Equiy marke inerdependence: he relaionship beween European and US sock markes SANVI AVOUYI-DOVI, DAVID NETO Direcorae General Economics and Inernaional Relaions Economic Analysis and Research Direcorae Economic Analysis and Research Cenre In his aricle, he degree of inerdependence beween European and US sock markes is measured by he condiional correlaion beween sock reurns: he correlaion coefficien is esimaed using a model describing he variaions over ime in a number of variables (reurns and volailiy, for example), and is esimae akes accoun of all available informaion a a given ime. We esimae condiional variance in he same way. Moreover, wo saisical ools, recenly inroduced in applied finance, are combined. The firs, developed by Engle in 00 an original specificaion of he condiional correlaions in mulivariae models enables us o describe ime-varying correlaions beween wo or more asses. The second ool, copula funcions, allows us o apply disribuions ha are more consisen wih he sylised facs observed on financial markes han hose commonly used. The approach used in his sudy is original in ha i combines boh he above ools. Using a mulivariae model implies rejecing he wo assumpions radiionally adoped in empirical sudies in finance: correlaions beween asses are presumed o be consan; asymmery or he presence of rare evens are no aken ino accoun in asse price disribuions. Consequenly, our empirical findings corroborae he assumpion ha correlaions vary over ime and validae he choice of an asymmeric join disribuion inegraing he presence of rare evens. We also observe he presence of periods of srong and weak correlaions and similar periods for volailiy. Furhermore, our resuls highligh a close link beween he correlaions and volailiies observed on he differen equiy markes: in phases of high volailiy, he correlaion ends o rise above is medium-erm average; inversely, in phases of low volailiy, markes seem o display greaer independence. Lasly, he correlaion coefficien of close o confirms ha French and German sock marke indices have been converging in recen years. This may reflec he growing inegraion of hese wo markes and of he economies of hese wo counries wihin Economic and Moneary Union. 08 Banque de France Financial Sabiliy Review No. 4 June 004
Equiy markes are assumed o be inerdependen bu he insrumens used for measuring his relaionship over ime are ofen unsophisicaed. Indeed, no only can several models be applied o monior and deermine variaions in volailiy over ime, in paricular when he laer is calculaed on he basis of available informaion (condiional volailiy), bu furhermore relaively few approaches exis for measuring ime-varying inerdependence beween markes. In keeping wih empirical lieraure available on volailiy (see Bollerslev, Engle and Nelson, 994, or Gouriéroux, 99), we observe ha he degree of inerdependence of markes may be higher during periods of crisis or euphoria han in normal imes. This appears o be due o he fac ha financial markes overreac, in general, o very bad or very good news (bu no necessarily in a symmerical way). Consequenly, when analysing several equiy markes (mulivariae analysis), we can reasonably assume ha srong and weak links alernae. Moreover, he globalisaion and marke inegraion process, which has been underway for he pas wo decades, suggess ha inerdependence beween sock markes reflecs srucural changes in he global financial sysem in he long erm and developmens in he financial environmen in he shor erm. This process may have heighened he risks of conagion beween financial markes and, more specifically, beween equiy markes. We use he condiional correlaion as a measure of he inerdependence or of he degree of linkage beween wo or more variables. In oher words, he correlaion coefficien is esimaed using a model ha describes noably sock reurn dynamics and ime-varying volailiy. This sudy ses ou o ascerain wheher he assumpion ha correlaions vary over ime is rue and wheher hey have similar properies o hose of condiional volailiies. In addiion o equaions describing ime-varying correlaions, we also analyse sock reurn equaions (calculaed as relaive changes in prices) and volailiy (here, condiional variances) on US and European sock markes. We can hus compare ime-varying volailiies esimaed on hese markes, on he one hand, and volailiies and correlaions, on he oher. To make hese comparisons, we sudy ime-varying condiional correlaions beween he wo main euro area sock markes (Paris and Frankfur) and he US sock marke by combining properies from he Dynamic Condiional Correlaion Mulivariae Model (see Appendix ), developed by Engle, wih hose obained using copula heory (see Appendix ), which make i possible o carry ou an appropriae decomposiion of he join disribuion of a number of variables. As we will see furher on, he approach developed by Engle (00, 00) elaboraes on his earlier work ha se ou o simulaneously explain sock reurn dynamics and ime-varying volailiy, for example. Engle s models, more commonly known as ARCH models (i.e. AuoRegressive Condiional Heeroscedasiciy), are exremely useful in applied finance sudies. Amongs oher hings, on he basis of hese models we can rejec he assumpion ha condiional variance does no vary over ime. ARCH models simulaneously describe he dynamics of sock reurns and ime-varying volailiy. To a cerain exen, he assumpion ha correlaions beween endogenous variables vary over ime exends ARCH models o he mulivariae case and enables us o add equaions describing relaionships of inerdependence beween hese variables. Unil recenly, due o he complexiy of he analyical expression of he join disribuions relaing o a mulivariae analysis, we had only used a limied range of join disribuions in empirical sudies. The copula funcion approach allows us o avoid his problem and access a broader range of join disribuions. This aricle is organised as follows: Secion One gives a brief overview of he means used o define he condiional correlaion; Secion Two ses ou he presenaion and reamen of daa and he presenaion and inerpreaion of he resuls; i also includes an example of he applicaion of he ime-varying correlaion coefficien assumpion using a capial asse pricing model (CAPM). Finally, he main conclusions are presened a he end of he aricle. Empirical resuls (see De Bond and Thaler, 985, for example) have revealed periods of equiy marke overreacion resuling, iner alia, from successive bous of opimism or pessimism on he par of marke paricipans. These resuls, which have given rise o much debae, can be found mainly in lieraure on marke efficiency ess. Banque de France Financial Sabiliy Review No. 4 June 004 09
Brief mehodological overview General framework A number of models can be used o calculae he volailiy on a given marke. They may be eiher of a srucural (explicaion by fundamenals), or a saisical naure. Wihou doub, in he pas few decades, he greaes developmens have been seen in saisical models. ARCH models come wihin his laer caegory. They have been applied o financial markes o ake accoun of cerain sylised facs (fa ails, asymmeric effecs, ec.). Moreover, hanks o recen progress in economerics heir esimaes have become more robus. ARCH models are a key focal poin of his sudy. When here is only one endogenous variable (one equiy marke for example), aside from he sock reurn equaion, he ARCH model (or GARCH model, i.e. generalised ARCH) allows us o define an explanaory equaion of he condiional variance based on hree facors: he lagged value of his variance, which inroduces a phenomenon of momenum (or persisence) in he equaion; recen shocks, represened here by he difference beween he esimaed and he observed values of he variable sudied; and a consan facor (in fac, he consan of he equaion). Therefore, if he condiional variance is assumed o be consan (i.e. if he equaion is reduced o a single consan), he coefficiens of he firs wo facors (he persisence effec and he recen shock effec) are zero. When we aemp o simulaneously analyse several variables or markes (mulivariae analysis), one of he rickies problems sems from he fac ha he number of unknown parameers increases in line wih ha of he variables or he markes. Furhermore, his analysis imposes addiional consrains, in paricular in erms of he signs or values of he parameers. This general difficuly relaing o mulivariae models also concerns ARCH or GARCH models. If we ake hree markes assumed o be dependen for example, in addiion o he parameers of he sock reurn equaions, i is also necessary o inroduce hree correlaion coefficiens, hree condiional variances as well as parameers specific o he join disribuions of hese variables. If we aemp o describe he variances and correlaions using equaions, i would be exremely difficul o simulaneously esimae all he equaions, unless we only use very simple explanaory equaions. In he case of mulivariae ARCH models, several sudies have focused on he paricular specificaions ha make i possible o boh reduce he number of parameers and limi he size of he consrains, while mainaining a relaively rich dynamic srucure of he model. One approach would be o assume ha here are one or more explanaory facors common o he differen markes (see Diebold and Nerlove, 989). The main difficuly of his approach, which is more cenred on finding a srucural explanaion, is o idenify he facors when hey can be observed and o esimae hem when hey canno. Consequenly, he complexiy of esimaion mehods of his kind of model does no generally yield resuls ha are as robus as evidence would sugges. Anoher approach would consis in using purely saisical models such as ARCH univariae models (Baba, Engle, Kraf and Kroner, 987, for example). The Consan Condiional Correlaions ARCH (CCC-ARCH) model developed by Bollerslev in 987 one of he approaches represenaive of his caegory akes condiional variances o be ime-varying bu mainains consan correlaions. This model considerably reduces he number of parameers o be esimaed bu he assumpion of consan correlaions does no reflec he realiy. Therefore, researchers have aimed o reain he main properies of Bollerslev s model (simpliciy of implemenaion, flexible framework, ec.) by adding a more realisic assumpion abou he behaviour of he correlaions. Engle (00, 00), Engle and Sheppard (00) and Tse and Tsui (00) proposed an original dynamic specificaion of he condiional correlaions in mulivariae GARCH or ARCH models, he DCC-GARCH model. In relaion o Bollerslev s approach, he DCC-GARCH model inroduces equaions ha describe ime-varying correlaion coefficiens similar in heir concepion o hose of he condiional variances discussed above (see Appendix ). Indeed, in he same way as wih he condiional variances, hese coefficiens can be explained by hree main facors: heir own lagged values, wih a view o aking accoun of persisence phenomena; a facor represening he effec of recen shocks; and 0 Banque de France Financial Sabiliy Review No. 4 June 004
a consan. If he assumpion ha correlaion coefficiens vary over ime is rejeced, heir equaions are hus reduced o consan parameers (his is hen anamoun o a CCC-ARCH model). This approach is more realisic han ha pu forward by Bollerslev, which does no sand up o empirical evidence, in paricular if i derives from equiy marke analysis. Moreover, he implemenaion of DCC-GARCH models is relaively simple hanks o recen advances in economerics. I is also adapable in respec o a cerain number of ess including ha o ascerain wheher correlaion coefficiens are consan. The conribuion of copula funcions Copula funcions have recenly been used in applied finance o obain a broader and more realisic range of join disribuions of sock reurns on a number of markes. Indeed, prior o his, in order o ake accoun of cerain sylised facs (he presence of rare evens and asymmeric effecs) in mulivariae models, i was necessary o know he analyical expression of disribuions or how easy or difficul hey were o implemen. This, for example, was he case for he Suden disribuion, which is symmeric bu does no ake accoun of he presence of rare evens. This grealy limied he scope of he simulaneous modelling of markes. A number of difficulies, arising in paricular from he choice of join disribuion, he sharp increase in he number of unknown parameers and, a imes, he decrease in he amoun of daa available, have penalised mulivariae empirical models. Copula funcions can be used, in par, o deal wih hese problems. Under easily-verifiable condiions, copula funcions make i possible o make a single decomposiion of a given join disribuion of a number of variables ino wo componens. The firs is a funcion, or srucure, of dependence, which is characerised by a se of parameers, ermed dependence measures or parameers. These parameers include he correlaion coefficien, which is one of he measures of inerdependence. The second componen is a erm corresponding o he produc of he marginal disribuions of he variables sudied; if we ake he case of he wo variables for example, his erm will correspond o he marginal disribuion of he firs muliplied by ha of he second, see Appendix or Paon (00) or Rockinger and Jondeau (00). Thanks o his decomposiion, if we know he dependence srucure and he marginal disribuions, we can obain ha of he join disribuion, which is defined as he produc of is wo componens. Consequenly, i is no longer necessary o know he exac analyical expression of his disribuion. We can for example choose asymmeric marginal disribuions and/or fa ails (presence of rare evens) combined wih a dependence srucure ha makes i possible o esablish links beween exreme evens (upward or downward price spikes). Furhermore, copula funcions faciliae mulivariae model esimaions; hey make he implemenaion of hese models more flexible. 3 Brief descripion of he esimaed model The model used for he applicaions conains equaions describing reurns, variances and condiional correlaions (see Appendix 3). In paricular, he variance equaion makes i possible o differeniae beween he posiive and negaive shock effecs (asymmeric effecs). This disincion was inroduced in order o ake ino accoun he sylised fac according o which financial markes reac more violenly o bad news. As we menioned above, copula funcions make i possible o decompose he join disribuion, which faciliaes he implemenaion of he model. In he conex of his sudy, afer preliminary ess (see Avouyi-Dovi and Neo, 003), he mos appropriae disribuion on each marke mus be asymmeric and allow he presence of rare evens. The Pearson-IV disribuion has he above properies and has recenly been esed wih success in oher sudies. We have used his disribuion here. Empirical resuls (see Avouyi-Dovi and Neo, 003, and Longin and Solnik, 998, or Mashal and Zeevi, 00) have shown ha he dependence srucure of he join disribuion of reurns should allow a marked dependence for boh faer and hinner ails, i.e. ha rare evens (upward or downward price spikes) mus be linked. The choice of his srucure mus herefore be resriced o he family of The asymmeric reacion o he signs of shocks could be explained by marke paricipans long posiions on equiy markes ha would make hem more sensiive o negaive shocks. Banque de France Financial Sabiliy Review No. 4 June 004
funcions liable o correspond o he previous propery. This is he case for he dependence srucure of he Suden disribuion. Moreover, he dependence parameers of he laer are correlaions (see Appendices and 3), which is exacly wha we se ou o define here using he DCC-GARCH model. A his sage, we should specify ha: he correlaion coefficiens analysed here are calculaed beween pairs of markes. They do no concern he relaionship beween he volailiies observed on he differen markes. For simpliciy s sake, hese coefficiens may be inerpreed as measures of he relaionship beween sock reurns; we can verify he exisence of he relaionship beween indices for insance by inroducing he reurn on he US marke in he equaion of is French counerpar. In order o es he variaion over ime of he inensiy of he relaionship, we would hen have o assume ha he coefficien of he US index, as an explanaory facor of he French index, varies over ime. We did no choose his opion here because i is no an easy exercise given ha series of variances and correlaions were no observable (ex ane). In his sudy, we assumed ha he lagged values of each reurn explains is curren dynamics. 3 The resuls and heir inerpreaion A brief descripive analysis of he daa The inensiy of he relaionship beween French, German and US sock markes (i.e. he CAC40, he DAX, and he Dow Jones) is sudied here using daily daa for he period from 3 December 993 o 30 July 00 (i.e.,38 poins for each series). For reasons of homogeneiy, we used he narrow indices of hese sock markes. The series are derived from Daasream daabases; closing (c) and opening (o) index values are available for he hree markes. The daa associaed wih paricular closing days such as public holidays specific o each counry have been replaced by moving averages cenred on he missing poins. In order o ake accoun of excepional closures (he hree days following Sepember 00 for example), dummy variables were inroduced ino he models. Sock reurns, calculaed as he firs difference of he log of he daily indices muliplied by 00 (i.e. 00 *(np np ) where n denoes he log), are analysed a he same ime as heir volailiies and correlaions. 4 Table Coefficiens of correlaion beween reurns DJ c DJ o DJ c o c DAX / DAX o o CAC / DAX 0.5838/0.5648 c c CAC / DAX 0.7497/0.307 0.309/0.33656 0.38640/0.39480 0.38640/0.39480 o c CAC / CAC 0.7353/0.7356 In he analyses of he relaionship beween US and European markes, we generally compare reurns on European markes wih ha on heir US counerpar lagged by one period in order o ake accoun of he ime difference beween Europe and he Unied Saes. By analysing he (non-condiional) correlaion coefficiens beween he reurns on he CAC, he DAX and he Dow Jones, esimaed a or a, wih he opening or closing indices (see Table ), we observe ha: he closing value of he US marke a appears o mos srongly influence European markes a opening a (Table ). The correlaion coefficiens beween he reurns on European and US markes 3 Our findings would have probably been more relevan had inraday daa been available, bu his was no he case. 4 The reurns are cenred (of zero average) o avoid he problem of idenifying consans in a rivariae model. These reurns have he saisic properies (saionariy) ha make i possible o preclude false relaionships. Moreover, as we menioned above, we rejec he assumpion ha he join disribuion is normal. Banque de France Financial Sabiliy Review No. 4 June 004
come o 0.58 for he CAC and he Dow Jones, and 0.56 for DAX and Dow Jones. These coefficiens show ha here is a relaively close relaionship beween he European indices a opening a and US indices a closing a ; he correlaion coefficiens calculaed beween he reurns observed a a closing in Europe and a opening and closing in he Unied Saes (0.39 a closing for he pairs CAC/Dow Jones and DAX/Dow Jones; 0.30 and 0.34 a opening for he same pairs) as well as hose esimaed a closing beween European reurns a and ha of Wall Sree a (0.8 and 0.3) are relaively weak and may indicae ha he inensiy of he relaionship beween hese markes is low. In general, when opening indices are no available a comparison is made beween he reurns a closing in Europe a and hose of he Unied Saes a. Clearly, his grealy underesimaes he relaionship beween he European and US markes. For he pair CAC and Dow Jones for example, he correlaion coefficien falls from 58% o 7%. Based on he resuls of he descripive saisics, reurns on European markes a opening a will herefore be compared wih he reurn on he US marke a closing a in he rivariae model used in his aricle. A sudy of condiional correlaions From analysing he variaion over ime of he correlaions beween he differen pairs of sock reurns (CAC-Dow Jones, CAC-DAX, DAX-Dow Jones, see Char ) we conclude ha: irrespecive of he reurn pair sudied, packes of srong and weak correlaions appear. This only reflecs he persisence phenomenon menioned above; he correlaion coefficiens calculaed for he pairs CAC-Dow Jones and DAX-Dow Jones are, unsurprisingly, very close (boh qualiaively and quaniaively speaking). For example, we observe peaks in he correlaions around he recen crisis periods (he Asian and Russian crises and he bursing of he ech bubble) and roughs a he firs signs of he cyclical urnaround in he Unied Saes in 000. The same is rue for 996 when he firs warning signs appeared poining o an overvaluaion of he US sock marke; aside from some rare excepions, correlaion coefficiens of he reurns on he CAC and on he DAX are compued a beween 70% and 80% over he enire sudy period wih a slighly more marked upward rend beween he hird quarer of 999 and he firs quarer of 00. Despie he fac ha we observe a sligh drop owards he very end of he period, no doub caused by cyclical differences beween he wo counries, he high levels of he correlaion coefficiens probably reflec he growing inegraion of hese wo markes and, beyond his, of he French and German economies wihin Economic and Moneary Union. Char Condiional correlaions (as a %; daily daa) 0.85 0.75 0.65 0.55 0.45 0.35 0.5 4 May 4 Jan. 4 Sep. 4 May 4 Jan. 4 Sep. 4 May 4 Jan. 4 Sep. 4 May 4 Jan. 4 Sep. 4 May 994 995 995 996 997 997 998 999 999 000 00 00 00 DAX-CAC CAC-DJ DAX-DJ Source: Daasream and Banque de France calculaions If we compare he paerns of correlaions wih hose of condiional variances (for example he variances of he CAC and of he Dow Jones and he correlaion coefficien beween he wo reurns), and if we ake he wo sub-periods (996-998 and 000-00), for greaer clariy, wih he conclusions remaining rue for he whole period, we noe ha (see Char ): he correlaion coefficiens increase as soon as one of he markes becomes relaively volaile; when boh markes display high levels of volailiy, he previous rend (increasing correlaions) becomes more pronounced (see Asian and Russian crises or Sepember 00). The magniude of he variaions of he correlaion coefficiens depends on he srengh of hese volailiies; Banque de France Financial Sabiliy Review No. 4 June 004 3
Char Condiional volailiies and correlaions (as a %; daily daa) CAC-DAX 5 5 4 4 3 3 0 s March 996 s July 996 s Nov. s March s July 996 997 997 Volailiy CAC Volailiy DAX Correlaion CAC-DAX s Nov. s March s July s Nov. 997 998 998 998 0 3 Feb. 000 3 April 000 3 June 000 3 Aug. 000 3 Oc. 000 3 Dec. 000 Volailiy CAC Volailiy DAX Correlaion CAC-DAX 3 Feb. 00 3 April 00 3 June 00 3 Aug. 00 3 Oc. 3 Dec. 00 00 CAC-Dow Jones 5 5 4 4 3 3 0 s March 996 s July 996 s Nov. s March s July 996 997 997 Volailiy CAC Volailiy Dow Jones Correlaion CAC-DJ s Nov. s March s July s Nov. 997 998 998 998 0 3 Feb. 000 3 April 000 3 June 3 Aug. 3 Oc. 3 Dec. 000 000 000 000 Volailiy CAC Volailiy Dow Jones Correlaion CAC-DJ 3 Feb. 00 3 April 3 June 3 Aug. 3 Oc. 3 Dec. 00 00 00 00 00 DAX-Dow Jones 5 5 4 4 3 3 0 s March 996 s July 996 s Nov. s March s July 996 997 997 Volailiy DAX Volailiy Dow Jones Correlaion DAX-DJ s Nov. s March s July 997 998 998 s Nov. 998 0 3 Feb. 000 3 April 3 June 3 Aug. 3 Oc. 3 Dec. 000 000 000 000 000 Volailiy DAX Volailiy Dow Jones Correlaion DAX-DJ 3 Feb. 3 April 00 00 3 June 3 Aug. 3 Oc. 3 Dec. 00 00 00 00 Source: Daasream and Banque de France calculaions. 4 Banque de France Financial Sabiliy Review No. 4 June 004
conversely, during periods of seady rises or falls in volailiy or of low volailiy, he correlaion coefficiens end o decline or sagnae. From his graphical sudy, i is difficul o subsaniae he assumpion ha correlaions do no vary over ime. Furhermore, in view of he fac ha agens operaing on hese markes may inerpre informaion differenly, he variaion in correlaions and in volailiies is no abnormal. We will now analyse he resuls of he esimaes in order o assess heir qualiy, in paricular in saisical erms. 3 Some commens on he resuls of he esimaes The esimae was carried ou in wo sages, using copula funcions (see Appendix 3): in he firs sage, we esimaed he marginal disribuion parameers as well as hose of he equaions describing he dynamics of sock reurns and volailiies (EGARCH process) and, in he second, we esimaed he coefficiens of he dependence srucure and he parameers of he equaions of he correlaions. The deailed findings are presened in Appendix 3. In general, he parameers esimaed in boh sages are all significanly differen from zero. For hose esimaed in he firs sage, we noe ha: he parameer of he sock reurn equaions (reduced here o a single endogenous variable coefficien lagged by one period, ϕ, i.e. he auoregression coefficien, see Appendix 3) is, generally, low in absolue erms. However, while i remains significanly differen from zero for he European markes, i is almos zero for he US marke. This means ha he weigh of pas reurns is less significan in he calculaion of reurns for he Dow Jones han for he European indices. This difference in he mehod of calculaing reurns on European and US indices may sem from he differences in he behaviour of marke paricipans in paricular in erms of he speed of reacion o informaion ha can influence he price discovery process. This observaion should however be reaed wih cauion as i only focuses on narrow indices sudied over a specific period; he condiional variance parameers of he hree markes are relaively close. In paricular, we noe a very srong persisence of volailiy (he coefficien β is close o in he hree cases and varies from 0.97 for he CAC o 0.986 for he DAX). This suggess he presence of a radiional persisence phenomenon, in paricular in he case of equiy markes. Moreover, he use of an EGARCH specificaion (see Appendix 3) appears o be relevan. Indeed, he impacs of he posiive and negaive shocks on volailiy seem asymmeric: he coefficien of he sensiiviy of volailiy o negaive shocks (γ α, see Appendix 3) amouns o -0.363 for he Dow Jones, and -0.44 and -0.30 for he CAC and he DAX respecively; he coefficien of he sensiiviy of volailiy o posiive shocks (γ + α, see Appendix 3) is around 0.0 for he European markes and only 0.05 for he US marke. As he confidence inervals of he coefficiens do no overlap, we can consider hem as saisically differen. As expeced, equiy markes herefore reac more srongly o negaive shocks. For example, a significan rise in unemploymen in he Unied Saes, perceived as a negaive signal, would lead o a relaively large increase in volailiy whereas a significan fall in unemploymen (posiive shock) would resul in a decline in volailiy of a lesser magniude. Furhermore, he asymmery seems much more marked in he Unied Saes. We also observe a similariy in he behaviour of French and German markes whose coefficiens of he sensiiviy of volailiy o shocks are very similar. We mus rejec he assumpion ha he empirical resuls relaing o he join disribuions of he reurns on he hree markes are symmerical. In order o make his assumpion, he parameer represening he degree of asymmery (or symmery checking parameer, δ, see Appendix 3) would have o be saisically zero, which is no he case. This confirms our assumpion as o he choice of an asymmeric disribuion in he specificaion of he model and shows ha modelling asymmery using a EGARCH process is no sufficien when analysing he reurns on hese indices. Likewise, we accep he presence of fa ails (rare evens) in he disribuions of reurns on European and US markes. As regards he coefficiens esimaed in he second sage, we can make he following commens: he averages of condiional correlaion coefficiens are no significanly differen from he values obained in Table. For he CAC and he Dow Jones, he averages were 0.59 compared wih 0.58, and 0.75 compared wih 0.74 for he CAC and DAX; lasly, hey were 0.56 compared wih 0.564 in he case of he DAX and he Dow Jones. In he long run, he impacs of hese posiive and negaive shocks migh herefore have been offse or adjused; Banque de France Financial Sabiliy Review No. 4 June 004 5
he presence of persisence phenomena in he ime-varying correlaion marix beween he reurns is confirmed. Indeed, he closer he parameer measuring he degree of persisence (here θ, see Appendix 3) is o, he longer he impac of shocks persiss in ime-varying correlaions (i.e. when a correlaion coefficien reaches a given level due o a shock, i remains here for a cerain ime). Here, his coefficien is 0.935. This corroboraes he empirical findings showing a marked persisence of volailiy, which is an indicaor of he same naure as covariance (or correlaions). I is no surprising ha persisence phenomena, considered o be sylised facs in he analysis of he dynamics of sock markes, are also borne ou by he correlaions; we noe he high significance of he parameers of recen shocks (θ, see Appendix 3) on he correlaions. As we have jus seen, he shocks alone do no explain he variaion of he correlaion coefficiens over ime. The resul we obained was herefore very much in line wih expecaions: he inensiy of he relaionship beween sock markes is no consan over ime. This finding can be se agains ha relaing o condiional volailiies. 4 Applicaion of he DCC-GARCH in he framework of he capial asse pricing model (CAPM) Adoping he assumpion ha correlaions vary over ime, we show ha he bea (i.e. he measure of he volailiy of a risky asse relaive o he overall marke), evaluaed by he CAPM, also varies over ime (see Box). To illusrae his, we ake an invesor who holds a risky asse (German sock marke index), a risk-free asse (seven-day money marke rae), and a benchmark asse (World ex. EMU-Daasream Marke Price Index 5 ). The daa are derived from Daasream. The condiional correlaions and variances are obained from he esimae of a DCC-GARCH model. 6 By aking only he curves reflecing he variaion over ime of he bea and he correlaion coefficien beween he reurn on a risky and a benchmark asse (Char 3), we noe ha here is a marked similariy beween hese wo curves. In paricular, he roughs and he peaks coincide. The periods in which he reurn on he risky asse amplifies (or dampens) significanly he shocks affecing he marke are associaed wih marked increases (or decreases) in he correlaions. This variaion of he marke bea is passed on o he variaion of he sysemaic risk (see Box) 7 which increases or decreases according o ha of he bea. This example shows ha, in he analysis of financial sabiliy hrough he sudy of risks, correlaions should be considered o be imevarying. We observed ha sysemaic risk is far from invariable as radiional analysis, in which variances and correlaions are assumed o be consan, may sugges. The assumpion ha correlaions are ime-varying offers a more dynamic and realisic inerpreaion of he bea and he risk. Indeed, in his conex, we can observe a series of phases of amplificaion (β > ) or damping (β < ), by he asse, of he shocks arising from he marke. 5 The index used is calculaed by Daasream for he world excluding European Moneary Union. The sum of he weighs of he Unied Saes, he Unied Kingdom and Japan makes up around 80% of his index. 6 The resuls of he esimaes may be obained from he auhors. 7 Sysemaic risk is he componen of oal risk aribuable o he sysem, ha is o say o he economic environmen i.e. ha which canno be diversified. 6 Banque de France Financial Sabiliy Review No. 4 June 004
Chars 3 Bea and correlaion From February 98 o 8 May 003..0 0.8 0.6 0.4 0. 0.0 Feb. 98 Feb. 984 Feb. 986 Feb. 988 Feb. 990 Feb. 99 Feb. 994 Feb. 996 Feb. 998 Feb. 000 Feb. 00 From s January 996 o 3 December 998..0 0.8 0.6 0.4 0. 0.0 3 March 996 3 July 996 3 November 996 3 March 997 3 July 997 3 November 997 3 March 998 3 July 998 3 November 998 From s January 000 o 8 May 003..0 0.8 0.6 0.4 0. 0.0 5 March 000 5 July 000 5 Nov. 000 5 March 00 5 July 00 5 Nov. 00 5 March 00 5 July 00 5 Nov. 00 5 March 003 Bea Rho Source: Daasream and Banque de France calculaions. Banque de France Financial Sabiliy Review No. 4 June 004 7
Applicaion of he DCC-GARCH in he framework of he CAPM The CAPM was developed by Sharpe (964) and Linner (965). I builds on Markowiz s porfolio selecion heory of 95. The CAPM is based on he following assumpions: invesors are risk averse and use he mean-variance crierion o selec an efficien porfolio; hey all op for he same probabiliy disribuion of reurns (informaional efficiency of he marke); he marke is perfec (here are no ransacion coss, asses are infiniely divisible, shor selling is allowed); he marke is compeiive (agens are price akers); here is a finie number of linearly independen asses. If r j, denoes he reurn on a risky asse j, r f he reurn on a risk-free asse, r m he reurn on he marke porfolio and if E [r j, ], V [r m, ] and COV [(r j, ),(r m, )] are respecively he operaors of expecaion, variance and covariance, he fundamenal resul of he CAPM is: COV [(r j, ),(r m, )] E [r j, ] = E [r m, ] V [r m, ] When he expecaion, variance and covariance of he spreads vary over ime, he CAPM is expressed as follows: COV [(r j, ),(r m, )] E [r j, ] = E [r m, ] V [r m, ] where he operaors E, V and COV are respecively he expecaion, variance and covariance condiional on he informaion se available a ime. Equaion [ ] is expressed as follows: E [r j, ] = β E [r m, ] β measures he relaive volailiy of he asse j o he marke. When i is higher (lower) han, asse j amplifies (dampens) he shocks ha affec he marke. When i is equal o, he flucuaions of he risky asse replicae hose of he marke. Under he assumpion ha correlaions vary over ime, β can be expressed as follows: β = COV [(r j, ),(r m, )] V [r j, ] V [r j, ] = ρ V [r m, ]V [r j, ] V [r m, ] V [r m, ] ρ, he correlaion beween asse j and he marke, may be generaed by a DCC-GARCH model (see Appendix ): ρ = ( θ θ )ρ + θ ψ + θ ρ The condiional sysemaic risk is expressed as σ m, = β V [r m, ]. 8 Banque de France Financial Sabiliy Review No. 4 June 004
By combining he condiional correlaions defined by Engle (00) wih copula funcions, we were able o sudy, in a flexible way, he dynamics of he dependence beween European and US equiy markes. The specificaion adoped allows us o easily model, despie he difficulies of mulivariae saisical analysis, he condiional correlaion beween he reurns on he hree markes, aken in pairs. Moreover, we were able o es and rejec he assumpion ha correlaions are consan over ime. Furhermore, hanks o he recen applicaion of copula funcions in empirical analysis in finance, a broader range of join disribuions was esed by using, iner alia, a copula ha allows dependence beween exreme evens (bubbles and crises). Several empirical sudies, noably on developed sock markes, corroborae he relevance of hese findings (Longin and Solnik, 998). The observaion ha correlaions vary over ime calls ino quesion many models in which hey are assumed o be consan. Such is he case, for example, in Markowiz s porfolio selecion model, he capial asse pricing model (CAPM) and mulivariae Value a Risk (VaR) models. In he case of he CAPM, for insance, if he correlaion and he bea are assumed o be consan, ceeris paribus, he risky asse may consanly amplify (or dampen) he shocks affecing he marke as a whole. If, however, he correlaion coefficien is assumed o vary over ime, he bea could flucuae and display phases corresponding o high values (i.e. amplificaion of shocks) or phases associaed wih lower values (i.e. dampening of shocks). Moreover, if we inegrae he dynamic inerdependence of sock markes in he previous models we should be able o ake beer accoun of spillover effecs, which are a significan componen of overall risk. Two key findings of his sudy could influence modelling in applied finance: firs, he assumpion ha correlaions are consan is oally ruled ou. While he inerdependence of markes is naurally aken ino accoun in he models of inernaional porfolio diversificaion, we should inroduce he noion ha his inerdependence varies over ime; a facor ha in iself could jusify he need for more or less large and frequen porfolio shifs. Ye, he variaion of correlaions over ime has been largely ignored in empirical sudies due o he complexiies i inroduces. Besides, by carrying ou a combined analysis of he volailiy and he condiional correlaion we observed a clear relaionship beween hese wo variables: in periods of high volailiy, he correlaion ends o rise above is normal level, symmerically, in periods of low volailiy, markes seem o be more independen; second, here is a marked persisence in ime-varying correlaions. This can be explained by he exisence of cycles (a succession of packes of phases of rises or falls) in he formaion dynamics of he inerdependence indicaor of equiy markes. Anoher explanaion for his persisence phenomenon is he heerogeneous behaviour of agens operaing on he markes sudied. Neverheless, o validae his assumpion more in-deph analysis is called for. Banque de France Financial Sabiliy Review No. 4 June 004 9
Bibliographie Avouyi-Dovi (S.) and Neo (D.) (004): Les foncions copules en finance, Banque & Marchés, 68 Avouyi-Dovi (S.) and Neo (D.) (003): Inerdépendance des marchés financiers : cas des marchés américain e européens, Banque de France, mimeo Baba (Y.), Engle (R.), Kraf (D.) and Kroner (K.) (987): Mulivariae simulaneous generalized ARCH, Universiy of California San Diego, Deparmen of Economics, Working paper Bollerslev (T.) (987): A mulivariae GARCH model wih consan condiional correlaions for a se of exchange raes, Norhwesern Universiy, D.P. Bollerslev (T.), Engle (R. F.) and Nelson (D. B.) (994): ARCH Models, Handbook of economerics, Vol. 4, chaper 49, pp. 96-303, Elsevier, Norh-Holland De Bond (W. M.) and Thaler (R.) (985): Does he sock marke overreac?, Journal of Finance, 40, pp. 793-805 Diebold (F.) and Nerlove (M.) (989): The dynamic of exchange rae volailiy: A mulivariae laen facor ARCH model, Journal of Applied Economerics, 4, pp. - Engle (R. F.) (00): Dynamic condiional correlaion: A simple class of mulivariae GARCH models, Universiy of California San Diego, Deparmen of Economics, Working paper Engle (R. F.) (00): Dynamic condiional correlaion: A simple class of mulivariae generalized auoregressive condiional heeroscedasiciy models, Journal of Business Economic Saisics, 0(3), pp. 339-350 Engle (R. F.) and Sheppard (K.) (00): Theoreical and empirical properies of dynamic condiional correlaion mulivariae GARCH, Naional Bureau Economic Research, Working paper, 8554 Gouriéroux (C.) (99): Modèles ARCH e applicaions financières, Economica Joe (H.) (997): Mulivariae models and dependence conceps, Monographs on Applied Probabiliy and Saisics, (73), Chapman and Hall Linner (J.) (965): The valuaion of risky asses and he selecion of risky invesmens in sock porfolios and capial budges, Review of Economics and Saisics, 47, pp. 3-37 Longin (F.) and Solnik (B.) (998): Correlaion srucure of inernaional equiy markes during exremely volaile periods, HEC, Working paper Markowiz (H. M.) (95): Porfolio selecion, Journal of Finance, 7(), pp. 77-9 Mashal (R.) and Zeevi (A.) (00): Beyond correlaion: exreme co-movemens beween financial asses, Columbia Universiy, mimeo Nelsen (R. B.) (998): An inroducion o copulas, Lecure Noes in Saisics, (39), Springer Verlag Paon (A. J.) (00): Modelling ime-varying exchange rae dependence using he condiional copula, Universiy of California San Diego, Deparmen of Economics, Working paper Rockinger (M.) and Jondeau (E.) (00): Condiional dependency of financial series: An applicaion of copulas, Banque de France, Noes d éudes e de recherche, 8 Sharpe (W.) (964): Capial asse prices: A heory of marke equilibrium under condiions of risk, Journal of Finance, 9, p. 45 Tse (Y. K.) and Tsui (A. K. C.) (00): A mulivariae GARCH model wih ime-varying correlaions, Journal of Business and Economic Saisics, 0(3), pp. 35-36 0 Banque de France Financial Sabiliy Review No. 4 June 004
Appendix Specificaion of dynamic condiional correlaions As an illusraion, le us ake wo dependen financial markes. On marke i, i=,, r j,, ε i,, m i and I, denoe he reurn, he random variable, he condiional expecaion and he informaion se available a ime respecively. For simpliciy s sake, le us assume ha he reurns follow a normal join disribuion, of dimension (bivariae) and wih a ime-dependen condiional variance-covariance marix H : ε, ( ε ) I ~ N(0,H )., For each i, i=,, r i, is generaed by an AR [] process. Therefore, for each, =,...,T (T being he oal number of observaions), he model is expressed as follows: r, =m +ϕ r, +ε, r, =m +ϕ r, +ε, ε, [] ( ) I ~ N(0,H ) avec H ( ) = h, h, ε h,, h, We shall now specify he equaions describing he elemens of he variance-covariance marix H (i.e. equaions describing he dynamics of he h i,, i=,, and of ρ,, respecively, he condiional correlaion and variances). H may be broken down ino a produc of marices: H =D R D where: D is a diagonal marix whose non-zero elemens are he square roos of he condiional variances (or volailiies) h i,, i=,; he definiion of D enables us o consider R as a marix of correlaions whose elemens of he main diagonal equal (if R =R, i.e. ρ, =ρ, we obain Bollerslev s consan condiional correlaion model, 987). More specifically, H is expressed as: [] = H h, 0 ρ, h, 0 ( ) ( ) ( ). 0 h ρ,, 0 h, h i,, i=,, are assumed o be described by GARCH (p, q) processes. If p = q =, we obain: h i, =α 0 +α i ε i, +β i h i, wih α 0 > 0, α i and β i 0 such ha α i e β i <. The condiional correlaions are described by an auoregressive process, he Dynamic condiional correlaions (DCC), originally developed by Engle and Sheppard (00) and subsequenly buil on by Tse and Tsui (00): P Q P Q [3] = ( θ θ )R + θ Ψ + θ R R i =,i j =,j i =,i i j =,j j where : R, he marix of he non-condiional correlaions calculaed over he period, is expressed as: = ρ, ρ, R ( ) ; Ψ is a marix whose elemens are empirical correlaions calculaed a on a window of given lengh m (,, 5,..., days): ',, = Q MMQ ; M =, ; h, h, 0,5 m ε, i m ε, i i, = ( εi, m,..., ε i, ); Q = diag, (Σi = i= h Σ, h ), Ψ ξ ξ ξ ( ) where diag denoes he operaor defining a diagonal marix. Engle and Sheppard showed ha if θ,i and θ,j 0 ( i, i P and j, j Q ) P Q Σ,i Σ, j < i = j = and θ + θ, he marix R is posiive a all poins in ime. As in he case of models, he sum of he parameers measures he degree of persisence of he correlaion. Commens The previous equaions of he condiional variances and correlaions define a DCC-GARCH (,) model. Banque de France Financial Sabiliy Review No. 4 June 004
In he bivariae case, he equaion describing he ime-varying correlaion marix can be reduced o an equaion explaining he variaion of he correlaion coefficien beween he wo markes. We can assume ha he join disribuion of reurns is non-normal and hus ake ino accoun he sylised facs (presence of fa ails and/or asymmery in he disribuion of reurns) of he financial markes. In heir sudies, Engle and Sheppard on he one hand, and Tse and Tsui on he oher assumed ha he join disribuion was normal. This assumpion is no realisic in he case of financial asse prices. Consequenly, in his sudy, we oped for a Pearson IV disribuion which enables us o verify boh he asymmery and he presence of exreme values. Tess carried ou ex pos show ha he adjusmen o his disribuion is of excellen qualiy. Banque de France Financial Sabiliy Review No. 4 June 004
Appendix Copula funcions Copula funcions have recenly been used in applied finance o obain greaer flexibiliy in mulivariae modelling (wider choice of join disribuions, larger diversiy of dependence funcions, increased choice of disribuion funcions, greaer ease of implemenaion, ec., see Nelsen, 998, or Avouyi-Dovi and Neo, 004). They make i possible o ake beer accoun of he evens observed on financial markes. We shall define hese funcions for he wo markes; he generalisaion o n markes is immediae. Le wo random variables X and X of disribuion funcions F e F, be defined by he vecor of parameers θ i, i=,. Le H be he join disribuion of X and X of he vecor of parameers θ H. The parameric copula of family Q, denoed C Q and of dependence parameer marix θ c, is a link funcion beween H and he marginal funcions F and F wih a value in he inerval [0,], defined by: [] H(X,X ;θ H )=C Q (F (X ;θ ),F (X ;θ );θ c ) According o Sklar s heorem if F and F are coninuous, hen he above decomposiion is unique. From equaion [], we derive an equivalen expression ha enables us o define he copula from he join disribuion (assuming ha u =F (X ;θ ) and u =F (X ;θ ) : [] C Q ( u, u ) [0,] (u, u ; θ c ) = H(F (u ; θ ), F (u ; θ ); θ By differeniaing H [] wih respec o each of he variables, we obain a relaionship beween he join densiy, h, (he derivaive of H) and he densiies C Q, (he derivaive of C Q ) and f i (i=,, he derivaives of funcions F i ). The join densiy funcion is herefore equal o he produc of he densiy funcions f i, i=,, and of a dependence funcion C Q, hus: H ) By definiion, c Q CQ (u,u ; θc ) = (u,u ; θc ) =, u u F i (X i; θi ) f i (X i; θi ) = X i H(X, X ; θh ) and h(x., X ; θh ) = X X This decomposiion of he join disribuion is paricularly appropriae: i enables us o carry ou an esimae in wo sages (known as he Inference Funcion for Margins approach, Joe, 997) which means ha we can solve, a leas in par, he problem of he number of unknown parameers. Moreover, i allows us o use a more general join disribuion because we are no longer limied by difficulies relaing o he analyical expression of his disribuion. We can herefore selec any disribuion funcions (provided ha hey are coninuous) combined wih a very general dependence srucure. In his empirical sudy, he marginal disribuions are Pearson IV disribuions, while he dependence funcion is a Suden copula ha allows dependence in ails (dependence beween rare evens of he same naure). Furhermore, he dependence parameer marix is he marix of he correlaions in his case. [3] c u, u ; θ ) f ( X ; θ ) f ( X ; θ ) = h( X, X ; θ ). Q ( c H Banque de France Financial Sabiliy Review No. 4 June 004 3
Appendix 3 The model: specificaions and esimaes As we noed in Appendix, copula funcions make i possible o separae margins and he dependence srucure corresponding o he join disribuion. More specifically, he wo sages (Inference Funcion of Margins) consis in firsly esimaing he parameers of he marginal funcions and hen hose of he dependence srucure, aking accoun of he parameers esimaed in he firs sage. Specificaion of he marginal funcions The reurns and condiional variances of he financial asses are modelled in such a way as o ake accoun of he sylised facs observed on he markes (presence of asymmery and fa disribuion ails, ec.). Asymmery is checked for in wo ways: differeniaing beween effecs of he shocks on variance by heir signs (using a Exponenial GARCH, EGARCH process); using an asymmeric disribuion. Rare evens are aken ino accoun using a fa ail disribuion. The Pearson IV disribuion (generalised Suden or gamma disribuions, for example) can be used o check for he presence of fa ails. This disribuion was used here due o he resuls in recen lieraure. Expeced reurns are deermined auoregressively and he variance of errors is modelled by a radiional : r i, = mi + ϕiri, + / β <, i =,,3 ε i, ε i, = ( hi, ) ηi, [ ] ln hi, = α 0 + β ln( hi, ) + γηi, + α ηi, π η i, ~ P IV (.; a, q, δ, σ ) Where: ϕ i (i=,,3) are he auoregressive coefficiens; β is he parameer ha measures he persisence effec of he variance; he influence of he posiive (negaive) shocks (represened by η i, ), on he variance is measured by γ+α (resp. γ α ), which represens he coefficien of he η i,. X denoes he absolue value of X and P IV (.;a,q,δ,σ) denoes he Pearson IV disribuion. Is reduced cenred densiy is expressed as: f ( η ;a,q, δ, σ) = κ IV [ i, σ h i, [ σ µ + ηi, + a a ] ( ) ] + q exp δan σ µ ηi, + a a where κ is a normalisaion consan π κ = π q δa a (q + δ ) a cos ( ω) exp( δω)dω, µ = and σ =. q q (q ) Commens Pearson IV has a parameer, δ, ha checks for asymmery and a parameer q ha provides indicaions of he hickness of disribuion ails; if δ=0, he disribuion is symmeric; if, furhermore, q, hen f IV (.;a,q,0,σ) N(0,) ; parameer a, which appears in he normalisaion consan, may have o be fixed a (due o difficulies in idenifying he parameers of he model). In fac, we can eiher fix he consan m i a zero or fix he parameer a a (log(a)=0). We oped for he former soluion; examples of Pearson IV densiy funcions defined wih respec o he values of he parameers q and δ (see Char) are given here in order o visualise he properies of his disribuion: PIV(δ =,5, q = 4) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. PIV(δ =,5, q = 4) PIV(δ = 0, q = 4) N(0,) PIV(δ = 0, q = 00) - 5-4 -3 - - 0 3 4 5 4 Banque de France Financial Sabiliy Review No. 4 June 004
Specificaion of he dependence srucure Recen empirical sudies (see Longin and Solnik 998, for example) have shown ha some financial markes, equiy markes for example, were characerised by a dependence beween exreme evens (or disribuion ails). We hus ake ino accoun he fac ha bubbles (posiive ails) or, inversely, crashes (negaive ails) observed on he differen markes are inerrelaed. A copula funcion ha allows dependence beween rare evens such as he Suden funcion (bu no he Gaussian funcion) is herefore appropriae. Furhermore, he dependence parameer marix of he Suden copula (as well as ha of he Gaussian funcion) is inerpreed as a correlaion marix. Consequenly, i seems appropriae o choose he laer for analysing he inerdependence of markes as i mees he crieria for boh he dependence in ails and he inerpreaion of he dependence parameer marix. In he case of wo markes, he Suden funcion akes he following sandard form (see Avouyi-Dovi and Neo (003) for a more general expression of his funcion): c S ν ν = s( ( u), ( u ); R, ν ) where: s( ν ( u), ν ( u ); R, ν ) denoes he densiy funcion of he Suden copula, R and ν represen he dependence parameer marix and he degree of freedom (when ν, hen he Suden copula converges owards a Gaussian copula); he inverse ν of he univariae Suden disribuion funcion and lasly u and u he disribuion funcions (empirical and heoreical) and he marginal funcions (here Pearson IV disribuions). The ime-varying correlaion marix is described by an Engle and Sheppard DCC compleed by Tse and Tsui (see Appendix ). In he applicaions presened here, we have used a DCC model wih an auoregressive process of order (Q=) and an empirical correlaion marix lag (P=). R is hus expressed as: R = ( θ θ )R + θψ + θr. The empirical correlaions are calculaed on a window wih a lengh of five working days (m=5). The esimae of he model is made in wo sages: esimae of he parameers of Par, followed by he esimae of he parameers of he dependence funcion aking ino accoun he resuls of he firs sage. We can check he consancy of condiional correlaions using he probabiliy raio es. The null hypohesis H 0 is defined by H 0 : H 0 : θ = θ = 0 R = R. Under H 0, es saisic (W, see able below) follows a Chi disribuion wih degrees of freedom. Esimae of he EGARCH (,) model wih a Pearson disribuion (a) ϕ i α 0 α β γ a q δ DJ c 0.0790 (0.0996) 0.09068 (0.05833) 0.938 (0.067) 0.97989 (0.00557) 0.6900 (0.05575) 0.8659 (0.5037) 7.5396 (.5960).8037 (0.545) CAC o 0.0507 (0.0330) 0.03599 (0.09894) 0.7430 (0.06808) 0.974 (0.0076) 0.074 (0.075).37343 (.37) 9.3660 (.37405).53695 (0.9643) DAX 0.05854 (0.0009) 0.089 (0.04358) 0.7036 (0.04540) 0.98598 (0.0043) 0.0663 (0.087).786 (.465).09497 (3.566).4463 (.4878) (a) (o) opening and (c) closing. The figures in brackes represen he sandard deviaions. Esimae of he Suden dependence srucure (a) m = 5 ρ CAC-DJ ρ DAX-CAC ρ DJ-DAX θ θ ν Moy(ρCAC-DJ.) Moy(ρDAX-CAC.) Moy(ρDJ-DAX.) o DJ c o, CAC, DAX 0.64606 (0.085) 0.77978 (0.048) 0.60997 (0.076) 0.0638 (0.00539) 0.93475 (0.059) 9.5637 (3.3547) 0.5933 0.7498 0.5664 W 79.8 P-Value 0.000 (a) Esimaes are obained using he IFM mehod (Inference Funcion for Margins) described in Joe (997). The values of he elemens of he correlaion marix associaed wih he consan correlaion assumpion H 0, (elemens ρ i, j, i j, i, j=cac, DAX, Dow Jones of marix R, see Appendix 3), are relaively srong (all greaer han 0.50) and significanly differen from zero. More specifically, hey are all greaer han or Banque de France Financial Sabiliy Review No. 4 June 004 5
equal o he averages of he ime-varying correlaions. If we had chosen consan correlaion marices, and did no rejec H 0, we would have obained de faco higher coefficiens and overesimaed, on average, he degree of correlaion beween markes. However, he conribuion of his marix (corresponding o H 0 ) o ime-varying correlaions is modes due o he weakness of is coefficien in he equaion describing R ( θ θ = 0.04). The commens regarding he similariy beween he averages of non-condiional correlaion coefficiens and hose of he condiional correlaion coefficiens confirmed he need o es he consan correlaion assumpion H 0. 8 The resuls of his es presened in he Appendix (see he saisic W and P -value) enable us o rejec H 0 a he % hreshold. 8 As consan correlaion models and ime-varying correlaion models are overlapping, we can use a probabiliy raio es o check H 0 (see Avouyi-Dovi and Neo, 003). 6 Banque de France Financial Sabiliy Review No. 4 June 004