DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń 2006

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1 DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus Universiy Toruń 6 Jacek Osiewalski, Anna Pajor, Maeusz Pipień Cracow Universiy of Economics (Kraków, Poland) Bayesian Analysis of Main Bivariae GARCH and SV models for PLN/USD and PLN/DEM (996-). Inroducion In a previous Bayesian comparison of 6 GARCH and SV (sochasic variance) bivariae models Osiewalski, Pajor and Pipień (6) have shown ha even simple SV specificaions fi he daa much beer han very sophisicaed GARCH srucures. This phenomenon can be aribued o describing volailiy by laen AR() processes, which is he main feaure of he SV class and yields is disribuional flexibiliy and ease in modelling ouliers. The aim of he paper is o presen and compare poserior inferences (for he main quaniies of ineres) obained using differen models. Here we ake ino accoun our previous resuls on model comparison and focus only on hree leading SV specificaions and wo represenaive GARCH srucures: he bes one, i.e. he -BEKK(,) model, and he parsimonious -DCC model, based on he one proposed by Engle (). As in our previous papers, we use Markov chain Mone Carlo (MCMC) echniques o conduc our Bayesian approach and, for he sake of comparison, he daily growh raes of wo exchange raes: PLN/USD and PLN/DEM ( ). We show ha sequences of esimaes of he condiional sandard deviaions and correlaion coefficiens can be moderaely similar for good SV and reasonable GARCH models, despie huge differences in model fi and incomparabiliy of he condiional covariance marices (we condiion on differen variables in GARCH and SV models). Due o space limiaions, we do no presen he Bayesian mehodology, described in our oher papers. In he nex secion we review he main models from previous sudies. In secion we summarise he resuls of he Bayesian model comparison. Secion is devoed o he presenaion and comparison of poserior resuls obained wihin paricular models.

2 6 Jacek Osiewalski, Anna Pajor, Maeusz Pipień. Main Bayesian Models from he SV and GARCH Classes We denoe by y =(y, y, ) bivariae observaions on growh (or reurn) raes, and we model hem using he basic VAR() framework: y δ = R( δ ) + ε () y wih ε described by compeing bivariae ime-varying volailiy processes. More specifically, y, δ R R y, δ ε, = +, =,...,T. y, δ R R y, δ ε, The elemens of δ and R are common parameers, which we rea as a priori independen of all oher (model-specific) parameers and assume for hem he mulivariae sandardised Normal prior N(, I 6 ), runcaed by he resricion ha all eigenvalues of R lie inside he uni circle... Bivariae Sochasic Volailiy Specificaions Assume ha ε in () is condiionally Normal (given parameers and laen variables in θ (i) ) wih mean vecor and covariance marix Σ, depending on laen variables, i.e. ε M, θ ~ N(, Σ ). i ( i) [ ] Thus, he corresponding condiional disribuion of y = (y, y, ) (given is pas and laen variables) is Normal wih mean μ = δ + R (y - δ) and covariance marix Σ. The compeing bivariae SV models M i are defined hrough differen specificaions of he laen processes and differen srucures of Σ (always, by consrucion, posiive definie symmeric).... The Sochasic Discoun Facor Model (SDF) This simples MSV specificaion (M ) uses jus one laen process g o describe he dynamics of he whole condiional covariance marix (see Jacquier, Polson and Rossi (995)): ε = ξ g, ln g = γ + φ(ln g γ ) + σ gη, ξ ~ iin([ ], H ), η ~ iin (,), ξ η s ;, s Z. The condiional covariance marix of ε akes he very simple form g h g h Σ = g H =, g h g h which leads o he invariable condiional correlaion coefficien ρ, = ρ = h hh. In order o ensure idenifiabiliy, he resricion γ= is imposed, while H is

3 Bayesian Analysis of Main Bivariae GARCH and SV Models... 7 a symmeric posiive definie marix consising of hree free enries. We assume independence among parameers and use he following prior disribuions: H ~ IW(I, ), i.e. inverse (or invered) Wishar wih degrees of freedom and parameer marix I ; σ g ~ IG(., ), an inverse Gamma disribuion; φ ~ N(, ) I (-,) (φ), i.e. Normal wih mean and variance, runcaed o (-, ); and ln(g ) ~ N(, ). Here we use he same parameerisaion of he inverse Wishar and Gamma class as O Hagan (99). Tha is, for he hyperparameer values chosen here, he prior expecaions of H and σ g do no exis. Equivalenly, H - has a Wishar prior wih mean I, and σ g - has a Gamma prior wih mean and variance....the JSV Model The SDF specificaion is very resricive since i assumes he same dynamics for all enries of he condiional covariance marix. The JSV model (M ), proposed by Pajor (5b) and based on he specral decomposiion of Σ = P Λ PP-, uses a separae laen process o describe each eigenvalue of Σ. More specifically, Λ is he diagonal marix of eigenvalues of Σ, Λ =Diag(λ,, λ,), and P is he orhogonal marix of eigenvecors of Σ, depending on p, a parameer from (, ]: p p P =. p p Hence, he condiional covariance marix of ε has a non-rivial srucure: λ, p + λ, ( p) ( λ, λ, ) p p Σ =, ( λ ) + ( ), λ, p p λ, p λ, p which leads o he varying condiional correlaion coefficien: ρ, ( λ, λ, ) p p =. ( λ λ ) p ( p ) + λ λ,, Noe ha we have a pair of posiive laen processes, Θ ln λ +,,, γ = φ(ln λ, γ ) σ η,,, γ = φ (ln λ, γ ) σ η, ( η,, η, )' η ~ iin([ ], I. ln λ +, η =, ) = ( λ,, λ, )', where We impose prior independence and assume he following priors: (γ ii, φ ii ) ~ N(, I ) I (-,) (φ ii ); σ ii ~ IG(., ); lnλ i, ~ N(, ), i =, ; p ~ U([, ]), where U(A) is he uniform disribuion over A.

4 8 Jacek Osiewalski, Anna Pajor, Maeusz Pipień...The TSV Model The non-rivial srucure of he JSV condiional covariance marix is based on as many laen variables as here are ime series under consideraion. Hence, he covariance dynamics is no compleely free, as i is relaed o volailiies. The hird MSV model (M ), proposed by Tsay () hus called TSV and used by Pajor (5a, 6), is based on as many separae laen processes as here are disinc elemens of he condiional covariance marix. The TSV model relies on he Cholesky decomposiion Σ = L G L, where q, L = G =, q, q, ln γ = φ (ln q γ + σ η, q,, ), ln q, γ = φ (ln q, γ ) + σ η,, q, γ = φ( q, γ ) + σ η,, ( η,, η,, η, )' ~ iin([ ], I η =, η ). Now Θ = (q,, q,, q, ) is a rivariae laen process wih wo posiive componens, and he condiional covariance marix of ε akes he form: q, q, q, Σ =, q, q, q, q, + q, which leads o he following variable condiional correlaion coefficien: q, q, ρ, =. q + q q,,, We make similar assumpions abou he prior srucure as previously: (γ ij, φ ij ) ~ N(, I ) I (-,) (φ ij ), σ ij ~ IG(., ) (i,j {,}, i j); lnq ii, ~ N(, ), i =, ; q, ~ N(, ). Finally, noe ha he diagonal enries of Σ are no modelled in a symmeric way and, hus, he order of appearance (numbering) of financial insrumens maers in he TSV specificaion, conrary o oher models... Bivariae GARCH Specificaions We assume ha he condiional disribuion of ε (given is pas, ψ -, and parameers) is Suden wih locaion vecor, inverse precision marix H and degrees of freedom ν >, i.e. h, h, ε M i, θ ( i), ψ ~ S([x], H, ν ), H =. () h, h, Noe ha we now use he Suden disribuion insead of condiional Normaliy assumed for he SV class. However, hese disribuional assumpions are no

5 Bayesian Analysis of Main Bivariae GARCH and SV Models... 9 direcly comparable as he condiioning variables are differen in he SV and GARCH classes. As regards iniial condiions for H, we ake H = h I and rea h as an addiional parameer. We assume prior independence for h, ν and he remaining parameers; h follows he Exponenial prior wih mean, Exp(), and ν has he Exp() prior, runcaed by he condiion ν >. The condiional covariance marix of ε given θ (i) and ψ - is (ν ) - ν H. The compeing bivariae GARCH models are defined by imposing differen srucures on H. Osiewalski, Pajor and Pipień (6) consider wo differen groups of mulivariae GARCH specificaions: he VECH(,) srucure ogeher wih is special cases, including a simple BEKK(,) model, and Bollerslev s CCC model and is generalisaions proposed by Engle () and Tse and Tsui (). Here we focus on Engle s DCC srucure, which can easily be used in higher dimensional problems, as well as on he BEKK(,) case, which was he winner among bivariae GARCH models in our previous Bayesian comparisons.... The -BEKK(,) Model This model (M ) is defined by he following srucure of H in (): a a b b b b c c c c H = + ε ε ' + H a a b b b b c c c c () ' i.e., H = A + B ε ε B + C H C, wih ρ h h as he ( ), = h,,, condiional correlaion coefficien. The parameers of he covariance srucure () have he following prior disribuions: a ~Exp(), a ~Exp(), a ~N(,), b ~N(.5,), b ~N(,), b ~N(,), b ~N(.5,), c ~N(.5,), c ~N(,), c ~N(,), c ~N(.5,), which are runcaed by he resricions of posiive semi-definieness of he symmeric (x) marix A and sabiliy of he general (x) marix C (all eigenvalues of C lie inside he uni circle). We also impose b > and c > in order o guaranee idenifiabiliy.... The -DCC model In Engle s dynamic condiional correlaion (DCC) model he diagonal elemens of H are described as in he CCC specificaion of Bollerslev (99): hii, = α i + α iε i, + β ihii, (i=,), () and for he off-diagonal elemen i is assumed h, = ρ, h, h,, (5) where ρ, is he ime-varying condiional correlaion coefficien, modelled as ρ q q (6), = q,,,

6 Jacek Osiewalski, Anna Pajor, Maeusz Pipień wih q ij, s being enries of a symmeric posiive definie marix Q of he same order as he dimension of ε. A simple specificaion for Q, considered by Engle (), assumes ha ' Q = ( b c) S + bξ ξ + cq, (7) where b and c are nonnegaive scalar parameers (b+c<), ξ is he vecor of sandardised errors and S is heir uncondiional correlaion marix. In he case of our bivariae condiionally Suden specificaion, we keep Engle s basic srucure and define S as a square marix wih ones on he diagonal and s =s =ρ, an unknown parameer from he inerval (-, ); his assures posiive definieness of S and Q. Also, in our case ξ = ε ν ) ( ν h ) (i=,). (8) i, i, ( ii, Thus, our DCC model (called -DCC, M 5 ) generalises he condiionally Normal srucure proposed by Engle () o he Suden condiional disribuion; see Osiewalski and Pipień (5). The original N-DCC model, corresponding o ν, is based on ξ =. The iniial condiion for Q is Q = q I, i, ε i, h ii, wih free q >. The -CCC model (M 6 ) is nesed in -DCC assuming b=c=. We follow he exac Bayesian approach, which is fully feasible in he bivariae case. So, we do no use he approximae wo-sep esimaion procedure suggesed by Engle (). We assume ha a priori (b c) is uniform over he uni simplex, α ~Exp(), α ~Exp(), (α,α,β,β )~U([,] ), ρ ~U([-,]) and q ~Exp().. The Daa and Resuls of Model Comparison In order o compare he main bivariae GARCH and SV srucures we use he same daa se as Osiewalski and Pipień (,5), Pajor (5b), and Osiewalski, Pajor and Pipień (6), who also presen our Bayesian approach and MCMC echniques used. The daa se consiss of 85 observaions on he zloy (PLN) values of he US dollar (x, ) and German mark (x, ). They are he official daily exchange raes of he Naional Bank of Poland (NBP fixing raes), which cover he period from February, 996 ill December,. The firs hree observaions (February, and 5, 996) are used o consruc iniial condiions, y (). Thus, T, he lengh of he modelled vecor ime series of daily growh raes of x, and x,, is equal o 8. We use he bivariae VAR() framework (), where y i, = ln(x i, / x i,- ) for i=,. Boh series (y, and y, ) are cenred abou zero, wih several ouliers and changing volailiy. Their sample correlaion coefficien (.567) indicaes posiive correlaion. The overall ranking of he compared models M i as well as log (BB,i), he decimal logarihms of he Bayes facors in favour of he TSV specificaion (M ), calculaed using he Newon and Rafery (99) mehod, are shown in Table. Since Bayes facors differ by many orders of magniude, he model ranking is numerically sable and robus wih respec o reasonable changes in he prior

7 Bayesian Analysis of Main Bivariae GARCH and SV Models... disribuions of he parameers. I is clear ha even he SDF model, a very simple SV specificaion, beas he GARCH srucures in erms of he marginal daa densiy value (a naural Bayesian measure of fi). The SDF specificaion is beer by orders of magniude han he bes GARCH model from previous sudies, i.e. he -BEKK(,) model; see Osiewalski, Pajor and Pipień (6). Using only one laen process (a he expense of common dynamics of he condiional variances and covariance) already helps a lo in modelling ouliers. Of course, he use of more laen processes improves fi enormously. Table. Logs of Bayes facors in favour of VAR() TSV (M ) Model (M i ) Number of parameers (and laen variables) Rank VAR() TSV (M ) 8 (+T) VAR() JSV (M ) 5 (+T) 5 VAR() SDF (M ) (+T) 9 VAR() -BEKK(,) (M ) VAR() -DCC (M 5 ) 8 6 VAR() -CCC (M 6 ) Inference on Volailiy and Condiional Correlaion log (BB,i) The aim of his secion is o compare poserior resuls for he dynamic correlaion coefficien and individual volailiies. In his comparison we keep M M 5, omiing he wors model M 6 (-CCC). I is imporan o know wheher models ha have so differen fi lead o similar poserior inference on quaniies of ineres. Since he condiional disribuions in GARCH and SV models condiion on differen variables, we inerpre he condiional covariance marices as based on he larges possible se of condiioning variables. Tha is, we formally condiion on pas observaions and all laen variables used in our hree SV models. The plos of he main poserior characerisics of ρ, (for each =,...,T; T=8) are presened in Fig. wih wo lines, showing E ρ y, y ) ± D( ρ y, ). We focus on ypical paerns, so only hree (, (), y() models are represened. I is clear ha consancy of condiional correlaions (assumed in SDF and CCC) is no suppored by he daa. Also, he poserior sandard deviaions of ρ,, D(ρ, y,y () ), are much higher in he case of TSV. This resul can be explained by addiional poserior uncerainy caused by he laen processes in case when he condiional correlaion coefficien depends on hese processes. In view of our model comparison, he high poserior precision in he GARCH cases is overly opimisic. See also Table, which shows he averages of he poserior means and sandard deviaions of ρ,.

8 Jacek Osiewalski, Anna Pajor, Maeusz Pipień Similariies in he dynamics of he condiional correlaion coefficiens are summarised in Table, which shows empirical correlaion beween differen sequences of T esimaes of ρ, ; i also shows empirical correlaion beween sequences of he condiional covariance esimaes. The poserior means of he condiional correlaion obained using eiher GARCH (-BEKK, -DCC) or SV (TSV, JSV) models are quie highly correlaed. However, he SDF specificaion (which fis he daa much beer han he GARCH models) assumes consancy of he condiional correlaion coefficien, conrary o he evidence from he bes SV models. Thus, he beer fi of SDF (as compared o -BEKK or -DCC) need no mean more reasonable inference on he condiional correlaion. As regards he volailiy esimaes, ploed in Fig. wih averages in Table, he average volailiies of he SV specificaions are much lower han in he case of he BEKK and DCC srucures. Similariy of volailiy dynamics of compeing srucures can be seen in Table, where he empirical correlaion coefficiens beween sequences of volailiy esimaes from differen models are presened. The hree SV models show almos he same dynamics of volailiies; he resuls in he wo GARCH models are also very highly correlaed. Wha is mos ineresing, he volailiy esimaes obained in models of differen ype show similar dynamics as well. The coefficiens for -BEKK and differen SV models always exceed.7. In our example, he dynamics of volailiy is esimaed quie similarly, despie huge differences in model fi. However, his is no he case for oher imporan aspecs of poserior inference. This means ha we should rely on rich enough bivariae SV srucures (TSV, JSV) in modelling pairs of financial ime series. Unforunaely, here are serious problems wih generalisaions o k- variae ime series (wih k>) as he JSV and especially TSV specificaions are hard o esimae in highly mulivariae cases. Since he GARCH models do no have enough flexibiliy o describe ouliers, he quesion of good specificaions for mulivariae financial modelling is sill open. Table. Correlaion coefficiens beween he poserior means of he condiional correlaions (upper par) and covariances (lower par) Model TSV JSV SDF -DCC -BEKK TSV.98 X JSV.9855 X SDF X X -DCC BEKK

9 Bayesian Analysis of Main Bivariae GARCH and SV Models... TSV BEKK(,) DCC Fig.. Condiional correlaion (poserior mean ± sandard deviaion) Table. Average poserior means of ρ, and average volailiy esimaes Model Average E(ρ, y, y () ) (and D(ρ, y, y () )) Average volailiy of PLN/USD Average volailiy of PLN/DEM TSV.59 (.) JSV.9 (.5) SDF.6 (.98) DCC.9 (.) BEKK(,).67 (.69)

10 Jacek Osiewalski, Anna Pajor, Maeusz Pipień Table. Correlaion coefficiens beween volailiy esimaes for PLN/USD (upper par) and for PLN/DEM (lower par) Model TSV JSV SDF -BEKK -DCC TSV JSV SDF BEKK DCC Condiional sandard deviaions for PLN/USD TSV SDF DCC Condiional sandard deviaions for PLN/DEM TSV SDF DCC Fig.. Volailiy esimaes in TSV, SDF and -DCC

11 Bayesian Analysis of Main Bivariae GARCH and SV Models... 5 References Bollerslev T. (99), Modelling he coherence in shor-run nominal exchange raes: A mulivariae generalised ARCH Model, Review of Economics and Saisics 7, Engle R. (), Dynamic condiional correlaion: A simple class of mulivariae generalized auoregressive condiional heeroskedasiciy models, Journal of Business and Economic Saisics, 9 5. Jacquier E., Polson N., Rossi P. (995), Models and prior disribuions for mulivariae sochasic volailiy, echnical repor, Universiy of Chicago, Graduae School of Business. Newon M. A., Rafery A. E. (99), Approximae Bayesian inference by he weighed likelihood boosrap (wih discussion), Journal of he Royal Saisical Sociey B 56, 8. O Hagan A. (99), Bayesian Inference, Edward Arnold, London. Osiewalski J., Pajor A., Pipień M. (6), Bayes facors for bivariae GARCH and SV models, Aca Universiais Lodziensis Folia Oeconomica, forhcoming. Osiewalski J., Pipień M. (), Bayesian comparison of bivariae GARCH processes. The role of he condiional mean specificaion, in: Welfe, A. (Ed.), New Direcions in Macromodelling, Elsevier, Amserdam, Osiewalski J., Pipień M. (5), Bayesian analysis of dynamic condiional correlaion using bivariae GARCH models, Aca Universiais Lodziensis Folia Oeconomica 9, -7. Pajor A. (), Procesy zmienności sochasycznej SV w bayesowskiej analizie finansowych szeregów czasowych (Sochasic Volailiy Processes in Bayesian Analysis of Financial Time Series), docoral disseraion (in Polish), published by Cracow Universiy of Economics, Kraków. Pajor A. (5a), Bayesian analysis of sochasic volailiy model and porfolio allocaion, Aca Universiais Lodziensis Folia Oeconomica, 9, 9-9. Pajor A. (5b), Bayesian comparison of bivariae SV models for wo relaed ime series, Aca Universiais Lodziensis Folia Oeconomica 9, Pajor A. (6), VECM - TSV Models for Exchange Raes of he Polish Zloy, Aca Universiais Lodziensis Folia Oeconomica, forhcoming. Tsay R.S. (), Analysis of Financial Time Series, Wiley, New York. Tse Y.K., Tsui A.K.C. (), A mulivariae generalized auoregressive condiional heeroskedasiciy model wih ime-varying correlaions, Journal of Business and Economic Saisics,, 5-6.