in each olling window o he o iginal and ans o med
p ices.14 As expec ed, he la ges numbe o cojumps is de ec ed in
13 Fo 3𝜎, logis ic, asinh, and mlog ans o ma ions he common s anda d-
iza ion has been applied o p ices. S a is ics o s anda dized p ices wi h he
median absolu e de ia ion do no change signi ican ly and a e no shown. They
a e a ailable upon eques .
14 These esul s a e o s anda dized p ices using he s anda d de ia ion. Re-
sul s o s anda dized p ices using he median absolu e de ia ion a e a ailable
upon eques .
he o iginal p ice se ies. Obse e ha he numbe o cojumps de ec ed
ises o almos 70 a he end o he pe iod in which he ma ke was in
place. Du ing pe iods o highly a iable enewable gene a ion, nega i e
elec ici y p ice spikes occu because p ices d op sha ply o eed ha
enewable gene a ion in o he g id. In pe iods o low enewable gen-
e a ion when he mal uni s ill he demand gap, p ices may spike as
a esponse o unexpec ed ossil uel luc ua ions, as happened in 2018.
The spikes obse ed may happen in adjacen hou s, so cojumps a e also
iden i ied. This seems o be pa icula ly so in ea ly 2017 and la e 2018.
Fo he ans o med se ies cojumps a e de ec ed in each olling
window, al hough hey a e ewe in numbe . In gene al, he T-PIT
is he ans o med p ice se ies wi h he la ges numbe o cojumps,
especially du ing he i s hal o he sample. This esul is expec ed,
as he p obabili y dis ibu ion o T-PIT ans o med p ices is hea y-
ailed. By con as , he numbe o cojumps de ec ed is lowe in he
N-PIT ans o med p ice se ies.
4. Es ima ion and o ecas esul s
4.1. Es ima ion esul s
Models (1),(2), and (3) a e es ima ed and he co esponding ad-
jus ed R-squa ed is calcula ed o each olling window. Table 2 epo s
he mean alues o he adjus ed R-squa ed o all hou s and each model,
and o o iginal and ans o med p ices. The compa ison wi hin each
ans o ma ion shows ha he bes model in e ms o goodness o i is
he ARX.
4.2. Fo ecas esul s
Models (1),(2), and (3) a e used o o ecas p ices o 1 o 7 days
ahead o each olling window.
Elec ic Powe Sys ems Resea ch 212 (2022) 108144
6
A. Cia e a e al.
Table 2
Mean adjus ed-R2 o all hou s.
O iginal 3𝜎13𝜎2Logis ic1Logis ic2Asinh1Asinh2Mlog1Mlog2N-PIT T-PIT
ARX 0.655 0.683 0.675 0.686 0.688 0.688 0.688 0.683 0.679 0.686 0.680
ARX-J 0.653 0.681 0.674 0.684 0.687 0.687 0.687 0.682 0.678 0.685 0.679
ARX-J-CJ 0.653 0.681 0.674 0.684 0.687 0.687 0.687 0.682 0.678 0.685 0.679
Mean adjus ed R-squa ed c i e ion o all hou s, models, and p ice ans o ma ions. Subsc ip s 1 and 2 indica e ha p ices a e
s anda dized using he s anda d de ia ion and he median absolu e de ia ion, espec i ely. A hea map is used o indica e highe
(g een) and lowe ( ed) alues wi hin each ans o ma ion.
Table 3
Mean MAE o all hou s.
Model T ans . H1 H2 H3 H4 H5 H6 H7
Nai e 8.396 10.754 11.297 11.530 11.603 10.706 9.389
ARX
O iginal 5.598 7.327 7.845 8.116 8.311 8.475 8.563
3𝜎15.584 7.153 7.639 7.908 8.119 8.290 8.382
3𝜎25.530 7.167 7.675 7.951 8.156 8.315 8.401
Logis ic15.722 7.270 7.734 7.993 8.180 8.343 8.444
Logis ic25.544 7.178 7.672 7.941 8.144 8.310 8.405
Asinh15.558 7.229 7.718 7.989 8.187 8.351 8.445
Asinh26.254 7.310 7.734 7.991 8.229 8.392 8.483
Mlog15.500 7.190 7.703 7.976 8.181 8.346 8.444
Mlog25.507 7.198 7.705 7.982 8.188 8.351 8.447
N-PIT 5.529 7.134 7.638 7.919 8.128 8.308 8.402
T-PIT 5.523 7.125 7.627 7.910 8.129 8.304 8.404
ARX-J
O iginal 5.623 7.316 7.848 8.114 8.274 8.428 8.520
3𝜎15.615 7.174 7.658 7.925 8.123 8.281 8.378
3𝜎25.553 7.167 7.671 7.943 8.115 8.261 8.361
Logis ic15.746 7.324 7.787 8.042 8.219 8.374 8.471
Logis ic25.579 7.207 7.701 7.967 8.159 8.314 8.409
Asinh15.590 7.275 7.770 8.025 8.214 8.366 8.460
Asinh26.285 7.349 7.785 8.038 8.259 8.411 8.496
Mlog15.533 7.226 7.728 7.982 8.177 8.331 8.435
Mlog25.532 7.209 7.716 7.973 8.169 8.319 8.424
N-PIT 5.550 7.171 7.672 7.959 8.164 8.338 8.445
T-PIT 5.548 7.137 7.631 7.911 8.120 8.305 8.413
ARX-J-CJ
O iginal 5.626 7.316 7.847 8.116 8.281 8.436 8.536
3𝜎15.614 7.178 7.666 7.931 8.129 8.288 8.385
3𝜎25.553 7.169 7.673 7.946 8.122 8.268 8.368
Logis ic15.747 7.326 7.786 8.039 8.218 8.373 8.472
Logis ic25.579 7.211 7.700 7.969 8.164 8.314 8.410
Asinh15.593 7.277 7.770 8.027 8.216 8.370 8.466
Asinh26.281 7.354 7.785 8.040 8.257 8.411 8.497
Mlog15.534 7.218 7.727 7.986 8.176 8.336 8.442
Mlog25.539 7.217 7.725 7.989 8.180 8.331 8.438
N-PIT 5.555 7.175 7.674 7.964 8.166 8.341 8.451
T-PIT 5.547 7.138 7.624 7.904 8.117 8.296 8.405
Mean MAE c i e ion (Eq. (4)) o all hou s pe ho izon (H1 o H7), model, and p ice
ans o ma ion. Subsc ip s 1 and 2 indica e ha p ices a e s anda dized using he s anda d
de ia ion and he median absolu e de ia ion, espec i ely. A hea map is used o indica e lowe
(g een) and highe ( ed) o ecas e o s wi hin each ho izon.
Model selec ion in ol es wo c i e ia. Fi s , we measu e he o e-
cas ing pe o mance o he models by so ing hem acco ding o he
alue o he MAE and RMSE c i e ia and hen we choose he ones wi h
he lowes alues o each c i e ion and o ecas ho izon. Second, we
un he mul i a ia e app oach o he DM es o de e mine whe he
o ecas s o each pai o models a e signi ican ly be e in one o hem.
Tables 3 and 4show he o ecas ing e o s o each model o o iginal
and ans o med p ices, o he nai e model and o he se en ho izons
(H1 o H7) using he MAE and RMSE c i e ia ac oss all hou s o he
day, i.e. Eqs. (4) and (5), espec i ely.15
Rega dless o he o ecas ing ho izon, esul s show ha he nai e
model p o ides he la ges e o s, which means ha a leas e ms
accoun ing o co ela ion be ween p ices a a gi en hou and day
and hei lags should be conside ed as explana o y a iables. Fo he
15 The esul s o he DM es o he MAE c i e ion a e epo ed in Appendix
and hose using he RMSE c i e ion a e a ailable om he au ho s upon
eques .
es o he models, he VST esul s show ha none o he models
a e selec ed wi h he o iginal p ices, so i is impo an o smoo h
p ice se ies so as o ob ain mo e accu a e o ecas s. Howe e , no all
he ans o ma ions a e equally good: logis ic1, logis ic2, and asinh1
ans o ma ions do no p o ide be e o ecas s as hey a e no selec ed
o he bes models. By con as , 3𝜎2, mlog1, mlog2, and T-PIT a e, in
gene al, he ans o ma ions wi h he bes o ecas pe o mances. These
a e he ans o ma ions o which p ice dis ibu ion has excess ku osis,
so i is impo an o accu a ely cap u e he beha io a he ails because
jumps a e obse a ions ha all a he ails o he dis ibu ion.
Ma ke pa icipan s ake decisions ha depend on he ime ho i-
zon unde conside a ion. Fo ecas ing is ele an in day- o-day ma ke
ope a ions o EPEX and isk managemen in he EEX u u es ma ke s
o di e en deli e y pe iods. We discuss esul s anging om he
closes - o-deli e y one day-ahead o ecas o a one week-ahead p ice
o ecas .
•Ho izon 1: Day-ahead o ecas ing is impo an o elec ici y ad-
ing and plan ope a ion scheduling decisions. The ARX model
Elec ic Powe Sys ems Resea ch 212 (2022) 108144
7
A. Cia e a e al.
Table 4
Mean RMSE o all hou s.
Model T ans . H1 H2 H3 H4 H5 H6 H7
Nai e 13.351 16.535 17.444 17.803 17.905 16.727 15.195
ARX
O iginal 8.959 11.235 11.859 12.163 12.397 12.558 12.636
3𝜎19.397 11.225 11.801 12.133 12.376 12.530 12.613
3𝜎29.121 11.141 11.776 12.113 12.360 12.513 12.592
Logis ic19.506 11.399 11.981 12.313 12.537 12.688 12.773
Logis ic29.148 11.234 11.840 12.178 12.423 12.574 12.660
Asinh19.030 11.245 11.850 12.195 12.446 12.601 12.695
Asinh29.623 11.252 11.769 12.084 12.349 12.527 12.603
Mlog18.891 11.150 11.771 12.106 12.359 12.514 12.613
Mlog28.894 11.143 11.755 12.094 12.342 12.505 12.599
N-PIT 9.081 11.211 11.830 12.185 12.435 12.598 12.688
T-PIT 8.981 11.159 11.786 12.133 12.388 12.543 12.642
ARX-J
O iginal 9.007 11.277 11.900 12.187 12.380 12.541 12.631
3𝜎19.445 11.264 11.840 12.159 12.386 12.525 12.614
3𝜎29.161 11.166 11.785 12.114 12.324 12.461 12.557
Logis ic19.550 11.490 12.057 12.373 12.580 12.719 12.806
Logis ic29.205 11.289 11.886 12.216 12.442 12.583 12.673
Asinh19.083 11.318 11.922 12.245 12.481 12.621 12.715
Asinh29.676 11.316 11.824 12.131 12.379 12.544 12.612
Mlog18.941 11.212 11.813 12.123 12.355 12.506 12.617
Mlog28.941 11.191 11.796 12.101 12.338 12.486 12.594
N-PIT 9.116 11.264 11.879 12.226 12.470 12.625 12.729
T-PIT 9.032 11.195 11.801 12.133 12.383 12.540 12.659
ARX-J-CJ
O iginal 9.022 11.273 11.904 12.196 12.390 12.558 12.651
3𝜎19.448 11.269 11.846 12.164 12.391 12.534 12.619
3𝜎29.163 11.165 11.789 12.120 12.329 12.467 12.568
Logis ic19.569 11.499 12.057 12.373 12.582 12.720 12.813
Logis ic29.207 11.292 11.888 12.221 12.449 12.584 12.670
Asinh19.085 11.322 11.924 12.251 12.488 12.626 12.725
Asinh29.655 11.324 11.826 12.134 12.376 12.542 12.617
Mlog18.940 11.207 11.814 12.126 12.362 12.508 12.624
Mlog28.942 11.198 11.810 12.122 12.357 12.503 12.612
N-PIT 9.126 11.273 11.886 12.237 12.477 12.632 12.740
T-PIT 9.039 11.200 11.799 12.133 12.378 12.538 12.650
Mean RMSE c i e ion (Eq. (5)) o all hou s pe ho izon (H1 o H7), model, and p ice
ans o ma ion. Subsc ip s 1 and 2 indica e ha p ices a e s anda dized using he s anda d
de ia ion and he median absolu e de ia ion, espec i ely. A hea map is used o indica e lowe
(g een) and highe ( ed) o ecas e o s wi hin each ho izon.
wi h mlog ans o med p ices ou pe o ms he es . Fu he mo e,
models ha include jumps and/o cojumps a e no selec ed. Thus,
he inclusion o jumps as explana o y ac o s does no imp o e he
o ecas .
•Ho izon 2: The esul s a e no ha conclusi e. Unde he MAE
c i e ion, ARX, ARX-J, and ARX-J-CJ a e candida e models o
selec ion using T-PIT ans o ma ion. Howe e , he DM es inds
no signi ican di e ences be ween hem o wi h espec o mlog
and N-PIT ans o ma ions. Fo he RMSE c i e ion, in gene al he
ARX model gi es he lowes e o and he DM es ne e selec s
models wi h jumps and cojumps. These esul s a e in line wi h
hose o ho izon 1, so in o ma ion on jumps does no help o
o ecas p ices wo days ahead.
•Ho izon 3: The esul s di e depending on he c i e ion. Unde
MAE he bes pe o ming model is he ARX-J-CJ wi h T-PIT
ans o med p ices, ollowed by he ARX model wi h he same
ans o ma ion. Howe e , he DM es inds no signi ican di e -
ences be ween hem. By con as , RMSE selec s he ARX model
o he mlog2 ans o med p ices. DM esul s show ha o he
ARX model he di e ence in e o measu es be ween he mlog2
and he asinh2 ans o ma ions is no signi ican . These esul s a e
quali a i ely simila o p e ious ho izons, so he e is no clea gain
om including jumps o cojumps.
•Ho izon 4: Unde he MAE c i e ion, ARX-J-CJ wi h he T-PIT
ans o ma ion is he bes model, bu he e o di e ence wi h
espec o he ARX model o 3𝜎1and T-PIT ans o ma ions is
no signi ican acco ding o he DM es esul s. The esul s o he
RMSE c i e ion di e because he model wi h he smalles e o is
he ARX wi h he asinh2 ans o med p ices. Howe e , DM esul s
show no signi ican di e ences be ween he o ecas ing accu acy
in his case and ha o he h ee models o 3𝜎2, mlog1and
mlog2 ans o ma ions. The e o e, esul s a e quali a i ely simila
o hose o p e ious ho izons.
•Ho izon 5: MAE and RMSE c i e ia selec he ARX-J model wi h
he 3𝜎2 ans o ma ion as he bes model. Howe e , he DM
es using MAE shows ha i does no ou pe o m he ARX-J-CJ
model wi h he T-PIT ans o ma ion. Mo eo e , he same es
using RMSE shows no clea e idence o supe io i y o any o
he h ee models. Fi e days ahead he e is some e idence ha
models ha inco po a e jumps and/o cojumps as explana o y
a iables p o ide be e p ice o ecas s. Fac o s ha con ibu e
o he occu ence o p ice shocks a e expec ed o become mo e
likely as he o ecas ho izon becomes longe .
•Ho izon 6: MAE and RMSE c i e ia selec he ARX-J model wi h
he 3𝜎2 ans o ma ion as he bes model. This esul is con i med
by he DM es using MAE, which shows his model o be signi -
ican ly supe io o he ARX-J-CJ model. Howe e , he DM es
using RMSE shows ha i does no ou pe o m he ARX-J-CJ
model wi h he same ans o ma ion. No is i supe io o any o
he models wi h he mlog ans o ma ion. In o ma ion on jumps
and/o cojumps is he e o e mo e signi ican in o ecas ing. These
esul s ein o ce he indings o ho izon 5 and show ha hedging
is impo an in longe ho izons oo.
•Ho izon 7: MAE and RMSE c i e ia selec he ARX-J model wi h
he 3𝜎2 ans o ma ion as he bes model. This esul is con i med
by DM es ing using ei he c i e ion, which shows i o be signi -
ican ly supe io o he ARX and ARX-J-CJ models. As expec ed,
hese esul s a e in line wi h hose o ho izons 5 and 6, and
Elec ic Powe Sys ems Resea ch 212 (2022) 108144
8
A. Cia e a e al.
highligh he gain om aking in o accoun he in o ma ion on
jumps and/o cojumps in managing isk. Gi en ha he Phelix
base p oduc is calcula ed as he mean o all 168 hou ly o ecas
p ices, including jumps in he models imp o es he Phelix base
weekly p oduc .
To summa ize, in e ms o o ecas accu acy, he ARX model ou pe -
o ms models ha inco po a e jumps and/o cojumps o he sho es
ho izons. The e o e, elec ici y ading and plan ope a ion schedul-
ing decisions do no bene i om in o ma ion on jumps and/o co-
jumps. Howe e , as he o ecas ho izon leng hens, inco po a ing jumps
and/o cojumps in o he es ima ion o he models imp o es o ecas ing
accu acy. This could be because jumps a e ex emely a e, sho -li ed
e en s, so he likelihood o hei occu ing inc eases wi h ime. These
esul s ha e implica ions o p icing weekly p oduc s in u u es ma -
ke s. Fo ins ance, he way in which he Phelix base p oduc o
one-week deli e y is calcula ed shows he impo ance o models ha
inco po a e jumps and cojumps as explana o y a iables.
5. Summa y and conclusions
P ice modeling and o ecas ing ha e become challenging since elec-
ici y ma ke s we e libe alized. This is especially ele an wi h he
la ge-scale deploymen o enewable ene gy p oduc ion, in eg a ion
wi h neighbo ing ma ke s, and inc eased use o inancial p oduc s.
Elec ici y p ices also exhibi unique cha ac e is ics ha make hese
asks mo e complex. One o hose cha ac e is ics is he p esence o
spikes.
We use day-ahead p ices om he Ge man-Aus ian elec ici y ma -
ke o he pe iod om Janua y 1, 2014 o Sep embe 30, 2018 o
analyze he ole o jumps and cojumps in p ice o ecas ing. I should
be no ed ha his is he ma ke wi h he g ea es liquidi y in Ge many
and Aus ia, e en a e ma ke decoupling. I is he e o e impo an
o model p ice dynamics accu a ely o o ecas p ices se e al pe iods
ahead.
P ice se ies o each hou o he day a e conside ed, leading o a
mul i a ia e amewo k. We speci y h ee models: The ARX model; he
ARX-J model, which includes jumps; and he ARX-J-CJ model, which
includes jumps and cojumps. Cojumps a e de ined as jumps ha occu
on he same day. We also ans o m he p ice se ies using se e al
a iance s abilizing ans o ma ions.
Ou esul s show ha using 3𝜎2, mlog1, mlog2, and T-PIT a iance
s abilizing ans o ma ions p o ides mo e accu a e o ecas s o p ices
han conside ing o iginal p ice da a. Fu he mo e, including jumps
and cojumps as co a ia es u he imp o es p ice o ecas ing only o
ho izons beyond ou days.
These conclusions a e also o in e es o pa icipan s in he u u es
ma ke . Elec ici y ma ke s a ound he wo ld a e encou aging ma ke
agen s o pa icipa e in u u es ma ke s, and he decision o do so
is aken a e p o i abili y analyses. In pa icula , Phelix u u es a e
aded on he EEX ma ke . Hence, day-ahead p ice o ecas ing helps
pa icipan s o op imize hei bidding s a egies o he ollowing days
and decide whe he o pa icipa e in he u u es ma ke o no .
Finally, we conside he ollowing lines o u u e esea ch. Fi s , i
should be no ed ha he e is an in aday auc ion ha se s p ices e e y
15 min, and a con inuous ma ke wi h se e al p oduc s (e e y 15 min,
e e y 30 min and hou ly). The inco po a ion o la ge-scale in e mi en
enewable gene a ion and he in eg a ion o he Eu opean ma ke
inc ease he impo ance o hese ma ke s. Ou amewo k o analysis
could also be ex ended o hese ma ke s. Howe e , he di e en e-
quencies o p ice o ma ion would need o be ca e ully conside ed in
speci ying he s a is ical models. Second, models including o he ac o s
ha may a ec elec ici y p ices, such as load, wea he o ecas s and
ese e ma gin, could also be conside ed as mo e sys em ope a o s
begin disclosing such in o ma ion. In his case, he day-ahead o ecas
should be conside ed o ensu e ha hese ac o s gi e he in o ma ion
closes o he ime o he o ecas . Thi d, aking in o accoun he
in eg a ion o he Eu opean ma ke , p ice o ecas ing could also be
assessed o se e al Eu opean elec ici y ma ke s o check o cojumps
be ween hem.
CRediT au ho ship con ibu ion s a emen
Ai o Cia e a: Concep ualiza ion, Me hodology, Fo mal analysis,
W i ing – e iew & edi ing. Pe u Muniain: Concep ualiza ion, Me hod-
ology, Fo mal analysis, So wa e, W i ing – e iew & edi ing. Ainhoa
Za aga: Concep ualiza ion, Me hodology, Fo mal analysis, W i ing –
e iew & edi ing.
Decla a ion o compe ing in e es
The au ho s decla e ha hey ha e no known compe ing inan-
cial in e es s o pe sonal ela ionships ha could ha e appea ed o
in luence he wo k epo ed in his pape .
Acknowledgmen s
The au ho s would like o hank wo anonymous e iewe s o
aluable commen s and sugges ions ha helped o imp o e he pa-
pe . Financial suppo om Dp o. de Educación del Gobie no Vasco,
Spain unde esea ch g an IT1336-19 and om Minis e io de Ciencia
e Inno ación, Spain unde esea ch g an PID2019-108718GB-I00 is
acknowledged. Open access unding p o ided by he Uni e si y o he
Basque Coun y. The au ho s a e g a e ul o aluable commen s om
pa icipan s in he Wo kshop on Fo ecas ing in Elec ici y Ma ke s held
in Bilbao in 2019. The au ho s also hank Susan O be, Luiggi G ossi and
Ra ał We on o help ul commen s.
Appendix. Resul s o he mul i a ia e DM es o he MAE c i e-
ion
See Tables A.1–A.7.
Elec ic Powe Sys ems Resea ch 212 (2022) 108144
9
A. Cia e a e al.
Table A.1
DM using MAE o H1.
–ARX– ARX-J –ARX-J-CJ–
O ig. 3𝜎13𝜎2Logis ic1Logis ic2Asinh1Asinh2Mlog1Mlog2N-PIT T-PIT O ig. 3𝜎13𝜎2Logis ic1Logis ic2Asinh1Asinh2Mlog1Mlog2N-PIT T-PIT O ig. 3𝜎13𝜎2Logis ic1Logis ic2Asinh1Asinh2Mlog1Mlog2N-PIT
3𝜎1−0.18
(0.43)
ARX
3𝜎2−1.16
(0.122) −1.99
(0.023)
Logis ic11.5
(0.933) 3.88
(>0.999) 4.16
(>0.999)
Logis ic2−0.83
(0.205) −1.47
(0.071) 0.57
(0.715) −6.55
(<0.001)
Asinh1−0.75
(0.228) −0.56
(0.289) 0.75
(0.774) −4.28
(<0.001) 0.57
(0.715)
Asinh28.81
(>0.999) 6.39
(>0.999) 8.01
(>0.999) 5.13
(>0.999) 7.68
(>0.999) 8.21
(>0.999)
Mlog1−2.96
(0.002) −1.51
(0.066) −0.75
(0.227) −3.92
(<0.001) −1.16
(0.123) −2.6
(0.005) −9.78
(<0.001)
Mlog2−3.34
(<0.001) −1.31
(0.095) −0.55
(0.291) −3.49
(<0.001) −0.88
(0.189) −1.8
(0.036) −9.9
(<0.001) 0.95
(0.829)
N-PIT −1.16
(0.123) −1.35
(0.089) −0.01
(0.496) −4.72
(<0.001) −0.53
(0.298) −1.1
(0.135) −8.1
(<0.001) 0.85
(0.803) 0.58
(0.718)
T-PIT −1.51
(0.065) −1.21
(0.113) −0.19
(0.423) −3.62
(<0.001) −0.54
(0.294) −1.05
(0.147) −8.73
(<0.001) 0.73
(0.767) 0.47
(0.681) −0.35
(0.361)
O ig. 2.22
(0.987) 0.5
(0.691) 1.59
(0.944) −1.2
(0.116) 1.22
(0.888) 1.22
(0.889) −8.31
(<0.001) 3.68
(>0.999) 4.17
(>0.999) 1.6
(0.945) 2.03
(0.979)
3𝜎10.21
(0.583) 3.78
(>0.999) 2.94
(0.998) −2.93
(0.002) 2.47
(0.993) 1.18
(0.882) −6.04
(<0.001) 2.01
(0.978) 1.79
(0.964) 2.05
(0.98) 1.78
(0.962) −0.11
(0.456)
ARX-J
3𝜎2−0.76
(0.225) −1.16
(0.124) 2.62
(0.996) −3.68
(<0.001) 0.31
(0.622) −0.15
(0.44) −7.64
(<0.001) 1.28
(0.899) 1.06
(0.856) 0.72
(0.763) 0.82
(0.794) −1.19
(0.116) −2.29
(0.011)
Logis ic11.77
(0.962) 4.7
(>0.999) 4.71
(>0.999) 2.18
(0.986) 7.41
(>0.999) 4.79
(>0.999) −4.82
(<0.001) 4.29
(>0.999) 3.85
(>0.999) 5.13
(>0.999) 3.97
(>0.999) 1.48
(0.931) 3.82
(>0.999) 4.28
(>0.999)
Logis ic2−0.28
(0.389) −0.2
(0.421) 1.84
(0.967) −5.12
(<0.001) 3.6
(>0.999) 0.75
(0.773) −7.15
(<0.001) 1.94
(0.974) 1.6
(0.946) 1.64
(0.95) 1.35
(0.911) −0.66
(0.253) −1.38
(0.084) 1.03
(0.847) −6.59
(<0.001)
Asinh1−0.15
(0.441) 0.13
(0.55) 1.58
(0.943) −3.46
(<0.001) 1.84
(0.967) 4.09
(>0.999) −7.72
(<0.001) 3.72
(>0.999) 2.77
(0.997) 2.25
(0.988) 1.93
(0.973) −0.62
(0.268) −0.53
(0.297) 0.98
(0.836) −4.15
(<0.001) 0.44
(0.669)
Asinh29.25
(>0.999) 6.76
(>0.999) 8.45
(>0.999) 5.48
(>0.999) 8.11
(>0.999) 8.66
(>0.999) 2.85
(0.998) 10.25
(>0.999) 10.38
(>0.999) 8.53
(>0.999) 9.18
(>0.999) 8.79
(>0.999) 6.42
(>0.999) 8.1
(>0.999) 5.2
(>0.999) 7.62
(>0.999) 8.22
(>0.999)
Mlog1−1.84
(0.033) −0.94
(0.174) 0.09
(0.536) −3.42
(<0.001) −0.31
(0.38) −1.14
(0.126) −9.22
(<0.001) 3.7
(>0.999) 2.2
(0.986) 0.11
(0.545) 0.34
(0.631) −2.63
(0.004) −1.49
(0.068) −0.5
(0.308) −3.87
(<0.001) −1.21
(0.114) −2.62
(0.004) −9.78
(<0.001)
Mlog2−2.27
(0.012) −0.9
(0.185) 0.05
(0.521) −3.11
(<0.001) −0.3
(0.382) −0.92
(0.178) −9.47
(<0.001) 2.92
(0.998) 2.89
(0.998) 0.06
(0.525) 0.27
(0.605) −3.28
(<0.001) −1.41
(0.08) −0.5
(0.308) −3.51
(<0.001) −1.09
(0.139) −2.02
(0.022) −10.03
(<0.001) −0.15
(0.439)
N-PIT −0.78
(0.217) −0.85
(0.198) 0.64
(0.739) −4.22
(<0.001) 0.2
(0.581) −0.3
(0.383) −7.72
(<0.001) 1.39
(0.917) 1.06
(0.856) 2.49
(0.994) 1.31
(0.905) −1.22
(0.112) −1.61
(0.053) −0.08
(0.47) −4.78
(<0.001) −1.01
(0.157) −1.48
(0.069) −8.18
(<0.001) 0.51
(0.696) 0.47
(0.682)
T-PIT −0.98
(0.164) −0.73
(0.233) 0.51
(0.694) −3.19
(<0.001) 0.08
(0.532) −0.31
(0.377) −8.29
(<0.001) 1.44
(0.925) 1.14
(0.873) 0.97
(0.835) 2.68
(0.996) −1.5
(0.067) −1.33
(0.091) −0.14
(0.444) −3.6
(<0.001) −0.78
(0.219) −1.23
(0.109) −8.79
(<0.001) 0.47
(0.681) 0.46
(0.679) −0.14
(0.445)
O ig. 2.5
(0.994) 0.54
(0.705) 1.65
(0.95) −1.17
(0.122) 1.27
(0.897) 1.29
(0.901) −8.29
(<0.001) 3.8
(>0.999) 4.33
(>0.999) 1.66
(0.951) 2.11
(0.982) 0.63
(0.735) 0.15
(0.558) 1.24
(0.893) −1.45
(0.073) 0.71
(0.761) 0.68
(0.751) −8.77
(<0.001) 2.73
(0.997) 3.42
(>0.999) 1.27
(0.898) 1.56
(0.941)
3𝜎10.2
(0.579) 3.73
(>0.999) 2.9
(0.998) −2.95
(0.002) 2.44
(0.993) 1.17
(0.879) −6.04
(<0.001) 2
(0.977) 1.78
(0.962) 2.03
(0.979) 1.76
(0.961) −0.12
(0.453) −0.42
(0.338) 2.25
(0.988) −3.84
(<0.001) 1.34
(0.911) 0.52
(0.697) −6.42
(<0.001) 1.47
(0.93) 1.39
(0.918) 1.59
(0.944) 1.32
(0.906) −0.16
(0.438)
ARX-J-CJ
3𝜎2−0.76
(0.224) −1.15
(0.125) 2.69
(0.996) −3.66
(<0.001) 0.31
(0.621) −0.15
(0.439) −7.65
(<0.001) 1.29
(0.901) 1.07
(0.858) 0.72
(0.763) 0.82
(0.794) −1.2
(0.115) −2.27
(0.012) −0.01
(0.497) −4.26
(<0.001) −1.02
(0.155) −0.98
(0.163) −8.11
(<0.001) 0.5
(0.693) 0.5
(0.693) 0.07
(0.53) 0.14
(0.556) −1.25
(0.105) −2.23
(0.013)
Logis ic11.76
(0.961) 4.7
(>0.999) 4.69
(>0.999) 2.21
(0.986) 7.19
(>0.999) 4.71
(>0.999) −4.78
(<0.001) 4.24
(>0.999) 3.81
(>0.999) 5.11
(>0.999) 3.96
(>0.999) 1.47
(0.93) 3.82
(>0.999) 4.26
(>0.999) 0.13
(0.551) 6.39
(>0.999) 4.07
(>0.999) −5.16
(<0.001) 3.82
(>0.999) 3.48
(>0.999) 4.76
(>0.999) 3.59
(>0.999) 1.44
(0.926) 3.84
(>0.999) 4.23
(>0.999)
Logis ic2−0.28
(0.389) −0.2
(0.419) 1.82
(0.965) −5.11
(<0.001) 3.46
(>0.999) 0.74
(0.769) −7.12
(<0.001) 1.92
(0.972) 1.59
(0.944) 1.63
(0.949) 1.34
(0.91) −0.66
(0.254) −1.39
(0.083) 1.02
(0.845) −6.57
(<0.001) −0.05
(0.479) −0.44
(0.331) −7.58
(<0.001) 1.19
(0.883) 1.07
(0.859) 1
(0.842) 0.77
(0.78) −0.71
(0.239) −1.35
(0.088) 1.01
(0.843) −6.37
(<0.001)
Asinh1−0.1
(0.462) 0.19
(0.575) 1.67
(0.953) −3.53
(<0.001) 2.04
(0.979) 3.98
(>0.999) −7.61
(<0.001) 3.58
(>0.999) 2.71
(0.997) 2.42
(0.992) 2.02
(0.978) −0.55
(0.29) −0.49
(0.313) 1.07
(0.858) −4.24
(<0.001) 0.57
(0.717) 0.76
(0.775) −8.11
(<0.001) 2.56
(0.995) 2.01
(0.978) 1.65
(0.95) 1.32
(0.907) −0.61
(0.271) −0.47
(0.319) 1.07
(0.858) −4.16
(<0.001) 0.57
(0.717)
Asinh29.18
(>0.999) 6.8
(>0.999) 8.5
(>0.999) 5.51
(>0.999) 8.16
(>0.999) 8.71
(>0.999) 2.2
(0.986) 10.28
(>0.999) 10.4
(>0.999) 8.6
(>0.999) 9.25
(>0.999) 8.73
(>0.999) 6.45
(>0.999) 8.16
(>0.999) 5.22
(>0.999) 7.68
(>0.999) 8.27
(>0.999) −0.97
(0.165) 9.82
(>0.999) 10.05
(>0.999) 8.25
(>0.999) 8.86
(>0.999) 8.71
(>0.999) 6.45
(>0.999) 8.17
(>0.999) 5.18
(>0.999) 7.64
(>0.999) 8.17
(>0.999)
Mlog1−1.75
(0.04) −0.93
(0.177) 0.13
(0.55) −3.45
(<0.001) −0.28
(0.391) −1.11
(0.133) −9.16
(<0.001) 3.45
(>0.999) 2.1
(0.982) 0.16
(0.562) 0.38
(0.649) −2.52
(0.006) −1.49
(0.068) −0.48
(0.315) −3.9
(<0.001) −1.2
(0.114) −2.64
(0.004) −9.72
(<0.001) 0.42
(0.662) 0.27
(0.606) −0.49
(0.313) −0.44
(0.33) −2.62
(0.004) −1.47
(0.07) −0.48
(0.315) −3.86
(<0.001) −1.19
(0.117) −2.6
(0.005) −9.76
(<0.001)
Mlog2−1.97
(0.024) −0.78
(0.216) 0.23
(0.59) −3.03
(0.001) −0.13
(0.447) −0.69
(0.244) −9.29
(<0.001) 3.56
(>0.999) 3.47
(>0.999) 0.25
(0.6) 0.49
(0.686) −2.96
(0.002) −1.3
(0.096) −0.34
(0.368) −3.44
(<0.001) −0.94
(0.173) −1.82
(0.034) −9.85
(<0.001) 0.71
(0.76) 2.43
(0.992) −0.3
(0.383) −0.26
(0.396) −3.09
(<0.001) −1.29
(0.098) −0.34
(0.368) −3.4
(<0.001) −0.93
(0.176) −1.82
(0.034) −9.87
(<0.001) 0.51
(0.693)
N-PIT −0.71
(0.24) −0.74
(0.23) 0.78
(0.783) −4.13
(<0.001) 0.37
(0.643) −0.13
(0.447) −7.65
(<0.001) 1.5
(0.933) 1.17
(0.879) 3
(0.999) 1.51
(0.934) −1.14
(0.128) −1.51
(0.066) 0.07
(0.528) −4.7
(<0.001) −0.85
(0.198) −1.31
(0.095) −8.11
(<0.001) 0.64
(0.74) 0.59
(0.721) 1.82
(0.965) 0.38
(0.648) −1.19
(0.117) −1.49
(0.068) 0.07
(0.528) −4.68
(<0.001) −0.85
(0.199) −1.47
(0.071) −8.18
(<0.001) 0.62
(0.734) 0.42
(0.661)
T-PIT −0.97
(0.166) −0.75
(0.228) 0.51
(0.693) −3.23
(<0.001) 0.07
(0.53) −0.32
(0.374) −8.23
(<0.001) 1.4
(0.92) 1.11
(0.867) 0.97
(0.834) 2.47
(0.993) −1.49
(0.069) −1.36
(0.087) −0.15
(0.439) −3.64
(<0.001) −0.79
(0.214) −1.24
(0.107) −8.72
(<0.001) 0.46
(0.676) 0.45
(0.673) −0.16
(0.436) −0.09
(0.466) −1.55
(0.06) −1.34
(0.09) −0.15
(0.439) −3.63
(<0.001) −0.79
(0.215) −1.34
(0.091) −8.79
(<0.001) 0.43
(0.665) 0.25
(0.599) −0.41
(0.341)
Mul i a ia e DM es s a is ic using MAE c i e ion o o ecas ho izon 1. P- alues in pa en heses. A 𝑝- alue lowe han 0.10 indica es ha he o ecas s o he model o he ow a e be e han hose o he model o he column a he 10% signi icance le el. A hea map is used o indica e lowe (g een) and g ea e ( ed) p- alues.
Elec ic Powe Sys ems Resea ch 212 (2022) 108144
16
A. Cia e a e al.
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