Volatility Threshold Dynamic Conditional Correlations: An International Analysis *

Transcription

1 Volailiy hreshold Dynamic Condiional Correlaions: An Inernaional Analysis * Maria Kasch Universiy of Bonn mkasch@uni-bonn.de Massimiliano Caporin Universiy of Padova massimiliano.caporin@unipd.i Firs version: January 2005 his version: Ocober 2007 * We would like o hank Rober Engle, ribukai-vasconcelos Hermann, Giampiero Gallo, Michael Groe, Ua Pigorsch, imo eräsvira, Erik heissen, as well as he paricipans of he 2005 Journal of Applied Economerics meeing in Venice, he 2005 EFMA Conference in Milan, he 2005 Inernaional Conference on Finance in Copenhagen, he 2006 Swiss Sociey for Financial Marke Research Conference in Zürich for helpful commens and suggesions. he second auhor acknowledges financial suppor from he Ialian Minisry of Universiy and Research projec PRIN2006 Economeric analysis of inerdependence, sabilisaion and conagion in real and financial markes. his paper was compleed while he firs auhor was visiing he NYU Sern School of Business. Corresponding auhor: Universiy of Bonn, BWL I, Adenauerallee 24-42, Bonn, Germany.

2 Absrac We exend he Dynamic Condiional Correlaion mulivariae GARCH specificaion o invesigae he dynamic conemporaneous relaionship beween correlaions and variances of he underlying asses. We presen a generalizaion of he DCC model where he dynamic behavior depends on he asses variances hrough a hreshold srucure. Our purpose is o analyse he behavior of correlaions in periods of high volailiy. he applicaion of he proposed specificaion o a sample of markes heerogeneous in he level of heir developmen allows he idenificaion of marke pairs he correlaions of which are less sensiive o high volailiy values JEL classificaion: C50, F37, G11, G15 2

3 1. Inroducion Undersanding he relaionship beween correlaions and volailiies is crucial for risk managemen and opimal porfolio allocaion sraegies. Correlaions which increase in volaile periods reduce he power of porfolio diversificaion when i is needed mos. his paper exends he mulivariae Dynamic Condiional Correlaion (DCC) model of Engle (2002) and is generalizaion by Cappiello, Engle and Sheppard (2006) (hereafer, CES (2006)) o invesigae he dynamic relaionship beween he correlaions and he volailiies of he underlying asses. In paricular we examine wheher high volailiy values of he asses, implied by he model, are associaed wih an increase in heir correlaion values. he resuling specificaion could be inerpreed as asymmeric in he level of volailiy. he early sudies on he relaionship of correlaions and volailiies in inernaional markes have ofen relied on he analysis of hese measures compued over differen sub-periods of he daa sample. In paricular, a range of sudies has been based on he comparison of he correlaion coefficiens during sable and volaile marke periods (e.g., Berero and Mayer 1990, King and Wadhwani 1990, Lee and Kim 1993, Erb e al. 1994, Calvo and Reinhar 1996). hese papers presen evidences ha inernaional correlaions increase significanly in urbulen imes. Sambaugh (1995), Boyer e al. (1999) and Forbes and Rigobon (2002) show ha ess of changing correlaions based on correlaion coefficiens condiional on differen levels of one or boh reurn variables are biased due o heeroskedasiciy of financial reurn series. A range of papers ake ino accoun he heeroskedasiciy propery of financial ime series when esing for changing correlaions in varying volailiy regimes. Longin and Solnik (1995) and Ramchard and Susmel (1998) are examples of sudies doing his in he framework of mulivariae ARCH-ype models. 3

4 Longin and Solnik (1995) es he hypohesis of higher correlaion during volaile periods, using a bivariae Consan Condiional Correlaion (CCC) GARCH model (Bollerslev, 1990) as a base specificaion. he auhors allow he esimaed correlaion value for he urbulen marke periods o differ from he consan correlaion coefficien for he res of he sample by inroducing a hreshold on he conemporaneous value of he volailiy. Differenly, Ramchard and Susmel (1998) propose a bivariae SWARCH model ha makes correlaions a funcion of variance regimes, wih differen correlaions for periods of high and low volailiy. While in Longin and Solnik he correlaion depend on an exogenous volailiy hreshold (he uncondiional variance of he process), in Ramchard and Susmel he volailiy regime is endogenously deermined wihin he model. Noe ha in boh Longin and Solnik (1995) and Ramchard and Susmel (1998) he correlaions are assumed o be consan wihin high and low volailiy saes. Boh sudies find ha marke correlaions rise when he condiional volailiies are high. Edwards and Susmel (2001) presen an analysis of he inernaional sock marke co-movemens sudying he codependence of volailiy regimes 1. hey also use a bivariae SWARCH model, bu wih he purpose of invesigaing wheher periods of high volailiy are correlaed across counries, and presen empirical evidence which confirms his hypohesis. In his paper we propose a generalisaion of he approach of Longin and Solnik (1995) in wo direcions: firs, we model correlaions in a dynamic way following he growing lieraure sared by Engle (2002) wih he Dynamic Condiional Correlaion model and hen generalised by Cappiello e al. (2006); second, aking advanage of he dynamic behaviour of correlaions, we explicily include in he model he volailiy hresholds. 1 An alernaive approach o he examinaion of his ype of asymmeric codependence srucure of asse reurns is proposed in Longin and Solnik (2001). he paper uses he exreme value heory o model he asympoic disribuion of mulivariae ail correlaion, and shows ha condiional correlaions in he inernaional markes increase in volaile bear markes, bu no in bull markes. Furher evidence ha he correlaions of inernaional markes increase condiional on large negaive reurns is presened, for example, in Karolyi and Sulz (1996), Solnik e al. (1996), De Sanis and Gerard (1997), Ang and Bekaer (2002) and Das and Uppal (2004). 4

5 o demonsrae he pracical relevance of our model we employ a sample of naional sock indices from markes heerogeneous in he level of heir developmen and inegraion ino inernaional securiies markes. While here is a considerable body of research invesigaing he Asian and Lain American emerging sock markes, he ransiion markes of Cenral Europe have seen much less aenion so far. Our sample includes sock indices from he major developed markes as well as hree larges ransiion sock markes of Cenral Europe: Hungary, Poland and he Czech Republic. 2 he empirical evidence indicaes ha he response of he ransiion markes o global marke evens is no always similar o ha of he developed markes. he resuls of he applicaion of he exended DCC specificaions o our sample delivers srong evidence ha he urbulen periods are associaed wih an increase in he correlaions among he developed markes. For he crosscorrelaions of he ransiion markes (in paricular of he Hungarian and he Czech markes) among each oher and wih he developed markes, however, his paern is by far no as pronounced. he Polish marke, on he oher hand, in many respecs behaves similar o he developed markes. he idenificaion of marke pairs he correlaions of which do no increase in volaile periods has poenial implicaions for leveraging he benefis of inernaional porfolio diversificaion. he invesigaion of hese implicaions iself is, however, ouside he scope of his paper. We proceed as follows. In Secion 2 we presen our modelling sraegy evidencing he differences wih respec o he acual approaches. In Secion 3 we presen he daase used in he empirical analysis of Secion 4. Secion 5 concludes. 2 Hungary, Poland and he Czech Republic are among he eigh Cenral and Easern European (CEE) counries which joined European Union in We choose Hungary, Poland and he Czech Republic because, among he CEE counries, hey represens he larges capial markes. 5

6 2. V-DCC: Volailiy hreshold Dynamic Condiional Correlaions Models he main innovaion included in his paper is given by he inroducion of a new class of Dynamic Condiional Correlaion models ha generalise he original conribuion of Engle (2002) and Cappiello e al. (2006). he models belonging o his class are named Volailiy hreshold Dynamic Condiional Correlaions models (wih a V- prefix henceforh) given ha he correlaion dynamic parially depends on variance values hrough a hreshold srucure. We firs presen he basic Variance hreshold generalisaion of he DCC model of Engle (2002) and in a following secion he addiional represenaions we propose. Consider an n -variae condiional process ε wih zero mean and covariance marix are idenically disribued following an un-specified densiy D(.): H which ε F D(0, H ) (1) 1 where F 1 denoes condiioning informaion se including he informaion up o ime -1. he vecor ε may represen eiher a zero mean reurns vecor or he residuals vecor of a reurn mean model. he V-DCC model has he following represenaion. he condiional covariance marix can be decomposed as H = D R D (2) where D is a diagonal marix of condiional volailiies { i, } D = diag h (3) and R { ρ ij, } = represens he ime-varying condiional correlaion marix. Furhermore, denoe by η he variance sandardised residuals 6

7 η = D ε (4) 1 and noe ha hey are correlaed and wih a uni variance. Following Bollerslev (1990), Engle (2002) and he several conribuions generalising heir models, he condiional variance h i, could be modelled by any univariae GARCH model. here are no reasons for requiring he use of a specific represenaion for all condiional variances which can be specified on a series-specific case. Furhermore, noe ha R corresponds o he condiional covariance marix of he variance sandardized residuals η and if i is assumed o be ime invarian he model collapses on he Consan Condiional Correlaion model of Bollerslev (1990). he V-DCC model specifies he dynamics of he correlaion marix as follows: ( ( )) ( ( )) R = diag Q Q diag Q (5) ( 1 β ) ( η η ) β 1 ( ) Q = R + + Q + V V (6) where, β and are scalar coefficiens, R is he uncondiional correlaion marix of η, R = E ηη, V is a dummy variable marix relaed o he volailiy hreshold srucure, V = E [ V ] and Ei [ ] denoes an uncondiional expecaion. he dummy variable marix V has he following srucure: ( ) ({ } ) { } 1 if h d h or h d h V = vij, v > > ij, = 0 oherwise i, i, = 1 j, j, = 1 (7) (, 1 ) where { } d h i = is a given hreshold for he condiional variances of variable i, deermined h i, = 1 using he condiional variance series { } (and similarly for { hj }, = 1 ). he definiion of hresholds and heir possible generalisaions will be exensively discussed in a following secion; 7

8 a his sage we only evidence ha he V dummy marix may be creaed using an and condiion insead of an or condiion, as follows: ( ) ({ } ) { } 1 if h d h and h d h V = vij, v > > ij, = 0 oherwise i, i, = 1 j, j, = 1 (8) Noe ha he V-DCC model collapses on he DCC model of Engle (2002) if he coefficien is zero. In order o avoid explosive paerns in he dynamic of Q we impose ha + β < 1. Furhermore, uncondiionally, he expecaion of Q is sill equal o he uncondiional correlaion marix R, meaning ha he V-DCC model is subjec o he uncondiional correlaion argeing consrain. his fac allows inerpreing he V-DCC model as a correlaion model similarly o he DCC model of Engle (2002). Given he quadraic srucure in (5), R is guaraneed o be posiive definie if Q is posiive definie. Differenly from he DCC model of Engle (2002), he choice of a suiable saring poin Q 0 is no sufficien o guaranee he posiive definieness if he dummy marix is defined as in (7). In his case, he posiive definieness mus be imposed in he esimaion sep of he model. If he V marix follows (8) i will be posiive semi-definie by consrucion and hus he choice of an appropriae Q 0 value guaranees he posiive definieness of Q (given ha i will be he sum of posiive definie marices, Q 1 and R, and posiive semi-definie marices, ( ) η η ). V, V and he V-DCC model could be used in several financial areas. he varying relaionship beween volailiy and correlaion values of he differen asse pairs in he porfolio, if presen bu ignored, could have serious consequences for porfolio hedging effeciveness. In paricular, he value of 8

9 porfolio diversificaion may be oversaed if he increase in co-movemen of asse prices in he periods of high volailiy, and, in paricular, high downside volailiy is no accouned for. he exension of he DCC model we propose ess he hypohesis wheher high volailiy values of he underlying asses are associaed wih an increase in heir correlaion values. An invesor rearranging his porfolio would appreciae he possibiliy of idenifying asses for which his associaion is relaively weak, as, oher hings being equal, one could consider such asses as poenially aracive arges for porfolio diversificaion. Furhermore, he V-DCC model could be useful in he conagion lieraure given ha i will enable he disincion of correlaion movemens associaed o volailiy spillover effecs from he changes in he correlaion levels associaed o pure conagion evens. In fac, once he correlaion dynamic has been esimaed, we could filer ou he effecs of he volailiy hreshold componen and analyse he remaining paerns in order o highligh jumps in he correlaions ha we could associae o conagion. 2.1 Model exensions A drawback of he V-DCC dynamic specified in (6) is ha all he elemens of he condiional correlaion marix are resriced o have he same behaviour. One srand of he DCC-relaed lieraure proposed exensions of he DCC model wih richer dynamic, Franses and Hafner (2003) and Billio e al. (2006), among ohers. A second possible approach has been followed by Cappiello e al. (2006) ha propose a BEKK 3 -ype generalizaion of he model. In he Generalised DCC (GDCC) model of Cappiello e al. (2006), he following equaion drives he correlaion dynamic: Q = ( Q AQA' BQB ') + A( η η ) A' + BQ B ' (9) ' he BEKK mulivariae GARCH model was firsly proposed by Engle and Kroner (1995). 9

10 where A and B are n n diagonal marices 4. As a resul, he dynamics of he individual elemens of he marix Q is specified as follows: q = (1 β β ) q + η η + β β q (10) ij, i j i j ij i j i, 1 j, 1 i j ij, 1 Alhough his generalized model adds flexibiliy o Engle s specificaion, he number of parameers o be esimaed increases considerably, bu remains feasible (hey are linear in he number of correlaions, which are, however, quadraic in he number of asses) 5. Wihin a Volailiy hreshold framework, he approach of Cappiello e al. (2006) allows for he inroducion of individual series specific volailiy impac parameers. We propose he following exension of he GDCC models in (9) inroducing a diagonal Volailiy hreshold componen: Q = ( Q AQA' BQB ' ΓV Γ ') + A( η η ) A' + BQ B ' + ΓV Γ ' (11) ' where V = E [ V ], and A, B and Γ are n n diagonal marices. Now, a sufficien condiion ensuring he posiive definieness of he covariance marix Q is ha ( Q AQA' BQB ' ΓV Γ ') is posiive definie 6 if definieness of V is creaed as in equaion (8), while if Q mus be imposed in he esimaion sep. V follows (7) he posiive Wihin he V-GDCC specificaion, he dynamics of he individual elemens of Q are hen specified as: q = (1 β β ) q v + η η + β β q + v (12) ij, i j i j ij i j ij i j i, 1 j, 1 i j ij, 1 i j ij, 4 he mos general model represenaion includes full parameer marices, bu his will raise he well know course of dimensionaliy. 5 he Asymmeric Generalized DCC in CES (2006) includes an addiional componen accouning for he asymmeric impac of he pas negaive shocks on he correlaion process. 6 he sufficien condiion is generally imposed in he opimizaion rouines. 10

11 In he empirical applicaions, he V-GDCC model allows for idenificaion of heerogeneiy in he response of he markes o, say, high volailiy, given he inroducion of differen coefficiens in he diagonal of marix Γ. On he oher hand, a resricion on he GARCH dynamics of he condiional correlaions (we may impose ha A and B diagonal elemens o be idenical, or similarly we may ransform he marices ino scalars 7 ) in some cases could be well jusifiable (we may have a se of inegraed financial markes wih common correlaion dynamic because hey reac in a similar way o he shocks), leading o a more parsimonious specificaion and/or making he model esimaion feasible also in large dimensions. 8 Inroducing he resricions on he GARCH correlaion dynamic in model (11), bu mainaining he heerogeneiy in he volailiy hreshold componen, lead o he following correlaion specific dynamic behaviour: q = (1 β ) q v + η η + β q + v (13) ij, ij i j ij i, 1 j, 1 ij, 1 i j ij, In he following, we refer o he specificaion in (6) as he Volailiy hreshold DCC (V-DCC), he one in (11)-(12) as he Volailiy hreshold GDCC (V-GDCC), and o he specificaion in (13) as he resriced V-GDCC. In he generalized versions of he model he producs of he coefficiens, i j, measure he sensiiviy of he correlaions beween markes i and j o he levels of volailiy in he underlying markes. herefore, hey are of direc ineres in he invesigaion of he relaion beween correlaion and volailiy. We sugges o direcly es he significance of hese producs raher han simply analyse he individual i coefficiens. Noe ha he V- componen can in general be added o differen dynamic correlaion specificaions proposed in he lieraure, creaing, as we suggesed, a new model class. Among he 7 If we impose he scalar resricions in he basic GDCC model in (8) we obain he DCC model of Engle (2002). 8 As shown in Engle and Sheppard (2001), he scalar DCC model leads o sub-opimal porfolio selecion in case of many asses (like 20 or 30) as i assumes he same GARCH-ype dynamics for all he asse-specific condiional correlaions. his assumpion becomes, however, increasingly more likely o be saisfied in case of small number of asses. 11

12 possible ineresing specificaions o be invesigaed in fuure conribuions, we menion he works of se and sui (2002) and of Pelleier (2006). In paricular, he join use of he Volailiy hreshold srucure and he Markov swiching dynamics may provide useful ools for he conagion analysis. A furher generalisaion of he V-DCC models refers o he relaion beween he Volailiy hreshold componen and he marix Q. In he previous dynamic equaions we have always assumed ha he V effec is conemporaneous o he correlaion marix, Q f ( V ) generalise his relaion allowing for lagged effecs, Q f ( V, V,... V ) 1 K =. We can = where K is he maximum lag. Noe ha by increasing he lags of V we may largerly increase he parameer se. For his reason, we sugges he inclusion of lagged effecs only in he V-DCC models in (6) and in he resriced V-GDCC of equaion (13). Posiive definieness of he correlaion marix is achieved as in he cases wihou lags in he dummy variable marix V. Model exensions relaed o he hresholds definiion and srucure are discussed in he following secion. 2.2 he volailiy hresholds he Volailiy hreshold model inheris is name from he presence of a hreshold based componen affecing he correlaion dynamic. We previously inroduced he differen model represenaions simply saing ha he hresholds are funcions of he condiional variance series. We now presen he possible approaches ha could be followed for he definiion of he hresholds. he firs mehod defines hresholds as fraciles of he condiional variance series. In his case, he hresholds may be defined given an esimae of he condiional variances and we may choose o fix for each h i, series a hreshold ha idenifies condiional variances in he upper k-h% of he 12

13 empirical densiy of h i,. However, his approach may raise a problem since he hresholds will be series specific, and he magniude of he hresholds beween counries may vary. A differen approach is o provide hresholds based on fraciles bu deermined on sandardised condiional variance sequences as follows: i) compue he mean and he variances of each condiional variance sequence υi = hi,, τ 2 1 i = i i = 1 ( h ) 2, υ, and compue he sandardised condiional variance sequences ( ) 1 h = h υ τ ; i, i, i i ii) compue he d hreshold on a common basis as fraciles of he densiy of he enire se of n ;, i= 1 sandardised condiional variances { hi } iii) ge hen back o he specific hresholds di = dτ i + υi. Wih his alernaive approach he hresholds are deermined on a common basis, aking ino accoun possible differences in erms of magniude and dispersion of he condiional variance sequences. In he case of boh sraegies oulined above he hresholds are based on fraciles in order o ensure he exisence of a minimum number of hreshold evens. Noe ha he choice of he preferred fracile could be solved by some calibraion exercises. he general approach we propose is close o he mehods of ong (1983). In he empirical applicaion we will presen a comparison of he wo alernaive ways for he hreshold definiion. A raher differen mehod which is no included in he presen paper is he endogenous esimaion of he hresholds. We may define he series i hreshold as an addiional parameer o be esimaed. In his las case, he model would require more compuaional inensive esimaion mehods. 1 = 1 he V-DCC models could be furher generalised by modifying he hreshold componen. In fac we could consider he inroducion of muliple hresholds in order o capure he possible changes 13

14 in correlaions associaed o differen changes in he variance levels, we may disinguish beween moderae and severe jumps in he volailiies. In his case, if we inroduce L hresholds, he V componen of he V-DCC model in (8) may be resaed as follow: L ( V V ) l ( Vl, Vl ) (14) l= 1 where Vl = E Vl,, ({, } ), 1 {, } + = 1 ( = 1) ({, } ), 1 ({, } = 1 + = 1) if dl hi < hi dl hi 1 Vl, = v l, ij, vl, ij, = or d h < h d h 0 oherwise l i j l j. (15) and { } ( i ) ({ } )... ({ } = 1 i = 1 L i = 1 ) d h < d h < < d h. Noe ha when l = L he condiions are one 1, 2,, sided only ( { } ( ) or ({ } = 1 = 1 ) d h < h d h < h ). Coninuing our previous example, he L i, i, L i, j, inroducion of wo-sided condiions allows separaing he effec of moderae from severe volailiy increase on correlaions. In fac, he correlaion effec of a variance change beween hresholds l and l+1 is compleely associaed o coefficien l and a direc significance es is available. Differenly, we could define he dummy variable marices as follows ({ } ) { } ( ) 1 if d h < h or d h < h Vl, = v l, ij, vl, ij, = 0 oherwise l i, = 1 i, l i, = 1 j, (16) 14

15 where we used everywhere one-sided if condiions. In his second hypohesis, he coefficien l can be inerpreed as an incremenal correlaion effec coming from variances above hreshold l wih respec o he effec coming from variance above he hreshold l-1 (because variances above l are also above l-1). he previous generalisaions of he V componen were all presened wih he or condiion. hey can be adaped for he inclusion of he and condiion as in equaion (8). he main difference beween he wo approaches ( or agains and condiion) is in he consrain needed for ensuring he posiive definieness of he correlaion marix: he or condiion requires a direc imposiion or check of posiive definieness of Q in he esimaion sep, while he and condiion requires eiher a consrain on he parameers in V-GDCC model or he choice of a suiable saring poin in he V-DCC model. As emphasized in he inroducion o his paper, a range of sudies have idenified ha he correlaions beween asses increase for downside moves, especially for exreme downside moves, raher han for upside moves. Below we propose a furher modificaion of he V componen which considers he case of exreme volailiy associaed wih bear markes. 9 In he framework of he DCC model his could, for example, be defined as he case when he fied volailiy for he period exceeds he pre-specified hreshold and a he same ime he observed reurn a ime 1 is negaive (which is equivalen o he corresponding sandardized residual being negaive). o inegrae his feaure ino our specificaion, he dummy variables marix, V, is redefined as follows: 9 In his conex, see CES (2006), who provide an exension of he GDCC model in (7), he Asymmeric Generalized DCC, o accoun for he asymmeric impac of he sign of he pas innovaions on he curren correlaion values. 15

16 v ij, ( hi, > d ({ hi, } ) ε, 1 ) 1 i < = hj, d ({ hj, } 1) ε = j, 1 if and 0 1 = or > and < 0 0 oherwise ( ) (17) We refer o he specificaions in (6) and (11), wih he elemens of he marix V defined in (17) as he Volailiy hreshold Asymmeric DCC (V-ADCC) and he Volailiy hreshold Asymmeric GDCC (V-AGDCC), respecively. All he models described in his secion could be modified in such a way ha he correlaion values are condiioned on he observed pas reurn series only (bu no on he fied volailiy values). In his furher case, he marix V would be defined in order o condiion he correlaion values on he pas reurns or squared reurns exceeding a pre-specified hreshold. Noe ha if we define he V marices using squared reurns we may also add an asymmeric effec as in he V- ADCC and V-AGDCC specificaions. he discussion on he or and and condiions and on he posiive definieness of he correlaion marix previously presened direcly exend o hese furher generalisaions of he V-DCC and V-GDCC models. 2.3 Model esimaion Dynamic condiional correlaion mulivariae GARCH models generally allow for wo-sage esimaion. Specifically, he likelihood funcion of he DCC models can be wrien as a sum of a volailiy par and a correlaion par. We can express he quasi normal likelihood of (1) as follows: 16

17 1 1 ( θ ) ( ln ε ε ) = 1 2 = ( ln D R D εd R D ε ) L = L H + H = = + 2 = 1 (18) Following Engle (2002), he esimaes of volailiy parameers can be found replacing R in (18) by an ideniy marix of size n. he firs sage log-likelihood is simply he sum of he individual series volailiy log-likelihoods. Given he sandardized residuals and he parameer esimaes from he firs sage of esimaion, he correlaion parameers are obained by maximizing he second-sage log-likelihood. Under a se of regulariy condiions, Engle and Sheppard (2001) demonsrae consisency and asympoic normaliy of he wo-sage esimaor. In our case, he second sage likelihood will have parameers and correlaion dynamic depending on he firs sage parameers boh via he firs sage sandardised residuals and hrough he firs sage esimaed condiional variances. However, as in Engle (2002), he parameers of he volailiy models are deermined exclusively in he firs sep. herefore, he fied volailiy series could be considered as given for he second sep of he esimaion, focusing on correlaion specific parameers. Furhermore, as we evidenced in he previous secion, he researcher s ineres may be relaed o funcions of parameers, like he produc of he volailiy-hreshold coefficiens. In his case, he coefficien funcion values will be deermined using he esimaion oupus while he sandard errors will be evaluaed using he dela mehod. 17

18 3. Daa Descripion he empirical par of his paper concenraes on he invesigaion of he ime-varying correlaion dynamics of he major Cenral Europe s ransiion markes. As we previously menioned, Hungary, Poland and Czech Republic represen he larges capial markes in he CEE area. he analysis we propose is based on he blue-chip indices of a group of markes. hese include developed markes, France (CAC 40 index), Germany (DAX 30), Unied Kingdom (FSE 100) and he Unied Saes (S&P 500), as well as he afore menioned ransiion economies, Hungary (BUX 30), Poland (WIG 20), and Czech Republic (PX 50). All he indices are observed a weekly frequency and have been expressed in Euro. he sample spans he period from January 1995 o July 2007, consiuing a oal of 655 weekly reurn observaions. he use of weekly daa is preferred because of he exisence of marke fricions, in paricular for ransiion economies, and for he differen rading hours wihin European counries and beween Europe and he US. [INSER ABLE 1 ABOU HERE] able 1 presens some descripive saisics of log reurns of he seven sock marke indices considered. All series show he ypical non-normaliy of financial ime series. hey are negaively skewed and display excess kurosis. he Ljung-Box saisics sugges serial auocorrelaion in he reurns of mos indices (wih excepion of German DAX). he squared reurns of all series are highly auocorrelaed, which can be aken as evidence of ARCH effecs in he considered series. In order o model he mean correlaion and he possible relaion beween markes we specified a VAR-ype srucure for he weekly indices reurns. Our mean model also includes a number of addiional explanaory variables, which have been useful o predic asse reurns, see he works by Ai-Sahalia and Brand (2001) and Pesaran and immerman (1995, 2000). hese variables we included are: (i) he shor erm ineres raes for he European marke, measured by he German 3 monh money rae for he period from January 1995 hrough 18

19 December 1998, and he Euro Inerbank Offered Rae (ERIBOR) for he res of he sample 10 ; (ii) he shor erm U.S. ineres raes, measured by he 3 monh U.S. reasury bill rae; (iii) he long erm European ineres raes, measured by he German 10 year governmen bond yield; (iv) he long erm U.S. ineres raes measured by he U.S. 10 year governmen bond yield; finally, (v) he OPEC oil price. [INSER ABLE 2 ABOU HERE] able 2 shows uncondiional correlaions of he reurn series. he highes correlaions are beween he hree developed European markes, CAC 40 and DAX 30 (0.88), CAC 40 and FSE 100 (0.80), and DAX 30 and FSE 100 (0.77). hese are followed by he correlaion beween S&P 500 and FSE 100 (0.74), S&P 500 and DAX 30 (0.71), and S&P 500 and CAC 40 (0.71). he correlaions wihin he ransiion markes range from for he Polish and Czech indices o for he Hungarian and Polish indices. I is ineresing o noe ha he correlaions among he ransiion markes are higher han he correlaion of hese markes wih he developed markes. V. Empirical Resuls In order o capure he lagged dependence srucure in he reurns of he analyzed daa series, he mean dynamics is specified as a VAR model: X µ X φxx ( L) φxz ( L) X ε x, Z = µ + + φ ( L) φ ( L) Z ε Z zx zz z, where X is he se of sock marke reurns and Z is he se of addiional economic and financial variables. hese addiional variables include changes in he shor and long ineres raes for he 10 he euro was inroduced on he 1 s of January

20 European and he US markes, as well as reurns on oil prices. o assess wheher he considered VAR specificaion is adequae we perform a Granger-causaliy es on he marix φ ( L ), in order o verify he null hypohesis of no effec from he sock markes o he variables in zx Z. he Wald coefficien es indicaes ha he null hypohesis of non-causaliy is rejeced. We herefore coninue o use he full VAR specificaion above for he mean esimaion. he residuals from he mean model are hen used for modelling he volailiy of he considered sock marke indices. For all he series we fi he sandard GARCH(1,1) specificaion of Bollerslev (1986) as well as he asymmeric generalizaion of Glosen e al. (1993) (he GJR(1,1) model). he choice of he volailiy models is performed on he basis of he Schwarz Informaion Crierion (SIC). Ineresingly, he esimaes indicae ha for all seven indices he model preferred by SIC is GARCH(1,1). [INSER ABLES 3 AND 4 ABOU HERE] able 3 presens he cross-counry correlaions of sandardized residuals, once he condiional heeroskedasiciy has been removed, while able 4 presens he correlaions of he fied volailiy series. In boh cases he correlaions among he developed markes are higher han he correlaions beween developed and ransiion markes, and among he ransiion markes hemselves. here is, however, an imporan difference beween hese wo ses of correlaions. he differences in correlaions among developed markes, and he correlaions beween ransiion and developed markes in he case of sandardized residuals are no as pronounced as in he case of volailiy correlaions. While he correlaions among he developed markes volailiies range from 0.85 for he pair S&P 500 and DAX 30, o 0.94 for he pair CAC 40 and DAX 30, he crossmarke correlaions of he developed and ransiion volailiy series range from 0.08 for he pair BUX 30 and S&P 500, o 0.29 for WGI 20 and FSE100. he volailiy correlaions among 20

21 ransiion markes are relaively high: 0.47 for he pair WGI 20 and PX50, 0.65 for BUX 30 and PX50, and 0.69 for WGI 20 and BUX 30. [INSER FIGURE 1 ABOU HERE] Figure 1 illusraes he developmen of he volailiies over he considered sample period. One observes ha he volailiies of he developed markes comove, and reac o significan inernaional evens in a similar manner. 11 In he case of ransiion markes, i is ineresing o noe ha while he reacion of hese markes o he Asian crisis in lae 1997, and he Russian defaul in Augus-Sepember 1998 was very srong, oher major inernaional evens like Sepember 11 or he new economy bubble burs in 2002 did no have a pronounced effec on hese markes. Volailiy hreshold Dynamic Condiional Correlaion Esimaes In secion 2.2 we discussed possible approaches for he definiion of he volailiy hresholds. In he empirical applicaion below we consider wo ypes of hresholds: (i) he series specific hresholds and (ii) he common hresholds, which are based on he sandardized condiional variances. 12 able 5 presens he specific and common volailiy hresholds esimaed for he 90% and 75% fraciles of he empirical densiy of he condiional variances. he choice of hese wo fraciles is moivaed by he fac ha he 90-h% fracile would capure he cases of exreme volailiy in he markes, while he 75-h% fracile would involve cases of relaively high, bu no only exreme volailiy. [INSER ABLE 5 ABOU HERE] 11 For he formal analysis of he cross-counry volailiy comovemens, paricularly focusing on he periods of high volailiy, see Edwards and Susmel (2001). 12 See secion 2.2. for he procedure employed for he calculaion of he hresholds. 21

22 he repored esimaes indicae only minor deviaions beween he hresholds calculaed on he series specific and he common basis. herefore, for breviy, we confine our presenaion below o he series specific hresholds. 13 We now urn o he esimaion of he Volailiy hreshold specificaions discussed in secion 2. As our major ineres is o explore poenially heerogeneous impac of high volailiy on correlaions of differen marke pairs, we concenrae on he wo exensions of he basic/resriced model in (6), wih (i) he unresriced series specific GARCH correlaion dynamic and series specific volailiy impac parameers referred o as he V-GDCC model (eq. (12)), and (ii) he resriced GARCH correlaion dynamic bu series specific volailiy impac parameers referred o as he resriced V-GDCC model (eq. (13)). We esimae a range of specificaions summarized in he able below: GARCH correlaion dynamic parameers Volailiy impac parameers Volailiy hreshold wih or condiion 1 if h d { h } or h d h V = vij, v > > ij, = 0 oherwise ( ) ({ } = 1 = 1 ) i, i, j, j, Specificaion 1 Unresriced Unresriced Conemporaneous Specificaion 2 Unresriced Unresriced Lagged Specificaion 3 Resriced Unresriced Conemporaneous Specificaion 4 Resriced Unresriced Lagged Volailiy hreshold wih and condiion 1 if h d { h } and h d h V = vij, v ij, = 0 oherwise Specificaion 5 Unresriced Unresriced Conemporaneous Specificaion 6 Unresriced Unresriced Lagged Specificaion 7 Resriced Unresriced Conemporaneous Specificaion 8 Resriced Unresriced Lagged ( ) ({ } = 1 = 1 ) i, i, j, j, able 6 repors he esimaes of Specificaion 1 for he 90% fracile of he condiional variances. his specificaion builds on unresriced GARCH correlaion dynamic and volailiy impac parameers, a conemporaneous volailiy hreshold and he condiion ha he volailiy exceeds 13 he resuls based on he common hresholds are qualiaively similar and are available upon reques. 22

23 he hreshold a leas in one of he wo markes in he pair ( or condiion). Significan ARCH ) and GARCH ( βiβ j ) effecs are presen in he correlaion dynamic of all marke pairs. I is ( i j imporan o noe, however, ha boh ARCH effecs and also he persisence in he correlaions are sronger for he developed marke pairs han for he pairs involving he ransiion markes. urning o he analysis of he effecs of high volailiy on he correlaion levels capured by he parameer producs, i j, he following observaions are worh noing: While here is a significan increase in he correlaions among he developed markes associaed wih high volailiy (a leas in one) of he underlying markes, he effecs for marke pairs involving he Hungarian (BUX) and he Czech (PX) markes are no significan a convenional levels. On he oher side, periods of high volailiy are associaed wih significan increase in he correlaions of he Polish (WGI) marke wih he developed markes. While Specificaion 1 invesigaes he conemporaneous relaion beween correlaions and volailiies in he underlying markes, Specificaion 2 explores wheher high volailiy in he markes affecs he level of heir correlaions wih a lag. he -saisics of he parameer producs,, in able 7 indicae ha he significance of he lagged volailiy hreshold effecs in he i j considered sample is on average higher han ha of he conemporaneous effecs in able 6. he esimaes for his specificaion indicae ha he posiive effecs of lagged high volailiy on he correlaions of he Hungarian (BUX) marke wih he developed markes are marginally significan. On he oher hand, he correlaions of he Czech (PX) marke wih he oher markes do no significanly increase following high volailiy in he underlying markes. Specificaions 3 and 4 in able 8 resric he GARCH correlaions dynamic as presened in (13), i.e he diagonal elemens of he parameer marices A and B are se o be idenical. Specificaion 3 considers he conemporaneous and Specificaion 4 he lagged volailiy hreshold effecs. he major difference beween he esimaes of hese models he unresriced and resriced GARCH dynamic lies in he magniude of he differences beween volailiy hresholds effecs. In he 23

24 resriced specificaions he significance of hese effecs for he pairs involving he Hungarian BUX and he Czech PX is much lower (in he Czech case his effec even changes is sign). he firs wo rows of able 15, Panel A, repor he likelihood raio ess beween he specificaions wih he resriced and unresriced GARCH correlaion dynamic. he specificaions wih he unresriced dynamic are preferred. his resul, as well as some variaion in he esimaes relaed o he volailiy hreshold effecs, emphasize ha in he heerogeneous sample, similar o ours, allowing for he series specific GARCH correlaion dynamic is imporan. able 15, Panels A and B, also repors he likelihood raio ess beween he resriced and unresriced versions of all oher model specificaions considered in he empirical par of his paper. In all cases he unresriced version is preferred. o conserve space for he furher specificaions we repor he esimaes of he unresriced versions only. 14 We now repea he esimaion of Specificaions 1 and 2, however wih he volailiy hreshold se a 75% fracile of he empirical densiy of he condiional variance series. he esimaes are presened in ables 9 and 10, respecively. he resuls indicae ha for boh specificaions considered, in mos cases here is no significan effec of volailiy on correlaions a he 75% fracile hreshold level. A few excepions include he correlaions of he Polish (WGI) wih he US (SP) marke, and he UK (FSE) wih he French (CAC) marke, where he correlaions increase wih he volailiies in a leas one of he underlying markes exceeding he 75% hreshold. Afer ha we urn o he resuls of he esimaion of he Specificaions 5 and 6, which consider he relaion beween he correlaion and he underlying volailiies wih a condiion ha volailiies in boh underlying markes exceed a cerain hreshold. he resuls for he 90% fracile of he condiional variance series are repored in able 11 for he conemporaneous hreshold (Specificaion 5) and in able 12 for he lagged hreshold (Specificaion 6). he resuls for hese wo specificaions are similar. hey indicae ha for all marke pairs involving he ransiion 14 he esimaes of he resriced versions are available on reques. 24

25 markes (including Polish (WGI) marke), he periods wih very high volailiy in boh markes are no associaed wih an increase in he correlaions beween hese markes. he -saisics reflecing he significance of his effec range from for he Hungarian (BUX) and he Czech (PX) markes pair, o for he French (CAC) and Polish (WGI) markes pair (Compare hese resuls o he corresponding esimaes for he Specificaions 1 and 2). On he oher hand, he high volailiy in boh underlying markes is associaed wih a significan increase in he correlaions beween he developed markes. he final se of esimaes is for he Specificaions 5 and 6, wih he volailiy hreshold se a 75% level. For he case of he conemporaneous hreshold (Specificaions 5), one observes a highly significan associaion beween high underlying volailiies and he correlaions of he Polish (WGI) marke wih he developed markes. he analysis of his resul in combinaion wih he corresponding resuls for he 90% hreshold level in able 11 indicaes ha hese significan effecs are generaed by he volailiies in he 75% - 90% range of he empirical densiy of he underlying condiional variance series. For he developed marke pairs he effec of volailiy hreshold on he correlaions is weakened a he 75% as compared o he 90% level. For he case of he lagged hreshold (Specificaion 6), he Polish effec is reduced o being now only marginally significan, wih he effecs for developed marke pairs remaining similar o he conemporaneous case. In boh Specificaions 5 and 6, for he pairs involving he Hungarian (BUX) and he Czech (PX) markes, here is no evidence of increased correlaions associaed wih volailiies exceeding 75% hreshold level. VI. Conclusion his paper inroduces a class of Volailiy hreshold Dynamic Condiional Correlaion models, in which he correlaion dynamic parially depends on variance values hrough a hreshold srucure. hese models allow an analysis of he dynamic behaviour of correlaions beween asses in he 25

26 periods of high volailiy, and, herefore, presen a ool, which could be applied o areas, like, for example, he porfolio hedging and he conagion analysis. he empirical applicaion of he proposed Volailiy hreshold specificaions o he sample of inernaional sock markes comprising developed and ransiion markes (Hungary, Poland and he Czech Republic) reveals heerogeneiy in he relaion beween correlaions and high volailiy values for differen marke pairs in he sample. For mos of he considered specificaions, high underlying volailiy implies an increase in he correlaions among he developed markes and in he correlaions beween he Polish and he developed markes. he effec of high volailiy on he correlaions of he marke pairs involving he Hungarian and he Czech markes is ypically insignifican. he sudy of he poenial implicaions of our findings for inernaional asse allocaion and porfolio consrucion consideraions is an ineresing opic for fuure research. 26

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30 able 1. Descripive saisics SP500 DAX30 CAC40 FSE100 BUX30 WIG20 PX50 Mean (1.5336) (1.5619) (1.5236) (1.4466) (2.2164) (1.0877) (1.6556) Max Min S.dev-n Skewness Kurosis JB LB(6) LBS(6) JB is Jarque-Bera es saisic, disribued χ 2. LB(6) and LBS(6) are Ljung-Box es saisics wih 6 lags for reurn levels and reurn squares, respecively, disribued 2 χ 6. he upper 1 and 5 percenile poins of he 2 disribuion are 9.21 and 5.99, respecively. he upper 1 and 5 percenile poins of he χ 6 disribuion are and 12.59, respecively. 2 χ 2 30

31 able 2. he cross-correlaions of sock marke reurns SP500 DAX30 CAC40 FSE100 BUX30 WIG20 PX50 SP DAX CAC FSE BUX WIG PX able 3. he cross-marke correlaions of he sandardized residuals SP500 DAX30 CAC40 FSE100 BUX30 WIG20 PX50 SP DAX CAC FSE BUX WIG PX able 4. he cross-marke correlaions of he GARCH volailiy SP500 DAX30 CAC40 FSE100 BUX30 WIG20 PX50 SP DAX CAC FSE BUX WIG PX