Multivariate Linear and Non-Linear Causality Tests

Transcription

1 Th Thal Economtrcs Soct Vol. No. (Januar ) Multvarat nar Non-nar Causalt Tsts Zhdong Ba a Wng-Kung Wong b Bngzh Zhang c a School of Mathmatcs Statstcs Northast Normal Unvrst Chna; Dpartmnt of Statstcs Appld Probablt Natonal Unvrst of Sngapor Sngapor b Dpartmnt of Economcs Hong Kong Baptst Unvrst Hong Kong c Dpartmnt of BoStatstcs Columba Unvrst USA ARTICE INFO ABSTRACT Kwords: lnar Grangr causalt nonlnar Grangr causalt U-statstcs JE classfcaton cods: C C G Th tradtonal lnar Grangr tst has bn wdl usd to amn th lnar causalt among svral tm srs n bvarat sttngs as wll as multvarat sttngs. Hmstra Jons (994) dvlop a nonlnar Grangr causalt tst n a bvarat sttng to nvstgat th nonlnar causalt btwn stock prcs radng volum. In ths papr w frst dscuss lnar causalt tsts n multvarat sttngs thraftr dvlop a non-lnar causalt tst n multvarat sttngs.. Introducton It s an mportant ssu to dtct th causal rlaton among svral tm srs t starts wth two srs s for ampl Chang t al () Qao t al (8 9) th rfrncs thrn for mor dscusson. To amn whthr past nformaton of on srs could contrbut to th prdcton of anothr srs Grangr causalt tst (Grangr 969) s dvlopd to amn whthr lag trms of on varabl sgnfcantl plan anothr varabl n a vctor autorgrssv rgrsson modl. * Corrspondng author. E-mal addrsss: awong@hkbu.du.hk. Acknowldgmnts Th scond author would lk to thank Profssors Robrt B. Mllr Howard E. Thompson for thr contnuous gudanc ncouragmnt. Ths rsarch s partall supportd b Northast Normal Unvrst th Natonal Unvrst of Sngapor Hong Kong Baptst Unvrst Columba Unvrst. nar Gangr causalt tst can b usd to dtct th causal rlaton btwn two tm srs. Howvr th lnar Grangr causalt tst dos not prform wll n dtctng nonlnar causal rlatonshps. To crcumvnt ths lmtaton Bak Brock (99) dvlop a nonlnar Grangr causalt tst to amn th rmanng nonlnar prdctv powr of a rsdual srs of a varabl on th rsdual of anothr varabl obtanng from a lnar modl. Hmstra Jons (994) hav furthr modfd th tst whch nabls acadmcs practtonrs to amn th bvarat nonlnar causalt rlatonshp btwn two srs. Nvrthlss th multvarat causal rlatonshps ar mportant but t has not bn wll-studd spcall for nonlnar causalt rlatonshp. Thus t s mportant to tnd th Grangr causalt tst to nonlnar causalt tst n th multvarat sttngs. In ths papr w frst dscuss lnar causalt tsts n multvarat sttngs thraftr dvlop a non-lnar causalt tst n multvarat sttngs. For an n

2 6 Z. Ba W.-K. Wong B. Zhang varabls nvolvd n th causalt tst w dscuss a n-quaton vctor autorgrssv rgrsson (VAR) modl to conduct th lnar Grangr tst tst for th sgnfcanc of rlvant coffcnts across quatons usng lklhood rato tst. If thos coffcnts ar sgnfcantl dffrnt from zro th lnar causalt rlatonshp s dntfd. W thn tnd th nonlnar Grangr tst from bvarat sttng to multvarat sttng. W notc that th bvarat nonlnar Grangr tst s dvlopd b manl applng th proprts of U-statstc dvlopd b Dnkr Kllr ( ). Cntral lmt thorm can b appld to th U- statstc whos argumnts ar strctl statonar wakl dpndnt satsf mng condtons of Dnkr Kllr ( ). Whn w tnd th tst to th multvarat sttngs w fnd that th proprts of th U-statstc for th bvarat sttngs could also b usd n th dvlopmnt of our proposd tst statstc n th multvarat sttngs whch s also a functon of U-statstc. Th papr s organzd as follows. W bgn n nt scton b ntroducng dfntons notatons statng som basc proprts for th lnar nonlnar Grangr causal tsts to tst for lnar nonlnar Grangr causal rlatonshps btwn two srs. In Scton 3 w frst dscuss th lnar Grangr causalt tsts n th multvarat sttngs thraftr dvlop th nonlnar Grangr causalt tsts n th multvarat sttngs Scton 4 gvs a summar of our papr.. Bvarat Grangr Causalt Tst In ths scton w wll rvw th dfntons of lnar nonlnar causalt dscuss th lnar nonlnar Grangr causalt tsts to dntf th causalt rlatonshps btwn two varabls.. Bvarat nar Grangr Causalt Tst Th lnar Grangr causalt s conductd basd on th followng twoquaton modl: Dfnton.. In a two-quaton modl: p = a + α + β + ε t t t t = = p (a) p = a + γ + δ + ε t t t t = = p (b) whr all { t } { t } ar statonar varabls p s th optmal lag n th sstm ε t ε t ar th dsturbancs satsfng th rgulart assumptons of th classcal lnar normal rgrsson modl. Th varabl{ t } s sad not to Grangr caus { t } f β = n (a) for an = p. In othr words th past valus of { t } do not provd an addtonal nformaton on th prformanc of { t }. Smlarl { t } dos not Grangr caus { t } f γ = n (b) for an = p. It s wll-known that on can tst for lnar causal rlatons btwn { t } { t } b tstng th followng null hpothss sparatl: H : β =... = β p = H : γ =... = γ p =. From tstng ths hpothss w hav four possbl tstng rsults: () If both Hpothss H H ar accptd thr s no lnar causal rlatonshp btwn { t } { t }. () If Hpothss H s accptd but Hpothss H s rjctd thn thr sts lnar causalt runnng undrctonall from { t } to{ t }.

3 Multvarat nar Non-nar Causalt Tsts 6 (3) If Hpothss H s rjctd but Hpothss H s accptd thn thr sts lnar causalt runnng undrctonall from { t } to{ t }. (4) If both Hpothss H H ar rjctd thn thr st fdback lnar causal rlatonshps btwn { t } { t }. Thr ar svral statstcs could b usd to tst th abov hpothss. On of th most commonl usd statstcs s th stard F-tst. To tst th hpothss H : β =... = β p = n (a) th sum of squars of th rsduals from both th full rgrsson SSE F th rstrctd rgrsson SSE R ar computd n th quaton (a) th F tst follows ( SSRR SSRF)/ p F = () SSR F /( n p ) whr p s th optmal numbr of lag trms of t n th rgrsson quaton on t n s th numbr of obsrvatons. If { t } dos not Grangr caus{ t } F n () s dstrbutd as F ( pn p ). For an gvn sgnfcanc lvlα w rjct th null hpothss H f F cds th crtcal valu F( α pn p ). Smlarl on could tst for th scond null hpothss H : γ =... = γ p = dntf th lnar causal rlatonshp from { t } to { t }.. Bvarat Nonlnar Causalt Tst Th gnral tst for nonlnar Grangr causalt s frst dvlopd b Bak Brock(99) latr on modfd b Hmstra Jons (994). As th lnar Grangr tst s nffcnt n dtctng an nonlnar causal rlatonshp to amn th nonlnar Grangr causalt rlatonshp btwn a par of srs sa { t } { t } on has to frst appl th lnar modls n (a) (b) to { t } { t } for dntfng thr lnar causal rlatonshps obtan thr corrspondng rsduals { ˆ ε t } { ˆ ε t }. Thraftr on has to appl a non-lnar Grangr causalt tst to th rsdual srs { ˆ ε } t { ˆ ε t } of th two varabls { t } { t } bng amnd to dntf th rmanng nonlnar causal rlatonshps btwn thr rsduals. W frst stat th dfnton of nonlnar Grangr causalt as follows: Dfnton.. For an two strctl statonar wakl dpndnt srs{ t } { Y t } th m-lngth lad vctor of t s gvn b m (... ) m=... t =... t t t+ t+ m -lngth lag vctor of t s dfnd as t (... ) t t t =... t = m Th m-lngth lad vctor Yt th - lngth lag vctor t of Y t can b dfnd smlarl. Srs{ Y t } dos not strctl Grangr caus anothr srs { t } nonlnarl f onl f: m m Pr( < < < ) t s t s t s m m = Pr( < < ) t s t s whr Pr( ) dnots condtonal probablt dnots th mamum norm whch s dfnd as Y = ma(... ) for an two vctors = (... n ) Y = (... n ). Undr Dfnton. th non-lnar Grangr causalt tst statstc s gvn b C ( m+ n) C3 ( m+ n) n () C( n) C4( n) n n

4 6 Z. Ba W.-K. Wong B. Zhang Whr m+ m+ C ( m+ n) I( t ) s nn ( ) t< s I( ) t s C( n) I( t ) s nn ( ) t< s I( ) t s m + m + C3 ( m+ n ) I( t ) s nn ( ) t< s C4( n) I( t ) s nn ( ) t< s f > I( ) =. f Th tst statstc s Hmstra Jons (994) posssss th followng proprt: Thorm.. For gvn valus of m > dfnd n Dfnton.undr th assumptons that { t }{ Y t } ar strctl statonar wakl dpndnt satsf th condtons statd n Dnkr Kllr (983) f { Y t } dos not strctl Grangr caus{ t } thn th tst statstc dfnd n () s dstrbutd as N( σ ( m )) asmptotcall th stmator of th varanc σ ( m ) s gvn b ˆT σ ( m ) ˆ ˆ = d d whr ˆ C ( m+ n) d = C( n) C( n) C3( m+ n) C4( n) C4( n) ˆ s a matr contanng lmnts K( n) ˆ j= 4 ωk( n) k = T ˆ ˆ ˆ ˆ ( At ( n) Ajt k+ ( n) + At k+ ( n) Ajt ( n)) ( n k+ ) t n whch K(n) = [n /4 [ s th ntgr part of f k = ωk ( n) = ( [( k) / K( n)) othrws ˆ m+ m+ A t = I( t ) ( ) s I t s n s t C ( ) m+ n ˆ A t = I( t ) ( ) s I t s n s t C ( ) n ˆ m+ m+ A3 t = I( t ) 3( ) s C m n n + s t ˆ A t = I( t ) 4( ) s C n n s t ts = ma( ) +... T m+ 3. Multvarat Grangr Causalt Tst In ths scton w frst dscuss th lnar Grangr causalt tsts n th multvarat sttngs thraftr dvlop th non-lnar Grangr causalt tsts from th bvarat sttngs to th multvarat sttngs. 3. Multvarat nar Grangr Causalt Tst W frst dscuss th lnar Grangr causalt tst from th bvarat sttngs to th multvarat sttngs. 3.. Vctor Autorgrssv Rgrsson Th lnar Grangr tst s appld n th vctor autorgrssv rgrsson (VAR) schm. For t = T th n- varabl VAR modl s rprsntd as:

5 Multvarat nar Non-nar Causalt Tsts 63 t A A( ) A( )... A n( ) t t A A( ) A( )... An( ) t = + A A ( ) A ( )... A ( ) nt n n n nn nt t t + (3) nt whr ( t... n s th vctor of n statonar tm srs at tm t s th backward opraton n whch t = t A ar ntrcpt paramtrs A j () ar polnomals n th lag oprator such that A j () = a j () + a j () + + a j (p) p t = ( t... n s th dsturbanc vctor obng th assumpton of th classcal lnar normal rgrsson modl. In practc t s common to st all th quatons n VAR to possss th sam lag lngth for ach varabl. So a unform ordr p wll b chosn for all th lag polnomals A j () n th VAR modl accordng to a crtan crtra such as Akak's nformaton crtron (AIC) or Schwarz crtron (SC). Along wth th Gauss-Markov assumptons satsfd for th rror trms ordnar last squar stmaton (OSE) s approprat to b usd to stmat th modl as t s consstnt ffcnt. Howvr long lag lngth for ach varabl wll consum larg numbr of dgrs of frdom. For ampl n th modl statd n quaton 3 thr wll b n(np + ) coffcnts ncludng n ntrcpt trms n varancs n(n - )/ covarancs to b stmatd. Whn th avalabl sampl sz T s not larg nough ncludng too man rgrssors wll mak th stmaton nffcnt thus caus th tst unrlabl. To crcumvnt ths problm on could adopt a Nar-VAR modl smngl unrlatd rgrssons stmaton tchnqu nstad of applng OSE to stmat th quatons smultanousl. W skp th dscusson of th Nar-VAR modl smngl unrlatd rgrssons stmaton n ths papr for smplct w onl us OSE to stmat th paramtrs n th VAR modl to dntf th causalt rlatonshp among vctors of dffrnt tm srs. 3.. Multpl nar Grangr Causalt Hpothss klhood Rato Tst To tst th lnar causalt rlatonshp btwn two vctors of dffrnt statonar tm srs t = ( t... n t = ( t... n whr thr ar n + n = n srs n total on could construct th followng n-quaton VAR as follows: [ ( )[ ( ) t A n A n n A [ n n t = + t A [ n A( ) [ n n A( ) [ n n t + (4) Whr A n [ A n [ ar two vctors of ntrcpt trms A ( ) [ n n A ( ) [ n n A ( ) [ n n A ( ) [ n n ar matrcs of lag polnomals. Smlar to th bvarat cas thr ar four dffrnt stuatons for th stnc of lnar causalt rlatonshps btwn two vctors of tm srs t t n (4): () Thr sts a undrctonal causalt from t to t f A () s sgnfcantl dffrnt from zro at th sam tm A () s not sgnfcantl dffrnt from zro; W sad A () s sgnfcantl dffrnt from zro f thr sts an trm n A () whch s sgnfcantl dffrnt from zro.

6 64 Z. Ba W.-K. Wong B. Zhang () thr sts a undrctonal causalt from t to t f A () s sgnfcantl dffrnt from zro at th sam tm A () s not sgnfcantl dffrnt from zro; (3) thr st fdback rlatons whn both A () A () ar sgnfcantl dffrnt from zro; (4) t t ar not rjctd to b ndpndnt whn both A () A () s not sgnfcantl dffrnt from zro. W not that on could consdr on mor stuaton as follows: (5) t t ar rjctd to b ndpndnt whn thr A () A () s sgnfcantl dffrnt from zro. Ths s th sam stuaton as thr () () or (3) s tru. To tst th abov statmnts s quvalnt to tst th followng null hpothss: () H : A () = () H : A () = (3) both H H : A () = A () =. On ma frst obtan th rsdual covaranc matr Σ from th full modl n (4) b usng OSE for ach quaton wthout mposng an rstrcton on th paramtrs comput th rsdual covaranc matr Σ from th rstrctd modl n (4) b usng OSE for ach quaton wth th rstrcton on th paramtrs mposd b th null hpothss H H or both H H. Thraftr bsds usng th F-tst n () on could us a smlar approach as n Sms (98) to obtan th followng lklhood rato statstc: ( T c)(log Σ log Σ ) (5) log Σ ar th natural logarthms of th dtrmnants of rstrctd unrstrctd rsdual covaranc matrcs rspctvl. Whn th null hpothss s tru ths tst statstc has a asmptotc χ dstrbuton wth th dgr of frdom qual to th numbr of rstrctons on th coffcnts n th sstm. For ampl whn w tst H : A () = on should lt c qual to np + thr ar n p rstrctons on th coffcnts n th frst n quatons of th modl. Hnc th corrspondng tst statstc ( T ( np+ ))(log Σ log Σ ) asmptotcall follows χ wth n n pdgrs of frdom. Th convntonal bvarat causalt tst s an spcal cas of th multvarat causalt tst whn n = n =. Bsds usng th F tst statd n () on could also us th lklhood rato tst n (5) ntroducd n ths papr to dntf th lnar causalt rlatonshp for two varabls n th bvarat sttngs ECM-VAR modl Consdr (Y Y nt ) to b a vctor of n non-statonar tm srs f contgraton sts wth rsdual vctor vcm t. t t = Y t for = n b th corrspondng statonar dffrncng srs. In ths stuaton on should not us th VAR modl as statd n (3) on should mpos th rror-corrcton mchansm (ECM) on th VAR to tst for Grangr causalt btwn ths varabls. Th ECM-VAR framwork s: t A A( ) A( )... A n( ) t t A A( ) A( )... An( ) t = + nt An An ( ) An ( )... Ann( ) nt whr T s th numbr of usabl obsrvatons c s th numbr of paramtrs stmatd n ach quaton of th unrstrctd sstm log Σ α t α + cm + α t t + n nt (6)

7 Multvarat nar Non-nar Causalt Tsts 65 whr cm t- s th rror corrcton trm. In partcular n ths papr w consdr to tst th causalt rlatonshp btwn two non-statonar vctor tm srs = (... ) t t n t Yt = ( Y t... Y n t = t t = Yt ar th corrspondng statonar dffrncng srs whr thr ar n + n = n srs n total. If t Y t ar contgratd wth rsdual vctor vcm t thn nstad of usng th n-quaton VAR n (4) on should adopt th n-quaton VAR n th followng: [ ( )[ ( ) t A n A n n A [ n n t = + t A [ n A( ) [ n n A( ) [ n n t α n [ + cmt + (7) α n [ whr α n [ α n [ ar two vctors of ntrcpt trms A( ) [ n n A ( ) [ n n A ( ) [ n n A ( ) [ n n ar matrcs of lag polnomals α n [ α n [ ar th coffcnt vctors for th rror corrcton trm cmt. Thraftr on should tst th null hpothss H : A () = or H : A () = to dntf strct causalt rlaton usng th R tst as dscussd n Scton Multvarat Nonlnar Causalt Tst In ths scton w wll tnd th nonlnar causalt tst for a bvarat sttng dvlopd b Hmstra Jons (994) to a multvarat sttng. 3.. Multvarat Nonlnar Causalt Hpothss As dscussd n Scton. to dntf an nonlnar Grangr causalt rlatonshp from an two srs sa { t } { t } n a bvarat sttng on has to frst appl th lnar modls n (a) (b) to { t } { t } to dntf thr lnar causal rlatonshps obtan thr corrspondng rsduals { ˆ ε t } { ˆ ε t }. Thraftr on has to appl a non-lnar Grangr causalt tst to th rsdual srs { ˆ ε } t { ˆ ε t } of th two varabls bng amnd to dntf th rmanng nonlnar causal rlatonshps btwn thr rsduals. Ths s also tru f on would lk to dntf stnc of an nonlnar Grangr causalt rlatons btwn two vctors of tm srs sa t = ( t... n t = ( t... n n a multvarat sttng. On has to appl th n-quaton VAR modl n (4) or (7) to th srs to dntf thr lnar causal rlatonshps obtan thr corrspondng rsduals. Thraftr on has to appl a non-lnar Grangr causalt tst to th rsdual srs nstad of th orgnal tm srs. For smplct n ths scton w wll dnot t = ( t... n Yt = ( Y t... Y n to b th corrspondng rsduals of an two vctors of varabls bng amnd. W frst dfn th lad vctor lag vctor of a tm srs sa t smlar to th trms dfnd n Dfnton. as follows. For t = n th m - lngth lad vctor th -lngth lag vctor of t ar dfnd rspctvl as m (... ) m =... t =... t t t + t + m t t t + t (... ) =... t = W n dnot M = ( m... m ) n = (... ) m = ma( m... m ) n m t Y t n l = ma(... ). Th m -lngth lad vctor Y th -lngth lag vctor of Y t M m l can b dfnd smlarl. Gvn m m w dfn th followng four vnts:

8 66 Z. Ba W.-K. Wong B. Zhang M () { } M t s < M { } M t s for an n < =... ; () { } t s < { } t s for an n < =... ; M (3) { } M Yt Ys < M { } M t s < = Y Y for an... n ; (4) { } Yt Y s < { } Yt Y s < for an = n... whr dnots th mamum norm dfnd n Dfnton.. Th vctor srs { Y t } s sad not to strctl Grangr caus anothr vctor srs { t } f: M M Pr < < Y Y < ) ( ) ( ) t s t s t s M M = Pr < < ) t s t s whr Pr( ) dnots condtonal probablt. 3.. Tst Statstc It's Asmptotc Dstrbuton Smlar to th bvarat cas th tst statstc for tstng non-stnc of nonlnar Grangr causalt can b obt as follows: CM ( + n ) C3 ( M+ n) n (8) C( ) n C4( ) n Whr CM ( ) + n nn ( ) n = n t< s = t s I( ) m + m + t s I( ) C ( ) n nn ( ) n = n t< s = t s I( ) C ( ) 3 M + n nn ( ) n t< s = t s I( ) m + m + t s I( ) C ( n) I( ) n 4 t s nn ( ) t< s = ts=ma( )+ T-m +n=t+-m - ma( ). Thorm 3.. To tst th null hpothss Y... Y dos not strctl H that { t n } t Grangr caus { t... n t} undr th assumptons that th tm srs... Y... Y ar strctl { t n t} { t n t} statonar wakl dpndnt satsf th mng condtons statd n Dnkr Kllr(983) f th null hpothss H s tru th tst statstc dfnd n (8) s dstrbutd as N( σ ( M )). Whn th tst statstcn (8) s too far awa from zro w rjct th null hpothss. A consstnt stmator of σ ( M z ) follows: ˆ σ ( M z ) = f ( θ) T Σ ˆ f ( θ) n whch ach componnt Σ j( j =...4) of th covaranc matr Σ s gvn b: Σ = 4 ω EA ( A ) j k t j t+ k k f k = ωk = othrws

9 Multvarat nar Non-nar Causalt Tsts 67 M+ ( ) A = h C ( M + ) t t t ( ) A = h C ( ) t t t M+ ( ) ( ) A = h C ( M + ) 3 t 3 t 3 A = h C ( ) 4 t 4 t 4 whr ( pctaton of ( ) z t as follows: h z =... 4 s th condtonal h z z gvn th valu of t s M M ( t ) t ( t t ) + + = ( t ) ( ) t = t t h E h h E h M+ M+ ( t ) ( ) t ( t ) = ( ). t h = E h 3 3 h E h 4 4 A consstnt stmator of Σ jlmnts s gvn b: Kn () Σ j = k k= ˆ 4 ω ( n) ( At n Ajt + k n At + k n Ajt n ) ˆ () ˆ () ˆ () ˆ + () ( n k+ ) t /4 Kn ( ) = [ n f k = ωk ( n) = ( [( k) / K( n)) othrws n m ˆ + m + A t = I( t ) s n s t = n ( I t ) s = C ( ) M + n n ˆ A t = I( t ) s n s t = n = t s I( ) C ( ) n n ˆ A4 t = I( t ) 4( ) s C n n s t = ts=ma( ) n n=t-m - ma( )+ a consstnt stmator of f ( θ ) s: T ˆ θ ˆ θ 3 f ( θ ) = ˆ ˆ ˆ ˆ θ θ θ4 θ4 C( m+ n) = C( n) C( n) T C3( M + n) C4( n ) C4( n ) n m ˆ + m + A3 t = I( t ) 3( ) s C m + n n s t = 4. Concluson Rmarks In ths papr w frst dscuss lnar causalt tsts n multvarat sttngs thraftr dvlop a non-lnar causalt tst n multvarat sttngs. Howvr thr s a dsadvantag of th Hmstra-Jons tst. For ampl Dks Panchnko (5) pont out that Hmstra-Jons tst mght hav an ovrrjcton bas on th null hpothss of Grangr non-causalt. Thr smulaton rsults show that rjcton probablt wll gos to on as th sampl sz ncrass. Dks Panchnko (6) addrss ths problm b rplacng th global tst b an avrag of local condtonal dpndnc masurs. Thr nw tst shows wakr vdnc for volum causng rturns than Hmstra-Jons tst dos. Bsds Hmstra-Jons tst othr forms of nonlnar causalt tst has also bn dvlopd. For ampl Marnazzo Pllcoro Stramagla (8) adopt thor of rproducng krnl Hlbr spacs to dvlop nonlnar Grangr causalt tst. And Dks DGod () dvlop an nformaton thortc tst statstcs for Grangr causalt. Th us bootstrap mthods nstad of asmptotc dstrbuton to calculat th sgnfcanc of

10 68 Z. Ba W.-K. Wong B. Zhang th tst statstcs. Thus furthr tnson of ths papr could nclud to dvlop multvarat sttngs for th mor powrful lnar non-lnar causalt tsts. Rfrncs Bak E.G. Brock W.A. 99 A Gnral Tst for Nonlnar Grangr Causalt : Bvarat Modl. workng papr Kora Dvlopmnt Insttut Unvrst of Wsconsn-Madson. Chang T.C. Qao Z. Wong W.-K. Nw Evdnc on th Rlaton btwn Rturn Volatlt Tradng Volum. Journal of Forcastng (forthcomng). Dnkr M. Kllr G. 983 On U- statstcs v. Mss' Statstcs for wakl Dpndnt Procsss Ztschrft fäur Wahrschnlchktsthor und vrwt Gbt Dnkr M. Kllr G. 986 Rgorous statstcal procdurs for data from dnamcal sstm. Journal of Statstcal Phscs Dks C. DGod J. A gnral nonparamtrc bootstrap tst for Grangr causalt. n: Global Analss of Dnamcal Sstms ds. H.W. Bror B. Krauskopf G. Vgtr IoP Publshng Brstol. Dks C. Panchnko V. 5 A not on th Hmstra-Jons tst for Grangr non-causalt. Studs n Nonlnar Dnamcs Economtrcs 9() -7. Dks C. Panchnko V. 6 A nw statstc practcal gudlns for nonparamtrc Grangr causalt tstng. Journal of Economc Dnamcs Control 3(9-) Grangr C.W.J. 969 Invstgatng causal rlatons b conomtrc modls cross-spctral mthods. Economtrca 37 (3) Hmstra C. Jons J.D. 994 Tstng for nar Nonlnar Grangr Causalt n th Stock Proc- Volum Rlaton. Journal of Fnanc 49(5) Kowalsk J. Tu.M. 7 Modrn Appld U-Statstcs. John Wl & Sons Nw York. Marnazzo D. Pllcoro M. Stramagla S. 8 Krnl Mthod for Nonlnar Grangr Causalt. Phscal Rvw ttrs (4) Artcl Nw W.K. Wst K.D. 987 A Smpl Postv Sm-Dfnt Htroskdastct Autocorrlaton Conststnt Covaranc Matr. Economtrca 55(3) Qao Z. Y. Wong W.-K. 8 Polc Chang ad-lag rlatons among Chna's Sgmntd Stock Markts. Journal of Multnatonal Fnancal Managmnt Qao Z. McAlr M. Wong W.-K. 9 nar Nonlnar Causalt of Consumpton Growth Consumr Atttuds. Economcs ttrs (3) Srflng R. 98 Appromaton Thorms of Mathmatcal Statstcs John Wl & Sons Nw York. Sms C.A. 98 Macroconomcs Ralt. Economtrca