Bayesian estimation and likelihood-based comparison of agent-based volatility models

Be schinge , Nils; Mozzho in, Iu ii A icle — Published Ve sion Bayesian es ima ion and likelihood-based compa ison o agen -based ola ili y models Jou nal o Economic In e ac ion and Coo dina ion P o ided in Coope a ion wi h: Sp inge Na u e Sugges ed Ci a ion: Be schinge , Nils; Mozzho in, Iu ii (2020) : Bayesian es ima ion and likelihood- based compa ison o agen -based ola ili y models, Jou nal o Economic In e ac ion and Coo dina ion, ISSN 1860-7128, Sp inge , Be lin, Heidelbe g, Vol. 16, Iss. 1, pp. 173-210, h ps://doi.o g/10.1007/s11403-020-00289-z This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/288932 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ Jou nal o Economic In e ac ion and Coo dina ion (2021) 16:173–210 h ps://doi.o g/10.1007/s11403-020-00289-z REGULAR ARTICLE Bayesian es ima ion and likelihood-based compa ison o agen -based ola ili y models Nils Be schinge 1,2 ·Iu ii Mozzho in2 Recei ed: 1 July 2019 / Accep ed: 26 May 2020 / Published online: 24 June 2020 © The Au ho (s) 2020 Abs ac The s a is ical desc ip ion and modeling o ola ili y plays a p ominen ole in econo- me ics, isk managemen and inance. GARCH and s ochas ic ola ili y models ha e been ex ensi ely s udied and a e ou inely i ed o ma ke da a, albei p o iding a phenomenological desc ip ion only. In con as , agen -based modeling s a s om he p emise ha mode n economies consis o a as numbe o indi idual ac o s wi h he - e ogeneous expec a ions and incen i es. Obse ed ma ke s a is ics hen eme ge om he collec i e dynamics o many ac o s ollowing he e ogeneous, ye simple ules. On he one hand, such models gene a e ola ili y dynamics, quali a i ely ma ching se e al s ylized ac s. On he o he hand, hey illus a e he possible ole o di e en mech- anisms, such as cha is ading and he ding beha io . Ye , igo ous and quan i a i e s a is ical i s a e s ill mos ly lacking. He e, we p opose Hamil onian Mon e Ca lo, an e icien and scalable Ma ko chain Mon e Ca lo algo i hm, as a gene al me hod o Bayesian in e ence o agen -based models. In pa icula , we implemen se e al models by Vik am and Sinha, F anke and Wes e ho and Al a ano, Lux and Wagne in S an, an accessible p obabilis ic p og amming language o Bayesian modeling. We also compa e he pe o mance o hese models wi h s anda d econome ic models o he GARCH and s ochas ic ola ili y amilies. We ind ha he bes agen -based models a e on pa wi h s ochas ic ola ili y models in e ms o p edic i e likelihood, ye exhibi challenging pos e io geome ies equi ing ca e in model compa ison and sophis ica ed sampling algo i hms. Keywo ds Agen -based models ·S ochas ic ola ili y models ·Bayesian es ima ion · Hamil onian Mon e Ca lo Elec onic supplemen a y ma e ial The online e sion o his a icle (h ps://doi.o g/10.1007/s11403- 020-00289-z) con ains supplemen a y ma e ial, which is a ailable o au ho ized use s. BNils Be schinge [email p o ec ed] 1F ank u Ins i u e o Ad anced S udies, F ank u am Main, Ge many 2Goe he Uni e si y, F ank u am Main, Ge many 123 174 N. Be schinge , I. Mozzho in JEL Classi ica ion C11 ·C52 ·C58 ·G12 I wonde who i was de ined man as a a ional animal. I was he mos p ema u e de ini ion e e gi en. Man is many hings, bu he is no a ional. Osca Wilde, The Pic u e o Do ian G ay and O he W i ings 1 In oduc ion Financial ma ke s exhibi some ema kable and o en su p isingly s able s a is ical signa u es, o en e e ed o as s ylized ac s (Con 2001;Lux2009). Mos no able and esea ched a e he p ope ies o asse p ice e u ns exhibi ing a - ailed dis ibu ions and ola ili y clus e ing. Vola ili y in pa icula has ecei ed much a en ion in he econophysics communi y o i s au oco ela ion decaying as a powe law sugges ing a long-memo y p ocess. Vola ili y also plays a p ominen ole in econome ics, isk managemen and inance. Co espondingly, phenomenological s a is ical models such as GARCH (Bolle sle 1986) and s ochas ic ola ili y models (Kim e al. 1998)a e ex ensi ely s udied and ou inely i ed o ma ke da a. Agen -based models conside he s a is ical signa u es o inancial ma ke s as eme - gen p ope ies, i.e., a ising om he collec i e ac ions o many in e ac ing ade s. They p o ide a complemen o s anda d economic models, which, p esuming a ional ac o s, a e o en unable o explain he apid changes in ola ili y be ween calm ma ke phases and highly ola ile episodes. Shille has coined he e m excess ola ili y, hin - ing a hese sho comings (Shille 1980). In con as , agen -based models allow o bounded a ional ac o s and can o en ep oduce he s ylized ac s p esuming cha is ading and/o he ding beha io (Samanidou e al. 2007). Agen -based models o specula i e beha io in inancial ma ke s a e nowadays able o eplica e many s ylized ac s simul aneously. They p o ide an al e na i e o s anda d econome ic models, o e ing beha io al explana ions o obse ed ma ke s a is ics (Lux 2009; LeBa on 2000). Ye , es ima ion o such models is s ill challeng- ing and has mos ly eso ed o simula ion-based me hods s i ing o ma ch selec ed momen s o he da a (F anke and Wes e ho 2011; Ghonghadze and Lux 2016). Mo e ecen ly, di ec compa isons be ween he p obabilis ic dynamics o simula ed and obse ed ime se ies ha e been p oposed. To his end, ansi ion p obabili ies o e u n ime se ies, obse ed and simula ed, a e disc e ized and compa ed in e ms o con ex - ee weigh ed Ma ko app oxima ions (Ba de 2016,2017) o JS di e gence (Lampe i 2018). Al e na i ely, p edic i e likelihoods a e es ima ed on he con inuous ime se ies o e u ns ia (Gue ini and Mone a 2017) i ing a VAR model o (Kukacka and Ba unik 2017) ke nel densi y es ima ion and again compa ed and ma ched wi h model p edic- ions. These me hods go beyond momen ma ching and allow o gene a e and e alua e model p edic ions. They s ill all sho o eco e la en dynamical s a es as es ima es a e no condi ional o da a bu me ely ma ched o hei p obabilis ic s uc u e. In o de o eco e la en dynamics, ei he maximum likelihood es ima ion o Bayesian me h- ods a e equi ed. Es ima ing agen -based models in his ashion is challenging as hei dynamics a e o en highly nonlinea . Ye ecen ly, sequen ial Mon e Ca lo me hods (Lux 2018) and he unscen ed Kalman il e (Majewski e al. 2018) ha e been suc- cess ully used in his con ex . 123 Bayesian es ima ion and likelihood-based compa ison o … 175 He e, we ollow his line o esea ch and u ilize mode n so wa e ools om machine lea ning and s a is ics o i agen -based ma ke models. In pa icula , we employ S an (2017), a p obabilis ic p og amming language o Bayesian modeling, o i se e al di - e en agen -based models, namely om Vik am & Sinha (2011), F anke & Wes e ho (2012) and Al a ano, Lux & Wagne (2008). We belie e ha Bayesian es ima ion has many ad an ages as i allows o access pa ame e unce ain ies as well as o gene a e model p edic ions. Fu he mo e, being based on he ull model p obabili y, including he likelihood, di e en models can be sys ema ically compa ed, e.g., based on hei p edic i e likelihood on held-ou da a. Indeed, o simila easons Bayesian es ima- ion is popula in mac oeconomics o es ima ing classical DSGE (An and Scho heide 2007) as well as agen -based models (G azzini e al. 2017). O e all, ou con ibu ion is h ee old: Fi s , we discuss se e al agen -based mod- els and he beha io al assump ions hey a e based on. In pa icula , his includes he model by Vik am & Sinha, which had no been i ed be o e, as well as a no el mo ing a e age speci ica ion o he model o F anke & Wes e ho . Secondly, we imple- men all models in S an, a mode n p obabilis ic p og amming language o Bayesian modeling. Thi dly, we p o ide a de ailed pai wise compa ison o all models based on c oss- alida ed p edic i e likelihoods. In pa icula , we ind ha he bes agen - based models a e compe i i e wi h s anda d econome ic models. While his has been obse ed p e iously o he F anke & Wes e ho model (Ba de 2016), we p o ide e idence ha o he he ding dynamics gi e compa able esul s p o ided ha he model allows o pe sis en misp icing be ween undamen al and obse ed p ices. Ou p esen a ion is s uc u ed as ollows: In Sec . 2, we in oduce Bayesian da a modeling and Ma ko chain Mon e Ca lo (MCMC) algo i hms. In pa icula , we sho ly explain Hamil onian Mon e Ca lo (HMC) and how i is implemen ed in S an. Then, in Sec .3we in oduce all conside ed models and exp ess hem as p obabilis ic models o e u n da a. Ou esul s on simula ed as well as ac ual S&P 500 s ock e u n da a a e summa ized in Sec .4. Finally, we conclude by discussing ou main indings in Sec .5. 2 S an and Hamil onian MCMC 2.1 Bayesian modeling In Bayesian modeling obse ed da a,1x=(x1,...,xN)a e ela ed o unobse ed pa ame e s/la en a iables θ=(θ1,...,θK)in e ms o a join p obabili y dis ibu ion wi h densi y p(x,θ). This densi y is usually ac o ized as p(x,θ)=p(x|θ)p(θ), i.e., in o he pa ame e likelihood and p io densi y. In e ence hen es s on Bayes ule o ob ain he densi y o he pos e io dis ibu ion p(θ|x)=p(x|θ)p(θ) p(x) 1Vec o s a e deno ed wi h bold symbols h oughou he ex . 123 176 N. Be schinge , I. Mozzho in whe e he no maliza ion is gi en by p(x)=p(x|θ)p(θ)dθ. The pos e io summa- izes he in o ma ion ob ained abou he unobse ed pa ame e s θand combines he a p io i assessmen o he modele , p(θ), wi h he in o ma ion ob ained om he da a p(x|θ). Concep ually, Bayesian es ima ion boils down o a a he mechanical appli- ca ion o Bayes ule, once he ull model p(x,θ)is speci ied. Below, we will explain how his applies o di e en agen -based models and discuss, in pa icula , he ole o p io choices in Bayesian modeling. In p ac ice, he no maliza ion cons an p(x)o he pos e io densi y is o en in ac able, in ol ing an in eg al o e he pa ame e space. Acco dingly, many app oxi- ma ion me hods ha e been p oposed which ei he aim o app oxima e i wi h a ac able densi y o allow o d aw pos e io samples om i s unno malized densi y. Hamil o- nian Mon e-Ca lo (HMC) sampling is an example o he la e app oach. As a Ma ko chain Mon e Ca lo me hod, i p oduces a sequence o possibly co ela ed samples. A comp ehensi e and eadable in oduc ion o HMC and i s p ope ies can be ound in Be ancou (2017). He e, a a he sho o e iew o he me hod should su ice. 2.2 Ma ko chain Mon e Ca lo (MCMC) Conside a a ge densi y p∗(θ), e.g., he pos e io dis ibu ion p(θ|x) om a Bayesian model. MCMC aims o cons uc a ansi ion densi y T (θ|θ)which lea es he a ge densi y in a ian , i.e., p∗(θ)=T(θ|θ)p∗(θ)dθ. Such a ansi ion densi y can hen be u ilized o d aw a sequence o samples θ1,θ2,... wi h p(θ2,...|θ1)=∞ i=1T(θi+1|θi). The Me opolis–Has ings algo i hm uses wo s eps in o de o compu e a sui able ansi ion s a ing a θi=θ: 1. D aw θ om a p oposal densi y2q(θ|θ) 2. Ei he e ain he cu en sample, i.e., θi+1=θo ansi ion o θi+1=θwi h accep ance p obabili y aθ|θ=min 1,p∗(θ)q(θ|θ) p∗(θ)q(θ|θ).(1) This so-de ined ansi ion densi y no only lea es he a ge densi y in a ian , bu , unde sui able condi ions, also ensu es ha he chain con e ges o i s unique in a ian densi y s a ing om any ini ial condi ion θ1(Bishop 2011). 2.3 Hamil onian Mon e Ca lo (HMC) While, in heo y, he Me opolis–Has ings algo i hm can p oduce samples om he desi ed a ge densi y, especially in high dimensions i can su e om slow con e - 2Choosing a sui able p oposal densi y is a c ucial s ep in he Me opolis–Has ings algo i hm as i con ols how e ec i ely he esul ing ansi ions can mo e ac oss he sampling space. 123 Bayesian es ima ion and likelihood-based compa ison o … 177 gence. The unde lying eason is ha mos o he p obabili y mass is con ined o a small subspace, he so-called ypical se . I he p oposal densi y is no well ma ched o he a ge densi y, many s eps o he Ma ko chain a e ei he ejec ed, e.g., when lea ing he ypical se , o slowly and andomly mo e along he ypical se . HMC u ilizes g a- dien in o ma ion in o de o gene a e long sweeps o he p oposed s a es which a e ne e heless accep ed. To his end, HMC cons uc s a g adien low which ollows he ypical se (Be ancou 2017). Fo mally, HMC samples om an augmen ed s a e space (θ,m)wi h densi y p(θ,m)=p(θ)p(m|θ) =elog p(θ)+log p(m|θ) =e−H(θ,m). The desi ed ma ginal densi y p(θ)can hen be ob ained by d opping he mcompo- nen o each join sample. In analogy wi h physical sys ems, mis conside ed as he momen um o a pa icle a posi ion θ. In ui i ely, mcon ols he speed a which he posi ion is mo ed along he ypical se . The g adien low ollowing he ypical se is ob ained by in eg a ing he Hamil onian H(θ,m)=−log p(θ)−log p(m|θ)as ˙ θ=∂ ∂mH(θ,m)=− ∂ ∂mlog p(m|θ) ˙ m=−∂ ∂θH(θ,m)=∂ ∂θlog p(θ)+∂ ∂θlog p(m|θ). Hamil onian dynamics ha e se e al well-known p ope ies. In pa icula , hey a e ime- e e sible and conse e olume and o al p obabili y p(θ,m). Thus, no u he co ec ions a e necessa y when using he s a e (θ,m)ob ained by in eg a ing he Hamil onian o some ime as a p oposal. Due o ime-symme y and conse a ion o p obabili y, he accep ance p obabili y in Eq. (1) educes o 1, i.e., he new s a e is always accep ed. HMC hen p oceeds in wo s eps. A each ansi ion, a new momen um is sampled acco ding o p(m|θ)which is commonly aken as a Gaussian dis ibu ion independen o he cu en s a e θ, i.e., m∼N(0,I). Then, he posi ion is mo ed om θwi h ini ial speed mby ollowing he g adien low o some ime. The inal posi ion (θ,m) is hen accep ed as he nex sample. By conse a ion o o al p obabili y, i holds ha p(θ,m)=p(θ,m)po en ially leading o a long sweep ac oss he ypical se along a le el con ou o he p obabili y densi y. Thus, a each s ep HMC i s jumps o a new p obabili y le el and hen ollows he g adien low a his le el, he eby allowing o an e icien explo a ion o he ypical se . In p ac ice, he di e en ial equa ion desc ibing he Hamil onian dynamics needs o be in eg a ed nume ically and ca e has o be aken ha nume ical e o s do no accumula e. Fo una ely, symplec ic in eg a o s can e icien ly in eg a e Hamil onian sys ems as nume ical e o s cancel and simula ed ajec o ies closely app oxima e he heo e ical dynamics. While ime-symme y and olume p ese a ion a e e ained by symplec ic in eg a o s, he o al p obabili y is only 123 178 N. Be schinge , I. Mozzho in app oxima ely conse ed along nume ical ajec o ies. Thus, in p ac ice, HMC uses a Me opolis–Has ings s ep o ei he accep o ejec he inal posi ion o a ajec o y. Especially in high-dimensional models, i.e., wi h many pa ame e s, he use o g a- dien in o ma ion o guide explo a ion is c ucial o ensu e e icien sampling. No e ha HMC is es ic ed o con inuous pa ame e spaces θ∈RK, bu could be combined wi h o he me hods when disc e e pa ame e s a e desi ed. O en, i is ad an ageous o ma ginalize o e disc e e pa ame e s as s ong co ela ions be ween hem can se e ely hinde e icien sampling.3In heo y, HMC is insensi i e o s ong co ela ions be ween pa ame e s θ. In p ac ice, he symplec ic in eg a o uses a ini e s ep size o nume - ically sol e he Hamil onian dynamics. In he case o pos e io densi ies wi h high cu a u e, his can p e en he sample o each ce ain pa s o he s a e space. Fu - he mo e, i makes HMC sensi i e o he scale o pa ame e s as he s ep size would need o be adjus ed acco dingly.4Fo una ely, by epa ame e izing he model i is o en possible o simpli y he geome y o he pos e io densi y. Ongoing esea ch explo es he geome ic aspec s o HMC bo h om heo e ical (Li ings one e al. 2019;Ma e al. 2015) and om p ac ical (Be ancou and Gi olami 2015) pe spec i es. Also, a ange o no el con e gence diagnos ics, e.g., based on he s abili y o he nume ical ajec o y, ha e been de eloped (Be ancou 2017). Nex , we u n o agen -based models o inancial ma ke s and show how hese can be exp essed as s a is ical models. All models a e hen implemen ed and i ed models wi h he p obabilis ic p og amming language S an (2017). Appendix Ap o ides a sho o e iew. The S an code o all models is a ailable in he online supplemen a y ma e ial. Implemen ing agen -based models in a well- es ed Bayesian amewo k has se e al ad an ages. On he one hand, in e ence algo i hms such as HMC suppo ed by S an a e well op imized and es ed. On he o he hand, model speci ica ions can be easily adap ed and explo ed wi h small changes o hei sou ce code. Ye , S an es ic s he use o models wi h con inuous andom a iables only. While many models, such as he ones p esen ed below, can be app oxima ed in his ashion in he limi o in ini ely many agen s, mo e de ailed simula ions in ol ing many disc e e agen s a e cu en ly beyond he scope o exis ing oolboxes o p obabilis ic modeling. In his sense, ou app oach can be seen as a p oo o concep which ne e heless co e s se e al models o in e es . 3 Ma ke models He e, we conside h ee models, in de ail, namely by Vik am & Sinha (2011), F anke & Wes e ho (2012) and Al a ano, Lux & Wagne (2008). In pa icula , we explain how hese models gi e ise o a la en s a e dynamics which can be simula ed and es ima ed wi h Bayesian me hods. 3This especially applies o Gibbs sampling. Despi e i s popula i y, Gibbs sampling is se e ely hinde ed by co ela ions in he pos e io which can ende i u e ly useless in high-dimensional p oblems. 4In con as , Gibbs sampling is se e ely e ec ed by s ong dependencies bu insensi i e o he scale o pa ame e s. 123 Bayesian es ima ion and likelihood-based compa ison o … 179 3.1 Model by Vik am & Sinha (VS) The ma ke in he VS model is popula ed by N ade s. A each ime s ep , a ade i ei he buys (Si( )=1), sells (Si( )=−1) o s ays inac i e (S( )=0). The no malized ne demand om all ade s is hen gi en as M =1 NN i=1Si( ), and he p ice adjus s as p +1=1+M 1−M p . An agen ’s decision o buy/sell o s aying ou depends on he pe cei ed misp icing be ween he cu en p ice p and i s unning a e age p∗ =p τ which is conside ed as a p oxy o he undamen al p ice o he asse . The p obabili y o an agen o ade is hen gi en by P(|Si( )|)=exp−μ p −p∗ p∗  and a ading agen buys Si( )=1 o sells Si( )=−1 a andom wi h equal p oba- bili y. In o de o ob ain a s a is ical model o ola ili y, in pa icula wi h a con inuous la en s a e as equi ed o HMC sampling, we ha e adap ed he model as ollows: – Fo a la ge numbe o agen s N→∞, he ne demand M con e ges o a Gaussian dis ibu ion wi h mean ze o (as E[Si( )]=0) and a iance P(|Si( )|=1) √N. – He e, we ha e used ha agen s ading decisions Si( )a e independen and E[Si( )]=1 2 P(|Si( )|=1)·1+1 2 P(|Si( )|=1)·(−1) +(1−P(|Si( )|=1)·0=0 Va [Si( )]=E[Si( )2] =1 2 P(|Si( )|=1)·12+1 2 P(|Si( )|=1)·(−1)2 +(1−P(|Si( )|=1)·02 =P(|Si( )|=1). – Nex , conside ing he numbe o agen s as unknown we in oduce a scal- ing pa ame e σ2 max o he a iance and model he demand as M ∼ N(0,σ2 maxP(|Si( )|=1)). – Finally, we app oxima e he log- e u n by linea izing he p ice impac 5 +1=log p +1 p =log 1+M 1−M ≈2M 5We ha e also i ed he exac model, i.e., pu ing a no mal dis ibu ion on he ans o med e u ns M = e +1−1 e +1+1wi hou any no iceable di e ence. 123 180 N. Be schinge , I. Mozzho in whe e we ha e used ha log(1+x)≈x o |x|1. O e all, we a i e a he ollowing model dynamics6 p τ=(1−τ)p +τp −1τ P(|S( )|=1)=e−μlog p p τ +1∼N(0,σ2 max ·4P(|S( )|=1)). (2) No e ha his is a s a e-space model wi h a con inuous la en s a e d i ing he ime- a ying ola ili y σ +1=σ2 max ·4P(|S( )|=1). Indeed, he amous GARCH(1, 1) (gene alized au o- eg essi e condi ional he e oscedas ic) model (Bolle sle 1986)is o a simila o m σ2 +1=α0+α1 2 +β1σ2 +1∼N(μ, σ 2 +1)(3) The main di e ence be ween he VS (in ou o mula ion) and he GARCH model is ha he ola ili y is a unc ion o pas p ices in he o me and pas e u ns in he la e model. Fu he mo e, due o being ounded in an agen -based model all pa ame e s o he VS model a e eadily in e p e able as he sensi i i y μo he agen s o misp icing and he weigh ing τo he unning p ice a e age. In con as , pa ame e s in he GARCH model a e mo i a ed pu ely om s a is ical g ounds and canno easily be ela ed o agen beha io s. F om a Bayesian pe spec i e, Eqs. (2) and (3) co espond o he likelihood p(x|θ), i.e., he condi ional p obabili y o he obse ed da a gi en he model pa ame e s. To comple e he model densi y p(x,θ), we need o speci y a p io dis ibu ion on he pa ame e s. The choice o a p io dis ibu ion is o en conside ed as subjec i e (whe eas he likelihood has an au a o objec i ism). A guably, om he pe spec i e o modeling he obse ed da a his dis inc ion is o limi ed ele ance. Ins ead, no e ha ixing he p io implici ly ixes a dis ibu ion on he da a space, i.e., ob ained as p(x)=p(x,θ)dθby ma ginalizing o e he pa ame e s. A model can be conside ed as misspeci ied when i assigns e y low p obabili y o he ac ual obse ed da a. In con as , a good model should be able o gene a e simila da a wi h easonable p obabili y. This iewpoin is in line wi h Gelman e al. (2017) who a gue ha he p io can only be unde s ood in he con ex o he likelihood. Indeed, p io and likelihood ac oge he in shaping he model and exp essing ou expec a ions o plausible da a. He e, we p opose he use o (weakly) in o ma i e p io s which ake in o accoun ou knowledge abou he ole played by he pa ame e s when gene a ing da a om he likelihood model. As an example, conside he pa ame e τ∈(0,1)o Eq. (2). While i migh be na u al o simply assign a uni o m p io ,7τcon ols he ime cons an o 6Simula ing his app oxima e model shows ha i p oduces simila p ice se ies wi h s ong ola ili y clus e ing as he o iginal model. 7No e ha uni o m p io s, especially on unbounded spaces, should no be conside ed as unin o ma i e. On he one hand, an imp ope uni o m p io , i.e., when i canno be no malized, exp esses a s ong belie abou ex eme pa ame e alues by assigning in ini e p obabili y mass o alues abo e any ini e h eshold. On he o he hand, hey a e no in a ian unde model epa ame e iza ion as shown in he abo e example. 123 Bayesian es ima ion and likelihood-based compa ison o … 187 Fig. 1 P io p edic i e checks, i.e., simula ed model e u ns wi h pa ame e s andomly d awn om he p io , o he FW ( op) and ALW model (bo om). Fo compa ison, an ac ual e u n ime se ies is included in he lowe igh panel (no e ha he scale o e u ns gene a ed by he ALW model is much la ge han obse ed in ac ual da a) 123 188 N. Be schinge , I. Mozzho in Fig. 2 Simula ed p ice and e u n se ies o FW model. No e ha a low ac ion o undamen al ade s n coincides wi h ola ile ma ke phases Fig. 3 T ace plo o model pa ame e s φ,ξ,α0,α n,αp,σ and σc. No e ha all chains appea o ha e con e ged o he same pos e io dis ibu ion a e jus abou 50 samples Indeed, o he e en mo e lexible models whe e he undamen al p ice is assumed o ollow an (unobse ed) andom walk, some pa ame e s could no be eco e ed a all om simula ed da a. Figu e5shows he ac ual and in e ed la en sen imen dynamics on da a simula ed om he ALW model. Es ima es a e shown as he pos e io mean oge he wi h he 95% c edibili y bands a ound i . He e, only one chain has success ully eco e ed he ac ual sen imen ime se ies, whe eas ano he chain exhibi s sho excu sions away om he ac ual sen imen . As he p obabili y o bo h chains 123 Bayesian es ima ion and likelihood-based compa ison o … 189 Fig. 4 Plo o pos e io densi ies o pa ame e s φ,ξ,α0,α n,αp,σ and σc. The ue alues a e well co e ed by he pos e io dis ibu ions Fig. 5 Plo o ma ke sen imen x o e ime. Chain 1 eco e s he ue sen imen dynamics o he simula ed da a, whe eas chain 2 shows empo a y de ia ions away om he ue dynamics. No e ha he likelihoods o hese wo pos e io modes a e ma kedly di e en is as ly di e en , wi h he co ec one being subs an ially highe , his appea s o be a p oblem o he sampling algo i hm which is ge ing s uck in a local mode o he pos e io . I migh well be ha o he algo i hms, such as he sequen ial Mon e Ca lo me hods employed by Lux (2018), a e less suscep ible o his p oblem. On he o he hand, HMC is highly e ec i e in sampling he global pa ame e s o he model which is a majo bo leneck o sequen ial Mon e Ca lo me hods (Li ings one e al. 2019; Monnahan e al. 2017). Fu he mo e, he FW model wi h he andom walk speci ica ion o he undamen al p ice shows an e en deepe non-iden i iabili y. The op panel o Fig. 6compa es he es ima ed ac ion o undamen al ade s o wo di e en chains o samples o he ac ual simula ed ime se ies. While one chain s ays close o he ac ual alues, he 123 190 N. Be schinge , I. Mozzho in Fig. 6 F ac ion o undamen al ade s n (uppe panel) and ola ili y es ima es σ o e ime. Chain 1 eco e s he ue ac ion o undamen al ade s o he simula ed da a, whe eas chain 2 shows a mi o image lipping he ole o undamen al and cha is ade s. No e ha he likelihoods and es ima ed ola ili ies o hese wo pos e io modes a e essen ially iden ical o he es ima e appea s o be a mi o image. In e es ingly, in his case, he likelihood o bo h chains is almos iden ical. Indeed, he lowe panel o Fig. 6shows ha he es ima ed ola ili ies a e almos iden ical as well and co espondingly bo h es ima es assign e y simila p obabili y o he obse ed e u ns. This sugges s a mul imodal pos e io whe e each chain samples om a well-de ined, ye di e en , mode o he dis ibu ion. Fu he mo e, in a leas one o he modes we ind ha σc<σ ! Acco dingly, he model o e s wo e y di e en explana ions o he obse ed ola ili y dynamics. In he i s scena io, he numbe o cha is s is usually low and ises sha ply in ola ile ma ke phases (as in ended by he model). In con as , in he second scena io he numbe o cha is s is usually high and d ops in ola ile ma - ke phases. Vola ili y is hen d i en by he high demand unce ain y o undamen al ade s. In e es ingly, bo h scena ios lead o e y simila es ima es and p edic ions o 123 Bayesian es ima ion and likelihood-based compa ison o … 191 he ola ili y σ . The model accomplishes his by assuming a e y di e en ajec o y o he unobse ed B ownian mo ion o he undamen al p ice leading in u n o e y di e en misp icings and demands o he undamen al ade s. Thus, he seemingly innocuous assump ion ha he undamen al p ice ollows a B ownian mo ion appa - en ly in oduces a symme y in o he model. This no only makes he unobse ed la en s a es o he model uniden i iable, bu also e eals ha ou unde s anding o agen -based models and hei gene a ed ime se ies dynamics is a om comple e. In pa icula , we canno use he model in o de o unambiguously unco e he undamen- al p ice as in Majewski e al. (2018) whe e he Kalman il e compu es a uni-modal pos e io app oxima ion. Fu he mo e, he he ding pa ame e s a e ha d o in e p e as hey ac di e en ly in di e en modes. Fu he wo k is needed in o de o cha ac e ize and ideally emo e his uniden i iabili y. Thus, cu en ly iden i ica ion o some o he models discussed he e is plaqued by wo majo issues: on he one hand, o algo i hmic na u e wi h he sample ge ing s uck in poo local op ima o he pos e io densi y; on he o he hand, mul imodali y o he model likelihood leading o se e al pos e io modes wi h as ly di e en in e p e a- ion o he model pa ame e s. He e, we lea e such in es iga ions o u u e s udies and ins ead ocus on model p edic ions. In pa icula , a de ailed compa ison o he e iciency, e.g., in e ms o e ec i e sample size pe wall clock ime, and eliabili y o di e en algo i hms o ola ili y models wi h complica ed pos e io geome ies is beyond he scope o his pape . Ins ead, we ocus on he models i sel and con inue wi h a p incipled model compa ison in he nex sec ion. 4.2 C oss- alida ion and handling o missing da a To ci cum en hese iden i ica ion issues, we now ocus on p edic ions whe e iden i i- abili y is o no conce n. In o de o also ci cum en he p oblem o he sample ge ing s uck in a local minimum, we un se e al Ma ko chains o each model and only use he chain wi h he highes in-sample likelihood o p edic ing. Then, we compa e models based on hei p obabili y assigned o held-ou da a. In he con ex o ime se ies, models a e commonly compa ed based on olling look-ahead p edic ions, i.e., using he p edic i e dis ibu ion o u u e e u ns T+τo ola ili ies σT+τbased on he Tp e ious ime poin s. Time is hen olled o wa d, and he model is e alua ed again. In gene al, his app oach equi es e i ing he model o each p edic ion. He e, o compu a ional easons, we ins ead eso o lea e-one-ou (LOO) p edic- ions, i.e., p edic ing cu en e u ns in he con ex o pas and u u e e u ns, excluding he cu en one. Wi h he me hod o Pa e o smoo hed impo ance sampling (PSIS), he co esponding p edic i e likelihoods p( i| 1,..., i−1, i+1,..., T) can be es ima ed om pos e io samples (Veh a i e al. 2017). No e ha in con as o he ull pos e io , he LOO likelihood is condi ioned on all bu he i h da a poin . In o de o es ima e he LOO likelihood om pos e io samples, he i h da a poin s need o be e ec i ely emo ed be o e e alua ing he p edic ion. In gene al, his is a om 123 192 N. Be schinge , I. Mozzho in Fig. 7 Schema ic o K- old c oss- alida ion o ime se ies da a (K=3). He e, each model is i ed K imes on he obse ed da a poin s (bold) and e alua ed in e ms o p edic ions on he held-ou se con aining e e y K h da a poin (i alic) i ial and we e e he eade o Veh a i e al. (2017) o de ails abou how PSIS es i- ma es LOO likelihood. He e, we jus no e ha he me hod smoo hens he es ima e by i ing a gene alized Pa e o dis ibu ion o he ail o obse ed log likelihoods. The eby, a mo e s able es ima e as well as a use ul diagnos ics is ob ained. In pa icula , la ge ail exponen s in he Pa e o dis ibu ion sugges ha es ima es could ha e unbounded a iance and should no be us ed. Indeed, his happens especially o in luen ial da a poin s and no eliable es ima e can be ob ained in his case. A leas o such da a poin s, esul s need o be alida ed by di ec es ima ion o he p edic i e likelihood, i.e., e i ing he model on some o he da a poin s and p edic ing he held-ou ones. To his end, we also implemen ed a di ec a ian o K- old c oss- alida ion whe e he da a a e pa i ioned in o Kpa s. LOO hen co esponds o N- old c oss- alida ion which would equi e i ing he model N imes on all bu one da a poin and p edic he le -ou one. He e o compu a ional easons, we ha e chosen o hold ou e e y 20 h da a poin ins ead. The model is hen i ed 20 imes, on all bu he emaining da a poin s and asked o p edic he unobse ed ones. (Figu e 7shows he esul ing schema o K=3.) The choice K=20, i.e., o hold ou e e y 20 h da a poin , p o ides a good comp omise be ween compu a ional pe o mance (low K) and almos independen p edic ions (high K). Using in o ma ion heo y, i has been shown ha ola ili y is only weakly dependen ac oss a wide ange o s ochas ic ola ili y models (P an e and Be schinge 2019). Thus, we can easonably expec ha ola ili ies a e nea ly independen a e a couple o weeks. Nume ical esul s in he nex sec ion con i m his assump ion wi h es ima es based on LOO and 20- old c oss- alida ion being s a is ically indis inguishable. Acco dingly, we ha e implemen ed all models in a way ha allows o unobse ed o missing e u ns. To his end, we pass in all e u ns 1,..., T, obse ed and unob- se ed, oge he wi h a bina y ec o miss_mask indica ing a each ime whe he he e u n is conside ed missing o no . F om a Bayesian pe spec i e, missing da a poin s a e simply unobse ed la en a iables ha can be in e ed oge he wi h o he la en a iables/pa ame e s. Thus, o each missing e u n a co esponding pa ame e miss is in oduced and sampled alongside he o he model pa ame e s. To enhance sampling e iciency, he missing e u ns a e ep esen ed in e ms o inno a ions which a e hen ans o med o ac ual e u ns ia =μ +σ miss akin o he non-cen e ed pa ame e iza ion explained in Eq. (9). As he model densi y is de ined on and no he unde lying pa ame e s miss, we need o accoun o he esul ing change o measu e by mul iplying he densi y wi h he absolu e Jacobian ∂ ∂miss o he ans o ma ion. 123 Bayesian es ima ion and likelihood-based compa ison o … 193 Fig. 8 GARCH model i on he S&P 500 4.3 Fi ing he S&P 500 Finally, we ha e i ed all models on p ice da a om he S&P 500 s ock ma ke index. As a benchma k, a s anda d GARCH(1, 1) and SV model ha e been included o compa ison. The co esponding i s om Janua y 2009 o Decembe 2014 and Janua y 2000 o Decembe 2010 a e shown in Figs. 8,9,10,11 and 12. The es ima ed model ola ili y is o e laid on he ac ual ma ke e u ns. Vola ili y es ima es a e shown as he pos e io mean oge he wi h he 95% c ed- ibili y bands a ound i . The pos e io o he ola ili y σ a ime s ep is based on da a poin s om e u ns obse ed o e all Tda a poin s, i.e., p(σ | 1,..., T) o =1,...,T. In he e minology o ime se ies models, his is known as he smoo h- ing dis ibu ion. Compa ing he ola ili y es ima es and p edic ions o he di e en models, a ew ema ks a e in o de : 123 194 N. Be schinge , I. Mozzho in Fig. 9 SV model i on he S&P 500 – The s ochas ic ola ili y models SV and VS, FW in andom walk speci ica ion exhibi highe unce ain y in hei ola ili y es ima es as compa ed o models assuming a de e minis ic ola ili y dynamics, e.g., GARCH. No e ha his does no imply ha p edic ions a e wo se, bu jus e lec s he in insic di icul y and imp ecision o ola ili y es ima ion. – The ALW model mos closely ma ches he ac ual e u ns, exhibi ing highly a iable ola ili y es ima es. The eby, he model appea s o o e i he ac ual e u n da a. – The i s o he VS and FW model wi h he mo ing a e age speci ica ion imp o es when some e u ns a e unobse ed, i.e., missing. We will discuss his issue in mo e de ail below. Ha ing compa ed he models g aphically, we p oceed wi h a mo e p incipled assessmen o model i . In pa icula , Table1con ains he p edic i e log likelihoods es ima ed using LOO and 20- old c oss- alida ion, espec i ely. As explained abo e, 123 Bayesian es ima ion and likelihood-based compa ison o … 195 Fig. 10 VS model i wi h mo ing a e age ( op) and andom walk (bo om) speci ica ion o he undamen al p ice on he S&P 500. Vola ili y es ima es a e shown wi h and wi hou missing da a. No e ha he es ima e wi h missing da a is ma kedly be e o he mo ing a e age speci ica ion o he undamen al p ice, whe eas no di e ence is isible o he andom walk speci ica ion each model has been i ed using 8 andomly ini ialized chains and p edic ions a e based on he bes chain only. As expec ed, o he GARCH and SV model he es ima es o p edic i e likelihood based on LOO and 20- old CV a e s a is ically indis inguishable. Simila ly, he es ima es o he VS and FW model using a andom walk speci ica ion o he undamen al p ice a e e y simila . Fo he o he models, he si ua ion is sligh ly di e en o se e al easons: 123 196 N. Be schinge , I. Mozzho in Fig. 11 FW model i wi h mo ing a e age ( op) and andom walk (bo om) speci ica ion o he undamen al p ice on he S&P 500. Vola ili y es ima es a e shown wi h and wi hou missing da a. No e ha he es ima e wi h missing da a is ma kedly be e o he mo ing a e age speci ica ion o he undamen al p ice, whe eas no di e ence is isible o he andom walk speci ica ion – VS_ma, FW_ma: The models using he mo ing a e age speci ica ion o he undamen al p ice show be e p edic i e log likelihoods when es ima ed wi h c oss- alida ion as compa ed o LOO. This is somewha su p ising, as he LOO es ima e is based on i ing all e u ns and hen co ec ing he likelihood ia PSIS which should lead o an o e es ima ion, i any hing. The solu ion is al eady shown in Figs. 10 and 11 which show ha he models wi h 123 Bayesian es ima ion and likelihood-based compa ison o … 203 in o ma ion needed in o de o in e he na u e o he he ding mechanism by empi ical means. Acknowledgemen s Open Access unding p o ided by P ojek DEAL. The au ho s hank D . H. C. Mauche o unding hei posi ions. We also hank Damian Challe o help ul commen s and sugges ing he andom walk a ian o he VS model. Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence, and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle a e included in he a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi line o he ma e ial. I ma e ial is no included in he a icle’s C ea i e Commons licence and you in ended use is no pe mi ed by s a u o y egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he copy igh holde . To iew a copy o his licence, isi h p://c ea i ecommons.o g/licenses/by/4.0/. A The S an language The p obabilis ic p og amming language S an (2017) allows he use o desc ibe he join p obabili y p(x,θ)o a model in a high-le el p og amming language. In u n, he p og am is hen compiled o C++ and se e al in e ence algo i hms, including Hamil onian Mon e Ca lo, a e buil -in. The equi ed g adien s a e compu ed ia a C++ lib a y o au oma ic di e en ia ion (Ca pen e e al. 2015), hus eeing he use om manually implemen ing and debugging g adien calcula ions. S an comes wi h an ex ensi e documen a ion which includes many example models and use ul icks o e icien ly implemen ing hem (S an De elopmen Team 2017). A minimal S an p og am consis s o h ee blocks 1. a da a block which decla es a iables co esponding o obse ed quan i ies x, 2. a pa ame e s block which decla es a iables co esponding o unobse ed pa am- e e s θand 3. a model block which con ains s a emen s compu ing he log densi y o he model, i.e., log p(x,θ). Fo ins ance, he ollowing S an p og am es ima es he mean o Gaussian obse a ions wi h a known s anda d de ia ion: 1da a { 2in <lowe =0> N; // numbe o da a poin s 3 ec o [N] x; // obse ed da a poin s 4} 5pa ame e s { 6 eal mu; // unobse ed mean 7} 8model { 9mu ∼no mal(0, 10); // compu es log p io densi y log p(mu ) 10 x∼no mal(mu, 1); // compu es log likelihood log p(x | mu) 11 // o al log densi y is log p(mu) + log p(x | mu) 12 } 123 204 N. Be schinge , I. Mozzho in The ∼s a emen s in he model block ela e a a iable wi h a densi y and a e sho - hand no a ion o he mo e basic s a emen s summing up log densi y con ibu ions, e.g., a ge += no mal_lpd (x | mu, 1). As shown in he example, all a iables a e yped and need o be decla ed be o e use. A pa icula ly con enien ea u e o S an is ha a iables can be gi en cons ain ypes, e.g., a s anda d de ia ion pa ame e could be decla ed as eal<lowe =0> sigma. In e nally, he a iable is hen au oma ically ans o med o an unbounded space and he log densi y is adjus ed o he esul ing change o measu e. Due o his me hod, many di e en da a ypes including ec o s, ma ices, bu also cons ain spaces such as simplices o co a iance ma ices a e eadily suppo ed. S an suppo s se e al in e ence algo i hms, namely g adien descen op imize s o maximum a pos e io i es ima ion, HMC sampling and s ochas ic g adien a ia ional Bayes. While HMC is he leas e icien o hese algo i hms, i usually p o ides he closes app oxima ion o he ue pos e io dis ibu ion. Fu he mo e, du ing wa m-up, also known as bu n-in, S an adap s se e al pa ame e s o he algo i hm such ha he algo i hm appea s essen ially pa ame e - ee o he use . This is especially e ec i e o he No U-Tu n Sample (NUTS) which au oma ically adjus s he leng h o simula ed ajec o ies. In a nu shell, NUTS in eg a es he Hamil onian dynamics un il i s a s u ning back owa d i sel which is locally decided based on he g adien di ec ion. Ca e needs o be aken o ensu e ha he esul ing ansi ions lea e he a ge densi y in a ian . To his end, ajec o ies a e expanded in a eelike ashion by successi ely doubling hei leng h o wa d and backwa d in ime. The nex s a e is hen sampled uni o mly om he esul ing o e all ajec o y. Fo u he de ails abou he S an p og amming language and he NUTS algo i hm, we e e he in e es ed eade o he S an manual (S an De elopmen Team 2017) and Be ancou (2017), espec i ely. B Pa ame e eco e y in he GARCH model He e, we p o ide addi ional eco e y expe imen s o he GARCH model. We ha e chosen his model as i is well unde s ood and pos e io sampling poses no p oblems o S an. Indeed, all chains quickly and eliably con e ge such ha he samples d awn ai h ully ep esen he ue pos e io dis ibu ion. Fu he mo e, as imp ope la p io s a eusedonα0,α 1and β1 hei pos e io jus e lec s he shape o he likelihood unc- ion. Fu he mo e, he s anda d no mal p io s on μand σ0a e only weakly in o ma i e conside ing he used pa ame e alues. Fo he expe imen shown in Fig.16, we ha e simula ed he GARCH model wi h pa ame e s μ=0.06 250 ,α 0=0.052 250 ,α 1=0.1,β 1=0.9 and σ0=0.1 o 5000 ime s eps, i.e., ading days. We ha e hen e i ed he model on he i s 2000 ime s eps as well as he las 2000 ime poin s. Figu e15 shows he co esponding e u n se ies. In Fig.16, he pos e io densi ies oge he wi h he ue pa ame e s ha gene a ed he da a and he pos e io means a e shown. In all cases, he pa ame e s a e eco e ed in he sense ha he ue pa ame e s all well wi hin a egion o high pos e io densi y. Ye , depending on he pa icula da a se he pos e io mean can subs an ially de ia e om he ue pa ame e . As he pos e io 123 Bayesian es ima ion and likelihood-based compa ison o … 205 Fig. 15 Simula ed e u ns om GARCH model wi h pa ame e s μ=0.06 250 ,α 0=0.052 250 ,α 1=0.1,β 1= 0.9andσ0=0.1. The simula ed e u ns exhibi clea ola ili y clus e ing as well as ealis ic magni udes Fig. 16 Pos e io dis ibu ions o he GARCH model i ed on he i s 2000, las 2000 and all 5000 simula ed e u ns o Fig.15. No e ha he ue alue is well co e ed by he pos e io dis ibu ion e en hough he pos e io mean can be subs an ially di e en mean is known o minimize he expec ed squa ed dis ance om he ue pa ame e s, his implies ha pa ame e es ima es exhibi a he high a iance in equen is e ms. I also shows ha e u n da a p o ide only limi ed amoun abou he unde lying model pa ame e s. Fo ins ance, especially he a e age e u n μis known o exhibi high unce ain y and also in ou example he in e al [−0.00065,0.001]con aining 95% o 123 206 N. Be schinge , I. Mozzho in Table 4 Model compa ison o di e en p io s based on lea e-one-ou (LOO) and c oss- alida ion (CV) bes chains p edic i e likelihoods Model LOO CV SV (o iginal) 4917 ±36 4910 ±37 SV ( la ) 4917 ±36 4910 ±37 FW_ma (o iginal) 4827 ±40 4901 ±35 FW_ma ( la ) 4761 ±40 4830 ±41 FW_ma (mixed) 4829 ±39 4874 ±36 FW_walk (o iginal) 4928 ±36 4927 ±36 FW_walk ( la ) 4981 ±41 4885 ±41 he pos e io mass when es ima ed on all 5000 da a poin s is no signi ican ly di e en om ze o. Fu he mo e, no ing ha he da a we e gene a ed wi h μco esponding o an a e age e u n o 6% pe yea , i co esponds o an es ima ed a e age yea ly e u n be ween −16% and 25%. C P io obus ness checks He e, we in es iga e he obus ness o model i s wi h espec o p io choices. As a benchma k, we e i ed he SV model wi h imp ope uni o m p io s12 on all pa ame- e s. Table 4shows he LOO and CV p edic i e likelihoods on he S&P 500 e u ns be ween 2009 and 2014. The e is no di e ence be ween he SV model wi h he weakly in o ma i e (o iginal) and imp ope ( la ) p io s. The co esponding esul s a e also shown o he FW model, in he andom walk and he mo ing a e age speci ica ion. In his case, he FW_walk model sligh ly o e i s when using imp ope la p io s. This is also isible in Fig.17 as he es ima ed ola ili y mo e closely acks he e u n da a unde he la p io . Table 4con i ms ha he model p edic ions a e s ill compe i i e wi h he SV model, bu he LOO es ima e becomes un eliable and de ia es clea ly om he CV es ima e. O e all, he p io sugges ed in he main ex has a weakly egula izing e ec and smoo hens he model es ima es and p edic ions. This e ec is e en mo e p onounced in he FW_ma model. In his case, he ola ili y es ima e esul ing om imp ope uni o m p io s is mos ly cons an and subs an ially wo se (see Fig. 18 and Table 4). Indeed, we ound ha he in o med p io s on σ and σcas well as he bounda y a oiding p io on he ime cons an lo he mo ing a e age a e c ucial. When hese a e combined wi h imp ope p io s on all o he pa ame e s (mixed), ola ili y es ima es and model p edic ions a e simila o he esul s a ising om p io s (o iginal) as sugges ed in he main ex . When weakening any o he h ee 12 No e ha S an ans o ms cons ained pa ame e s, e.g., σh>0, o an uncons ained space. Thus, he imp ope uni o m p io is imposed on log σhco esponding o p(σh)∝1 σhwhich is he Je eys p io o he s anda d de ia ion o a no mal dis ibu ion wi h ixed mean. Fu he mo e, his p io is in a ian wi h espec o mul iplica ion wi h posi i e eals and hus imposes no in o ma ion on he scale o σh. 123 Bayesian es ima ion and likelihood-based compa ison o … 207 Fig. 17 FW_walk model i on he S&P 500 (2009–2014) wi h di e en p io s Fig. 18 FW_ma model i on he S&P 500 (2009–2014) wi h di e en p io s p io s o an imp ope one, he model essen ially beha es as he one wi h la p io s on all pa ame e s (no shown). O e all, hese expe imen s con i m he in ui ion ha wen in o he choice o p io s as explained in Sec .3. In addi ion, a weakly egula izing e ec is ound in all cases and p o ides u he jus i ica ion o he use o Bayesian me hods o model i ing, in pa icula leading o imp o ed s abili y o ola ili y es ima es and model p edic ions. 123 208 N. Be schinge , I. Mozzho in D Iden i iabili y o he FW model In Sec .4.1, we obse ed wo modes in he pos e io o he FW model which lead o e y simila p edic ions. He e, we show ha unde ce ain condi ions his is indeed app oxima ely he case. In pa icula , we cons uc wo se s o pa ame e s σ ,σ c,α 0,α n,αpand ˜σ ,˜σc, ˜α0,˜αn,˜αpsuch ha ˜σ =σc,˜σc=σ and he model ola ili ies a e exac ly equal, i.e., (n −1)2σ2 +(nc −1)2σ2 c=(˜n −1)2˜σ2 +(˜nc −1)2˜σ2 c. By ˜σ =σc,˜σc=σ his implies ha ˜n −1=nc −1,˜nc −1=n −1co esponding o he obse ed lip in he ac ion o cha is and undamen al ade s. F om Eq. (6), we ha e n =1 1+e−βa −1 =˜nc =1−˜n =e−β˜a −1 1+e−β˜a −1 =1 1+eβ˜a −1. Thus, we need o choose pa ame e s such ha a −1=−˜a −1. Assuming ha he undamen al p ice is always co ec ,13 i.e., p∗ =p ,weha e a =α0+αn(n −nc )+αp(p∗ −p )2 =α0+αn(˜nc −˜n ) =α0−αn(˜n −˜nc ) =−−α0+αn(˜n −˜nc )+αp(p∗ −p )2 =−˜a when choosing ˜α0=−α0,˜αn=αnand ˜αp=αp. No e ha he mean e u n n −1φ(p∗ −p −1)+nc −1ξ(p −1−p −2) can usually no be ma ched exac ly in his ashion excep when φand ξ anish. On he o he hand, in s ochas ic ola ili y models he likelihood is commonly domina ed 13 The ac ual symme y obse ed in Sec .4.1 is mo e complex wi h subs an ially di e en undamen al p ices be ween he wo modes. Ou heo e ical unde s anding is a bes incomple e a his poin . 123 Bayesian es ima ion and likelihood-based compa ison o … 209 by he ola ili y wi h only a weak e ec o he mean e u n. Indeed, assuming a mean e u n o ze o usually leads o almos iden ical i s and p edic ions o ola ili y compa ed wi h mo e elabo a e models o he mean e u n. Nume ically, we obse e a simila esul in he wo pos e io modes o he FW model, whe e he likelihood is almos iden ical, despi e sligh ly di e en dynamics o he mean e u n in he wo modes. Re e ences Al a ano S, Lux T, Wagne F (2008) Time a ia ion o highe momen s in a inancial ma ke wi h he e o- geneous agen s: an analy ical app oach. J Econ Dyn Con ol 32(1):101–136 An S, Scho heide F (2007) Bayesian analysis o DSGE models. 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