Bayesian Analysis of Dynamic Conditional Correlation Using Bivariate GARCH Models

A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA OECONOMICA 192, 2005

J a c e k Os i e w a l s k i * , M a t e u s z Pi p i eń* *

BAYESIAN ANALYSIS OF DYNAM IC CO N D IT IO N A L CORRELATION U SIN G B1VARIATE GARCH M O DELS***

Abstract. Multivariate ARCH-typc specifications provide a theoretically promising frame­ work for analyses of correlation among financial instruments because they can model time-varying conditional covariance matrices. However, general VechGARCH models are too heavily parameterized and, thus, impractical for more than 2- or 3-dimensional vector lime series. A simple i-BEKK(l.l) specification seems a good compromise between parsimony and generality. Unfortunately, Bollerslev’s constant conditional correlation (CCC) model cannot be nested within VECH or BEKK GARCH structures. Recently, Engle (2002) proposed a par­ simoniously parameterized generalization of the CCC model; this dynamic conditional cor­ relation (DCC) specification may outperform many older multivariate GARCH models. In this paper we consider Bayesian analysis of the conditional correlation coefficient within different bivariate GARCH models, which are compared using Bayes factors and posterior odds. For daily growth rales of PLN/USD and PLN/DEM (6.02.1996-28.12.2001) we show that the £-BEKK(l, 1) specification fits the bivariate series much better than DCC models, but the posterior means of conditional correlation coefficients obtained within different models are very highly correlated.

Keywords: model comparison, Bayes factors, multivariate GARCH processes, BEKK models, DCC models, exchange rates.

JEL Classification: C li, C32, C52.

1. INTRODUCTION

A p p ro p ria te statistical m odelin g o f c o rre la tio n am o n g Financial in ­ stru m e n ts is cru cial fo r any ap p lica tio n o f p o rtfo lio analy sis and for em pirical research on dependencies between financial m ark e ts. M u ltiv ariate

* Prof. dr hab. (Professor), Department of Econometrics, Cracow University of Economics.

** Dr (Ph.D., Assistant Professor), Department of Econometrics, Cracow University of Economics.

A R C H -ty p e specifications provide a theoretically prom ising fram ew ork as they can m odel tim e-varying conditional covariance m atrices. H ow ever, general V echG A R C H m odels presented by Engle and K ro n e r (1995) and G o u rie ro u x (1997, C h a p te r 6) are too heavily param eterized . T h e num ber o f free p aram eters o f m u ltiv ariate A R C H -ty p e m odels can increase very fast as the dim ension к o f the vector tim e series grow s. In the general version o f the /с-variate V echG A R C H (p, q) (or VECH(/>, q)) m odel, this n u m b e r is a fo u rth o rd e r polynom ial o f k, m a k in g even V E C H (1 , 1) im practical for к > 2 . T h u s, w ithin A R C H -ty p e m odels, interest focuses on restricted G A R C H specifications o r on factor A R C H m odels (e.g. D iebold and N erlo v e 1989, K in g, et al. 1994), G o u rie ro u x 1997, C h a p te r 8). H ow ever, facto r G A R C H m odels can be no t only difficult to estim ate (due to the presence o f latent variables), b u t also in ad eq u a te (inflexible in m o d elin g co m p licated dynam ics o f th e c o n d itio n a l co v a rian ce m atrix ). A sim ple t-B E K K (l, 1) m odel, based on specifications p ro po sed by B aba et al. (1989) and corresponding to certain n on -lin ear restrictions in t- V E C H (1, 1) - cf. Osiewalski and Pipień (2002), seems a good com prom ise betw een parsim ony and generality. H ow ever, this B E K K (1, 1) m odel in­ herits som e inflexibility o f the V E C H (1, 1) covarian cc stru ctu re; nam ely, Bollerslev’s (1990) co n stan t co nditional correlatio n (C C C ) m odel ca n n o t be nested w ithin V E C H o r B E K K G A R C H structures. R ecently, Engle (2002) p ro p o se d a parsim oniously param eterized generalization o f th e C C C m odel; his dynam ic co nditional correlatio n (D C C ) specification m ay o u tp erfo rm m an y o lder m u ltiv ariate G A R C H m odels. H ence, it is o f g reat interest to em pirically check the explanatory pow er o f D C C m odels.

In o rd e r to illu strate a form al Bayesian com p arison o f various bivariate A R C H -ty p e m odels th ro u g h their Bayes factors, Osiew alski and Pipień (2004a, b) used tw o exchange rates th a t were m o st im p o rta n t for th e Polish econom y till the end o f 2001, nam ely the zloty (P L N ) values o f the US d o llar and G e rm an m ark. T he d a ta consisted o f the official daily exchange rates o f the N a tio n al Bank o f Poland (N B P fixing rates), startin g from F e b ru a ry 1, 1996. By restricting to only bivariate V A R (I) m odels with G A R C H (1 , 1) d isturbances, it was possible to estim ate unparsim oniously param eterized specifications, such as V echG A R C H m odels. T h o se first co m p ariso n s focused on older m ultiv ariate G A R C H stru ctu res, proposed p rio r to 2001. T hus, the class o f m odels did n o t co n tain m o re recent m odels p rop osed by T se and T sui (2002), van der W eide (2002) and , in p articu lar, the D C C m odels o f Engle (2002).

T h e m ain result o f Osiewalski and Pipień (2004a, b) is th a t the simple i-B E K K (l, 1) m odel wins m odel com parison. In its /с-variate version, it has 0 ( k 2) free param eters, m uch less th a n the /с-variate general version o f V E C H (1, 1), requiring 0(/c4) free p aram eters. In this p ap e r we focus on

0 ( k 2) specifications, in p articular on variants o f E ngle’s D C C structures. O u r aim is to com pare these m odels, which were n o t considered in our previous Bayesian w orks, to the w inner from those studies. W e show th at, for o u r d a ta set, the unrestricted £-B E K K (l, 1) m odel describes th e tim e- varying co n d itio n al covariance m atrix still m uch , m u ch b etter th a n quite sophisticated (and very elegant) D C C structures, specially designed to m odel dynam ic co n d itio n al correlation.

In Section 2 we briefly present o u r Bayesian statistical m eth o d o lo g y and the num erical too ls we use. Section 3 is m ainly devoted to the description o f the com peting m odel specifications and the results o f their form al com p ariso n using Bayes factors. In Section 4 the sequences o f estim ates of th e conditional correlation coefficients (representing dynam ics o f the relation ­ ship betw een o u r tw o series) and stan d ard deviatio ns (m easuring volatility o f each series) are presented and com pared.

2. STATISTICAL METHODOLOGY AND NUMERICAL TOOLS

W e consider several com peting param etric Bayesian m odels for the same o bservation m atrix y. T h e i-th Bayesian m odel (M f) is characterised by the jo in t density function:

(1) p ( y A i ) \ M l, ym ) = p ( y \ M l,0{[), y(o))pi0l()\ M ^ ( i = l , ..., m),

w here y(0) den o tes initial conditions and p ( y \ M t,0w , ym ), p(0w \ M ^ are the sam pling density function and the prior density function u nder M f, respec­ tively. 0(I), the p aram eter vector in M „ groups p aram eters com m o n to all m m odels an d m odel-specific p aram eters. F o r th e p urpo ses o f inference w ithin M ; an d m odel com parison, we use the obvious deco m p osition

р(у,0щ \ М (,ут ) = р ( у \ М „ у 10др(От \у,МьУ«>д,

where p(0(i)\ y , M t, yi0)) is the posterior density function in and

Р(У\ Mi , y (0)) = $ p ( y \ M h 0(i), y(O))p(0M)d0Vl) o.

is th e m arg in al d a ta density in th e i-th B ayesian m o del. C o m peting m o d els a re co m p a re d pair-w ise th ro u g h th e Bayes fa c to r В и = — p ( y \ M i, y i0)) / p ( y \ M j , y l0)), w hich, to g e th e r w ith th e p rio r o d d s ra tio Р (М ;)/Р (М 7), determ ines the p osterior odds o f M , against M ■.

(2) Р(М,\У,Ут ) = Р ( Щ B P ( M j \ y , y m ) P( Mj ) ,J’

w here P ( M h) and P ( M h\ y, yw ) are, respectively, the p rio r an d posterior p rob ab ility o f M h (e.g. O ’H agan 1994). T he crucial role o f the Bayes factor in m odel com parison m eans th a t com puting m arginal d a ta densities under com p etin g m odels is the m ain num erical task. D irect evaluation o f the integral defining the m arginal d a ta density (as well as o f integrals related to p o sterio r inferences) - thro u g h either num erical q u a d ra tu re s o r M on te C arlo sam pling from the prior density - is not efficient (or even n o t feasible) w hen the dim ension o f the p aram eter space is as high as in th e m odels considered in this paper. T h u s we have to reso rt to o th er num erical tools, based on good exp lo ratio n o f the param eter space th ro u g h sam pling from the posterior. H ere we use M etropo lis-H asting s (M -H ) M ark o v chains (e.g. O ’H a g a n 1994), G am erm an (1997).

Using simple identities, we can write the m arginal d a ta density in the form

w here Р(0^ \ М1, у, ут ) denotes the p o sterior cum ulative d istrib u tio n function.

T his fo rm u la is the basis o f the m ethod by N ew ton and R aftery (1994), w hich ap p ro x im ates the m arginal d a ta density by the h arm o n ic m ean of the values р ( у |М ;,0(0,у{о)), calculated for the observed у an d fo r 0(i) draw n from the p o ste rio r distribution. T h e N -R h arm o n ic m ean estim ato r is consistent, b u t w ith o u t finite asym ptotic variance. D espite this serious theoretical w eakness, the N -R estim ato r (very easy to co m p u te) was quite stable fo r all o u r models; (rf. Osiewalski and Pipień 2004a fo r m ore discussion o f c o m p u ta tio n a l aspects).

In o rd e r to sam ple from the p o sterio r d istrib u tio n in a m odel with the p a ra m e te r vector 0, we use a sequential version o f the M -H algorithm , w here the p ro p o sal density q(0\0(m~ i)) for the next value o f 0 given the previous d raw 0(m~ 1> is p ro p o rtio n al to / s(0 |3 ,0 (m~1),C ), a S tu d en t t density w ith 3 degrees o f freedom , m ean 0(m~ l) an d a fixed covariance m atrix С (a p p ro x im a tin g the p o sterio r covariance m atrix). T h is S tu d en t-i density (sym m etric in 0 and is truncated by the inequality restrictions described in Section 3, i.e.

T his leads to th e M -H M ark o v chain w ith the follow ing acceptance probability:

(5) a ( 0 ; r - ł >) = m i n { ( g y(0)aq( 0) )l (9y( 0 * - " ) a t ( 0 * - " ) ) , l} ,

where gy(.) denotes the kernel o f the posterior density. T h u s, given the previous state o f the chain, 0(ffl_1), the cu rren t state 0(m) is equal to the candidate value 0* (draw n from the truncated Student-i d istrib ution discussed ab o v e ) w ith p ro b a b ility a o r 0<m> = 0< '"-1> w ith p ro b a b ility 1 — a(0*;0'm~ 1)). O u r results, presented in next sections, are based on 500 000 states o f the M ark o v chain, generated after 10 000 b urn t-in states.

In order to com pare com peting bivariate A R C H -type specifications we use the grow th rates o f P L N /U S D and P L N /D E M . O u r original d a ta set consists o f 1485 daily observations on the exchange rates them selves, P L N /U S D ( x u ) and P L N /D E M (x 2t). It covers the period from 1.02.1996 till 28.12.2001. T he first three observations from 1996 (F ebruary 1, 2 and 5) are used to construct initial conditions. T hus T, the length o f the m odeled vector tim e series o f daily grow th rates o f x 1( and x 2l is equal to 1482.

W e den o te o u r m odeled bivariate observatio ns as y, = ( У и У и ) \ where y u is the daily grow th (or return) rate o f the P L N value o f US do llar and y 2t is the daily g row th (or re tu rn ) ra te o f the P L N value o f G erm an m ark, b o th expressed in percentage points and obtain ed from the daily exchange rates x lt(i = 1, 2) by the form ula yif = 100 H x j x ^ - i ) . Osiewalski and Pipień (2004a) used only a sh o rt p a rt o f this bivariate series, till the end o f 1997 ( 7 = 475). N ow we base o u r results on all T = 1482 observations, as Osiewalski and Pipień (2004b), but we do no t use any exogenous variables in th e c o n d itio n a l m ean specification. T h u s we stay w ithin the p ure V A R -G A R C H fram ew ork, like Osiewalski and Pipień (2004a).

W e m odel th e d a ta using the basic V A R (l) fram ew ork:

w ith the e rro r described by com peting b ivariate G A R C H specifications. M o re specifically,

3. THE DATA AND COMPETING MODEUS

y t - ô = R ( y i- l - ô ) + Et (7)

h u

(

:)

T h e elem ents o f 6 and R are com m on param eters, which we treat as a priori independent o f all o ther (m ainly m odel-specific) param eters and assum e for them the m ultiv ariate standardized norm al p rio r iV(0, I 6), tru n cated by the restriction th a t all eigenvalues o f R lie inside the unit circle. W e assum e th a t the conditional d istrib u tio n o f e, (given its past, is Studcnt-£ with zero location vector, inverse precision m atrix H, and u n k n o w n degrees o f freedom v > 2, i.e.

As regards initial condition s for //,, we take H0 = /i0 / 2 an d treat h0 as an add itio n al p aram eter. We assum e prior independence for v, h0 (which are com m on) and the rem aining param eters; v follow s the exponential distrib u tio n w ith m ean 10, Exp( 10), truncated by the co n d itio n v > 2; h0 has the exponential p rio r with m ean 1, Exp(l).

T h e co nditio n al covariance m atrix o f e, given y/t_ x is (v — 2) l v Ht. C o m peting bivariate G A R C H m odels are defined by im posing different structures on H t. T h a t is, m odel-specific param eters are the ones describing II, in a given m odel. T h e sam pling density function in each m odel is always the p ro d u c t o f T co nditional bivariate S tudent-i densities (for y(r)) with v d eg rees o f freed o m , m ean ó + R ( y t- t — <$) an d c o v a ria n c e m a trix

T h e first specification considered here is the very parsim o n io u s co n stan t conditio n al correlatio n (C C C ) m odel o f Bollerslev (1990); it im poses the follow ing stru ctu re on H,:

where p1 2 is th e tim e-invariant conditional co rrelatio n coefficient. This sim ple stru ctu re o f H, am o u n ts to m odeling each conditional variance by a differen t G A R C H (1 , 1) process and m aking the co nd itio n al covariance a sim ple function o f the variances. In its /с-variate version, th e C C C m odel describes et using only 2 + 3/c 4- k ( k - \ ) / 2 free p aram eters; so we have 9 p aram eters w hen к = 2. F o r the m odel-specific p aram eters we take the follow ing priors:

(8)

(v — 2 ) _1vH (.

(9) / i t 1,1 — £iio + f l l l f i l . t - 1 "t" ^ l l ^ l l , t - l > ^ 2 2 , ( = a 20 + a 22e2 , t - l + ^ 2 2 ^ 2 2 , t - l >

(1 0) a l 0 ~ E x p ( 1), a2o ~ E x p ( l ) , (a t 1.0 2 2. ^1 1.6 2 2) ~ t/([0, l]4),

P i 2 ~ U ( [ - l , l ] ) ,

w here U(A) d en otes the uniform distribution over A. Osiewalski and Pipień (2004a, b) show th at, for our d ata, the C C C m odel is inad equ ate - it is m uch worse th a n heavily param eterized V echG A R C II specifications and th an m ore p arsim o n io u s B E K K structures, which all assum e tim e-varying conditional correlatio ns. It seems th a t m odeling dynam ic co rrelatio n w ith alm ost as few p aram eters as in the C C C m odel would be the m o st w elcom e solution.

T h e sim ple C C C specification (under conditio n al n o rm ality , i.e. with v = + oo) has been generalized by Engle (2002) in such a way as to m ake co n d itio n al co rrelatio n s fully dynam ic, keeping the co n d itio n al covariancc fo rm u la basically unchanged. Engle’s dynam ic con ditio n al correlation (D C C ) m odels describe the diagonal elem ents o f H, in the sam e way as in C CC , bu t assum e th a t

( 1 1 ) ^12,1 = P l2 ,t \ A l l .t^ 2 2 ,t >

where p 12it is the tim e-varying conditional co rrelatio n coefficient, m odeled as Pl2,t — 1 2, f / > /11, 1 <722,0

w ith qiJt's being entries o f a sym m etric positive definite m atrix Q, o f the sam e o rd e r as the dim ension o f e(. A sim ple specification fo r Qt, considered in Engle (2002), assum es th a t

( 12)

Q, = ( i - * - № + aŁl- l ęt- l +fiQt - l

w here a and ß are nonnegative scalar param eters (a + ß < 1), ę t is the vector o f stand ard ized errors and S is their unco nd itio nal co rrelatio n m atrix. In th e case o f o u r bivariate cond itionally S tu d e n t-г specification, we keep E ngle’s basic stru ctu re and define S as a square m a trix w ith ones on the diagonal and s1 2 = s2 1 = p 12, an unknow n p aram eter from th e interval ( -1, 1); this assures positive definiteness o f S and Q,. A lso, in o u r case

0 3) 6 , = E i J (v - 2)/(v/tiil() (i = 1, 2).

T h u s, o u r second specification (called D C C 0) generalizes th e conditionally no rm al basic stru ctu re proposed by Engle (2002) to th e S tu d en t-i conditional e rro r d istrib u tio n . T h e initial co n d itio n for Q, is Q0 = q012, where q0 is a free p a ra m e te r. In its /c-variate version, D C C 0 d esc rib es e, using

5 + 3/c + k(k — l)/2 free p aram eters - only three (q0, ot an d ß) m o re th an the C C C m odel (irrespective o f k). O f course, C C C corresponds to a = ß = 0, so it is nested in DCCO. We follow the exact Bayesian ap p ro ach , which is fully feasible in the bivariate case. T h u s we d o n o t use th e appro xim ate tw o-step estim atio n procedu re suggested by Engle (2002). T h e three new p aram eters are assum ed independent a priori o f the rem ainin g ones. T he p rio r fo r q0 is E xp ( 1), while the one for (a, ß) is uniform over the unit simplex.

T h e third m odel (called D C C 1) is also o f the D C C fo rm , but the specification o f Q, is different. T h e previous period e rro r term s arc not stan d ard ized and th ere are less restrictions:

(14) Q, = V -l-a e ,-iß, ' - 1 + ß Q t ~ i ,

V consists o f Vn ~ E x p ( l ) , v2 2 ~ E x p ( l ) and v1 2 = v2 1 = P i2V vnv 2 2 p l2 ~ L/([—1,1]), so V is positive definite w ith prior p ro b ab ility 1, and (a, ß) ~ l/([0, l]2).

T h e fo u rth m odel (D C C 2) generalizes the stru ctu re o f Q, by replacing the tw o scalar p aram eters (a and ß) by tw o sym m etric, n o nn egativ c definite m atrices (A an d B):

(15) Q, = 7 + A ° e t- x V - i + Bo Qt_ b

w here C ° D is the H a d am ard p ro d u c t o f tw o m atrices o f the sam e size (i.e., the elem ent-by-elem ent m ultiplication). T his e q u a tio n resem bles (24) in Engle (2002), b u t (as in D C C 1) the previous period e rro r term s are no t stand ard ized and th ere are no restrictions on A + B. O u r Bayesian D C C 2 specification uses the sam e V as in DCC1 and assum es th a t A consists of: a i2 = a2i = ®,Va n a2 2» «1 1.«2 2~ u g o , 1]) and ocr ~ £/([—1,1]); sim ilarly for ßij in B: ß 12 = ß 2l = ß j ß n ß z z , A i A a ~ П) and ßr ~ U( [ -1 ,1]). So Q, is positive definite with prio r prob ab ility 1. In its /с-v ariate version D C C 2 has 3 + 3{2k + k(k — l ) /2} free p aram eters th a t enter the co n ditio nal dist­ rib u tio n o f e, (18 for к = 2).

A s we have already noted, the i-C C C specification was strongly rejected by o u r d a ta w hen com pared to V EC H and B E K K b ivariate r-G A R C H structures. N ow we show the results o f o u r B ayesian co m p ariso n between the t - C C C and each f-D C C m odel. T he decim al lo garith m o f the Bayes facto r in favor o f the D C C 0 m odel is 46.60, in fav or o f the DCC1 specification is 45.15, while for the D C C 2 stru ctu re we o b tain 46.65. All three D C C m odels are ab o u t 45 orders o f m ag n itu d e b etter (i.e., m ore p ro b a b le a posteriori under equal p rio r m odel prob abilities) th a n the C C C

m odel ! H igh and alm ost equal values o f the Bayes facto rs fo r DCCO and D C C 2 indicate th a t these tw o m odels describe the d a ta equally well. T he pricc wc pay for no t using standardized residuals in Q, am o u n ts to estim ating m ore param eters in D C C 2 th an in DCCO. T h e p aram eteriz atio n in DCC1 seems no t rich enough, bu t the difference betw een the Bayes factors (of this m odel against C C C and o f DCCO against C C C ) is not large w hen we take into consid eratio n sensitivity with respect to the p rio r d istrib u tio n and num erical stability issues.

T h e results o b tain ed for the D C C m odels seem encouraging. H ow ever, o u r previous results (cf. Osiewalski and Pipień 2004b) show th a t th a t the decim al log o f the Bayes factor for a sim ple t-B E K K (l, 1) m odel (against C C C ) is even m uch higher, equal to 64.13. T h is B E K K specification is defined by the follow ing stru ctu re o f H t:

(16)

Я . - Г - ' " ! 1 + Г * “ ‘ , , Т < « , - л - 1 ' ) Г ! ’“ M + p “ c , 2 T » . - , p “ c ' [_a l 2 a 22_\ L " 21 2 2 J L " 21 " 2 2 J \_C2 l C 2 2 J L C 21 c 2

i.e. H t = A 4- + С 'Я ,_1С.

T h e p aram eters o f this stru ctu re have the follow ing p rio r d istribu tio ns: a n ~ E x p ( 1), a22 ~ E x p ( \ ) , a i 2 ~ N ( 0 ,1 ), b n ~ N ( 0 .5 ,1 ), b n (0 ,1 ) , Ь2 1~ Л Г (0,1), b22~ N ( 0 .5 ,1 ), c u ~ J V (0 .5 ,l),

c 1 2 ~ N ( 0 ,1 ), c 2 l ~ N ( 0 ,1 ), c2 2~ N ( 0 .5 ,1 ),

w hich are tru n ca ted by the restrictions o f positive sem i-definiteness o f the sym m etric (2 x 2) m atrix A and stability o f the general (2 x 2) m atrix С (all eigenvalues o f С lie inside the unit circle). A lso, the conditions: Ь ц > 0 and с и > 0 are im posed in o rd e r to g u aran tee identifiability, since В and - B as well as С and - C lead to the sam e H t, an d th u s are o bservationally equivalent. In the /с-variate version, o u r г-В Е К К (1, 1) m odel describes the co n d itio n al d istrib u tio n o f e, (given its past) using 2 + k ( k + l) / 2 + 2k 2 free

param eters (13 fo r к = 2).

T h e success o f the i-B EK JC (l, 1) m odel (its clear su periority over the D C C m odels and o th er specifications in explaining the tim e-varying co n ­ ditio n al covariance stru ctu re) suggests fu rth e r search fo r m ore p arsim o nio us special cases o f i-B E K J C (l,l) th a t would hopefully keep its ex planatory pow er. Som e m odels, like i-B E K K (l,0 ) th a t assum es С = 0, have already been tried (cf. Osiewalski and Pipień 2004b). T h e decim al log o f the Bayes facto r o f i - B E K K ( l,0) relative to t-C C C is -23.71 (!). T h u s, th e BEKJC(1, 0)

m odel (w ith an A R C H (l) stru ctu re only) is even m uch w orse the th an the C C C specification, so it will n o t be discussed fu rther. H ere we propose a simple “ scalar i-B E K K (l, 1)” structure, which am ounts to assum ing R = h l 2 and С = c l 2, where h and с arc independent scalar param eters with N ( 0.5, 1) p rio r d istrib u tio n s, tru n cated by the restrictions: b > 0 and 0 < c < l . So we consider

(17) II, = A + + с2Я ,_ ъ

w hich is m uch sim pler th an DCC1 (it uses the DCC1 stru ctu re o f Qt at th e level o f H t). T h e dccim al log o f the Bayes fa c to r o f this scalar t- B E K K ( l,l ) relative to t-C C C is 48.75, indicating th a t this restricted, extrem ely sim ple B E K K form ulation can co m pete in d yn am ic correlation m odeling w ith m o re sophisticated D C C structures, designed for this purpose. In fact, o u r scalar B E K K is a b o u t two or th ree ord ers o f m ag n itu d e m ore p ro b a b le a posteriori th an DCCO or D C C 2 (assum ing equal p rio r m odel probabilities). O f course, the unrestricted B E K K specification undoubtedly wins o u r m odel com parison for the analyzed d a ta set, being a b o u t 15 o rders o f m ag n itu d e better th an the second best specification.

All o u r results, those presented previously in Osiew alski and Pipień (2004b) and the new ones given here, indicate th a t the g ro w th rates of P L N /U S D and P L N /D E M strongly reject the con stan t conditional correlation hypothesis. T hese exchange rates form a bivariate tim e series with strong co rrelatio n dynam ics, w here B E K K m odels can (and sho uld) be used. T he fact th a t B E K K m odels do no t nest the C C C case is n o t a problem for the B ayesian ap p ro ach , which can deal with testing non-nested specifications using Bayes factors and po sterio r m odel probabilities. T h e results o f this section are sum m arised in T able 1, w here we ra n k the m odels by the increasing value o f the decim al logarithm o f the Bayes fa cto r o f B E K K (1 ,1 ) against the altern ativ e m odels.

Table 1. Logs o f Bayes factors in favor o f i-BEK K (l, 1) and average posterior means o f p i2t

Model Number of

parameters Rank lo8 10(ß i,)

Average E (РШ \У) M „ BEKK(1,1) 19 1 0 0.162 M 2, scalar BEK.K(1,1) 13 2 15.38 0.122 M 3, DCC2 24 3-4 17.48 0.132 M4, DCCO 18 3-4 17.53 0.132 M s, DCCI 20 5 18.99 0.132 M 6, CCC 15 6 64.13 0.237

4. POSTERIOR INFERENCE ON CONDITIONAL CORRELATION COEFFICIENTS AND VOLATILITIES

In this section we com pare m ain results for individual volatilities and the dynam ic correlation structure, obtained within each m odel. It is im portant to know w hether m odels th a t have different ex p lan ato ry pow er describe this stru ctu re in a sim ilar way.

T h e plots o f the sam pling conditional co rrelatio n coefficients p 12i, (for each ( = 1 , T; T = 1482) are presented in F igure 1, w here we d raw two lines: the u p p er one representing the p o sterio r m ean plus tw o p osterior stan d ard deviations and the lower one - the p o sterio r m ean m inu s two p o sterio r stan d ard deviations. We focus on typical p a tte rn s, so only two m odels arc represented in F igure 1. It is clear th a t con stan cy o f conditional co rrelatio n s, which are quite tightly co n cen trated aro u n d their abru ptly changing p o sterio r m eans, is n o t supported by the d a ta . T h is explains why the C C C m odel receives negligible posterior prob ab ility w hen com pared to D C C or B E K K specifications. T h e last colum n o f T ab le 1 presents tim e averages for the sequences o f posterio r m eans o f the co n d itio n al co rrelatio n coefficient in each m odel, while T ab le 2 gives th e em pirical co rrelation coefficients betw een these sequences (the num bers above the diagonal). Table 2. Correlation coefficients between the posterior means o f the conditional correlations

(upper part) and covariances (lower part)

Specification BEKK Scalar BEKK DCC2 DCCO DCC1

BEKK X 0.9113 0.9180 0.9094 0.9125

Scalar BEKK 0.9152 X 0.9837 0.9810 0.9826

DCC2 0.9299 0.9987 X 0.9767 0.9959

DCCO 0.9172 0.9982 0.9982 X 0.9643

DCC1 0.9192 0.9996 0.9993 0.9983 X

T hese results show th a t the m odels o f com p arab le ex p lan a to ry pow er lead to alm ost the sam e inference on the dynam ics o f co n d itio n al correlation. F o r the scalar B E K K and all three D C C m odels, averages o f E (p1 2,t |y) (£ = 1, ..., T) are ab o u t 0 . 1 2 - 0 .1 3 and the em pirical co rrelatio n coefficients betw een p airs o f E (p1 2tl|y) sequences are bigger th a n 0.96. H ow ever, the sequence o f E (p 12,,|>’) com ing from the unrestricted B E K K is slightly less correlated w ith th e o thers (a b o u t 0.91) and has a som ew hat higher average (0.16). A lso, the plot obtained for the unrestricted B E K K looks som ew hat different (cf. F igure 1).

Unrestricted BEKK(1, 1)

DCCO

Fig. 1. Conditional correlations (posterior mean + 2 standard deviations)

Very sim ilar results (as for the co nditio nal co rrelatio n p 12,t) have been obtained for the sequences of the posterior m eans o f the conditional covariance h l2y, em pirical co rrelatio n s are given in T ab le 2 (the num bers below the diagonal).

Indiv idual tim e-varying volatility o f each tim e series is m easured by the co n d itio n al stan d ard deviation s / ( v— 2) ~ 1 (i = 1 ,2 ). T h e sequences oi 1482 p oint estim ates, obtained by inserting the p o sterio r m eans o f the m odel p aram eters, are plotted in F igure 2 for two m odels. T hese estim ates exhibit the sam e dynam ic p attern for all m odels o f the sam e explanatory pow er (scalar B E K K , D C C 2, DCCO, D C C 1) - the em pirical correlation coefficients (T able 3) are basically equal to 1. T he results o b tained in C C C and all D C C arc also highly correlated. T h e em pirical co rrelatio n coefficients are som ew hat lower (especially for P L N /D E M ) when we co m p are the unrestricted B E K K specification to the rem aining m odels. T h u s the best m odel leads to slightly different inference on volatility. As regards time averages o f the sequences o f estim ated in-sam ple volatilities, they are alm ost the sam e in all m odels (including CCC).

O u r conclusion is th a t inferences from the best fitting m odel can be ap p ro x im ated by the scalar B E K K or D C C specifications. Since the scalar B E K K m odel is the sim plest, b u t it does n o t nest th e C C C case, one should estim ate and com pare these tw o non-nested m odels. T his seems a feasible strategy even for /с-variate tim e series w ith k > 2 .

Conditional standard deviations for PLN/USD unrestricted BEKK(1,1)

Conditional standard deviations for PLN/DEM unrestricted BEKK( 1,1)

DCCO DCCO

Fig. 2. Point estimates o f the conditional standard deviations in two main models

B ay es ia n A n a ly sis of D y n a m ic Co nditional C o rr el at io n

Table 3. Correlation coefficients between the estimates o f the conditional standard deviations for PLN/USD (upper part) and for PLN/DEM (lower part)

Specification BEKK Scalar BEKK DCC2 DCCO DCC1 CCC

ВЕК К X 0.9217 0.9234 0.9219 0.9233 0.9420 Scalar BEKK 0.8826 X 0.99988 0.99997 0.99988 0.9930 DCC2 0.8854 0.99988 X 0.99990 0.99999 0.9941 DCCO 0.8826 0.99992 0.99989 X 0.99988 0.9933 DCC1 0.8860 0.99986 0.99999 0.99986 X 0.9940 CCC 0.8828 0.99966 0.99973 0.99974 0.99967 X REFERENCES

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Jacek Osiewalski, Mateusz Pipień

BAYESOWSKA ANALIZA DYNAMICZNEJ KORELACJI WARUNKOWEJ Z WYKORZYSTANIEM DWUWYMIAROWYCH MODELI GARCH

(Streszczenie)

Wielowymiarowe specyfikacje ty^pu ARCH stanowią teoretycznie obiecujące ramy dla analiz skorelowania instrumentów finansowych, ponieważ umożliwiają modelowanie zmiennych w czasie macierzy warunkowych kowariancji. Jednak ogólne modele VechGARCH mają zbyt wiele parametrów, są więc niepraktyczne w przypadku więcej niż 2- lub 3-wymiarowych wektorowych szeregów czasowych. Prosta specyfikacja /-BEKK(1,1) wydaje się dobrym kompromisem pomiędzy oszczędnością parametryzacji i ogólnością modelu. Niestety model stałych korelacji warunkowych (CCC) Boilersleva nie jest szczególnym przypadkiem struktur VECH czy BEKK. Ostatnio Englc (2002) zaproponował oszczędnie sparametryzowane uogólnienie modelu CCC; ta specyfikacja o dynamicznej korelacji warunkowej (DCC) może zdominować wiele starszych wielowymiarowych modeli GARCH . W artykule rozważamy bayesowską analizę warunkowego współczynnika korelacji w ramach różnych dwuwymiarowych modeli GARCH, które są porównywane przy użyciu czynników Bayesa i ilorazów szans a posteriori. D la dziennych stóp zmian kursów PLN /U SD i PLN/DEM (6.02.1996 - 28.12.2001) wykazuje się, że specyfikacja t-B E K K (l.l) opisuje dwuwymiarowy szereg czasowy znacznie lepiej niż modele DCC. Jednak wartości oczekiwane a posteriori warunkowych współczynników korelacji, uzyskane w ramach różnych modeli, są bardzo silnie skorelowane.