FORECASTING EXCHANGE RATES: A ROBUST REGRESSION APPROACH

Transcription

1 FORECASTING EXCHANGE RATES: A ROBUST REGRESSION APPROACH Arie PREMINGER a, and Raphael FRANCK b Absrac The leas squares esimaion mehod as well as oher ordinary esimaion mehod for regression models can be severely affeced by a small number of ouliers, hus providing poor ou-ofsample forecass. This paper suggess a robus regression approach, based on he S-esimaion mehod, o consruc forecasing models ha are less sensiive o daa conaminaion by ouliers. A robus linear auoregressive (RAR) and a robus neural nework (RNN) models are esimaed o sudy he predicabiliy of wo exchange raes a he -, 3- and 6-monh horizon. We compare he predicive abiliy of he robus models o hose of he random walk (RW), he sandard linear auoregressive (AR) and neural neworks (NN) models in erms of forecas accuracy and sign predicabiliy measures. We find ha robus models end o improve he forecasing accuracy of he AR and of he NN a all ime horizons, and even of he RW for forecass carried ou a he -monh horizon. Robus models are also shown o have significan marke iming abiliy a all forecas horizons. Keywords: Exchange raes; Forecasing; Neural neworks; Ouliers; Robus regression approach; S-esimaion. JEL classificaion: F3, C45, C53. (a) Cener of Economerics and Operaion Research, Universiy de Caholique, LLN, Belgium (b) Deparmen of Economics, Bar Ilan Universiy, Rama Gan, Israel. (*) Corresponding auhor: Arie Preminger, Universiy de Caholique, Voie du Roman Pays 34, B-348 Louvain-La-Neuve, Belgium address: preminger@core.ucl.ac.be; Tel ; Fax: The auhors hank Paolo Colla and Shinichi Sakaa and paricipans a he economerics group in CORE, Universie de Caholique for heir commens, which improved his paper. Preminger graefully acknowledges research suppor from he Ernes Foundaion.

2 . Inroducion Building forecasing models for asse reurns is of pracical as well as of heoreical imporance. The pracical value lies in ha good forecass can provide useful informaion o invesors for asse allocaion, firms in risk hedging and cenral banks in policy making. From a heoreical poin of view, reurns predicabiliy has imporan implicaions for he efficien marke hypohesis, and more generally, for he heoreical modeling in finance. Since he breakdown of he Breon-Woods sysem, here has been more ineres in predicing exchange raes. The lieraure shows, however, ha exchange raes are largely unpredicable. In heir seminal work Meese and Rogoff (983) find ha a simple random walk model performs no worse han a range of represenaive imeseries and srucural exchange rae models. Alexander and Thomas (987), and Wolff (987) furher show ha he random walk model ouperforms hese economeric models even when ime-varying parameers are incorporaed ino he models. Wolff (988) reaches a similar conclusion using ime-varying auoregressive models. Since hese empirical resuls rely mainly on linear ime series models, i migh be reasonable o conjecure ha he unpredicabiliy of exchange raes may be due o he limiaions of linear models. Indeed, several sudies, noably by Baillie and McMahon (989), Hsieh (989) and Hong and Lee (2003), have shown ha changes of exchange rae are nonlinearly dependen, alhough hey are ofen serially uncorrelaed. A number of researchers have ried o capure nonlineariies in exchange raes, bu wih lile success. For example, Diebold and Nason (990) and Messe and Rose (990) used non-parameric kernel regression and were no able o improve upon a simple random walk. Meese and Rose (99) examine several srucural exchange rae models o accoun for poenial nonlineariies. They conclude ha he incorporaion of nonlineariies ino srucural models of exchange raes do no improve our abiliy o predic exchange rae movemens. Also, using he Markov-swiching model Engel and Hamilon (990) and Engel (994) were no able o produce forecass ha are superior o a random walk model Conrary o hese findings, exchange rae forecasing using neural neworks (NN) provides some evidence ha hey are beer han oher nonlinear models in erms of ouof-sample forecasing abiliy. Kuan and Liu (995) repor ha he use of NN for daily exchange raes generaes significanly lower ou-of-sample mean squared forecas errors relaive o he random walk model. Brooks (997) and Gencay (999) also documen some predicabiliy of daily exchange raes using hese models. However, Qi and Wu (2003) who employ a neural nework wih moneary fundamenals, find ha heir model canno bea he random walk model, alhough i occasionally shows a limied marke iming abiliy. Lisi and Schiavo (999) perform a deailed comparison of neural neworks and chaoic models for predicing monhly exchange raes. They find ha neural neworks compare favorably wih chaoic models and ha boh perform significanly beer han random walk model. In his paper, we ake anoher approach o exchange rae forecasing as we ake ino accoun he presence of ouliers in he daa and heir effec on he performance of he forecass. Exchange raes as well as oher financial asses are characerized by dramaic

3 changes over ime, as a resul of marke crashes or rallies, changes in economic policy and business cycles. Such changes can be viewed as inroducing ouliers (and leverage poins) ino he ime series of ineres. I is well known ha he presence of ouliers can have a dominaing and deleerious effec on sandard locaion model esimaors such as he leas squares regression, he leas absolue deviaion regression, or even he generalized M-esimaors (Maronna e al. (979), Donoho and Huber (983) and He (99)). Furhermore, Sakaa and Whie (995) show ha quasi-maximum likelihood (QML) regression esimaors are in general vulnerable o ouliers. The exisence of ouliers is also corroboraed by numerous sudies (see e.g. Hsieh (989), Baillie and Bollerslev (989) and Andersen e al. (200)). They have shown ha he disribuion of he changes in exchange raes is far from being normal, mainly due o he fac ha exchange raes are conaminaed by some ouliers or exreme values. Such ouliers may no be helpful in predicing fuure reurns. However, hey may unduly influence he esimaion and forecasing of financial ime series. Balke and Fomby (994) and Dijk e al. (999) find ha neglecing ouliers can erroneously sugges misspecificaion or inadequae modeling. Their sudies sugges ha ouliers can accoun for some nonlineariy deeced in he daa. Ledoler (989) and Hoa (993) noe ha ouliers lead o bias parameer esimaes and inflae he esimaed variance of he series, hus harming he qualiy of ou-of-sample forecass. Consequenly, hese findings lead o he percepion ha he regression model approach for forecasing may have unavoidable limiaions in predicing exchange rae reurns. In order o overcome his problem, a robus regression approach is developed. I is based on consrucing a regression model using he S-esimaion mehod, which limis he influence of oulying observaions on he esimaed parameers. The S- esimaors are robus o daa conaminaion by ouliers and are shown o be asympoically more efficien han he common leas squared (LS) esimaors when he daa is generaed from fa ails disribuions (Sakaa and Whie (200)). As was poined ou by Rousseeuw and Yohai (984), hese esimaors are also much more efficien and less compuaionally demanding han oher robus esimaors such as he leas rimmed squares (LTS) or he leas median of squares (LMS) esimaors. We apply our approach o explore he predicabiliy of currency prices of he Japanese yen and Briish pound vis-à-vis he US dollar a -, 3- and 6- monh horizons using monhly daa. The AR and he random walk are used as linear predicors while he single layer feedforward nework (NN) is used as a nonlinear predicor. Our approach is used o esimae a robus AR model (RAR) and a robus NN model (RNN). The choice of NN models was moivaed by he relaive success of his class of models in predicing exchange raes, as discussed above. In order o assess forecasing accuracy, we use measures such as he mean squared error, he mean absolue error, he median of he square error and direcion of change accuracy. We apply he Pesaran and Timmermann (992) es o check for marke iming abiliy of he sign predicions. The Diebold and Mariano (995) saisic is used o es he null hypohesis of no difference in he forecas accuracy of wo compeing models. This es is used o evaluae he saisical significance of he mean squared error and he mean absolue error of our robus forecasing models relaive o ha of he AR, NN and he random walk models. For a more deailed discussion on forecasing when ouliers are presen, see also Chen and Liu (993), Trivez (993) and Phillips (996). 2

4 We find ha robus models ends o improve he forecass of he AR and of he NN a all ime horizons, and even of he RW for forecass carried ou a he one-monh horizon. Robus models are also shown o provide good sign predicions and o have srong marke iming abiliy for all forecas horizons. The res of he aricle is organized as follows. The nex secion suggess a robus esimaion procedure for he parameer of regression model, which is based on he S- esimaion mehod. Secion 3 inroduces he daa and he forecasing models. We also describe various saisics o be employed o es he predicabiliy of exchange raes. Secion 4 repors he empirical resuls. Finally, Secion 5 concludes. 2. An oulier robus regression model 2. The forecasing problem A commonly used approach o forecas a ime series variable y is by esimaion of a regression model h ( x, θ ) where x includes lagged values of y as well as exogenous variables and θ Θ is a bounded parameer se. The esimaion procedure is based on finding he bes parameric model, which minimizes some scale (dispersion) measure of he errors { u ( θ ) = y h( x, θ )}. For example, suppose ha we observe realizaions of z = ( y, x ), and ha we wish o consruc a regression model in order o forecas y on he basis of x (which can include lagged variables of y ). We would usually employ he mean squared error (MSE) measure and solve he following Θ T T = T 2 2 min u = y h x )) θ ( θ ) ( (, θ () T = Under mild regulariy condiions we can show ha he LS esimaor he soluion o he following opimizaion problem LS θˆ T converges o T min Θ E T u 2 θ ( θ ) (2) = q Insead of using he MSE, we may use he L esimaors, which are scale measures based on he of q h absolue momens ( q ) of he error erm. In his case, we T choose θ o minimize he sample analog of q E T u ( θ ). This class of scale = measures belongs o a more general class called regular scale abou he origin, which was inroduced by Sakaa and Whie (200). This class basically measures he degree of concenraion of he disribuion of he error erm abou he origin. For q = 2 he opimal regression model for predicion is E ( y I ) where I is he σ algebra induced by all variables ha are observed a ime (hus, I conains he lagged values of y and oher predeermined variables). If insead q = he opimal predicor equals o he condiional median. Now, given a specified parameric regression model h (, θ )} : θ Θ, we can ry o approximae he opimal predicor funcion. { x Ν 3

5 As was poined ou by Sakaa and Whie (200), here is a clear relaionship among he opimum parameer values deermined by he various scales. Suppose ha here exiss θ 0 Θ such ha for Ν he condiional disribuion of y h( x, θ ) given x is symmeric abou he origin and non-increasing on he posiive real line 2. Then he scale measure is minimized a h ( x, θ 0 ) regardless of which scale measure is employed. Therefore, if we assume he following regression model 3 y = h(, θ ) + ε (3) x and esimae i using he LS esimaion mehod, we would obain he same predicion as if we were using esimaes which are based on oher regular scales abou he origin. q However, he scale measures employed by he L esimaors are very sensiive o ouliers which are frequen in financial daa. A measure of he sensiiviy of he esimaors o ouliers is he breakdown poin, which can be defined as he minimum proporion of he daa for which conaminaion by ouliers can lead o compleely noninformaive esimaion resuls. Several measures of he breakdown poin have been suggesed, noably by Hampel (968, 97), Donoho and Huber (983), Sromberg and Rupper (992) and Sakaa and Whie (995). Under all measures i has been shown q ha he L esimaors for regression models end o be easily affeced by a small number of ouliers. More specifically, hese esimaors are shown o have a breakdown poin of / T. Thus, a single bad observaion can already cause a breakdown 4. Furhermore, many oher familiar regression esimaors are also vulnerable o ouliers. Sakaa and Whie (995, 998) show ha he familiar quasi-maximum likelihood (QML) regression esimaors including he Laplace, Cauchy and Huber leas informaive densiies are vulnerable o daa conaminaion by ouliers. 2.2 The robus regression approach In order o overcome he problem of low breakdown poin, we should use scale measures ha are resisan o ouliers in he fied errors. Such alernaive scale measures wih high breakdown poin have been suggesed, noably, Rousseeuw's (983) leas rimmed sum of squares (LTS), Rousseeuw's (984) leas median of squares (LMS) 5 and he Rousseeuw and Yohai's (984) S-esimaion mehod. The esimaors based on hese scale measures are called high breakdown esimaors, because hey can resis up o 50% oulier conaminaion in cross secion daa. In he case of ime series daa, he regressors ofen include lagged variables so ha a single conaminaed value may appear muliple imes in he regressor vecor. Therefore, esimaors having a low breakdown poin in cross secion daa are even more vulnerable o daa conaminaion in he presence of lagged variables. On he oher hand high 2 This assumpion includes a wide range of disribuions for he error erm, such as he normal, suden, Laplace and also includes he case of condiional heeroskedasic errors) 3 Noe ha he regression model does no have o be correcly specified (i.e. σ ( x ) I ) ). 4 Precise resuls can be found in Rousseeuw (984), Rousseeuw and Leroy (987) Sromberg and Rupper (992), Sakaa and Whie (995) among ohers. 5 The scale is measured by he median of he absolue value of he random variable. 4

6 breakdown esimaors can sill provide good proecion as long as he maximum number of lags is no oo large. Suppose m is he maximum lag order of he regression funcion for each Ν, Sakaa and Whie (998, 200) show ha he lower bound for he breakdown poin is approximaely 50% ( + m). The smaller he maximum lag order is, he higher his lower bound. Rousseeuw (983) shows ha he sochasic 2 convergence rae of he LTS esimaor is O ( T / p ) while Kim and Pollard (990) and 3 Zinde-Walsh (2002) show ha he convergence rae of he LMS is O ( T / p ). Unforunaely, he LTS mehod is compuaionally much more demanding han he LMS mehod. However, he S-esimaion mehod is compuaionally as convenien as he LMS mehod while being asympoically as efficien as he LTS. Therefore in his paper we use he S-esimaion mehod o obain robus forecass as an alernaive o common LS esimaion mehod. Furhermore, i was shown by Sakaa and Whie (200) ha his scale measure is a regular scale abou he origin, as discussed above. Hence, in he absence of daa conaminaion, he forecass obained by using he S-esimaors are he same as hose obained by using he common leas squares mehod. However, he S-esimaion mehod provides robus forecass in he case of daa conaminaion. In order o define he S-esimaor, we use he funcion ρ ( ) which is even, bounded, coninuously differeniable wih ρ ( 0) = 0 and sricly increasing a every posiive poin where i has no achieved is supremum ρ. Now, for each θ Θ and a sample ( y, x ), =,2, KT we define he scale measure of he residuals as follows 6 T S( θ ) = sup s R++ : T ρ(( y h( x, θ )) / s) = K (4) = where R + + = ( 0, ) and 0, S K ρ ). Then he S-esimaor θˆ T is defined ( 2 ˆ S θ = arg minθ S( θ ) (5) T Θ S The esimaed θˆ T is used o consruc a robus regression model. The breakdown poin of his esimaor is deermined by he raio K ρ. For example, in order o obain an esimaor which is resisan o 0% daa conaminaion in he i.i.d. case, we se K equal o 0. ρ. In ime series conex a lower bound for he breakdown poin is he breakpoin in he i.i.d. case divided by he number of lags plus one. We wish o cauion he reader ha using he S-esimaor wih he highes breakdown poin is no necessarily advisable, since downweighing he exreme observaions may resul in loss of efficiency. In he absence of daa conaminaion i will be preferable o consider he use of he LS esimaion mehod which is simpler and more sraighforward. 6 Noe ha when here is no soluion o he equaion above we se S ( θ ) = 0. 5

7 In order o esimae our models, we employ an S-esimaor where he ρ ( ) funcion is derived from Tukey's biweigh funcion (see Rousseeuw and Yohai (984) and Beaon and Tukey (974)) and defined as w w w c 2c for w c ρ ( w) = (6) 2 c for w > c 6 This funcion is shown in Figure below ( c = ). ρ(w) w Figure : Tukey 'Bisquare' funcion ( ρ = 6 ). Since he objecive funcion in equaion (5) is no convex and has several local minima. We need o use a global minimizaion algorihm such as he simulaed annealing algorihm or he geneic algorihm. In addiion, he evaluaion of equaion (4) requires a nonlinear equaion solver ha uses some ieraive mehod. This compuaion akes more ime han he evaluaion of he sum of he squared residuals in he LS esimaion. We use he adapive simulaed annealing (ASA) algorihm (Ingber (989, 993)), as Sakaa and Whie (998, 200) did. In simulaed annealing algorihms including ASA, each ieraion draws a poin from a cerain disribuion on he parameer space, and decides if he algorihm should move o he new poin. If he value of he objecive funcion a he new poin is lower han he minimum value found by he algorihm, he new poin is acceped. Oherwise, he new poin is acceped if his poin gives a value of he objecion funcion lower han a cerain hreshold which is deermined depending on he previous lowes value of he objecive funcion and a random number. Nex, 6

8 local search algorihms are employed where he parameer values obained by he global opimizaion procedure are hen used as saring poins. 3. Empirical applicaion 3. The Daa We apply our approach o forecas exchange raes. The daa employed are he Japanese Yen (JPY) and he Briish Pound (GBP) raes agains he US dollar. All he daa are monhly and are obained from he Federal Reserve Board daabase. Our sample sars a January 97 and ends in Ocober 2004 wih 406 observaions. To avoid problems arising from non-saionariy observed in exchange rae daa, we compue he differences beween naural logarihms of he original exchange rae series. Le e denoe he exchange rae a ime, he reurn series is defined as r = 00 log( e e ). The reurn series is graphed in Figure 2. This plo shows episodic occurrences of crashes and rallies, as can be seen by he exreme posiive and negaive values. Hence, here may be added value in applying he robus regression approach for forecasing JPY/USD GBP/USD Figure 2: Exchange rae monhly reurns To obain more deailed informaion on he daa, he descripive saisics of he monhly reurns are presened in Table. I is eviden ha he mean reurn is quie small while he range of he reurn series is relaively large. This is especially rue for he GBP whose range is beween -0.5% and 9.9%. For boh series he esimaed skewness is small and negaive and he kurosis is significanly higher han ha of a normal disribuion, which is hree. These saisics along wih he Bera-Jarque es indicae ha he reurn series are characerized by fa ailed disribuions, as is usually he case in such financial daa. This non-normaliy can have a serious effec in finie samples on he esimaed regression parameers and he forecass, as was noed above. 7

9 We also repor in Table he firs hree auocorrelaion coefficiens and heir Barle sandard errors. The firs coefficien for he JPY and GBP are saisically differen from zero a 5% significance level while he res are no saisically significan from zero. Finally, we compue he Ljung-Box Q saisic for he firs en lags. This es is asympoically disribued χ 2 (0) under he null hypohesis. The resuls of his es lead us o rejec he null hypohesis of idenical and independen observaions for boh exchange raes. Table : Summary saisics for he reurn series of he monhly exchange raes from January 97 o Sepember GBP JPY Sample size 406 Mean Sandard deviaion Skewness Kurosis BJ es Max Min ρ ρ ρ Barle sandard errors Q(0) es Noes: GBP and JPY refer o he Briish Pound and he Japanese Yen. The BJ es is he Bera and Jarque 2 normaliy es which has a χ disribuion wih 2 degrees of freedom under he null hypohesis of normally disribued errors. ρ, ρ 2, ρ are he firs hree auocorrelaions of each series. The Q(0) 3 es refers o he Ljung-Box Q saisic for he firs 0 lags and i is disribued chi-square wih en degree 2 of freedom under he null hypohesis of idenical and independen observaions ( χ (0 ) 8. 3 ). 3.2 The forecasing models 0.05 = In order o obain forecass of he reurn series of he exchange raes we esimae regression models based on he lagged reurns. We consider he linear auoregressive (AR) model, which is given by r p = α + β ir i + ε i= (7) We also consruc forecass based on a nonlinear auoregressive (NAR) model called a feedforward nework regression model (NN), which is given as follows 8

10 r d p α + β G α + γ r + ε = 0 j j ij i (8) j= i= where d is he number of hidden unis in he neworks and G ( ) is known acivaion funcion, which is chosen o be he logisic funcion, as is common in he arificial neural neworks lieraure. I has been shown ha, under mild regulariy condiions, his class of funcions can approximae any (measurable) funcion o any desired degree of accuracy 7 (see e.g. Funanhashi (989) Hornik e al. (989), Whie (990) and Gallan and Whie (992)). Furhermore, Barron (993) and Hornik e al. (994) have esablished ha hese funcions can approximae unknown funcions wih error 2 decreasing a raes as fas as d independen of he dimension of he inpu space, p, whereas he sandard kernel regression, spline and oher rigonomeric expansions p require exponenially O ( d ) erms o achieve he same approximaion. Thus, he neural neworks are (asympoically) relaively more parsimonious in approximaing unknown funcions. This is an advanage in erms of having a desirable esimaor in small samples. However, i was shown by Connor and Marin (994) and Connor (996) ha ouliers can cause significan corrupion in he approximaion abiliy of he neural nework model. We also esimae he naive random walk (RW) model. The AR model is esimaed by he ordinary leas squares (OLS) while he parameers of he NN are esimaed by minimizing he sum of squared errors in equaion (8). Since he objecive funcion is non convex, we use he ASA algorihm, as described above, o obain iniial esimaes for he model parameers. Given hese values we use he Newon-Rampson algorihm for he local search procedure. Due o he sensiiviy of he LS-esimaion mehod o ouliers, we esimae a robus AR model (RAR) and a robus neural nework model (RNN) using he S-esimaion procedure as described in secion 2. For each model, we employ hree values 0.0ρ, 0.05ρ, 0. ρ for K o produce robus forecass wih a low breakdown poin, an inermediae breakdown poin and a high breakdown poin. The robus AR models are respecively denoed RAR(%), RAR(5%) and RAR(0%) while he robus neural nework models are respecively denoed RNN(%), RNN(5%) and RNN(0%). The values, which were chosen, reflec our assumpion ha he exisence of ouliers in he daa is a rare even. Oher values may be used, bu one should noe ha a low value for K should be chosen o avoid low efficiency in esimaion, while K close o ρ 2 affords more proecion agains ouliers. The opimal choice of K, given he possible rade-off beween he breakdown poin and he efficiency of he esimaor is ou of he scope of his paper. 3.3 Forecas evaluaion and comparison The ou-of-sample predicive performances of he robus and non-robus regression model are examined for each exchange rae. We employ he radiional measures of forecasing accuracy such as he Roo Mean Square Error (RMSE), he Mean Absolue Error (MAE) and he Median of Absolue Deviaion (MAD) abou he median. We also 7 For an excellen survey on arificial neural neworks, he reader may refer o Kuan and Whie (994). 9

11 employ he success raio (SR), which compues he number of imes a given model correcly predics he sign of he acual reurn. Le r τ and rˆ τ be he acual reurn and he prediced reurn a ime τ, respecively, wih a forecas period going from + o + n. The forecas error saisics are hen defined as: RMSE = + n ( r rˆ ) 2 n τ τ = + τ (9) MAE = + n r rˆ n τ τ = + τ (0) MAD = median( rτ median( rτ )) () SR = + n n I( r rˆ τ > 0) τ = + τ (2) where I ( ) is he indicaor funcion, ha is I ( a > 0) = if 'a ' is posiive and zero oherwise. The RMSE and MAE are he usual measures of predicion error performance, bu may be generally less robus o he possible presence of ouliers. Hence i is useful o include he MAD saisic, which is no inflaed by exreme observaions, and gives a more accurae indicaion of predicion performance for he bulk of he daa ha is free of ouliers. For hese hree saisics, he lower he oupu is, he beer he forecasing accuracy. These saisics are imporan measures for forecasing accuracy of he model concerned. Ye, hey may no consiue he bes crierion for invesors who are rying o maximize profis and no minimize forecas errors, as was poined ou by Diebold and Mariano (995) and Dunis and Haung (2002) amongs ohers. The SR as a measure of sign predicabiliy addresses his issue. We apply he Pesaran-Timmermann (992) (PT) non-parameric marke-iming es o check he null hypohesis ha a given model has no economic value in forecasing he direcion of he exchange rae. The PT es is given by PT SR Pˆ a = ~ N(0,) (3) V V 2 where Pˆ is an esimae of he probabiliy of correcly predicing he direcion of change assuming independence beween he acual and he prediced direcions, and V and V 2 are consisen esimaes of he variances of SR and Pˆ, respecively. In order o es wheher he forecass from wo compeing models are equally accurae, we use he Diebold and Mariano (995) (DM) es. This saisic is designed as follows: le us assume ha a pair of models produces he h -sep ahead forecas () (2) errors { ˆ ε, ˆ } and ha he qualiy of he forecass is measured by a specified loss + h ε + h g ( ˆ+ h funcion ε ) of he forecas error. We can define he "loss differenial" beween 0

12 () (2) he wo compeing forecass as = g( ˆ ε ) g( ˆ ε ). The es is hen based on he following large sample saisic: d + h} + h} DM d a = ~ N(0,) T (4) 2 π fˆ (0) d where d is he sample average of d, and π fˆ (0) is he specral densiy a 2 d frequency zero which is esimaed in he usual way as wo sided weighed sum of available auocorrelaions (see e.g. Newey and Wes (987)). We use Andrews (99) approximaion rule o se he runcaion lag. We define our loss funcions as () 2 (2) 2 () (2) = ε ) ( ε for he MSE es ; and = ε ε for MAE es. d ( + h + h ) 4. Empirical resuls d + h + h In order o apply our mehodology we use monhly reurns for he Briish Pound (GBP) and he Japanese Yen (JPY) vis-à-vis he US dollar from January 97 o Sepember 2004, a oal of 406 observaions. The forecas horizons, h, are chosen o be, 3 and 6 monhs. In all he experimens, he esimaion has been carried ou using a moving (or rolling) window of he mos recen ( 307 h) observaions, based on he esimaed model in each window he h-sep forecass were generaed. The ou-of-sample forecass include he las 00 observaions for all he forecas horizons 8.For each window and model we use up o six lags of reurns o predic he curren reurn, where a model selecion approach is applied o deermine he lag order. This approach was moivaed by Granger e al. (995) and Sin and Whie (996). I is based on he usage of informaion crieria, which ake ino accoun he rade off beween he model goodnessof-fi measure and model complexiy. We use he Schwarz Informaion Crierion (SIC) in selecing forecasing models 9. For a model wih k parameers esimaed on a window of size n, he SIC is SIC = log( S) + k (log( n)) 2n (5) where S is he scale measure used. The firs erm is a goodness-of-fi-measure, and he second erm is a penaly due o model complexiy. However, here is no economeric heory supporing he usage of he SIC as well as oher informaion crieria for selecion of he number of hidden unis 0, even hough 8 Noe ha economeric heory provides lile guidance as how o choose an opimal ou-of-sample period. As a rule of humb we use roughly abou ¾ of he sample o esimaion for he firs rolling regression and he remaining ¼ for he forecasing period. This choice reflec he need o use a sufficien number of observaions in order o obain precise parameer esimaes and he desire o have a reasonable sample size in order o repor a reliable ou-of-sample comparison beween he models. 9 This informaion crierion is widely used and srongly consisen in he sense ha i chooses he bes model wih probabiliy approaching one as he sample size increases. 0 p In his case he "nuisance parameers" γ } for some j are no idenified under he null hypohesis { ij i = (see Kuan and Whie (994) for deails). There are oher works, which moivae he usage of he model selecion approach in such cases such as Alissimo and Corradi (2002) or Preminger and Wesein

13 oher heurisic mehods can be applied for deermining his number. Therefore, we choose o fix he number of he hidden unis o five. The same choice was made by Whie (998) who claims ha a nework wih five hidden unis "represens a compromise beween he necessiy o include enough hidden unis so ha a leas simple nonlinear regulariies can be deeced by he nework and he necessiy o avoid including so many hidden unis ha he nework is capable of memorizing he enire raining sequence". Recenly he same NN model wih five hidden unis was employed by Hong and Lee (2003) o forecas weekly exchange raes. Furhermore, he S-esimaion mehod is a compuaionally inensive procedure and by simplifying he model selecion sage for each window, we make he compuaion burden of producing he ou-of-sample forecass, feasible. Tables 2 and 4 repor resuls on he forecas performance of he models for he Japanese Yen. Similarly, Tables 3 and 5 presen resuls for he Briish Pound. For he Pesaran Timmermann (PT) and he Diebold Mariano (DM) ess, we display he p- values, defined as he significance levels a which he null hypohesis under invesigaion can be rejeced. In each able, panels A, B, and C respecively repor he resuls for all he models a he -, 3-, and 6-monh forecas horizons. Wihin each panel of Tables 2 and 3, columns (2) o (4) repor he RMSE, MAE and MAD forecas accuracy measures. Columns (5) and (6) presen he SR saisic along wih he PT es for marke iming abiliy for all models excep he RW. This is because he RW can eiher yield all posiive or all negaive resuls and as such has no marke iming abiliy by definiion. Therefore, he PT es canno be compued and no p-value is lised in he ables. In calculaing he DM saisic, he null hypohesis of equal predicive abiliy is relaed o he hree benchmark models: he RW, NN and AR models. The es resuls are presened in Tables 3 and 5. We respecively repor in columns (2) and (3) of each panel he resuls of he DM es under he null hypohesis ha he square forecas error and he absolue forecas error produced by he RW are smaller han hose obained using each oher model. Columns (4) and (5) and columns (6) and (7) are organized in he same manner and show he es resuls when he benchmark models are respecively he AR and NN models. The forecas accuracy measures, which are presened in Tables 2 and 4 for each currency, indicae ha he ou-of-sample performances of each model vary wih he forecas horizon. A he one-monh horizon, he bes forecass are hose given by he robus models. Indeed, we find ha he RNN (0%) is he bes model over he GBP in erms of MAD and MAE, and over he JPY in erms of RMSE and of MAD. I also appears ha, in erms of RMSE, he bes model over he GBP is he RAR(0%) and in erms of MAE, he RNN(5%) is he bes model over he JPY. (2004). Bu heir approach does no cover esimaion using robus scale measures like he S-esimaion mehod and i is limied for parameric models. See e.g. Arifovic and Gencay (200), and Nag and Mira (2002). 2

14 Table 2: Ou-of-sample forecasing performances of models for he GBP. () (2) (3) (4) (5) (6) Models MSE MAD MAE SR PT Panel A: monh horizon RW AR NN RAR(%) RAR(5%) RAR(0%).837 * RNN(%) * * RNN(5%) RNN(0%) *.443 * Panel B: 3 monh horizon RW.864 * * AR NN * RAR(%) RAR(5%) RAR(0%) RNN(%) * * RNN(5%) RNN(0%) Panel C: 6 monh horizon RW.86 * * AR * NN *.095 * RAR(%) RAR(5%) RAR(0%) RNN(%) RNN(5%) RNN(0%) *.095 * Noes: RMSE MAD and MAE are he forecas accuracy measures. SR and he PT are marke iming abiliy saisics. All hese measures are defined in Secion 3.3. I mus be noed ha, when a model eiher yields only posiive or negaive values, like he random walk model, he PT es canno be compued and herefore, no p-value is lised in he able. The AR and NN respecively refer o he auoregressive linear model and he neural nework models. RAR(%), RAR(5%) and RAR(0%) refer o he AR model which is robus o exisence of %, 5% and 0% ouliers in (cross secion) daa. Similarly, he RNN(%), RNN(5%) and RNN(0%) refer o he NN model which is robus o exisence of %, 5% and 0% ouliers in (cross secion) daa. The aserisks ( * ) represen he smalles values over each column for he forecas accuracy measures (columns (2)-(4)) and he larges values for each column for he sign ess (columns (5)-(6)). 3

15 Table 3: Ou-of-sample forecasing performances of models on he JPY () (2) (3) (4) (5) (6) Models MSE MAD MAE SR PT Panel A: monh horizon RW AR NN RAR(%) *.04 * RAR(5%) *.04 * RAR(0%) RNN(%) * RNN(5%) * 0.55 * RNN(0%) *.738 * * Panel B: 3 monh horizon RW AR * *.82 * NN RAR(%) * *.82 * RAR(5%) * *.82 * RAR(0%) * RNN(%) RNN(5%) *.053 RNN(0%) Panel C: 6 monh horizon RW * AR NN RAR(%) RAR(5%) *.922 * RAR(0%) 2.75 * *.885 RNN(%) * RNN(5%) RNN(0%) Noes: RMSE MAD and MAE are he forecas accuracy measures. SR and he PT are marke iming abiliy saisics. All hese measures are defined in Secion 3.3. I mus be noed ha, when a model eiher yields only posiive or negaive values, like he random walk model, he PT es canno be compued and herefore, no p-value is lised in he able. The AR and NN respecively refer o he auoregressive linear model and he neural nework models. RAR(%), RAR(5%) and RAR(0%) refer o he AR model which is robus o exisence of %, 5% and 0% ouliers in (cross secion) daa. Similarly, he RNN(%), RNN(5%) and RNN(0%) refer o he NN model which is robus o exisence of %, 5% and 0% ouliers in (cross secion) daa. The aserisks ( * ) represen he smalles values over each column for he forecas accuracy measures (columns (2)-(4)) and he larges values for each column for he sign ess (columns (5)-(6)). 4

16 The resuls indicae he superioriy of robus esimaion over non-robus esimaion for one-sep ahead predicions since for boh currencies and all forecas accuracy crieria, he RW, he AR and he NN models are ouperformed a his forecas horizon. The resuls are, however, less clear-cu for forecass a he 3- and 6-monh horizon. Indeed, Table 2 shows ha over he GBP, he RW, AR and NN models ouperform he 3- and 6-monh forecass of he robus models. Bu i appears in Table 3 ha, over he JPY, robus models provide beer forecass, or a leas, do an equally good job a forecasing han he non-robus models. For insance, a he 6-monh horizon, he RAR(0%) and he RNN(%) ouperform oher models in erms of RMSE and MAD respecively. Sill, a his ime horizon, he random walk remains he bes predicor in erms of MAE. Tables 2 and 3 also display he ou-of-sample resuls for sign predicabiliy measured by he SR, where he PT saisic is used o es for marke iming abiliy. For he GBP, a he -monh horizon, we find ha he RNN(%) bes predics he direcion of change, wih saisically significan marke iming abiliy. A he 3-monh horizon, RNN(%) coninues o exhibi srong marke iming abiliy and high sign predicabiliy in comparison o he oher models. A he 6- monh, he PT es resuls indicae ha he sign predicions produced by all models do no have a significan marke iming abiliy. The same conclusion can be drawn for he JPY a -, 3- monh horizons. However, a he 6-monh horizon, he RAR(5%) and RAR(0%) provide he bes sign predicors, which are saisically significan, and wih he SR measures equaling 59%. On he whole, we can observe ha robus models improve he sign predicabiliy of non-robus models over boh currencies and all forecas horizons, and ha robus models end o have significan marke iming abiliy. Tables 4 and 5 display he resuls of he Diebold-Mariano (DM) es when he RW, AR and NN models are compared o each of he oher models considered in he sudy. We denoe hese ess DM, DM2 and DM3, respecively. In each able, columns (2) and (3) presen he es resuls for he RW, where a p-value no greaer han 0.05 indicaes ha he RW yield a lower forecas error (in erms of squared error or absolue error) relaive o he compeing model a 5% significance level, while a p-value no smaller han 0.95 means ha he benchmark model produces a higher forecas error a 5% level. The same inerpreaion is given for he p-values repored in columns (4)-(7). The es resuls for he GBP a he -monh horizon indicae ha we canno rejec he hypohesis of equal accuracy beween he RW and he oher models. However, given he DM2 es resuls, we observe ha in erms of MSE, he RAR models as a group is significanly beer han he AR model (a he 5% significan level). The NN model is significanly ouperformed by RNN(0%) and RAR(0%) models for boh he MSE and MAE loss funcions. Alhough, he RW model is he bes model in erms of MAE and MSE, a he 3-, 6- monh horizons, only a few cases are saisically significan. The DM2 es resuls show ha he usage of RAR models yields a lower MSE in comparison o he AR model a he 3-monh horizon. Noe ha as he forecas horizon lenghens he ou-of-sample performances of boh NN and RNN models deeriorae. The DM3 es resuls show ha RNN models end o be significanly ouperformed by NN model a he 3-, 6-monh horizons. 5

17 Table 4: Diebold Mariano es on he GBP: comparisons beween he AR model, he NAR and he random walk model and all he oher models. () (2) (3) (4) (5) (6) (7) Models DM DM2 DM3 MSE MAE MSE MAE MSE MAE Panel A: monh horizon RW AR NN RAR(%) RAR(5%) RAR(0%) RNN(%) RNN(5%) RNN(0%) Panel B: 3 monh horizon RW AR NN RAR(%) RAR(5%) RAR(0%) RNN(%) RNN(5%) RNN(0%) Panel C: 6 monh horizon RW AR NN RAR(%) RAR(5%) RAR(0%) RNN(%) RNN(5%) RNN(0%) Noes: DM, DM2 and DM3 are he p-values for Diebold-Mariano (995) es when he benchmark models are he random walk, AR and NN models, respecively. For each es we consider he MAE and MSE loss funcions. P-values no greaer he 0.05 indicae ha he benchmark model yields a lower forecas error (in erms of squared error or absolue error) relaive o he compeing model a 5% significance level, while p-values no smaller hen 0.95 mean ha he benchmark model produces a higher forecas error a he 5% level. 6

18 Table 5: Diebold Mariano es on he JPY: comparisons beween he AR model, he NAR and he random walk model and he oher models. () (2) (3) (4) (5) (6) (7) Models DM DM2 DM3 MSE MAE MSE MAE MSE MAE Panel A: monh horizon RW AR NN RAR(%) RAR(5%) RAR(0%) RNN(%) RNN(5%) RNN(0%) Panel B: 3 monh horizon RW AR NN RAR(%) RAR(5%) RAR(0%) RNN(%) RNN(5%) RNN(0%) Panel C: monh horizon RW AR NN RAR(%) RAR(5%) RAR(0%) RNN(%) RNN(5%) RNN(0%) Noes: DM, DM2 and DM3 are he p-values for Diebold-Mariano (995) es when he benchmark models are he random walk, AR and NN models, respecively. For each es we consider he MAE and MSE loss funcions. P-values no greaer he 0.05 indicae ha he benchmark model yields a lower forecas error (in erms of squared error or absolue error) relaive o he compeing model a 5% significance level, while p-value no smaller hen 0.95 mean ha he benchmark model produces a higher forecas error a he 5% level. 7

19 From Table 5, we observe ha for he JPY a he -monh horizon, he forecas accuracy of he RNN(0%) significanly oudoes ha of he RW when he chosen forecas accuracy measure is he MSE. Furhermore, he DM3 ess show ha he RNN models improve he NN model ou-of-sample forecasing for boh he MSE and MAE a 0% significan level. A longer forecas horizons, he RW does no perform significanly beer han he robus models as a group. On he oher hand, a he 3-monh horizon, he RAR(5%) and RAR(0%) produce a lower MSE han he RW a 0% significan level. The RAR models also yield a lower MSE and MAE han he AR model, a 3-, 6-monh horizons. However, according o DM2 es resuls, none of hese cases are saisically significan. The DM3 ess indicae ha for 3-monh horizon he RNN models have beer ou-of-sample forecasing performances in comparison o he NN model, while for 6-monh horizon hese models perform more poorly compared o he NN model. In summary, i appears ha a he -monh horizon, robus models do a beer job a predicing exchange raes han he AR or NN in erms of forecas accuracy as suppored by he DM es resuls. The performance of robus models in comparison o he RW is less clear. The robus models ouperform he RW for boh currencies, bu he DM ess show ha hese resuls are only significan for he JPY. The longer he forecas horizon is, he poorer he robus models perform compared o he RW, however only some of hese resuls are saisically significan. In general, he findings on he forecasabiliy of exchange raes using robus models are mixed as he forecas horizon lenghens. I appears, however, ha in some cases, he ou-of-sample performances of he robus models are significanly beer han hose of non-robus models in erms of forecas accuracy measures. In addiion, robus models produce good sign predicions and show srong marke iming abiliy a all ime horizons. 5. Conclusion There has been a growing ineres in modeling and forecasing foreign exchange rae movemens over he las wo decades. Researchers for forecasing have used various regression models for forecasing. However, exchange rae daa are characerized by he presence of ouliers, which are exreme values ha were no generaed by he underlying process responsible for he bulk of he daa. As a resul, regression models ha are esimaed by ordinary esimaion mehods such as he leas squares (LS) mehod end o provide poor ou-of-sample forecass. In order o reduce he impac of ouliers on he regression esimaors, we propose a robus regression approach, which is based on he usage of he S-esimaion mehod. The S-esimaors provide high resisance o ouliers and have he poenial o provide beer predicions. We invesigae he predicabiliy of wo exchange raes a he -, 3- and 6-monh horizon using linear and nonlinear specificaions of he condiional mean. Our approach is applied o he consrucion of robus forecasing models based on hese specificaions where he non-robus models are esimaed by using he ordinary LS esimaion mehod. We compare he robus models o non-robus models by using forecas accuracy measures, he Diebold-Mariano (995) es of equal forecas accuracy and he nonparameric es of Pesaran-Timmermann (992) for marke iming abiliy of he sign predicions. The resuls indicae ha robus models provide beer ou-of-sample performances han non-robus models in mos insances a he -monh horizon. The robus models, however, fail o sysemaically bea non-robus models a he 3-, 6-8

20 monh horizons in erms of forecas accuracy measures. Sill robus models exhibi srong marke abiliy across all ime horizons. Therefore, he use of he robus regression approach should be generalized o sudy he predicabiliy of exchange raes, even hough i is more compuaionally demanding han oher esimaion mehods ha are no robus o ouliers. Several exensions are possible for fuure research. Firs, oher high breakdown esimaion mehods, such as he LTS esimaors, can be considered for obaining robus forecass. Second, we use in he S-esimaion procedure several values for he uning parameer K in order o provide proecion agains several percenages of ouliers in he daa. Since here is a rade-off beween he breakdown poin and he efficiency of he esimaors, some procedures for he opimal selecion of K should be developed. However, such selecion procedures have o be suppored by heoreical argumens or exensive simulaion sudy. Finally, he robus regression approach may be applied o oher forecasing issues where ouliers are known o be par of he daa a hand for example predicing exchange raes wih moneary fundamenals or predicing oher financial reurns series. These opics, as well as ohers, are beyond he scope of his paper and are lef for fuure research. 9

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