Support Vector Regression Based GARCH Model with Application to Forecasting Volatility of Financial Returns

Transcription

1 Suppor Vecor Regression Based GARCH Model wih Applicaion o Forecasing Volailiy of Financial Reurns SHIYI CHEN * AND KIHO JEONG ** April, 007 ABSTRACT In recen years, suppor vecor regression (SVR), a novel neural nework (NN) echnique, has been successfully used for financial forecasing. This paper deals wih he applicaion of SVR in volailiy forecasing. Based on a recurren SVR, a GARCH mehod is proposed and is compared wih a moving average (MA), a recurren NN and a parameric GACH in erms of heir abiliy o forecas financial markes volailiy. The real daa in his sudy uses Briish Pound-US Dollar (GBP) daily exchange raes from July, 003 o June 30, 005 and New York Sock Exchange (NYSE) daily composie index from July 3, 003 o June 30, 005. The experimen shows ha, under boh varying and fixed forecasing schemes, he SVR-based GARCH ouperforms he MA, he recurren NN and he parameric GARCH based on he crieria of mean absolue error (MAE) and direcional accuracy (DA). No srucured way being available o choose he free parameers of SVR, he sensiiviy of performance is also examined o he free parameers. KEY WORDS: recurren suppor vecor regression; GARCH model; volailiy forecasing * (Corresponding auhor, Poenial speaker) China Cener for Economic Sudies (CCES), Fudan Universiy, Shanghai 00433, China; Shiyichen@fudan.edu.cn; TEL: ** School of Economics and Trade, Kyungpook Naional Universiy, Daegu 70-70, Korea khjeong@knu.ac.kr; TEL:

2 I. INTRODUCTION Volailiy is imporan in financial markes since i is a key variable in porfolio opimizaion, securiies valuaion, and risk managemen. Much aenion of academics and praciioners has been focused on modeling and forecasing volailiy in he las few decades. So far in he lieraure, he predominan model of he pas was GARCH model by Bollerslev (986), who generalizes he seminal idea on ARCH by Engle (98), and is various parameric exensions. The populariy of GARCH model is due o is abiliy o capure many of he empirically sylized facs of financial ime series, such as ime-varying volailiy, persisence and volailiy clusering (Marcucci, 005); see Bollerslev, Chou and Kroner (99) for lieraure surveys. Evidence on he forecasing abiliy of GARCH model is somewha mixed. Anderson and Bollerslev (998) show ha GARCH model provides good volailiy forecas. Conversely, some empirical sudies show ha GARCH model ends o give poor forecasing performances; for example, Figlewski (997), Cumby e al. (993), Jorion (995, 996), Brailsford and Faff (996), and McMillan e al. (000). To improve he forecasing abiliy of GARCH model, some alernaive approaches have been advocaed from he perspecive of esimaion and forecasing. Neural nework (NN) is one such mehod. In recen years, NNs have been successfully used for forecasing financial ime series; for recen work, see Fernandez-Rodriguez e al. (000) and Refenes and Whie (998). The main appeal of NNs is heir flexibiliy in approximaing any non-linear funcion arbirarily well wihou a priori assumpions abou he properies of he daa; see Hornik e al. (989) for a discussion of he NN universal approximaion propery. Moivaed by heir good propery and promising resuls in a broad range of financial applicaions, various NN-based GARCH models have been suggesed and applied o forecasing volailiy. The basic finding suppors ha NN-based GARCH ouperforms radiional GARCH models in forecasing condiional volailiy; see Donaldson and Kamsra (997), Schienkopf e al. (000), Taylor (000), Dunis and Huang (00). However, NN suffers from a number of weaknesses including he need for a large number of conrolling parameers, difficuly in obaining a global soluion and he danger of over-fiing (Tay and Cao, 00). The over-fiing problem is a consequence of he opimizaion algorihms used for parameer selecion and he saisical measures used o selec he bes model. Recenly, a novel NN algorihm, called suppor vecor machine (SVM), was developed by Vapnik and his co-workers (995, 997) and is gaining populariy due o many aracive feaures. While he radiional NN implemens he empirical risk minimizaion (ERM) principle, SVM implemens he srucural risk minimizaion (SRM) principle which seeks o minimize an upper bound on he Vapnik-Chervonenkis (VC) dimension (generalizaion error), as opposed o ERM ha minimizes he error on he in-sample esimaing daa;, refer o Gunn (998) for a good inroducion o SVM and relaed conceps. Based on SRM principle, SVM achieves a balance beween he raining error and generalizaion error, leading o beer forecasing performance han radiional NN. Selecing he bes model in SVM is equivalen o solving a quadraic programming, which gives SVM anoher meri of a unique global soluion. SVM was originally developed for classificaion problems (SVC)

3 and hen exended o regression problems (SVR). The main purpose of his paper is o formulae some ypes of SVR-based GARCH models and o compare he forecasing performance o he resuls obained from a moving average (MA), a recurren NN and a parameric GACH (MLE). Recenly, Pérez-Cruz e al. (003) also proposed a SVR-based GARCH(,) model and showed ha he proposed mehod provided beer volailiy forecass han parameric GARCH model. However, hey used feedforward SVR procedure which has he same srucure as auoressive (AR) process and has poor abiliy o model he long-ime memory (Haykin, 999). In his paper, we apply he recurren SVR procedure, firsly proposed by Chen and Jeong (005), which can inroduce ARMA srucure ino eiher mean funcion or condiional variance. The crieria of mean absolue error (MAE) and direcional accuracy (DA) reveal ha recurren SVR-based GARCH model ouperforms MA, MLE and NN-based GARCH model in he one- and muli-period-ahead forecass of volailiy. This paper is organized as follows. Secion inroduces he heory of SVR. Secion 3 specifies he empirical model and forecasing scheme. Secion 4 uses he Mone Carlo Simulaion o evaluae how he models perform under conrolled condiions. Secion 5 describes he GBP exchange raes and NYSE composie index daa and discusses he volailiy forecasing performance of all models for real daa. This paper concludes in secion 6. II. SUPPORT VECTOR REGRESSION SVR performs by nonlinearly mapping he inpu space ino a high dimensional feaure space and hen runs he linear regression in he oupu space. Thus, linear regression in he oupu space corresponds o nonlinear regression in he low dimensional inpu space. As he name implies, he design of he SVR hinges on he exracion of a subse of he raining daa ha serves as suppor vecors and ha herefore represens a sable characerisic of he daa. N { } Given a raining daa se ( x, y ), wih vecor inpus m0 x =, scalar oupus y, and unknown funcion g( x ), we need o esimae a decision funcion f ( ) approximaes g( x ) as below. x ha m j j () j= T ( ) ϕ ( ) ϕ( ) f x = w x + b= w x + b T T where ϕ( x) = ϕ ( x),, ϕ ( x), w= w,, w. The nonlinear funcion ϕ ( x) m m is he feaures of he inpu space, in SVR jargon. The dimension of he feaure space is m which is direcly relaed o he capaciy of he SVR o approximae a smooh inpu-oupu mapping; he higher he dimension of he feaure space, he more accurae he approximaion j will be. Parameer w i denoes a se of linear weighs connecing he feaure space o he 3

4 oupu space, and b is he bias. Using he decision funcion f ( x ), we can achieve he bes generalizaion capabiliy in forecasing y on new inpus. In order o derive he decision funcion, coefficiens w i and b have o be esimaed from he daa. Firs, we define a linear ε -insensiive loss funcion, Lε ( xyf,, ( x) ) proposed by Vapnik (995),, originally (,, ( )) L x y f x ε ( ) ε ( ) y f x for y f x ε = 0 oherwise () Under his loss funcion, errors belowε are no penalized; we can ignore he error and say he prediced f ( x ) has no loss. The derivaion of SVR follows he principle of srucural risk minimizaion ha is rooed in VC dimension heory. The primal consrained opimizaion problem of ε -SVR is obained as below. N ' ' min Φ ( wb,, ξ, ) ( ) () ' N ξ = w + C ξ ξ ξ N + = N w,, b (3) ( ) T s.. w ϕ x + b y ε + ξ =,,, N (4) ( ) ' T y w ϕ x b ε + ξ =,,, N (5) ' 0, ξ 0,,, N ξ = (6) The formulaion of he cos funcion ( wbξ,,, ' ξ) Φ in equaion (3) is in perfec accord wih he principle of srucural risk minimizaion; see Figure (in which he dark circles are daa poins exraced as suppor vecors). In equaion (3), he firs erm, w, is a measure of he funcion flaness, minimizing wha is relaed o maximizing he margin of separaion / w, i.e., indicaes maximizing he generalizaion abiliy. The second erm describes he ε -insensiive loss funcion (denoed by he nonnegaive slack variables similar o, alhough no idenical wih, he empirical risk employed in NN ' ξ i and ξ i ) and is [Figure ] 4

5 The corresponding dual problem of nonlinear SVR can be derived using he Karush-Kuhn-Tucker condiions as follows: N N N N ' ' ' ' min ( )( ) ( ) ( ) ( ) () ' α N s αs α α K xs x + ε α + α y α α α s= = = = (7) N = ( ' ) s.. α α = 0 (8) C α α s = N (9) N ' 0,,,,, where, ' α and α are he Lagrange mulipliers. The dual problem is easier o solve han he primal problem by relying on a quadraic programming (QP) scheme (Scholkopf and Smola, 00; Deng e. al., 004). We can hen use hem o obain he soluion of he primal problem: N = ' ( α α) ϕ( x ) w= (0) N ' ' ( α α) ( x x ) ( α α) ( x x ) N b= y + y K + K () j k j k = =, for ' C α j, α k 0, N Subsiuing equaion (0) and equaion () ino equaion (), he decision funcion can be obained: N N, () = = ' T ' ( ) ( α α) ϕ ( ) ϕ( ) ( α α) (, ) f x = x x + b= K x x + b T where K( x, x) ϕ ( x ) ϕ( x) = is he kernel funcion. The SVR heory considers he form of K( x, x ) in he feaure space wihou specifying ( x) ϕ explicily and wihou compuing all he corresponding inner producs. Therefore, kernels provide he flexibiliy of he high dimensional feaure space for low compuaional coss and are a crucial par of SVR. No analyical mehod is currenly available o deermine he mos suiable kernel for a paricular daa se. This paper experimens wih hree differen kernels o invesigae he effec of a kernel ype. 5

6 Linear: K( x, x) T Polynomial: K( x x) ( x x ) = x x (3) T, = + d (4) Gaussian: K( x x) x x, = exp σ (5) III. EMPIRICAL MODELING AND FORECASTING SCHEME In his paper, he daa we analyze is jus he daily financial reurns, y, convered from he corresponding price or index, I, using coninuous compounding ransformaion as ( ) y = 00 log I log I (6) + A GARCH (, ) specificaion is he mos popular form of condiional volailiy. As such, hroughou he paper he analysis is resriced o he case of GARCH (,) process. The Linear and Nonlinear GARCH (, ) Models For he parameric GARCH model, GARCH (,) model is usually specified as follows: y = c+ φ y + u (7-) h = κ + δ h + α u (7-) The imporan poin is ha he condiional variance of u is given by h = E u = u. Thus, he condiional variance of u is he ARMA process given by he expression h in equaion (7-) (Bollerslev, 986; Hamilon, 997; Enders, 004). ( ) u = κ + δ + α u + w δ w (8) = w u u u h 6

7 where w is whie noisy errors. The parameers, κ δ and α mus saisfy κ > 0, δ 0, and α 0 o ensure ha he condiional variance is posiive. Togeher wih he nonnegaive assumpion, if δ+ α <, hen u is covariance saionary. For recurren SVR and NN mehods, he nonlinear AR() GARCH(,) model has he following form: ( ) y = f y + u (9-) (, ) u = g u w + w (9-) Recurren SVR-based GARCH Modeling and Forecasing Scheme The algorihm of he recurren SVR-based GARCH model is described as follows: STEP : run he SVR-based AR() model for reurns y in he full sample period T, ( ) ( ) y = f y + u =,,, T, o obain residuals, u, u,, ut. STEP : recursively run he recurren SVR for squared residuals, u u u ( T < T) updaing, (, ) u = g u w + w o obain 60 one-period-ahead forecased volailiies. s sample : =,,, T u T + nd sample : =,,, T + u T h sample : =,,, T + 59 u T + 59+,,, T wih STEP 3: run he recurren SVR for squared residuals, u, u,, u ( T = T 0) updaing o obain 0 muli-period-ahead forecased volailiies. esimaing sample : =,,, T u, u,, u T T+ T+ T+ 0, wihou For each of 60 esimaions, he recurren SVR procedure proposed by Chen and Jeong (005) is run as follows; he residuals of w in equaion 9- are firs se o be zero series; hen 7

8 run he feedforward SVR o obain esimaed residuals; using he esimaed residuals as new w inpus we can carry ou his process repeaedly unil he sopping crierion is saisfied. The appropriae parameer ε dramaically depends on he given daa bu is no very sensiive o he same daa; afer invesigaion, we choose ε = for simulaion daa and ε = 0.05 for real daa. The value of C has been se o 0. for boh daa because invesigaion reveals ha he soluion is no very sensiive o C for a wide range. In addiion, he fixed widh value of 0. for Gaussian kernels and d= for polynomial kernels, are also se for he convenience of comparison. Evaluaion Measures and Proxy of Acual Volailiy We evaluae he forecasing performance using wo sandard saisical crieria: mean absolue forecas error (MAE) and direcional accuracy (DA), expressed as follows (Brooks, 998; Moosa, 000): n i i n i = MAE = u u (0) DA (%) n 00 = a () n i= ( i+ i ) where i u u ( u ) i+ ui 0 ai = 0 oherwise MAE measures he average magniude of forecasing error which disproporionaely weighs large forecas errors more genly relaive o MSE; and DA measures he correcness of he urning poins forecass, which gives a rough indicaion of he average direcion of he forecased volailiy. The fundamenal problem wih he evaluaion of volailiy forecass of real daa is ha volailiy is unobservable and so acual values, wih which o compare he forecass, do no exis. Therefore, researchers are necessarily required o make an auxiliary assumpion abou how he acual ex pos volailiy is calculaed. In his paper, we use square of he reurn minus is mean value as he proxy of acual volailiy agains which MAE and DA can be calculaed. This approach is similar o he sandard one, squared reurns, because he mean of reurns is usually close o zero. The proxy of acual volailiy in real daa is expressed as ( ) u = y y () where y : reurns; y : mean of reurns. 8

9 This proxy has been used in many recen papers such as Pagan and Schwer (990), Day and Lewis (99), Chan e al. (995), Wes and Cho (995), Chong e al. (999), Brooks (00), and Brooks and Persand (003). Specificaion of oher Mehods The MA mehod uses weighed moving averages of pas squared innovaions o forecas volailiy (Niemira and Klein, 994). For simulaed daa, he MA forecas for he nex-day volailiy, using he 5 mos recen observaions, is expressed as = (3) u5, + uj 5 j= 4 For real daa, he MA forecas for he nex-day volailiy is expressed as (Engle e al., 993) u ( y y ) (4) 5, + = j 5, 5 j= 4 where y5, = yj 5. j= 4 The recurren neworks experimened in his sudy are mulilayer perceprons (MLP) and a radial-basis funcion (RBF) nework wih he addiion of a global feedback connecion from he oupu layer o is inpu space. We specify a MLP nework wih he following archiecures: one nonlinear hidden layer wih 4 neurons using a an-sigmoid differeniable ransfer funcion, and one linear oupu layer wih neuron. The fas raining Levenberg-Marquard algorihm is chosen and designed ino a raining funcion. The value of he learning rae parameer is The RBF nework used in his sudy is a generalized regression neural nework (GRNN) which also has wo layers: he firs layer is a radial basis layer whose weighs are se o ransposed inpus, and he oupu layer is a special linear layer whose weighs are se o arge vecors. The spread of he radial basis funcion is.0. These specificaions and choices are quie sandard in he lieraure of neural neworks. Empirical Framework In his sudy he forecass are obained by applying he Mone Carlo mehod, following he suggesions in Andersen and Bollerslev (998), and Clemens and Smih (999, 00). The main moivaion for conducing a simulaion experimen is ha, since he rue volailiy is known, he candidae volailiy measures can be compared wih cerainy. We also fi each of Each poin forecas is obained as he average over all replicaions. 9

10 he models o he daily reurns on he GBP exchange rae and NYSE sock indexes and forecas heir respecive volailiy. The empirical modeling and forecasing scheme described above are employed for boh simulaion and real daa. The resuls in his paper are calculaed via MATLAB 7.0 sofware. IV. MONTE CARLO SIMULATION In his secion we invesigae he forecasing performance of all candidaes using arificial simulaed daa under conrolled condiions. To generae he daa, we firs need o parameerize he GARCH (,) model in equaions (7) wih he following seings (,,,, ) ( 0,0.5,0.0005,0.8,0.) c φ κ δ α = for medium persisence and a disurbance erm u disribued firs as Gaussian and hen as a Suden's wih five degrees of freedom (kurosis = 5). The second disribuion ries o model he excess of kurosis ha usually appears in real financial series. Using he same specified models, wo arificial samples wih sizes, 500 and 000, are creaed under wo disribuions assumpion, giving a oal of four siuaions. To limi he compuaional burden, each siuaion is replicaed only 50 imes. Then he muliple simulaed y and h are 500-by-50 and 000-by-50 elemen marices for differen disribuion. For each replicaion, he recurren SVR-based GARCH(,) model and ohers are esimaed and forecas errors are calculaed using he forecasing schemes described in Secion 3. We have repored he resuls of one-period-ahead forecasing measures for four siuaions, respecively, in Tables I (a) and (b) and 0-period-ahead forecasing resuls in ables II (a) and (b). The repored resuls are he mean values of 50 independen replicaions. [Tables I (a) and (b)] Le us firs evaluae he one-period-ahead forecass using he updaing forecasing scheme described in Table I. In erms of he sum rankings of MAE measures, he order of he forecasing abiliy of he differen mehods from he highes o lowes is displayed in urn as follows: poly-svr, rbf-svr, linear-svr, RBF, MLE, MA and MLP. Concreely, in he siuaion of normal disribuion, MLE which is ranked fourh behaves no bad and is only inferior o he hree kinds of SVR mehods in he 500 sizes, and even ranked hird, is only defeaed by polyand rbf-svr in he 000 sizes. As well, he MAE values of rbf-, poly-, and linear-svr and MLE are also close, for example, , , and , respecively, for he 500 sizes, and , , and for he 000 sizes; he laer of which becomes smaller for all mehods when he sizes of he sample increase. However, one hing noeworhy here is ha even hough he daa saisfy he Gaussian assumpion, he SVR mehods sill ouperform MLE in forecasing volailiy error magniude. In he siuaion of disribuion, he forecasing performance of MLE grows i.e., denoes SVR wih differen kernel funcion (polynomial, Gaussian and linear). 0

11 poorer, only ranked second las for he 500 sizes and hird las for he 000 sizes, and he difference of MAE values beween SVR and MLE becomes larger. For insance, he values of hree kinds of SVR are all below for he 500 sizes and below for he 000 sizes, while hose of MLE are and , respecively. The performance of NN is confusing: MLE is beer han MLP bu worse han RBF. According o he sum rankings of he DA measures, rbf-svr ranks highes for all four siuaions; linear- and poly-svr are ranked second side by side; now, MLE is ranked fourh and MA las among all candidaes. In he siuaion of normal disribuion, MLE behaves beer han in forecasing error magniude, and is ranked second for he 500 sizes (76.7%) only inferior o rbf-svr (86.44%) bu equal o linear- and poly-svr (76.7%), and also ranked second for he 000 sizes (79.83%) only inferior o rbf-svr (83.05%) bu beer han linear- and poly-svr (74.58% and 7.9%). These values of SVR and MLE are close and, paricularly higher han he NN and MA mehods. Alhough good, MLE canno defea rbf-svr even in he case of normal disribuion and large sample sizes. As for he siuaion of disribuion, he parameric GARCH is ranked las for he 500 sizes (55.9%) and second las for he 000 sizes (59.3%); while for any of he sizes, he hree kinds of SVR are always he op hree mehods in forecasing he volailiy direcion, all higher han 70% which none of he oher mehods can reach. This ime he NN defeas MLE. As for MLE and MA, in he siuaion of he 500 sizes and disribuion MLE performs worse han MA in erms of boh MAE and DA measures. [Tables II (a) and (b)] We now assess he performance of he fixed forecasing rule shown in Table II. Based on he sum rankings of he MAE measures, he order of forecasing abiliy of he differen mehods from he highes o lowes is displayed in urn as follows: rbf-svr, poly-svr, linear-svr, MLP, RBF and MLE. The special hing impressing us is ha he MLE has almos he wors forecasing performance. In he siuaion of Gaussian disribuion, for he 500 sizes, he MAE value of MLE is and ha of rbf-svr, wors among he hree kinds of SVR, is , he difference beween hem is For he 000 sizes, he MAE value of MLE decreases o and ha of linear-svr, he wors among he hree SVR, also decreases o , he difference reduces o Obviously, even if in he case of normal disribuion and large sample sizes, SVR sill ouranks MLE in forecasing volailiy error magniude. In he siuaion of disribuion, for he 500 sizes, he MAE value of MLE increases o bu ha of linear-svr, he wors among he hree kinds of SVR, decreases o , he difference is much larger han ha in normal disribuion. For he 000 sizes, he MAE value of MLE is and ha of linear-svr, he wors among he hree kinds of SVR, is , he difference reduces o which is smaller han he previous one in he 500 sizes bu larger han he corresponding one in normal disribuion. In a word, in he case of disribuion, he hree SVR are grealy superior o MLE, he laer of which grows poorer as i always does. As for NN, SVR also ouperforms hem for all four siuaions wih jus a few excepions (normal disribuion and he 500 sizes for RBF, he 000 sizes and boh disribuions for MLP). According o he sum rankings of he DA measures, rbf-svr also ranks highes for all four siuaions, he same as wih he updaing forecasing scheme; linear-svr is ranked second; bu

12 poly-svr is inferior o MLP and RBF; again, MLE is he wors one, he same as ha in erms of he MAE crieria. The DA value of MLE in he siuaion of he 500 sizes and disribuion is only 6.36%, he lowes among all he values; alhough in he siuaion of he 000 sizes and disribuion is 68.4% equal o he linear- and poly-svr. The wors performance of MLE in he case of boh measures denoes ha MLE is no suiable for long-run ex ane forecass. Two NNs hold he same highes posiion as rbf-svr in he siuaion of he 500 sizes and boh disribuion and as linear-svr in he 500 sizes and disribuion. Bu, according o he sum rankings of boh MAE and DA measures, hree kinds of SVR sill ouperform wo NNs under he fixed forecasing scheme. Taken ogeher, if you wan o forecas long-run volailiy using a fixed forecasing rule, he recurren SVR is he firs choice, followed by NN, and MLE is he final one. V. REAL DATA ANALYSIS The aim of his secion is o compare he volailiy forecasing performance of differen mehods for wo kinds of financial reurns: GBP/USD exchange raes and he NYSE sock index. Daa descripion The firs daa se consiss of he daily nominal bilaeral exchange raes of Briish Pounds (GBP) agains he U.S. dollar for he period of July, 003 o June 30, 005. The daa are obained from a daabase provided by Policy Analysis Compuing and Informaion Faciliy in Commerce (PACIFIC) a he Universiy of Briish Columbia, which conains he closing raes for a oal of 8 currencies and commodiies. The second daa se consiss of he daily closing price of he New York Sock Exchange TM (NYSE) composie sock index for he period of July 3, 003 o June 30, 005. The daa are downloaded direcly from he Marke Informaion secion of he NYSE TM web page. Boh ses of raw real daa are ransformed ino daily reurns via equaion 6, giving a reurns series of 503 observaions and hen a residual series of he same size is obained from a fied condiional mean equaion of he GARCH model. For he squared residuals of he 503 observaions, he recursive esimaing samples for he condiional volailiy funcion are updaed from he former 44 observaions hrough he former 483 and hen 60 numbers of one-period-ahead forecass are obained which correspond o an ou-of-sample evaluaion sample spanned from he 45h hrough he 484h daa poins. The muli-period-ahead evaluaion sample is he las 0 observaions which span from he 484h daa poin o he end. [Figure ] [Figure 3] The daily series for he log-levels and he reurns of he GBP and NYSE (503 observaions) are depiced in Figure and 3, respecively. Boh figures show ha he reurns series are mean-saionary, and exhibi he ypical volailiy clusering phenomenon wih periods of unusually large volailiy followed by periods of relaive ranquiliy. Table III repors he summary

13 of he descripive saisics for he GBP and NYSE reurns. The GBP series are ypically characerized by excessive kurosis and asymmery. The Bera-Jarque (98) es srongly rejecs he normaliy hypohesis for GBP. Bu he NYSE series canno rejec he normaliy hypohesis. For boh series, he Ljung-Box Q(6) saisics of raw reurns indicae no significan correlaion; bu he Q(6) values of he squared reurns reveal ha here is significan serial correlaion in he squared reurns. Engle s (Engle, 98) ARCH ess show significan evidence in suppor of GARCH effecs (i.e., heeroscedasiciy) for boh series. This examinaion of daily reurns on he GBP and NYSE daa reveals ha reurns can be characerised by heeroscedasiciy and ime-varying auocorrelaion, herefore, we expec he GARCH models o capure i adequaely. [Table III] Evaluaing he forecasing performance of each mehod The resuls of forecasing accuracy for each model using real daa are shown in Table IV for boh one- and muli-period-ahead forecass; where, (a) repors he values of he evaluaion measures and (b) is he ranking of all he models. [Table IV (a) and (b)] We firs evaluae one-period-ahead forecass of volailiy, as described in Table IV. According o he sum rankings of he MAE of wo reurns, hree kinds of SVR are he op hree mehods, followed by he MLE mehod which is ranked fourh for he NYSE, characerized by Gaussian disribuion, bu fifh for GBP reurns which show a high excess kurosis. Obviously, he values of MAE of he hree kinds of SVR are below 0. and 0.44, respecively, for he GBP and NYSE reurns, while hose of MLE are and RBF and MA mehod which are ranked fifh perform equally well while he MLE mehod is ranked las. Therefore, he values of MAE indicae he smalles deviaion beween he acual and forecased volailiy for he recurren SVR mehod as opposed o he compeing mehods. In erms of he sum rankings of he DA crieria, he hree kinds of SVR also rank he highes among all he mehods and perform equally well, followed by MLP, RBF and MA mehod which are equally ranked fourh side by side. For example, he DA values of he hree SVR for boh reurns all exceed 50%; among he oher values, only ha of he RBF for he NYSE daa is a lile more han 50%. MLE mehod performs wors for boh GBP and NYSE reurns, he DA values of which are only 8.8% and 30.5%, respecively. Here, MA almos ouranks MLE based on wo measures only excep for he MAE value in he NYSE. Nex, we consider he siuaion of muli-period-ahead forecass of volailiy. Based on he sum rankings of MAE for he wo reurns, he hree kinds of SVR are also he bes mehods (he MAE values of which are below 0.9 and 0. for GBP and NYSE), followed by MLE (0.099 and for boh daa), NN is he wors one (more han 0.4 and 0.39 for he wo reurns). There is a change in erms of he DA measure. Two kinds of NN rank in he firs class, he correcness raio of which is higher han 78%. Now, he hree kinds of SVR rank in he second class in he sum rankings of he wo reurns. Also, MLE ranks he lowes as i does 3

14 in one-period-ahead forecasing. However, in erms of he sum rankings of he MAE and DA measures, he recurren SVR is sill beer han he oher mehods while NN beas back MLE. Taken ogeher, he recurren SVR consisenly being he bes in error magniude and direcion forecass while MLE is always he wors in forecasing he urning poins of volailiy 3 and is only inferior o SVR in forecasing error size. We canno conclude ha NN ouperforms MLE overall as argued in oher sudies. [Figure 4 (a) and (b)] We have ploed he forecased volailiy from hree recurren SVR- and MLE based GARCH (,) models, along wih he acual ex-pos volailiy measures based on he squares of reurns minus heir means 4, for GBP exchange raes and he NYSE sock index in Figures 4 and 5, respecively; in which (a) graphs 60 one-period-ahead forecass for he ou-of-sample period of March 0, 005 o June 3, 005 resuling from he updaing forecasing scheme, and (b) displays 0 muli-period-ahead forecass for he ou-of-sample period of June 3, 005 o June 30, 005 from he fixed rule. [Figure 5 (a) and (b)] Seen from (a) and (b) in he wo Figures, i is clear ha he updaing forecasing scheme racks he ex-pos acual volailiy beer han he fixed one for all he models. For boh forecasing schemes, overall, he hree kinds of recurren SVR-based GARCH (,) models seem o do a remarkable job of capuring he fuure volailiy clusering effec in wo reurns in he ou-of-sample. CONCLUSIONS In many applicaions, SVR has shown excellen forecasing performance due o is paricular design of minimizing srucural risk raher han empirical risk employed by neural neworks and radiional mehods (Vapnik, 995, 997). This inspires us o use i o improve he forecasing abiliy of he radiional GARCH models. In erms of he MAE and DA measures, in his sudy, we invesigae he forecasing abiliy of he recurren SVR-based GARCH (,) models as compared wih MA, recurren NN and MLE mehods by using a Mone Carlo simulaion and real daa of he Briish Pound-US Dollar (GBP) daily exchange raes. The real daa resuls, ogeher wih he simulaion evidence, consisenly suppor he use of he hree recurren SVR-based GARCH models in forecasing one- and muli-period-ahead volailiy error magniude and direcion; alhough, in he case of forecasing long-erm volailiy direcion, neural neworks also perform equally well. As for he performance of he differen kinds of SVR, simulaion suppors poly-svr bu real daa analysis favors linear-svr in one-period-ahead error 3 I is noeworhy ha, DA provides a measure of he consisency in he predicion of he volailiy direcion which may yield imporan informaion for financial decisions in risk managemen field. Therefore, he problem should be considered seriously when using he MLE mehod. 4 The forecased volailiy from MLP, RBF and MA mehods are no ploed jus o no make he figure complicaed. 4

15 size forecass. Simulaion also favors rbf-svr in shor-erm volailiy direcion forecass and in long-erm error size and direcion forecass, he conclusions of which are confirmed by he real daa only wih he one excepion of long-erm direcion forecass. Since no single kernel funcion dominaes all predicions, praciioners should ry more han one kernel funcion. MLE canno always perform beer han SVR even wih he required assumpions of Gaussian disribuion and large sample sizes being saisfied for daa. NN is also inferior o SVR in volailiy forecass. Therefore, i is concluded ha he problem of good esimaion and poor forecass can be resolved using our recurren SVR mehod. In muli-period-ahead volailiy forecass, paricularly noeworhy, MLE almos ranks he lowes among all mehods, which indicaes ha MLE is no suiable for long-run volailiy forecasing. Due o he inroducion of global feedback loops and he corresponding richer dynamic srucures, SVR looks more promising in doing so. Using he squares of he reurns as he proxy of acual daa would grealy influence he forecasing evaluaion resuls; herefore, we have lef for fuure work he invesigaion of an alernaive use of cumulaive squared reurns from high frequency inraday daa as he proxy of ex pos volailiy, following Anderson and Bollerslev (998). The relaive accuracies of he various mehods are also highly sensiive o he saisic measures used o evaluae hem; herefore, i is generally impossible o specify a forecas evaluaion crierion ha is universally accepable. This problem is especially acue in he conex of nonlinear volailiy forecasing (Engle e al. 993; Diebold and Mariano 995; Wes 996; Andersen e al. 999; Dacco and Sachell 999; and Clemens and Smih 00), which should promp us o consider more appropriae evaluaion crieria ha are linked direcly o our fuure applicaions. REFERENCES Daabase of Exchange Raes: hp://pacific.commerce.ubc.ca/xr, Policy Analysis Compuing and Informaion Faciliy In Commerce (PACIFIC) a Universiy of Briish Columbia. Daabase of NYSE sock index: hp:// he Marke Informaion secion of he NYSE TM web page. Andersen TG, Bollerslev T, 998, Answering he Skepics: Yes, Sandard Volailiy Models do Provide Accurae Forecass, Inernaional Economic Review, 39, Andersen TG, Bollerslev T, Lange S, 999, Forecasing financial marke volailiy: Sample frequency vis-a-vis forecas horizon, Journal of Empirical Finance, Bera, A.K. and C.M. Jarque, 98, An efficien large-sample es for normaliy of observaions and regression residuals, Ausralian Naional Universiy Working Papers in Economerics, 40, Canberra. Bollerslev, T., 986, Generalized auoregressive condiional heeroskedasiciy, Journal of Economerics 3, Bollerslev, T., Chou R.Y., and Kroner K.F., 99, ARCH modeling in finance: A review of he heory and empirical evidence, Journal of Economerics 5, Brailsford T.J. and R.W. Faff, 996, An evaluaion of volailiy forecasing echniques. Journal of Banking and Finance 0, Brooks C., 998, Predicing sock index volailiy: can marke volume help? Journal of 5

16 Forecasing, 7, Brooks C., 00, A Double-hreshold GARCH Model for he French Franc / Deuschmark exchange rae, Journal of Forecasing, 0, Brooks C. and G. Persand, 003, Volailiy Forecasing for Risk Managemen, Journal of Forecasing,, -. Cao L, Tay F. 00. Financial forecasing using suppor vecor machines. Neural Compuaion and Applicaion 0: Chan K.C., W.G Chrisie and P.H. Schulz, 995. Marke srucure and he inraday paern of bid-ask spreads for NASDAQ securiies. Journal of Business 68(): Chen S.Y. and K.H. Jeong, 005, Forecasing Exchange Raes Using Feedback Suppor Vecor Regression: Nonlinear ARIMA Model, forhcoming. Chong C.W., M.I Ahmad, and M.Y Abdullah, 999. Performance of GARCH models in forecasing sock marke volailiy, Journal of Forecasing, 8, Clemens M.P, J.P. Smih, 999, A Mone Carlo sudy of he forecasing performance of empirical SETAR models, Journal of Applied Economerics 4: 3-4. Clemens M.P, J.P. Smih, 00, Evaluaing forecass from SETAR models of exchange raes. Journal of Inernaional Money and Finance 0: Cumby R., S. Figlewski and J. Hasbrouck, 993, Forecasing volailiy and correlaions wih EGARCH models. Journal of Derivaives winer: Dacco R. and S. Sachell, 999,. Why do regime-swiching models forecas so badly? Journal of Forecasing, 8: -6. Day, T.E. and C.M. Lewis, 99, Sock marke volailiy and informaion conen of sock index opions, Journal of Economerics, 5, Deng N.-Y and Y.-J. Tian (004), New Mehods in Daa Mining: Suppor Vecor Machine, Science Press, Beijing. Diebold F.X. and R.S. Mariano, 995, Comparing predicive accuracy, Journal of Business and Economic Saisics, 3, Donaldson RG., Kamsra M, 997, An arificial neural nework-garch model for inernaional sock reurn volailiy, Journal of Empirical Finance, 4, Dunis C.L., X.H. Huang, 00, Forecasing and rading currency volailiy: an applicaion of recurren neural regression and model combinaion, Journal of Forecasing,, Enders W., 004, Applied Economeric Time Series, nd ed.,, John Wiley & Sons, Inc., New York. Engle RF, 98, Auoregressive condiional heeroskedasiciy wih esimaes of he variance of UK inflaion, Economerica, 50, Engle R.F., C-H. Hong, A. Kane, and J. Noh, 993, Arbirage valuaion of variance forecass wih simulaed opions, in Chance, D.M. and Trippi, R.R. (eds), Advances in Fuures and Opions Research, Greenwich, CT: JAI Press, 6, Figlewski, S., 997. Forecasing volailiy, Financial Markes, Insiuions and Insrumens, 6, 88. Hamilon JD Time Series Analysis, Princeon Universiy Press. Haykin, S. (999), Neural Neworks: a comprehensive foundaions, nd ediion, Prenice 6

17 Hall, New Jersey. Jorion P., 995, Predicing volailiy in he foreign exchange marke, Journal of Finance, 50, Jorion P., 996, Risk and urnover in he foreign exchange marke, In The Microsrucure of Foreign Exchange Markes, Franke JA, Galli G, Giovannini A (eds). Chicago Universiy Press: Chicago. McMillan D.G., A.E.H. Speigh and O. Gwilym, 000, Forecasing UK sock marke volailiy: a comparaive analysis of alernae mehods. Applied Financial Economics 0, Moosa IA Exchange Rae Forecasing: Techniques and Applicaions, Macmillan Press LTD, Lonon. Niemira MP, Klein PA Forecasing Financial and Economic Cycles, John Wiley & Sons, Inc., New York. Pagan, A.R. and G.W Schwer, 990, Alernaive models for condiional sock volailiy, Journal of Economerics, 45, Pérez-Cruz F., J.A. Afonso-Rodr ıguez and J. Giner, 003, Esimaing GARCH models using SVM, Quaniaive Finance, 3, Schienkopf C., Dorffner G., Dockner EJ, 000, Forecasing ime-dependen condiional densiies: a semi-non-parameric neural nework approach, Journal of Forecasing, 9, Scholkopf B. and A. Smola, 00, Learning wih Kernels, Cambridge, MA: MIT Press Taylor JW, 000, A quanile regression neural nework approach o esimaing he condiional densiy of muliperiod reurns, Journal of Forecasing, 9, Vapnik, V. N. (995), The Naure of Saisical Learning Theory, Springer, New York. Vapnik, V. N. (997), Saisical Learning Theory, Wiley, New York. Wes KD. And D. Cho, 995, The predicive abiliy of several models of exchange rae volailiy, Journal of Economerics, 69, Wes K.D. 996, Asympoic inference abou predicive abiliy. Economerica 64:

18 Appendix 8

19 Table I (a) One-Period-Ahead Forecasing Accuracy for Mone Carlo Simulaion Mehods used for Sample Numbers = 500 Sample Numbers = 000 GARCH(,) Normal Disribuion Suden's T Disribuion Normal Disribuion Suden's T Disribuion Model MAE DA MAE DA MAE DA MAE DA MA MLE MLP RBF SVR-linear SVR-poly SVR-rbf Noe: The laer 5 mehods, excep for MA, are used for esimaing and forecasing GARCH(,) model. Table I (b) Rankings of One-Period-Ahead Forecasing Accuracy for Simulaion Daa MAE DA Mehods used for Normal T Normal T GARCH(,) Model Sum Sum MA MLE MLP RBF SVR-linear SVR-poly SVR-rbf

20 Table II (a) Muli-Period-Ahead Forecasing Accuracy for Mone Carlo Simulaion Daa (horizon=0) Mehods used for Sample Numbers = 500 Sample Numbers = 000 GARCH(,) Normal Disribuion Suden's T Disribuion Normal Disribuion Suden's T Disribuion Model MAE DA MAE DA MAE DA MAE DA MLE MLP RBF SVR-linear SVR-poly SVR-rbf Table II (b) Rankings of Muli-Period-Ahead Forecasing Accuracy for Simulaion (horizon=0) MAE DA Mehods used for Normal T Normal T GARCH(,) Model Sum Sum MLE MLP RBF SVR-linear SVR-poly SVR-rbf

21 Figure Briish Pounds Exchange Raes: (a) Log-Levels (b) Reurns Figure 3 New York Sock Exchange Composie Index: (a) Log-Levels (b) Reurns

22 Table III Descripive Saisics for Daily Financial Reurns Reurns GBP NYSE Mean Variance Skewness Kurosis Normaliy [0.0008].5555 [ ] Q(6) [ ] 4.06 [0.6690] Q(6)* [ ] [ ] ARCH(6) [ ] 5.00 [0.0347] Noes: Kurosis quoed is excess kurosis; Normaliy is he Bera-Jarque (98) normaliy es; Q(6) is he Ljung-Box Q es a 6 order for raw reurns; Q(6)* is LB Q es for squared reurns; ARCH(6) is Engle's (98) LM es for ARCH effec. Significance levels (p-values) are in brackes.

23 Table IV (a) One- and muli-period-ahead Forecasing Accuracy for Real Daa Mehods used for GBP NYSE GARCH(,) One- Muli- One- Muli- Model MAE DA MAE DA MAE DA MAE DA MA MLE MLP RBF SVR-linear SVR-poly SVR-rbf Noe: Moving average mehod is no included in muli-period-ahead forecasing evaluaion. Table IV (b) Rankings of One- and muli-period-ahead Forecasing Accuracy for Real Daa Mehods used for One- Muli- GARCH(,) MAE DA MAE DA Model GBP NYSE sum GBP NYSE sum GBP NYSE sum GBP NYSE sum MA MLE MLP RBF SVR-linear SVR-poly SVR-rbf

24 Figure 4 Volailiy Forecass of Briish Pounds Exchange Raes Reurns (a) One-Period-Ahead Forecass of Volailiy 0.8 Acual MLE SVR-lin SVR-poly SVR-rbf Figure 4 Volailiy Forecass of Briish Pounds Exchange Raes Reurns (b) Muli-Period-Ahead Forecass of Volailiy Acual MLE SVR-lin SVR-poly SVR-rbf

25 Figure 5 Volailiy Forecass of NYSE Composie Index Reurns (a) One-Period-Ahead Forecass of Volailiy Acual MLE SVR-lin SVR-poly SVR-rbf Figure 5 Volailiy Forecass of NYSE Composie Index Reurns (b) Muli-Period-Ahead Forecass of Volailiy Acual MLE SVR-lin SVR-poly SVR-rbf