A Nonparametric Test for Granger Causality in Distribution With Application to Financial Contagion

Transcription

1 Journal of Business & Economic Saisics ISSN: Prin) Online) Journal homepage: hps://amsa.andfonline.com/loi/ubes A Nonparameric es for Granger Causaliy in Disribuion Wih Applicaion o Financial Conagion Berrand Candelon & Sessi okpavi o cie his aricle: Berrand Candelon & Sessi okpavi 16) A Nonparameric es for Granger Causaliy in Disribuion Wih Applicaion o Financial Conagion, Journal of Business & Economic Saisics, 34:, 4-53, DOI: 1.18/ o link o his aricle: hps://doi.org/1.18/ he Auhors). Published wih license by aylor & Francis 16 Berrand Candelon and Sessi okpavi Acceped auhor version posed online: 8 Apr 15. Published online: 17 Mar 16. Submi your aricle o his journal Aricle views: 111 View Crossmark daa Ciing aricles: 5 View ciing aricles Full erms & Condiions of access and use can be found a hps://amsa.andfonline.com/acion/journalinformaion?journalcode=ubes

2 A Nonparameric es for Granger Causaliy in Disribuion Wih Applicaion o Financial Conagion Berrand CANDELON Insi7/IPAG Chaire in Financial Sabiliy and Sysemic Risks, IPAG Business School, Paris, France candelonb@gmail.com) Sessi OKPAVI Economi-CNRS, Universiy of Paris Oues, France sessi.okpavi@u-paris1.fr) his aricle inroduces a kernel-based nonparameric inferenial procedure o es for Granger causaliy in disribuion. his es is a mulivariae exension of he kernel-based Granger causaliy es in ail even. he main advanage of his es is is abiliy o examine a large number of lags, wih higher-order lags discouned. In addiion, our es is highly flexible because i can be used o idenify Granger causaliy in specific regions on he disribuion suppors, such as he cener or ails. We prove ha he es converges asympoically o a sandard Gaussian disribuion under he null hypohesis and hus is free of parameer esimaion uncerainy. Mone Carlo simulaions illusrae he excellen small sample size and power properies of he es. his new es is applied o a se of European sock markes o analyze spillovers during he recen European crisis and o disinguish conagion from inerdependence effecs. KE WORDS: Financial spillover; Kernel-based es; ails. 1. INRODUCION Analysis of causal relaionships is an imporan aspec of heoreical and empirical conribuions in quaniaive economics see he special issues of he Journal of Economerics in 1988 and 6). Alhough he concep of causaliy as defined by Granger 1969) is broad and consiss of esing ransmission effecs beween he whole disribuion of random variables, exensions of his concep have recenly been proposed, such as causaliy in he frequency domain or for specific disribuion momens. For insance, Granger causaliy in he mean Granger 198, 1988) is widely used in macroeconomics. For example, Sims 197, 198) es for Granger causaliy in he mean of money and income. Granger, Robins, and Engle 1986) also inroduced he concep of Granger causaliy in variance o es for causal effecs in he second-order momen beween financial series. his concep was furher explored by Cheung and Ng 1996), Kanas and Koureas ), and Hafner and Herwarz 8), among ohers. A unified reamen of Granger causaliy in he mean and variance is formalized by Come and Lieberman ). More recen conribuions have focused on he concep of Granger causaliy in quaniles, a paricularly imporan issue for non-gaussian disribuions ha exhibi asymmery, fa-ail characerisics, and nonlineariy Lee and ang 1; Jeong, Härdle, and Song 1). Indeed, for hese disribuions, he dynamic in he ails can differ subsanially from ha of he cener of he disribuion. In his case, he informaion conen of he quaniles provides greaer insigh ino he disribuion han he conen provided by he mean. Lee and ang 1) developed a parameric mehodology for Granger causaliy in quaniles based on he condiional predicive abiliy CPA) framework of Giacomini and Whie 6). Jeong, Härdle, and Song 1) inroduced a nonparameric approach o es for causaliy in quaniles and apply i o he deecion of causal relaions beween he price of crude oil, he USD/GBP exchange rae, and he price of gold. A closely relaed bu differen concep is Granger causaliy in ail evens by Hong, Liu, and Wang 9). A ail even occurs when he value of a ime series is lower han is value-a-risk a a specified risk level. Hence, he es deermines if an exreme downside movemen for a given ime series has predicive conen for an exreme downside movemen for anoher ime series, wih numerous poenial applicaions in risk managemen. All ess of causaliy in quaniles and ail evens share he limi ha saisical inference is exclusively performed a a specific fixed level of he quanile. A his given level, he null hypohesis should no be rejeced, while he opposie conclusion should hold for anoher quanile level. Indeed, as emphasized by Granger 3) and Engle and Manganelli 4), he imeseries behavior of quaniles can vary considerably across he disribuion because of long memory or nonsaionariy. Hence, a Granger causaliy es in quaniles or ail evens ha does no consider a large number of quaniles simulaneously over he disribuion suppor would be oo resricive. Because he predicive disribuion of a ime series is enirely deermined by is 16 Berrand Candelon and Sessi okpavi. Published wih license by aylor & Francis. his is an Open Access aricle disribued under he erms of he Creaive Commons Aribuion License hp://creaicecommons.org/license/by/4./), which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Journal of Business & Economic Saisics April 16, Vol. 34, No. DOI: 1.18/

3 Candelon and okpavi: A Nonparameric es for Granger Causaliy in Disribuion 41 quaniles, esing for Granger causaliy for he range of quaniles over he disribuion suppor is equivalen o esing for Granger causaliy in he disribuion. esing procedures for Granger causaliy in he whole disribuion in a ime series conex are developed only in Su and Whie 7, 8, 1, 14), Bouezmarni, Rombous, and aamoui 1), and aamoui, Bouezmarni, and El Ghouch 14). For example, Su and Whie 1) inroduced a condiional independence specificaion es ha can be used o es for Granger causaliy in quaniles over a coninuum of values of quanile levels beween, 1). Bouezmarni, Rombous, and aamoui 1) consruced a nonparameric Granger causaliy es in disribuion based on condiional independence in he framework of copulas. aamoui, Bouezmarni, and El Ghouch 14) also developed alernaive Granger causaliy ess using he copulas heory. he presen aricle adds o his lieraure by proposing a new nonparameric es for Granger causaliy in he whole disribuion beween wo ime series. o summarize, our esing procedure consiss of dividing he disribuion suppor of each series ino a mulivariae process of dynamic inerquanile even variables. he es for causaliy in he disribuion beween he wo series is enabled by an analysis of he cross-correlaion srucures of he mulivariae processes and relies on he generalized pormaneau es for independence beween mulivariae processes developed by Bouhaddioui and Roy 6). Alhough our approach examines he srong version of he Granger causaliy concep Granger 1969), i is sufficienly flexible ha i can be used o es for causaliy in specific regions on he disribuion suppors, such as he cener or he ails lef or righ). While Candelon, Joës, and okpavi 13) inroduced a parameric es o check for Granger causaliy in disribuion ails, ha mehodology, in conras o he ess developed in his aricle, does no apply o oher regions of he disribuion such as he cener. For example, he es can be used o es for causaliy in he lef-ail disribuion for wo ime series. In his case, he mulivariae process of inerquanile even variables should be defined o focus he analysis exclusively on his par of he disribuion. his flexibiliy consiues a clear advanage of our mehodology compared o hose based on copulas heory Bouezmarni, Rombous, and aamoui 1; aamoui, Bouezmarni, and El Ghouch 14) and allows us o go beyond he simple rejecion of he null hypohesis of Granger causaliy for he whole disribuion because i idenifies he specific regions for which Granger causaliy is rejeced. Our es saisic is also a mulivariae exension of he kernel-based nonparameric Granger causaliy es in ail evens by Hong, Liu, and Wang 9) and herefore shares is main advanage: he abiliy o examine a large number of lags by discouning higher-order lags. his characerisic is consisen wih empirical evidence in finance ha recen evens have a greaer influence on curren marke rends han older ones. hus, our Granger causaliy es in disribuion is differen from hose available in he lieraure ha checks for causaliy uniformly for a limied number of lags. echnically, we demonsrae ha he es has a sandard Gaussian disribuion under he null hypohesis, which is free of parameer esimaion uncerainy. Mone Carlo simulaions confirm ha he Gaussian disribuion provides a good approximaion of he disribuion of our es saisic, even in small samples. Moreover, he es has he power o rejec he null hypohesis of causaliy in disribuion semming from differen sources, including linear and nonlinear causaliy in he mean and causaliy in he variance. o illusrae he imporance of his es for he empirical lieraure, we explore he spillovers ha have occurred wihin European sock markes during he recen crisis. Our Granger causaliy es in disribuion allows us o consider asymmery beween markes which is no possible using correlaion), o ake ino accoun a break in volailiy as suggesed by Forbes and Rigobon ) and o disinguish beween conagion and inerdependence. Indeed, inerdependence is a long-run pah ha occurs during normal periods and herefore concerns he cener of he disribuion exclusively. By conras, conagion is deeced by a shor-run abrup increase in he causal linkages ha occur during crisis periods, ha is, only in he ails of he disribuion. Because our es is designed o check for causaliy in specific regions of he disribuion, i can be used o check for inerdependence or conagion. Anicipaing our resuls, we find weak respecively, srong) suppor for inerdependence respecively, conagion) during he recen crisis. Ineresingly, we observe a srong asymmery beween causal ess in he righ and lef ails: whereas spillovers are imporan in crisis periods, hey are only weakly presen during upswing periods. Such a resul consiues an imporan feaure of European sock markes. he aricle is organized as follows: he second secion presens he Granger causaliy es in disribuion. he properies of his es are analyzed in Secion 3 via a Mone Carlo simulaion experimen. Secion 4 proposes he empirical applicaion, and Secion 5 concludes.. NONPARAMERIC ES FOR GRANGER CAUSALI IN DISRIBUION his secion presens our kernel-based es for Granger causaliy in disribuion beween wo ime series. Because his es is a mulivariae exension of he Granger causaliy es in ail evens inroduced by Hong, Liu, and Wang 9), we firs presen heir es and hen inroduce he new approach..1 Granger Causaliy in ail Even For wo ime series and, he Granger causaliy es in ail evens developed by Hong, Liu, and Wang 9) deermines wheher an exreme downside risk from can be considered a lagged indicaor for an exreme downside risk for. Hong, Liu, and Wang 9) idenified an exreme downside risk as a siuaion in which and are lower han heir respecive value-a-risk VaR) a a prespecified level α. VaR is a risk measure ofen used by financial analyss and risk managers o measure and monior he risk of loss for a rading or invesmen porfolio. he VaR of an insrumen or a porfolio of insrumens is he maximum dollar loss wihin he α%-confidence inerval Jorion 7). For he wo ime series and,wehave Pr [ < VaR ) ] F 1 = α, 1) Pr [ < VaR ) ] F 1 = α, ) ) ) where VaR and VaR are he VaR of and, respecively, a ime ; and and are he rue unknown

4 4 Journal of Business & Economic Saisics, April 16 finie-dimensional parameers relaed o he specificaion of he VaR model for each variable. he informaion ses F 1 and F 1 are defined as F 1 = { l,l 1}, 3) F 1 = { l,l 1}. 4) In he framework of Hong, Liu, and Wang 9), an exreme downside ) risk occurs a ime for if he ail even variable is equal o one, wih Z Z Z ) = 1 if < VaR else. ) Similarly, ) an exreme downside risk for corresponds o aking a value of one, wih Z ) = 1 if < VaR else. ) Hence, he ime series does no Granger-cause in downside risk or ail even a level α) he ime series if he following hypohesis holds wih H : E [ Z ) F & 1 5) 6) ] = E [ Z ) F 1 ], 7) F 1 & = { l, l ),l 1}. 8) Under he null hypohesis and a he risk level α, spillovers of exreme downside movemens from o do no exis. Hong, Liu, and Wang 9) proposed a nonparameric approach for esing for he null hypohesis in 7) based on he cross-specrum of he esimaed bivariae process of ail even variables { } Ẑ, Ẑ, wih componens Ẑ Z ), Ẑ Z ), 9) where and are consisen esimaors of he rue unknown parameers and, respecively. o presen heir es saisic, le us define he sample cross-covariance funcion beween he esimaed ail even variables as 1 Ẑ )Ẑ ) α j α, j 1 =1+j Ĉ j) = 1 Ẑ )Ẑ ) +j α α, 1 j, =1 j 1) where is he sample lengh and α and α are he sample mean of Ẑ and Ẑ, respecively. he sample cross-correlaion funcion ρ j) is hen equivalen o ρ j) = Ĉ j), 11) S S where S and S are he sample variances of Ẑ and Ẑ, respecively. Using he cross-correlaion funcion, he kernel esimaor for he cross-specral densiy of he bivariae process of ail even variables corresponds o f ω) = 1 π 1 κ j M) ρ j) e ij ω, 1) where κ.) is a given kernel funcion and M is he runcaion parameer. he runcaion parameer M is a funcion of he sample size such ha M and M/ as. he kernel is a symmeric funcion defined on he real line and aking value in [ 1, 1]. I mus be coninuous a zero, wih a mos a finie number of disconinuiy poins such ha κ ) = 1, 13) κ z) dz <. 14) Under he null hypohesis of non-granger causaliy in ail evens from o, he kernel esimaor for he cross-specral densiy is equal o f 1 ω) = 1 π κ j /M) ρ j) e ij ω. 15) his equaion suggess ha he disance beween he wo esimaors f ω) and f 1 ω) can be used o es for he null hypohesis. Hong, Liu, and Wang 9) considered he following quadraic form L π f, f 1 ) = π f ω) f 1 ω) dω, 16) π which is equivalen o L f, f 1 ) 1 = κ j/m) ρ j). 17) he es saisic is a sandardized version of he quadraic form given by 1 / U = κ j/m) ρ j) C M) D M) 1, 18) and follows under he null hypohesis a sandard Gaussian disribuion, wih C M) and D M) as he locaion and scale parameers 1 C M) = 1 j/) κ j/m), 19) 1 D M) = 1 j/)1 j + 1) /) κ 4 j/m).). Granger Causaliy in Disribuion In his secion, we presen our mulivariae exension of he es of Hong, Liu, and Wang 9); his exension permis he idenificaion of Granger causaliy in he whole disribuion beween wo ime series.

5 Candelon and okpavi: A Nonparameric es for Granger Causaliy in Disribuion Noaions and he Null Hypohesis. he seing of our esing procedure is as follows. We consider a se A = {α 1,...,α m+1 } of m + 1 VaR risk levels ha covers he disribuion suppor of boh variables and, wih α 1 < < α m+1 1%. For he firs ime) series, he corresponding VaRs a ime are VaR,s,α s, s = 1,...,m+ 1, wih ) ),α 1 < < VaR,m+1,α m+1, 1) VaR,1 where he vecor is once again he rue unknown finiedimensional parameer relaed o he specificaion of he VaR ) model for. We adop he convenion ha VaR,s,α s = ) for α s = % and VaR,s,α s = for αs = 1%. We divide he disribuion suppor of ino m disjoin regions, each relaed o he indicaor or even variable ) 1 if VaR Z,s ),s,α s ) and < VaR =,s+1,α s+1 ) else, for s = 1,...,m. For illusraion, le m + 1 = 5, and suppose ha he se A = {α 1,α,α 3,α 4,α 5 } = {%, %, 4%, 6%, 8%}. Figure 1 displays he suppor of, along wih he VaRs and he even variables defining he m = 4 disinc regions. We do no consider he even variable corresponding o he ) exreme m + 1 region idenified by VaR,m+1,α m+1 ; his variable is implicily defined by he firs m even variables. ) Now, le H be he vecor of dimension m, 1) wih componens of he m even variables H ) ) ) )) = Z,1,Z,,...,Z,m. 3) We similarly define for he second ime ) series hese even variables colleced in he vecor H, wih ) ) ) )) = Z,1,Z,,...,Z,m. 4) H he ime series does no Granger-cause he ime series in disribuion if he following hypohesis holds H : E [ H ) ] [ ) ] F & 1 = E H F 1. 5) herefore, Granger causaliy in he disribuion from o) corresponds o Granger causaliy in he mean from H ) o H. When he null hypohesis of noncausaliy in disribuion holds, he even variables defined for he variable along is disribuion suppor do no have any predicive conen for he dynamics of he same even variables over he disribuion suppor of. Our null hypohesis is sufficienly flexible ha i can be used o check for Granger causaliy in specific regions on he disribuion suppors, such as he cener or ails lef or righ), by resricing he se A = {α 1,...,α m+1 } of VaR levels o seleced values. For insance, we can check for Granger causaliy in he lef-ail disribuion by seing A o A = {%, 1%, 5%, 1%}.In his case, he rejecion of he null hypohesis is of grea imporance in financial risk managemen because i indicaes spillover effecs from o in he lower ail. Similarly Granger causaliy in he cener of he disribuion can be checked by seing, for example, A = {%, 4%, 6%, 8%}. In he nex subsecion, we consruc a nonparameric kernel-based es saisic o es for our general null hypohesis of noncausaliy 5) and analyze is asympoic disribuion... es Saisic and Asympoic Disribuion. he consrucion of his es saisic is closely relaed o he aricle of Bouhaddioui and Roy 6), which proposes a generalized pormaneau es for he independence of wo infinie-order vecor auo regressive VAR) series. Neverheless, he asympoic analysis differs because i) we are no in a VaR framework, ) ii) and he even variables we are considering, Z,s and ) Z,s, are indicaor variables, which are herefore no differeniable wih respec o he unknown parameers and, respecively. o address his lack of differeniabiliy, we consider several asympoic resuls derived in Hong, Liu, and Wang 9). o presen he es saisic for our general ) null hypohesis ) of noncausaliy, consider Ĥ H and Ĥ H he esimaed counerpars ) of he mulivariae processes of even and H ) variables H, respecively, wih and as consisen esimaors of he rue unknown parameer vecors and.le j) denoe he sample cross-covariance marix beween Ĥ and Ĥ, wih j) 1 1 =1+j =1 j Ĥ )Ĥ j ) j 1 Ĥ +j )Ĥ ) 1 j, 6) Figure 1. Disribuion suppor of and localizaion of VaRs and even variables.

6 44 Journal of Business & Economic Saisics, April 16 where he vecor respecively, ) of lengh m is he sample mean of Ĥ respecively, Ĥ ). As in he univariae seing of Hong, Liu, and Wang 9), we can replace and by = E )) H and = E )) H, respecively, wihou affecing he asympoic disribuion of our es saisic. he corresponding sample cross-correlaion marix R j) equals R j) = D ) 1/ j) D ) 1/, 7) where D.) represens he diagonal form of a marix and and are he sample covariance marices of Ĥ and Ĥ, respecively. he es saisic can hus be expressed as he following weighed quadraic form ha accouns for he dependence beween he curren value of Ĥ and he lagged values of Ĥ 1 = κ j/m) Q j), 8) where κ.) is a kernel funcion, M is he runcaion parameer, and Q j) is equal o Q j) = vec R j) ) Ɣ 1 Ɣ 1 ) vec R j) ), 9) where Ɣ and Ɣ are he sample correlaion marices of Ĥ and Ĥ, respecively. he resricions on he runcaion parameer M and he kernel funcion κ.) are he same as hose considered by Hong, Liu, and Wang 9) in heir univariae seing see Secion.1). Mos common kernels used in specral analysis Daniell, Parzen, Barle, runcaed uniform) saisfy hese resricions. Moreover, as discussed by Hong, Liu, and Wang 9), he choice of kernel is no imporan because hey lead o comparable powers excep for he uniform kernel, which does no discoun higher-order lags. See Bouhaddioui and Roy 6) for he same conclusion in a mulivariae seing. Following Bouhaddioui and Roy 6), our essaisicisa cenered and scaled version of he quadraic form in 8), ha is, V = m C M) m D M) ) 1/, 3) where C M) and D M) are as defined in 19) and ), respecively. he above es saisic generalizes he one in Hong, Liu, and Wang 9) in a mulivariae seing. When m is equal o one, which corresponds o he univariae case in which each of he vecors Ĥ and Ĥ has only one even variable, he es saisic V in 3) is exacly equal o he es saisic in 18). he following proposiion yields he asympoic disribuion of our es saisic. Proposiion 1. Suppose ha he assumpions of heorem 1 in Hong, Liu, and Wang 9) hold. hen, under he null hypohesis of no Granger causaliy in disribuion as saed in 5), we have V = m C M) m D M) ) 1/ d N, 1). he assumpions of heorem 1 in Hong, Liu, and Wang 9) impose several regulaory condiions on he ime series and ; on he VaR models used, including smoohness, momen condiions, and adequacy; on he kernel funcion κ.); and on he runcaion parameer M. he laer should be equal o) M = c v wih <c<, <v<1/, v<min d, 3 d 1 if d max d,d ) > and d respecively, d ) is he dimension of he parameer respecively, ). See Hong, Liu, and Wang 9, pp. 75) for a complee discussion of hese assumpions. he proof of Proposiion 1 proceeds as follows. Consider he following decomposiion of our es saisic V = m C M) m D M) ) 1/ + m D M) ) 1/, 31) where is he pseudo version of he weighed quadraic form in 8) and 9) compued using he rue correlaion marices Ɣ and Ɣ, ha is, 1 = κ j/m) Q j), 3) Q j) = vec R j) ) Ɣ 1 ) Ɣ 1 vec R j) ). 33) Under he decomposiion in 31), he proof of Proposiion 1 is given by he following wo lemmas: Lemma 1. Under he null hypohesis of no Granger causaliy in disribuion and he assumpions of heorem 1 in Hong, Liu, and Wang 9), we have m C M) m D M) ) 1/ d N, 1). 34) Lemma. Under he assumpions of heorem 1 in Hong, Liu, and Wang 9), we have m D M) ) 1/ p. 35) he proofs of hese wo Lemmas are repored in Appendix A. 3. SMALL SAMPLE PROPERIES In his secion, we sudy he finie sample properies of our es via Mone Carlo simulaion experimens. We analyze he size in he firs par of he secion, while he remainder of he secion is devoed o an analysis of he power. 3.1 Empirical Size Analysis We simulae he size of he nonparameric es of Granger causaliy in disribuion assuming he following daa-generaing process DGP) for he second ime series : = u,, u, = σ, v,, σ, =.1 +.9σ 1, +.8u 1,, 36) N, 1), v, iid

7 Candelon and okpavi: A Nonparameric es for Granger Causaliy in Disribuion 45 which corresponds o an AR1)-GARCH1,1) model. We make he assumpion ha he firs ime series follows he same process. Because he wo processes are generaed independenly, here is no Granger causaliy in disribuion beween hem. For a given value of sample size {5, 1, }, for each simulaion we compue our es saisic in 3) and make inferences using he asympoic Gaussian disribuion. For he compuaion of he es saisic, we need o specify a model o esimae he VaRs a he risk level α 1,...,α m+1 ) and he m even variables for each variable and.hem + 1VaRs are compued using an AR1)-GARCH1,1) model esimaed by quasi-maximum likelihood. he esimaed values of he m + 1 VaRs a ime are VaR,s = μ, + σ, q v,,α s ), s = 1,...,m+ 1, 37) where μ, and σ, are he fied condiional mean and sandard deviaion a ime, respecively, and q v ),,α s is he empirical quanile of order α s of he esimaed sandardized innovaions. We proceed similarly o compue he m + 1 VaRs and he corresponding m even variables for he second ime series. Noe ha we se he parameer m + 1 o 14 and he se A o A = {α 1,α,...,α 14 } = {%, 1%, 5%, 1%, %,...,9%, 95%, 99%}, which covers regions in he ails and he cener of he disribuion suppor of each ime series. For α s = %, he VaR corresponds o.we mus also make a choice abou he kernel funcion o compue our es saisic. We consider he four differen sandard kernels, ha is, he Daniell DAN), he Parzen PAR), he Barle BAR), and he runcaed uniform R) kernels. See Appendix B for he descripion of he four kernel funcions. Finally, for he choice of runcaion parameer M, weuse hree differen values: M = [ln )], M = [ [ ] 1.5.3], and M =.3, where [.] is he ineger porion of he argumen. hese raes lead o he values M = 6, 1, 13 for = 5, M = 7, 1, 16 for = 1, and M = 8, 15, for =. hese values cover a range of lag orders for he sample sizes considered. able 1 displays he empirical sizes of our es over 1 simulaions and for wo differen significance levels η 5%, 1%). he resuls in able 1 indicae ha our es is adequaely sized. Indeed, he rejecion frequencies are close o he significance levels. Hence, he sandard Gaussian disribuion asympoically provides a good approximaion of he disribuion of our es saisic. his resul appears o hold regardless of he kernel funcion used and he value of he runcaion parameer M. 3. Empirical Power Analysis We now simulae he empirical power of our es. Because causaliy in disribuion springs from causaliy in momens such as mean or variance, we assume differen DGPs corresponding o hese cases. he firs DGP assumes he exisence of a linear Granger causaliy in he mean o generae daa under he alernaive hypohesis. Hence, we assume ha he second ime series has he DGP in 36) and ha he firs ime series is he able 1. Empirical sizes of he Granger causaliy es in disribuion M η DAN BAR PAR R 6 5% % % % % % % % % % % % % % % % % % NOES: he able displays he empirical sizes in %) of he Granger causaliy es in disribuion. Rejecion frequencies are repored over 1 simulaions for wo significance levels η, where is he sample size and M is he runcaion parameer. DAN, BAR, PAR, and R refer o he Daniell, Barle, Parzen, and runcaed uniform kernels, respecively. following = u,, u, = σ, v,, σ, =.1 +.9σ 1, +.8u 1,, N, 1). v, iid 38) he empirical powers of our es are compued over 1 simulaions for {5, 1, }. As in he analysis of size, we consider hree values of he runcaion parameer M,and wo significance levels η = 5%, 1%. he resuls for he four kernels used in he analysis of size are repored in able. For comparison, able also displays he resuls of he Granger causaliy es in mean in parenheses. o ensure an appropriae comparison, we do no use he usual parameric Granger causaliy es in mean derived from a vecor auoregressive model bu consider insead he kernel-based nonparameric Granger causaliy es in mean inroduced by Hong 1996). he resulsinable indicae ha our kernel-based nonparameric es for Granger causaliy in disribuion has appealing power properies. For insance, wih he Daniell kernel, he rejecion frequencies of he null hypohesis for,m) = 5, 6) equal 84.4% and 89.8% for η = 5% and 1%, respecively. For =, he powers are, in mos cases, equal o one. he rejecion frequencies of he Granger causaliy es in mean are always equal o or close o 1% and hence are higher han he ones obained by applying our Granger causaliy es in disribuion for he smalles sample size. his resul is expeced because he assumed causaliy in disribuion springs from causaliy in he mean. In all configuraions, he uniform kernel leads o he smalles powers because is uniform weighing scheme does no discoun higher-order lags. Moreover, we observe as Hong, Liu, and Wang 9) did

8 46 Journal of Business & Economic Saisics, April 16 able. Empirical powers of he Granger causaliy es in disribuion: DGP1 M η DAN BAR PAR R 6 5% ) 1% ) 5 1 5% ) 1% ) 13 5% ) 1% ) 7 5% ) 1% ) 1 1 5% ) 1% ) 16 5% ) 1% ) 8 5% 1. 1.) 1% 1. 1.) 15 5% 1. 1.) 1% 1. 1.) 5% 1. 1.) 1% 1. 1.) ) ) ) ) ) ) ) ) ) ) ) ) 1. 1.) 1. 1.) 1. 1.) 1. 1.) 1. 1.) 1. 1.) ) ) ) ) ) ) ) 1. 1.) ) ) ) ) 1. 1.) 1. 1.) 1. 1.) 1. 1.) 1. 1.) 1. 1.) ) ) ) ) ) ) ) ) ) ) ) ) 1. 1.) 1. 1.) ) ) ) ) NOES: he able displays he empirical powers in %) of he Granger causaliy es in disribuion. Rejecion frequencies are repored over 1 simulaions for wo significance levels η, where is he sample size and M is he runcaion parameer. For comparison, we also repor in parenheses) he rejecion frequencies of he kernel-based nonparameric es in mean. DAN, BAR, PAR, and R refer o he Daniell, Barle, Parzen, and runcaed uniform kernels, respecively. Daa are generaed under he alernaive hypohesis assuming linear Granger causaliy in mean. able 3. Empirical powers of he Granger causaliy es in disribuion: DGP M η DAN BAR PAR R 6 5% ) 1% ) 5 1 5% ) 1% ) 13 5% ) 1% ) 7 5% ) 1% ) 1 1 5% ) 1% ) 16 5% ) 1% ) 8 5% 1. 5.) 1% ) 15 5% ) 1% ) 5% ) 1% 1. 6.) ) ) ) 1. 5.) ) ) 1. 5.) ) ) ) ) ) 1. 5.) ) ) ) ) ) ) ) ) ) ) 1. 5.) 1. 5.) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 1. 5.) ) ) ) ) ) NOES: he able displays he empirical powers in %) of he Granger causaliy es in disribuion. Rejecion frequencies are repored over 1 simulaions for wo significance levels η, where is he sample size and M is he runcaion parameer. For comparison, we also repor in parenheses) he rejecion frequencies of he kernel-based nonparameric es in mean. DAN, BAR, PAR, and R refer o he Daniell, Barle, Parzen, and runcaed uniform kernels, respecively. Daa are generaed under he alernaive hypohesis assuming nonlinear Granger causaliy in mean. ha he rejecion frequencies of he null hypohesis decrease wih he runcaion parameer M. o sress he relevance of our esing approach, we consider a second ype of alernaive DGP ha assumes causaliy in disribuion semming from a nonlinear form of causaliy in he mean. Precisely, we generae daa for he ime series using he specificaion in 36), and he firs ime series is generaed as follows: = u,, u, = σ, v,, σ, =.1 +.9σ 1, +.8u 1,, iid v, N, 1). 39) able 3 repors he rejecion frequencies over 1 simulaions. he presenaion is similar o ha in able. We observe ha our es coninues o exhibi good power in deecing his nonlinear form of causaliy. Indeed, he rejecion frequencies are in all cases close o or even equal o one, even when considering he uniform kernel. By conras, he Granger causaliy es in mean fails o rejec he null hypohesis for approximaely half of he simulaions, and he rejecion frequencies do no seem o increase significanly wih he sample size. For illusraion, he rejecion frequency of he null hypohesis for,m) = 5, 6) amouns o 1% for he Daniell kernel and η = 5%, while i is only equal o 48.6% for he causaliy es in mean. Finally, we generae daa under he alernaive hypohesis, assuming Granger causaliy in variance. Formally, we suppose once again ha has he specificaion in 36), and is generaed as = u,, u, = σ, v,, σ, =.1 +.8σ 1, +.8u 1, +.8u 1,, 4) N, 1). v, iid Frequency rejecions are displayed in able 4 and are qualiaively similar o hose repored in able 3. Our causaliy es in disribuion performs quie well in rejecing he null hypohesis, while he causaliy es in mean is less robus and rejecs he null in few cases. Finally, we observe ha he rejecion frequencies are lower han hose repored in ables and 3. his resul occurs because i) causaliy in variance occurs mainly in he ails and ii) he dynamics of he ails are more difficul o fi due o he lack of daa. For he DGP in 4), we slighly decrease he persisence of volailiy for he process, which is equal o α + β = =.88. his calibraion differs from hose considered in previous simulaions, in which he persisence was se o

9 Candelon and okpavi: A Nonparameric es for Granger Causaliy in Disribuion 47 able 4. Empirical powers of he Granger causaliy es in disribuion: DGP3 M η DAN BAR PAR R 6 5% ) 1% 46.6.) 5 1 5% ) 1% ) 13 5% 4. 1.) 1% ) 7 5% 48..4) 1% ) 1 1 5% ) 1% ) 16 5% ) 1% ) 8 5% ) 1% ) 15 5% ) 1% ) 5% ) 1% ) ) 45.4.) 39.6.) 5.6.8) 4.8.4) ) ) ) ) ) 5. 7.) ) 77.8.) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 5.8.4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) NOES: he able displays he empirical powers in %) of he Granger causaliy es in disribuion. Rejecion frequencies are repored over 1 simulaions for wo significance levels η, where is he sample size and M is he runcaion parameer. For comparison, we also repor in parenheses) he rejecion frequencies of he kernel-based nonparameric es in mean. DAN, BAR, PAR, and R refer o he Daniell, Barle, Parzen, and runcaed uniform kernels, respecively. Daa are generaed under he alernaive hypohesis assuming Granger causaliy in variance. α + β =.98. We decrease he persisence of volailiy because if he volailiy of he process is highly persisen, he curren volailiy σ, mus be mainly driven by is pas values raher han u 1,, lowering he effec of causaliy. o provide furher insigh on his poin, Figure displays he power curve: we fix he parameer α o.8 and consider differen values of β, wih β =.5,.6,.7,.8,.9. We observe indeed ha he rejecion frequencies decrease wih persisence. 4. EMPIRICAL PAR Recen financial crises have been characerized by rapid, large regional spillovers of negaive financial shocks. For example, consecuive o he Greek disress, souhern European counries have been conaminaed and face skyrockeing refinancing raes. Norhern European saes have been impaced in an opposie manner. Considered safe harbors for invesors, hese counries were able o refinance heir deb on markes a lower raes. I is obvious ha he degree of globalizaion wihin he European Union as well as he low degree of fiscal federalism has fosered he speed as well as he ampliude of he ransmission mechanism of such a shock. Because souhern European counries used foreign capial markes o finance heir domesic invesmens and boos heir growh, hey have been highly subjec o financial insabiliy. Empirical sudies mus evaluae he imporance of hese spillovers. heoreically i relies on crisis-coningen heories, which explain he increase in marke cross-correlaion afer a shock issued in an origin counry as he resul of muliple equilibria based on invesor psychology; endogenous liquidiy shocks causing a porfolio recomposiion; and/or poliical disurbances affecing he exchange rae regime. By conras, according o noncrisis-coningen heories, he propagaion of shocks does no lead o a shif from a good o a bad equilibrium; he increase in cross-correlaion is he coninuaion of linkages rade and/or financial) ha exised before he crisis. he presence of spillovers during a crisis can hus be esed empirically by a significan and ransiory increase in crosscorrelaion beween markes see iner alia King and Wadhwani 199; Calvo and Reinhar 1995; Baig and Goldfajn 1998). Neverheless, his inuiive approach, which has he advanage of simpliciy because i avoids he idenificaion of ransmission channels, presens many shorcomings. Firs, Forbes and Rigobon ) demonsraed ha an increase in correlaion can be exclusively driven by higher volailiy during crisis periods. In such a case, he increase in correlaion could no be aribued o sronger economic inerdependence. o correc for his poenial bias, hey propose he use of he uncondiional correlaion raher han he condiional one and es for is emporary increase during crisis periods. Figure. Rejecion frequencies when considering DGP3 and several values of β.

10 48 Journal of Business & Economic Saisics, April 16 Second, correlaion is a symmerical measure: an increase in he correlaion beween markes i and j does no provide any informaion on he direcion of he conagion from i o j, fromj o i, or boh). For his reason, Bodar and Candelon 9) preferred o consider an indicaor of causaliy o measure spillovers. I is hus possible o evaluae asymmerical spillovers ha can hen move from i o j, j o i, or in boh direcions. In addiion, using he Granger causaliy approach requires he esimaion of mulivariae dynamic models, which are less prone o poenial misspecificaion issues. Addressing boh hese shorcomings in a classical framework is relaively feasible. However, alhough comparing causaliy beween precrisis and crisis periods permis he evaluaion of spillovers, i does no permi he separaion of inerdependence from conagion. Inerdependence addresses he long-run srucural causaliy beween markes and hus provides informaion on he exen o which markes are inegraed. herefore, inerdependence should be esed independenly of exreme posiive or negaive evens. By conras, conagion addresses shor-run abrup increases in causal linkages and occurs exclusively during crisis periods. hus, esing for conagion requires an exclusive focus on he exreme lef ail of he disribuion, as in exreme value heory see Harmann e al. 4). Considering he whole disribuion o evaluae conagion would hence aler he conclusions. Our new causaliy es allows all hese issues o be addressed because causaliy can be esed for he whole disribuion as well as for specific perceniles of he disribuion. As an illusraion, we analyze he recen European crisis considering a se of 1 European daily sock marke indices Ausria, Belgium, Finland, France, Germany, Greece, Ireland, Ialy, Luxemburg, he Neherlands, Porugal and Spain), yielding 13 pairwise sysems. Daa are downloaded from Daasream ranging from January 1, 7 o May 6, 11 i.e., = 1134 observaions). he firs empirical illusraion consiss of esing for inerdependence, which is performed by implemening he pairwise Granger causaliy for he cener of he disribuion, ha is, removing exreme evens locaed on he righ and lef ails. A large share of rejecion of he noncausaliy null hypohesis would suppor he hypohesis of inerdependence. hen, in a second analysis, we implemen he causaliy es exclusively for he lef ail o es for conagion during crisis. Such a hypohesis would be suppored if we observed a higher percenage of noncausaliy rejecion han ha previously obained when considering he cener of he disribuion. Similarly, he es is conduced for he righ ail, ha is, he upswing period. We can hen compare he srengh of conagion during crises and boom periods and deermine which period conagion is he mos significan. 4.1 he General Design of he Granger Causaliy es in Disribuion o es for Spillover o implemen he Granger causaliy es in disribuion in our empirical illusraion, we firs need o compue for each index m + 1 series of VaRs corresponding o m + 1risklevel α s s = 1,...,m+ 1, which cover is disribuion suppor. As for he Mone Carlo simulaions, we consider he following se for he VaR levels A = {%, 1%, 5%, 1%,...,9%, 95%, 99%} wih m + 1 = 14. o compue he VaRs, we use a semiparameric model. Formally, we suppose ha each index reurns series R i, i = 1,...,1, following an AR m)-garch p, q) model, wih R i, = m φ i,jr i, j + ε i,, 41) ε i, = σ i, v i,, 4) σi, = κ i + q γ i,jε i, j + p β i,jσ i, j, 43) and v i, is an iid innovaion wih mean zero and uni variance. he choice for an AR m)-garch p, q) is consisen wih he Forbes and Rigobon ) correcion and accouns for a volailiy increase ha biases he causaliy analysis. For each index, his model is esimaed using he quasi-maximum likelihood mehod. Hence, he m + 1 series of VaRs are obained as VaR i,s = m φ i,j R i, j + σ i, q v ) i,,α s, s = 1,...,m+ 1, 44) where σ i, is he fied volailiy a ime for he index number i and q v ) i,,α s is he empirical quanile of order αs of he esimaed sandardized innovaions v i,. able 5 displays he esimaion resuls of he ARm)-GARCHp, q) models for he indices. As shown by he Ljung Box es applied o he residuals and heir squares, he reained specificaions successfully capure he dependence in he firs wo momens. Wih he fied series of VaRs a hand, we calculae for each index he mulivariae process of dynamic inerquaniles evens variables and compue for each couple i, j) of indices our kernel-based nonparameric es saisic V j i as defined in 3). For he compuaion, we use he Daniell kernel and se he runcaion parameer M o [ 1.5.3], which yields a value of M = 1 for he whole sample of lengh = able 5. Esimaion resuls of he AR-GARCH models Index φ i,1 κ i γ i,1 γ i, γ i,3 β i,1 LB vi, 6) LB v i, 6) A ) BEL ) FI. 3.59) FR. 4.31) GER ) GRE.7.184). 3.57) IE. 3.79) I. 4.67) LU. 3.89) NL ) P ) ES ) ) ) ) ) ) ) )..) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) NOES: For each index, he able displays he esimaion resuls of he AR-GARCH model in Equaions 41) 43). We repor he parameer esimaes followed in brackes by he suden saisics. he wo las columns give he resuls of he Ljung Box es applied o he series of he sandardized innovaions v i, and is square, respecively, wih 6 as he number of lags. he criical value for he rejecion of he null hypohesis a he 5% significance level is 1.59.

11 Candelon and okpavi: A Nonparameric es for Granger Causaliy in Disribuion 49 able 6. Resuls of bilaeral ess of Granger causaliy in he cener of he disribuion A BEL FI FR GER GRE IE I LU NL P ES SUM A BEL FI FR GER GRE IE I LU NL P ES SUM NOES: Each enry of he able gives he p-value in %) of he es of causaliy in he cener of he disribuion from he index j in he column oward he index i in he row. Enries wih he rejecion of he null hypohesis a he 5% significance level are emphasized in bold. he las column, labeled Sum, indicaes he number of imes a given index in a row is Granger-caused by he ohers. Similarly, he las row, labeled Sum, indicaes he number of imes a given index in a column Granger-causes he oher indices. he enry corresponding o he las row and he las column gives he oal number of significan Granger causaliies in he sysem. he ess are performed over he period ranging from January 1, 7, o May 6, 11, wih a oal of = 1134 observaions. 4. esing for Inerdependence o es for inerdependence, we follow he general design of he pairwise es of Granger causaliy in disribuion as described above, excep ha we remove he exreme evens from he disribuion. he new se A of VaRs risk levels is equal o A = {%, 3%,...,7%, 8%} wih m + 1 = 7. able 6 displays he resuls of he es. he repored values are he p-values in percenages. Hence, for a significance level of 5%, we rejec he null hypohesis of no causaliy from index j o index i when he repored value is lower han 5%. p-values corresponding o he rejecion of he null hypohesis of no causaliy are shown in bold. he las column, labeled Sum, indicaes he number of imes a given index in a row is Granger-caused by he ohers. Similarly, he las row labeled Sum indicaes he number of imes a given sock marke index Granger-causes oher sock marke indices. Finally, he enry corresponding o he las row and las column repors he oal number of rejecions of he null of no causaliy for our se of counries. hus, inerdependence defined as causaliy in he cener of he disribuion) is suppored in only 9.8% of he cases 13 cases ou of 13). his resul indicaes ha European sock marke inegraion is far from being achieved. Among he counry resuls, we observe ha he Ausrian and French sock markes are he mos inegraed because hey are each affeced by hree oher European markes. By conras, Greece, Ireland, Ialy, Luxembourg, and he Neherlands appear o be independen from he oher markes. Ineresingly, he causal marix is no symmeric: France, which is among he mos caused markes, does no affec any marke. his resul suppors our choice of causaliy raher han correlaion as a measure of spillover. he mos causal markes are he Neherlands, Greece, and Porugal. he idenificaion of hese wo las counries is ineresing because hey were among he main drivers of he European crisis. heir causal imporance, which can be qualified as sysemic for he res of Europe, should have consiued a signal of alarm a he edge of he crisis. able 7. Resuls of bilaeral ess of Granger causaliy in he lef-ail disribuion A BEL FI FR GER GRE IE I LU NL P ES SUM A BEL FI FR GER GRE IE I LU NL P ES SUM NOES: Each enry of he able gives he p-value in %) of he es of causaliy in he lef-ail disribuion from he index j in he column oward he index i in he row. Enries wih he rejecion of he null hypohesis a he 5% significance level are indicaed in bold. he las column, labeled Sum, indicaes he number of imes a given index in a row is Granger-caused by he ohers. Similarly, he las row, labeled Sum, indicaes he number of imes a given index in column Granger-causes he oher indices. he enry corresponding o he las row and he las column gives he oal number of significan Granger causaliies in he sysem. he ess are performed over he period ranging from January 1, 7, o May 6, 11, wih a oal of = 1134 observaions.

12 5 Journal of Business & Economic Saisics, April 16 able 8. Resuls of bilaeral ess of Granger causaliy in he righ-ail disribuion A BEL FI FR GER GRE IE I LU NL P ES SUM A BEL FI FR GER GRE IE I LU NL P ES SUM NOES: Each enry of he able gives he p-value in %) of he es of causaliy in he righ-ail disribuion from he index j in a column oward he index i in a row. Enries wih he rejecion of he null hypohesis a he 5% significance level are indicaed in bold. he las column, labeled Sum, indicaes he number of imes a given index in a row is Granger-caused by he ohers. Similarly, he las row, labeled Sum, indicaes he number of imes a given index in a column Granger-causes he oher indices. he enry corresponding o he las row and he las column gives he oal number of significan Granger causaliies in he sysem. he ess are performed over he period ranging from January 1, 7, o May 6, 11, wih a oal of = 1134 observaions. 4.3 esing for Conagion As explained previously, conagion is apprehended by implemening our Granger causaliy es in he lef-ail disribuion. he se A of VaRs risk levels is now se as A = {%, 1%, 5%, 1%} wih m + 1 = 4. able 7 displays he oucomes of he ess. he pairs for which we find a rejecion of he null of no causaliy in he lef ail of he disribuion amoun o 35.6% of he cases 47 rejecions over 13 cases). his resul is clearly higher han ha obained considering he cener of he disribuion, hence supporing he presence of conagion. Moreover, we observe ha he mos causal markes are Porugal, Ialy, he Neherlands, Greece, and Ireland, and excep for he Neherlands, his group includes all counries in urmoil Porugal, Ialy, Greece, and Ireland) around which he crisis was buil. By conras, he mos caused markes are Ausria, Belgium, Ialy, France, Luxembourg, and Greece. Remark he predominan role in he sysem of Ialy and Greece, which cause and are caused in many cases. he Granger causaliy es is now repeaed for he righ-ail disribuion wih A = {9%, 95%, 99%, 1%}, ha is, m + 1 = 4. he resuls are repored in able 8. Conagion in posiive periods is only suppored in 7.5% of he cases 1 rejecions over 13 cases) and concerns mainly Spain as he driver of spillover and Luxembourg, Germany, and Belgium as spillover receivers. he huge difference in causal links for he righ 7.5%) and lef ails 35.6%) is sriking. Whereas spillovers are imporan in crisis periods, hey are only weakly presen during upswing periods. his feaure emphasizes he subsanial vulnerabiliy of European sock markes o negaive shocks. European policy makers should acknowledge his vulnerabiliy and implemen srucural measures o limi i. 5. CONCLUSION A kernel-based nonparameric es for Granger causaliy in disribuion beween wo ime series is proposed in his aricle. he es checks for spillovers beween he mulivariae processes of dynamic inerquanile even variables are associaed wih each variable. Our esing approach has wo main advanages over exising approaches. Firs, i can be used o es for Granger causaliy in specific regions of he disribuions, such as he cener or he ails lef and righ). Second, i checks for a large number of lags by discouning higher-order lags and hence is consisen agains causaliy, which carries over long disribuional lags. We demonsrae ha he es has a sandard Gaussian disribuion under he null hypohesis, which is free of parameer esimaion uncerainy. A Mone Carlo simulaion exercise revealed ha he Gaussian disribuion is valid in small samples. he es also has very appealing power properies in various seings, including linear and nonlinear causaliy in mean and causaliy in variance. In he empirical secion, we implemen our esing procedure for 1 European daily sock marke indices o analyze spillover during he recen European crisis. Because our es is designed o check for causaliy in specific regions of he disribuion cener or ails), i can be used o es for he presence of inerdependence as well as conagion. Indeed, inerdependence can be deermined hrough Granger causaliy in he cener of he disribuion because inerdependence is a long-erm pah ha occurs during normal periods. By conras, conagion refers o a shor-erm, abrup increase in he causal linkages occurring exclusively during crisis periods and can be esed via he Granger causaliy in he disribuion s ails. he empirical resuls indicae ha European sock marke inegraion is far from achieved because we observe few cases of an inerdependen pairwise relaionship. By conras, our resuls suppor he presence of conagion, wih a srong difference beween conagion in he righ and lef ails. More precisely, conagion is frequen among counries during crisis periods and comparaively infrequen during upswing periods. his resul reveals an imporan feaure of he European sock markes, and policy makers should acknowledge