Vol. LIII (2012) PL ISSN 0071-674X
o r t h o g o n a l
t r a n s f o r m a t i o n
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c o o r d i n a t e s
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g a r c h
m o d e l s
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b a y e s i a n
a n a l y s i s
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w iG 2 0
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M A T E U S Z P IP IE Ń
D ep artm en t o f Econom etrics an d O perations R esearch, C racow U n iversity o f Econom ics e-mail: eepipien@cyf-kr.edu.pl
ABSTRACT
We ch eck th e em pirical im portan ce o f so m e generalisation s o f th e co n dition al distribu tion in M -GARCH case. A cop ula M -GARCH m o d e l w ith coord inate free cond ition al distribution is c o n sid ered , as a co n tin uation o f research c o n cern in g specification o f th e cond itional distribution in m ultivariate volatility m o d e ls, see P ip ień (2007, 2010). The m a in a d van tage o f th e p ro p o sed fam ily o f probability distributions is that the coordin ate axes, a lo n g w h ic h h e a v y tails a n d sy m m etry can b e m o d e lle d , are subject to statistical inference. A lo n g a set o f sp ecified coord inates b o th , linear a n d n on lin ear d e p e n d e n c e can b e exp ressed in a d e c o m p o se d form.
In th e em pirical part o f th e p ap er w e co n sid ered a problem o f m o d e llin g th e d yn am ics o f the returns o n the sp o t a n d future q u otation s of th e WIG20 in d ex from the W arsaw Stock Exchange. O n th e basis of the posterior o d d s ratio w e ch eck ed th e data su pp ort o f co nsid ered generalisation, co m p a rin g it w ith BEKK m o d e l w ith th e cond ition al distribution sim p ly constructed as a product o f th e univariate sk ew ed co m p onen ts. Our exam p le clearly s h o w e d the em pirical im portance of th e p ro p o sed class o f the coord in ate free con d ition al distributions.
STRESZCZEN IE
M. Pipień. Wielowymiarowe modele Copula M -G A R C H o rozkładach niezm ienniczych na transformacje ortogonalne — bayesowska analiza dla notowań spot i futures indeksu WIG20. Folia O econ om ica Craco- v ien sia 2012, 53: 21-40.
W artykule p r z e d sta w io n o u o g ó ln ie n ie rozk ład u w a r u n k o w e g o w w ie lo w y m ia r o w y m m o d e lu ty p u GARCH, oraz p o d d a n o em p irycznej w eryfikacji sk o n stru o w a n y m od el. Praca sta n o w i k o n tynu ację b a d a ń p r o w a d zo n y ch p rzez Pipienia (2007, 2010) n ad w ła ściw ą specyfikacją rozk ład ów w a r u n k o w y c h w ek tora stóp zm ia n in str u m e n tó w fin ansow ych . Z asad n iczym ele m en tem okre ślającym giętkość rozw ażan ej klasy w ie lo w y m ia r o w y c h ro zk ła d ó w jest m o ż liw o ś ć zm ia n y układu w s p ó łrz ę d n y c h , i - tym sam ym — k ieru n k ó w w p rzestrzen i obserw acji, w e d łu g których grube o g o n y i asym etria rozkładu m o g ą w y s tę p o w a ć em pirycznie. Z g o d n ie z przyjętą orientacją w p rze strzeni obserwacji, m o ż liw e jest m o d e lo w a n ie zależn ości p o m ię d z y elem en tam i w ektora lo so w e g o , za ró w n o o charakterze lin io w y m (stosow ana transformacja liniow a) jak i n ielin io w y m (funkcja p o w ią za ń , copula).
W części em p iryczn ej p rzed sta w ia m y w y n ik i m o d e lo w a n ia d y n a m ic z n y c h zależn o ści p o m ię d z y zw rotam i z n o to w a n ia sp o t i futures in d ek su WIG20. U zy sk a n e rezultaty w skazu ją na zasadn ość p r o p o n o w a n e g o u o g ó ln ie n ia sto s o w a n e g o w m o d e lu BEKK. M o d el z p ro p o n o w a n y m ty p em roz k ład u w a r u n k o w e g o u zysk u je silne p o tw ie r d z en ie em p iry czn e, m ierzo n e ilorazem szans a p o s te riori i w artością b rzegow ej gęstości w ek tora obserwacji.
KEY WORDS — SŁOWA KLUCZOWE
Bayes factors, m ultivariate GARCH m o d els, coord in ate free distributions, H o u se h o ld er matrices c zy n n ik Bayesa, w ie lo w y m ia r o w e m o d e le GARCH, m acierze H o u seh old era
1. IN T R O D U C T IO N
M ost of contributions involved w ith m ultivariate GARCH (M -G ARCH ) m odels — for a su rv ey see B au w en s, L a u re n t an d R om b ou ts (2006) — rely on the as su m p tion of the conditional G aussian distribution. In spite of the fact th at the M -G A R C H m od els are applied in m od ellin g an d p red ictin g tem p oral d e p en d ence in the se co n d -o rd e r m om en ts, som e o th er properties of the conditional dis tribution, like for exam ple fat tails an d skew ness, are also v e ry im p ortan t. This result w as confirm ed by Bayesian com p arison of G ARCH -type m od els w ith n or m al an d Student-i conditional distributions p resen ted by Osiewalski an d Pipień (2004). In term s of the m od el data su p p ort, m easu red by posterior od d s ratio and p osterior probabilities, th e y clearly sh o w ed th at conditional norm ality is co m pletely unrealistic in m odelling financial tim e series. H en ce, lon g jo u rn ey beyond n orm ality is n ecessary — see G enton (2004) — for b etter u n d erstan d in g the d e p en d en ce stru ctu re b etw een related tim e series in general, an d b etw een financial re tu rn s particularly.
In the p resen ce of em pirical analyses decisively rejectin g conditional n orm al distribution, a few studies co n ce n tra te d on the application of the conditional distributions th at allow b oth for h e a v y tails an d asy m m etry w ithin M -G ARCH m odels. Som e d evelopm ents on this subject p resen t B au w en s an d L au ren t (2005). M odern propositions of m odelling volatility an d conditional d ep en den ce betw een financial retu rn s try to resolve the problem b y com plicating stochastic stru ctu re of the m od el rather, th an generalising explicitly conditional distribution. R ecently Osiewalski an d Pajor (2009, 2010) p ropose M SF-BEKK m odel, as an exam ple of the process attributed w ith both, the flexibility of the Stochastic Volatility family of m odels, an d p arsim on y of p aram eterisation of simple M -G ARCH covariance structures. Som e other, m ore com p licated m ultifactor processes h as b een recently p ro p o sed by Osiewalski an d Osiewalski (2011, 2012). Those hybrid processes ca n o u tp erfo rm p u re M -G A R C H specification, even in the case of conditional norm ality. As an alternative to a p p ro ach in v estigated by Osiewalski an d Pajor (2009, 2010) one m a y con sid er an explicit generalisation of the conditional distribution, also leading to m ore em pirically im p ortan t specifications.
In m odelling volatility an d dyn am ic d ep en d en ce of retu rn s of different finan cial assets, a linear d ep en d en ce is econom ically interpretable an d popular. Stand ard em pirical exercises in financial econ om etrics, like con trollin g an d pricing risks, optim al portfolio allocation, analysing volatility transm ission m ech an ism or con tagion an d building h ed gin g strategies, rely on solutions th at are strictly co n n ecte d w ith m easu res of stoch astic d ep en d en ce of the linear n atu re. H o w e v e r last d ecad e h ave seen particularly stro n g attention in m odelling d ep en d en ce in a n onlinear setting. O ne of the im p ortan t topic of financial econ om etrics th at m ad e substantial p rogress d u rin g last d ecad e, relates to m ak ing inference about m eas u res of stochastic d ep en d en ce th at are alternatives to the conditional correlation.
It seem s th at both, definition of a n o n stan d ard distribution of observables, an d a m o re detailed analysis of d ep en d en ce are crucial in p ro p er m odelling of financial retu rn s. O ne of the ap p roach es th at m ay resolve to som e extent b oth is sues involves cop u la functions. The ap p roach w as intensively d eveloped by Pat ton (2001, 2009), Jo n d eau an d R ockinger (2006) and, in the case of Polish financial m arket, by D om an (2008), D om an an d D om an (2009), Jaw orski an d Pitera (2012) an d others. Vast em pirical literature clearly indicate th at volatility m od els built w ithin fram ew o rk of cop u la functions contribute substantially to stan d ard em pirical issues in financial econ om etrics stated above; see E m b rech ts, M cN eil and Straum ann (2002), Bradley an d Taqqu (2004), R odriguez (2007), C havez-D em oulin an d E m b rech ts (2010), Balkem a, N olde, E m b rech ts (2012).
The m ain goal of this p ap er is to ch eck the em pirical im p ortan ce of som e generalisations of the conditional distribution in M -G A R C H case. We generalise the M -G ARCH m od el p rop osed an d em pirically analysed by Pipień (2006, 2007) w h o applied a novel class of probability distributions, w hich is coord in ate free in the sense form u lated by Fang, Kotz an d N g (1990). Pipień (2010) considered a m u ltivariate distribution w ith in d e p e n d e n t co m p o n e n ts, w ith skew ness im p o se d a cco rd in g to th e in v erse probability in teg ral tran sfo rm atio n s, discussed in details by F erreira an d Steel (2006) an d Pipień (2006). In the next step, o rth o go n al tran sfo rm atio n w as in co rp o rated in o rd er to assure th at fat tails an d also possible skew ness can be im p osed alon g a set of coord in ate axes. Consequently, the co n stru ct p ostu lated the existence of a set of coord in ate axes, along w hich the univariate co m p o n en ts are in d ep en d en t an d the densities of the m argin al distributions are k n ow n analytically. N o w w e additionally con sid er a generalisation, by im posing copula function th at cap tu res possible d ep en d en ce of nonlinear n atu re b etw een elem ents of the ra n d o m vector. The m ain ad van tage of the p ro p o sed family of probability distributions is th at the coord in ate axes are subject to statistical inference an d can be v e ry different from the ones defined by can on ical basis. A long a set of coord in ates, su p p o rted by the data, both, linear an d nonlinear d ep en d en ce can be m odelled.
In the em pirical p a rt of the p a p e r w e con sid er the bivariate series of the re tu rn s on the sp ot an d fu tures quotations of the W IG 20 ind ex (W IG 20 and
F W IG2 0 i nstr u m e n ts) c ov erin g t h e p erio d fro m 2 7.0 2 .2 0 08; t = 2053 observ ation s . In m ndelh n g tfee co n d to o n e1 d e . e n d e n c e vl: the c om p o n en t r= fri e b rvariate tim e series w e con sid er C opula-B EK K (1,1) m od el w ith coord in ate free eon ditit n a l divfrtoe tion acco r d i n g t o t f e e p o etulates e^f: .i h e c e n t tru e t iloe a com p arison w e also consider som e restricted cases, leading to the m u ch sim pler conditional distribution. We apply form al ap p roach to test exp lan atory p o w e r of a set of co m p etin g specifications, based on the p osterior o dd s ratio, an d discuss su p eriority an d possible p ractical usefulness of the con sid ered coord in ate free c c n dttion e l dierribuOo n . A ddieio n allo tn e p o s ler i o r i n f eeen ee ab ou t l oovCinato vnncis elso p re oen te d.
2. A C L o e s n
oF
c <e^c^]fiD tN ior e f v e e c o N D m o N A L D i s e R iB U T IO N ST h e m oi n g o al of t hi s c h pptar i s t o p re sen t a family o f m u lTpariate ske w ed distributions an d apply it in the m ultivariate GARCH setting. The basic notion eo n eid ere d h ece rn frie n nif i e d vep eeeent a t ^ n of frie u nre ariate skc w n ers friat a p ^ e s i n o eree peub aH lity i n t e ^ e l f r an ef o r m a t io n ^ r o p ove. i m .o U y b y F e oeefre an d Steel (2006). We follow the settin g p resen ted in the univariate case by Pipień (2006, 2007) an d by Pipień (2010) in m ultivariate case. The skew ed version of originally sym m etric an d u n im od al density f(.| 0 ) (w ith cum ulative distribution function F(.| 0 )) can be defined as follows:
s(x | 0,")= fx| &)"p(F(x\ 0) |" ) , 0ve x # R , (1)
w h ere p(.| h) d en otes the density of the distribution defined on the unit interval. The asym m etric distribution s(.| 0 ,h ) is obtained by application of the density p(.| h) as a w eigh tin g function of the density f(.| 0). The case, w h e n p(.| h) = 1, resto res sym m etry. A n y fam ily of densities p(.|h), for h £ H, defined o ver unit interval, is called skew ness m echanism . For a review of skew ing m echan ism s th a t in corp o rate h id d en tru n catio n m ech an ism , som e a p p ro a ch e s b ased on the inverse scale factors, o rd er statistics co n cep t, Beta or Bernstein distribution tran sfo rm atio n or a co n stru ctive m e th o d see Pipień (2006). The em pirical im p ortan ce of the conditional skew ness in m odelling the relationship b etw een risk an d re tu rn w as also stu d ied in the univariate case by Pipień (2007). Some re ce n t d evelop m en ts confirm results p resen ted by Pipień (2007) th a t it is possible to resto re the relationsh ip , m en tio n e d above, o n ce a high ly n o n sta n d a rd stochastic process is con sid ered in volatility m odelling; see for exam ple M arkov sw itch in g-in -m ean Stochastic Volatility m odel, p ro p o sed b y Kwiatkowski (2010).
N o w let con sid er m -dim ensional ra n d o m v e cto r £ = (f j ,...,e m) ' an d let denote byfi(.| 0 i),...,/m (.| im) a set of un im od al (w ith m od e at zero) univariate densities,
im p o se skew n ess m ech a n ism s py . l a l 0 11 d ensities ( ■(. | (9,) . N ote th at in g e n er al the co n stru ct d oes n o t re qdire im p osin g the sa m d type of skew noso m e dh s e ism fc>r oec h z = 1 , . . .,s j. I;or sim p lisiOy, in th s ( m hicic s l p a r O ot th t f>^e>es. e ts co nsiSer thk ca se, w h r s e th e oa m t skew n ees m e sh aniom is c a n sid osen foc each cS th e c c ornin t tot. Ps soiblo ditfenen t ae ( m m etoy effcc t s w ili stoiiit lsrom ditfsre n i v t i u es
2. y a rc m sCors bTlmti os^kitidijo S an dity s,(. | <t;rkit ts kes rquSsoen pm s n seb in
eo u ati on (10:
s ,( x \6 ln d = f I(x\dI)-pI(FI(x\6l) |n)t fso x # R rs d z = 1 ,...,e ,
w h ere F,(. | i ) d en otes cum ulative distribution function. Initially, for the ran d om
v ector a = ( s c,...,£ mk svs defins Oke dieSeibutioo aeTidCi m d e p e n d ebO asy m m e eho c om h e n ents:
m
( c o o n )= H # ( e , \ 0 , , (2)
i = l
w h ere a = (s>a\ ...,e my , = ( m ,...,h m' ) ' .
Pipien (2010) sh o w s exam p les of distributions in bivariate cases indicating th a t possible outliers an d asy m m etry can be ca p tu re d by distribution (2 ) only if those featu res of the d ata will o ccu r alon g original co o rd in a ts axes, defined
by can on ical basis in R m. Also, an y fam ily of distributions (2) is n ot closed w ith
resp ect to the orth ogon al tran sform ation s of the com p onen ts. H en ce, in o rd er to im p rove flexibility of o u r class of distributions, a special m ech an ism th a t w ould m ake the coord in ate axes v a ry in g is in corp o rated accordin g to the idea p roposed b y y er r e i r a a n y yte e l (2 0 0 6 ). We pocrncis ft o n th e bos^ o f t hie ft l low i n ^ m e a c (s ffine) tra n ste r m hh ok of tn a ra n q om v e cto r f :
n = A £ + n (3
fot e n o n sin g u lar ( K teD. 2 ^,,,,,] t n d loe stion = ector / / .^ je,2 0 . Tine d en s.ty os t y s dii tribution o ffli e oan cio m v i c t o ^ i s definen by the follow ing form ula:
m
p O i a, o S aA)n |dc s r ' 1| r i ! ( 0 ' - / i ) ,4'1 1 ", , " ( a (4)
i= 1
w h ere A f 1 d en otes the i-th co lu m n of A -1. If the densities f,(. 10,) are unim odal,
w ith m od e at zero, th en the distribution the v e cto r of y in (4) is unim odal, with
m o d e defined by n an d skew ing m ech an ism s p,(.| h ). Transform ation m atrix A
in tro d u ces the d ep en d en ce b etw een co m p o n en ts of y, while h d eterm in es the
skew ness of the in d ep en d en t co m p o n en ts of £. Assuring the variability of the p a
ram eters, equation (4) g en erates a flexible class of m ultivariate distributions that is closed u n d e r orth o go n al transform ations. H en ce, the co n stru ct (4) is co o rd i
nate free, in the sense defined in Fang, Kotz an d N g (1990). In our ap p roach we
do n ot restrict the distribution to the case th at A is a square ro o t of the sy m m et
ric an d positive d efinite co v arian c e m atrix. C onsequ e ntly, p ra rtical ajcp ticat i o n c f
sp ecific f am ilies of m ultivariate disM tm tinn s ^ oequires in terp retin g the effect
of the tran sform ation m atrix A. W ith no loss of generality let assum e in (3) that
n 0 [mxl]*
n = Gf' a
-A cco rd in g to the th e o re m p resen ted in Golub an d Van L o s n (1993) an y
n o n sin g u lapm a to n A j^ ^ non b e w ritts n as the p ro d u ct of mxm orthogonal m atrix
O m an d u p p e r triangular m atrix U[mxm] w ith positive diagonal elem ents:
A n o w e,
an d sn cWe d ecom p o t iU a n (sp lled th e Q n d s com position) is unique. N o w the re
sults of the tran sform ation m atrix A can be con sid ered in tw o steps:
y = A ° e = {Oml U ' £ = U ' Om £ . (5)
Initi ally th t oa n Cam v nctor £ i n t 5 t i s n i ^l^ S ^tt tf n aofattom )ii d efOq = l ) oc i otoin varslon (it d atO mA- l ). T h s n t h e v e c t o c £ = O m' P ( t tean sia r m e d a c c o adin g to t t n c o o ^ji^ni^^t a i i s hn c a r b r onat^^matinn. T h e dieoribo tion e f th o v e cto r f postulates th at there exist a set of coordinate axes, along w hich the co m p o n en ts of p are in d ep en d en t an d the densities of the m arginal distributions are k now n analytieally n he m oan diffaren o e b atw een drnfrtoa ^ on n ^^i^<f. ^ i o thiaM h ose c o ord in a ta aees can v a ry ^l:nnc l n ^a a^^^i o eun e d b n t anoniao( n asis i n . R ^ 'I h e c^ntri^ A o n o- y is t h e n on am e n by im ° osin° scal e manafosmat:ion oi l ih e d^^tcif^u tion c^f ^, b ecau ce m atclo I ca o t e m ferp rere d a a t he q holesby eq uarr aaat o t c he syw m eteio a n n n o s itiv e d onnite m a trix d e fin in g covarian ce structure.
A p ara m a tiip t a m p h n g m o d e( t n q t i q c o rp or s t s s alpfrinu tions Oetc r ib e d b y
anu a a a n ( У ) r eqn(aer u niou e )o n e -to -a n e i p ar bm a ta n sbtie n o f tna law Uy o Co o
th o g o n a i m aaric e s Om rn ta^ Alea co m t resSrietie n s n ave t o U a im o osed , i n l :>cd ea t o a s s nae inontifica(ion. Thr on e-to o in e p a ^^^e^^risatiun w a s o ^c^vid ^c i by Srew c and cc a o -) aaU - e rnelm a a o t t e el IUO^^A I^^ an ap pneali o n e t t i e H o u se h oCdes
m atrices decom position. L et d en ote V = (V1,...,Vm)' e R m, the m -dim pnsion al col
u m n vector. The H ou seh o ld er m atrix H(V) (H o u seh old er reflection or H o u se h old er tran sform ation ) is defined as follows:
2
H ( v ) = I v v '.
m v v
Golub an d Van L o a n (1983) sh o w som e useful p roperties of H(V). Firstly, for each
V s R m H(V) is o rth ogon al, an d secon d ly H(V) = H(-V) = H(a V), for an y scalar a^ 0.
(d en o ted by HSm-1) w e will keep the co v erage of the w hole family of H ou seh o lder m atric e s .P a r a m e terisation of th e u nit h alf s p h e re i s easily obtain ed t f w e w rite d ow n the v e cto r V~ = (V1r ..,Vm) ' e H S m-1 in polar coordinates:
j m-1
ai=rin(a»i), aj= rm (oj) ! cos(rns) , for j< m , am= n cos( ms) , (6)
s=1 s=l
w here
( % / 2, % /2 ) i f m = 2
(0,%/2) x ( %/2, % /2 )m-3 x (%,%) i f m > 2.
N ow , for an y [mxm] o rth ogon al m atrix O m w ith detO m = - l m+1, there exist unique
decom position:
Om = H ( ~ m)-... H (~ 2), (7)
to m -1 H ou seh old er reflections H ( ~ ) defined by v ectors ~ [ mx1] of the form:
~ j (om-j, V~j) ,j 2,...,m,
for m-j dim ensional v e cto r of zeros, om-j = (0,...,0 ) ' if j < m an d for an em p ty v ector
for j = m. The v ecto rs V ~ je H S j-1 are p aram eterised in term s of the polar co o rd in a
tes applied in (6 ). The interesting case is m = 2, w h ere the class of H ou seh old er
reflections provide p aram etric family of o rth ogon al m atrices of dim ension [2x 2 ] w ith identification restrictions im p osed; see Stew ard (1980), Golub an d Van L oan (1983).
3. A N O T H E R S T E P — IN T R O D U C IN G C O P U L A F U N C T IO N S
Distribution of y, defined by the density (4), w here A = O mU, w ith orthogonal m a
trix O m, param eterised accord in g to d ecom position (7), is obtained on the basis of
the linear tran sform ation of a ra n d o m v e cto r £ w ith the density (2). C on sequ en tly, only linear d ep en d en ce b etw een ra n d o m variables, rep resen tin g coordinates, can be m odelled. Possible ch an g es in co ord in ates th at m ay be subject to statistical in feren ce, en rich ed flexibility of the family, h o w ev er the n atu re of d ep en d en ce of
elem en ts of the v e cto r y m ay still be linear. In ord er to m od el a m ore com plicated
d ep en d en ce stru ctu re in v e cto r y w e follow the ap p roach th at in volves copula
functions.
L et consider a bivariate ra n d o m variable z = (z1,z2) ' , w ith cum ulative density
function (cdf) F an d density function f, an d w ith f i an d Fi the density an d cdf of
the m argin al distribution of Zi respectively (i = 1,2). A cco rd in g to Sklar (1959),
1. C(uyu2) is increasin g in u1 an d u2
2. C(0,u2) = C (u1,0) = 0, C(1,u2) = u2, C(u1,1) = u1
3. For each (u1,u1',u 2,u2' ) e [0,1]4, su ch u1< u 1' an d u2< u 2' : C(u1',u 2')-C (u 1',u 2)-C(u1,u2' ) + C(u1,u2)>0, such:
F(Z1,Z2) = C ( F ^ ) , F 2(Z2)).
The density of the joint distribution of z (if exist) is defined as follows: f(z1,z2) = f1(z1) f2(z2) cd (F1(z1^ F2(z2))/
w here:
cd( « i , “ 2 ) = ^ — ( « i , « 2 ) -dul du2
Function C is called copula, an d restores d ep en d en ce reflected in the joint distri
bution F, w h en m arginal distributions F 1 an d F 2 are considered. F un ction cd(•,•)
is called the density of the cop u la C. In the case w ith C(u1,u2) = u1u2, w e have F (z1,z2) = F 1(z1)F2(z2), Cd(u1,u2) = 1 an d f f o ^ ) = f1(z1) fj(z2), h en ce C(u1,u2) = u1u2 defines stochastic in d ep en d en ce b etw een z1 an d z2. For detailed th e o ry of copula functions an d of the co n ce p t of m easu rin g stochastic d ep en d en ce w ithin copula fram ew o rk see Joe (1997) an d N elsen (2006).
N ow , in the bivariate case (m = 2), w e generalise o u r distribution of y, defined by the density (4), by in corp o ratin g cop u la function in the distribution of the
ra n d o m v e cto r f . We consider a ra n d o m v e cto r y of the form :
y = U 'O m 'z , (8)
w ith u p p e r trian gu lar m atrix U an d the o rth ogon al m atrix O m defined by (7) and
the bivariate ra n d o m variable z w ith the follow ing density:
p(z | e ,h ,9 cop) = S1(z1 | 01,h 1) S2(z2 | 02,h2) Cd(S1(21), S2(z2)| 6 cop), (9)
for the density cd of a particular cop u la function param eterised by the v e cto r Qcop,
an d skew ed univariate densities s{, con sid ered initially in (2). In (9) b y S1 an d S2
w e denote cdf functions of those skew ed univariate distributions. In trod u cin g co
pula function in the distribution of y, accordin g to (9), p rovid es an oth er source of
possible stochastic d ep en d en ce in the ra n d o m v e cto r y, n ot involved w ith linear
tran sform ation w ith m atrix A, con sid ered initially. The case w ith C(u1,u2) = u1u2
the distribution is d efin ed just like fo r £ in (2 ). In th i s c a s e on ly a ^^ne^a r i i ef^e^i^-i^en^ce l^ie^Ta^i^^nc c^ord ^nate s o f y c od be m odelled.
4. T H E SE T O F C O M P E T IN G S P E C IF IC A T IO N S
By yyw e d an oOe the tw o -d im en sio n ti v ee^^r of log r rith m is r etu rn s nt tim e t’, l.e .
y = (yw,h r ) r, v\^her^ cy, = 100ln(xji/xj_= . ae^dt ^^^,cle]^oees S^hevalue c^tztl^r i fina^ci^l
i n^trux ^i^nt at tim e /. I n x r d e y to m o 0 a le y n 0 ibon c i d ap c n S o R a e bs toreen c n d d o-n e o-n ts s f t s w e assum e the follow io-ng structure:
h = H$ < f i t-efj_1y H{v 0J) > 7 = 1< . ..,ti (210
w h e=e y/j_e = e...ja;ae1/ П) d en otea^:he Inform ation set e t t^r^ey. R sn d o m b criablh2
Zo= (a?■icZJ2b fc>Пaw t h a dis^yi2 u t<orl d r finc S i n j9ij w here c o m p o eLehts s,-(. | C^ia ^fea^^ tn e r a e w ad ¥ 1X31 /^ o f f ice ^t^nC^rdi^<^c0 Sfu d r nlW derLsitia r w (t h ^d■> 0 e r g^i^i^jiof
freed om p aram eter (hence 0 i = Oi), an d skew ness p aram eters hi- M atrix H(V~) in
(10) is a H ou seh old er reflection defined by: V V '
H " ) = I m 2 - — ,
V 'V
! !
w h ere V~ = (s in ~ 1,c o s ~ 1), an d ~ 1e ( - r / 2 ; r /2 ) . Sym m etric an d positive definite
m atrix H j( b ,} j - i ) follows BEKK (1,1) specification:
H j(b ,}j-i) = A + B • yr iyr { - B ' + C - H j-i(b ,}j-2) • C ', an d b g rou p s all required p aram eters, nam ely
b = (a11,a12,a22,b11,b12,b21,b22,c11,c12,c21,c22,) . R ew riting (10) in the follow ing form:
yj = W zp j = 1,...,t,
w h ere Wj = H(V~) Hj°'5( b , } j -1), just like in (8), w e can form ulate the conditional
distribution of yp (w ith resp ect to } j -1) as a result of linear tran sform ation of dis
tribution of Zj, w ith tran sform ation m atrix Wj.
P (yj I }-1 ,0 1 ,0 2 ,h 1 ,h 2 ,~ 1 ,b , ^ 1) =
= |detWj I -1S1(y; ' W j ) I V1,h 1) 32(yj' W j ) I V2,h2) C d S y W-1m ) S 2 ( y f W-1p(2))| 9 Cop),
w h ere W^jo d en otes i-th colum n of W j 1, an d s i(. | Oi,h i) are skew ed Student-t d en
si(z 1 ° i h i ) = f st(z l0,1,°i)-P(FSt(z |0 ,1,° i )| hi), z e R
w ith the density an d cdf of the stan d ard ised S tu d en t-t distribution w ith zero m o d e , u nit p recision a n d d eg r ees of fre e d om p ara m e ter l# > 0 d en o ted by / o(. 10,1 , 0 an d Fst( . | 0, l ,Vf) re sp ectid'dy.
W e co n sid ered five different single p a ra m e te r cop u la fu nctions, n am ely G aussian, C layton, Frank, Plackett an d G um bel, to g e th e r w ith the ca se of no c op u l a fun ction. This g c v es u s s ix c o m p etic g sam p l in g m o d els eollecte d i n the se^ledefiote^ b jt F^s. t ^e e o n e ^]^ftt^(i e c^em of c o p u la s an O )ts d eosiiies c a n b e foim d in Joe iSct^ ) a n O N esd^ O O d^ T h e c^i; of copula funcCon c p ph^ce ^n^^tj ^ mg^ir^^^l p ast od t h e p a p nr i e r nstg a ied to on fy t o ih e eases w d cse ooly a l ieig l e p a fa m s iee ie iS t0e d e^scrines cSer)ee^iie^irte^it l ^hs j ^s^i^ddm dtr^(^1tes. t^oo^^ otO^^e i ^^pi^l^ Ou n rd on t ateriboced w ith ricS^e^rrJnri^mete^i^^^^l^^n ca n b e found i n Cnei(a99O0
Tl e te s^m p lin g m e d e i ic detcoesee^^^d be Sh s follow in g p ro d u c t of the c c^e^di^^ist^ti d ^o ^i^Sie^s:
t+k
pS y,yf\vr,vo ,er,eo ,m rAM r) r n PSyj\-j-r,vr,vo,er,eo,wr,$,M r), (11)
w h e re y = (y1,...,y t) d e n o te s the m atrix of o b serv ed daily re tu rn s, while
yf = (yt+1,...,yt+k) g rou p s forecasted observables. In o rd er to com p lete Bayesian
m odels, the prior distributions of all p aram eters m u st be stated. For the v e cto r b w e ad op ted p rior u sed in Osiewalski an d Pipien (2004), for skew ness p aram eters
hi an d d egrees of freed o m p aram eters Vi w e applied p rior distribution studied by
Pipien (2007). Since the orth ogon al co m p o n e n t H(V~) in (10) is param eterised by a single p aram eter ~ 1e ( - r / 2; r / 2), w e assu m ed for simplicity uniform prior over the w hole interval. L ess trivial probability distributions, w ith som e interesting to pological properties, ad op ted for a subset of the orth ogon al m atrices, w ere p ro p osed by Stew ard (1980).
All p rior densities, excep t the one im p osed on the p a ra m e te r ~ 1e ( - r / 2 ; r / 2 ) , w ere in v estigated p reviou sly in o u r p apers. As it w as clearly sh o w n by Osiewalski an d Pipien (2004) ran d tPipien (2007) the p rior inform ation included in the Bayesian m odels is v e ry w eak, as the p rior distributions of p aram eters are v e ry diffuse. For p aram eters in copula functions w e im p osed n orm al distributions tru n ca te d to the ap propriate dom ain, w ith the prior m o d e at the p oin t assuring ind ep en d en ce. Consequently, we do n ot specify any type of d ep en den ce betw een co ord in ates an d im p osed ap p rop riately diffused distributions. Consequently, the conclusions d raw n from the em pirical analysis does n ot seem to be biased by the p rior kn ow led ge, w hich is vagu e an d n o t precisely stated in o u r case.
The m ain goal of the em pirical p art of the p ap er is to discuss the im p ortan ce of orth o go n al co m p o n e n t H(V~) an d its form w ith resp ect to the typ e of the cop u la function in clud ed in the sam pling m odel. As an alternative to m odels
in class H 1 w e also con sid ered C opula-B EK K (1,1) specifications w ritten in the follow ing w ay:
yj = Hj05(b ,}j-i)'Z j, j= 1 ,...,t, (12)
w ith no o rth ogon al m ech an ism , ch an g in g coord in ates, included. The assu m p
tions co n ce rn in g Zj an d H j ( b ,} j -1) are rem ain ed u n ch an ged . In p articu lar the
distribution of Zj m ay involve five different form of copula function an d also m ay n o t involve copula. This gives us additional set of six com p etin g specifications, d en oted by H0. The m od el 0 can be in terp reted as a special case of (10), obtained
b y im p osin g zero restriction on H ou seh o ld er v e cto r = ( 0 ,0 ) ', leading to the
case, w h ere H (y~) = I2.
5. E M P IR IC A L A N A LYSIS
In the em pirical p a rt of the p a p e r w e analyse bivariate tim e series of the logarith m ic re tu rn s of the sp ot an d futures quotations of the W IG 20 ind ex, co v erin g the p eriod from 21.12.1999 till 27.02.2008; t= 2 0 5 3 observations. The dataset, depicted on Figure 1, tog eth er w ith som e descriptive statistics, exemplifies ra th e r com pli cated n atu re of the d ep en d en ce b etw een both univariate tim e series. The possible d ep en d en ce is clearly d eterm in ed by the coincidence of outliers, m aking the em pirical distribution considerably m ore dispersed along first an d secon d quarter of the C artesian p ro d u ct, as co m p ared w ith relative stro n ger co n cen tratio n of daily re tu rn s of sp ot an d futures quotations w ith different sing at the sam e day. The m odelled time series covers rath er long h istory of spot an d futures trad in g on the W arsaw Stock E xch an ge. But, w e cu t the dataset at the en d of the F ebruary 2008 in o rd er to co m p are o u r results of m od el co m p ariso n w ith those p resen ted in a m u ch sim pler m od el setting by Pipien (2010).
A n o th er re a so n to focus on the co n sid ered tim e series is th a t possible em pirical im p ortan ce of cop u la function in sam pling m od el receiv ed so far atten tion only d u rin g the financial crisis. There is v a st literature su ggestin g th a t d u rin g last global financial crisis, the d ep en d en ce b etw een financial tim e series b ecom e v e ry com p licated an d n on stan d ard . H en ce, m a n y au th ors clearly ind icated th at cop u la functions are a prom isin g tool in m odelling tim e series d u rin g crises an d m arket crashes; see Bradley an d Taqqu (2004), R odriguez (2007), P atton (2009). H ow ever, there is a little evid en ce in favou r of the existence of n onlinear d ep en d en ce prior to the latest financial crisis.Consequently, w e did not u p d ated o u r dataset an d focus on the pre-crisis period. The em pirical im p ortan ce of cop u la co n stru ct in the sam pling m od el p resen ted in this p ap er will be m u ch greater, if the data su p p o rt will be obtained on the basis of the time series th at en d s before global financial crisis.
Table 1 pr esen ts th e r e s u lts o f m od e l co m p ariso n .W e c onsid e r e d l 2 co m p et in g specifications, im posing 5 different copula functions (N orm al, C layton, Frank, Plackett an d Gum bel) an d no cop u la function. In all cases respectively, w e co n sid ered existence of orthogonal tran sfo rm atio n against conditional distribution
w ith m arginal densities for both series defined as sim ply skew ed Stu d en t-t
dis-frfoutiom . We d enute by H 1 th e t uns r t go m y p e l s fret h o st2 og s n a l lr en sfer m ar hn in clu d ed , w H le by1 H0 a class of C opula-B EK K m odels w ith no free coord in ates in the conditional distribution. In Table 1 w e p u t decim al logarithm s of the m
ar-D escriptive statistics WIG20 FWIG20 M ean 0.0215 0.0284 Std. Dev. 1.557 1.579 Skew 0.1612 0.1149 Kurt 4.5503 4.8788 Max 7.3724 9.8815 M in -6.3286 -7.7057 Correlation 0.3738
Figure 1. The plot o f th e daily returns o n WIG20 (vertical coordinate) an d o n FWIG20 (horizontal coordinate) from 21.12.1999 till 27.02.2008; t= 2 0 5 3 observations.
Table 1
D ecim al logarithm s o f th e m arginal data d e n sity va lu es in all c o m p e tin g specifications, a n d o f the Bayes factor in favour o f the existen ce o f o rthogon al co m p o n e n t in m o d e l C op ula fun ction
a p p lied in sa m p lin g m o d e l O rth ogon al co m p o n e n t in clu d ed (Hi) N o orthogon al c o m p o n e n t (H0)
B ayes factor in favour o f m o d e l from H 1 against m o d e l from H 0 N o Copula -2974.9263 -2977.5126 2.5863 N orm al -2971.2150 -2976.2267 5.0117 Clayton -2972.2007 -2977.7896 5.5889 Frank -2970.3253 -2973.0979 1.7726 Plackett -2966.0346 -2968.1112 2.0766 G um bel -2973.3153 -2975.0409 4.1973
ginal data density v alu es in case of all m odels, an d also decim al logarith m s of Bayes factors in favou r of the existence of orth o go n al co m p o n en t. The results clearly indicate the em pirical im p ortan ce of cop u la functions in sam pling m odel.
The m od el w ith ou t the co n stru ct receives a little d ata su p p o rt in both subsets H1
an d H0 invariantly w ithin H 1 an d H0 subset. The g reatest data su p p o rt both, in case of H 1 an d H 0, receives m od el w ith Plackett cop u la in corp o rated in sam pling function.
A n o th er interesting issue co n cern in g m od el co m p ariso n is th at orth ogon al co m p o n e n t alw ays im p rov es the exp lan atory p o w e r of m odels. In case of no cop u la sam pling m odels, an d also for all cop u la functions, decim al logarithm of the Bayes factor against p u re C opula-BEK K specification is greater th an one, indicating in m o st cases the decisive su p p o rt of this co m p o n e n t in the sam pling m odel. This result seem s to be invariant w ith resp ect to all rem ain ed p arts of the sam pling m odel, an d w as su ggested previously by Pipien (2010). Table 2 presen ts the results of p osterior inference about tail p aram eters in all m odels. W e focus on p osterior m ean an d stan d ard deviations of the d egrees of freed om p aram eters of
the conditional distributions of univariate series. W ithin subsets of m odels H1 and
H 0, the inference about the tails of the conditional distribution is relatively the sam e. In case of m odels, w h ere orth ogon al co m p o n e n t excluded in the sam pling m od el, p osterior m ea n s of p aram eters V1 an d V2 indicate th a t the conditional distribution is n o t of G aussian typ e, h o w e v e r the p osterior u n certainty, as m ea su re d by the p osterio r stan d ard deviation, d oes n o t p reclu d e stron gly
Table 2 Posterior inferen ce about tails of th e con d ition al distribution in all co m p e tin g specifications
C op ula fu n ction a p p lied in sa m p lin g m o d e l O rth ogon al c o m p o n e n t in clu d ed (subclass of m o d e ls H 1) N o o rthogon al c o m p o n en t (subclass of m o d e ls H 0) N o Copula V1 5.64 (1.03) V2 18.93 (3.45) Vj 7.49 (1.98) V2 10.85 (1.98) N orm al V1 6.94 (1.26) Vj 7.49 (1.35) V2 18.37 (3.40) V2 10.84 (1.97) V1 5.77 (1.05) Vj 7.25 (1.32) V2 19.13 (3.49) V2 11.00 (2.01) V1 6.93 (1.27) Vj 8.57 (1.59) V2 19.65 (3.59) V2 11.42 (2.08) Plackett V1 6.61 (1.21) Vj 8.82 (1.61) V2 18.95 (3.43) V2 12.00 (2.20) G um bel V1 5.51 (1.01) Vj 7.46 (1.33) V2 19.34 (3.55) V2 10.36 (1.83)
the sam e type of tail b eh aviou r for b oth coordinates. If we include orth ogon al tran sform ation in sam pling m odel, the posterior inference ch an g es substantially, but in the sam e w a y in case of all copula functions an d also in n o-co p u la case. If w e con sid er the m ech an ism th at enables search for a set of coord in ates, along w h ich possible h e a v y tails an d asy m m e try can be m o d elled , the results of estim ation of the p roperties of the conditional distribution in tails are changing. In all cases in subset H 1, invariantly w ith resp ect to the type of copula function, tails of the conditional distribution of univariate co o rd in ates are different. The d ata clearly su p p o rt h e a v y tails for the first co o rd in ate, while the secon d one exhibit the G aussian type tails.
In ord er to illustrate ch an g es in conditional distribution, w h e n orth ogon al
m ech an ism is includ ed in the sam pling m odel, w e plotted the isodensities of zj
in case of m odels from subset H 0 (Table 3) an d isodensities of a ra n d o m variable H(V~)'zj in case of m od els from subset H 1 (Table 4). All p aram eters required to d raw the plots we ch o sen as p osterior m eans. O n the plots in Table 3 an d 4, we d raw v ectors rep resen tin g coord in ates ap propriate in sam pling m odels. In case of m od els from subset H 0 w e d raw v e cto rs p rop ortion al to the v e cto rs from canonical basis in R2, n am ely e1 = (10,0) an d e2= (0 ,1 0 ). In case of m od el from H 1 (Table 4) a set of co o rd in ates are subject to p osterior inference an d h en ce we p resen t p osterior m ean s, to g e th e r w ith the b an d s of the 95% H PD (H ighest Posterior D ensity) intervals for H (V ~ )'e1 an d H (V ~)'e2 respectively.
Analysing isodensities plotted in Table 3 an d 4 it is clear th at the data su p p o rt different directions, th a n can on ical, alon g w h ich h e a v y tails an d possible a sy m m etry can be m odelled. C opula functions ch an ge the shape of isodensities strongly. H o w e v e r the m ost im p ortan t feature of the sam pling m od el seem s to be the existence of the o rth ogon al m ech an ism ch an g in g coordinates. O nly in case of m odels from subset H 1, a m o re com p licated d ep en d en ce b etw een observed tim e series can be d iscovered, as the sh ap es of isodensities in Table 4 exhibit considerable excess from reg u lar "elliptical" shape. For m od els from subset H0, w ith ou t orth ogon al m echan ism , differences b etw een shap es of isodensities of the
distribution of zj are ra th e r m in or a m o n g m odels. N ew , estim ated, directions in
the sam pling m odels from subset H 1 (Table 4) are different from initial, canonical, ones. Taking into acco u n t dispersion of the p osterior distribution, the b an d s of the H PD intervals for H (V ~ )'e1 an d H (V ~)'e2 are located far aw ay from the case, w h ere H(V~) = I2. This clearly m ak es m od els w ith ou t o rth o go n al co m p o n e n t im probable in the v iew of the data. Additionally, ch an g in g directions in m odels
from subset H1 is nontrivial an d d oes n o t only involve rotation . C om p arin g
v e cto rs e1 an d e2 w ith its co rre sp o n d in g im ages, w e see th at can on ical basis is
subject to inversion an d th en to appropriate clock-wise rotation. This is due to the p roperties of the H ou seh old er reflections applied in the con stru ct. It enables to search for optim al orientation in a m ore co m p o sed way.
The plots of th e iso d en sities of Zj in m o d e ls from class H 0/ i.e. in sa m p lin g m o d e ls w ith
n o o rth ogon al c o m p o n e n t in clu d ed . Isoden sities are p lo tte d o n the basis of v a lu es of param eters equal to posterior m e a n s
No Copula
Normal
Clayton
Tue Sw 0? U:2<;55. 2003
The p lots of the iso d en sities of in m o d e ls from class H 0/ i.e. in sa m p lin g m o d e ls w ith n o orthogon al c o m p o n e n t in clu d ed . Isod ensities are p lo tted o n the basis of va lu es of param eters equal to posterior m ea n s
Posterior in feren ce a b ou t lin ear con d ition al d e p e n d e n c e obtain ed o n th e basis o f th e e le m en ts o f matrix i) in case o f th e b e st cop ula fu n ctio n (Placket). All param eters a ssu m e d to b e e qu al to posterior m e a n s
Linear conditional dependence in the best m odel in Linear conditional dependence in the best m odel in H0
20001010 20010730 20020521 20030307 20031222 20041007 20050725 20060511 20070223 20071211 -1
A v e ry im p o rta n t qu estion co n ce rn in g d iscu ssed em pirical analysis involves possible conclusions about ch an g es of the linear d ep en d en ce b etw een m odelled univariate series, w h e n orth ogon al co m p o n e n t an d cop u la function is incorporated . In Table 5 w e p resen t plots of posterior exp ectation s of conditional co rrelation s b etw een re tu rn s of sp ot an d futures q uotations of W IG 20. Since the results are practically the sam e in case of all pairs of m odels, w e focus our atten tion on the best m od els in H 1 an d H0 respectively, b oth based on Plackett
cop u la function. In case of the best m od el from the set H1 the variability of the
conditional correlation coefficient seem s to be only slightly less variable during the w hole tim e interval coverin g m odelled tim e series.
Existence of orth o go n al m ech an ism in sam pling m od el d oes n ot seem to influence the d yn am ics of conditional linear d ep en d en ce strongly. Both series of p osterior expectations exhibit the sam e d yn am ic p attern , w ith stro n g variability arou n d value 0.4, starting from August the 1st 2001, w h en W arsaw Stock Exchange quoted W IG 20 index officially for the first time.
6. C O N C L U D I N G R EM A R K S
The m ain goal of this p a p e r w as to ch eck the em pirical im p ortan ce of som e generalisations of the conditional distribution in M -G ARCH case. We considered cop u la M -G A R C H m o d el w ith co o rd in ate free con d ition al distribution. We co n tin u e research co n ce rn in g specification of the conditional distribution in m ultivariate volatility m odels started by Pipien (2007, 2010). The m ain ad van tage of the p ro p o sed fam ily of probability distributions is th at the coord in ate axes, along w hich h e a v y tails an d sy m m etry can be m odelled, are subject to statistical inference. A lon g a set of specified co o rd in ates b oth , linear an d n on lin ear d ep en d en ce can be expressed in form al an d co m p o sed form .
In the em pirical p a rt of the p ap er w e con sid ered a problem of m odelling the d yn am ics of the re tu rn s on the sp ot an d future quotations of the W IG 20 index from the W arsaw Stock E xch an ge. O n the basis of the p osterior od d s ratio we ch eck ed the data su p p o rt of con sid ered generalisation, co m p arin g it w ith BEKK m od el w ith the conditional distribution sim ply co n stru cted as a p ro d u ct of the univariate skew ed com p onen ts.
O u r exam ple clearly sh o w ed the em pirical im p ortan ce of the p ro p o sed class of the coordinate free conditional distributions. Both, orth ogon al co m p o n en t, and copula function, are necessary in p ro p er m odelling of the conditional distribution of the v e cto r financial retu rn s. The existence of the o rth ogon al tran sform ation of co o rd in ates in observation space receives decisive d ata su p p o rt invariantly w ith resp ect to the existence cop u la function in the sam pling m od el an d to the typ e of specified copula. The dataset su p p o rt m u ch different orientation in the sam ple sp ace along w hich h e a v y tails, asy m m etry an d d ep en d en ce b etw een
coord in ates, can be discovered. A m o n g the class of cop u la function Plackett one receiv ed the g reatest data support. Generally, p resen ted in the em pirical p a rt of the p a p e r noticeable flexibility of the class in directional m od ellin g of the tails an d a sy m m etry suggests th at possible applications, co n cern in g futures h ed gin g or Value-at-Risk calculation, are v e ry prom ising.
R E F E R E N C E S
A zzalini A. (1985), A Class o f Distributions w h ic h In clud es the N orm al O nes, Scandinavian Journal of Statistics 12, 171-178.
Balkema G., N o ld e N ., Em brechts P (2012), The sh a p e o f asym p totic d e p e n d e n c e , [in:] Prokhorov and Contemporary Probability Theory [eds.:] A. Shiryaev, S. Varadhan, and E. Presm an, Springer, Berlin.
B a u w en s L., Laurent S. (2005), A N e w Class o f M ultivariate S k ew D en sities w ith A pplication to G eneralised A utoregressive C on dition al H eteroscedasticity M od els, Journal of Business and Eco nomic Statistics 23, 346-254.
B a u w en s L., Laurent S., R om b outs J. (2006), M ultivariate GARCH M odels: A Survey, Journal of Applied Econometrics 21, 79-109.
Bradley B. O., Taqqu M.S. (2004), Fram ew ork for A n a ly zin g Spatial C on tagion b e tw e e n Financial Markets, Finance Letters 2, 8-16.
C h a v e z -D e m o u lin V , Em brechts P (2010), R evisitin g th e e d g e , te n years o n , Communications in Statistics — Theory and Methods 39, 1674-1688.
D o m a n R. (2008), M o d e llin g C on dition al D e p e n d e n c ie s b e t w e e n P olish Financial R eturns w it h M arkov S w itch in g Copula M odels, Dynam ic Econometric Models 8, 21-28.
D o m a n R., D o m a n M. (2010), Copula Based Im p ulse R esp o n e A nalysis o f Linkages b e tw e e n Stock Markets, R isk M anagem ent eJournal 05/2010; D0I:10.2139/ssrn.1615108.
Em brechts P, M cN eil A., Straum ann, D. (2002), C orrelation a n d D e p e n d e n c e in Risk M anagem ent: Properties an d Pitfalls [in:] R isk Management: Value at R isk and Beyond, [ed.] D em p ster M.A.H., Cam bridge U n iversity Press, C am bridge, 176-223.
Fang K.-T, Kotz S., N g K.-W (1990), Sym m etric M ultivariate and Related Distributions, C h ap m an and Hall, N e w York.
Ferreira J. T. A. S, Steel M. F. J. (2006), A C onstructive R ep resen tation o f U nivariate S k ew ed Distri b u tion s, Journal of the American Statistical Association 101, 823-839.
Ferreira J. T. A. S, Steel M. F. J. (2007), M o d el C om p arison for Coordinate-Free M ultivariate S k ew ed D istributions w ith a n A pplication to Stochastic Frontiers, Journal of Econometrics 137, 641-673. G en to n M. G. (2004), Skew Elliptical Distributions and Their Applications: A Journey Beyond Norm ality,
C h ap m an & Hall, London.
G olub G.H., Van L oan C.F (1983), M atrix Computations, John H o p k in s U n iversity Press, Baltimore. Jaworski P, Pitera M. (2012), O n Spatial C on tagion a n d m GARC H M od els, Institute of Mathematics,
University of Warsaw W orking Paper.
Joe H. (1997), Multivariate Models and Dependence Concepts, Chapm an& H all, London.
Jon d eau E., R ockinger M. (2006), The C opula-G A R C H M o d el o f C on d ition al D e p en d en cies: A n International Stock M arket A p p lication, Journal of International M o ney and Finance 25, 827-853.
K w iatkow ski L. (2010), M arkov S w itch in g In M ean Effect. B ayesian A nalysis in Stochastic Volatility Fram ew ork, Central European Journal of Economic M odelling and Econometrics 2, 59-94.
O siew alski J., O siew alski K. (2011), M o d ele h y b ry d o w e MSV-MGARCH z trzem a p rocesam i uk ry tym i w b a d an iu z m ien n o ści c en na ró żn y ch rynkach, Folia Oeconomica Cracoviensia 52, 71-85. O siew alsk i J., O siew alsk i K. (2012), M o d ele h y b r y d o w e z d w o m a p rocesam i uk rytym i, Z eszyty
Naukowe UEK, Seria Finanse 895, (w druku).
O siew alsk i J., Pajor A. (2009), B ayesian A nalysis for H ybrid MSF-SBEKK M od els of Multivariate Volatility, Central European Journal of Economic M odelling and Econometrics 1, 179-202.
O siew a lsk i J., Pajor A. (2010), B ayesian Value-at-Risk for a Portfolio: M ulti- a n d U nivariate A p p roach es u s in g MSF-SBEKK M odels, Central European Journal of Economic M odelling and Econometrics 2 (2010), 253-277.
O siew alsk i J., P ip ień M. (2004), B ayesian C om p arison o f Bivariate ARCH-Type M odels for th e M ain E xchange Rates in Poland, Journal of Econometrics 123, 371-391.
Patton A. (2009), C opula-B ased M od els for Financial Time Series. [in:] A n d e rsen TG., D avis R.A., Kreiss J. P, M ikosch T., [eds], Handbook of Financial Time Series, Springer, Berlin, 767-785. Patton A. J. (2001), Applications of Copula Theory in Financial Econometrics, U n p u b lish ed Ph.D. disser
tation, U n iversity o f California, San D iego.
P ip ień M. (2006), B ayesian C om p arison o f GARCH Processes w ith S k ew n ess M echan ism in C on d i tional D istributions, Acta Physica Polonica B 37, 3105-3121.
P ip ień M. (2007), A n A p p roach to M ea su rin g th e R elation b e tw e e n Risk a n d Return. B ayesian A n alysis for WIG Data, Folia Oeconomica Cracoviensia 48, 97-119.
P ip ień M. (2010), A C oordinate Free C on d ition al D istributions in M ultivariate GARCH M odels, [in:] Financial Markets. Principles of M odelling Forecasting and Decision M aking, Fin dE con Confer en ce M on o g r a p h Series N o. 8, Ł ód ź U n iversity Press, Ł ódź, 99-111.
R od rigu ez J. C. (2007), M easu rin g Financial Contagion: a C opula A pproach. Journal of Empirical Finance 14, 401-423.
Sklar A. (1959), Fonctions d e repartition & n d im en sio n s et leurs m arges, Publications de l' In stitu t Statistique de l.U niversite' de Paris 8, 229-231.
Stew ard G. W (1980), The Efficient G en eration o f R andom O rth ogon al Matrices w ith a n A p plica tio n to C on d itio n Estimators, S IA M Journal of Numerical Analysis 17, 403-409.
