BAYESIAN ESTIMATION AND MODEL SELECTION FOR

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1 RODRÍGUEZ: BAYESIAN ESTIMATION AND MODEL SELECTION REVISTA FOR THE DE WEEKLY ECONOMÍA COLOMBIAN DEL ROSARIO, EXCHANGE 4 (DICIEMBRE RATE ) 43-7 BAYESIAN ESTIMATION AND MODEL SELECTION FOR THE WEEKLY COLOMBIAN EXCHANGE RATE Norbero Rodríguez Niño * Banco de la República nrodrini@banrep.gov.co ABSTRACT Tis documen reviews and applies recenly developed ecniques for Bayesian esimaion and model selecion in e conex of Time Series modeling for Socasic Volailiy. Afer e lieraure review on Generalized Condiional Auoregressive models, Socasic Volailiy models, and e relevan resuls on Markov Cain Mone Carlo meods (MCMC), an example applying suc ecniques is sown. Te meodology is used wi a series of Weekly Colombian-USA Excange Rae on seven differen models. Te GARCH model, wic uses Type-IV Pearson disribuion, is favored for e selecing ecnique, Reversible Jump MCMC, over oer models, including Socasic Volailiy Models wi a Suden- disribuion. Keywords: Bayesian Esimaion, Markov Cain Mone Carlo Meods (MCMC) GARCH Models, Socasic Volailiy, Model Selecion. JEL Classificaion: C, C3, C5, C, C5 I. INTRODUCTION Two differen and compeing ecniques are nowadays used for economericians and saisicians o model volailiy, as in reurn asses or excange raes: One of em, Auoregressive Condiional Heeroscedasic (ARCH), is generalizaion, GARCH, and muliple exensions ave proven o be very successful in modeling financial ime-varying volailiy series. Te compeing alernaive o GARCH models are Socasic Volailiy models, mainly reaed in e frequenis framework, wic ave more eoreical background. Tey appear in e financial lieraure on opion pricing as a generalizaion of e Black-Scoles model. * Tis work is based on e Maser Repor presened o Te Universiy of Texas a Ausin, by e auor, as a requiremen for e degree of Maser of Science in saisics.

2 44 REVISTA DE ECONOMÍA DEL ROSARIO, 4 (DICIEMBRE ) Usually e researcer faces e quesion of wa model o use. Several alernaives ave been proposed in e frequenis saisical framework o deal wi is, ranging from R, e radiional and exensively used Akaike Informaion Crierion (AIC), PRESS saisic, and many oers. In ose alernaives e model residuals are obained (usually observed minus adjused or peraps deviance residuals) and aggregaed o form e measures of adequacy. Unforunaely, classical approaces o model coice are limied. Te wellknown sandard Neyman-Pearson eory provides an opimal es roug e likeliood raio for e limied case of e comparison of wo disinc models. More generally, e likeliood raio es enables a coice beween models only in e nesed case, were ere is an unambiguous null ypoesis. Selecion is based upon an asympoic χ approximaion, wic usually is poor for small sample sizes. Frequenis eory does no offer muc for model selecion of non-nesed models, wic are no rare in pracice. (See Piorier, 995, and Gelfand, 995, for references). Tis documens uses MCMC, an inuiive, compuaionally easy-o-implemen, and inexpensive Bayesian alernaive, o decide among suiable models. Tis documen is organized as follows: Secion II presens a review of GARCH models and Socasic Volailiy models, especially ose models o be used laer in is paper; en i moves on o MCMC meods and Bayesian model selecion ecniques. Secion III deals wi ow o use MCMC o implemen esimaion of GARCH models, as Vronos e al. () sugges, and ow o esimae SV models. Secion IV deals wi e resuls and Secion V presens e main conclusions, some suggesions and limiaions. II. BACKGROUND Tis caper presens a sor review of some saisical and economerics model proposed in e lieraure o model ime-varying volailiy series. Afer a, Bayesian ecniques for model esimaion and selecion are briefly presened. A. Generalized Auoregressive Condiional Heerocedasic Models Time series models, radiionally fied in pracice, are supppose o ave consan variance, bu Wen working wi ig frequency ime series a is seldom e case, as can be seen in Figure, an Auoregressive Condiional Heerocedasic models (ARCH) ave been proposed in e lieraure o deal wi is problem, in e spiri of Engel (98) could be ried; is is y = ε or y = c + ε or y = c + ε () as some model for e levels of e observed ime serie, wi some funcion of lags of y or oer exogenous variables, or even an ARMA erm (see Box and Jenkins, 976); were ε ~ N(, ) and r = α + α ε =, for α i, i =,,..., r. i i i

3 RODRÍGUEZ: BAYESIAN ESTIMATION AND MODEL SELECTION FOR THE WEEKLY COLOMBIAN EXCHANGE RATE 45 Hence, e condiional variances are oug of as a funcion of e square of e previous observaional residuals. In e original and simple case a normaliy is assumed, e likeliood is given by: l T / ( y α,..., αr, ) = ( π ) exp ( ε / ) = () wen, in reurn financial ime series analysis, e usual mos imporan unknown o esimae and forecas is e volailiy, A Generalized ARCH (GARCH), a usually resuls in more parsimonious represenaions, as Bollerslev (986) proposed, assumes a e condiional variances follow an ARMA process, us: r s = + α αiε i + i = j = β j j (3) wi e analogous likeliood, as in e ARCH case, if normaliy for ε is assumed. Wi resricions α i, i =,,..., r, and β j, i =,,..., s, o guaranee a r s for all, and Ó i = α i + Ó j = β j < in order o assure saionariy. Weaker resricions can be required, in pracice, oug (See Nelson and Cao, 99). Anoer exension of (3) is using a Suden- disribuion, wi degrees of freedom o accoun for e eavier ails of e disribuion of e error process {ε } as was inroduced by Bollerslev (987), and by Baillie and Bollerslev (989). Te likeliood en is l( y α,..., α, β,..., β, r s n, ) = T = Γ Γ ( n / ) n+ ( ) [ π n ] / ε + n ( n+ ) / (4) An Exponenial GARCH (EGARCH), is inroduced by Nelson (99) o avoid imposing resricions on α i, β j in (3) wi ln( r s ε i ε i ε i ) = α + ϕ i E + α i + j ln( j ) β (5) i = i i i j = ε E for e condiional variance of {ε } and [ ] i i = Γ ( / ν ) Γ (/ ν ) Γ ( 3 / ν

4 46 REVISTA DE ECONOMÍA DEL ROSARIO, 4 (DICIEMBRE ) Addiionally, assuming a ε follows a Generalized Gaussian Disribuion, GED, en e likeliood is: l( y α, α, ϕ, β, ν, i i j ) = T = c exp.5 ε λ ν (6) were c = λ ν Γ ν ( + / ν ) and λ = / ( / ν ) Γ (/ ν ) Γ ( 3 / ν ) Te use of Bernoulli-Mixures of wo normal disribuions, proposed by Ball and Torous (983), was successfully implemened by Vlaar and Palm (993): r s = + α α i ε i + i = j = β j j for e condiional variances. Tey used an MA() erm o model e canges in levels of several excange raes, as: ε = λν θ ε y (7) and eac ε is disribued as a mixure of wo normals, ence ε ~ ( λ )N( λν, ) + λn (( λ ) ν, + δ ) were λ is e jump inensiy, v is e expecaion in e jump size, and δ e expeced cange in variance. Tis represenaion is useful and inuiive for economies wi arge or bands for eir excange raes, like Colombia or e European Economic Union. Te MA erm, θ ε is explained as allowing for mean reversion. Terefore, e likeliood is expressed as: l( y, θ, α, β, λ, δ, ν, i j ) T = = / λ ( ε + λν ) ( π ) exp + λ ( ε ( ) ) exp λ ν / + ) δ ( + δ ) ( (8) More recenly, Bera and Premarane () propose e use of e Pearson Type- IV Family Disribuions in order o model skewness and lepokurosis a are larger an usual. For a GARCH(,), e log-likeliood is:

5 47 RODRÍGUEZ: BAYESIAN ESTIMATION AND MODEL SELECTION FOR THE WEEKLY COLOMBIAN EXCHANGE RATE = = T ) y ( l ) y ( l Θ Θ = y y ln r y y arcan C ) ln( ).5 ln( ) ln( y l( α µ α µ δ α Θ (9) wi { }ψ δψ ψ π d exp sin C r = and ( ),,, r,,,, µ δ β α α Θ =. Tey used y y ε + + = and + + = β ε α for eir empirical applicaion, bu, of course, e model on y can be exended. Bu according o Nagaara (999), (9) sould be = y y ln r y y arcan ) r ( C ) ln( ).5 ln( ) ln( y l( α µ α µ δ α Θ wi [ ] [ ] + + = π ψ ψ δ ψ δ π r d ) r ( exp sin / ) r ( exp C Wen informaion arrives a random order and daa refers o close-o-close periods, Hsie (989) sowed a e use of a normal-lognormal mixure disribuion improves e fi over oer GARCH alernaives. Ta disribuion is no ried ere because i requires e compuaion of a defined inegral over one mus rely on ig numerical inegraion, owever, o provide a suiable soluion o a problem. ()

6 48 REVISTA DE ECONOMÍA DEL ROSARIO, 4 (DICIEMBRE ) B. Sae Space Models A differen alernaive in ime-series analysis is an Sae-Space (S-S), model, as Wes and Harrison (997) sae, ey exend and updae e seminal paper by Harrison and Sevenson (976). Te model is ypically represened by wo equaions: ' Observaion Equaion: Y = F θ + ν, v ~ N(, V ), () Sysem Equaion: θ = G θ + w, w ~ N(, W ) () were y is e observed (someimes laen) variable; θ is a vecor of unknown parameers, wic follows e firs order Markov process (); v is a vecor of unobserved and uncorrelaed socasic error erms; is a marix of known coefficiens; and w is an unobserved socasic erm, generally assumed uncorrelaed wi v. I is wor noing a any daa series for wic ere exiss a naural ordering of observaion, fis ino e dynamic framework, so e ime series no need be equally spaced, and missing daa problems can easily be andled in is conex. Pole e al. (994) presen applied meodology, as well as muliple examples. Usually, and wiou muc loss, V can be considered consan, and working in erms of e precision = V, i is possible o ge esimaions of W. Te likeliood for e S-S model is given below: l( y Θ ) = T l( y θ ) α T = = ( πv ) ( exp ' y F θ ) V (3) wic is e densiy of a normal wi mean ' F θ and variance V. Prior probabiliies can be se up on θ and a fully Bayesian analysis of e Sae-Space model can be run. A non-linear (non-gaussian) S-S model can be se up as follows, (See Harvey e al., 994 for deails): y = ε exp ( /) (4) as e non-linear observaion equaion, and η = γ + + η, η ~ N(, σ ), (5) σ η as e sysem equaion, were = ln ( )

7 RODRÍGUEZ: BAYESIAN ESTIMATION AND MODEL SELECTION FOR THE WEEKLY COLOMBIAN EXCHANGE RATE 49 Tis model, wic is known in e financial and economeric lieraure as e Socasic Volailiy model (SV for sor) can be ransformed o ge a linear observaional equaion, as ln ( y ) = + ln ( ε ) (6) were y is e mean correced reurn a ime ; is e log-volailiy a ime, wic is assumed o follow a saionary process, wi drawn from a saionary disribuion; ε and η are uncorrelaed sandard normal wie noise socks; is e persisence in e volailiy (wen for saionariy resricio < ); and σ η is e volailiy of e log-volailiy. Te likeliood, assuming normal disribuion, can be obained by using a Kalman Filer (See Jaquier e al, 994, Ap. B.). Te use of a v degrees of freedom Suden disribuion on as been considered, and e Kalman Filer needs some minor modificaions (See Ruiz, 994). C. Markov Cain Mone Carlo Meods Wen doing fully Bayesian analysis of complex or ig-dimensionaliy models, e researcer usually faces e problem of non-conjugacy, meaning a non-exac analyical poserior disribuion can be acieved. Tis leads o e necessiy of using simulaion approaces. Direc simulaion is ofen impossible, due o e complicaed maemaical form of e poserior disribuion in many applied models. Because of a, an exponenial rise in e ineres and applicaion of Markov Cain Mone Carlo (referred o by is acronym, MCMC) as a ool for numerical compuaion of complex inegrals, paricularly in Bayesian analysis, as emerged. Te key o Markov Cain simulaion is o creae a Markov process wose saionary disribuion is a specified π(θ y) and o run e simulaion long enoug a e disribuion of e curren draws is close enoug o e saionary disribuion. Once e simulaion algorim as been implemened, i sould be ieraed unil convergence as been reaced, or, if convergence is painfully slow, e algorim sould be alered. Hence, e sudy of MCMC as seen a corresponding ineres in e convergence properies of e resulan cains, wic may be assessed roug a suie of diagnosics borrowed from diverse areas suc as ime series, exploraory daa analysis (EDA), and cenral limi eory. Te mos widely used Markov Cain Mone Carlo meods are e Gibbs Sampler and e group of Meropolis-Hasings algorims and a good descripion of em can be found in Gamerman (997) or Gelman e al. (995), among many oers.. E.g., e use of digamma and rigamma funcions; Abramowiz and Segun (967) offer compuaional deails.

8 5 REVISTA DE ECONOMÍA DEL ROSARIO, 4 (DICIEMBRE ) D. Model Selecion Te mos damaging commen on e sandard pracice of coosing a single model, and en proceeding condiional on i, is a e researc s uncerainy is undersaed. Piorier (995, p. 65) For frequeniss e model selecion problem reduces o coosing one from a se of M models. Tis is usually e main aim of e analysis, and is done according o some model selecion crierion, as saed above. Boosrap meods ave been used for model selecion (See Maddala and Li, 996, sec 5, for references). For anoer perspecive see Poski and Tremayne (983). Essenially, wo alernaive approaces in e Bayesian conex are presened. Te firs was inroduced by Carlin and Cib (995) and considers all models in a formaion, called ere a supermodel. Te Markov Cain simulaion sceme for is supermodel is presened below. Te second approac presens sopisicaed simulaion ecniques using Markov cain wi jumps beween e differen models; i is referred o as Reversible Jumping, and i was inroduced by Green (995). I will be assumed rougou is secion a y is observed and i can be described according o a model M j wi parameers θ j of dimension d j, aking values in a parameer space Θ j R dj, j =,,..., M. Te value of M could be as, for insance, wen considering counable classes of models. m serves e purpose of indicaing a specific model. Assume for e momen a e poserior disribuion π(θ, j), e join disribuion of e super-parameer and e model indicaor are o be obained. However, e main ineres in inference is o obain e poserior disribuion of θ j m = j, j =,,...,M. Tese disribuions respecively provide e poserior inference wiin eac of e models and e poserior probabiliies of e models. Te supermodel approac provides a sample from is more general, peraps unnecessary poserior disribuion wereas e approac wi jumps only provides samples from θ j m = j, j =,,...,M, and m. Te presence of common parameers does no pose any problem ere.. Markov Cains for Super-Models Te join disribuion of all random quaniies is given by π(y, θ, j) = π(y θ,j) π(θ j)π j (7) were j is e value of m and π j = P(m = j). Given a m = j, e disribuion of y depends on θ only roug θ j, or maemaically, π(y, θ, j) = π(y θ,j) (8)

9 RODRÍGUEZ: BAYESIAN ESTIMATION AND MODEL SELECTION FOR THE WEEKLY COLOMBIAN EXCHANGE RATE 5 Assume also a e θ j are condiionally independen, given e value of m. Hence, M π( θ j ) = π( θ j i = i ) Noe a e prior disribuion π(θ i j), for i j does no make muc sense. I specifies e disribuion of e parameers of model i, condiioned on e fac a is is no e rue model. Carlin and Cib (995) refer o ese as pseudo-prior or linking disribuions. Due o e condiional independence (8), ese priors do no inerfere in e expressions of e marginal predicive densiies for eac model. Nevereless, ey are relevan for e consrucion of e cain and mus be specified. I follows from e above specificaion a M j, j ) π( θi j ) j = π( y, θ, j ) = π( y θ π wic is proporional o e join poserior disribuion of θ and m. A naural blocking is formed by grouping eac model s parameers and m. Te full condiional disribuions for θ, θ,..., θ M and m are obained as follows: For block θ j, j =,..., M, j π ( p j( θ j ) y θ, j ) π( θ j π( θ i ), i j j ) for m = j for m = i j For block m p M ( j ) = k π( y θ, j ) j M j = π( θ j ) π, j i j =,..., M a is a discree disribuion wi proporionaliy consan k = M M π( y θl, l ) π( θi l ) π l l = i =

10 5 REVISTA DE ECONOMÍA DEL ROSARIO, 4 (DICIEMBRE ) m can always be sampled direcly because i as a discree disribuion. Direc sampling from blocks θ j will depend on e conjugacy srucure for model m = j and e form of e pseudo prior disribuion. Wen direc sampling for some of e θ j s is no possible, Meropolis-Hasings seps may be used. Te above sceme saisfies e condiions of a convenional Markov Cain and erefore converges o e arge disribuion given by e poserior. Comparison beween models is based on e marginal poserior disribuion of m, p(j), j =,..., M. Tese probabiliies are esimaed by e proporion of values of m equal o j in e sample of size n. Te pseudo prior disribuions mus be carefully cosen, as ey affec e rae of convergence of e cain. Carlin and Cib (995) recommend e use of simple sandard approximaions based on univariae esimaes obained from pilo cains. Te same auors sugges using fairly vague prior disribuions, bu i is well-known a wen using is pracice on models wi differen dimensions e Bayes facors urn ou o be very sensiive. So, is prior seing may need furer jusificaion o saisfy poenial users. Finally, is approac is no applicable o e case of counable number of models under consideraion. Hence, e number of pracical and eoreical difficulies of is approac sugges i sould be used wi care. See Gamerman (997), were more deails can be found.. Markov Cains wi Jumps Green (995) inroduced a reversible-jump MCMC sraegy for generaing from e join poserior π(m,θ m y), based on e sandard Meropolis-Hasings approac. Te reversible-jump MCMC was also applied by Ricardson and Green (997) for an analysis of univariae normal mixure; by Nobile and Green (), for facorial experimens using mixure modeling; and Dellaporas and Foreser (999), for analysis of coningency ables. During reversible-jump MCMC sampling, e consruced Markov Cain moves wiin and beween models, so a e limiing proporion of visis o a given model is e required π(m y) In general, suppose a e curren sae of e Markov Cain a ime is (m, θ m ) were θ m as dimension d(θ m ) and a move is proposed a ime + o a new model m wi probabiliy j(m, m ) and corresponding parameer vecor θ m. Ten, a vecor is generaed from a specified proposal densiy q(u θ m, m, m ) and (θ m,, u ) = g m,m (θ m, u) is se for a specified inverible funcion g m,m suc a g m,m = g Noe m,m a d(θ m ) + d(u) = d(θ m ) + d(u ). Green (995) sowed a, if e new move is acceped as e nex realizaion of e Markov Cain wi probabiliy α = min{,r}, were π( y m', θ' m' ) π( θ' m' m' ) π( r = π( y m, θ ) π( θ m ) π( m m m' ) j( m', m )q( m ) j( m, m' u' θ' )q( u θ m m', m', m ), m, m' ) J

11 RODRÍGUEZ: BAYESIAN ESTIMATION AND MODEL SELECTION FOR THE WEEKLY COLOMBIAN EXCHANGE RATE 53 wi J = (θ m, u )/ (θ m, u) denoing e Jacobian of e ransformaion, en e cain saisfies e condiion of deailed balance and as e required limiing disribuion π(m, θ m y). Te condiion of deailed balance requires a e equilibrium probabiliy of moving from a sae (m,θ m ) o (m,θ m ) equals a of moving from (m,θ m ) o (m,θ m ) for deails, see Green (995). To implemen e reversible-jump MCMC, e probabiliies j(m,m ) need o be specified for every proposed move, as well as e proposal disribuions q(u θ m, m, m ), q(u θ m, m, m) and e funcion g m,m Tese coices do no affec e resuls in erms of models seleced bu may affec crucially e convergence rae of e Markov Cain. For e probabiliy j(m,m ) one non-informaive alernaive is j(m,m ) = (M ), for all m, m M, wen a eac sae of e cain a move from one model o oer one is always proposed. Vronos e al. () proposed a modificaion of Green s ecnique, wic ey ave successfully implemened in a series of experimens wi GARCH and EGARCH models; is is described as follows: Firs, ey sugges a all e parameers of e proposed model be generaed from a proposal disribuion. Consequenly, (θ m, u ) = (u,θ m ) wi d(θ m ) = d(u ) and d(θ m ) = d(u), q(u θ m, m, m ) = q(u m ), q(u θ m, m, m) = q(u m), and e Jacobian in (9) is. In is case, e probabiliy of accepance of e new move as e nex realizaion of e Markov cain is given by α = min{,r}, were π( y m', θ' m' r = π( y m, θ m ) π( θ' ) π( θ m m' m' ) π( m ) π( m' ) j( m', m )q( u' m ) m ) j( m, m' )q( u m' ) () Te proposal densiies q(u m ) and q(u m) can be cosen by invesigaion of a pilo run Tey sar e cain from e bes available saring values (e.g., e maximum likeliood esimaes wen available) and simulae e wiin-model Markov Cain many imes o obain approximae marginal poserior means and covariance marices for eac model parameer vecor. Tese esimaes are en used o consruc proposal densiies q(u m ) and q(u m) aken as mulivariae normal densiies. III. METHODOLOGY A. Te Daa In order o illusrae e esimaion and selecing meodologies, e weekly observaions of e US/Colombian (on spo) excange rae are used. Le er represen e Fridays excange raes, running from Ocober, 99, roug December 9 of 999; a made T = 48 observaions. Te daily excange rae corresponds o e weig average of rading, selling and buying, of U.S. dollars in e open marke. In e even of e marke being closed on a Friday, e observaion on e previous Tursday was used.

12 54 REVISTA DE ECONOMÍA DEL ROSARIO, 4 (DICIEMBRE ) Figure sows e raw daa; from is and wi e elp of a uni roo es, (See Enders, 995), i is easy o conclude a e excange rae as a uni roo, so, as usual, one works wi e firs difference of e naural logarim of excange raes, reurns, also known as coninuously compounded raes of reurn, (r = ln (er ) ln (er ) for =,..., T), wose represenaion is sown in Fig.. Figure Weekly Colombian excange rae From Figure 3, wic sows e isogram of reurns, i is noewory a e skewness and kurosis coefficiens for r wic are.858 and 5.687, respecively, are bo significanly posiive and muc larger an common, us sowing asymmery and lepokurosis. As poined ou by Vlaar and Palm (993), e skewness could be e resul of e asymmery in e movemens of e pariy adjusmens, and a ig kurosis could resul from a ime varying-variance. Tese wo resuls lead o considering ose disribuions differen from e normal, wose use is unlikely o yield appropriae resuls. Also, i is clear a e variance is no consan a all, wic can be seen from e Lunjg-Box s auocorrelaion saisic of e squared reurns: Q () = 8,3 and Q (4) = 44,86 for lags and 4, respecively.

13 RODRÍGUEZ: BAYESIAN ESTIMATION AND MODEL SELECTION FOR THE WEEKLY COLOMBIAN EXCHANGE RATE 55 B. Esimaion Based on e eoreical jusificaion of several models and aken in accoun e paricular Colombian economy, seven differen models were proposed for esimaion and selecion among em; Tey are described below..8 Figure Weekly reurn from colombian excange rae Wi e aim o implemen a fully Bayesian analysis, MCMCs were used o esimae e models, muc of em are esimaed by e firs ime from e Bayesian poin of view, wic is a new developmen of is work. Firs, an ARMA(,) model wi normal disurbances was fied; is model could be ried in pracice wen e scolar ignores e fac a e variance is no consan. In is repor, suc a model is also used as e reference model, agains wic oers are o be compared. Non-informaive Bea(,) from - o 3 priors were cosen for and θ. A N(,5) as prior for was used; finally, a nearly noninformaive bu proper Inverse-Gamma(.,.), (So, Expeced value and variance equal o /) was used for σ. 4. Maximum Likeliood Esimaion (MLE) was ried, bu problems geing precise esimaions of e variance-covariance marix, for some models, resriced is use. 3. In e sense a if X ~ Bea(,) en Y = X ~ Bea(,) from o. See Jaquier e al (999) 4. /σ as a prior for σ was ried oo, wi similar resuls

14 56 REVISTA DE ECONOMÍA DEL ROSARIO, 4 (DICIEMBRE ) 6 Figure 3 Hisogram of reurns. 4 # Frequency Midpoin Second, an ARMA(,) for e levels of reurn plus a GARCH(,) for e variance, so, c in () comes from an ARMA(,), and V(ε ) = from (3) wi r =, and s = ; sill assuming normaliy, e likeliood funcion is given by (). Noninformaive U(,) prior for and θ ; /α for α ; a N(,5) for α ; U(,) for α, and β, and in order o assure saionariy in (3), any proposal a did no fi α + β < is rejeced. Te same priors on as before was used. From iniial runs urned ou o be saisically equal o zero, us i is se o α / ( α ) as Nelson and Cao (99) propose for compuaional convenience. Tird, an ARMA(,) plus a GARCH(,) wi Suden- disribuion, and α + β >= are rejeced in (3) for saionariy purposes; n > in (4); was se as in before. Priors: /α for α ; /(n ) on n and α, β as in e second model,, and θ were reaed as in e firs model. Four, an ARMA(,) plus an EGARCH(,), insead of wi GED disribuion, erefore, e likeliood is given by (6). Priors:,, and θ as above; a N(,) was used as prior for α ; N(,5) for α for β an U(-,); ϕ is assumed idenically o zero; and U(,8) for υ. Any proposal wi β > was rejeced for saionariy purposes. Fif, an ARMA(,) plus an GARCH(,) wi a Mixure of wo normals as e error erm disribuion using (8) as e likeliood. Priors:,, and θ as before. /α for α for α, and β U(,); N(,) for υ; /(δ ) for δ ; finally, U(,) for λ. Six, an ARMA(,) plus an GARCH(,) wi Type-IV Pearson disribuion, ence, e likeliood is compued by using (). Priors: Inverse-Gamma(.,.) was used for α, U(,) for α, and β ; (r ) following an Inverse-Gamma (.,.); N(,) for δ and N(,5) for µ. Te disribuions used previously for,, and θ were used ere oo. Te resricion r > 3 is imposed, so e firs four momens from e Type-IV Pearson disribuion exis. Again, any proposal wi α + β >= is rejeced. α <. were rejeced for compuaional reasons.

15 RODRÍGUEZ: BAYESIAN ESTIMATION AND MODEL SELECTION FOR THE WEEKLY COLOMBIAN EXCHANGE RATE 57 Seven, and finally, a Socasic Volailiy model wi a Suden- disribuion on e error erm, expressed in (6) and (5). Non-informaive prior U(,) for, a N(,5) for γ; Inverse-Gamma(.,.) for σ ; and for υ a U(5,8) was used as η Jaquier e al. (999) did o assure e -Suden as a leas e firs-four momens. Aloug e use of single versus muliple (parallel) cains in MCMC is an open discussion, single cains for eac model are used in is job because of ime and compuing resources limiaions. For a recen discussion of is dilemma, e reader is referred o Mengersen e al. (999), wic conains poins in favor of eac alernaive. Rafery and Lewis (996) sraegy was implemened ere wi consan seleced in suc a way a e proporions of e proposals acceped were beween and 5%, as as become common pracice. Te updae is elemen-by-elemen and in random order. However, wen ig correlaions beween parameers in conjuncion wi slow convergence were found e blocking updae was implemened o improve convergence. In every case, a final cain of 8, was run and en seps of 5 o 3 were aken o avoid large auocorrelaions in e cains. In a way, firs-order auocorrelaions no larger an.55 were guaraneed. Convergence of eac cain is assessed by applying e Geweke s (99) crierion, wic null ypoesis is a saionariy as been reaced, and e es-saisic is suppose o follow a sandard normal disribuion under H. Tis es is implemened by using CODA (See Bes e al, 997). Te mean-vecor and Variance-covariance marices are o be obained in order o feed or implemen e RJMCMC, as explained laer. C. Model Selecion Te model selecion exercise consiss of applying e Reversible Jump MCMC algorim and e poserior probabiliies, running, ieraions and sowing e proporion of eac, a model m (m=,,... 7) is seleced. For cecking sabiliy, visual analysis is used. Aloug ere are some fres resuls abou assessing convergence in RJMCMC, eir value are no well-known ye, as menioned by Brooks and Guidici (999). For all seven models e same priors menioned in Secion B are o be used. Te proposal densiies q(u/m ) and q(u /m) for eac parameer were consruced by using e MCMC oupu of e separae model runs described above. Tese densiies are aken as mulivariae normals wi mean vecors, consising of e sample mean values and covariance marix equal o e corresponding sample mean vecor and covariance marix of e parameers in eac model.

16 58 REVISTA DE ECONOMÍA DEL ROSARIO, 4 (DICIEMBRE ) IV. RESULTS A. Esimaion 5 In e following presenaion e reurn raes are expressed in - scale, wic was used because of compuaional and presenaional avenues; oerwise, models wic use GARCH componen ge suck, i seems because values for go so close o zero a e algorim ge overwelmed. Table Geweke s convergence z scores for e seven models Model θ α α β σ n ARMA(,)Normal ARMA(,)+ GARCH(,):N ARMA(,)+ GARCH(,): Model θ α α β v ARMA(,)+ EGARCH(,):GED Model θ α α β λ σ n ARMA(,)+ GARCH(,):MixN Model θ α α β r σ n ARMA(,)+ GARCH(,):T-IV P Model γ υ σ η STOCHASTIC VOL.: Te Geweke s Convergence z Scores for e seven models are presened in Table, looking o e numbers i is clear a convergence as been reaced for almos every parameer in all models. Te esimaion resuls are presened in Table wi sandard errors in pareneses. 6 Aloug, parameer ransformaions 7 were ried for some models, convergence was no improved, ence i use was discarged. From Table i sould 5. I ook beween ours 4 minues and 56 ours and 48 minues, from e fases o e slowes model, o made all e ieraions, using a compuer wi a Penium I 33 MHz processor, and 64 MB RAM, running under WINDOWS- 98 Second Ediion. Times reduce o one ird using a Penium III 7 MHz processor, wi e same sofware and 9 MB RAM.

17 RODRÍGUEZ: BAYESIAN ESTIMATION AND MODEL SELECTION FOR THE WEEKLY COLOMBIAN EXCHANGE RATE 59 be said a excep for e non-inclusion of some parameers in mos of e models, no addiional work for e exclusion of non-significan parameers in any model was aemped because ime limiaions, and because i is no e main purpose of is work o improve every and/or one specific model. Fig. 4 presens e resuling cains diagrams and e isograms of e poserior sample of e parameers of e ARMA(,) model. Te sape of e poserior disribuion of and σ parameers indicae asymmery, ence deviaion from normaliy. Figs. 5 o do e same for e GARCH-N,GARCH-, EGARCH- GED, GARCH-MixN and GARCH Type-IV Pearson models, respecively. Table Esimaion resuls of e seven compeing models Model θ α α β σ n One (6.6e-3) (.) (.) (3-E-3) Two.95 Two (.4) (.79) (.6) (.4) (.4) (.4) Tree.53 Tree (.5) (.4) (.) (.8) (.) (.6) (.6) Model θ α α β v Four (.3) (.3) (.9) (.9) (.7) (.9) (.5) Model θ α α β λ υ δ Five (.3) (.5) (.44) (.9) (.) (.7) (.5) (.46) (.8) Model θ α α β r δ µ Six e-3..9e e-3 (.9) (-5) (.3) (3.8e-4) (.e-5) (.8e-5) (8.e-3) (4.3e-3) (7.4e-4) Model γ υ σ η Seven (.3) (.9) (.53) (.6) B. Model Selecion Te processing ime for, ieraions, using e same compuer as for esimaion, was 6 ours and 3 minues, for all e seven models, a oal of 47 parameers wic use above 3 MB of disk-space. Noe a, according o Fig., is is a conservaive run leng, and less an one-four of e run could be sufficien o acieve e same poserior disribuions. 6. Suc sandard errors refer o e ime-series esimaes wic are asympoic, e square roo of e specral densiy esimae divided by e sample size. 7. Like logarim or = ln(+)/( ), wen < ence R

18 6 REVISTA DE ECONOMÍA DEL ROSARIO, 4 (DICIEMBRE ) Figure 4 ARMA(,) Model θ σ Ieraions

19 RODRÍGUEZ: BAYESIAN ESTIMATION AND MODEL SELECTION FOR THE WEEKLY COLOMBIAN EXCHANGE RATE 6 Figure 5 GARCH-Normal Model θ α α β Ieraions

20 6 REVISTA DE ECONOMÍA DEL ROSARIO, 4 (DICIEMBRE ) Figure 6 GARCH- Model θ α α β η Ieraions