The 7 th Intenatinal Days f Statisti and Ecnmi, Pague, Septembe 9-2, 23 THE COUPLING SPECTRUM: A NEW METHOD FOR DETECTING TEMPORAL NONLINEAR CAUSALIT IN FINANCIAL TIME SERIES Ami Reza Alizad-Rahva Masud Adakani Iv Cibben Abstact Identifying dynamic causal elatinships between financial time seies may help explain maket dynami. The Gange causality (G-causality) test is a methd t detect linea causal elatinships between time seies. Hweve, thee exists significant evidence f nnlinea causality between financial time seies. Hence, seveal nnlinea extensins f G-causality (NLG-causality) wee ppsed. Meve, a new methd called the cupling spectum (CS) was ecently ppsed t find the nnlinea causal elatinship between tw time seies. In many financial cases, the diectin f causality is changing ve time. In this wk, we adapt the NLG-causality and CS methds by using a mving windw technique t identify pssible causality changes ve time. We cmpae the pefmance f the adapted CS and NLG-causality methds n a simulated tempal nnlinea causal system and a eal data set - the stck pices f Apple Inc. and Mift Cpatin. The simulated and empiical esults shw that the CS methd is me bust than the NLG-causality methd and that CS is capable f dealing with time-vaying nnlinea causality between financial time seies. Key wds: Causality infeence, nnlinea causality, Gange causality, time seies analysis, stck dynami JEL Cde: C9, C5, C58 Intductin The Gange causality (G-causality) test (Gange, 969) is a statistical hypthesis test f identifying causal elatinships between time seies. This methd estimates a linea egessin mdel with lagged values f the time seies {d t } (the dive time seies) used t pedict the futue values f { t } (the espnse time seies) in the pesence f lagged values f { t }. If the e f pedictin is educed by inclusin f {d t }, {d t } is the Gange-cause f { t }. The assumptin f lineaity in G-causality test can be vilated in eal applicatins and it cannt detect nnlinea causal elatinships (Bck, 99). Many investigatins in the liteatue pvide evidence f linea and nnlinea causality between financial time seies
The 7 th Intenatinal Days f Statisti and Ecnmi, Pague, Septembe 9-2, 23 (Hiemsta & Jnes, 994; öük, 26). Hence, diffeent nnlinea extensins f G-causality (NLG-causality) wee ppsed t detect nnlinea causality in financial data (Hiemsta & Jnes, 994; Diks & Panchenk, 26; Dhamala et al., 26; Papadimitiu et al., 23). Recently, a statistical methd called the cupling spectum (CS) (Alizad-Rahva & Adakani, 22) was ppsed t detect nnlinea causality between tw time seies. This nnpaametic methd can identify causality in diffeent scenais including unidiectinal and bidiectinal causality, linea and nnlinea causality, and time seies with small and lage sample sizes. In this wk, we use the CS methd t identify causality between financial time seies. In many financial data sets, the diectin f causality changes ve time. T deal with tempal causality, causality infeence methds can be cmbined with mving windw techniques t identify pssible causality changes ve time. In this pape, we extend the CS methd by using a mving windw technique and cmpae its pefmance n a simulated tempal nnlinea causal system t a mving windw adaptatin f the NLG-causality methd ppsed in Hiemsta & Jnes (994). We als cmpae thei pefmance n a eal data set -the stck pices f Apple Inc. and Mift Cpatin. The simulated and empiical esults shw that the CS methd is me bust than NLG-causality methd. The est f the pape is ganized as fllws: in Sectin, we begin by intducing sme ntatin and by setting the famewk f the NLG-causality and CS methds. The esults fm the simulatin and eal data examples ae pesented in Sectin 2. Finally, we cnclude with a discussin. Backgund. Ntatin Cnside a cause-effect elatinship as a cupled system cnsisting f a dive system D and a espnse system R, dented by D R. The samples f D ae dented by a time seies {d t }, cnsisting f n time pints. Nw, define D ( d, d 2,..., d ) epesenting L d lagged t t t t L d values f {d t }. Similaly, f { t } we can define R t with the length f lagged values L. Using the maximum nm, we define the distance f the pints cespnding t times t and t' D D max d d ; (.a) D tt ' t t ' Ld t t ' R tt ' R R ' max ; (.b) t t t t ' L (.c) tt ' t t '. 2
The 7 th Intenatinal Days f Statisti and Ecnmi, Pague, Septembe 9-2, 23.2 Causality and clseness In this sectin, we explain the cmmn undepinnings f bth the NLG-causality and CS methds. If we have D sufficiently lage values f L d and L ), i.e., R, t shuld be pedictable by the lagged values f {d t } and { t } (f f( R, D ). (2) t t t Theefe, pvided that D R, the clseness f the pints Dt and D t ' in the dive space and Rt and R t ' in the espnse space imply the clseness f t and t'. T quantify the dependency between the clseness f the pints in dive and espnse spaces, we can define a cnditinal pbability based n the distance f the pints. If the distance f Rt fm R t ' is smalle than δ R (i.e., tt ' δ ), and pvided that the distance between Dt and D t ' is d smalle than δ D (i.e., ' δ d ), then the pbability that the distance between t and t' is tt smalle than (i.e., tt' ) is dented by P( δ,δ ) P ( δ, δ ). (3) d R D d tt ' tt ' tt ' Bth the NLG-causality methd (Hiemsta & Jnes, 994) and the CS methd ae based n d the calculatin f P ( δ,δ )..3 Nnlinea Gange causality The Hiemsta-Jnes (HJ) methd (Hiemsta & Jnes, 994) is a nnlinea extensin f G- causality methd. The HJ test is a hypthesis test f the fllwing hypthesis H : D des nt Gange cause R. d d If we define P ( δ,δ ) P ( δ,δ ), the HJ test states that the null hypthesis H is tue if we have f all d P ( δ,δ ) P ( δ ). (4) In the wds, if D des nt cause R, the distance f the pints in the espnse space is independent f the cespnding distances in the dive space. Unde the assumptin that {d t } and { t } ae stictly statinay, Hiemsta and Jnes intduce the test statistic TVAL and its asympttic distibutin unde the null hypthesis H a d 2 TVAL ( δ,δ ) ( δ ) ~ (, ( d,, ) n P P N L L ) (5) whee the vaiance f the nmal distibutin and its estimated value is pesented in thei appendix. By using the bseved value f TVAL, we can make a cnclusin abut H. 3
d The 7 th Intenatinal Days f Statisti and Ecnmi, Pague, Septembe 9-2, 23.4 Cupling spectum (CS) methd As it is mentined in Sectin.2, pvided that D R, the clseness f the pints in the dive space implies the clseness f the pints in the espnse space. Theefe, accding t equatin (2), by inceasing D tt ', the pbability that t stays in the neighbhd f t ' educes. Hence, f fixed values f and δ d, P( δ,δ ) deceases mntnically as δ d inceases. On the the hand, pvided that D des nt cause R, dented by D R, d d P( δ,δ ) des nt vay by δ, d i.e., P( δ,δ ) P ( δ ). Theefe, by investigating d the changes f P( δ,δ ) with δ d, we can detect the causal elatinship D R. Cnside a causal elatinship D Rsimulated by equatin (7) in Sectin 2, whee and d epesent D and R systems. Figs. (a) and (b) visualize P( δ,δ ) by a cl map f d each pai f (δ,δ ). This epesentatin is called the cupling spectum (CS), dented by CS( D R ). If we bseve a change f cl in each clumn f the CS, this means that D R exists. Othewise, if all the clumns f the CS lack the cl change, we can cnclude d d D R. The standad deviatin f P( δ,δ ) f diffeent values f δ, dented by, can d be used t measue the changes f P( δ,δ ) with δ d in each clumn f CS( D R ). Fig. (c) depicts cespnding t Figs. (a) and (b) whee ( D R ) is cnsideably geate than ( R D ). T evaluate the significance f f each diectin f causality individually, we pemute the dive time seies t desty any dynamical causality and dente the pemuted time seies by p { d }. By using the pecentile btstap methd (Efn, 982), we btain the % cnfidence t inteval ( CI % ) f ( p ) D R. As an example, Fig. (c) shws the uppe bund f the Fig. : The cupling spectum (CS). (a) D R: the cl f each clumn changes with δ d ; (b) R D: the cl f each clumn is fixed; (c) σ and UCI 9%. CS(D R) CS(R D).8.6 -.4.2.25.2.5..5 (D R) (R D) UCI 9% (D p R) UCI 9% (R p D) -2-4 -2 (a) - -4-2 d (b) 4-4 -2 (c) Suce: wn
The 7 th Intenatinal Days f Statisti and Ecnmi, Pague, Septembe 9-2, 23 CI 9% ( UCI 9% ) f bth diectins f D R and R D. As ( D ) R lies utside the 9% p UCI ( D R) f sme values f δ, we cnfim that ( D ) R is significant, i.e., D causes R. Cnvesely, f all values f δ d, ( R D) is belw p UCI 9% ( R D ) ; hence, we cnclude D R. T evaluate the significance f ( D R ) elative t p UCI 9% ( D R ), we use the fllwing measue SIG ( D R ) (δ ) UCI (δ ) I (δ ) UCI (δ ) (6) 9% 9% δ whee I( ) is an indicat functin and I( x ) ; I( x ). If D R, f sme values f δ, ( D R ) is geate than p UCI 9% ( D R ) ; theefe, SIG ( D R ). On the the hand, pvided that D R, ( D R ) will be smalle than p UCI 9% ( D R ) f all δ ; cnsequently, SIG ( D R ). d d It is ntewthy that P ( δ,δ ) used in the HJ methd is a specific case f P ( δ,δ ) d in the CS methd whee = δ = δ =. In the wds, the HJ methd cnsides the d cupling spectums in Fig. nly f ne pai f (δ,δ ) (, ) and shuld als be equal t. As we see in Sectin 2., the value f has a sevee impact n the esults f the HJ d methd. Hweve, in the CS methd, we investigate P ( δ, δ ) f the whle ange f δ and δ d values and is detemined independently f δ and δ d (f me details abut detemining the value f, see (Alizad-Rahva & Adakani, 22)). In Sectin 2, we illustate hw the flexibility and geneality f the paametes in the CS methd makes it me bust than the HJ methd. 2 Results and applicatins In this sectin, the gal is t discve causal elatinships between tw time seies whee the diectin f causality is changing ve time. T find the tempal changing causality, we use the mving windw technique t detect the diectin f causality in a small peid f time. Nw we cmpae the HJ methd with the CS methd f simulated and empiical data. 5
The 7 th Intenatinal Days f Statisti and Ecnmi, Pague, Septembe 9-2, 23 2. Simulatin esults We evaluate the pefmance f the HJ and CS methds n simulated data t detect tempal changing causality. Cnside tw time seies {x t } and {y t } having a causal elatinship by the Hénn map (Wiesenfeldt et al., 2) 2 2 2 xt a xt bxt 2 cy x( xt y t ) (7-a) 2 2 2 y a y by c ( y x ) (7-b) t t t 2 x y t t whee a.4, b.3 and the initial values f x and y ae unifmly distibuted in [,.5]. Each time seies is assciated with the squaed lagged vesin f the the ne. The stength f causalities between and ae cntlled by c x y and c y x, espectively. T have tempal causality in the mdel, cx yand c y x change with time as shwn in Fig. 2(a). Hee, we use the velapping windw with windw length N w. In each step, the windw mves N N time pints futhe. In the simulatin, we used N 3 and N 6. The f w w f lag-lengths L d and L ae set t 2. A significant level f 5% is used f the HJ methd and we estimate the UCI9% f the CS methd. Figue 2 shws the cmpaisn f the CS and HJ methds f 5 tials with diffeent initial values f x and y in equatin (7). The mean f the SIG and TVAL values ve the 5 tials ae pltted. Figue 2(b) shws that the utcme f the CS methd is cnsistent with the eal causal elatinships. In the wds, the CS methd ) cectly detects the diectin f causality in all thee pats (the detected causality in pat (i) is vey weak); 2) distinguishes the stng and weak causality in the bidiectinal scenais; 3) finds f each diectin f causality the cect atis f causality stengths in diffeent pats that ae pptinal t eal atis. 6
Causality stength The 7 th Intenatinal Days f Statisti and Ecnmi, Pague, Septembe 9-2, 23 Fig. 2: Finding tempal causality f the simulated data. (a) The eal stength f the tempal causality; (b) CS methd; (c) and (d) HJ methd f diffeent values f. Real causality stength.2 (a) TVAL TVAL. (i) (ii) (iii) c c SIG 5 5 2 25 (b).5 CS (c) 5 5 5 2 25 HJ ( =.2) 4 (d) 2 5 5 2 25 HJ ( = ) 5 5 2 25 Blck # Suce: wn Figues 2(c) and 2(d) shw the pefmance f the HJ methd f.2 and, espectively. F.2, HJ des nt detect any causality in pat (i). In pats (ii) and (iii), HJ pefms as well as CS. Hweve, Fig. 2(d) illustates that inceasing affects the HJ methd. In this case, the advesely causality in pat (i) is nt detected; weak and stng causalities in bidiectinal scenais ae nt distinguishable; and f each diectin f causality, the atis f causality stengths in diffeent pats ae nt pptinal t the eal d atis. Indeed, as HJ cnsides P ( δ, δ ) just f a specific value f, the validity f the HJ esults depends seveely n and in all pactical applicatins will be unknwn. 2.2 Empiical esults In this sectin, we investigate the tempal causality between the stck pices f Apple Inc. (AAPL) and Mift Cpatin (MSFT). A ttal f 399 daily stck pices duing the time between Januay 2 and August 22 ae used. T ende each time seies weakly statinay, we cay ut a piecewise linea detending. We select a windw length f five mnth and N f is the duatin f ne mnth. The lag-lengths L d and L ae set t 5, i.e., we investigate the causal effect f the stck pices f past five business days n the pice f the next business day. In additin, a test significant level f 5% and methd (this value f esults in lage TVALs). UCI9%.7 ae used in the HJ is estimated f the CS methd. The tempal causalities AAPL MSFT and MSFT AAPL deived by the CS and HJ methds ae pltted in Figs. 3 and 4, espectively. The mnths in these figues epesent the middle mnth f each blck. As an evidence f detected causality, the timeline f AAPL and 7
SIG (MSFT AAPL) Win 2.Net/Pcket PC Win ME Office P Win P-manufactue Win P Pcket PC 22 Win Seve 23 Win Mbile 3 Mift Office 23 Win Mbile 3 SE Win Mbile 5 Win Mbile 6 Win Vista/ Office 27 Win Mbile 6. WinSeve 28-R2 Win 7 Office 2 bx 36/Win Mbile 7 WinSeve 28 SQL/Visual studi SIG (AAPL MSFT) Pw Mctsh G4 ibk PwBk G4 ibk Pw Mctsh G4 Seve G4 ipd G4/iBk seve G4 Pw Mctsh G4 ibk G4 G4 Cinema Disp. G5 Mac mini McBk P-5/Mac mini McBk P-7/Mac mini iphne ipd /Mac mini Pw Mctsh G5 McBk Ai/Mac P McBk/McBk P/iPd iphne 3G McBk/McBk P Mac Mini//Mac P/iPd seve McBk P/Ai/iPhne 3GS Mac OS ipd imac/mcbk ipad McBk Mac P McBk Ai Mac Mini/iPhne 3GS/4 iphne 4/McBk P ipad 2 McBk Ai/Mac Mini/ Mac OS ios5/iclud/iphne4 The 7 th Intenatinal Days f Statisti and Ecnmi, Pague, Septembe 9-2, 23 MSFT maj pducts ae depicted by aws in subplts (a) and (b), espectively. Figues 3 and 4 eveal the fllwing esults: The diectin f causality between these tw cmpanies changes ve time. Theefe, t investigate causality between financial time seies ve a lng peid f time, we have t use a mving windw t deal with this time-vaying causality. Mst f the pducts f each cmpany affect the the ne's stck pice immediately a cuple f mnths afte each pduct elease. Hweve, the numbe f the causal elatinships detected by the HJ methd is less than that f the CS methd. Thee ae detected causalities that culd be due t the facts the than pducts eleases, e.g., detected causalities in the secnd half-yea f 28 in MSFT AAPL. In geneal, f bth methds, it can be cncluded that the causal effect f AAPL n MSFT's stck pice is geate ve time than vice vesa. Fig. 3: Tempal causality between the stck pices f Apple (AAPL) and Mift (MSFT) detected by the CS methd. (a) AAPL MSFT, (b) MSFT AAPL..5 (a).5 Jan-.5 (b) Jul- Jan- Jul- Jan-2 Jul-2 Jan-3 Jul-3 Jan-4 Jul-4 Jan-5 Jul-5 Jan-6 Jul-6 Jan-7 Jul-7 Jan-8 Jul-8 ipd McBk Jan-9 Jul-9 Jan- Jul- Jan- Jul- Jan-2.5 Jan- Jul- Jan- Jul- Jan-2 Jul-2 Jan-3 Jul-3 Jan-4 Jul-4 Jan-5 Jul-5 Jan-6 Jul-6 Jan-7 Jul-7 Jan-8 Jul-8 Jan-9 Jul-9 Jan- Jul- Jan- Jul- Jan-2 8
TVAL(AAPL MSFT) Pw Mctsh G4 PwBk G4 Pw Mctsh G4 Seve G4 ipd G4/iBk seve G4 Pw Mctsh G4 Cinema Disp. G5 ibk ibk ibk G4 G4 Mac mini Pw Mctsh G5 McBk P-5/Mac mini McBk P-7/Mac mini The 7 th Intenatinal Days f Statisti and Ecnmi, Pague, Septembe 9-2, 23 Fig. 4: Tempal causality between the stck pices f AAPL and MSFT deived by the HJ methd (.7). (a) AAPL MSFT, (b) MSFT AAPL. 45 4 35 3 McBk Ai/Mac P McBk/McBk P/iPd ipd iphne 3G ipd McBk McBk/McBk P Mac Mini//Mac P/iPd seve McBk P/Ai/iPhne 3GS Mac OS ipd imac/mcbk TVAL(MSFT AAPL) Win 2.Net/Pcket PC Win ME Win P Pcket PC 22 Win Seve 23 Win Mbile 3 Win Mbile 3 SE Win Mbile 5 Win Mbile 6 Win Vista/ Office 27 WinSeve 28/SQL/VS Win Mbile 6. WinSeve 28-R2 Win 7 Office 2 bx 36/Win Mbile 7 Office P Win P-manufactue Mift Office 23 ipad ipd/apple TV McBk Ai McBk Ai/Mac Mini/ Mac OS ios5/iclud/iphne4 (a) 25 2 5 5 Jan- 45 4 (b) Jul- Jan- Jul- Jan-2 Jul-2 Jan-3 Jul-3 Jan-4 Jul-4 Jan-5 Jul-5 Jan-6 Jul-6 Jan-7 iphne /Mac mini Jul-7 Jan-8 Jul-8 Jan-9 Jul-9 Jan- McBk Mac Mini/iPhne 3GS/4Mac P iphne 4/McBk P ipad 2 Jul- Jan- Jul- Jan-2 35 3 25 2 5 Suce: wn Cnclusin The dynamic causal elatinships between many financial time seies have a nnlinea and time-vaying natue. In this pape, we extended a ecently ppsed appach called the cupling spectum (CS) t detect tempal nnlinea causalities between financial time seies. We cmpaed tw nnlinea causality infeence methds, the HJ and CS methds, and used the velapping mving windw technique t deal with tempal causalities. Examinatin f these tw methds n a simulated nnlinea causal elatinship shwed that due t the geneality f the CS paametes ve the HJ paametes, the pefmance f the CS methd is me bust than the HJ methd. In the wds, HJ can be seveely affected by its paamete value selectin. 5 Jan- Jul- Jan- Jul- Jan-2 Jul-2 Jan-3 Jul-3 Jan-4 Jul-4 Jan-5 Jul-5 In the final sectin we applied the CS and HJ methds t the stck pices f tw cmpanies, Apple Inc. and Mift Cpatin, ve a decade t detect the tempal causal effects f thei stck pices n each the. We fund that the diectin f causality changes ve time, especially aund the advent f new pducts. Hence, in cnclusin, in analyzing causality between financial time seies ve lng peids f time, we have t use mving windw techniques t deal with the time-vaying causality. Jan-6 Jul-6 Jan-7 Jul-7 Jan-8 Jul-8 Jan-9 Jul-9 Jan- Jul- Jan- Jul- Jan-2 9
The 7 th Intenatinal Days f Statisti and Ecnmi, Pague, Septembe 9-2, 23 Refeences [] Alizad-Rahva, A. R., & Adakani, M. (22). Finding weak diectinal cupling in multiscale time seies. Phys. Rev. E, 86, 625. [2] Bck, W. A. (99). Causality, chas, explanatin and pedictin in ecnmi and finance. (pp. 23-279). Bca Ratn, FL: CRC Pess. [3] Dhamala, M., Rangaajan, G., & Ding, M. (26). Estimating Gange causality fm Fuie and wavelet tansfms f time seies data. Phys. Rev. Lett., (), 87. [4] Diks, C., & Panchenk, M. (26). A new statistic and pactical guidelines f nnpaametic Gange causality testing. Junal f Ecnmic Dynami & Cntl, 3, 647-669. [5] Efn, B. (982). The jackknife, the btstap and the esampling plans. Philadelphia, PA, USA: SIAM. [6] Gange, C. W. J. (969). Investigating Causal Relatins by Ecnmetic Mdels and Css-spectal Methds. Ecnmetica, 37 (3), 424 438. [7] Hiemsta, C. & Jnes, J. D. (994). Testing f linea and nnlinea Gange causality in the stck pice-vlume elatin. Junal f Finance, 49, 639 664. [8] Papadimitiu, S., Bckwell, A., & Falutss, C. (23). Adaptive, hands-ff steam mining. Vldb, Belin, Gemany. [9] Wiesenfeldt, M., Palitz, U., & Lautebn, W. (2). Mixed state analysis f multivaiate time seies. Int. J. Bifucatin Chas, (8), 227-2226. [] öük, N. (26). Testing f linea and nnlinea Gange causality in the stck picevlume elatin: Tukish banking fims' evidence. Applied Financial Ecnmi Lettes, 2(3), 65-7. 2
The 7 th Intenatinal Days f Statisti and Ecnmi, Pague, Septembe 9-2, 23 Cntact Ami Reza Alizad-Rahva Depatment f Electical and Cmpute Engineeing, Univesity f Albeta 2 nd fl, ECERF Building, Univesity f Albeta, Edmntn, AB, Canada, T6G 2V4 a.alizad@ualbeta.ca Masud Adakani Depatment f Electical and Cmpute Engineeing, Univesity f Albeta, 2 nd fl, ECERF Building, Univesity f Albeta, Edmntn, AB, Canada, T6G 2V4 adakani@ualbeta.ca Iv Cibben Depatment f Finance and Statistical Analysis, Albeta Schl f Business, Univesity f Albeta 2-32G Business Building, Univesity f Albeta, Edmntn, AB, Canada, T6G 2R6 cibben@ualbeta.cabagm 2