Modelling he dependence of he UK sock marke on he US sock marke: A need for muliple regimes A J Khadaroo Deparmen of Economics and Saisics Universiy of Mauriius Redui Mauriius Email: j.khadaroo@uom.ac.mu Absrac Through he use of regime-swiching models, recen empirical research has essenially shown ha he dynamics of sock reurns depend on he sae of one sock marke. The presen paper exends his analyical framework by allowing he dynamics of reurns o depend on he join-saes of wo differen sock markes. Such an exension is naural given he globalisaion of financial markes and he rapid ransmission of news from one inernaional sock marke o anoher. In an applicaion involving he S&P500, he FTSE00 and he NIKKEI5 over he period January 984 Ocober 003, UK sock reurns are found o depend on he join-saes of he US and UK sock markes hree monhs back. Moreover he conemporaneous dependence of UK sock reurns on US sock reurns increases wih a rising US marke and a falling UK marke bu decreases wih a falling US marke and a rising UK marke. This is consisen wih a rappor de force effec whereby he relaive srenghs of he US and UK sock markes maer in deermining he degree of conemporaneous dependence of he UK sock marke on he US sock marke. Keywords: Sock Reurns, Muliple Regimes, Smooh Transiion I hank wo anonymous referees for useful commens. Any remaining error is my own responsibiliy.
. Inroducion Finance heory has long recognised he relevance of regime-swiching behaviour in explaining he dynamics of sock markes. For insance, noise rading and ransacions coss in financial markes make arbirage aciviy profiable only when he magniude of he deviaion from equilibrium is large enough o warran adjusmen. Differing sock marke dynamics for small and large disequilibria implies nonlinear reversion owards equilibrium. McMillan (00) finds ha an exponenial smooh ransiion regression (STR) equaion, which is consisen wih he exisence of ransacions coss and noise rading, produces improved in-sample performance and marginally superior forecas resuls compared o he corresponding single-regime linear equaion. McMillan (003) also finds ha an exponenial STR equaion for he FT-All index ouperforms a logisic STR equaion boh in-sample and ou-of-sample. A differen and more general explanaion for regime-swiching sock marke dynamics relaes o marke senimen, such ha he reurns process differs for bear and bull markes. This reasoning flows from he business cycle lieraure which documens ha he dynamics of oupu and unemploymen during expansions and conracions are differen [see Terasvira and Anderson (99), Beaudry and Koop (993), Peel and Speigh (998), Skalin and Terasvira (00)]. Such general nonlineariy implies ha he speed of adjusmen owards he equilibrium sock reurn is differen for reurns above equilibrium and reurns below equilibrium. Saranis (00) shows ha he sock reurns
of five ou of he G7 economies are beer described by logisic STR equaions, consisen wih he idea of differing dynamics in bear and bull markes. He also finds ha he logisic STR equaions provide superior ou-of-sample forecass o he corresponding single-regime linear equaions. Aslanidis, Osborn and Sensier (003) esimae a wofuncion logisic STR equaion indicaing ha UK sock reurn dynamics depend on pas changes in he dividend yield, wih conemporaneous US sock reurns providing a second nonlinear influence. The above hrows ligh on he heoreical as well as empirical significance of regimeswiching in sock marke dynamics. However i is noed ha he empirical analysis has so far resriced sock dynamics o depend on he regime prevailing in only one sock marke. The presen paper proposes o invesigae FTSE00 dynamics in a mulipleregime framework ha essenially allows sock reurn dynamics o depend on he joinregimes prevailing in wo differen sock markes. Such an exension is raher naural and worh invesigaing given he globalisaion of inernaional financial markes and he rapid ransmission of news from one inernaional sock marke o anoher. The STR framework, unlike he Markov-swiching framework, allows for explici ransiion variables and for he ransiion beween regimes o be smooh as well as abrup. The modeling approach here sars from a single-regime framework and expands owards a muliple-regime framework. Secion describes he daa used in his paper. Secion 3 esimaes a parsimonious single-regime linear equaion for FTSE00 reurns. Secion 4 describes and esimaes a wo-regime equaion whereby reurns depend on he sae of 3
one sock marke. Secion 5 proposes, moivaes and esimaes a muliple-regime equaion whereby he regimes in FTSE00 reurns are characerised by he saes of wo differen sock markes. Secion 6 concludes.. The Daa The raw daa consis of 38 monhly observaions on he FTSE00 (FT), he NIKKEI5 (NK) and he S&P500 (SP) indices over he period January 984 Ocober 003. Monhly reurns for he hree indices were hen compued. Table implies ha based on he sandard deviaion of reurns, he SP is he leas risky while he NK is he mos risky of he hree porfolios. Moreover, he SP has he highes average reurn of 0.773% while he NK has he lowes average reurn of 0.06%. This indicaes ha he efficien fronier of he SP, as well as he efficien fronier of he FT, in principle lies above he efficien fronier of he NK. The skewness and kurosis saisics reveal ha all hree porfolio reurns are negaively skewed and lepokuric, as shown in Figure. The correlaion marix in Table also implies ha US and UK reurns are more srongly relaed o each oher han o Japanese reurns. 3. Single-Regime Analysis A parsimonious single-regime equaion for FT reurns is firs esimaed. Since he ADF es showed ha he hree sock indices have a uni roo in levels bu no in firsdifference, such ha hey are I(), he Johansen (988) procedure was applied o invesigae he presence of coinegraion. Allowing for lag orders of, and 3 in a vecor The monhly reurn for a given sock index is compued as 00*[ln(P /P - )], where P is he average of he daily closing prices in monh. 4
error-correcion model (VECM) of FT, NK and SP reurns, he evidence clearly suppored he absence of coinegraion beween he hree indices a he 0% significance level. 3 This reveals ha inernaional porfolio diversificaion in hese hree indices is poenially beneficial, especially o long-erm invesors. In he absence of coinegraion, he VECM reduces o a vecor auoregression (VAR) in firs-difference and he singleregime economeric specificaion for FT reurns urns ou o be an auoregressive disribued lag (ARDL) equaion. The adoped ARDL equaion for FT reurns ( y ) here is: y x () / where NID(0 ) x y y z z / ( k m ) is a vecor of I(0) variables p / ( 0 ) is a parameer vecor, wih p k m The iniial ARDL equaion () for FT reurns conains a consan, lagged FT reurns, and conemporaneous and lagged SP and NK reurns. The inclusion of he conemporaneous SP and NK reurns in he ARDL equaion for FT reurns is o accoun for he rapid informaion ransmission beween inernaional sock markes. The coefficiens on he reurn variables SP and NK hus reflec he conemporaneous dependence of FT reurns on SP and NK reurns. The higher he magniude of hese coefficiens he higher is such dependence. 3 To save on space, he ADF and Johansen coinegraion resuls are no included bu are available upon reques. The VECM conains unresriced inerceps bu no ime rends, since he reurns series are no rended. Experimenaion showed ha higher lag orders in he VECM confirmed more srongly he absence of coinegraion. 5
A general-o-specific mehodology is applied o find a parsimonious linear equaion for FT reurns. Beginning wih he iniial general ARDL equaion (), successive resriced ARDL equaions are esimaed by excluding he mos insignifican variable each ime, based on he -raio. 4 The AIC, BIC, adjused R, join significance, residual auocorrelaion and ARCH are also moniored hroughou his general-o-specific exercise. A parsimonious ARDL equaion, where all righ-hand-side variables are individually significan, is hus obained. The resuling ARDL equaion, which appears in Table, explains 57% of he variaion in he FT reurns and passes auocorrelaion ess of order 3. The equaion also passes condiional heeroscedasiciy ess of order and bu no of order 3. The ARDL residuals are highly non-normal. I is ineresing o noe ha he conemporaneous SP reurns have a much sronger effec han he conemporaneous NK reurns on FT reurns. 4. Two-Regime Analysis In line wih recen evidence poining o he exisence of wo regimes in he dynamics of sock reurns, a wo-regime smooh ransiion regression (STR) equaion for FT reurns is here considered. 4 A variable wih -raio of less han.6 in magniude is considered insignifican. The consan is mainained in he ARDL equaion. 6
y x x F ( s c ) () / / where NID (0 v) / / / x ( y y z z ) ( x ) is a vecor of I(0) variables k m p / ( 0 ) and p / ( 0 ), where p k m, are parameer vecors F ( s c ) is a ransiion funcion bounded beween zero and one s is a ransiion variable is a posiive ransiion parameer and c is a locaion parameer Unlike hreshold and Markov-swiching models, he STR model can accommodae a nonabrup and smooh ransiion from one regime o anoher and is hus preferred here. A smooh ransiion beween regimes is relevan in he case of heerogeneous raders who do no all reac o news a one and he same ime. An addiional reason for preferring he STR model o he Markov-swiching model here is ha he variable governing he ransiion beween regimes is observable in he former bu no in he laer. I should be noed ha, since he invesigaion for wo regimes sars from he parsimonious ARDL equaion appearing in Table, some elemens of and in he STR equaion () have a pre-assigned value of zero. Equaion () becomes he logisic STR (LSTR) model when: F ( s c ) [ exp{ ( s c )}] 0 (3) The logisic ransiion funcion in (3) is monoonically increasing in he ransiion variable s. The value of, he ransiion parameer, indicaes he smoohness of 7
ransiion from he lower regime o he upper regime and vice-versa. When, he logisic funcion urns ino a sep funcion and equaion () becomes a hreshold regression implying an abrup regime swich a s c. Equaion () becomes he exponenial STR (ESTR) model when: F ( s c ) [ exp{ ( s c ) }] 0 (4) The exponenial ransiion funcion in (4) is symmeric abou s c. The value of here indicaes he smoohness of ransiion beween he inner regime and he ouer regimes. When, he exponenial funcion in (4) ends oward a linear funcion, hereby making i hard o disinguish beween an ESTR equaion and a single-regime linear equaion in case of high-speed ransiion beween he inner and ouer regimes. Wih a lagged sock reurn as he ransiion variable, a STR equaion allows for differen ypes of marke behaviour depending on he naure of he ransiion funcion. The LSTR equaion is consisen wih invesor behaviour depending on he sign of reurns and hus he direcion of he marke. The lower and upper regimes of he LSTR equaion capure reurn dynamics in bear and bull markes respecively. The ESTR equaion is consisen wih invesor behaviour depending on he size of reurns regardless of sign and is moivaed by marke fricions such as ransacions coss and noise rader risk. The parsimonious ARDL equaion appearing in Table is evaluaed agains he woregime STR equaion () using a hird-order Taylor approximaion o he ransiion 8
funcion F abou 0, in line wih Luukknonen, Saikkonen and Terasvira (988). This procedure leads o he auxiliary equaion: y x ( x s ) ( x s ) ( x s ) w (5) / / / 3 / 0 3 where w is a composie error process i (i = 0,,, 3) are parameer vecors y, x, x and s are as defined in equaion () The null hypohesis of a single-regime is expressed as: H : 0 (6) 0 3 Granger and Terasvira (993, chaper 7) propose a convenional F es o invesigae he above null hypohesis. The F es is performed wih a range of poenial ransiion variables and he seleced ransiion variable is he one which, subjec o a given significance level, produces he minimum p-value for he F es (as recommended by Terasvira, 994). The following sequence of null hypoheses is applied o discriminae beween he LSTR and ESTR models: H H H 0 03 3 0 0 0 3 0 0 0 3 (7) If he p-value for H 0 is considerably smaller han he p-values for H 0 and H 03 ESTR model is seleced. On he oher hand, if he p-values for H 0 and H 03 are considerably smaller han he p-value for H 0, an LSTR model is seleced. When he p- value crierion does no clearly poin o eiher he ESTR or he LSTR specificaion bu, an 9
he p-values are significan, boh specificaions could be esimaed and he selecion made afer model evaluaion. Equaion (5) is esimaed by ordinary leas squares for all poenial ransiion variables. The conemporaneous SP and NK reurns as well as he SP, FT and NK reurns lagged up o hree monhs are allowed o consiue he poenial ransiion variables in he woregime STR equaion (). Table shows ha, a he 5% significance level, here is ample ground for a second regime in FT reurns. The seleced ransiion variable, based on he minimum p-value crierion, is he SP reurns hree monhs ago wih a p-value of 0.4%. Moreover he very low p-value of 0.% for he H 0 hypohesis compared o a highly insignifican p-value of 3% for he H 0 hypohesis clearly suppors he LSTR as opposed o he ESTR specificaion for FT reurns. Given he presence of severe condiional heeroscedasiciy of order 3 in he residuals of he parsimonious ARDL equaion as shown in Table, his paper also carries ou he heeroscedasiciy-robus nonlineariy es in line wih procedure 3. of Wooldridge (99) for he seleced ransiion variable (here SP reurns hree monhs ago) and seleced STR specificaion (here LSTR) so as o ascerain ha he rejecion of lineariy is no spurious. The mehodology of he heeroscedasiciy-robus es for omied nonlineariy in he single-regime ARDL equaion is provided in Appendix. The p-value for he H 0 hypohesis now increases o 4.%, up from 0.% obained in he case of he non-robus es. The heeroscedasiciy-robus p-value of less han 5% implies ha he finding of 0
LSTR nonlineariy in FT reurns, wih SP reurns hree monhs ago being he ransiion variable, is no spurious. A wo-regime LSTR equaion () is esimaed by nonlinear leas squares, wih saring values based on a -dimensional grid search for and c. 5 For given values of and c, he LSTR equaion becomes linear and may be consisenly esimaed by condiional leas squares. The grid search applies his echnique o find an inerior-soluion combinaion of and c which minimises he residual sum of squares of he linearised LSTR equaion. The locaion parameer c is allowed o vary beween he 0 h and 90 h perceniles of SP reurns a quarer ago. An inerior-soluion combinaion of he parameers acs as he saring values for he nonlinear esimaion of he LSTR equaion. The Newon algorihm leads o a convergen soluion shown in Table 3. The esimaed value of is.5 implying a smooh ransiion beween he lower and upper regimes. The esimaed locaion parameer c implies ha he regime-swich in he dynamics of FT reurns occurs a SP monhly reurns of.56% a quarer ago. The esimaed LSTR equaion is evaluaed agains he parsimonious ARDL equaion on he basis of he adjused R, variance raio, residual auocorrelaion, residual ARCH and residual normaliy. 6 Table 3 shows ha he LSTR equaion explains 60% of he variaion in FT reurns and reduces he ARDL residual variance by 0%. The LSTR residuals are 5 In line wih usual pracice, he LSTR ransiion variable is normalised by is sandard deviaion o make scale-free. 6 The variance raio is a raio of he residual variance of an unresriced equaion o ha of a resriced equaion.
sill non-normal bu much less han he ARDL residuals. There is no evidence of auocorrelaion of order 3. Like he ARDL equaion, he LSTR equaion passes condiional heeroscedasiciy ess of order and bu no of order 3. The finding of SP reurns being he mos significan ransiion variable governing FT reurn dynamics in he LSTR equaion reveals he crucial role of he sae of he US sock marke in deermining he dynamics of he UK sock marke. An imporan implicaion of he globalisaion of financial markes is ha he dynamics of reurns in a given sock marke could be governed by he saes of more han one sock marke. This paper hus proceeds o invesigae he usefulness of he sae of eiher he UK or Japanese sock marke as a second influence on FT reurn dynamics in a muliple-regime STR framework conaining wo LSTR funcions. 5. Muliple-Regime Analysis The proposed muliple-regime STR (MRSTR) equaion for FT reurns ( y ) is: y x x F ( s c ) x F ( s c ) u (8) / / / where u NID (0 u) / / / x ( y y z z ) ( x ) is a vecor of I(0) variables k m p / ( 0 ), p / ( 0 ), and p / ( 0 ) are parameer vecors F ( s c ) and F ( s c ) are LSTR ransiion funcions s and s are ransiion variables from wo differen sock markes and are posiive ransiion parameers
Marke Regime c and c are locaion parameers The MRSTR is here a wo-funcion LSTR equaion ha enables he dynamics of FT reurns o depend on he saes of wo differen sock markes. As explained above, such an exension o he single-funcion LSTR equaion is here desirable and worh invesigaing in ligh of he globalisaion of financial markes. The MRSTR equaion leads o four join-marke regimes as follows. Join-Marke Regimes Marke Regime Bear ( F 0 ) Bear ( F 0 ) F 0 F 0 Bull ( F ) F 0 F Bull ( F ) F F 0 F F A single sock marke i is bullish when he LSTR ransiion funcion F i is in he upper regime, corresponding o highly posiive reurns. On he oher hand, a single sock marke i is bearish when he LSTR ransiion funcion F i is in he lower regime, ha is, when reurns are highly negaive. A join-marke regime is joinly defined by he regimes prevailing in wo differen sock markes and is here used o assess wheher he dynamics 3
of FT reurns are beer described by he saes of wo, raher han jus one, sock markes. The presen MRSTR framework is able o capure he effecs of converging (bear-bear and bull-bull) as well as diverging (bear-bull and bull-bear) inernaional marke senimens on he dynamics of FT reurns. I should be noed ha, based on he LSTR equaion, he ransiion variable s in he MRSTR equaion is already known o be he SP reurns lagged hree monhs. To invesigae he imporance of he sae of eiher he Japanese or UK sock marke as a second influence on he dynamics of FT reurns, as posulaed by he muliple-regime MRSTR equaion (8), he following equaion esing for remaining nonlineariy in he LSTR residuals (see van Dijk and Franses, 999) is employed: / y / v ( x s ) z (9) where z NID (0 z) v is he residual from LSTR equaion () y is he esimaed gradien vecor from LSTR equaion () / (,,, c ) concaenaes he esimaed parameers in LSTR equaion () x is as defined in equaion () and s is as defined in equaion (8) and are parameer vecors 4
Remaining nonlineariy in he wo-regime LSTR equaion () is presen when he null hypohesis H : 0 0 is rejeced, using he F es. Equaion (9) is esimaed by ordinary leas squares several imes o uncover he poenial exisence of a second ransiion variable s. Given ha SP reurns lagged hree monhs already consiues he firs ransiion variable in he MRSTR equaion, he search for a second ransiion variable s is limied o NK and FT reurns in order o be consisen wih he noion of join-marke regimes as defined above. The seleced second ransiion variable is he one which, subjec o a given significance level, leads o he minimum p-value for he remaining nonlineariy es. Table 3 shows ha FT reurns lagged hree monhs is a very significan second ransiion variable [p-value = 0.03%] in he MRSTR equaion. 7 Given he presence of severe condiional heeroscedasiciy of order 3 in he residuals of he wo-regime LSTR equaion as shown in Table 3, his paper also performs he heeroscedasiciy-robus nonlineariy es in line wih procedure 3. of Wooldridge (99) for he seleced ransiion variables (here SP reurns and FT reurns hree monhs ago) so as o ascerain ha he rejecion of addiional lineariy is no spurious. The mehodology of he heeroscedasiciy-robus es for omied nonlineariy in he woregime LSTR equaion is provided in Appendix. The p-value for he null hypohesis now increases o 3.0%, up from 0.03% obained in he case of he non-robus es. The heeroscedasiciy-robus p-value of less han 5% implies ha he finding of MRSTR nonlineariy in FT reurns, wih SP reurns and FT reurns hree monhs ago being he 7 Ineresingly, he saisical ess revealed ha he conemporaneous and lagged SP reurns do no consiue a significan second ransiion variable, hus reinforcing he noion of join-marke regimes. 5
ransiion variables, is no spurious. Thus UK sock reurn dynamics do depend on he recen join saes of he US and UK sock markes. The muliple-regime MRSTR equaion (8) is esimaed by nonlinear leas squares, wih saring values based on a 4-dimensional grid search for, c, and c. 8 For given values of, c, and c, he MRSTR equaion becomes linear and may be consisenly esimaed by condiional leas squares. The grid search uses his approach o find an inerior-soluion combinaion of, c, and c which minimises he residual sum of squares of he linearised MRSTR equaion. The locaion parameers c and c in he MRSTR equaion are allowed o vary beween he 0 h and 80 h perceniles of SP reurns lagged hree monhs and FT reurns lagged hree monhs respecively. An ineriorsoluion combinaion of he parameers is obained and acs as he saring values for he nonlinear esimaion of he MRSTR equaion. The Broyden, Flecher, Goldfarb and Shanno (BFGS) algorihm leads o a convergen soluion shown in Table 4. 9 The ransiion funcion plos in Figure 4 and Figure 5 demonsrae ha curren FT reurn dynamics swich rapidly around a SP reurn of.% and a FT reurn of -0.8%, boh lagged hree monhs. The MRSTR equaion explains 63% of he variaion in FT reurns and reduces he LSTR residual variance by 0%. The MRSTR residuals are non-normal bu o a lesser exen 8 In he esimaion process, he MRSTR ransiion variables are normalised by heir respecive sandard deviaion o make and scale-free. 9 The high sandard errors for and arise because when ransiion parameers are large, a large number of observaions around he hreshold poins are needed for precise esimaion. This argumen is well documened in he smooh ransiion lieraure (see van Dijk, Terasvira and Franses, 00). 6
han he LSTR residuals. There is no evidence of auocorrelaion of order 3 a he 5% significance level. The MRSTR equaion passes condiional heeroscedasiciy ess of order and bu no of order 3. Imporanly he MRSTR equaion displays no evidence of significan remaining nonlineariy, suggesing ha FT reurn dynamics are here adequaely capured by he join-saes of he US and UK sock markes and are no dependen on he sae of he Japanese sock marke. Table 4 also conains esimaes of he conemporaneous dependence of FT reurns on SP reurns in he four regimes of he MRSTR equaion. The conemporaneous dependence of he UK sock marke on he US sock marke is sronger when boh markes were bearish (0.85) as compared o bullish (0.69) hree monhs ago. This is in line wih Longin and Solnik (00) who find ha he correlaion beween inernaional sock markes is sronger in bear markes han in bull markes. The conemporaneous dependence esimaes in Table 4 reveal a novel finding. Saring from a bear-bear join-marke regime, he UK dependence increases when he US marke urns bull bu decreases when he UK marke urns bull. Saring from a bull-bull joinmarke regime, he UK dependence decreases when he US marke urns bear bu increases when he UK marke urns bear. The implicaion of such a regime-specific dependence paern is ha he dependence of he UK sock marke on he US sock marke increases wih a rising US marke and a falling UK marke bu decreases wih a falling US marke and a rising UK marke. This finding, which ineresingly seems inuiive, is consisen wih a rappor de force effec whereby he relaive srengh of he 7
US and UK sock markes maer in deermining he degree of conemporaneous dependence of he UK sock marke on he US sock marke. 5. Conclusion The recen widespread applicaion of regime-swiching models o he leading inernaional sock markes has indeed improved our undersanding of he dynamics of sock reurns in he presence of changing marke senimens and marke fricions like noise rading and ransacions coss. However such analysis has ypically resriced sock reurn dynamics o be deermined by he sae of a single sock marke. In ligh of he globalisaion of inernaional financial markes and he rapid ransmission of news from one inernaional sock marke o anoher, he presen paper has proposed a mulipleregime framework ha enables sock reurn dynamics o be deermined by he saes of wo differen sock markes. An applicaion suppors he exisence of four regimes governing UK sock reurns, where each regime is characerised by he join-saes of he US and UK sock markes hree monhs ago. The esimaed muliple-regime equaion exhibis regime-specific dependence and reveals ha he conemporaneous dependence of he UK sock marke on he US sock marke increases wih a rising US marke and a falling UK marke bu decreases wih a falling US marke and a rising UK marke. This indicaes ha a rappor de force effec operaes in deermining he degree of conemporaneous dependence of he UK sock marke on he US sock marke. 8
Fuure research may exend he daa period and invesigae he effecs of he prevailing economic and financial crisis, which originaed in he US sub-prime morgage marke in Augus 007, on he findings of his paper. 9
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van Dijk, D, Tearsvira, T and Franses, P H (00). Smooh ransiion auoregressive models: A survey of recen developmens. Economeric Reviews,, p.-47
Table : Descripive Saisics Sample period: Feb 84 Oc 03 FT NK SP Mean 0.596 0.06 0.773 Sandard Deviaion 3.99 4.98 3.56 Minimum -4.5-7.86-3.4 Maximum 9.99 4. 0.7 Skewness -.0-0.37-0.8 Excess Kurosis 5.67 0.87.53 Reurns Correlaion Marix FT NK SP FT.000 NK 0.44.000 SP 0.73 0.447.000 Noe: - The monhly reurns are in percen
Table : Parsimonious ARDL equaion (Single Regime) Variable Coefficien Sandard Error -raio Inercep -0.08 0.838-0.5889 NK 0.0795 0.039.0337 SP 0.766 0.0559 3.643 FT - -0.09 0.0636-3.7 SP - 0.089 0.053.7037 SP - 0.544 0.075.85 Adjused R = 0.567 Jarque-Bera LM es saisic = 38.75 [Criical chi-sq a 5% significance level = 5.99] AIC = 0.88, BIC = 0.89 LM es for Auocorrelaion: AR() AR() AR(3) p-value 0.457 0.63 0.77 LM es for Condiional Heeroscedasiciy: ARCH() ARCH() ARCH(3) p-value 0.74 0..478 0.000 Tesing for wo regimes FTSE00 reurn dynamics (in order of H 0 significance): Transiion Variable p-values [Null hypoheses in equaions (6) and (7)] (S ) H H 0 H 03 H 0 0 SP -3 0.0044 0.585 0.3986 0.0005 SP - 0.037 0.40646 0.57 0.00658 NK -3 0.039 0.4007 0.0458 0.03858 NK 0.05 0.875 0.035 0.0487 FT - 0.04644 0.5389 0.03807 0.0889 SP 0.06986 0.7745 0.6764 0.000 FT -3 0.808 0.5936 0.6657 0.0056 SP - 0.4989 0.647 0.9868 0.568 FT - 0.47354 0.34637 0.999 0.06 NK - 0.674 0.658 0.930 0.886 NK - 0.7788 0.405 0.49349 0.9546 3
Table 3: LSTR equaion (Two Regimes) Variable Coefficien Sandard Error -raio Inercep -0.77 0.8330-0.60806 NK 0.53 0.05809.9854 SP 0.69963 0.0796 8.8759 FT - -0.0 0.0890 -.585 SP - -0.0869 0.066-0.87 SP - 0.08497 0.007 0.84744 Inercep* F 0.39074 0.63034 0.6989 NK * F -0.4577 0.944 -.048 SP * F 0.3870 0.9689 0.70446 FT - * F -0.437 0.0774 -.704 SP - * F 0.6740 0.545.6379 SP - * F 0.06056 0.636 0.97 F [ exp{.5 *( SP.56) / }] 3 SP (.684) (.475) 3 Adjused R = 0.599 Variance raio beween LSTR and ARDL equaions = 0.90 Jarque-Bera LM es saisic = 9.69 [Criical chi-sq a 5% significance level = 5.99] LM es for Auocorrelaion: AR() AR() AR(3) p-value 0.7 0.76 0.00 LM es for Condiional Heeroscedasiciy: ARCH() ARCH() ARCH(3) p-value 0.56 0.64 0.000 Tesing for muliple regimes in FTSE00 reurn dynamics (in order of significance): Transiion Variable (S ) p-value for F es [refer o equaion (9)] FT -3 0.00033 NK 0.84 FT - 0.0000 NK -3 0.39785 FT - 0.449 NK - 0.46873 NK - 0.98870 4
Table 4: MRSTR equaion (Muliple Regimes) Variable Coefficien Sandard Error -raio Inercep -0.564 0.30496-0.59 NK 0.06684 0.06700 0.99754 SP 0.846 0.0887 0.3358 FT - -0.33 0.09060 -.3607 SP - 0.0407 0.070 0.55780 SP - 0.083 0.0745.87840 Inercep* F 0.58973 0.4633.3837 NK * F -0.7737 0.0890 -.395 SP * F 0.0434 0.347.65496 FT - * F -0.776 0.30 -.64944 SP - * F 0.4099 0.60 3.6859 SP - * F 0.444 0.653.49476 Inercep* F -0.9089 0.38767-0.494 NK * F 0.546 0.0878.75556 SP * F -0.3643 0.447 -.9678 FT - * F 0.07856 0.988 0.60484 SP - * F -0.6 0.055 -.5366 SP - * F -0.3300 0.4686 -.5383 F [ exp{ 34 *( SP.09) / }] 3 SP (43.73) (0.30) 3 F [ exp{ 0 *( FT 0.79) / }] 3 FT (50.67) (0.044) 3 Adjused R = 0.634 Variance raio beween MRSTR and LSTR equaions = 0.889 Jarque-Bera LM es saisic =.55 [Criical chi-sq a 5% significance level = 5.99] LM es for Auocorrelaion: AR() AR() AR(3) p-value 0.47 0.77 0.067 LM es for Condiional Heeroscedasiciy: ARCH() ARCH() ARCH(3) p-value 0.588 0.549 0.000 5
US Marke Regime (hree monhs back) Table 4 (coninued) Tesing for remaining nonlineariy in FTSE00 MRSTR equaion (in order of significance): Transiion Variable (S 3 ) p-value for F es NK -3 0.595 FT - 0.7563 SP -3 0.83 SP 0.37 SP - 0.339 FT -3 0.30069 NK 0.38947 SP - 0.46893 FT - 0.59848 NK - 0.85446 NK - 0.98 Conemporaneous dependence of FTSE00 reurns on S&P500 reurns under differen US-UK join-marke regimes (Sample frequency in brackes): UK Marke Regime (hree monhs back) Bear 0.85 Bull 0.48 Bear (64) (63) Bull.05 (4) 0.69 (93) 6
984 985 986 987 988 989 990 99 99 993 994 995 996 997 998 999 000 00 00 003 984 985 986 987 988 989 990 99 99 993 994 995 996 997 998 999 000 00 00 003 984 985 986 987 988 989 990 99 99 993 994 995 996 997 998 999 000 00 00 003 Figure : Monhly Reurns on FTSE00, NIKKEI5 and S&P500 Feb 84 Oc 03 5 Monhly Reurns on FTSE00 0 5 0-5 -0-5 -0-5 5 Monhly Reurns on NIKKEI5 0 5 0-5 -0-5 -0-5 5 Monhly Reurns on S&P500 0 5 0-5 -0-5 -0-5 7
Frequency Frequency Frequency Figure : Hisograms for Monhly Reurns on FTSE00, NIKKEI5 and S&P500 0.5 Hisogram for reurn on FTSE00 (wih Normal Curve superimposed) 0.0 0.05 0.00-3.0-8.46-3.9-9.356-4.805-0.543 4.97 8.847 3.4 0.0 Hisogram for reurn on NIKKEI5 (wih Normal Curve superimposed) 0.08 0.06 0.04 0.0 0.00-8.93-4.65-0.37-6.097 -.8.457 6.734.0 5.9 9.57 0.5 Hisogram for reurn on S&P500 (wih Normal Curve superimposed) 0.0 0.05 0.00-4.3 -.0-7.794-4.577 -.359.858 5.075 8.9.5 4.73 8
984 985 986 987 988 989 990 99 99 993 994 995 996 997 998 999 000 00 00 003 984 985 986 987 988 989 990 99 99 993 994 995 996 997 998 999 000 00 00 003 984 985 986 987 988 989 990 99 99 993 994 995 996 997 998 999 000 00 00 003 Figure 3: Time Plo of ARDL, LSTR and MRSTR Residuals 0 Residuals from ARDL equaion 5 0-5 -0-5 5 Residuals from LSTR equaion 0 5 0-5 -0 0 Residuals from MRSTR equaion 5 0-5 -0 9
984 985 986 987 988 989 990 99 99 993 994 995 996 997 998 999 000 00 00 003 Figure 4: Transiion Plos for F in MRSTR equaion Plo of Transiion Funcion F agains Transiion Variable S 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 0-5 -0-5 0 5 0 5 Time Plo of Transiion Funcion F 0.5 0 30
984 985 986 987 988 989 990 99 99 993 994 995 996 997 998 999 000 00 00 003 Figure 5: Transiion Plos for F in MRSTR equaion Plo of Transiion Funcion F agains Transiion Variable S 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 0-5 -0-5 -0-5 0 5 0 5 Time Plo of Transiion Funcion F 0.5 0 3
Appendix The heeroscedasiciy-robus es for omied nonlineariy in he single-regime ARDL equaion is in line wih procedure 3. of Wooldridge (99) and is based on he auxiliary equaion (5) of his paper: y x ( x s ) ( x s ) ( x s ) w (5) / / / 3 / 0 3 Given ha he convenional es for STR nonlineariy repored in Table poins o a LSTR equaion wih SP reurns hree monhs ago as he ransiion variable and given ha he ARDL residuals show evidence of condiional heeroscedasiciy, he null hypohesis H 0 ( 0 3 0) is evaluaed in a heeroscedasiciy-robus manner using his paricular ransiion variable as follows: - Regress each variable in xs on x and save he x Q vecor of residuals, say r - Regress on er, where e is he residual from he parsimonious ARDL equaion - (T SSR) is asympoically Q under H 0, where SSR is he sum of squared residuals 3
Appendix The heeroscedasiciy-robus es for remaining nonlineariy in he wo-regime LSTR equaion is in line wih procedure 3. of Wooldridge (99) and is based on equaion (9) of his paper: / y / v ( x s ) z (9) Given ha he convenional es for remaining nonlineariy in he wo-regime LSTR equaion repored in Table 3 poins o a MRSTR equaion wih SP reurns and FT reurns hree monhs ago as he ransiion variables and given ha he LSTR residuals show evidence of condiional heeroscedasiciy, he null hypohesis H : 0 0 is evaluaed in a heeroscedasiciy-robus manner using hese paricular ransiion variables as follows: - Regress each variable in xs on y / and save he x Q vecor of residuals, say r - Regress on vr, where v is he residual from he wo-regime LSTR equaion - (T SSR) is asympoically Q under H 0, where SSR is he sum of squared residuals 33