SMC in Estimation of a State Space Model

Transcription

1 SMC in Esimaion of a Sae Space Model Dong-Whan Ko Deparmen of Economics Rugers, he Sae Universiy of New Jersey December 31, 2012 Absrac I briefly summarize procedures for macroeconomic Dynamic Sochasic General Equilibrium DSGE model esimaion. I hen review he basic Sequenial Mone Carlo SMC mehods. Especially, I review Boosrap Filer and Auxiliary Paricle Filer for sae filering mehod and Liu and Wes filer and Sorvik Filer for parameer learning as a building block for Paricle Leaning. I consruc simple linear sae space model. I hen obain he likelihood using differen filering mehod: Kalman filer and Auxiliary Paricle Filer. I also esimae he parameers using Markov Chain Mone Carlo MCMC and Paricle Learning. Comparing he resuls, I confirm ha SMC mehod can be alernaive o MCMC mehod for DSGE model esimaion. 1 Inroducion In Dynamic Sochasic General Equilibrium DSGE model, we have a sysem of non-linear firs order condiions. In order o solve he model, researchers usually log-linearize he model and consruc linear Gaussian sae space model. A ypical way o esimae he srucural parameers is o calculae likelihood wih Kalman filer a firs and hen esimae he srucural parameers wih random walk Meropolis-Hasing algorihm. Rubio-Ramirez and Fernndez-Villaverde 2005, however, ried o solve he sysem of equaions non-linearly wih finie-elemen mehod and hen applied he paricle filers o obain he likelihood. They argued ha for boh of simulaed and real daa, paricle filer mehod delivers a subsanially beer fi of he model o he daa. Among many sampling mehods of he proposal disribuion, Rubio-Ramirez and Fernndez-Villaverde 2005 use 1

2 2 he boosrap filer proposed by Gordon e al in which hey chose he ransiion kernel as he proposal disribuion, accordingly he weigh is simplified o he likelihood a ime. 1 This is no an opimal choice for he proposal, however, because he informaion of curren observaion is no used for proposal densiy, as Creal 2012 poined ou. In addiion, Rubio-Ramirez and Fernndez-Villaverde 2005 used paricle filer no for esimaing he parameers bu for calculaing he likelihood: hey used MCMC mehod for esimaion. On he oher hand, Carvalho e al presen paricle learning hereafer PL in which one direcly samples from he paricle approximaion o he join poserior disribuion of saes and condiional sufficien saisics for fixed parameers in a fully-adaped resample-propagae framework. I will give more deail laer. Chen, Peralia and Lopes 2010 argue ha even for a case in which MCMC mehod is available, he paricle filer have several advanages. Firs, MCMC mehods rely on Markov chain convergence which migh no be geomerically ergodic for srucural models such as DSGE models Papaspiliopoulos and Robers 2008, whereas consisency and asympoic normaliy of paricle filers have been proven in Douc e al Second, when we use paricle filer, poserior approximaions of he parameers and saes can be obained in on-line manner, ha is, i can be calculaed a each ime period. To obain he same amoun of informaion wih MCMC, one would have o resor o repeaed implemenaion of MCMC a each ime period, which is more inefficien in erms of running ime and compuing resources. In line wih his argumen, I consruc simple linear Gaussian sae space model and see wheher SMC can compee wih MCMC in esimaion of he parameers. For he sake of comparison, I obain he likelihood and esimaes of parameers using differen mehodologies; Kalman Filer and Auxiliary Paricle Filer for calculaing he likelihood; Random Walk Meropolis-Hasing algorihm and Paricle Learning for he esimaes. A general form of sae space model is given by y = fx, θ where ν N0, Σ ν x = gx 1, θ where w N0, Σ w 1.1 for any given = 1,..., T. Where f, and g, could be non-linear funcions of sae variables and he parameers. In addiion, i could be including ime varying parameers. However, I as- 1 Deails on Paricle filer mehod as one of he Sequenial Mone Carlo SMC mehods are well explained in Douce and Johansen 2009

3 3 sume for my example ha hose are linear funcions wih fixed parameers. The disribuion of observaions and sae variables is deermined by he shocks, ν and w, and he corresponding densiies can be wrien as p y x, θ and p x x 1, θ respecively. The sequence of sae variables 2, x = x 0,..., x and parameers, θ, are unobservable, hus i has o be esimaed using observaions. For his, here are hree imporan densiies: he one sep ahead predicive disribuion, p x y 1 ; θ, he filering disribuion, p x y ; θ, and he smoohing disribuion, p x y T ; θ. Generally speaking, given iniial disribuion of he sae variable, one sep ahead predicive disribuion for sae variable, p x 0 ; θ, can be obained wih Chapman-Kolmogorov Equaion: p x y 1 ; θ = p x x 1 ; θ p x 1 y 1 ; θ Updaing wih observaion a, y, Bayes heorem gives he filering disribuion: p x y ; θ = p y, x y 1 ; θ p y y 1 ; θ = p y x, y 1 ; θ p x y 1 ; θ p y y 1 ; θ Since we assume Markovian process in observaion and sae variables, finally we can ge p x y ; θ = p y x ; θ p x y 1 ; θ p y x ; θ p x y 1 ; θ dx 1.2 However, mos macroeconomic researchers have ineres in pθ y T and py T raher han p x y ; θ. In paricular, he denominaor in Eq 2, called normalizing consan or marginal daa densiy, can be used for model assessmen such as poserior odd es. In linear-gaussian case i is possible o obain i wih Kalman filer. Oherwise, i could be approximaed wih paricle filer. 2 Typical Esimaing Mehod for DSGE Model In macroeconomics, researchers usually use log-linearized Gaussian model of he economy. Accordingly, hey calculae he likelihood for he observaions hrough he Kalman Filer and hen esimae he parameers based on Meropolis-Hasing MH Algorihm. 3 Since his ypical mehod 2 A variable wih superscrip denoes a sequence of he variables from iniial value o hose a ime, for insance, x = x 0, x 1,..., x, and y = y 1, y 2,..., y. 3 see Guerrón-Quinana and Nason 2012 for deails

4 4 is well known, I briefly describe he process of he Kalman Filer and MH algorihm in below. In doing his, he likelihood can be obained as a by-produc of he Kalman Filer. 2.1 Kalman Filer Given x 0 Nm 0, C 0, Predicive Sep for sae variable: x x 1, θ Na, R where a = G m 1, R = G C 1 G + Σ w Predicive Sep for observaion: y x, θ Nf, Q where f = F a, Q = F R F + Σ ν Filering Sep: x y, θ Nm, C where m = a + A y a C = R A Q A A R F Q 1 likelihood wih Kalman Filer ln θ y T 1 2 T =1 ln Q + y f Q 1 y f 2.1 Once hey have Kalman filer, hey use Random Walk Meropolis-Hasing algorihm for he esimaion of he parameers. The likelihood obained from Kalman filer is used o choose desirable draws. 2.2 Algorihm 1: Random Walk Meropolis-Hasing For i = 1,, M 1. Iniializaion: se i = 0 and iniial θ 0. Solve he model for θ 0 and calculae f θ 0 and g θ 0 wih Eq. 1. Then evaluae pθ 0 and θ 0 y T from Eq. 3. Se i = i Proposal draw: ge a proposal draw θ i = θ i 1 + ε i, where ε i N0, Σ ε. Σ ε is a scaling marix ; when accepance rae is oo high we can scale up i by muliplying consan.

5 5 A recommended opimal accepance rae is around 23% 3. Solving he model: solve he model for θ i and calculae f θ i and g θ i wih Eq. 1 building new sae space represenaion. 4. Evaluaing he proposal: Evaluae pθ i and y T θ i from Eq Accep/Rejec: Draw ρ i U0, 1. If ρ i θi y T pθ i θ i 1 y T pθ i 1 se θi = θ i oherwise θ i = θ i 1. If i M se i = i + 1 and go o 2. Oherwise sop. Once I ge he draws, I can approximae expeced value of a funcion of he parameers, hθ, by 1 M M hθi. 3 Paricle Filers In his secion, I briefly review he four paricle filers. Firs wo filers are for sae filering and he ohers are for parameer learning. I assume all parameers are known for sae filering and omi i from mos of he equaions unil I deal wih parameer learning. The idea behind of paricle filer is he imporance sampling. In order o approximae he arge disribuion p x y, one samples from proposal disribuion q 0: x y wih imporance weigh w = px y q 0: x y. Define q 0: x y q x x 1, y q 0: 1 x 1 y 1. We can compue imporance weigh recursively as following way: w = p y x p x x 1 p x 1 y 1 p y y 1 q x x 1, y q 0: 1 x 1 y 1 w p y x p x x 1 1 q x x 1, y 3.1 The raio of he densiies in second par of Eq. 4 is called incremenal imporance weigh. Given { } he draws x,i, w i N, we can approximae expecaions of a funcion f x of sae variables as: E q [f x ] = f x p x y q 0:n x y q 0: x y dx E q [f x ] N f x,i ŵ i where ŵ i = w i N wi However, paricle filering has he weigh degeneracy problem ha only one paricle is lef as number of ieraions increases. In order o miigae his problem, Resampling sep can be added 3.2

6 6 end of an ieraion in which effecive number of paricle is less han hreshold. The effecive sample size can be calculaed by ESS = 1 N ŵ i 2. Thus, given known parameers, sae filering can be done following he mehod described above. Accordingly, when i comes o sae filering, choosing proposal densiy is crucial. I inroduce Boosrap Filer and Auxiliary Paricle filer which adop differen proposal densiies. On he oher hand, in case wih unknown parameers, hings become lile bi complicaed. This is because simply including parameers ino paricle se would no successful. Applying a sandard SMC algorihm o he Markov process of {x, θ } means ha he parameer space would only be explored a he iniializaion of he algorihm. As a resul of he successive resampling seps, afer a cerain ime, he approximaion o pθ y will only conain a single unique value for θ 4. There have been wo imporan aemps for his difficuly: arificial dynamics based approach Liu and Wes 2001 and sufficien saisics based approach Sorvik 2002 as explained below. 3.1 Paricle Filers wih known parameers Boosrap filer Transiion densiy is used as he proposal densiy in boosrap filer, ha is, q x x 1, y = p x x 1, hen incremenal imporance weigh becomes he likelihood, p y x, and he weigh evolves w w 1 p y x. As I menioned in Inroducion, however, his mehod does no ake ino accoun curren observaion, y, when drawing a sample from proposal disribuion. We can use condiional ransiion densiy, p x y, x 1 o supplemen boosrap filer, bu i is available only if he measuremen equaions are linear and Gaussian. If i is case, w w 1 p y x 1, ha is, imporan weigh independen from curren paricles. Accordingly, resample sep can be done before sampling new paricles: propagae-resample sep can be reversed o resample-propagae sep. The paricles would be improved because now he proposal disribuion reflecs he informaion of curren observaion. Auxiliary paricle filers are arising from his propery. Auxiliary Paricle Filer Pi and Shephard 1999 inroduced proposal densiy q x, k y where k is an auxiliary variable ha indexes he paricles in exisence from ime 1. The purpose of his auxiliary variable is o find a way o use he informaion in he curren observaion, y, 4 Kanas e al.

7 o find he good paricles wihin he exising se { x i 1, } N disribuion. Noice, from Eq. 2 and Eq. 4, p x y N p y x p 7 in order o form a beer proposal x x i 1 w i 1. Pi and Shephard 1999 idea is newly inroduce arge and proposal disribuion as follows: p x, k y p y x p x x k 1 w k 1 q x, k y = p y gx k p x x k 1 w k 1 where gx 1 = Ex x 1 usually, and hen he imporance weigh is expressed as w p y x 3.3 p y gx k 1 Douce and Johansen 2009, however, show ha he performance of he APF will depend upon he signal o noise raio in he sae space model. When he raio is low, i can mislead he se of paricles away from ineresing areas of suppor. Followings are he algorihm for auxiliary paricle filer presened by Carvalho e al Algorihm 2: Auxiliary Paricle Filer APF 1. Resample: { } x i N { 1 from 2. Propagae x i 1 xi : 3. Resample: { x i } N { from { } x i N } x i N wih weighs wi p x x i 1 x i } N w i wih weighs p p y +1 x i y +1 g x i p y g x i 1 Log-likelihood from APF Noice ha T ln py 1,..., y T θ = ln p y y 1, θ n=1

8 8 Once we have done wih sampling from he proposal according o he above algorihm, we can approximae he marginal disribuion by: ˆpy y 1, θ = 1 N px i y q n x i y = 1 N N w i hen, ln θ y T T ln ˆpy y 1, θ = =1 [ T 1 N ln N =1 w i ] Creal 2009, however, also menion ha for a case in which resampling is performed no in every period, log-likelihood can be obained as: ˆpy y 1, θ N ŵ i 1 wi Noicing ha ŵ i is normalized weigh, ln θ y T T ln ˆpy y 1, θ = =1 3.2 Paricle Filers wih unknown parameer [ T N ln =1 ŵ i 1 wi So far, he paricle filering is used for he sae filering and likelihood for observaion under he assumpion ha all parameers are known. In his subsecion, I relaxed his assumpion and briefly inroduce wo imporan filers by which one can esimae he parameers as well as approximae sae disribuion. ] Liu and Wes 2001 Filer They assume ha he poserior disribuion of he parameers can be approximaed by mixure disribuion: p θ y N N m i ; h 2 V where m i = aθ i + 1 a θ, θ = 1 N N θi, and V = 1 N N θ i θ θ. θ i A uning parameer a deermines he shrinkage and smoohness of he normal approximaion and he desired value for a is suggesed higher han 0.98 bu less han 1. LW filering algorihm

9 9 presened by Carvalho e al is following: 1. Resample: { x, θ i } N from {x, θ i} N w i +1 y p +1 g wih weighs x i, m i 2. Propagae θ i 3. Propagae x i 4. Resample: i ˆθ {ˆθi } N +1 : +1 p m i, V { } ˆx i +1 : ˆx i N +1 p x +1 x i, i ˆθ +1 {x +1, θ +1 i} { } N ˆx from +1, ˆθ i N +1 w i p y +1 ˆx i i +1 +1, ˆθ +1 p y +1 g, m i x i wih weighs As we have seen above, he imporance weigh can be obained before sampling sae variables. Thus i makes resample-propagae framework as in APF. Sorvik 2002 Filer He assume ha he poserior disribuion p θ x, y depends on a low dimensional se of sufficien saisics, s, ha can be recursively updaed: s = S s 1, x, y. As in below algorihm, Sorvik filer can be considered as a Boosrap filer wih some addiional seps because resampling is conduced afer propagaing. In oher words, i uses blind proposal in ha i does no ake ino accoun curren observaion, y. The algorihm suggesed by Sorvik 2002 is following: 1. Propagae 2. Resample: 3. Propagae { } x i x i +1 : x i N +1 q x i, θ i, y +1 {x +1, s i} N { from x +1, s i} N wih weighs p w i +1 = s i s i +1 : s i +1 = S y +1 x i +1, θ p s i q x i x i, θ, y +1, x i +1, y xi, θ

10 10 4. Sample: θ i p θ s i +1 4 Paricle Learning: PL Carvalho e al inroduce paricle learning mehod which is aking boh of sufficien saisics of sae variables and resample-propagae framework a he same ime. PL direcly samples from he paricle approximaion o he join poserior disribuion of saes and condiional sufficien saisics for fixed parameers in a fully-adap resample-propagae framework. Carvalho e al shows ha PL ouperforms Liu and Wes 2001 in regards o accuracy and argues PL is comparable o MCMC samplers. Le s and s x denoe he parameer and sae sufficien saisics saisfying deerminisic updaing rules s = Ss 1, x, y as in Sorvik 2002 and s x = K s x 1, θ, y where K is he Kalman filer recursions. Algorihm 3: PL 1. Resample θ, s x 1, s 1 2. Sample x from p x s x 1, θ, y. from θ, s x 1, s 1 wih weighs w p y s x 1, θ 3. Updae parameer sufficien saisics: s = S s 1, x, y 4. Sample θ from pθ s. 5. Updae sae sufficien saisics: s x = K s x 1, θ, y. 5 Applicaion o he simple model In order o compare SMC mehod o MCMC, I ry o replicae Johannes and Polson s Example using boh MCMC and PL. The model is called AR1 wih noise and is given by: x = α + βx 1 + ε ε N0, τ 2 y = x + ν ν N0, σ I se he rue parameers θ = α β σ 2 τ 2 = I hen generae T = 200 observaions, y for = 1,..., T using Eq. 7. Wih known parameers and x 0 = 0, I compue log-likelihood from

11 11 APF wih paricles as well as Kalman filer which is presened in Table 1. I also calculae he likelihood wih 1000 and paricles. I improves he likelihood only in 1 decimal place as number of paricles increases bu i creaes huge difference in running ime. In addiion, I compare hose wo filering mehods in erm of sae filering. Figure 1 shows prediced sae variable for each filering mehod wih he rue values. The grey lines in upper panel show 5% and 95% quaniles of normal disribuion wih predicive mean, m, and variance, C, in Kalman Filer. The grey lines in lower panel show 5% and 95% quaniles of he paricles in APF. I confirms ha APF can compee wih Kalman Filer. I hen assume prior disribuion for θ = α β σ 2 τ 2 o be pθ = pσ 2 pτ 2 pα, β τ 2 where σ 2 IGψ 1, Ψ 1, τ 2 IGψ 2, Ψ 2 and α, β τ 2 MN ψ 3, τ 2 Ψ 3. The weigh used in iniial resampling sep is he one sep ahead predicive likelihood py +1 x, θ N α + βx, σ 2 + τ 2 which is given by w x, θ i 1 σ 2 i + τ 2 i exp 1 2 y +1 α i β i x i σ 2 i + τ 2 i The updaed sae disribuion is px x 1, θ, y py x, θpx x 1, θ N µ, Ω 2 where µ Ω 2 = y σ 2 + α + βx 1 τ 2 and 1 Ω 2 = 1 σ τ 2. Given sufficien saisics, s, he poserior disribuion for parameers is now pθ s 1 = pα, β τ 2, s 1 pσ 2 s 1 pτ 2 s 1. pσ 2 ψ1, s IG 2, Ψ 1, 2,, pτ 2 ψ2, s IG 2, Ψ 2, 2 pα, β τ 2, s N 2 ψ 3,, τ 2 Ψ 1 3, 5.3 Table 1: Log-likelihood Kalman Filer Auxiliary Paricle Filer Log-likelihood

12 12 Sae Filering KF Acual x Time APF Acual x Time Figure 1: Simulaed Sae Variables Le Z = [1 x ] and he vecor of sufficien saisics s = Ψ 1, Ψ 2, ψ 3, Ψ 3, hen, corresponding ransiion kernel S s 1, x, y is given by: Ψ 1, = y x 2 + Ψ 1, 1 Ψ 2, = Ψ 2, 1 + ψ 3, 1Ψ 3, 1 ψ 3, 1 + x x ψ 3,Ψ 3, ψ 3, ψ 3, = Ψ 1 3, Ψ 3, 1ψ 3, 1 + Z 1 x 5.4 Ψ 3, = Ψ 3, 1 + Z Z

13 13 The hyperparameers are deerminisic and given by ψ 1, = ψ 1, and ψ 2, = ψ 2, For he parameer esimaion, I addiionally se he iniial prior parameers Θ 0 = ψ 1,0 Ψ 1,0 ψ 2,0 Ψ 2,0 ψ 3,0 Ψ 3,0 = , 0.9 I 2. Accordingly, he algorihm for paricle learning can be wrien as 1. Resample θ, s x 1, s 1 2. Sample x from N from θ, s x 1, s 1 wih weighs w N α + βx i Ω 2 y σ 2 + α + β xi τ 2 1, Ω 2 3. Updae parameer sufficien saisics: s = S s 1, x, y 1, σ2 + τ 2 4. Sample θ from pθ s Eq.9. Figure 2. shows he esimaed parameers based on PL which is geing closer o he rue value of he parameer as i is updaed; Red dashed lines indicaes he rue parameer value, black lines represen median of he esimaed parameer value and he grey lines are corresponding and quaniles. For he sake of comparison, I also esimae he parameers wih Random Walk MH algorihm raher han Gibbs sampler even hough, in my example, he disribuions can be obained analyically. This is because I inend o compare he SMC mehod wih a mehod ypically used in DSGE model esimaion which is explained above. I run MH algorihm 100,000 imes and burn firs 50,000 ieraions. I draw each of he parameers α, β, σ, and τ from he proposal disribuion which is univariae normal disribuion cenered a previous acceped parameer values. I chose jumping parameer c = 0.083, hen he accepance rae become 24.33%. In order o ge he parameers σ 2 and τ 2, I squared he acceped draws for σ and τ. I summarize poserior quaniles obained by PL as well as MCMC in Table 2. For all he parameers, rue parameer values are wihin 95% credible inervals. For all parameers bu τ 2, he median of PL closer o rue value han hose of Random walk MH algorihm o he rue values. 6 Conclusion In his paper I have summarized paricle filers as an alernaive o MCMC mehod. Depending on he order of sampling and resampling, he filers can be caegorized in wo. One is propagae-

14 14 alpha bea sigma^2 au^ Figure 2: Median and 95% credible inervals of he parameers resampling filer such as he Boosrap filer and Sorvik filer. The oher one is resample-propagae filers such as APF and LW filer. The laer is more desirable in ha he informaion of curren observaion is used o sample draws. In addiion, I have learned PL suggesed by Carvalho e al for parameer esimaion. The purpose of presen paper is o see wheher PL ouperforms MCMC as hey argue in heir paper. To do his, I consruc simple linear Gaussian sae space model called AR1 wih noise by Johannes and Polson. Through Kalman filer and APF, I obained he likelihood for he observaions and prediced saed variables. According o he likelihood, Kalman filer is sill beer han APF for linear case, bu i is comparable. I hen esimae he parameers using random walk MH algorihm and PL. Based on poserior quaniles, PL gives beer

15 15 Table 2: Comparison of quaniles beween PL and MCMC Parameers MCMC Paricle Learning True Value 5% Median 95% 5% Median 95% α β σ τ approximaion for he rue parameers and i is also beer in running ime.

16 16 References Carlos M Carvalho, Michael S Johannes, Hediber F Lopes, Nicholas G Polson, e al. Paricle learning and smoohing. Saisical Science, 251:88 106, Drew Creal. A survey of sequenial mone carlo mehods for economics and finance. Economeric Reviews, 313: , Randal Douc, Arnaud Guillin, J-M Marin, Chrisian P Rober, e al. Convergence of adapive mixures of imporance sampling schemes. The Annals of Saisics, 351: , Arnaud Douce and Adam M Johansen. A uorial on paricle filering and smoohing: Fifeen years laer. Handbook of Nonlinear Filering, 12, Neil J Gordon, David J Salmond, and Adrian FM Smih. Novel approach o nonlinear/non-gaussian bayesian sae esimaion. In IEE Proceedings F Radar and Signal Processing, volume 140, pages IET, Pablo A Guerrón-Quinana and James M Nason. Bayesian esimaion of DSGE models Michael Johannes and Nick Polson. Paricle filering and parameer learning. Nicholas Kanas, Arnaud Douce, Sumeepal Sindhu Singh, and Jan Marian Maciejowski. An overview of sequenial mone carlo mehods for parameer esimaion in general sae-space models. Jane Liu and Mike Wes. Combined parameer and sae esimaion in simulaion-based filering. In Sequenial Mone Carlo mehods in pracice, pages Springer, Omiros Papaspiliopoulos and Gareh Robers. Sabiliy of he gibbs sampler for bayesian hierarchical models. The Annals of Saisics, pages , Michael K Pi and Neil Shephard. Filering via simulaion: Auxiliary paricle filers. Journal of he American saisical associaion, 94446: , Juan F. Rubio-Ramirez and Jesus Fernndez-Villaverde. Esimaing dynamic equilibrium economies: linear versus nonlinear likelihood. Journal of Applied Economerics, 207: , 2005.

17 17 Geir Sorvik. Paricle filers for sae-space models wih he presence of unknown saic parameers. Signal Processing, IEEE Transacions on, 502: , 2002.

18 Appendix I: Resuls from he MCMC alpha bea Frequency Frequency MCMC[burn:Nsim, i] MCMC[burn:Nsim, i] sigma^2 au^2 Frequency Frequency MCMC[burn:Nsim, i] MCMC[burn:Nsim, i] Figure 3: MCMC: Poserior Disribuion of he Parameers Figure 3. represens he poserior disribuion of he parameers obained by MCMC. Red verical lines indicae rue parameer values. Mode of he draws are no coinciden wih he rue values bu as in Table 2, α and τ are well capured relaive o ohers. 15

19 alpha bea Cumulaive Mean Cumulaive Mean Ieraions Ieraions sigma^2 au^2 Cumulaive Mean Cumulaive Mean Ieraions Ieraions Figure 4: MCMC: Cumulaive mean of he parameers I plo Figure 4. o see he convergency of he algorihm. As i shows, cumulaive mean of he parameers are converges o some poin. However, i seems o be far from is rue values excep for τ 2. 16

20 alpha bea Draws Draws Ieraions Ieraions sigma^2 au^2 Draws Draws Ieraions Ieraions Figure 5: MCMC: Draws of he parameers 17