Using the correlation dimension to detect non-linear dynamics: Evidence from the Athens Stock Exchange. David Chappell University of Sheffield

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1 Using he correlaion diension o deec non-linear dynaics: Evidence fro he Ahens Sock Exchange David Chappell Universiy of Sheffield & Theodore Panagioidis Loughborough Universiy Absrac : The sandardised residuals fro GARCH odels fied o hree sock indices of he Ahens Sock Exchange are exained for evidence of chaoic behaviour. In each case he correlaion diension is calculaed for a range of ebedding diensions. The resuls do no suppor he hypohesis of chaoic behaviour; i appears ha each se of residuals is iid. Key words: Non-linear Dynaics, Sock Indices, Chaos, Correlaion Diension Corresponding auhor: Professor David Chappell Deparen of Econoics Universiy of Sheffield 9, Mappin Sree Sheffield S 4DT, UK Eail: D.Chappell@Sheffield.ac.uk Tel +44 (0)4 347 Fax +44 (0)4 3458

2 INTRODUCTION A large body of lieraure has accuulaed over he pas hree decades concerning he validiy of he weak-for efficien arkes hypohesis (EMH) in financial econoics. The weak-for of he EMH posulaes ha successive oneperiod sock reurns are independen and idenically disribued (iid), i.e. he price levels reseble a rando walk. A he sae ie i is well known ha sock reurns are characerised by volailiy clusering. Addiionally we usually observe large reurns o be followed by large reurns and sall reurns o be followed by sall reurns, leading o coniguous periods of volailiy and sabiliy. Alhough os of he epirical ess of he efficien arkes hypohesis are based on linear odels, ineres in nonlinear processes has experienced a reendous rae of developen over he las few years (for an excellen review see Barne and Serleis 000). In his paper, we will exaine how he inroducion of he single European currency has affeced earlier clais in he lieraure ha he Ahens Sock Exchange (ASE) is characerised by deerinisic chaos as he ASE is in he process of becoing a fully developed capial arke. A liied nuber of sudies have appeared in he lieraure providing epirical resuls for he ASE (for a review see Panagioidis 003). None has esed for he presence of nonlinear dynaics (oher han GARCH) afer he inroducion of he coon currency. Siriopoulos (996) used onhly observaions of he ASE General Index fro 974: o 994:6. Using he BDS es saisic and he correlaion diension, i was concluded ha a GARCH odel could no explain he nonlineariies of he series ha igh be generaed by sei-chaoic behaviour. Barkoulas and Travlos (998) used daily observaions of he ASE30, he 30 os

3 arkeable socks, fro January 98 o Deceber 990. Models including an AR(p) and a GARCH (,) were eployed and diagnosic ools such as BDS, correlaion diension and Kologorov enropy were esiaed. They concluded ha he BDS es deecs reaining unspecified hidden srucure in he Greek sock reurns bu do no find evidence in suppor of a chaoic srucure in he Ahens Sock Exchange. Niarchos and Alexakis (998) followed a differen ehodology o es he EMH in he Ahens Sock exchange. They used error correcion odels and copared he speed of adjusen. Their evidence rejeced he EMH. More recenly, Apergis and Elepheriou (00) exained arke volailiy using daily observaions of he ASE General Index for he period January 990 o July 999. They copared differen GARCH odels based on he log likelihood and concluded ha he presence of persisence in volailiy clusering iplies inefficiency of he ASE arke. Lasly, Siourounis (00) eploys GARCH ype odels and ess for heir validiy using a daa se of daily closings of he ASE General Index for he period of 4 h January 988 unil 30 h Ocober 998. The Ljung-Box es saisic is eployed as a diagnosic ool and i was found ha he GARCH(,) and LGARCH(,) odels can explain quie saisfacorily he dependencies of he firs and second oens. CORRELATION DIMENSION Grassberger and Procaccia (983) suggesed he correlaion diension as a ool for disinguishing rando fro chaoic ie series. To briefly discuss his, le us sar wih he -diensional series, { } x n =, and fro his for he sequence of N = n + -diensional vecors X s = { x s, x n + s+,..., xs+ } s=. The seleced In July 000 Morgan Sanley announced he change in he classificaion of he MSCI Greece Index fro an eerging o a developed arke index wih effec fro he s of June 00 (see hp://

4 value of is called he ebedding diension and each X s is known as an -hisory of he series { } x n =. This convers he original scalar series ino a shorer series of N (diensional) vecors wih overlapping enries. Assuing ha he rue, bu unknown, syse which generaed { } x n = is θ-diensional and provided ha θ +, hen he se of -hisories recreaes he dynaics of he daa generaion process and can be used o analyse he dynaics of he syse - see Takens (98). The correlaion diension is based on he correlaion funcion (or correlaion inegral), C( N,, ε ) = #{( s, ) s, N}, where # denoes he nuber of N( N ) eleens in he se. The correlaion diension is defined as D C ( C( N,, ε ) log = li. ε 0 log( ε) In pracice, one esiaes D c for =,, 3,.,k for k no larger han around 0. If, as increases, D c coninues o rise hen his is sypoaic of a sochasic syse. If, however, he daa are generaed by a deerinisic process (consisen wih chaoic behaviour), hen D c will reach a finie lii a soe relaively sall. The correlaion diension can herefore be used o disinguish rue sochasic processes fro deerinisic chaos (which ay be low-diensional or high-diensional). Figure illusraes he heoreical relaionship beween log( ( N,,ε ) log(ε ) (see Chappell & Eldridge, 977). For a C and log(ε ) b, ε is oo sall and very few -hisories lie wih a disance ε of each oher. For log(ε ) > c, ε is oo large and all -hisories will lie wihin a disance ε of each oher. For b < log(ε ) < c, C ( N,,ε ) increases as increases; ( N,,ε ) C is he slope of he line for b < log(ε ) < c. This slope will increase iniially as is increased 3

5 log Figure : Theoreical relaionship beween log( ( N,,ε )) ( C ( N,,ε )) C and log(ε ) c d 0 a b log(ε ) Figure shows he heoreical relaionship beween D C and for a purely rando series and a possibly chaoic series (Chappell & Eldridge, 997). Figure : Theoreical relaionship beween D C and for a purely rando series and a possibly chaoic series. D C Rando Possibly chaoic: Necessary condiion 0 4

6 While he correlaion diension easure is herefore poenially very useful in esing for chaos, he sapling properies of he correlaion diension are, unforunaely, unknown. As Barne e al (997, pp. 306) pu i if he only source of sochasiciy is observaional noise in he daa, and if ha noise is sligh, hen i is possible o filer he noise ou of he daa and use he correlaion diension es deerinisically. However, if he econoic srucure ha generaed he daa conains a sochasic disurbance wihin equaions, he correlaion diension is sochasic and is derived disribuion is iporan in producing reliable inference. Moreover, if he correlaion diension is very large as in he case of highdiensional chaos, i will be very difficul o esiae i wihou an enorous aoun of daa. In his regard, Ruelle (990) argues ha a chaoic series can only be disinguished if i has a correlaion diension well below log 0 N, where N is he size of he daa se, suggesing ha wih econoic ie series he correlaion diension can only disinguish low-diensional chaos fro high-diensional sochasic processes - see also Grassberger and Procaccia (983) for ore deails. This paper will invesigae he following conflicing clais. Panagioidis (003) used a baery of ess which signalled ha he sandardised residuals of he preferred GARCH odels are iid processes. However, his does no exclude he case of deerinisic chaos (looks rando, bu isn ). On he oher hand, here are clais in he lieraure (see Barkoulas and Travlos 998, and Siriopoulos 996)) ha he ASE is being deerined by chaoic dynaics. To proceed, he sandardised residuals fro he esiaed GARCH odels in Panagioidis (003) are exained and he correlaion diension is calculaed for a range of ebedding diensions. As enioned above, if he daa under consideraion conain a deecable non-linear deerinisic coponen, he correlaion diension should increase wih 5

7 increasing values of he ebedding diension. However, his should level off a soe poin and reain consan for all furher values of he ebedding diension (see figure ). On he oher hand, if he rue daa generaing process is purely rando, hen we would expec he correlaion diension always o increase wih he ebedding diension. The oucoe is presened in Figures 3, 4 and 5. In each of hese, he ebedding diension is on he horizonal axis and he correlaion diension is on he verical axis and he calculaions were carried ou using a progra by Spro (998). I is clear in each of hese figures ha he correlaion diension keeps on increasing as a funcion of he ebedding diension and here is no sign ha his levels off a soe poin in any of he series. Consequenly, we could argue ha here is no evidence o sugges ha any for of chaoic non-linear deerinisic process is presen in he sandardised residuals of he preferred GARCH odels for he hree indices. This finding furher reinforces our arguen ha he series under invesigaion are iid. CONCLUSIONS In his paper, we have exained he clai ha here is chaoic behaviour in he ASE. We feel ha his was an ineresing exercise since his sock arke has recenly joined he Euro zone. To su up, we argued agains he chaos hypohesis in he case of he ASE. Firsly, here are heoreical reasons, which are explained in Lalley (999). Lalley discusses he resriced nuber of cases where i is ipossible o recover he original ie series when here is an added noise coponen. Secondly, on an epirical level, he correlaion diension failed o provide any evidence in favour of chaoic dynaics. The hree esiaed equaions are reproduced in he appendix. 6

8 Figure 3: Correlaion Diension for he sandardised residuals of he preferred GARCH odel of he ASE FTSE Correlaion Diension - FTSE Figure 4: Correlaion Diension for he sandardised residuals of he preferred GARCH odel of he ASE FTSE Mid 40 Correlaion Diension - FTSE Mid

9 Figure 5: Correlaion Diension for he sandardised residuals of he preferred GARCH odel of he ASE FTSE Sall Cap Index Correlaion Diension - FTSE Sall Cap REFERENCES Apergis, N. and Elepheriou, S. (00), Sock Reurns and Volailiy: Evidence fro he Ahens Sock Exchange, Journal of Econoics and Finance, 5, Barkoulas, J. and Travlos, N. (998), Chaos in an eerging capial arke? The case of he Ahens Sock Exchange, Applied Financial Econoics, 8, Barne, W.A., A.R. Gallan, M.J. Hinich, J.A. Jungeilges, D.T. Caplan, and M.J. Jensen (997), A single-blind conrolled copeiion aongs ess for nonlineariy and chaos, Journal of Econoerics, 8, Barne, W.A and Serleis, A. (000), Maringales, nonlineariy and chaos, Journal of Econoics Dynaics and Conrol, 4, Chappell, D., and Eldridge, R.M. (997), Nonlinear characerisics of he Serling/ECU exchange rae: , The European Journal of Finance, 3,

10 Grassberger, P. and Procaccia, I. (983), Measuring he Srangeness of Srange Aracors, Physica, 90, Lalley, S.P. (999), Beneah he noise, chaos, The Annals of saisics, 7,, Panagioidis T. (003), Marke Efficiency and he Euro: The case of he Ahens Sock Exchange, Discussion paper 03-08, Deparen of Econoics and Finance, Brunel Universiy. Ruelle, D., (990), Deerinisic Chaos: The Science and he Ficion, Proc. R.Soc. London A 47 (873), Siourounis, G. (00), Modelling volailiy and esing for efficiency in eerging capial arkes: he case of he Ahens sock exchange, Applied Financial Econoics,, Siriopoulos, C. (996), Invesigaing he behaviour of aure and eerging capial arkes, Indian Journal of Quaniaive Econoics,,, Spro, J.C. (998), Chaos Daa Analyzer, Professional Version., Universiy of Wisconsin a Madison, Madison, WI 53706, USA. 9

11 APPENDIX The hree esiaed equaions fro Panagioidis (003) are given below. Daa are daily reurns for he hree sock indices, calculaed fro daily closing prices, and he saple period is fro s June 000 o 3 s Deceber 00. R is he daily reurn, σ he condiional variance and ε he lagged squared residual. Nubers in parenheses are he corresponding saisics. I is he sandardised residuals fro each of hese equaions for which correlaion diensions are calculaed, and illusraed in Figures 3 5 in he ain ex 3.. The ASE FTSE 0 Index R ASEFTSE 0 = σ (.3) (.67) σ = ε σ (3.54) (5.73) (.88). The ASE FTSE Mid 40 Index R = R σ ASEFTSEMID 40 ASEFTSEMID 40 (.54) (.9) (.79) σ = ε σ (.33) (6.84) (46.0) 3. The ASE FTSE Sall Cap Index R ASEFTSESMALLCAP = R (.) (.34) σ = ε σ (.) (5.8) (8.78) ASEFTSESMALLCAP 3 For ore inforaion on he indices and heir coposiion hp:// and hp:// The daa are available free fro hp:// 0