Mas e Deg ee P og am in
Da a Science and Ad anced Analy ics
Compa a i e Analysis o GDP Fo ecas ing using Ensemble T ee
Reg ession Models:
Machine Lea ning s. Econome ic Models
Rica do Paulo Ba bosa de Ca alho Almeida Coelho
Mas e Thesis
p esen ed as pa ial equi emen o ob aining a Mas e ’s Deg ee in Da a Science and Ad anced Analy ics
NOVA In o ma ion Managemen School
Ins i u o Supe io de Es a ís ica e Ges ão de In o mação
Uni e sidade No a de Lisboa
MDSAA
NOVA In o ma ion Managemen School
Ins i u o Supe io de Es a ís ica e Ges ão de In o mação
Uni e sidade No a de Lisboa
Compa a i e Analysis o GDP Fo ecas ing using Ensemble T ee Reg ession Models:
Machine Lea ning s. Econome ic Models
by
Rica do Paulo Ba bosa de Ca alho Almeida Coelho
Mas e Thesis p esen ed as pa ial equi emen o ob aining he Mas e ’s deg ee in Da a
Science and Ad anced Analy ics, wi h a specializa ion in Business Analy ics.
Supe ised by
P o . Jo ge Miguel Ven u a B a o, PhD
NOVA In o ma ion Managemen School &
Uni e si é Pa is-Dauphine PSL
July, 2024
i
STATEMENT OF INTEGRITY
I he eby decla e ha ing conduc ed his academic wo k wi h in eg i y. I con i m ha I ha e no
used plagia ism o any o m o undue use o in o ma ion o alsi ica ion o esul s along he
p ocess leading o i s elabo a ion. I u he decla e ha I ha e ully acknowledged he Rules
o Conduc and Code o Hono om he NOVA In o ma ion Managemen School.
Lisbon, 07/07/2024
ii
ABSTRACT
This hesis e alua es whe he machine lea ning can ha e be e esul s han well-es ablished
ins i u ions in o ecas ing Po ugal’s GDP g ow h using ensemble ee eg ession models, wi h
he OECD’s economic ou look o ecas s o Po ugal se ing as a benchma k based on da a
om 1962 o 2022 eco e ed om OECD and he Fede al Rese e’s economic da a. The
indings e eal ha , in gene al, machine lea ning did no su pass he o ecas s o he OECD.
Howe e , machine lea ning demons a ed he po en ial o be e accu acy a many poin s,
despi e a highe p opensi y o la ge e o s compa ed o adi ional me hods. Among he
ensemble ee eg ession models es ed, he g adien boos ing eg esso consis en ly
p o ided he bes o ecas s ac oss all ho izons, ou pe o ming he andom o es , ex eme
g adien boos ing, and ligh g adien boos ing machine models. The esul s also sugges ha
machine lea ning pe o ms be e wi h a la ge olume o da a han wi h highe
dimensionali y, e en i some da a poin s seem i ele an o o ecas u u e alues. This hesis
highligh s he po en ial o machine lea ning in ime se ies o ecas ing as a complemen a y ool
o adi ional me hods, a he han a comple e eplacemen .
KEYWORDS
Machine lea ning; Fo ecas ing; GDP; Po ugal; G adien Boos ing Reg esso
Sus ainable De elopmen Goals (SDG):

iii
TABLE OF CONTENTS
1. In oduc ion ............................................................................................................. 1
2. Li e a u e e iew ..................................................................................................... 3
3. Me hodology ........................................................................................................... 6
3.1. Me hodology o he hesis ................................................................................ 6
3.2. OECD’s Me hodology ........................................................................................ 7
4. Da a ......................................................................................................................... 8
5. Machine lea ning echniques and model................................................................ 11
5.1. Ou lie s ........................................................................................................... 11
5.2. Fea u e selec ion ............................................................................................ 12
5.3. G adien Boos ing Reg esso ........................................................................... 13
5.4. Rolling window echnique ............................................................................... 15
5.5. Mul iple s eps ahead app oach ....................................................................... 18
6. Resul s ................................................................................................................... 22
7. Conclusions ............................................................................................................ 26
Bibliog aphical Re e ences.......................................................................................... 27
Appendix A ................................................................................................................. 32
Appendix B ................................................................................................................. 33
Appendix C ................................................................................................................. 34
Appendix D ................................................................................................................. 35
Appendix E ................................................................................................................. 40
Appendix F ................................................................................................................. 42
Appendix G ................................................................................................................. 43
Appendix H ................................................................................................................. 44
i
LIST OF FIGURES
Figu e 1 – T ain, alida ion, es spli ....................................................................................15
Figu e 2 – C oss alida ion ....................................................................................................16
Figu e 3 – Ancho ed olling window app oach ......................................................................17
Figu e 4 – Non-ancho ed olling window app oach ..............................................................17
Figu e 5 – Linea g aph o OECD’s and Machine Lea nings o ecas s o eal GDP g ow h o
Po ugal compa ed wi h eal alues ..............................................................................22
LIST OF TABLES
Table 1 - Table o me ics o RMSE, MPE, and R2 o OECD’s and Machine Lea nings o ecas s o
eal GDP o Po ugal g ow h compa ed wi h eal alues ..................................... 23
i
LIST OF ACRONYMS
ADE A bi a ed Dynamic Ensemble
AR Au o-Reg essi e
ARIMA Au o-Reg essi e In eg a ed Mo ing A e ages
BMA Bayesian Model A e aging
CPI Consume P ice Index
CRISP-DM C oss Indus y S anda d P ocess o Da a Mining
GDP G oss Domes ic P oduc
IMA In eg a ed Mo ing A e age
LGBM Ligh G adien Boos ing Machine
MIMO Mul i-Inpu Mul i-Ou pu
ML Machine Lea ning
MLP Mul i-Laye Pe cep on
MPE Mean Pe cen age E o
NiGEM Na ional Ins i u e Global Econome ic
NN Neu al Ne wo k
OECD O ganiza ion o Economic Coope a ion and De elopmen
R2 R-squa ed
RF Random Fo es
Ac onym RMSE Roo Mean Squa ed E o
Ac onym SARIMA Seasonal Au o eg essi e In eg a ed Mo ing A e age
Ac onym SVM Suppo Vec o Machine
Ac onym US Uni ed S a es
Ac onym VAR Vec o Au o-Reg essi e
Ac onym XGB Ex eme G adien Boos ing
7
A e a comp ehensi e analysis o he indings and he de elopmen o six unique models o
each subsequen s ep, i was de e mined ha he mos e ec i e model, which op imizes
RMSE, MPE, and R2, is he GBR ac oss all six s eps ahead. The me hodology employed o
cons uc hese models adhe ed o he CRISP-DM. This in ol ed he use o ailo ed echniques
o ime se ies o ecas ing wi hin supe ised models. These echniques included me hods such
as he olling window and he mul iple s eps-ahead di ec s a egy. A mo e de ailed
explana ion o hese echniques will be p o ided in sec ion 5.
3.2. OECD’S METHODOLOGY
The OECD gene a es wo annual o ecas s, which encompass no only Po ugal's GDP g ow h
bu also a a ie y o mac oeconomic a iables o all membe coun ies. These o ecas s a e
p esen ed in epo s known as "Economic Ou look". The c ea ion o hese o ecas s is based
on a combina ion o model-based analyses, s a is ical indica o models, and e en he
judgmen o expe s and pee e iew p ocesses (OECD, 2013). In ecen yea s, he OECD has
enhanced i s o ecas ing capabili y by inco po a ing nowcas ing o he cu en and
subsequen qua e s. The p ima y ool o o ecas ing consis s o he Na ional Ins i u e Global
Econome ic (NiGEM) model. This model is p e e ed o i s eliabili y o e s a is ical models
ha u ilize high- equency indica o s o nea - e m o ecas ing (Tu ne , 2016). The da a ha
he OECD u ilizes comp ises bo h "so " indica o s, such as business sen imen and consume
su eys, and "ha d" indica o s, which include commodi y p ices, exchange a es, and
indus ial p oduc ion (OECD, 2013).
The o ecas ing p ocess o he OECD is la gely au oma ed, acili a ed by a Fo ecas En y
Sys em ha cen alizes he necessa y da a. This sys em enables expe s associa ed wi h
di e en coun ies o e i y assump ions, ou pu s, and new in o ma ion. I also allows hem
o adjus necessa y pa ame e s o achie e he bes possible esul s. This p ocess akes in o
conside a ion bo h da a-d i en p ocesses and expe knowledge o each coun y's cu en
economic and geopoli ical en i onmen (OECD, 2013). This combina ion o da a-d i en
p ocesses and expe knowledge ensu es a comp ehensi e and accu a e o ecas ing p ocess,
conside ing bo h quan i a i e da a and quali a i e insigh s.
a ailable. We a e mind ul o hese a ia ions and hei implica ions ( e e o Appendix B o guidance on accessing
he la es his o ical da a ele an o OECD o ecas s a he ime).

8
4. DATA
The o ecas ing exe cise is based on a comp ehensi e se o qua e ly mac oeconomic da a
pe aining o Po ugal, which spans om he hi d qua e o 1962 o he second qua e o
2022. The a iables a e adjus ed o calenda e ec s and seasonali y, linked o he 2016
olume, and measu ed in he na ional cu ency
4
. In o al, he e a e nine a iables, each wi h
a ange o 86 o 227 obse a ions. These nine a iables we e selec ed om an ini ial se o 51
h ough a igo ous ea u e selec ion p ocess (see subsec ion 5.2) aimed a maximizing model
accu acy. Mos o hese a iables we e d awn om he OECD’s economic ou look da abase
(OECD, 2019), as well as o he s udies, including Bolhuis & Rayne 's (2020) o ecas o Tu key’s
g ow h and Haque's (2023) o ecas o Uni ed S a es GDP g ow h. The selec ed a iables a e:
 Real GDP g ow h o Po ugal.
 Real GDP g ow h o Spain.
 Real GDP g ow h o F ance.
 Real GDP g ow h o he Uni ed Kingdom.
 Real GDP g ow h o he Uni ed S a es o Ame ica.
 Cu ency spo exchange a e o he US dolla o he na ional cu ency o Po ugal.
 Final Consump ion expendi u e o Po ugal.
 Recession.
 Yea .
These a iables s ood ou p ima ily due o hei signi ican ela ionship wi h he a ge . In
o ecas ing mos ime se ies, pa icula ly hose exhibi ing s ong au oco ela ion in he a ge
a iable, his o ical da a i sel is c ucial (Beaumon e al., 1984). Conside ing ha o he s udies
ha e e i ied ha GDP g ow h ha ing a s ong au oco ela ion e ec (A bia & Pi as, 2005), we
ocused on u ilizing a iables such as he lag o Real GDP g ow h o Po ugal, ou a ge
a iable, while also conside ing pas ecessions and he yea o enhance ou models'
unde s anding o he empo al e olu ion o he a ge .
Fo he o he a iables, nume ous s udies, bo h empi ical and heo e ical, including hose by
Doyle & Faus (2002), demons a e a obus co ela ion be ween GDP g ow h among
coun ies, which is inc easingly s eng hened by he apid globaliza ion o he wo ld economy.
This co ela ion is pa icula ly e iden among ading pa ne s, as no ed by Jouini (2015),
whe e ade plays a pi o al ole in d i ing his associa ion. Gi en ha Spain, F ance, he Uni ed
4
The e a e excep ions o he abo e-men ioned indica ions. In e ms o chain linked alues, OECD u ilizes
coun ies' s a is ics, consequen ly, Spain's da a is linked o 2015 olume, F ance's o 2014 he Uni ed Kingdom’s
o 2018, and he Uni ed S a es o 2017, a he han being ied o 2016 olume. (OECD ECONOMIC OUTLOOK
Da abase In en o y, 2019).
9
Kingdom, and he Uni ed S a es (US) a e Po ugal’s majo expo des ina ions (OECD, 2017),
i 's clea why hese ou a iables we e p ominen ly ea u ed.
Addi ionally, Anchelache e al. (2015) iden i ies inal consump ion expendi u e as a signi ican
p edic o o GDP g ow h in mul iple linea eg ession models. Mo eo e , exchange a es ha e
been ecognized as in eg al o a coun y's economic g ow h, s imula ing i s economy
(Macdonald, 2010).
All he a iables men ioned abo e we e sou ced om he OECD da abank due o hei
accessibili y and o ensu e di ec compa abili y wi h hei o ecas s. The only excep ion is he
cu ency spo exchange a e o he US dolla o he na ional cu ency o Po ugal, which was
e ie ed om he Fede al Rese e Economic Da a.
While mac oeconomic da a is eadily a ailable and comes om eliable sou ces, i poses a
signi ican challenge o ML due o i s low equency. Typically, his da a is only a ailable down
o a qua e ly basis and lacks an ex ensi e his o ical ange. Al hough mos o he a iables
used in he inal o ecas da e back o he hi d qua e o 1962, many a iables analysed in
ea u e selec ion, such as inal consump ion expendi u e, he ne alue o p ima y income,
and many mo e, s a la e . This limi a ion hinde s o ecas ing e o s, especially conside ing
ha hal o he models being used, namely GBR and RF, canno handle null alues (Ben éjac
e al., 2021). Fu he mo e, ML h i es in da a- ich en i onmen s (Jean e al., 2016). To add ess
his issue, h ee con igu a ions o he da a we e de eloped: one including a iables da ing
back o he hi d qua e o 1962, ano he om he i s qua e o 1998, and ano he om
he i s qua e o 2001
5
. This will allow us o unde s and which is he op imal ade-o
be ween obse a ions and he numbe o a iables.
Fu he mo e, o gain a deepe unde s anding o how he models espond o he emo al o
da a poin s, we will conduc a sensi i i y analysis on he bes models wi h espec o a ious
his o ical pe iods. This analysis ocuses on he ime pe iods a he han he exis ence o
absence o a iables, as abo e men ioned, o de e mine i concep d i should be
inco po a ed in o ou o ecas . Concep d i being he idea ha ecen da a is mo e ele an
o unde s anding e ol ing economic dynamics, and emo ing ou da ed da a poin s can
imp o e accu acy (Woloszko, 2020).
The his o ical pe iods conside ed s a om 1962, he ea lies a ailable da a poin , du ing
Po ugal's ascis dic a o ship e a. We also examine da a s a ing om 1979, a ew yea s a e
he es ablishmen o he cu en democ a ic s a e. Ano he key pe iod begins in 1998, when
Po ugal was well-es ablished in he Eu opean Union. Addi ionally, we conside da a om
2002, ma king he u n o he new cen u y, he implemen a ion o he eu o, and ollowing he
5
I 's impo an o no e ha he a iables used in he da ase da ing back o he hi d qua e o 1962 a e also
p esen in he pe spec i es o 1998 and 2001. Addi ionally, he a iables om 1998 a e included in he 2001
con igu a ion. Da a is included in he analysis i i is a ailable.
10
11 h o Sep embe o 2001, e o is a acks in he US. Las ly, we analyse da a s a ing om
2009, in he a e ma h o he 2008 economic ecession (Eu ydice, 2024).
A e ex ensi e ea u e selec ion and model e alua ion, he op imal ade-o be ween
obse a ions and he numbe o a iables led us o choose he abo e-men ioned ea u es.
The insu iciency and low equency o da a conce ning Po ugal's mac oeconomic a iables
will be u he elabo a ed upon in sec ion 6 o he pape .
Be o e u ilizing he da a o modelling, se e al ans o ma ions we e pe o med, including
no maliza ion using he s anda d scale . While his me hod isn' ideal due o one o ou
o ecas ing objec i es being he a oidance o un ealis ic assump ions, pa icula ly ega ding
he s a iona i y o inancial and economic ime se ies, i is necessa y because no maliza ion
imp o es machine lea ning o ecas s. The o he commonly used me hods, such as min-max
scaling and decimal scaling, a e e en less sui able as hey equi e knowledge o he minimum
and/o maximum alues o a ime se ies, which is imp ac ical o ou ea u es and a ge s since
u u e da a may all ou side he ange o he aining da a as indica ed by Ogasawa a e al.
(2019).
Fu he mo e, we make he con e sion o absolu e alues o hei pe cen age change o
acili a e he o ecas ing o he a ge , which in his case is he GDP g ow h a e ( o mo e
de ailed in o ma ion ega ding he ea u es, please e e o Appendices C and D).
The modelling was de eloped using Py hon, wi h he Sciki -lea n lib a y being u ilized.
This lib a y is a popula choice o ML due o i s e iciency and e sa ili y, o e ing a ange o
supe ised lea ning algo i hms.
11
5. MACHINE LEARNING TECHNIQUES AND MODEL
In his sec ion, we will p o ide a de ailed o e iew o he p ocedu es, me hodologies, and
amewo ks ha we e u ilized in he de elopmen o ou models. The sec ion will be di ided
in o se e al key componen s. Fi s , we will discuss he handling o ou lie s, o a he , he
decision o no handle hem in his con ex . Nex , we will del e in o he p ocess o ea u e
selec ion, whe e we will elabo a e on he me hodologies ha we e employed o de e mine
he a iables ha we e used in he a ious models. Following his, we will discuss he speci ic
model ha was employed in ou s udy. We will hen ou line he hype - uning echnique ha
was applied, which in ol ed he use o a olling window app oach. Finally, we will explain how
we o ecas ed mul iple s eps. Each o hese componen s plays a c ucial ole in he o e all
modelling p ocess, and we will p o ide a comp ehensi e explana ion o each o p o ide a clea
unde s anding o ou app oach.
5.1. OUTLIERS
In he ealm o ML, a c i ical s ep in he p ep ocessing phase is he iden i ica ion and
subsequen handling o ou lie s. These unique da a poin s, despi e hei po en ial a i y, can
exe a signi ican in luence on o ecas ing ou comes, o en disp opo iona ely compa ed o
o he obse a ions. This in luence becomes pa icula ly p onounced when dealing wi h
complex da ase s, such as hose in ol ing mac oeconomic a iables. Ou lie s ha e he
po en ial o skew subsequen obse a ions and can p o ide aluable insigh s in o he complex
in e play be ween di e en a iables. Fu he mo e, ou lie s ha may seem highly unlikely
could s ill mani es in he u u e, he eby a ec ing he accu acy o o ecas s (Boje , 2022;
Rožanec e al., 2021).
In he con ex o ou esea ch, i 's impo an o no e ha ou aining da a does no include
pe iods ha we e in luenced by unp edic able ex e nal economic e en s such as he COVID-
19 pandemic o he con lic in Uk aine. Gi en he low equency o ou da a, we made he
conscious decision o e ain all obse a ions, choosing no o emo e any da a poin s based
on hem being ou lie s. This decision was made wi h he unde s anding ha hese ou lie s
could po en ially p o ide aluable insigh s and con ibu e o he obus ness o ou o ecas ing
model.
12
5.2. FEATURE SELECTION
In he con ex o ea u e selec ion, we me iculously c a ed a comp ehensi e p ocess. This
p ocess was mul i ace ed, in ol ing he e alua ion o ea u e impo ance ou comes de i ed
om a a ie y o me hods, he in eg a ion o hese ou comes, he comple e dis ega d o hese
ou comes, and he exclusi e conside a ion o he lagged a ge . This in ica e app oach was
implemen ed ac oss h ee sepa a e da ase s, each cha ac e ized by di e en quan i ies o
obse a ions and ea u es, as desc ibed in he six h pa ag aph o sec ion 4. The need o such
di e en ia ion a ose om he di e se ela ionships ha exis be ween ea u es, especially
when ewe a iables o obse a ions come in o play (Jo ić e al., 2015).
The me hods we employed in ou app oach included Spea man's co ela ion, embedded
echniques, and w appe me hods ha le e age decision ees. These me hods we e chosen
o e al e na i es such as ecu si e ea u e selec ion, Lasso embedded me hods, and
dimensionali y educ ion echniques. The p ima y eason o his selec ion was ha he la e
g oup o me hods elies on linea ela ionships be ween a iable (Jo ić e al., 2015).
The main d i ing o ce behind ou decision o use ML o e econome ics was ML's abili y o
ci cum en he assump ions ha a e deeply ing ained in adi ional s a is ical me hods.
The e o e, i we we e o e ain echniques ha a e cen ed a ound linea ela ionships, i
would be in di ec con adic ion wi h he undamen al objec i e o his hesis. This objec i e
is o explo e and le e age he powe o ML in unde s anding and in e p e ing complex
ela ionships be ween a iables, which adi ional s a is ical me hods migh no be able o
ully cap u e due o hei inhe en assump ions (Kaushik, 2020).
In essence, ou ea u e selec ion p ocess was designed o be obus and lexible, capable o
adap ing o he unique cha ac e is ics o each da ase and he complex ela ionships be ween
hei ea u es. This app oach allows us o ully ha ness he po en ial o ML in ou esea ch.

13
5.3. GRADIENT BOOSTING REGRESSOR
The GBR is a supe ised ML model ha is speci ically designed o eg ession asks. I ope a es
using a sequen ial ensemble app oach, which in ol es he u iliza ion o base-lea ne s o
i e a i ely e ine i s p edic ions.
The unique s eng h o GBR lies in i s abili y o enhance i s o ecas s wi h each i e a ion by
lea ning om he e o s o p e ious p edic ions. This i e a i e lea ning p ocess allows GBR o
con inuously imp o e i s p edic i e accu acy, making i a powe ul ool o eg ession asks.
This p ocess o lea ning om pas e o s and e ining u u e p edic ions is a de ining
cha ac e is ic o GBR and is wha dis inguishes i om o he ensemble models (Na ekin &
Knoll, 2013).
Models like RF, while also being ensemble models, do no possess his i e a i e lea ning
capabili y. Ins ead, hey ely on he simple agg ega ion o he ou pu s o hei base-lea ne s,
wi hou any mechanism o lea ning om pas e o s o e ining u u e p edic ions. This
di e ence in app oach is wha se s GBR apa and con ibu es o i s supe io pe o mance in
eg ession asks (Na ekin & Knoll, 2013).
The equa ion below illus a es he sum o es ima ion unc ions, he key componen desc ibed
abo e. This equa ion ep esen s he cumula i e e ec o he es ima ions, which a e e ined
i e a i ely o imp o e he model's p edic ions. Howe e , i 's impo an o no e ha his
equa ion does no ye show he sum o he base-lea ne s, which is ano he c ucial pa o he
GBR model's ope a ion.
𝑓󰆹(𝑥)= ∑𝑓𝑖

𝑀
𝑖=0 (𝑥),
(1)
Whe e 𝑓󰆹(𝑥) deno es he es ima ion unc ion, 𝑀 is numbe o i e a ions, and he i s i e a ion
is he ini ial guess o he model.
Building on he p e ious discussion, he GBR model's ope a ion is unde pinned by se e al key
componen s. These include he loss unc ion, he base lea ne , and he numbe o i e a ions.
The loss unc ion se es as a me ic, quan i ying he dispa i y be ween he p edic ions
gene a ed by he model and he ac ual da a. The base lea ne s, which a e inco po a ed
i e a i ely manne o enhance he model's o ecas s, can be cons uc ed using a a ie y o
me hods, including decision ees and splines, among o he s (Na ekin & Knoll, 2013).
This i e a i e lea ning p ocess is encapsula ed in he equa ion ha ollows. This equa ion
demons a es how each i e a ion o he model is compu ed, wi h each new p edic ion being
14
he sum o he p e ious es ima ion unc ion and he p oduc o he base lea ne and he
lea ning a e. This equa ion, he e o e, p o ides a ma hema ical ep esen a ion o he GBR
model's i e a i e lea ning p ocess, highligh ing how i con inuously e ines i s p edic ions o
imp o e accu acy (Na ekin & Knoll, 2013).
𝑓𝑖
= 𝑓𝑖−1
 + ρ𝑖ℎ(𝑥,θ𝑖),
(2)
Whe e 𝑓𝑖
 deno es he es ima ion unc ion o he i e a ion 𝑖, 𝑓𝑖−1
 he esul o he p e ious
i e a ion o he es ima ion unc ion, ρ𝑖 being he lea ning a e associa ed o he base-lea ne
o said i e a ion, and ℎ(𝑥,θ𝑖) he base lea ne o he i e a ion.
The ope a ional p ocess o he GBR algo i hm can be b oken down in o he ollowing s eps:
1. Ini ializa ion: The es ima ion unc ion is ini ialized wi h a cons an alue, ypically
he mean o he a ge a iable.
2. I e a ion Loop:
a. Calcula e Nega i e G adien : The nega i e g adien o he loss unc ion
wi h espec o he p edic ed alues is compu ed a each i e a ion.
b. Fi a New Base Lea ne : A new base lea ne is i ed o he nega i e
g adien calcula ed in he p e ious s ep.
c. Upda e Es ima ion Func ion: The es ima ion unc ion is upda ed by
adding he p edic ion o he new base lea ne , which is mul iplied by a
lea ning a e, o he p e ious es ima ion unc ion.
GBR s ands ou as an ensemble model ha is highly p o icien a cap u ing complex non-linea
unc ions. I s e ec i eness in o ecas ing eal-wo ld da a has been ex ensi ely documen ed.
One o he key s eng hs o GBR is i s excep ional lexibili y in cus omiza ion, pa icula ly in
he selec ion o loss unc ions and base lea ne s. Howe e , his e sa ili y comes wi h a ade-
o in e ms o compu a ional cos . Due o i s i e a i e op imiza ion app oach, GBR is mo e
compu a ionally in ensi e compa ed o o he ensemble models such as RF (Na ekin & Knoll,
2013).
In ou o ecas ing model, we u ilized he mean squa ed e o loss unc ion. Al hough his is
no one o ou p ima y me ics, i assis s in op imizing he RMSE and R2, while also penalizing
signi ican e o s. Fu he mo e, we chose decision ees as ou base lea ne s. This choice was
15
made because decision ees do no ely on assump ions o linea i y, hey con ibu e o he
di e si y o he model, and hey p o ide compu a ional e iciency (Chicco e al., 2021).
5.4. ROLLING WINDOW TECHNIQUE
When i comes o ine- uning pa ame e s o ML models, he e a e a ious app oaches. In his
subsec ion, we will s a by desc ibing one o he simples me hods: he ain, alida ion, and
es me hod. This will be ollowed by an explana ion o he c oss- alida ion me hod. Finally,
we will discuss he olling window echnique, including i s wo a ian s, which is he me hod
we u ilized.
In he p ocess o enhancing he pe o mance o a model ia hype pa ame e uning, a
p e alen me hodology in ol es he di ision o he da ase in o h ee dis inc segmen s:
aining, alida ion, and es se s ( e e o Figu e 1). The aining da a se es he pu pose o
i ing he model, u ilizing he pa ame e s ha ha e been de ined. On he o he hand, he
alida ion se is exclusi ely se aside o he ask o choosing he mos e ec i e model, he eby
a oiding any bias ha could po en ially be in oduced i he es se was used o his pu pose.
The inal segmen , he es se , is hen employed o assess he pe o mance o he selec ed
model when i is exposed o da a ha i has no encoun e ed be o e (Fi & Ga iga, 2010). This
app oach ensu es a obus and unbiased e alua ion o he model's abili y o gene alize om
he aining da a o unseen da a, which is a c i ical aspec o model pe o mance in p ac ical
applica ions.
Figu e 1 – T ain, alida ion, es spli .
16
In ci cums ances whe e he a ailable da a is sca ce, such as in ou case, he adi ional
app oach o di iding he da a in o aining, alida ion, and es se s can esul in a subs an ial
educ ion o aining da a. To coun e ac his issue, he echnique o k- old c oss- alida ion is
equen ly u ilized. This me hod de ia es om he igid pa i ioning in o aining, alida ion,
and es se s. Ins ead, i in ol es he di ision o he aining da a in o k- olds, which a e
gene ally subse s o equal size. Each o hese olds is subsequen ly used as a alida ion se ,
while he model is ained using he emaining k-1 olds. This p ocedu e is ca ied ou o each
old, ensu ing ha e e y da a poin is used o bo h aining and alida ion pu poses ( e e o
Figu e 2). Consequen ly, his allows o a mo e comp ehensi e assessmen o he model's
pe o mance, wi hou he need o o ei aluable da a (Fi & Ga iga, 2010).
Figu e 2 – C oss- alida ion echnique.
23
Table 1 – Table o me ics o RMSE, MPE, and R2 o OECD’s and Machine Lea nings o ecas s
o eal GDP g ow h o Po ugal compa ed wi h eal alues.
Fo ecas
Me ic
All ins ances
(10 ins ances)
1-2 s eps-ahead
Q3-Q4 2019 &
Q3-Q4 2022
(4 ins ances)
3-6 s eps-ahead
Q1-Q4 2022
(4 ins ances)
OECD
RMSE
5.8373
1.5432
3.6095
MPE
-94.58%
45.99%
-282.65%
R2
0.0891
-4.0518
-0.0041
ML
RMSE
10.0153
0.3279
2.3132
MPE
-163.14%
19.59%
-448.86%
R2
-0.5628
-0.0735
0.5876
No e: he eal alues o Po ugal’s eal GDP g ow h we e e ie ed om he Economic Ou look o Decembe
2023 o OECD.
Upon examina ion, i was ound ha each o he six indi idual models, ained o s ep-by-
s ep o ecas ing om one o six s eps ahead, achie ed hei highes accu acy using he GBR
model. Addi ionally, ou ou o he six models used da a s a ing om 1962. while wo models
used da a s a ing om 1998 when only conside ing he ime pe iods o 1962, 1998, and 2001.
I was also e i ied ha ou o he six models employed a non-ancho ed me hod o he olling
window echnique, whe eas he emaining wo used he ancho ed me hod. Rega ding he
model pa ame e s, no speci ic pa e n was obse ed, as each indi idual model p edominan ly
used unique se s o maximize hei accu acy in es ing (see Appendix E).
Rega ding he sensi i i y analysis ac oss he ime pe iods o 1962, 1979, 1998, 2002, and 2009
o he bes models and espec i e pa ame e s in each s ep-ahead, we obse e ha o he
i s s ep-ahead, he bes me ics we e 0.60 o RMSE, -7% o MPE, and 0.29 o R², s a ing
om 1962. Fo he second s ep-ahead, he bes me ics we e 0.31 o RMSE, -55% o MPE,
and -0.47 o R², s a ing om 1998. Fo he hi d s ep-ahead, he bes me ics we e 0.76 o
RMSE, 14% o MPE, and -0.12 o R², s a ing om 1962. Fo he ou h s ep-ahead, he bes
me ics we e 0.85 o RMSE, -8% o MPE, and -0.42 o R², s a ing om 1962. Fo he i h
s ep-ahead, he bes me ics we e 0.80 o RMSE, 12% o MPE, and -0.25 o R², also s a ing
om 1962. Fo he six h s ep-ahead, he bes me ics we e 0.81 o RMSE, -30% o MPE, and
-0.28 o R², s a ing om 1998 (see Appendix F). We can e i y ha he op imal ime pe iods
a e he same as when analysing wi h 1962, 1998, and 2001, as seen abo e.
Al hough i was e i ied ha o mos s eps he bes ime pe iod o conside is s a ing om
1962, we mus conside ha no single ime span maximizes all h ee me ics in es o any o
he s eps-ahead. Fo ins ance, in he i s s ep, s a ing om he mos ecen pe iod, ha being
2009, we achie e he lowes RMSE wi h a alue o 0.25. The bes MPE is wi h he da a s a ing
in 1962 wi h -7%, and o R², i is s a ing om 1979, wi h a alue o 0.36. We a i ed a he
abo e-men ioned conclusions using he same me hod de eloped o ind he op imal

24
pa ame e s and bes model o he o iginal o ecas (see Appendix G). I we we e o only
conside RMSE, 2009 would be he bes s a ing poin o all he s eps, showcasing how much
he e alua ion me hod can a ec he conclusions o analysis and showing how pa amoun i ’s
o ha e an a ay o me ics o analyse (see Appendix F).
Gi en he ou comes discussed ea lie we can now discuss an a ay o conclusions and
ques ions. We will s a o wi h he mos p ominen ques ion ega ding he inal o ecas and
ha will a ec di ec ly i ou hypo hesis was con i med, how did he OECD manage o
ou pe o m ML so signi ican ly in he o e all assessmen ? The disc epancy s ems om he
p edic ions o he hi d qua e o 2021 when employing one s ep-ahead o ecas . While he
OECD nea ly achie ed a pe ec o ecas , displaying a RMSE o 0.27 and a MPE o 2%, he ML
model's o ecas s ell sho . The ML model exhibi ed signi ican unde pe o mance wi h a
RMSE o 8.71 and a MPE o 70%.
So, a e hese esul s, we mus conside why exac ly ML was unable o o ecas he hi d
qua e o 2021 in such an e ec i e way as he ones o 2019 and 2022 ela i e o OECD’s
o ecas .
In add essing he ini ial inqui y, i 's impe a i e o acknowledge ha ML excels in da a- ich
se ings. Ou o ecas ing elies on mac oeconomic da a a ailable only qua e ly, which can
hinde i s o ecas ing accu acy compa ed o mo e con en ional me hods like hose employed
by he OECD (Lim & Zoh en, 2021). In ce ain cases, ML o ecas s may e en pe o m wo se
han simple models (Mak idakis e al., 2018). Mo eo e , ML ou comes a e highly sensi i e o
a ailable da a; a ious s udies indica e ha he same model can ou pe o m econome ic
me hods in speci ic scena ios while unde pe o ming in o he s, depending on he da ase
u ilized (Boje , 2022; Woloszko, 2020). I 's wo h men ioning ha while ML was no able o
ou pe o m hose o he OECD, as e idenced in ou esul s (see g aph 1 and able 1), hey
we e signi ican ly penalized o hei inaccu a e p edic ion o he i s qua e o 2021 ela i e
o he OECD's o ecas .
Rega ding he o he conclusions, in ou analysis o a ious o ecas ing models, i was
consis en ly e iden ha GBR ou pe o med o he models ac oss all o ecas ho izons in e ms
o RMSE, MPE, and R2 me ics. Addi ionally, i was e i ied a e sensi i e analysis ha , o he
mos pa , he models pe o med bes when u ilizing he g ea es numbe o ins ances as
possible, using da a s a ing om 1962 and he use o non-ancho ed olling window
echnique. These obse a ions unde sco e he signi icance o concep d i by p io i ising a
highe numbe o ins ances o e only conside ing mo e ecen da a. Such happens due o ML
models exhibi ing a p e e ence o highe equency da a, as lowe ing he numbe o
obse a ions u he de e io a es o ecas accu acy gi en he al eady low equency o
economic da a.
Despi e he ad an age o ML in iden i ying complex pa e ns wi hou elying on linea
assump ions, ou ex ensi e ea u e selec ion p ocess educed he ini ial pool o 51 a iables
25
o jus a ew essen ial ones. Mos o he a iables used demons a ed a s ong linea
ela ionship wi h he a ge , as e idenced by Spea man’s co ela ion (check Appendix H).
O he s udies, such as Woloszko (2020), ha e eached simila conclusions, no ing ha
iden ical ML models pe o m be e wi h a iables ha exhibi linea ela ionships wi h he
a ge .
In essence, he hesis did no con i m i s hypo hesis o "Can ML echniques, speci ically
ensembles o eg ession ees, a i e a be e esul s han OECD’s o ecas o he GDP
g ow h o Po ugal?” by e i ying ha ML did no ou pe o m OECD's o ecas s in all h ee
me ics (RMSE, MPE, and R2) o en di e en o ecas s spanning he las wo qua e s o 2019,
he las wo qua e s o 2021, and he en i e y o 2022 (wi h he las wo qua e s o ecas ed
wice). While ML o en su passed OECD p edic ions in a ious cases, i gene ally
unde es ima ed o e all economic g ow h, consis en ly displaying lowe MPE alues compa ed
o OECD o ecas s. Addi ionally, ML o ecas s occasionally di e ged signi ican ly om ac ual
alues, no ably obse ed in he hi d qua e o 2021. Hence, i 's c ucial o subjec ML
o ecas ing echniques o ho ough e alua ion agains es ablished benchma ks such as OECD
be o e widesp ead adop ion in economic ins i u ions, as ad oca ed in ecen li e a u e (Boje ,
2022).
26
7. CONCLUSIONS
The objec i e o his hesis was o e alua e whe he ML could su pass o ecas s p oduced by
es ablished ins i u ions in o ecas ing Po ugal’s GDP g ow h using ensemble eg ession ee
models whils using OECD’s economic ou look as a benchma k. The esul s e eal a complex
pic u e o ML's s eng hs and limi a ions in economic o ecas ing.
The p ima y indings indica e ha while ML achie ed supe io o ecas s compa ed o he
OECD in mos cases, i ul ima ely unde pe o med due o one signi ican ou lie ha de ia ed
subs an ially om he ac ual alue. This ou lie unde sco es he sensi i i y o ML models o
speci ic da a poin s and hei po en ial o la ge e o s in ce ain scena ios, sugges ing ha
ML is, o e all, less eliable han o ecas s p oduced by es ablished ins i u ions.
Se e al ac o s con ibu ed o hese esul s. ML models h i e in da a- ich en i onmen s and
pe o m be e wi h high equency da a. Howe e , he eliance on qua e ly mac oeconomic
da a cons ained hei pe o mance. Addi ionally, al hough ML can cap u e complex pa e ns,
he bes -pe o ming models we e hose wi h ewe a iables, ypically exhibi ing s ong linea
ela ionships wi h he a ge , hus diminishing ML's ad an ages.
Fu he mo e, he s udy ound ha GBR was he mos e ec i e model o o ecas ing
Po ugal’s GDP g ow h ac oss all ime ho izons. I also showcased a p e e ence o mo e
obse a ion poin s o e a la ge numbe o a iables sugges ing ha ML models alue
his o ical da a e en when i appea s less ele an o u u e o ecas s.
In conclusion, while ML models ha e shown po en ial in speci ic ins ances, hei o e all
pe o mance in his s udy was no be e han o ecas s p oduced by es ablished ins i u ions
o GDP g ow h o ecas ing. The indings sugges ha ML should be iewed as a
complemen a y ool a he han a subs i u e o es ablished echniques. The mixed esul s
highligh he impo ance o benchma king ML o ecas s agains adi ional me hods be o e
conside ing hei b oade applica ion in economic o ecas ing. Fu u e esea ch should explo e
in eg a ing ML wi h o ecas s p oduced by es ablished ins i u ions o le e age he s eng hs
o bo h app oaches, po en ially leading o mo e obus and accu a e o ecas s.
27
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32
APPENDIX A
Table wi h he da a a ailable a he ime o he issue ega ding eal GDP g ow h a e o
Po ugal o OECD om pe iod o pe iod, exp essed annually.
Da e
No embe
2019
Decembe
2021
Decembe
2022
No embe
2023
Di e ence be ween No embe
2023 and alues used o he
o ecas
Q1-2019
2.33%
3.57%
3.58%
3.52%
1.19%
Q2-2019
2.30%
2.28%
2.30%
2.37%
0.06%
Q3-2019
-
1.86%
1.82%
1.82%
-
Q4-2019
-
3.34%
3.32%
3.38%
-
Q1-2020
-
-16.52%
-16.32%
-16.40%
-
Q2-2020
-
-48.41%
-48.09%
-48.06%
-
Q3-2020
-
72.92%
72.41%
72.50%
-
Q4-2020
-
1.10%
1.49%
1.41%
-
Q1-2021
-
-12.48%
-9.89%
-9.39%
3.09%
Q2-2021
-
19.32%
18.91%
19.03%
-0.29%
Q3-2021
-
-
11.60%
12.47%
-
Q4-2021
-
-
7.98%
7.79%
-
Q1-2022
-
-
9.88%
9.57%
-0.31%
Q2-2022
-
-
0.51%
0.23%
-0.28%
Q3-2022
-
-
-
1.98%
-
Q4-2022
-
-
-
2.05%
-
No e: The only alues used o he o ecas we e he wo lags be o ehand, he e o e, he only
ele an alues o he o ecas a e he i s and second qua e s o 2019, 2021, and 2022.
39
Enginee ed
Taxes on p oduc ion and
impo s less subsidies:
axes and subsidies on
p oduc ion and impo s
ne pe cen age
Pe cen age change om pe iod o
pe iod; adjus ed o calenda and
seasonal ac o s; chain-linked 2016
Calcula ion o :
(Taxes on p oduc ion and impo s
less subsidies: axes on p oduc ion
and impo s - Taxes on p oduc ion
and impo s less subsidies: subsidies
on p oduc ion and impo s) / Taxes
on p oduc ion and impo s less
subsidies
1998-Q1 o
2022-Q2
No
Enginee ed
Qua e
The qua e being analysed
1960-Q2 o
2022-Q2
No
Enginee ed
Mon h
The mon h being analysed
1960-Q2 o
2022-Q2
No

40
APPENDIX E
Table wi h he de ails o he models de eloped conside ing each s ep-ahead.
S ep-
ahead
Model
Time
pe iod
Rolling
Window
me hod
Va iables
Pa ame e s
1s
GBR
1962-Q3 o
2019-Q2
Ancho ed
[Real GDP g ow h o Po ugal,
Real GDP g ow h o F ance,
Real GDP g ow h o Spain,
Real GDP g ow h o Uni ed S a es o
Ame ica,
Real GDP g ow h o Uni ed Kingdom,
Cu ency spo exchange a e o he US
dolla o he na ional cu ency o
Po ugal,
Recession,
Yea ]
es ima o s: 150
lea ning_ a e: 0.1
max_dep h: 3
min_samples_spli : 3
min_samples_lea : 3
2nd
GBR
1998-Q1 o
2019-Q2
Non-
ancho ed
[Real GDP g ow h o Po ugal]
es ima o s: 50
lea ning_ a e: 0.05
max_dep h: 1
min_samples_spli : 2
min_samples_lea : 6
3 d
GBR
1962-Q3 o
2019-Q2
Non-
ancho ed
[Real GDP g ow h o Po ugal,
Real GDP g ow h o F ance,
Real GDP g ow h o Spain,
Real GDP g ow h o Uni ed S a es o
Ame ica,
Real GDP g ow h o Uni ed Kingdom,
Cu ency spo exchange a e o he US
dolla o he na ional cu ency o
Po ugal,
Recession,
Yea ]
es ima o s: 50
lea ning_ a e: 0.2
max_dep h: 1
min_samples_spli : 3
min_samples_lea : 6
4 h
GBR
1962-Q3 o
2019-Q2
Ancho ed
[Real GDP g ow h o Po ugal,
Real GDP g ow h o F ance,
Real GDP g ow h o Spain,
Real GDP g ow h o Uni ed S a es o
Ame ica,
Real GDP g ow h o Uni ed Kingdom,
Cu ency spo exchange a e o he US
dolla o he na ional cu ency o
Po ugal,
Recession,
Yea ]
es ima o s: 50
lea ning_ a e: 0.05
max_dep h: 5
min_samples_spli : 4
min_samples_lea : 6
41
5 h
GBR
1962-Q3 o
2019-Q2
Non-
ancho ed
[Real GDP g ow h o Po ugal,
Real GDP g ow h o Spain,
Real GDP g ow h o Uni ed S a es o
Ame ica,
Cu ency spo exchange a e o he US
dolla o he na ional cu ency o
Po ugal,
Recession,
Yea ,
Da e]
es ima o s: 150
lea ning_ a e: 0.3
max_dep h: 3
min_samples_spli : 3
min_samples_lea : 3
6 h
GBR
1998-Q1 o
2019-Q2
Non-
ancho ed
[Real GDP g ow h o Po ugal,
Final Consump ion expendi u e o
Po ugal,
Real GDP g ow h o Spain,
Real GDP g ow h o Uni ed S a es o
Ame ica]
es ima o s: 100
lea ning_ a e: 0.1
max_dep h: 3
min_samples_spli : 2
min_samples_lea : 1
42
APPENDIX F
Table wi h he sensi i e analysis o he ime pe iods o 1962, 1979, 1998, 2002, and 2009 o
each s ep-ahead u ilizing he same models and pa ame e s as in he inal o ecas .
S ep-ahead
Time pe iod
RMSE
MPE
R2
1s
1962-Q3 o 2019-Q2
0.6024
-6.70%
0.2925
1979-Q1 o 2019-Q2
0.5364
-1196.64%
0.3594
1998-Q1 o 2019-Q2
0.6017
328.51%
-4.4709
2002-Q1 o 2019-Q2
0.4491
333.08%
-2.2401
2009-Q1 o 2019-Q2
0.2513
71.62%
-2.8569
2nd
1962-Q3 o 2019-Q2
0.741
-1871.51%
-0.0705
1979-Q1 o 2019-Q2
0.5463
-2630.83%
0.3355
1998-Q1 o 2019-Q2
0.3121
-54.9%
-0.4715
2002-Q1 o 2019-Q2
0.4395
220.1858%
-2.1016
2009-Q1 o 2019-Q2
0.1899
260.33%
-1.2019
3 d
1962-Q3 o 2019-Q2
0.7573
14.2%
-0.1182
1979-Q1 o 2019-Q2
0.6697
-1157.53%
0.6975
1998-Q1 o 2019-Q2
0.6975
183.06%
-6.3522
2002-Q1 o 2019-Q2
0.7735
-43.17%
-8.6078
2009-Q1 o 2019-Q2
0.3637
-125.2%
-7.0804
4 h
1962-Q3 o 2019-Q2
0.8534
-8.24%
-0.4206
1979-Q1 o 2019-Q2
0.7732
-673.07%
-0.3311
1998-Q1 o 2019-Q2
0.7421
178.26%
-7.3226
2002-Q1 o 2019-Q2
0.8424
234.64%
-10.397
2009-Q1 o 2019-Q2
0.1232
-141.15%
0.0729
5 h
1962-Q3 o 2019-Q2
0.8018
11.59%
-0.2536
1979-Q1 o 2019-Q2
0.9297
-1225.96%
-0.9244
1998-Q1 o 2019-Q2
1.2379
1074.24%
-22.154
2002-Q1 o 2019-Q2
0.7725
177.58%
-8.5816
2009-Q1 o 2019-Q2
0.5383
604.34%
-16.6966
6 h
1962-Q3 o 2019-Q2
0.9544
340.63%
-0.7763
1979-Q1 o 2019-Q2
0.7210
-1152.94%
-0.3272
1998-Q1 o 2019-Q2
0.8093
-30,00%
-0.2771
2002-Q1 o 2019-Q2
0.7250
679.96%
-8.5831
2009-Q1 o 2019-Q2
0.5660
612.86%
-18.5687
43
APPENDIX G
Me hod used o calcula e he bes model conside ing he me ics RMSE, MPE, and R2.
Gi en he di e en scales and complexi ies o he me ics u ilized, we employed a sys em o
no malize hem by using an a bi a y benchma k alue: an RMSE o 0.6618, an MPE o 73.33%,
and an R2 o -0.1959, which we e he bes alues om he o iginal op imal o ecas om
ea lie model de elopmen . I a model ma ches hese alues, i sco es 1; below 1 is be e ,
and abo e 1 is wo se. Al hough no pe ec due o he a bi a y alues, his me hod helps
de i e sensible conclusions.
44
APPENDIX H
Spea man’s co ela ion be ween Po ugal’s GDP g ow h and he a iables used in he
models.
Va iable
Co ela ion
Final Consump ion expendi u e o Po ugal
0.7065
Real GDP g ow h o F ance
0.5748
Real GDP g ow h o Spain
0.5451
Recession
-0.5421
Yea
-0.5114
Real GDP g ow h o he Uni ed S a es o Ame ica
0.2174
Real GDP g ow h o he Uni ed Kingdom
0.1406
Cu ency spo exchange a e o he US dolla o he na ional cu ency o Po ugal
0.0663

45