STAR and ANN models: Forecasting performance on the Spanish Ibex-35 stock index

Transcription

1 STAR and ANN models: Forecasing performance on he Spanish Ibex-35 sock index Jorge V. Pérez-Rodríguez Deparmen of Quaniaive Mehods Universiy of Las Palmas de Gran Canaria Campus de Tafira, Tafira Baja. E-3507, Las Palmas. Spain Phone: y Fax: / 829 ( jorge@empresariales.ulpgc.es) Salvador Torra Deparmen of Economerics, Saisics and Spanish Economy Universiy of Barcelona ( sorra@eco.ub.es ) and Julian Andrada-Félix Deparmen of Quaniaive Mehods Universiy of Las Palmas de Gran Canaria ( julian@empresariales.ulpgc.es) Corresponding auhor: Dr. Jorge V. Pérez-Rodríguez Deparmen of Quaniaive Mehods Universiy of Las Palmas de Gran Canaria Campus de Tafira, Tafira Baja. E-3507, Las Palmas. Spain

2 Absrac: This paper examines he ou-of-sample forecas performance of smooh ransiion auoregressive (STAR) models and arificial neural neworks (ANNs) when applied o daily reurns on he Ibex-35 sock index, during he period from 30 December 989 o 0 February The forecass are evaluaed wih saisical crieria such as goodness of forecas, including ess of forecas encompassing, direcional accuracy and he equaliy of mean squared predicion error; he relaive forecas performance is assessed wih economic crieria in a simple rading sraegy including he impac of ransacion coss on rading sraegy profis. In erms of saisical crieria, he resuls show ha differen arificial neural nework specificaions forecas beer han he AR model and smooh ransiion non-linear models. In erms of he economic crieria in he ou-of-sample forecass, we assess profiabiliy and combine a simple rading sraegy known as he filer echnique by using a range filer percenage and rading coss. The resuls indicae a beer fi for ANN models, in erms of he Sharpe risk-adjused raio. These resuls show here is a good chance of obaining a more accurae fi and forecas of he daily sock index reurns by using nonlinear models, bu ha hese are inherenly complex and presen a difficul economic inerpreaion. Keywords : Non-lineariies, saisical crieria, rading sraegies. JEL classificaions: C22, C45, C52.

3 . Inroducion Some researchers have quesioned he hypohesis of Efficien Markes (HEM), i.e. ha he random walk model is a reasonable descripion of asse price movemen and ha linear models successfully describe he evoluion of such prices. For example, Hinich and Paerson (985), Cochrane (988), Fama and French (988), Lo and McKinlay (988), Whie (988), Sheinkman and LeBaron (989), Hsieh (99), Granger (992), Gençay (996), Campbell, Lo and McKinlay (997), De Lima (998), Fernández, García and Sosvilla (999) and García and Gençay (2000) have raised he quesion of wheher he behaviour of asse reurns is compleely random; wheher linear modelling echniques are appropriae o capure some of he complex models ha chariss have observed in he evoluion of asse prices and he marke negoiaion process; wheher i is possible o idenify and exploi he behaviour of asse reurns over ime; or wheher he adjusmens made in he marke in response o price deviaions and heir heoreical value migh no be proporional o he quaniy by which prices deviae from heir real value. Theoreical and pracical ineres in non-linear ime series models has increased rapidly in recen years. Various facors migh accoun for non-lineariy. On he one hand, we could admi he possibiliy ha no all he agens simulaneously receive all he informaion; here may be imporan differences in arges and in negoiaion ime; or hose agens wih more complex algorihms migh be able o make beer use of he available informaion. However, here are several reasons why non-linear modelling is no easy. Firs, because here exis a grea number of opions, i.e. bilinear models, ARCH and is exensions, smooh ransiion auoregressive models (STAR), arificial neural neworks (ANN), waveles and even chaoic dynamics. Second, because he flexibiliy inheren in is use can creae spurious fis [Granger and Teräsvira (993)]; and hird, because when considering a long period of ime here can appear he problem of srucural change and he exisence of more ouliers, which makes model esimaion difficul [De Lima (998)]. For example, Granger (992) argues ha if we spread he ime horizon, use seasonally adjused daa, give a suiable reamen o excepional evens and ouliers and, in paricular, consider non-lineariy, we can achieve beer reurns. However, if here is no rule abou profis and no profis are made over a long period, hen he weak hypohesis of Efficien Marke (WHEM) should no be rejeced. 2

4 Due o heir variey and flexibiliy, one class of regime swiching models and ANN models has become popular in he class of non-linear models. The regime swiches in economic ime series can be described by STAR models. These models imply he exisence of wo disinc regimes, wih poenially differen dynamic properies, bu wih a smooh ransiion beween regimes. On he oher hand, ANN is considered o be a universal approximaor in a wide variey of non-linear paerns, including regime swiches and oher non-lineariies. Boh models are examined in his sudy. The purpose of his aricle is o evaluae heir adequacy and validiy and o compare he forecasing performance of differen STAR and ANN models in predicing Ibex-35 Spanish sock index reurns 2. In his sense, he work in his paper is empirical, and we do no aemp o explain he resuls obained, or hose claimed by oher researchers, on heoreical grounds. The ou-of sample one-sep-ahead forecass from differen models are evaluaed using saisical crieria such as mean squared predicion error (MSPE), ess for forecas encompassing [Chong and Hendry (986)] 3, equaliy of accuracy of compeing forecass or MSPE of compeing models [Diebold and Mariano (995)] and direcional accuracy [DA, Pesaran and Timmermann (992)]. We examine wheher ou-of-sample forecass generaed by he non-linear models are more accurae and preferable o ou-of-sample forecass generaed by linear ARMA models for sock index reurns. We also analyse wheher nonlinear ANNs really are superior o linear and STAR models in pracice, assessing he relaive forecas performance wih economic crieria. For example, we use he reurn forecass from he differen linear and non-linear models in a simple rading sraegy and compare pay-offs o deermine if ANNs are useful forecasing ools for an invesor. As shown by Leich and Tanner (99) and Sachell and Timmermann (995), he use of saisical or economic crieria can lead o very differen oucomes. The correlaion beween MSPE and rading profis, for example, is usually quie small. The performance of a paricular model in erms of DA is ofen a beer indicaor of is performance in a rading sraegy. However, given ha some papers find ha neural neworks do no perform much 2 We analyze one of he official indexes of he Madrid Sock Marke: he Ibex35, an index composed of he 35 mos liquid values lised in he Compuer Assised Trading Sysem (CATS). This index was designed o be used as a reference value in he rade of derivaives producs, i.e. opions and fuures. This coninuous sysem was inroduced ono he Madrid Exchange Marke in December

5 beer han linear and STAR models in erms of DA, we would no find i surprising if i urned ou ha ANNs do no offer significanly higher rading profis. Finally, i would also be useful o examine he impac of ransacion coss on he profis of rading sraegies. This paper is srucured as follows. Secion 2 describes he main characerisics of STAR and ANN non-linear models. In Secion 3, we describe he Ibex35 daa and some saisical properies. Secion 4 shows non-linear model esimaes. In Secion 5, we examine he predicive capaciy of some non-linear models over a long period, and in Secion 6, we examine he rading sraegy profis. Finally, Secion 7 summarises he mos imporan conclusions of his sudy. 2. Regime swiching models and ANNs for sock index reurns In his secion we briefly explain regime swiching models such as STAR and a class of flexible non-linear models inspired by he way in which he human brain processes informaio n. Le us consider an asse which provides a daily reurn equal o r, =,...,T. Consider he asse marke as an informaion processing sysem. The informaion se consanly changes, and he processing of marke informaion produces a fiing of prices owards he perceived marke value. The marke is considered o form an expecaion for he nex period, depending on curren informaion, which could be wrien mahemaically as: [ r ] f ( ) E ψ where ψ is he informaion se during period and he f( ) funcion = could be eiher linear or characerised by complex non-linear funcions. Some auhors argue ha he asse marke has he capaciy o be a non-linear dynamic sysem. In his sense, we could say ha he reurn during he period is equal o: r ( ψ ) + ε = f [] where ε is a predicion error. In he following sub-secions, we assume ha ( ) f ψ can be modelled by p-order AR in a non-linear way. We also assume ha lagged reurns are needed in he condiional mean specificaion, because auocorrelaion in sock reurns can appear because of non- 3 A se of forecass is said o encompass a compeing se if he laer should opimally receive a zero weigh in a composie predicor ha is a weighed average of he wo individual predicors. 4

6 synchronous rading effecs. In his sense, we describe he STAR and ANN models ha are compared in his paper. 3.. Specificaion and esimaion of STAR models for sock index reurns Non-linear ime series models have become very popular in recen years. Regime swiching models are very popular in he class of non-linear models and hey are an alernaive way o invesigae poenial non-lineariies and cyclical behaviour in sock reurns. Esimaes based on non-linear models sugges ha sock price growh raes are characerised by asymmeric cycles in mos counries, wih he speed of ransi ion beween expansion and conracion regimes being relaively slow. The regime swiching models we consider here are known as smooh ransiion regression (STAR), and hey are a flexible family of non-linear ime series models ha have also been used for modelling economic daa. STAR models have been described by Teräsvira, Tjosheim and Granger (994). This paper evaluaes he saisical adjusmen and he forecas performance of differen STAR models using he Ibex-35 index of sock reurns. A simple firs-order STAR model wih wo regimes is defined as follows: r p p = φ 0 + φir i + φ 20 + φ2ir i F, d ( s; γ, c) + ε [2] i= i= where r are he reurns, φ ij, (i=,2, j=0,,2..,p) are he unknown parameers ha correspond o each of he wo regimes. F ( s ; c) d,, γ is he ransiion funcion, assumed o be wice differeniable and bounded beween 0 and, γ is he ransiion rae or smoohness parameer, c is he hreshold value which represens he change from one regime o anoher, and d is he number of lags of ransiion variable. This funcion inroduces regime swiching and non-lineariy ino he parameers of he model. Alhough here are few heoreical resuls regarding he saionariy of he STAR model, a sufficien condiion is φ ij <, i, j. The ransiion variable, s, is usually (bu no always) defined as a linear combinaion of he lagged values of r, as: s α r. = d i = i i 5

7 Regarding he choice of ransiion funcion, he wo mos widely used in he lieraure are he firs-order logisic funcion: ( s ; γ, c) = { + exp[ γ ( s c) ]}, γ 0 F, d >, [3] in which case he model is called logisic STAR or LSTAR(p;d); and by he firs-order exponenial funcion, for which: 2 { [ ]}, 0 ( s ; γ, c) exp γ ( s c) F, d = γ >, [4] and in his case, he model is called exponenial STAR or ESTAR(p;d). In boh cases, he ransiion variable can be any variable in he informaion se ψ. In order o use his model effecively, i is imporan o choose he appropriae ransiion funcion and hreshold variable. There exis many LM-ype ess o deermine he appropriae choice of ( s ; c) F, d γ, and s. However, LSTAR and ESTAR models describe differen ypes of dynamic behaviour. The LSTAR model allows he expansion and conracion regimes o have differen dynamics, wih a smooh ransiion from one o anoher. On he conrary, he ESTAR model suggess ha wo regimes have similar dynamics, while he behaviour in he ransiion period (middle regime) may be differen. Boh models characerise asymmeric cycles. Such models are ofen esimaed by non-linear leas squares (NLS) or by maximum likelihood esimaions (MLE). If ε is normal, NLS is equivalen o MLE (bu no generally), oherwise i can be inerpreed as QMLE. Under suiable regulariy condiions, NLS is consisen and asympoic normal. Afer many ieraions we would probably reach he opimal value of he arge funcion Specificaion and esimaion of ANN models for sock index reurns Though no wihou heir criics, ANNs have come ino wide use in recen years, due o he advanages such models offer analyss and forecasers in he financial markes 4. In paricular, non-parameric and non-linear models can be rained o map pas values of a ime series for purposes of classificaion or funcion esimaion, and allow us o depic non- 4 For a deailed discussion of ANNs and heir economeric applicaions, see Kuan and Whie (994). 6

8 linear complex relaionships auomaically; hey are universal approximaors; hey describe various forms of regime swiching, and hus differen asymmeric effecs, which leads us o sugges ha some subperiods are more predicable han ohers; finally, hey are good predicors 5 [see Swanson and Whie (997)]. Perhaps, he ANN mehodology is preferred o oher non-linear models because i is non-parameric. This echnique consiss of modelling in a non-linear fashion he relaionships beween variables o consruc a forecas. An ANN is a collecion of ransfer funcions which relae he dependen variable, r, o cerain vecors of explanaory variables, R, which can even be funcions of oher explanaory variables. In his sense, ANNs are a class of non-linear regression models and in paricular mechanisms for non-parameric saisical inference. Two basic aspecs characerize hem: a parameric specificaion or nework opology, and esimaion mechanism or nework raining. These represenaions nes many familiar saisical models, such as linear and non-linear regressions, classificaion (i.e. logi and probi), laen variable models (MIMIC), principal componen analysis and ime series analysis (ARMA, GARCH). Three ANN models are examined in his paper. The lagged sock reurns are aken as explanaory variables, because we assume ha a forecasing relaionship for r can be derived from he informaion revealed by p inpus, R (, r, r,, r ) = 2 p L, including a consan erm. Thus, an ANN model for r can be aken as an exension of a basic linear regression. Like he STAR model, he ANN model can describe regime swiches in economic ime series, a leas when hese are confined o he inerceps. The ANN models are he mulilayer percepron model (MLP), jump connecion nes (JCN) and a parial recurren nework by Elman (990). Such neworks are capable of rich dynamic behaviour. MLP and JCN neworks are referred o in he lieraure as feedforward 5 Their applicaion in economics is mainly in managemen. For example, in he areas of cross-secional daa, bankrupcy predicion [Tam and Kiang (992)], he raings of corporae bonds [Surkan and Singleon (990), Moody and Uans (995)], in he area of ime series predicion, he sudy of asse reurns [Whie (988)], and decision-relaed opics [Sharda and Pail (992) and Hill, Márquez, O Connor and Remus (994), among ohers. In general, almos every sudy has analysed he predicive capaciy of neworks by comparing several models, boh linear and non -linear. The resuls obained have shown he moderae advanage of ANN predicion agains any of he linear ARIMA and non-linear GARCH ime -series models analysed. 7

9 neworks, while Elman is designaed as a recurren nework, because i exhibis memory and conex sensiiviy. The firs model ha we buil was a mulilayer percepron model (MLP) wih a single hidden layer, and q hidden unis. This is he mos commonly found neural model in he specialized lieraure. In general, a non-linear regression model which represens he MLP(p;q) has he following form for a single hidden layer nework: q p r = β + Σ β g φ r + φ + ε 0 j ij i 0 j [5] j= i= where r is he reurn in or sysem oupu 6 ; he parameer vecor is ( ) ( β, ) θ = β, φ, where β = L, β q and φ = ( φ j, L, φpj ), j=,..,q, brings ogeher all he nework weighs, wih β j represening he weighs from he hidden o he oupu uni and φ ij he weighs from he inpu layer o he hidden uni j; g(.) can ake several funcional forms, such us he hreshold funcion, which produces binary ( ± ) or (0/) oupu, or he sigmoid funcion, which produces an oupu beween 0 and 7. This funcion deermines he connecions beween nodes of he hidden layer, and i is used as he hidden-uni acivaion funcion o enhance he non-lineariy of he model; and ε is a resid ual i.i.d.. 6 Hornik, Sinchcombe and Whie (990) showed ha he ANNs of he ype defined in Eq.[5] are universal approximaors in a wide variey of funcion spaces of pracical ineres. We specified one hidden layer on he basis ha single hidden layer MLPs possess he universal approximaion propery, namely hey can approximae any nonlinear funcion o an arbirary degree of accuracy wih a suiable number. 7 Funcion g(.) is sigmoid if g: R [0,]; g(a) 0 when a - ; g(a) when a. For example, g can be g a = + exp a. I could also be a bipolar he logisic acivaion cumulaive disribuion funcion: ( ) [ ( )] funcion: ( a) = 2g( a) anh( a) = [ exp ( a) exp ( a) ] [ exp ( a ) + exp ( a )] h. Or i could be defined by he hyperbolic angen funcion:. There are some heurisic rules for he selecion of he acivaion funcion. For example, Klimasauskas (99) suggess logisic acivaion funcions for classificaion problems and hyperbolic angen funcions if he problem involves learning abou deviaions from he average, such as he forecasing problem. However, i is no clear wheher differen acivaion funcions have a greaer effec on he performance of he neworks [see Zhang e al. (998)]. 8

10 r β j z z 2... z q β 0 φ ij r r 2 r 3... r p φ 0 j Figure. Hidden layer nework for sock index reurns. The nework inerpreaion of Eq. [5] is as follows (see Figure ). The explanaory variables (or inpu unis) defined in R send signals o each of he hidden unis, z j, ha represen he oupu vecors of hidden unis. The signal from he i-h inpu uni o he j-h hidden uni is weighed, denoed by φ ij, before i reaches he hidden uni number i. All signals arriving a he hidden unis are firs summed and hen convered o a hidden uni acivaion by he operaion of he hidden uni acivaion funcion g(.) ha ransforms he signal ino a value beween 0 and. The nex layer operaes similarly wih connecions sen o he dependen variable (or oupu uni). As before, hese signals are aenuaed or amplified by weighs β j and summed. The second model ha we use is a nework wih direc connecions beween he inpus and oupus, called jump connecion nes (JCN). According o Kuan and Whie (994), he parameric specificaion for he oupu of he model adds he p-order AR o he MLP nework. In his sense, he ANN wih a single hidden layer has a linear componen augmened by non-linear erms, and i is wrien as JCN(p;q) by: r α φijr i + φ0 j + ε p q p = ir i + β 0 + Σ β jg j= i= i= [6] where α,...,α p are direc inpu-oupu weighs (see Figure 2). Eq.[6] ness he linear model because i includes he erm α p i= r i i as a linear auoregressive componen. 9

11 r β j z z 2... z q β 0 φ ij α i φ 0 j r r r r p Figure 2. Augmened hidden layer nework for sock index reurns. The nework inerpreaion of Eq. [6] is similar o ha of Eq. [5], bu wih one added aspec. Also, signals are sen direcly from all he explanaory variables o he dependen variable wih weighs α. The laer signals effecively consiue he linear par of his JCN i model 8. This model ness he linear model wihin he JCN, and ensures ha he JCN will perform in-sample a leas as well as he linear model. Finally, he hird model we use is a parially recurren nework, as proposed by Elman (990). This has he abiliy o recognize and, someimes, o reproduce sequences. This ype of ANN is somewha more complex han he unidirecional ANNs defined by Eq. [5] and by Eq. [6]. In he specific case of a recurren Elman(p,q) ype nework, his is characerized by a dynamic srucure where he hidden layer oupu feeds back ino he hidden layer wih a ime delay. This model can ake he form in he single hidden layer as: r = β z j, 0 q + Σ β z j= j j, + ε p = g φ + + ijr i φ0 j δijz j, i= where z j is he oupu vecor of he hidden unis, and δ ij are he weighs beween he hidden unis evaluaed in and -. In economeric erms, a model of he form Eq. [7] can be viewed as a non-linear dynamic laen variable model [see Kuan and Whie (994)]. Elman [7] 8 In is mos complex version, he opology allows us o inroduce one or more hidden layers beween he oupu and he inpus. The main advanage of his model is is capaciy o ac as an approximaion of nonlinear complex relaionships. Is main disadvanage is is saic naure, which is overcome by oher opologies ha incorporae he dynamics of inpu-oupu relaionships wih ime. 0

12 has inroduced an archiecure called he simple recurren nework where he inpu layer can be considered o be divided ino wo pars, rue inpu unis R and conex unis, z j. The feedback beween hem is represened schemaically in Figure 3. The conex unis simply hold a copy of he acivaions of he hidden nodes from he previous ime sep, z j, by recursive subsiuion 9. The nework inerpolaion of Eq. [7] is similar o Eq. [5], bu adds he possibiliy ha hidden unis can be conneced wih lagged hidden unis by he weighs δ, which inroduce a recursive updae. ij r β j β 0 z z2... z q φ ij δ ij φ 0 j r r z z p, q, Figure 3. Elman nework for sock index reurns. The expressions of Eq. [5], [6] or [7], and he flexibiliy of specificaions defined in foonoe 7, show ha when q akes a large enough value, he ANN model can approximae any arbirarily close funcion [see Kuan and Whie (994)]. The mos widely used esimaion mehod (or so-called learning rule) of he neural nework is error backpropagaion. Backpropagaion is a recursive gradien descen mehod ha mimics a learning behaviour. In his mehod, he weighs of he signals are updaed. Using he firs se of observaions, a he iniial sage he mehod does a forward and backward pass hrough he nework, iniially compues he weighs, and deermines he value of he error 9 These ypes of nework have cerain feaures which make hem especially suiable for modelling ime series because hrough he feedback, nework oupu depends on he iniial value and he enire hisory of sysem inpus. These neworks are capable of rich dynamic behaviour, exhibiing memory and conex sensiiviy by he presence of inernal feedbacks [see Gençay (997)].

13 funcion, recompues he weighs, and redeermines he value of he error he arge values of he oupu variable. A he nex sage, i uses he second se of observaions, and so on. This esimaion procedure is characerized by he recursive process. The learning algorihm converges and hus he process sops when he value of he error funcion is lower han a predeermined convergence crierion. More specifically, he nework weigh vecor θ is chosen o minimize he sum of he squared-error loss: min θ T [ r rˆ ] = 2, where T is he sample size, and rˆ is he calculaed oupu value from Eq.[5], Eq.[6] and Eq.[7]. Then he ieraive sep of he gradien descen algorihm akes θ o θ + θ, and ( R, θˆ ) εˆ θ = η f, where η is he learning rae ; f (, θˆ ) is he gradien of f ( θˆ ) R R, wih respec o θ (a column vecor of parameers); and ε ˆ = r rˆ is he nework error beween he compued r oupu and he arge reurn value, r. For recurren neworks, he nework oupu depends on θ direcly and indirecly hrough he presence of lagged hidden-uni acivaions. For his reason, he model can be esimaed by he recurren backpropagaion algorihm and by he recurren Newon algorihm [see Gençay (997) for deails]. 3. Daa and preliminary saisics This sudy uses he daily closing prices of he Spanish Ibex-35sock index, from 30 December 989 o 0 February 2000, wih a oal of 2520 observaions. The Ibex-35 index (I ) comprises he 35 mos liquid values negoiaed in he coninuous sysem which during he conrol period had he highes rading volume in cash peseas. The Ibex-35 is a composie index which is highly represenaive and is fied by capialisaion and dividends of he asses included, bu no by expansions in capial. The series is ransformed ino logarihms o compue coninuous reurns, according o he following expression: r I = log I, where log is he naural logarihm. 2

14 Figure 4 shows he evoluion of he index and is daily reurns. Is sharp increase since 996 is due o he downurn in risk-free ineres raes and he sequenial move by invesors owards he sock marke. The period of special ineres for he evoluion of prices and reurns is he Asiaic crisis of Ocober 997, when reurns fell abruply. Figure 4. Time evoluion of daily closing Ibex35 and reurns. (i) Closing prices(ii) Reurns //92 4//94 22//96 6//98 4// //92 4//94 22//96 6//98 4//00 We sudy some saisical properies of he Ibex35 index, shown in Tables AI., AI.2 and AI.3 (see Appendix I). Table AI. repors he augmened Dickey-Fuller (ADF) and Phillips and Perron (PP) saisics for non-saionariy for he logarihm index and reurns. The saisics indicae ha log I is non-saionary and ha he index of reurns is saionary. Anoher saisical es for he null hypohesis of saionariy, Kwiakowski, Phillips, Schmid and Shin (KPSS), obains he same resuls. Table AI.2 repors he variance-raio es. The resuls show he exisence of negaive auocorrelaions or mean reversion (for values beween 2 q 45 days). This es rejecs he null hypohesis of random walk a he 0% significance level. Finally, he Table AI.3 repors he Brock, Decher and Scheinkman (BDS) es. This es indicaes he presence of non-lineariies and, herefore, of complex models in he daa 0. The main conclusion is ha Ibex35 sock index reurns may be prediced using non-linear models. 0 In order o avoid possible rejecions of he null hypohesis due o non-saionariy he BDS es is commonly applied o he esimaed residuals of he ARIMA process. The asympoic disribuion of BDS is no affeced when linear filers are applied o daa. Table AI-3 also shows shuffled residuals, i.e., recreaed randomly as if hey were sample daa wihou replacemen. We use his echnique following Scheinkman and LeBaron (989) in order o reinforce he resuls, so ha in his case we should no rejec he null 3

15 4. Non-linear model esimaes The BDS saisic reveals considerable evidence of non-lineariy, and he variance raio es shows ha mean reversion exiss. In his secion, we analyse he non-linear model esimaes from he STAR and ANN models, which we consider o represen some sylised facs of he shor-erm dynamics of sock index reurns. The fied period is 30 December 989 o 30 April 999 (T=2320 observaions). We did no include GARCH models in he se of forecasing models because hese models parameerise he condiional variance, whereas he objec o be forecas is he sock index reurns, no heir volailiy. However, we esed for any omied ARCH non-lineariy. 4.. STAR Models This secion invesigaes empirical issues regarding STAR models wih Gaussian errors. In his paper we do no disinguish beween regimes of low and high volailiy, because our aim is o analyse sock index reurns, no heir volailiy. Therefore, we evaluaed differen models ha show regime swiching, for example, Eq. [2] wih he ESTAR and LSTAR funcion. The modelling procedure for building STAR models is carried ou in hree sages [see Granger and Teräsvira (993, pp.3-24), Teräsvira (994), and Eirheim and Teräsvira (996)]. The firs sage is o specify a linear AR(p) model. We esimaed differen AR models and chose p on he basis of he AIC, SBIC and Ljung-Box (LB) saisics for auocorrelaion. The AR model has a relaively shor order. We chose p=2 on he basis of AIC and SBIC equal o and LB()=0.0 (P-value is.0), LB(5)=-0.00 (Pvalue is 0.98) and LB(0)=0.039 (P-value is 0.78), which indicaes ha he AR(2) model has whie noise residuals. The second sage is o es he lineariy agains STAR models, hypohesis of he i.i.d. linear process. Thus, we will be able o prove ha here is a non-linear srucure in he original daa which has been removed by he shuffling. In boh siuaions, we use m=2 o 8 and a value for ε beween 0.5 σ and 2σ, using σ= The resuls show ha here are non -linear srucures in daa in he logarihm of he Ibex35 index, since he ess applied o residuals and o shuffled residuals show he rejecion of he null hypohesis in he firs case and is non-rejecion in he second case. Also, STAR models have been esimaed assuming condiional heeroscedasiciy or GARCH errors, bu he resuls are worse han hose obained wihou considering such an assumpion. For his reason, we have no shown i. Lundbergh and Teräsvira (998) made an exensive sudy of STAR models wih GARCH errors. 4

16 for differen values of he delay parameer d, using he linear model specified a he firs sage. This sage ess he parameer consancy, such as esing wheher STAR is more appropriae han a single AR model. Therefore, we esed wheher non-linear funcions of lagged regressor variables conribue significanly o he fi (afer correcion for a linear AR par), using s =. The lineariy es is based on he auxiliary regression: r d r = φ p p p p φir i + β jr jr d + β2 jr jr d + β3jr jr d + i= j= j= j= To specify he value of he delay parameer d, he esimaion of he auxiliary regression is carried ou for a wide range of values, appropriae value of d. The null hypohesis is d D, given uncerainy abou he mos β ij = 0, i, j. The F-es values for he significance of he regressor added o he linear AR regressions can be used o es he null hypohesis of lineariy. We can obain a firs impression of he d value by looking a he relaive value of he F-es saisics, ha is, he d for which he corresponding P-value is smalles may be seleced, and his corresponds o he larges 2 R of he regression model. In carrying ou lineariy ess, we considered values for he delay parameer over he range d 2 (Table ). The d value seleced is 6, because i has he lowes P-value. The lineariy is rejeced a he 5% level of significance because he minimum P-value is v Table. P-values for lineariy es and sequenial procedure. Lineariy es Choosing beween ESTAR and LSTAR Delay H 0 : β = β2 = β3 = 0 H 0 : β3 = 0 H 02 : β2 = 0 β3 = 0 H 03 : β = 0 β 2 = β3 = ª b Noe: a indicaes lowes P-value for he null hypohesis of lineariy over he inerval 0 d 2. b indicaes lowes P-value when d=6. 5

17 The hird sage is o choose beween ESTAR and LSTAR models where lineariy is rejeced. Teräsvira (994) suggess applying he following sequence of nesed ess: (i) es wheher all fourh-order erms are insignifican, β 3 j = 0, j ; (ii) condiional on all fourhorder erms being zero, es he join significance of all hird-order erms β 2 j 0 β3 j = = 0, j, and (iii) condiional on all hird and fourh-order erms being zero, es he significance of he second-order erms, β j 0 3 j β 2 j = β = = 0, j. If he es in (i) does no rejec he null hypohesis, we choose he LSTAR model. If we accep (i) and rejec (ii), we choose he ESTAR model. Finally, acceping he null hypohesis in (i) and (ii), bu rejecing (iii), we can choose an LSTAR model. We used P-values for he F-ess and made he choice of he STAR model on he basis of he lowes P-value. The P-values obained were (i) , (ii) and (iii) Thus, we chose o fi an LSTAR model (Table ). The nex sep is o esimae he parameers in he STAR models. Table summarises he esimaion resuls, including he ESTAR model. We used he MLE and BFGS numerical algorihms, which saisfy various regulariy condiions (such as saionary, ergodiciy, consisency and asympoic normaliy). We now commen on some specific aspecs of he wo models, alhough he model seleced was LSTAR in erms of he sequenial procedure suggesed by Teräsvira (994) 2. The esimaed coefficiens are lower han uniy, φ ij <, i, j. The ML esimaions of he STAR model parameers of he wo regimes are similar. The -saisics, repored in Table 2, are adjused for heeroskedasiciy using Whie heeroskedasiciy-consisen sandard errors o assess he significance of he parameer esimaes. Wih respec o he smoohness parameer (γ),his is always posiive, small in he case of ESTAR and large in he case of LSTAR. The LSTAR esimaion suggess ha regime shifs or ransiions beween he regimes are smooh. 2 In general, he LSTAR and ESTAR models have he same number of parameers, and he comparison of heir log-likelihoods may be meaningful. In his sense, he resuls show ha boh esimaions are possible bu, saisically, LSTAR(2;3) seems o fi beer han ESTAR(2;3), in erms of he log-likelihood value. However, similar likelihood values migh sugges ha hese models are likely o produce a similar forecas performance. 6

18 Table 2. MLE (BFGS) for STAR models. Period from o T=2350. ESTAR (.26) LSTAR (.72) φ 0 φ φ 2 φ 20 φ (5.5) (3.0) (-.0) (-.29) (0.32) (0.8) (-4.99) (-.77) φ γ c LogL (.76) (2.) (0.59) (5.74) (.75) (7.84) Noe: The -Suden values for he null hypohesis ha he parameer is equal o zero are given in parenheses. These values are calculaed using Whie heeroskedasic-consisen sandard errors. The regime shif or hreshold parameer (c) indicaes he halfway poin beween he expansion and conracion regimes. This is posiive and saisically significan a he 5% level of significance in he ESTAR model and a 0% in he LSTAR model. Boh models are in he range of he ransiion variable s = r 6 (which varies beween abou and 0.06). The ransiion is slow a he values for r of cˆ, wih ransiion probabiliies ˆ 6 ( s ; c) F, γ ˆ, ˆ swiching from 0 o a his poin. The wo regimes can be described as follows: when F = 0, which we migh refer o as he lower regime in he LSTAR model and he middle regime in he ESTAR model, he mean process for r is an AR(2) wih complex roos (i.e. for he ESTAR model he LSTAR model i is 0.09 ± 0.08i, having a modulus of 0.2; and for ± 0.22i wih a modulus equal o 0.22). When F =, we migh refer o i as he upper or expansion regime in he LSTAR model and he ouer regime (expansion and conracion regime) in he ESTAR model. In his case, he mean process for r is also an AR(2) wih complex roos (i.e. for he ESTAR model having a modulus of 0.2; and for he LSTAR model i is 0.06 ± 0.0i 0.02 ± 0.42i wih a modulus equal o 0.42). So, if r 6 exceeds in he ESTAR model and in he LSTAR ˆ, 6 γ ˆ, ˆ can ake values close o one, bu wih differen dynamic properies. ˆ, 6 γ ˆ, ˆ versus ime in days, and Figure 5b displays model, F ( s ; c) Figure 5a shows he graph of F ( s ; c) ˆ 6 ( s ; c) F, γ ˆ, ˆ versus r 6 for he ESTAR model. In his model, high values for he ransiion probabiliies imply ha sock index reurns are eiher in an expansion or in a conracion regime (ouer regime). From Figure 5b we observe ha he behaviour of sock index reurns in he ransiion period or middle regime is differen, bu ha he wo regimes have similar dynamics. In his sense, we canno idenify expansionary and conracionary phases, bu we can disinguish beween he ouer regime and he middle regime. Figures 6a and 6b show 7

19 he same relaionships when he model is LSTAR. In his model, he cyclical behaviour of sock reurns can be inferred from he esimaes of ransiion probabiliies. When he ˆ 6 ( s ; c) F, γ ˆ, ˆ ransiion probabiliies are greaer han 0.5, he sock marke could be considered o be in an expansion regime. Figure 6a shows he probabiliy of an expansion ˆ, 6 γ ˆ, ˆ. We can clearly observe he periods of high and low reurns and, from regime F ( s ; c) Figure 6b, we see ha he ransiion beween high and low reurns (expansion and conracion regimes) is reasonably smooh, alhough here are no many daa poins for which r 6 exceeds cˆ Figure 5a. F ( s ; c) ˆ, 6 γ ˆ, ˆ ESTAR versus ime Figure 5b. F ( s ; c) ˆ, 6 γ ˆ, ˆ ESTAR vs. 6 r Figure 6a. F ( s ; c) ˆ, 6 γ ˆ, ˆ LSTAR versus ime Figure 6b. F ( s ; c) ˆ, 6 γ ˆ, ˆ LSTAR vs. 6 r. 8

20 To evaluae he wihin-sample performance of he esimaed STAR mode ls, we used some misspecificaion ess, only for he LSTAR model. We did no use he Ljung-Box es for serial correlaion because simulaion sudies sugges ha 2 χ asympoic disribuion may no be valid. Mehods of esing he adequacy of fied STAR models are discussed in Eirheim and Teräsvira (996). These auhors conribue o he evaluaion sage of a proposed specificaion, esimaion and evaluaion of hese models. To deermine wheher such a model is adequae, we firs esed he hypohesis of no error auocorrelaion or serial independence, bu we did no rejec he null hypohesis a he 5% level of significance. Secondly, o es agains general negleced non-lineariy or remaining lineariy, second and hird-order erms of he form r r i j for i=,..,p and j=i,...,p, and i j k r r r for k=j,..,p may be added o he LSTAR model and esed for significance. Doing so for he fied LSTAR(2;3) model leads o a saisic ha is significan a any level (P-value equal o 0.00), which confirms he possibiliy of addiive non-lineariy, alhough a rejecion as such in general does no give much orienaion as o wha o do nex [see Eirheim and Teräsvira (996)]. In his sense, if he non-lineariy is manifess in he condiional variance, hen we would expec o find significan ARCH effecs. Using he Lagrange muliplier es for ARCH effecs, we obained a P-value ha suggesed he presence of his ype of nonlineariy (i.e. ARCH() Lagrange Muliplier es wih P-value equal o 0.00, ARCH(5) equal o 0.00 and ARCH(0) equal o 0.00). Anoher imporan assumpion is es parameer consancy. Eirheim and Teräsvira (996) posulae a parameric alernaive parameer consancy in STAR models, which explicily allows he parameers o change smoohly over ime. These ess are monoonic parameer change, a symmeric non-monoonic change and a more flexible es ha allows monoically and non-monoonically changing parameers. For hese ess, he P-values are , and , respecively. In none of he hree cases do we rejec he null hypohesis a he 5% level of significance. Thus, he validiy of he LSTAR model for sock reurns depends on he exisence of remaining non-lineariy and ARCH errors. We rea hese empirical facs by reducing he magniudes of exreme observaions and ouliers, and explore he esimaion of he MLE for GARCH and STAR-ype models by using wo highly flexible non-linear models, namely STAR-GARCH and STAR-Smooh Transiion GARCH [see Lundbergh and Teräsvira 9

21 (998)]. These auhors have exended he STAR model by incorporaing he concep of smooh ransiion ino he GARCH componen (STGARCH). This model is non-linear no only in he condiional mean, bu also in he condiional variance. Moreover, ε is assumed o follow a GARCH(,) process ha is useful for capuring volailiy clusering, while he hreshold variables are useful if he daa exhibi regime swiching behaviour for varying sock reurns and ε. In he case of STGARCH-ype models, we consider H( p, e) ransiion funcion which saisfies he same condiions as F ( s ; c) d, ;ξ as a, γ. We assume here exis wo regimes wih he ransiion variable p = ε ; ξ is he ransiion rae; e is he hreshold value, and regarding he choice of ransiion funcion, we employ he firs-order logisic funcion. The following implicaions follow from he esimaes in he LSTAR- GARCH(,) and LSTAR-STGARCH(,) models. Firs, he MLE is exremely sensiive o he choice of iniial values. Second, convergence is achieved afer very ieraions. In he case of LSTAR-GARCH(,), i is easily achieved bu he ransiion variable seleced for he mean process is r 2. The esimaed coefficiens γ ˆ = 3. 4 and c ˆ = are significan a he 5% level of significance, bu he AR(2) parameers are no significan a any level. The GARCH coefficiens mee he sufficien condiions for sric posiive condiional variance ( ω ˆ = , α ˆ = 0. and β ˆ = 0. 82, respecively). However, he resuls for he LSTAR-STGARCH(,) model appear o be worse afer such adjusmens. We conclude ha he esimaed models based on adjused daa perform similarly and do no improve on he wihin-sample esimaes in Table 2 for sock index reurns. Alhough he effecs of misspecificaion of non-linear models are generally unknown, i is difficul o draw firm conclusions abou he effecs of ouliers and remaining non-lineariy, because here are difficulies in fiing he LSTAR-STGARCH model for wo regimes Arificial neural nework models In his secion we employ he echnique of ANN esimaion o obain ou-of-sample forecass. An imporan feaure of ANNs is ha hey are non-parameric models. We do no wan o rea he ANN as a black box, in he sense ha no analysis of he characerisics 20

22 and properies of he esimaed neworks is performed and no explanaion is given as o why hese models perform quie well in he forecasing exercise. The specific ypes of ANN esimaed in his sudy are MLP(p,q), JCN(p,q) and Elman(p,q) as discussed in Secion 2. The archiecure of hese models includes one hidden layer and various hidden unis or elemens of he single hidden layer (q). The oupu variable is he daily sock reurn. The inpu variables seleced in he inpu layer include lagged sock index reurns, p (he number of lags in he auoregressive par), and are scaled assuming a uniform disribuion wihin he inerval [-;]. The p-order lagged reurns are calculaed by sequenial validaion, and so we esimae ANN models wih differen values of p and q. The rank of he erms employed is p,q=,,5. The inclusion of hese lags is based on he evidence in Secion 4. ha suggess lagged reurns are needed in he condiional mean specificaion, while auocorrelaion in he sock index reurns can appear because of non-synchronous rading effecs. Moreover, a link was inroduced beween he inpu variables and he oupu variable. As here is no reliable mehod of specifying he opimal number of hidden layers, we specified one hidden layer. This choice was made because many sudies ha carry ou sensiiviy analysis o deermine he opimal number of hidden layers have found ha one hidden layer is generally effecive in capuring non-linear srucures [see Adya and Collopy (998) for an overview]. The hidden uni acivaion funcion g(.) is he hyperbolic angen funcion [see foonoe 7], because i produces a beer fi. We did no choose p, q and g(.) a priori. The ANN models were rained over a raining period (i.e. raining sample) using 500 raining cycles and crossvalidaion. The raining se was used o esimae he neural nework weighs. To improve on he in-sample fiing performance of he ANN models, he esimaed se of weighs was used as a se of iniial values for raining. We used crossvalidaion sraegy in raining o avoid overfiing (good in-sample, bu poor ou-sample performance). The raining phase of he ANN was performed wih 855 observaions, whereas in he es phase 463 observaions were used, boh sizes being randomly deermined. The wo hundred final observaions were se aside o make predicions. The decrease in he error rae in he raining and es phases was hen esed. The oupu was compared o he sample of original values of he oupu by comparing he roo mean squared error (RMSE). We observed as he RMSE declines over successive raining (i.e., 2

23 n). When RMSE reaches a minimum and hen sars increasing, his indicaes ha overfiing may occur. On he basis of he esimaed weighs from n-h raining over he raining period, ou-of-sample forecass were generaed for subsequen es periods. Table 3. MSE and MAE saisics of he ANN models wih a single hidden layer during he raining phase (period from o ) and he es phase (period from o ). MLP(p,q) JCN(p,q) Elman(p,q) Training Tes Training Tes Training Tes p q MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE Noe: Bold ype denoes he MAE and MSE in he raining and es phases which correspond o he bes MSE in he ou -of-sample phase. Table 3 shows he final resuls in he raining and es phases in he las ieraion for mean squared error (MSE) and mean absolue error (MAE) saisics. The esimaes of nework paerns presen he following aspecs. In erms of he minimum MSE in he ouof-sample phase, he bes adjused model holds wo-explanaory variables, p=2 (i.e. he one-period and wo-period lagged sock index reurns, r and r 2, as ESTAR and LSTAR esimaed models in Secion 4.), and q=4 hidden unis in he single hidden layer in Eq. [5] and Eq. [6], and q=3 in Eq. [7]. We can wrie hese models as MLP(2,4), JCN(2,4) and 22

24 Elman (2,3) arificial neural neworks. For hese seleced models, he MLP model has a lower MSE and MAE han he JCN and Elman models in he raining phase (wihinsample). If we compare he ANN resuls wih he AR and STAR models, he ANN models fi he wihin-sample daa beer han he oher models (i.e. regarding he MSE and MAE saisics, AR(2) has.526 and.098; ESTAR(2;3) has.73 and 0.79; and LSTAR(2;3) has.7 and 0.788). We do no repor he esimaed weighs from raining he ANN model given in Eq.[5], Eq.[6] and Eq.[7] for he raining period. However, here are some similariies regarding he magniudes and signs of he weighs ha appear in all hese models, such as i=,,p, j=,..,q. β ˆ, βˆ, φˆ φˆ, 0 j 0j i, j Le us consider wha kind of non-linear relaionships beween he reurn and pas reurns are picked up by ANNs. Like Qi and Maddala (999), o visualize wha relaionship beween reurns and he underlying predicing variables has been capured by he neural nework, we repor he resuls of sensiiviy analysis and compare i wih he observed reurns. As an illusraive graph of possible non-lineariy, le us consider Figure 7, which plos he observed reurns ( r ) agains he one-period lagged reurn ( r ) and he woperiod lagged reurn ( r 2 ) in hree samples: (i) he firs sample is similar o he raining se; (ii) he second is similar o he es phase; and (iii) he hird is equivalen o he forecas phase. Figure 8 conains various groups of graphs (Figures 8a, 8b and 8c), which show he esimaed reurns in he raining, es and forecas phases from he neural nework for MLP (Figure 8a), JCN (Figure 8b) and Elman (Figure 8c) aga ins he observed r and r 2. In Figures 8a, 8b and 8c, case (c) plos he simulaed sock reurn ( r ) from he neural nework for MLP, JCN and Elman in he forecas phase agains r and r 2. From hese graphs, we can observe a complex non-linear relaionship beween reurns and lagged reurns, showing ha his series displays a cyclical behaviour around poins ha shif over ime when hese shifs are endogenous, i.e., caused by pas observaions on r hemselves, which can be viewed as a ypical feaure of non-linear ime series. MLP and Elman ANNs perform beer han JCN. 23

25 Figure 7. Observed reurns. (i) Firs sample (ii) Second sample (iii ) Third sample Figure 8. Sensiiviy analysis Figure 8a. MLP(2,4) esimaed reurns and lagged observed reurns. (a) Training (b) Tes (c) Forecas Figure 8b. JCN(2,4) esimaed reurns and lagged observed reurns. (a) Training (b) Tes (c) Forecas 24

26 Figure 8c. Elman(2,3) esimaed reurns and lagged observed reurns. (a) Training (b) Tes (c) Forecas The beer fi of he neural nework model repored above is no surprising given is universal approximaion propery. 5. Saisical assessmen of he ou-of-sample forecas This secion focuses on he ou-of-sample forecasing abiliy of he STAR and ANN models in erms of saisical accuracy. The randomly seleced predicion period corresponds o he las 200 periods of he sample. This forecas period was from 3 May 999 o 0 February One-sep-ahead forecass were generaed from all models. I is generally impossible o specify a forecas evaluaion crierion ha is universally accepable. In order o assess he predicive abiliy of he differen models, we use various saisics of predicion accuracy. The measures of accuracy used in his paper are based on h=,...,h predicion periods for r h, called rˆ h. Alhough ANN is expeced o have a superior in-sample performance, since i ness he AR linear model and STAR model, here is no guaranee ha i will predominae in he ou-of-sample period. The relaionship beween sock reurns and lagged sock reurns was invesigaed by comparing he predicions of AR and non-linear models ha can be used for reurn predicion. The forecas evaluaion was made beween he resuls from he following models for sock index reurns: AR(2), LSTAR(2;3), ESTAR(2;3), MLP(2,4), JCN(2,4), and Elman(2,3), sricly for he predicion period. We did no include GARCH models in he se of forecasing models because hese models parameerise he condiional variance, whereas he objec o be forecas is he sock index reurns, no is volailiy. 25

27 We compared he ou-of-sample forecass using wo differen esing approaches. Firs, we examined he forecas accuracy from all he esimaed models by calculaing he MAE, mean absolue percenage error (MAPE), RMSE, U-Theil and he proporion of imes he signs of reurns are correcly forecased (Table 4, Panel A). In erms of classic forecas evaluaion crieria, he bes resuls are he lowes values. As indicaed in his able, he MAE, RMSE and U-Theil of he forecass from he ANN models are lower han hose of he linear model, excep in he Elman ne for RMSE. In erms of MAPE, AR is beer han he oher models. However, he signs correcly esimaed are slighly superior in ANNs, wih 55% success in he MLP model. This resul implies ha he ANN-based forecass are in general more accurae han hose of he linear and STAR models. Second, o examine he direcional predicion of changes, he forecas encompassing and o analyse wheher he difference beween he RMSEs is saisically significan for our ou-of-sample forecass, we employed various ess of hypoheses, such as he Pesaran and Timmermann (DA, 992) es, which was used as a direcional predicion es of changes. Under he null hypohesis, he real and prediced values are independen. The disribuion of he DA saisic is N(0,), and i has he following srucure: DA = [ ( SR) ( SRI )] 0. 5 var var ( SR SRI ), where SR = H I [ y yˆ > 0] ( p )( p ) ˆ ˆ H i h= h. and SRI = p p +, where SRI is he success raio in he case of independence beween he real and prediced values under he null hypohesis. The oher elemens H are: p = H I i [ y h > 0 ], pˆ = H I i [ yˆ h > 0], var( SR) = H [ SRI ( SRI )] var h= H h= [ 2 ˆ ˆ ˆ ˆ ˆ ] ( SRI ) = H H ( p ) p ( p ) + ( 2 p ) p ( p ) + 4 p p ( p )( p ) h and. The resuls are repored in Table 4, Panel B. A he 5% significance level hese resuls do no rejec he null hypohesis ha forecass and realizaions are independen, which indicaes ha independence is no rejeced for all he linear and non-linear models analysed. We employed he forecas encompassing esing approach for our ou-of-sample forecass. In forecas encompassing, he crierion is ha he i-h model should be preferred o he j h model if he former can explain wha he laer canno. Le ( f ) f, be wo i j compeing forecass of sock reurns. When fi encompasses f 2, Chong and Hendry (986) 26