THE HORIZON OF PREDICTION FOR EXCHANGE RATE EUR-LEU

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1 Aal of he Uiveriy of Peroşai Ecoomic 12(2) THE HORIZON OF PREDICTION FOR EXCHANGE RATE EUR-LEU DUMITRU CIOBANU ABSTRACT: For he chaoic yem he fac ha hey are deermiiic doe o make hem predicable However he predicive power i he cae of chaoic yem ca be improved ad hi ca be illuraed by weaher yem for which predicio for hor period have reached o a very good accuracy A poiive Large Lyapuov Epoe idicae he preece of chao i he evoluio of a ime erie ad i i ued o eablih a correc horizo of predicio KEY WORDS: chao; Lyapuov Epoe; echage rae; horizo of predicio JEL CLASSIFICATION: C53; G17 1 INTRODUCTION Chaoic yem are i fac comple deermiiic yem wih a large umber of variable ha ifluece he evoluio of he proce makig i impoible for huma o imulae i ad herefore makig hem o eem upredicable Thi alo make i impoible o deermie he iiial ae of he yem kowig ju he fial ae Mo procee ad yem foud i aure ivolve he ieracio of may facor ha allow u o caalog hem a chaoic yem Thu chao i me i olar yem dyamic evoluio of populaio he weaher chemical reacio ec I addiio he ecoomy ca be ee a a chaoic yem a facor ha brig a huge umber of variable i direc ivolveme of people The chao from comple yem i kow a chao deermiiic For he chaoic yem he fac ha hey are deermiiic doe o make hem predicable However he predicive power i he cae of chaoic yem ca be improved ad ca be illurae by weaher yem for which predicio for hor period have reached o a very good accuracy PhD Sude Uiveriy of Craiova Romaia ciobaubebedumiru@yahoocom

2 86 Ciobau D We mu emphaize he fac ha he emergece ad developme of chao heory could o have o ake place before he iveio of compuer a imulaio of comple yem wih may variable could o have doe wihou heir help A impora feaure of chaoic yem i Seiive Depedece o Iiial Codiio (SDIC) Thi ell u ha wo iiially cloe rajecorie depar epoeially i a fiie umber of ieraio omeime very quickly I uch a yem predicio i impoible ecep maybe he predicio for very hor period The mo ued ool for ideifyig hee procee from dyamical yem heory or eperimeal erie i Lyapuov characeriic epoe (LCE) Alhough chao i fudameal deermiiic i realiy i upredicable ecep for hor period Approimae ime limi ha ca ge accurae predicio for a chaoic yem i a fucio of he large Lyapuov epoe (Frio & Abarbael 1997) 1 Δma λ ma 2 MAXIMAL LYAPUNOV EXPONENT Coider a model ad wo eighborig poi 1 2 a he ime = arig poi for wo rajecorie i phae pace Deoe he diace bewee hee wo poi d() A he ime ha i afer movig he wo poi alog heir rajecorie diace bewee poi i meaured agai ad deoed d() Uig a differe ermiology we ca ay ha we applied a flow Φ o boh poi ad afer he ime period we meaured he diace bewee he wo poi d() The evoluio of he relaiohip bewee he wo diace i moiored d d Whe ed o ifiiy χ coverge o a value The value of hi limi i Lyapuov characeriic epoe If χ i i aid ha he wo orbi iiially cloe diverge epoeially uder he acio of he flow I alo ay ha he Lyapuov characeriic epoe meaure he rae of divergece of he yem (Georgecu 212) Afer calculaig he Lyapuov maimum epoe or he deermiaio of i approimaio we make aumpio abou he aure of he yem: λ The yem geerae a able fied poi or a able periodic orbi Negaive value of Lyapuov epoe are characeriic o o-coervaive or diipaive yem The higher he abolue value of he Lyapuov epoe he more able i he yem A uperable fied poi will have a Lyapuov epoe ha ed o miu ifiiy λ A yem wih uch a epoe i coervaive χ e

3 The Horizo of Predicio for Echage Rae EUR-LEU 87 λ I hi cae he orbi are uable ad chaoic Poi iiially very cloe diverge o arbirary value over ime A graphical repreeaio i imilar o a cloud of poi wihou a diic pah 21 Maimal Lyapuov Epoe for a differeiable fucio Lyapuov epoe λ meaure he gap bewee he rajecorie Le ad ε be wo eighborig poi Lyapuov epoe aifie he equaliy εe λ f ε f Separaig Lyapuov epoe ad paig o he limi we obai ε f 1 df 1 f λ lim lim l lim l ε ε d i 1 Becaue i f ad f f f df d we have f ' f ' f f ' ' ad for λ he calculaio formula i wrie 1 λ 1 ' i lim l f 22 Maimal Lyapuov Epoe for a rajecory For a rajecory of poi from d deermied by a ukow i differeiable fucio Φ : d d ca be eimaed Lyapuov maimum epoe of Φ a equece of average by ieraive elecio of o-egaive ieger 1 2 uil he

4 88 Ciobau D 1 1 l 1 1 Show ig of covergece The limi of hi row would repree a eimae for he large Lyapuov coefficie of he rajecory The row of average eimae he global Lyapuov epoe of applicaio Φ a relaive o he direcio ha i mo likely he large Lyapuov epoe for rajecory Wih he oaio y we have y Φ 1 1 Φ The followig pree he way how ca be choe he value We coider wo hrehold oe egaive y z ad oe poiive we ideify he diace oo mall ad oo large I i choe o be a mall a i ca bu greaer ha he miimum hrehold I choe differe from ad o ha δ 1 1 wih δ mall eough i abolue value o allow he eimaio Φ Φ J Φ z by which o ha he diace bu large eough ha he diace bewee ad eceed he miimum hrehold for oie removal z I i coidered he e C z z k k If 1 1 C δ 1 i mall eough o aify he codiio ad chooe 11 Oherwie coider all he cadidae from he e 1 1 C From hi e i choe o ha approimae a muliple of 1 1 which i mall i abolue value All muliple of 1 1 are o a lie paig hrough he origi Alo all muliple of k are o aoher lie paig hrough he origi Coie of he agle bewee he wo lie i z

5 The Horizo of Predicio for Echage Rae EUR-LEU 89 c 1 k 1 1 k 1 Thi quaiy ha value bewee -1 ad 1 The choice of k for which k i a good approimaio of 1 1 mu be uch ha he value of coie o be a cloe a poible o 1 uig a parial order defied by he relaio I ordered cadidae for ad c i c j i j i j accordig o hi parial order 23 Maimal Lyapuov Epoe for a ime erie k I choe a he large eleme For a fiie equece N of poi from d comig from a ukow differeiable fucio Φ : d d i proceeded a decribed previouly oly i o loger epeced ha he equece of average 1 1 l 1 1 o how ig of covergece bu imply he eleme of hi row are coidered a approimaio of he maimum Lyapuov epoe for he fucio Φ a 3 MAXIMAL LYAPUNOV EXPONENT FOR THE TIME SERIES OF EXCHANGE RATE EUR-LEU I ued he ime erie of echage rae EUR-LEU Hiorical value of foreig echage are available for dowload o he webie of Naioal Bak a hp://wwwbrro/baza-de-dae-ieraciva-64ap For ime erie modelig ad imulaio I ued MATLAB (R211 a) ad ool Toolbo (ime erie ool) The coidered ime erie coai 3881 record durig ad coi of echage rae quoaio of EUR-LEU eablihed by Naioal Bak of Romaia for weekday

6 9 Ciobau D From he graphical repreeaio (Figure 1) we ca ee ha i he cae of he echage rae EUR-LEU we deal wih rogly oliear proce Nolieariy doe o ecearily imply chao bu ay chaoic proce i oliear The workig mode for highlighig chao i ime erie i ikig hem io a mulidimeioal pace Traiio from oe-dimeioal ime erie o he correpodig d- dimeioal erie i ae pace i doe uig Take heorem We embed oe-dimeioal erie i a d-dimeioal pace by buildig vecor of legh d a follow: τ τ d 1 d 12 N τd 1 where τ i he umber of ime delay Value of EUR i LEI Time (repreeed a umber of obervaio) Figure 1 The evoluio of echage rae EUR-LEU over ime Ideificaio of a uiable ime delay i doe by buildig auo-muual iformaio fucio ad fidig hi fir local miimum I have coduced everal imulaio for differe value of he embeddig dimeio ad I have obaied value bewee 12 ad 21 The mo commo value for he delay ime wa τ=19 Uig Cao mehod for deermiig he embeddig dimeio I obaied value bewee 5 ad 7 I decided ha miimum embeddig dimeio i 6 he value mo ofe idicaed by e

7 The Horizo of Predicio for Echage Rae EUR-LEU Cel mai mare epoe Lyapuov Decalajul i imp Figure 2 The Large Lyapuov Epoe veru ime delay Cel mai mare epoe Lyapuov Dimeiuea paiului de cufudare Figure 3 The Large Lyapuov Epoe veru embeddig pace dimeio

8 92 Ciobau D Figure 2 ad 3 how he evoluio of he large Lyapuov epoe veru ime delay repecively veru embeddig pace dimeio Poiive value obaied are a idicaio of chao for echage rae ime erie of EUR-LEU 4 CONCLUSIONS I he cae of he ime erie of he echage rae bewee he euro ad leu imulaio idicae he preece of chao Wih a maimum Lyapuov epoe of abou 25 heoreically accepable predicio are poible for a umber of abou 4 ep Thu remai ope he problem of deermiig he model ha imulae reaoably well he ime erie of he echage rae o ha he predicio for fir ep o be wihi accepable error margi Deermiaio of chaoic behavior i impora o eablih a correc predicio horizo REFERENCES: [1] Abarbael H (1996) Aalyi of oberved chaoic daa Spriger New York [2] Cao L; Mee A; Judd K (1997) Modelig ad Predicig No-Saioary Time Serie Ieraioal Joural of Bifurcaio ad Chao Vol 7 No 8 pp [3] Frio T; Abarbael H (1997) Ocea graviy wave: A oliear aalyi of obervaio Joural of Geophyical Reearch Vol 12 No C1 pp [4] Georgecu V (212) Noliear Dyamic Chao Theory Applicaio i Fiace prepri [5] Goldmih M (29) The Maimal Lyapuov Epoe of a Time Serie A Thei i The Deparme of Compuer Sciece Cocordia Uiveriy Moreal Caada [6] Lych S (24) Dyamical yem wih applicaio uig MATLAB Birkhauer 24 [7] Roeei M; Colli J; De Luca C (1992) A pracical mehod for calculaig large Lyapuov epoe from mall daa e Phiica D Vol 65 pp [8] Suparu D; Vaile T (29) The Elecroic Commerce i he Globaliaio Era Aal of he Uiveriy of Peroşai Ecoomic 9(2) Uiveria Publihig Houe Peroşai pp [9] Turke (Vi) MC (21) The Globalizaio of he Bakig ad Fiacial Crii o Ieraioal Level Aal of he Uiveriy of Peroşai Ecoomic 1(1) Uiveria Publihig Houe Peroşai pp [1] Vailecu M; Mugiu-Pupăza MC (21) Iflaio Targeig - Bewee Theory ad Realiy Aal of he Uiveriy of Peroşai Ecoomic 1(3) Uiveria Publihig Houe Peroşai pp