PREDICTION OF HIGH-FREQUENCY DATA: APPLICATION TO EXCHANGE RATES TIME SERIES

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1 PREDICTION OF HIGH-FREQUENCY DATA: APPLICATION TO EXCHANGE RATES TIME SERIES Milan Marček Medis, spol. s r.o., Pri Dobroke, 659/8, Nira, Silesian Universiy, Insiue of compuer Science, Opava, Czech Republic Dušan Marček Universiy of Žilina, Faculy of Manageme Science and Informaics, Žilina, Slovak Republic Absrac This paper considers he use of ANN mehodology for parameers esimaion of he auoregressive condiional heeroscedasic (ARCH) processes. The paper provides heurisic approach of ARCH processes modelling. This approach is ofen employed o esimae he values of financial variables as raes of reurn, exchange raes, means and variances of inflaion, sock marke reurns and price indexes and also o predic heir variances. Keywords Ekonomerics, forecasing, RF neural nework economic hypohesis ARCH/GARC models. INTRODUCTION The usual assumpions for he linear regression model are ha he noise or disurbance ermsε have mean zero [ E( ε ) = 0 ], consan bu unknown variance σ ε = V ( ε ), and ha ε s are uncorrelaed. Someimes hese assumpions are unreasonable. In his paper we will briefly consider he case when he symmeric variance - covariance marix V of ε in Ordinary Leas Squares (OLS) procedure have unequal diagonal elemens, ha is, he errors ε do no have equal (nonconsan) variances (heeroscedasiciy), while some of he off-diagonal elemens of V are zero. Heeroscedasiciy can be caused by incorrec specificaion or by he use of he wrong funcional form. To remove he heeroscedasiciy i is necessary eiher o inroduce or o eliminae some variables or o inroduce some ransformaion so ha he variance errors will be uniform across all observaions. Financial variables as raes of reurn, exchange raes, inflaion, sock marke reurns, price indexes and oher economic variable of generally high frequency, are likely o originae from low complexiy chaos and respond no only o he mean, bu also o he variance and o he higher momens of random variables. I is widely know ha high frequency economic, especially financial daa change heir variances over ime. Deecion of implausible variance behaviour in such ime series provides imporan informaion which can improve he forecasing abiliy over shor ime period. The auoregressive condiional heeroscedasic process is based on such assumpion ha he recen pas gives informaion abou he one-period forecas variance. I is assumed ha he variance of a random variable forecas y is depend on he value of he condiioning variable y ha is V( y y ). ARCH models are going widely applied o series, where hourly, daily or weekly observaions are of ineres. Over he pas en years academics of compuer science have developed new sof echniques based on laes informaion echnologies such as sof, neural and granular compuing o help predic fuure values of high frequency financial daa. A he same ime, he field of financial economerics has undergone various new developmens, especially in finance models, sochasic volailiy, and sofware availabiliy.

2 This paper discusses and compares he forecass from volailiy models which are derived from compeing saisical and Radial Basic Funcion (RBF) neural nework (NN) specificaions. Our moivaion for his comparaive sudy lies in boh he difficuly for consrucing of appropriae saisical Auoregressive/Generalised Condiionally Heeroscedasic (ARCH-GARCH) models (so called hard compuing) o forecas volailiy even in ex pos simulaions and he recenly emerging problem-solving mehods ha exploi olerance for impression o achieve low soluion coss (sof compuing). The paper is organized as follows. In Secion we briefly describe he basic ARCH-GARCH model. In Secion 3 we presen he daa, conduc some preliminary analysis of he ime series and demonsrae he forecasing abiliies of ARCH-GARCH modes of an applicaion. In Secion 4 we inroduce he archiecure of fuzzy logic RBF neural nework for ime series forecasing. Secion 5 compares boh approaches and briefly concludes. ARCH MODELS As we menioned above, he basic idea of ARCH models is o incorporae in sandard regression or economeric model an independen variable, which evaluaed or predics he variance. The sample iniial model inroduced by Engle [4] is y = ε h h = a0 + ay or in erms of ψ he informaion se avaliable a ime, using condiional densiies y ψ ~ Ν ( 0, h ) () where ε is random error componen (whie noise) wih V ( ε ) = σ and normally disribued, a0, aare unknown parameers o be esimaed and h is he variance funcion. Bollerslev [] proposed a useful exension of Engle s ARCH model known as he generalised ARCH (GARCH) model for ime sequence { y } in he following form y = ν h, m s h = α + α y h () + β 0 i i i= = Model () forms he underlying basis for analysis of many financial processes. In [6] he models of dependence of exchange rae changes for four currencies (Deusche Mark, Swiss Franc, French Franc, Japanise Yen) agains US dollar are described by using six order of he ARCH model, represened in he following form ^ ^ 6 Δlog S = ε a + a( Δlog S ) 0 τ ττ =, =,,..., N where Δ log S = y is he weekly end rae of change of he log exchange rae by aking he firs logarihmic difference, N is sample size. ARCH-GARCH models are also designed o capure cerain characerisics ha are commonly associaed wih financial ime series. They are among ohers: fa ails, volailiy clusering, persisence, mean-reversion and leverage effec. As far as fa ails, i is well know ha he disribuion of many high frequency financial ime series usually have faer ails han a normal disribuion. The phenomena of faer ails is also called excess kurosis. In addiion, financial ime series usually exhibi a characerisic known as volailiy clusering in which large changes end o follow large changes, and small changes end o follow small changes. Volailiy is ofen persisen, or has a long memory if he curren level of volailiy affecs he fuure level for more ime periods ahead. Alhough financial ime series can exhibi excessive volailiy from ime o ime, volailiy will evenually sele

3 down o a long run level. The leverage effec expresses he asymmeric impac of posiive and negaive changes in financial ime series. I means ha he negaive shocks in price influence he volailiy differenly han he posiive shocks a he same size. Anoher varians ARCH-GARCH models can be found in [9], [0], [7]. 3 AN APPLICATIONOF ARCH-GARCH MODELS We illusrae he ARCH-GARCH mehodology on he developing a forecas model for he currency SKK/CZK (Slovak crown/czech crown) during he period from January, 007 o December, 008. The lengh of he ime series is 70 observaions. The daa are available a hp:// To build a forecas model he sample period (raining daa se denoed Α ) for analysis r,..., r 670 was defined, and he ex pos forecas period (validaion daa se denoed Ε ) r 67,..., r 70 The ime plo of he daa are shown in Fig.. Figure : The evoluion of cross rae CZK/SKK using he R sysem sofware [].36 0/0/ /0/008 CZK/SKK M0 007M07 008M0 008M07 We assume ha he generaing mechanism is probabilisic. The main purpose of ime series analysis is o undersand he underlying mechanism ha generaes he observed daa and, in urn o forecas he fuures values. In order o capure variance-nonlineariy an ARMA-GARCH model of changes raes was esimaed. Te specificaion of he esimaed model () wih Gaussian error disribuion resuled ino AR(7)+GARCH(,) process wih mean equaion []. r = r r r r 7 + ε r r r and he variance equaion (3) h ε = h The fied model vs. he original daase drawn by Eviews sofware [] is shown in Figure. 5 + Figure : The esimaed ARMA(7,0)+GARCH(,) model agains he original daases on whole range. (Residuals are a he boom) ARMA(7,0)+GARCH(,) ResidualS Acual CZKSKK Fied ARMA+GARCH M07 008M0 008M07

4 4 Soaf RBF Neural worapproach In his secion we show an approach of funcion esimaion for ime series modelled by means a granular fuzzy logic RBF neural nework based on Gaussian acivaion [5]. We proposed he neural archiecure according o Figure 3. Figure 3: Sof (fuzzy logic) NN The srucure of a neural nework is defined by is archiecure (processing unis and heir inerconnecions, acivaion funcions, mehods of learning and so on). In Figure 3 each circle or node represens he neuron. This neural nework consiss an inpu layer wih inpu vecor x and an oupu layer wih he oupu value ŷ. The layer beween he inpu and oupu layers is normally referred o as he hidden layer and is neurons as RBF neurons. Here, he inpu layer is no reaed as a layer of neural processing unis. One imporan feaure of RBF neworks is he way how oupu signals are calculaed in compuaional neurons. The oupu signals of he hidden layer are o =ψ ( x w ) (4) where x is a k-dimensional neural inpu vecor, w represens he hidden layer weighs, ψ are radial basis (Gaussian) acivaion funcions. Noe ha for an RBF nework, he hidden layer weighs w represen he cenres c of acivaion funcions in he hidden layer. To find he weighs w or cenres of acivaion funcions we used he adapive (learning) version of K-means clusering algorihm for s clusers. The second parameer of he RB funcion, he sandard deviaion is esimaed as K, (K ) muliple of he mean value of quadraic disance among he paerns and heir cluser cenres. The oupu layer neuron is linear and has a scalar oupu. The RBF nework compues he oupu daa se as s s ŷ = G ( x, c, v) = v ψ ( x, c ) =, v o (5), Α = where s denoes he number of he hidden layer neurons. The hidden layer neurons receive he Euclidian disances ( x c ) and compue he scalar values o, of he Gaussian funcion ψ ( x, c ) ha form he hidden layer oupu vecor o. Finally, he single linear oupu layer neuron compues he weighed sum of he Gaussian funcions ha form he oupu value of ŷ. If he scalar oupu values o, from he hidden layer will be normalised, where he normalisaion means ha he sum of he oupus from he hidden layer is equal o, hen he RBF nework will compue he normalised oupu daa se ŷ as follows s o s ψ ( x, c ), ŷ = G( x, c, v) = v s =, v, Α (6), s = = o, ψ ( x, c ) = = The nework wih one hidden layer and normalised oupu values o, is he fuzzy logic model or he sof RBF nework. To adap he weighs of v, we use he firs-order gradien procedure. In our case, he subecs of learning are he weighs v, only. These weighs can be adaped by he error back-propagaion algorihm. In his case, he weigh updae is paricularly simple. If he esimaed oupu for he single

5 oupu neuron is ŷ, and he correc oupu should be y, hen he error e is given by e = y - ŷ and he learning rule has he form v, v, + η e, =,,..., s; Α (7) o, where he erm, η (0,) is a consan called he learning rae parameer, o, is he normalised oupu signal from he hidden layer. Typically, he updaing process is divided ino epochs. Each epoch involves updaing all he weighs for all he examples. 5 Empirical Comparison The RBF NN was rained using he variables and daa ses as each ARCH-GARCH model above. The archiecure (7--) of he nework, repored in he las column of Table, consiss of hree layers and seven neurons: five neurons in inpu layer, one in hidden and oupu layer. Model AR(7)+ Sof RBF NN GARCH(,) Disribuion Topology (7--) Gaussian η = 0.03, K=.6 λ 0 () = 0.0 Table : Ex pos forecas RMSEs for ARCH-GARCH model and sof RBF NN. See ex for deails In he fuzzy logic RBF NN, he non-linear forecasing funcion f(x) was esimaed according o he expressions (6) wih Gaussian funcion ψ ( x, c ). The forecasing abiliy of paricular neworks was measured by he RMSE (Roo Mean Square Error) crierion of ex pos forecas periods (validaion daa se ). The deailed compuaional algorihm for he forecas RMSE values and he weigh updae rule for he nework is shown [8]. The resul of his applicaion is shown in Table. A direc comparison beween saisical (ARCH-GARCH) and neural nework models shows ha he saisical approach is beer han he neural nework compeior. The achieved ex pos accuracy of RBF NN (RMSE = ), bu is sill reasonable and accepable for use in forecasing sysems ha rouinely predic values of variables imporan in decision processes. Moreover, as we could see, he sof RBF NN has such aribues as compuaional efficiency, simpliciy, and ease adusing o changes in he process being forecas. Acknowledgemens This work was suppored by Czech gran foundaion under he gran GACR No.: 40/08/00 References [] BABEL, J., MECIAROVA, Z., PANCIKOVA, L.: Saisical and Sof Compuing Mehods in Cross Raes Modeling. Proceedings of seveh IC on SC Applied in Compuer and Economic Environmens.ICSC 009, European Polyechnical Insiue Kunovice, January 3, 009, Czech Republic [] BOLLERSLEV, D.: Generalized Auoregressive Condiional Heeroscedasiciy. Journal of Economerics 3 (986) [3] DING, Z., GRANGER, C.W., and ENGLE, R F.: A Long Memory Propery of Sock Marke Reurns and a New Mode. Journal of Empirical Finance,, (993) [4] ENGLE, R.F.: Auo regressive Condiional Heeroscedas-iciy wih Esimaes of he Variance of Unied Kindom Inflaion. Economerica 50 (98) [5] KECMAN, V.: Learning and sof compuing: suppor vecor machines, neural neworks, and

6 fuzzy logic. Massachuses: The MIT Press, 00 [6] KOCENDA, E.: A Tes for idd Residuals Based on Inegraing over he Correlaion Inegral. CERGE - EI, Working Paper 0, 996 [6] LI, D., and DU, Y.: Arificial inelligence wih uncerainy. Boca Raon: Chapman & Hall/CRC, Taylor & Francis Group, 008 [7] MARCEK, D., and MARCEK, M.: Time Series Analysis, Modelling and Forecasing wih Applicaions in Economics. The Universiy Press, Zilina, 00 [8] MARCEK, M., PANCIKOVA, L. and MARCEK, D.: Economerics & Sof Compuing. The Universiy Press, Zilina, 008 [9] NELSON, D.B.: Condiional Heeroskedasiciy in Asse Reurns: a New Approach. Economerica 59 () (99) [0] ZIVOT, E., WANG, J. (005): Modeling Financial Time Series wih S-PLUS, Springer Verlag, 005 [] hp:// [] hp:// Milan Marček milan.marcek@fri.uniza.sk Dušan Marček dusan.marcek@fri.uniza.sk