Comparative Analysis of GDP Forecasting using Ensemble Tree Regression Models: Machine Learning vs. Econometric Models

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 BIBLIOGRAPHICAL REFERENCES Asho eh, A., B a o, J. M., & Ayuso, M. (2022). 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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