Forecasting Bitcoin returns: Econometric time series analysis vs. machine learning

Be ge , Theo; Koubo á, Jana
A icle — Published Ve sion
Fo ecas ing Bi coin e u ns: Econome ic ime se ies
analysis s. machine lea ning
Jou nal o Fo ecas ing
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Sugges ed Ci a ion: Be ge , Theo; Koubo á, Jana (2024) : Fo ecas ing Bi coin e u ns: Econome ic
ime se ies analysis s. machine lea ning, Jou nal o Fo ecas ing, ISSN 1099-131X, Wiley, Hoboken,
NJ, Vol. 43, Iss. 7, pp. 2904-2916,
h ps://doi.o g/10.1002/ o .3165
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RESEARCH ARTICLE
Fo ecas ing Bi coin e u ns: Econome ic ime se ies
analysis s. machine lea ning
Theo Be ge
1,2
| Jana Koubo 
a
3
1
Depa men o Business and Compu e
Science, Uni e si y o Applied Sciences
Hanno e , Hano e , Ge many
2
Depa men o Business and
Adminis a ion, Uni e si y o B emen,
B emen, Ge many
3
Spa kassen Ra ing und Risikosys eme,
Be lin, Ge many
Co espondence
Theo Be ge , Depa men o Business and
Compu e Science, Uni e si y o Applied
Sciences Hanno e , Ricklinge S ad weg
120, D-30459 Hanno e , Ge many.
Email: [email p o ec ed]
Abs ac
We s udy he s a is ical p ope ies o he Bi coin e u n se ies and p o ide a
ho ough o ecas ing exe cise. Also, we calib a e s a e-o - he-a machine
lea ning echniques and compa e he esul s wi h econome ic ime se ies
models. The empi ical assessmen p o ides e idence ha he applica ion o
machine lea ning echniques ou pe o ms econome ic benchma ks in e ms
o o ecas ing p ecision o bo h in- and ou -o -sample o ecas s. We ind ha
bo h deep lea ning a chi ec u es as well as complex laye s, such as LSTM, do
no inc ease he p ecision o daily o ecas s. Speci ically, a simple ecu en
neu al ne wo k desc ibes a sensible choice o o ecas ing daily e u n se ies.
KEYWORDS
o ecas ing, machine lea ning, isk measu emen , ime se ies analysis
JEL CLASSIFICATION
C33, C58, G17, G23
1|INTRODUCTION
Finding he adequa e me hodological app oach o o e-
cas indi idual inancial e u n se ies desc ibes a s agge -
ing ask. In o de o achie e p ecise ou -o -sample
o ecas s, i is necessa y o unde s and he p ope ies o
he unde lying ime se ies. In his ein, he s a is ical
p ope ies o inancial ime se ies ha e been widely dis-
cussed and he applica ion o Au o eg essi e Mo ing
A e age (ARMA) Models is widely accep ed (Be ge &
Gencay, 2018; Halbleib & Pohlmeie , 2012).
Due o s eadily g owing compu a ional powe , as well
as inc easing da a a ailabili y, o ecas ing economic ime
se ies ia machine lea ning echniques desc ibes a no el
s ing o esea ch. As discussed by K aus e al. (2020),
machine lea ning is less es ic i e ega ding he
assump ions on he unde lying da a and cu en machine
lea ning app oaches can adjus o p ope ies o economic
ime se ies indi idually. The e o e machine lea ning
desc ibes a ui ul al e na i e o econome ic modelling.
Al hough machine lea ning echniques a e cha ac e ized
as black boxes, ecen s udies p o ide empi ical e idence
ha machine lea ning achie es highe o ecas ing p eci-
sion han widely accep ed econome ic app oaches. Gu
e al. (2020) p o ide a ho ough empi ical assessmen and
discuss compe ing machine-lea ning echniques applied
o economic da a se s. As a esul , adequa ely calib a ed
machine lea ning app oaches ou pe o m in e p e able
econome ic models in e ms o o ecas ing accu acy.
Feng e al. (2020) con i m hese esul s o s ock e u ns,
Longo e al. (2022) o GDP o ecas s, and Mak idakis
e al. (2018) o a ious economic da a se s.
Recei ed: 5 June 2023 Re ised: 28 Ma ch 2024 Accep ed: 19 May 2024
DOI: 10.1002/ o .3165
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion-NonComme cial-NoDe i s License, which pe mi s use and dis ibu ion in any
medium, p o ided he o iginal wo k is p ope ly ci ed, he use is non-comme cial and no modi ica ions o adap a ions a e made.
© 2024 The Au ho (s). Jou nal o Fo ecas ing published by John Wiley & Sons L d.
2904 Jou nal o Fo ecas ing. 2024;43:2904–2916.
wileyonlinelib a y.com/jou nal/ o
This s udy adds o his s ing o li e a u e, and we
ocus on o ecas s o inno a i e economic e u n se ies,
namely Bi coin e u ns. As desc ibed in Alessand e i
e al. (2018) and Tandon e al. (2019), Bi coins belong o
he asse class o c yp ocu encies and a e cha ac e ized
by highe ola ili y han classical cu encies. Fu he -
mo e, as c yp ocu encies a e no con olled by na ional
cen al banks, Bi coins a e exposed o di e en economic
de e minan s and hence his o ic e u n se ies can exhibi
di e en ime se ies cha ac e is ics. Alessand e i e al.
(2018) p o ide empi ical e idence ha he applica ion o
neu al ne wo ks o ime se ies p edic ions is a ui ul
app oach, especially when i comes o p edic ing
Bi coins. Also, Lahmi i and Beki os (2019) p o ide
empi ical e idence ha o ecas ing Bi coin e u n se ies
ia machine lea ning ou pe o ms ypical econome ic
benchma ks, such as ARMA models. Typically, ecen
s udies assess o ecas ing pe o mance ia MAE (Mean
Absolu e E o ) o RMSE (Roo Mean Squa ed E o ) o
discuss he p ecision o compe ing machine lea ning
models (see Jang & Lee, 2018; Lahmi i & Beki os, 2019;
Phaladisailoed & Numnonda, 2018; Tandon e al., 2019).
Ou empi ical assessmen p o ides a ho ough o e-
cas ing s udy wi h an exclusi e ocus on daily Bi coin
p ices and he con ibu ion o ou s udy is wo old. Fi s ,
we p o ide an economic assessmen o s a is ical ime
se ies p ope ies o daily Bi coin p ices o assess i he
iden i ied cha ac e is ics a e in line wi h he unde lying
assump ions o econome ic benchma k models. Second,
we apply s a e-o - he-a machine lea ning app oaches
and analyze he pe o mance o deep lea ning
ne wo k a chi ec u es and complex ecu en neu al
ne wo k (RNN) laye s, namely ecu en Long Sho -
Te m Memo y (LSTM). Also, we compa e he o ecas ing
pe o mance o bo h econome ic ime se ies models and
machine lea ning echniques o discuss s a e-o - he-a
machine lea ning echniques agains solid econome ic
benchma ks. Speci ically, in addi ion o he widely used
ARMA(1,1) app oach, we also ake in o accoun condi-
ional ola ili y clus e ing ia gene alized au o eg essi e
condi ional he e oscedas ici y (GARCH) and assess a
a ie y o compe ing ARMA-GARCH app oaches.
The emainde o his pape is s uc u ed as ollows.
Sec ion 2p o ides he ele an li e a u e, Sec ion 3gi es
an o e iew o he me hodology, and Sec ion 4p esen s
he in es iga ed da a. The esul s o he empi ical assess-
men a e in Sec ion 5and Sec ion 6concludes.
2|LITERATURE REVIEW
The s ing o li e a u e, dealing wi h ime se ies o ecas s
o c yp ocu encies can be sepa a ed in o wo subs ings.
One s ing deals wi h he applica ion o black-box
machine lea ning app oaches and one deals wi h in e -
p e able econome ic app oaches.
2.1 |Machine Lea ning and Bi coins
An ex ensi e s udy on machine lea ning-based o ecas s
is p esen ed by Alessand e i e al. (2018). The au ho s
s udy he pe o mance o h ee models and p edic daily
c yp ocu ency p ices o 1,681 cu encies om ime
pe iod be ween No embe 11 h 2015 and Ap il 24 h 2018.
Two o he applied app oaches a e based on g adien -
boos ing decision ees and one is based on ecu en
Long Sho -Te m Memo y (LSTM) neu al ne wo ks.
Then, in es men po olios a e buil based on he p e-
dic ions, and he compa ison o hei pe o mance in
e ms o e u n on in es men is assessed. The au ho s
p o ide e idence ha all models ou pe o m he base-
line “simple mo ing a e age”app oach. The au ho s
ind ha he ecu en LSTM app oach is cha ac e ized
by supe io o ecas ing pe o mance. Also, Jang and
Lee (2018) p o ide an empi ical ho se ace be ween
Bayesian Neu al Ne wo ks (BNNs) wi h compe ing lin-
ea and non-linea benchma k models o p edic
Bi coin p ice p ocesses. They co e he daily da a om
Sep embe 11 h 2011 o Augus 22 h 2017. As a esul ,
BNNs pe o m well in p edic ing Bi coin p ice ime se ies
and explaining ele an ola ili y componen s o his o ic
p ices. Based on log- ans o med ma ke p ices and ola-
ili y p ocesses, he au ho s p o ide expe imen al e i-
dence ha he p edic i e pe o mance o BNNs
ou pe o ms compe ing benchma k me hods. Ka asu
e al. (2018) also assess Bi coin p ice p edic ions ia
machine lea ning using daily da a be ween Janua y 2012
and Decembe 2018. The au ho s s udy Suppo Vec o
Machines (SVM) and linea eg essions o assess daily
closing p ices. As a esul , SVM models a e cha ac e ized
by highe o ecas ing p ecision han linea eg ession
models.
Lahmi i and Beki os (2019)s udy heapplica iono
sophis ica ed machine lea ning applica ions o c yp ocu -
ency p edic ion o he pe iod be ween July 10 h 2010 and
Oc obe 1s 2018 and ind ha he accu acy o mo e com-
plex RNN laye s, namely Long Sho -Te m Memo y
(LSTM), signi ican ly inc eases o ecas ing p ecision. As
a benchma k, he au ho s apply a gene alized neu al
eg ession a chi ec u e benchma k. In his ein, Muniye
(2020) compa es wo deep lea ning echniques, LSTM
and Ga ed Recu en Uni (GRU) using daily da a
be ween Janua y 1 h 2014 and Feb ua y 20 h 2018. The
esul s sugges ha he GRU model desc ibes an ade-
qua e app oach o p edic ing Bi coin p ices as i equi es
BERGER and KOUBOV
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less compila ion ime han LSTM. Phaladisailoed and
Numnonda (2018) p o ide simila conclusion based on
high- equency, 1-minu e in e al ading da a om
Janua y 1 h 2012 o Janua y 8 h 2018. The au ho s com-
pa e he pe o mance o compe ing eg ession models
wi h RNNs wi h bo h LSTM and GRU laye s and ind
ha GRU esul s in supe io o ecas ing accu acy. Also,
Tandon e al. (2019) p o ide a ho ough s udy on Bi coin
p ice p edic ions o he pe iod be ween 2013 and 2019
and compa e RNN wi h LSTM wi h 10- old c oss alida-
ion, linea eg ession and andom o es s. The esul s
indica e ha c oss- alida ion in RNN wi h LSTM con ib-
u es o he imp o emen o he e iciency o he model
o Bi coin p edic ion. The p oposed RNN wi h LSTM
using 10- old c oss alida ion leads o signi ican ly lowe
MAE (mean absolu e e o ) han andom o es and lin-
ea eg ession models.
2.2 |Econome ic Time Se ies Analysis
and Bi coins
In he con ex o econome ic modeling, Chu e al. (2017)
p o ide an ex ensi e s udy on GARCH modeling dealing
wi h he se en mos popula c yp ocu encies using da a
be ween June 22nd 2014 and May 17 h 2017. They ind
ha IGARCH and GJR-GARCH models esul in supe io
model i o condi ional ola ili y modeling. Among he
assessed GARCH- ype models, i is he IGARCH(1,1)
model ha p o ides he bes i o Bi coins. T os e e al.
(2019) compa e he GARCH and GAS model's o ecas ing
p ecision o p edic he condi ional mean and ola ili y o
he Bi coin e u n se ies. They co e he daily da a om
July 19 h 2010 o Ap il 16 h 2018. The au ho s ake a
ails in o accoun and demons a e ha he assump ion
o no mally dis ibu ed e u ns ge s ou pe o med by he
hea y- ailed GAS app oach, measu ed ia Value-a -Risk
(VaR) o ecas s. By i ing nonlinea econome ic models
o his o ical da a, Fig
a-Talamanca and Pa acca (2019)
s udy he ela i e impac o a en ion measu es on bo h
he mean and he a iance o Bi coin e u ns using da a
be ween Janua y 1 h 2012 and Decembe 31 h 2017. The
au ho s s udy compe ing models belonging o he amily
o ARMA(p,q)-X (E)GARCH(1,1)-X nonlinea models
and include a en ion- ela ed explana o y a iables. The
esul s p o ide e idence ha a en ion measu es ha e a
signi ican impac on he condi ional mean and condi-
ional a iance o Bi coin e u ns, especially when a en-
ion is measu ed in e ms o ading olume. The au ho s
assess he Akaike In o ma ion C i e ion (AIC) and Bayes-
ian In o ma ion C i e ion (BIC) as well as o ecas
pe o mance.
2.3 |Machine Lea ning and
Econome ic Time Se ies Analysis
Shen e al. (2021) p o ide a pionee ing s udy o compa e
bo h econome ic and machine lea ning app oaches. The
au ho s s udy GARCH models and RNN wi h GRU laye s
o o ecas Bi coin's e u n ola ili y and VaR igu es on
da a be ween Ap il 30 h 2013 and May 21 h 2021. The
au ho s p o ide empi ical e idence ha RNN ou pe -
o ms GARCH and EWMA in e ms o a e age o ecas -
ing pe o mance. Fu he mo e, RNN shows poo
pe o mance in VaR o ecas ing, indica ing ha econo-
me ic models ou pe o m machine lea ning in dealing
wi h ex eme ola ili y. Fu he mo e, his s udy sugges s
an al e na i e me hod o Bi coin ola ili y analysis and
inds ha machine lea ning me hods pe o m well in less
ola ile inancial ma ke condi ions. McNally e al. (2018)
compa e bo h econome ic and machine lea ning ech-
niques and compa e Bayesian op imized RNN, LSTM,
and ARIMA models o p edic ing his o ic ma ke p ices
o Bi coins using da a be ween Augus 19 h 2013 and July
19 h 2016. The au ho s demons a e ha non-linea deep
lea ning me hods ou pe o m he ARIMA p edic ion.
Also, Co ez e al. (2021) compa e he p edic ions o he
ARMA-GARCH model wi h he K-Nea es Neighbou
app oach (KNN) applied o ma ke liquidi y in c yp ocu -
encies om Feb ua y 9 h 2018 o Feb ua y 8 h 2019. The
au ho s p o ide empi ical e idence ha KNN app oaches
ou pe o m ARMA and GARCH models in p edic ing he
log a es o he bid-ask sp eads. Fu he mo e, he au ho s
demons a e ha , compa ed o he ARMA and GARCH
models, he KNN app oach is mo e e ec i e a cap u ing he
sho - e m ma ke liquidi y o c yp ocu encies. Fu he
s udies ha con i m he supe io i y o Neu al Ne wo ks
when i comes o p edic ing Bi coin p ices a e gi en by
Alessand e i e al. (2018), Jang and Lee (2018), Lahmi i and
Beki os (2019), Phaladisailoed and Numnonda (2018), and
Tandon e al. (2019). Mos o he pape s apply MAE o
RMSE o assess he pe o mance o applied models, see
Tandon e al. (2019). Fu he mo e, he e a e also s udies ha
in es iga e o he ea u es, which could ha e an e ec on he
beha io o Bi coin (Balcila e al., 2017;Fig

a-Talamanca &
Pa acca, 2019;Saad&Mohaisen,2018;Sin&Wang,2017;
Velanka e al., 2018). As well, Taskaya-Temizel and Casey
(2005) also discuss he idea o hyb id app oaches and com-
bine ime se ies models wi h machine lea ning models.
3|METHODOLOGY
In his sec ion, we in oduce he no a ion used h ough-
ou he pape , de ine he desi able p ope ies o
2906 BERGER and KOUBOV
´
A
econome ic ime se ies models and ecu en neu al ne -
wo ks, speci y long sho - e m memo y, and p esen he
pe o mance me ics.
3.1 |Econome ic Time Se ies Models
In o de o adequa ely cap u e he ele an cha ac e is ics
o Bi coin e u ns, we apply an au o eg essi e mo ing
a e age (ARMA) app oach o model he condi ional
mean o he daily e u n se ies. This app oach desc ibes a
combina ion o he au o eg essi e model wi h plags and
he mo ing a e age model wi h qlags. Fu he mo e, we
also ake in o accoun ime- a ying condi ional ola ili y
and au o eg essi e ola ili y clus e ing and apply he
gene alized au o eg essi e condi ional he e oscedas ici y
(GARCH) model o he squa ed esiduals o he
ARMA app oach, wi h mand nlags, espec i ely. Then
he ARMA(p,q)-GARCH(m,n) model is de ined as
ollows:
y ¼X
p
i¼1
ϕiy iþX
q
j¼1
θju jþu , wi h :u iid D 0,σ2

,
ð1Þ
σ2
¼α0þα1u2
1þα2u2
2þ…þαmu2
mþβ1σ2
1
þβ2σ2
2þ…þβnσ2
n,
ð2Þ
wi h: α0>0,αi≥0i¼1,…,mðÞ,βj≥0j¼1,…,nðÞand
P
max m,nðÞ
i¼1
αiþβi
ðÞ<1:Whe e u is whi e noise (has ze o
mean, cons an a iance σ2, and is unco ela ed in ime)
and he ime se ies y a e he Bi coin e u ns a ime .
Also, mdesc ibes he o de o he ARCH e ms and n he
o de o he GARCH e ms. The esiduals u a e cha ac-
e ized by dis ibu ion D0,σ2
ðÞ. We will assess bo h a
Gaussian and a -dis ibu ion. Also, we apply a nai e
benchma k, which is simply he e u n om he p e ious
pe iod.
Fu he , his amewo k allows us o also assess com-
pe ing app oaches, ha a e nes ed in equa ion (2),
namely ARMA(p,q), ARMA(p,q)-GARCH(m,n), and
di e en pa ame e iza ions o p,q,m
,
and n. The pa ame-
e s a e es ima ed ia he maximum likelihood me hod.
3.2 |Recu en Neu al Ne wo ks
Ou baseline machine-lea ning app oach is a simple
RNN. In compa ison o ypical Feed-Fo wa d-Ne wo ks
(FFNs), RNNs include backwa d connec ions and
he e o e his app oach is p edes ined o cap u e ime-
dependen ea u es o a ime se ies. Also, by s acking
mo e han one RNN laye , we a e able o s udy Deep
RNNs. Le he inpu and ou pu a ime s ep be
desc ibed by X and y , he weigh s o he hidden laye
by Whx and bias by bx. The in o ma ion om he p e i-
ous ime s ep is h 1wi h weigh s Whh and bias bh. Then,
in o de o calcula e he hidden s a e a he ime s ep h ,
ha is he in o ma ion ha will be passed o þ1, is
desc ibed as ollows:
h ¼ Wh 1,X
ðÞ,
wi h :
¼1,…,Mand h0¼0:
ð3Þ
The hidden s a e a ime can be desc ibed as a unc-
ion wi h pa ame e s Wo he p e ious hidden s a e and
he inpu a ime . In ou s udy, we apply he s anda d
hype bolic angen unc ion ( anh) and W. The amoun
o uples o he ain se is M. Then, we apply RNN and
s acked RNNs which a e hen de ined as ollows:
h ¼ anh Whhh 1þWhxX þbh
ðÞ,ð4Þ
y ¼ anh Wyhh þby

,
wi h :
¼1,…,M:
ð5Þ
Fo ins ance, o a Deep RNN wi h 2 hidden laye s,
5 neu ons, and a o ecas ing ho izon o 10 days, he ou -
pu o he las hidden laye (h o ime s ep ) is a ma ix
105, as i con ains he esul s o each neu on. The ou -
pu laye is a simple laye and ans o ms his ma ix in o
101 ec o o o ecas s wi h he weigh s Wyh,as
desc ibed in equa ion 5. F om his, he numbe o neu-
ons can also be seen as a dimension o he hidden s a e
o one laye . In his pape , we s udy 1, 2, and 3 hidden
simple RNN laye s in combina ion wi h 1, 5, and 10 uni s
and we e e o Gé on (2020) o a ho ough in oduc ion
o RNNs and machine lea ning.
3.2.1 | Long Sho -Te m Memo y
In addi ion o simple RNN, we also apply RNN wi h long
sho - e m memo y (LSTM) laye . LSTM was in oduced
by Hoch ei e and Schmidhue (1997) in o de o o e -
come he p oblem o long- e m dependency da a causing
a anishing g adien wi hin a simple RNN amewo k. In
compa ison o RNN, he aining o LSTM con e ges
BERGER and KOUBOV
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as e and he iden i ica ion o long- e m dependencies in
he da a is possible. In con adic ion o RNN, whe e
e e y laye ge s 2 inpu s, a he ime s ep :X and he
ou pu o he p e ious ime s ep 1, namely h 1, LSTM
adds an addi ional inpu o each laye . Tha is, he cu -
en long- e m memo y o he ne wo k, known as he cell
s a e c 1and he hidden s a e h becomes he sho - e m
s a e, see Gé on (2020). The pa ame e iza ion o he
LSTM app oach is gi en as ollows:
¼σW xX þW hh 1ðÞþb

ð6Þ
g ¼ anh WgxX þWghh 1ðÞþbg

ð7Þ
i ¼σWixX þWihh 1ðÞþbi
ðÞð8Þ
o ¼σWoxX þWohh 1ðÞh 1ðÞþbo
ðÞð9Þ
c ¼ Oc 1þi Og ð10Þ
y ¼h ¼o O anh c
ðÞ ð11Þ
The i s s ep in he LSTM cell is he Fo ge Ga e ( )
as desc ibed in equa ion (6). Th ough his ga e, ce ain
in o ma ion is dele ed ( o go en) and he ou pu o he
Fo ge Ga e ( Nc 1) is hen added o he ou pu o
he Inpu Ga e (i Ng ). The esul o c ge s s o ed wi h-
ou u he ans o ma ion as desc ibed in equa ion (10).
Hence, a each ime s ep, i ele an in o ma ion is dis-
ca ded and ele an his o ical in o ma ion is added, see
equa ion (11). The long- e m s a e c is copied a e his
addi ion and passed h ough he anh unc ion. As
desc ibed in Yu e al. (2019). The numbe o neu ons pe
LSTM laye deno es he dimensions o he hidden s a e
and o he ou pu s a e o one laye . Simila o he RNN
amewo k, a dense laye wi h 1, 5, and 10 neu ons is
used as an Ou pu Laye o he RNN wi h LSTM.
In o de o ain he ne wo ks, we apply Adam
(Adap i e Momen Es ima ion) op imiza ion. As
desc ibed in Gé on (2020) Adam op imiza ion combines
he idea o Momen um Op imiza ion and RMSP op.
3.3 |Pe o mance Me ics
In addi ion o he assessmen o au oco ela ed esidual
and pa ial au oco ela ion unc ions, we d aw on exis -
ing s udies and assess he p ecision o ou -o -sample o e-
cas s ia bo h oo mean squa ed e o (RMSE) and mean
absolu e e o s (MAE). The me ics a e calcula ed as
ollows:
RMSET
B¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PM
i¼1PT
j¼1yB, o ecas
ij yB, ue
ij

2
MT
u
u
,ð12Þ
MAET
B¼PM
i¼1PT
j¼1jyB, o ecas
ij yB, ue
ij j
MT:ð13Þ
Whe e Mdeno es he numbe o uples, B he ba ch
size, 100, 250, o 500 days, and T he numbe o days ha
desc ibe he o ecas ing ho izon, 1, 5, o 10 days. Fu he -
mo e, in ou s udy, nai e o ecas ing deals as a bench-
ma k o bo h app oaches. Tha is, he las day o he
aining sample is used as he o ecas o he nex con-
secu i e day. Fu he mo e, in o de o assess compe ing
neu al ne s, we apply a nai e ully connec ed neu al ne
as an addi ional benchma k.
4|DATA
The da a s a s wi h he i s a ailable ma ke p ice o Bi -
coin, Ap il 28 h 2013, and anges un il Decembe 12 h in
2021.
1
The g aphical ep esen a ion o he p ices and log-
a i hmic e u ns is displayed in Figu e 1.
Figu e 1illus a es bo h daily ma ke p ices and daily
log e u ns o e ime. Table 1p esen s he desc ip i e s a-
is ics o he da a. We ind ha daily e u ns a e cha ac-
e ized by a s anda d de ia ion o 4.1698 and a mean o
0.1884. These esul s a e in line wi h Cunha and Sil a
(2020). Howe e , in compa ison o o he cu encies, as
epo ed by Migno and Wes e ho (2024), daily Bi coin
e u ns a e cha ac e ized by highe ola ili y.
2
Table 2gi es he Ljung-Box (LB) es s a is ics o
assess au oco ela ion o he loga i hmic e u ns and
squa ed loga i hmic e u ns, he Augmen ed Dickey-
Fulle (ADF) es o in es iga e s a iona i y, and he
Ja que-Be a (JB) es o s udy no mali y o he loga i h-
mic e u ns. The ADF Tes shows, ha he ime se ies o
Bi coin daily p ices is no s a iona y. As a esul , loga i h-
mic e u ns a e c ea ed (he ea e e e ed o only as
e u ns). As can be seen in Table 2, he Bi coin daily
e u ns a e s a iona y. As discussed in Box e al. (1994),
we se he numbe o in es iga ed lags o he Ljung-Box
es o 20. The s anda d de ia ion o Bi coin e u ns is
desc ibed as 4.1698 and, as indica ed by he LB es s a is-
ic is cha ac e ized by s a is ically signi ican ola ili y
1
Da a sou ce is wwww.coinpap ika.com and he applied da a se is also
a ailable upon eques o he au ho s.
2
Fo an in-dep h discussion on s ylized ac s o exchange a es, we e e
o Guillaume e al. (1997) and G auwe and G imaldi (2006).
2908 BERGER and KOUBOV
´
A
clus e s. Howe e , as illus a ed in Figu e 2 he au oco -
ela ion o e u ns decays quickly.
These indings a e consis en wi h Hu e al. (2019),
Mom az (2021), and Zhang e al. (2018) who epo simi-
la indings on ola ili y and ola ili y clus e s. Fu he -
mo e, as p esen ed by Zhang e al. (2018) and Bo i
(2019), based on he JB es , we can con i m ha he dis-
ibu ion o Bi coin e u ns is non-no mal and exhibi s
a ails, a s ylized ac ha is well-known om he s ock
ma ke and can also be seen in Figu e 3.
As desc ibed abo e, Bi coin e u ns a e in line wi h
s ylized ac s o inancial e u n se ies, as also shown in
he li e a u e. The ac s ha a e shown in his pape a e
high ola ili y, ola ili y clus e ing, and a ails.
In o de o assess o ecas ing accu acy, we sepa a e
he da a in o o e lapping uples comp ising 110, 260, and
510 days. Hence, he aining se o he uple consis s o
100, 250, and 500 days, he addi ional 10 days a e used
o assessing he goodness o a o ecas . All in all, he e
a e 3,032 uples o 110 days, 2,882 uples o 260 days, and
2,632 uples o 510 days o T¼100,250
,
and 500
and 10 days ahead o ecas .
An example o his sepa a ion is displayed in
Figu e 4, he unde lying da a o he machine lea ning
exe cise is sepa a ed in o ain, alida ion, and es da a
se s. Following he s udy om Shen e al. (2021), he
uples a e cha ac e ized as ollows. The ain da ase con-
ains he i s 70% o he uples, he alida ion da a se
he nex consecu i e 20% and he es da a se he emain-
ing 10%. Then, based on he ained model, he ain and
alida ion da a se desc ibe he in-sample da a, and he
es da a is he ou -o -sample da a se . As illus a ed in
Figu e 4, he da a can hen be spli in o di e en da a
se s. In ou s udy, we assess aining and alida ion se s
comp ising T¼100,250
,
and 500 days and s udy o ecas
o 1,5 and 10 days ahead.
FIGURE 1 Bi coin p ice and e u ns (in %) in ime.
TABLE 1 Desc ip i e s a is ics.
Bi coin daily p ices Bi coin daily e u ns
n 3145.00 3143.00
Mean 8818.48 0.19
S d 14568.24 4.17
Min 67.81 43.37
25% 442.82 1.37
50% 3410.45 0.21
75% 9239.90 1.90
Max 67617.02 28.71
No e: The able p esen s desc ip i e s a is ics o he daily ma ke p ices and
e u n se ies o Bi coins om Ap il 28, 2013 o Decembe 12, 2021. The
amoun o obse a ions is n, Min and Max a e he minimum and maximum
alue o each a iable. 25%, 50% and 75% a e he espec i e qua iles o he
empi ical e u n dis ibu ion and S d is he s anda d de ia ion o he mean.
TABLE 2 Ljung-Box, ADF and Ja que-Be a es s.
Ljung-Box es s a is ic P- alue
Bi coin e u ns 53.70 0.00
Bi coin squa ed e u ns 356.65 0.00
ADF es s a is ic P- alue
Bi coin p ices 0.39 0.91
Bi coin e u ns 14.71 0.00
Ja que-Be a es s a is ic P- alue
Bi coin e u ns 21.02 0.00
No e: This able p o ides he esul s o he Ljung-Box (LB), Augmen ed
Dickey Fulle (ADF) and Ja que-Be a (JB) es . The Ljung-Box (LB) es is
pe o med wi h 20 lags wi h H0: The da a is independen ly dis ibu ed and
H1: The da a is no independen ly dis ibu ed, and exhibi s se ial
co ela ion. The Augmen ed Dickey Fulle (ADF) es assesses H0: The
se ies is non-s a iona y. H1: The se ies is s a iona y. And he hypo hesis o
he Ja que-Be a (JB) es es a e H0: The se ies is no mally dis ibu ed. H1:
The se ies ollows a non-no mal dis ibu ion. Fo all es , he p- alue is
p esen ed.
BERGER and KOUBOV
´
A2909
FIGURE 2 Au oco ela ion unc ion and pa ial au oco ela ion unc ion o daily e u ns and squa ed Re u ns. The uppe g aphs
isualize he Au oco ela ion unc ion (ACF) o daily Bi coin e u ns (le ) and squa ed Bi coin e u ns ( igh ). The lowe g aphs p o ide
in o ma ion on he pa ial au oco ela ion o Bi coin e u ns (le ) and squa ed Bi coin e u ns ( igh ).
FIGURE 3 His og ams o Bi coin Re u ns (in %) and compa ison wi h No mal Dis ibu ion.
FIGURE 4 Sepa a ing he da a in o uples o ecu en neu al ne s (RNN). Each uple con ains aining da a (yellow) and labeled
o ecas ing da a ( ed). The da a is a ime se ies and is desc ibed by consecu i e daily Bi coin e u ns. In o de o ain a neu al ne , 90% o
he uples a e used o ain and alida e he neu al ne , and 10% a e applied o es he esul s. The aining and es da a is spli in o 70%
aining and 20% es ing da a he es da a p esen s he basis o he e alua ion o ou -o -sample o ecas s.
2910 BERGER and KOUBOV
´
A
5|EMPIRICAL ASSESSMENT
In his sec ion, we discuss he empi ical esul s o ou
o ecas ing s udy. Due o he ac , ha we compa e wo
compe ing me hodological app oaches, namely econo-
me ic modeling and machine lea ning, we p o ide a dis-
cussion on he op imal pa ame e iza ion o bo h
app oaches, econome ic and machine lea ning,
TABLE 3 AIC alues o ARMA p,qðÞand GARCH m,nðÞmodel.
ARMA
Gaussian dis ibu ion
p=q012345
111,046.28 11,044.39 11,043.19 11,046.69 11,047.48 11,046.82
211,044.72 11,043.24 11,044.73 11,076.25 11,052.37 11,073.12
311,043.90 11,047.88 11,050.74 11,051.88 11,067.92 11,067.21
411,044.79 11,048.39 11,053.40 11,035.87 11,044.69 11,067.93
511,045.94 11,048.40 11,051.41 11,077.58 11,035.10 11,035.78
-dis ibu ion
p=q012345
112,055.74 12,054.43 12,055.72 12,054.39 12,056.16 12,086.41
212,055.07 12,055.66 12,058.30 12,085.20 12,089.96 12,083.32
312,054.64 12,054.46 12,089.46 12,060.73 12,090.63 12,088.63
412,054.49 12,056.17 12,085.86 12,090.29 12,094.27 12,089.88
512,053.56 12,088.41 12,056.28 12,092.36 12,092.34 12,091.47
GARCH
m=n012345
112237.04 12547.45 12529.13 12551.13 12549.13 12531.89
212310.91 12543.04 12527.54 12548.28 12538.42 12531.98
312361.34 12525.42 12526.80 12548.17 12529.90 12525.40
412421.43 12539.38 12473.29 12524.98 12534.34 12543.49
512452.35 12536.82 12539.46 12536.74 12144.82 12538.43
No e: This able p esen s he AIC c i e ion o di e en combina ions o lags unde he assump ion o Gaussian and -dis ibu ed esiduals.
TABLE 4 Fo ecas ing pe o mance:
Econome ic app oach. Nai e benchma k ARMA-GARCH
RMSE 1 day 5 days 10 days 1 day 5 days 10 days
100 days 0.0585 0.0578 0.0576 0.0427 0.0431 0.0427
250 days 0.0567 0.0558 0.0556 0.0405 0.0402 0.0401
500 days 0.0563 0.0555 0.0554 0.0398 0.0401 0.0408
Mean 0.0572 0.0564 0.0562 0.0410 0.0411 0.0412
MAE 1 day 5 days 10 days 1 day 5 days 10 days
100 days 0.0391 0.0391 0.0391 0.0284 0.0282 0.0279
250 days 0.0382 0.0381 0.0381 0.0265 0.0263 0.0262
500 days 0.0380 0.0379 0.0379 0.0259 0.0260 0.0260
Mean 0.0384 0.0384 0.0384 0.0269 0.0268 0.0267
No e: This able p esen s he ou -o -sample o ecas ing pe o mance o he nai e benchma k is he e u n om
he p e ious pe iod and he applied ARMA(4,4)-GARCH(1,3) model. The o ecas ing ho izon is 1 day, 5 days,
and 10 days and he me ics a e he oo mean squa ed e o s (RMSE) and mean absolu e e o (MAE).
BERGER and KOUBOV
´
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