A Quantile Regression Neural Network Approach to Estimating. the Conditional Density of Multiperiod Returns. James W. Taylor. Saïd Business School

Transcription

1 A Quanile Neural Nework Approach o Esimaing he Condiional Densiy of Muliperiod Reurns James W. Taylor Saïd Business School Universiy of Oxford Journal of Forecasing, 000, Vol. 9, pp Address for Correspondence: James W. Taylor Saïd Business School Universiy of Oxford 59 George Sree Oxford OX BE, UK Tel: +44 (0) Fax: +44 (0) james.aylor@sbs.ox.ac.uk

2 A Quanile Neural Nework Approach o Esimaing he Condiional Densiy of Muliperiod Reurns Absrac This paper presens a new approach o esimaing he condiional probabiliy disribuion of muliperiod financial reurns. Esimaion of he ails of he disribuion is paricularly imporan for risk managemen ools, such as value a risk models. A popular approach is o assume a Gaussian disribuion, and o use a heoreically derived variance expression which is a nonlinear funcion of he holding period, k, and he -sep-ahead volailiy forecas, σ. The new mehod avoids he need for a disribuional assumpion by applying quanile regression o he hisorical reurns from a range of differen holding periods o produce quanile models which are funcions of k and σ. A neural nework is used o esimae he poenially nonlinear quanile models. Using daily exchange raes, he approach is compared o GARCH-based quanile esimaes. The resuls sugges ha he new mehod offers a useful alernaive for esimaing he condiional densiy. Key words: quanile regression; neural neworks; muliperiod reurns; condiional densiy

3 . INTRODUCTION This paper aims o improve he esimaion of he condiional probabiliy disribuion of financial reurns. Accurae esimaion of he ails of he disribuion are of paricular imporance for risk managemen ools, such as value a risk models, which have received considerable aenion in recen years (see Duffie and Pan, 997). Value a risk calculaions aim o measure he wors expeced loss over a given ime inerval under normal marke condiions a a given confidence level (see Jorion, 997). The condiional densiy is no only needed for risk managemen. Hansen (994) noes ha he densiy is ofen imporan for opion pricing, and Baillie and Bollerslev (99) highligh he necessiy for accurae confidence inerval esimaion o accompany forecass of he condiional mean. A popular procedure for esimaing he disribuion of -period reurns is o forecas he volailiy and hen o make a Gaussian assumpion. However, reurns are no always normally disribued which has promped alernaives, such as he use of a -disribuion and nonparameric mehods. Boudoukh e al. (997) observe ha, even if he -period reurn is Gaussian, he disribuion may well be much more complicaed for he muliperiod reurn. This provides addiional moivaion for he use of a nonparameric mehod. In his paper, we propose a quanile regression approach o he esimaion of he disribuion of muliperiod reurns. This nonparameric approach uses hisorical reurns from a range of differen holding periods and produces quanile models which are funcions of he lengh, k, of he holding period and he -sep-ahead volailiy forecas, σ, as suggesed by heoreically derived variance expressions. The θh quanile of a variable y is he value, Q(θ), for which P(y<Q(θ))=θ. An approximaion of he full probabiliy disribuion can be produced from he quanile esimaes corresponding o a range of values of θ (0<θ<). The funcional form of muliperiod volailiy forecass varies grealy depending upon he choice of forecasing mehod. For example, wih moving average mehods, he k-period

4 volailiy forecas is usually calculaed as he -sep-ahead forecas, σ, inflaed by k. By conras, he GARCH(,) volailiy forecas is a much more complex nonlinear funcion of k and σ. Whils he moivaion for using k and σ as explanaory variables in he quanile regression models is apparen, he appropriae nonlinear specificaion is much less clear. In his paper, we overcome his problem by using a neural nework o perform he quanile regression. This compuaionally inensive approach o modelling enables he esimaion of poenially nonlinear models, wihou he need o specify a precise funcional form. The nex secion of he paper considers he exising approaches o esimaing he quaniles of he muliperiod reurns. In he secion ha follows, we describe he heory of quanile regression, and discuss how an arificial neural nework can be used o perform quanile regression. We hen presen our new proposal. In he nex secion, we use daily exchange rae daa o compare he quanile esimaes of our new approach wih hose of wo commonly used GARCH-based mehods. The final secion provides a summary and conclusion.. TRADITIONAL APPROACHES TO MULTIPERIOD QUANTILE ESTIMATION The radiional procedure for esimaing quaniles of muliperiod reurns consiss of wo sages. Firsly, he volailiy is esimaed for he periods under consideraion, and secondly, a probabiliy disribuion is assumed. This procedure is used by Alexander and Leigh (997) and proposed by Kroner e al. (995). In his secion, we discuss he wo sages, and highligh poenial improvemens... Disribuional Assumpion The quaniles of -sep-ahead reurns are usually consruced using a Gaussian disribuion (see Duffie and Pan, 997). This is consisen wih an assumpion of Gaussian log reurns in he finance lieraure. I is also consisen wih ARCH models, provided he parameers have been

5 esimaed using maximum likelihood based on Gaussian disurbances. However, marke reurns are frequenly found o have excess kurosis relaive o a normal disribuion (see, for example, he analysis of Hull and Whie (998)). Baillie and Bollerslev (989) sugges he use of a - disribuion for he esimaion of ARCH models in order o accommodae fa ails. If a - disribuion is used for parameer esimaion, i would be consisen o use a -disribuion o esimae he quaniles. However, log reurns do no always have a normal or -disribuion, and so a nonparameric approach o esimaing quaniles has srong appeal. A normal disribuion is also ofen used for esimaing he quaniles of muliperiod reurns. If he reurns a differen lead imes are assumed o be uncorrelaed and normally disribued, hen perhaps i seems reasonable o assume normaliy for he disribuion of he muliperiod reurns, since hey are he sum of he reurns over a holding period. However, his assumpion is inappropriae because, alhough he reurns a differen lead imes are uncorrelaed, hey are no necessarily independen. This inerdependence is apparen from he variance which would be unpredicable by moving average or ARCH mehods, if i did no depend on is pas values. Indeed, he hypohesis underlying ARCH processes is ha he variance is auoregressive. Furhermore, Baillie and Bollerslev (99) discuss how he higher order momens (such as he skewness and kurosis) of he reurns disribuion are also inerdependen for ARCH models. Recognising his, Hansen (994) presens a nonparameric auoregressive condiional densiy (ARCD) model which aims o model he full probabiliy densiy in an auoregressive framework. Hansen s work produces k-sep-ahead densiy esimaes bu his work has no been exended o muliperiod esimaion (which focuses on he sum of he reurns over a holding period). In summary, as Boudoukh e al. (997) noe, even if he -period innovaion is normally disribued, he muliperiod innovaion will have a much more complicaed disribuion. There is hus srong moivaion for using a nonparameric approach o esimaing he quaniles of muliperiod reurns. 3

6 .. Funcional Form of he Volailiy Forecass The funcional form of muliperiod volailiy forecass varies grealy depending upon he choice of forecasing mehod. Moving average mehods predic volailiy using simple or weighed moving averages of pas volailiy. One of he mos popular approaches is he exponenially weighed moving average esimaor, which is used in he RiskMerics TM Technical Documen (996). Wih moving average mehods, he usual approach for calculaing he volailiy of he k-period reurn (i.e. he reurn over a holding period of lengh k) is o assume ha he reurns are uncorrelaed wih he same variance. The forecas for he k-period variance is hen jus he -sep-ahead forecas, σ, muliplied by k. A radiional esimae of he θh quanile would hen be Q ˆ ˆ, k ( θ ) = Z σ, k = Zθ k σ + θ () where Z θ is he θh quanile of he sandard normal disribuion. Here he quanile is a simple funcion of k and σ. Auoregressive Condiional Heeroskedasiciy (ARCH) models provide esimaes of he variance of he reurn, r, a ime condiional upon I -, he informaion se of all observed reurns up o ime - (see Engle, (98)). This can be viewed as he variance of he error erm, e =r -E(r I - ). Bollerslev (986) exended he ARCH class of models o Generalised Auoregressive Condiional Heeroskedasic (GARCH) models which enables a more parsimonious represenaion in many applicaions. GARCH models express he condiional variance as a linear funcion of lagged squared error erms and also lagged condiional variance erms. For example, he -sep-ahead GARCH(,) variance forecas is given by ˆ σ + ˆ + = 0 + e βσ The s-sep-ahead forecas is given by he following recursive expression for s> ˆ σ + s = 0 + ( + β) σ + s ˆ 4

7 5 Using his, we can wrie he k-period GARCH(,) variance forecas as + + = = + = + 0 0, ) ( ˆ ˆ ˆ β β β σ β σ σ k k i i k k If we assume he condiional mean of he reurns is zero, a radiional esimae of he θh quanile is hen 0 0, ) ( ˆ ) ( + + = + β β β σ β θ θ k k k Z Q () The quanile is a complicaed nonlinear funcion of k and σ. Indeed, he same is rue for oher GARCH volailiy models. Expressions () and () sugges ha he funcional form of he volailiy, and hence he quanile, is open o debae. Indeed, as Kroner e al. (995) poin ou, he volailiy modelling lieraure indicaes ha volailiy is mean revering a a hyperbolic rae which is slower han GARCH models permi. In addiion, Diebold e al. (998) show ha scaling -sep-ahead volailiy forecass by k ½ is inappropriae and overesimaes he volailiy a long horizons. Furhermore, as we discussed earlier, he Gaussian assumpion is ofen inappropriae. There is hus srong poenial for a nonparameric approach ha allows more flexible modelling of he muliperiod quanile as a funcion of he holding period, k. In his paper, we presen a new nonparameric approach which uses quanile regression. Before describing he procedure, we firs inroduce he quanile regression heory of Koenker and Basse (978, 98).

8 3. QUANTILE REGRESSION This secion consiss of wo pars. Firsly, we presen he linear quanile regression heory of Koenker and Basse (978, 98). Secondly, we describe Whie s (99) proposal for he use of quanile regression wihin an arificial neural nework for nonlinear quanile modelling. 3.. Linear Quanile Koenker and Basse (978, 98) developed heory for he esimaion of he quaniles of a variable y which is assumed o be a linear funcion of oher variables. In order o provide some inuiion, le us firs consider he simple case of he consan model y =β 0 +e, where β 0 is a consan parameer and e is an i.i.d. random error erm. Koenker and Basse began by noing ha he θh quanile of y can be derived, from a sample of observaions, as he soluion β 0 (θ) o he following minimisaion problem: min θ y β0 + ( θ ) β 0 y β0 y < β0 y β 0 The case of he median (θ=½) is well known, bu he general resul is no. The minimisaion problem, as a means for finding he θh sample quanile, readily exends o he more general case where y is a linear funcion of explanaory variables. Consider he following raher general model of sysemaic heeroscedasiciy, y = µ ( x ) + σ ( x ) e where x is a row vecor of explanaory variables, µ (x ) may be hough of as he condiional mean of he regression process, σ (x ) as he condiional scale, and e as an error erm independen of vecor x. The θh quanile of e is defined as he value, Q e (θ), for which P(e <Q e (θ))=θ. Noe ha having µ and σ depend on he same vecor x is solely for noaional convenience. The condiional quanile funcions of y are hen 6

9 Q y ( θ x ) = µ ( x ) + σ ( x ) Q ( θ ) Consider he case where µ and σ are linear funcions of x which has as firs elemen, Q y e ( θ x ) = x β + (+ x γ ) Q ( θ ) (3) where β and γ are vecors of parameers. (Seing all he elemens of γ o zero is equivalen o assuming ha he error erm of y is i.i.d.) (3) can be rewrien as Q y e ( θ x ) = x β ( θ ) (4) where β(θ) is a vecor of parameers dependen on θ. Koenker and Basse (978) defined he θh regression quanile (0<θ<) as any soluion, β(θ), o he quanile regression minimisaion problem min β y θ y x + β ( θ ) y x β (5) xβ y < xβ Koenker and Basse (98) showed ha if y and x are seleced as dependen and independen variables respecively, hen quanile regression delivers parameers ha asympoically approach he parameers, β(θ), in (4) as he number of observaions increases. The common procedure for building an explanaory model for a variable is o look for a relaionship beween pas observaions of ha variable and pas observaions of poenial explanaory variables. This is no a feasible procedure for building a model for he quaniles of a variable because pas observaions of he quaniles will no be available, as hey are unobservable. The appeal of quanile regression is ha pas observaions of he quaniles are no required. Insead, he variable iself is regressed on explanaory variables o produce a model for he quanile. 7

10 3.. A Quanile Arificial Neural Nework Arificial neural neworks allow he esimaion of possibly nonlinear models wihou he need o specify a precise funcional form. The mos widely-used neural nework for forecasing is he single hidden layer feedforward nework (Zhang e al., 998). I consiss of a se of n inpus, which are conneced o each of m unis in a single hidden layer, which, in urn, are conneced o an oupu. In regression erminology, he inpus are explanaory variables, x i, and he oupu is he dependen variable, y. The resulan model can be wrien as f ( x, v, w) m = g v j g n j= 0 i= 0 w ji x i where g ( ) and g ( ) are acivaion funcions, which are frequenly chosen as sigmoidal and linear respecively, and w ji and v j are he weighs (parameers) o be esimaed. Whie (99) presens heoreical suppor for he use of quanile regression wihin an arificial neural nework for he esimaion of poenially nonlinear quanile models. The only oher work ha we are aware of, ha considers quanile regression neural neworks, is ha of Burgess (995) who briefly discusses he appeal of he procedure. Insead of fiing a linear quanile funcion using he expression in (5), a quanile regression neural nework model, f(x,v,w), of he θh quanile can be esimaed using he following minimisaion min v, w θ y f ( x, v, w) + ( θ ) y f ( x, v, w) + λ w ji + λ vi y f ( x, v, w) y < f ( x, v, w) j, i (6) where λ and λ are regularisaion parameers which penalise he complexiy of he nework and hus avoid overfiing (see Bishop, 997, 9.). The opimal values of λ and λ and he number, m, of unis in he hidden layer can be esablished by cross-validaion (see Donaldson and Kamsra, 996; Bishop, 997, 9.8). In he nex secion, we describe how a quanile regression neural nework can be used o esimae he probabiliy disribuion of muliperiod reurns. i 8

11 4.ESTIMATING THE MULTIPERIOD DISTRIBUTION USING QUANTILE REGRESSION Our proposal is o use quanile regression o consruc quanile funcions for he muliperiod reurns. As dependen variable, we use a series of muliperiod reurns corresponding o various holding periods, k. In view of he quanile expressions in () and (), candidaes for he explanaory variables could be simple, linear and nonlinear, funcions of k and σ. However, selecing appropriae explanaory variables is no sraighforward and so, in his paper, we use an arificial neural nework o esimae he nonlinear quanile models. I is imporan o noe ha our proposal is very differen from sandard quanile regression described in he previous secion. The sandard approach involves he esimaion of a model for he quanile of a variable in period, condiional upon informaion available up o period. Our procedure aims o esimae a condiional quanile model which describes he evoluion of he quanile over he nex k periods. 4.. Implemenaion of he Quanile Approach The mehod proceeds by collecing ogeher, as a single series, muliperiod reurns corresponding o various holding periods, k. For illusraive purposes, consider he case where we are ineresed in consrucing he reurns disribuions for holding periods of lengh, 3, 5, 7, 0, and 5 periods. Suppose we have a housand reurns available for each of hese holding periods. The single series would have he housand -period reurns firs, followed by he housand 3-period reurns, hen he housand 5-period reurns, ec. This is he reurns series. We hen define he elemens of he k series as aking a value of k when he corresponding elemen of he forecas error series is a k-period reurn. If esimaing quaniles for holding periods of lengh, 3, 5, 7, 0, and 5, he k series will consis of a housand s, followed by a housand 3 s, hen a housand 5 s, ec. The hird series o be consruced is he volailiy series. This series conains -sep-ahead volailiy forecass, σ, which have been esimaed by any mehod, such 9

12 as a GARCH model. The forecas origin of hese forecass is se a he same origin as he muliperiod reurn in he corresponding elemen of he muliperiod reurns series. If he firs - period reurn has he same origin as he firs 3-period reurn, he firs 5-period reurn, ec., hen he volailiy series will have he same forecas in he s enry as in he 00s enry, and he 00s enry, ec. We hen carry ou quanile regressions wih he muliperiod reurns series as dependen variable. Earlier work invesigaed he use of simple funcions of he k vecor and he vecor of volailiy forecass, σ, as explanaory variables (Taylor, 999a). For example, he 95h quanile, Q,k (0.95), was esimaed by using θ=0.95 in he quanile regression minimisaion in (5) wih he reurns series as dependen variable and k, kσ and k ½ σ as regressors. (The vecor k σ was consruced wih ih elemen equal o he produc of he ih elemen of he k vecor and he ih elemen of he volailiy vecor.) The resul was a model of he form Q ˆ ˆ, k ( 0.95) = a + bk + ckσ + + d k σ + where a, b, c and d are parameers esimaed by quanile regression. The choice of explanaory variables was based on an inspecion of coefficien boosrapped sandard errors and a pseudo R saisic. A reasonable approximaion o he correc quanile expression should be obained if he explanaory variables are well chosen. In his paper, we ake he view ha a more efficien approach o quanile model specificaion is o use an arificial neural nework. Neural neworks avoid he laborious, and poenially inefficien, procedure of selecing ransformed nonlinear variables for he linear regression. We use a quanile regression neural nework o fi a nonlinear quanile model as a funcion of k and σ. 0

13 4.. Addiional Feaures of he Quanile Approach The moving average and ARCH muliperiod volailiy forecass are esimaed solely from -period reurns. Therefore, he corresponding quanile forecass in () and () are also based on jus -period reurns. Our new proposal has he appeal of using muliperiod reurns from several differen holding periods in consrucing he muliperiod quaniles. If he reurn is correlaed wih oher marke reurns, hen, ideally, his should be accommodaed in a quanile esimae. This is an imporan issue for value a risk applicaions. Our proposal can be adaped o allow for correlaion by simply including an esimae of he correlaion as an exra inpu variable o he neural nework. Gibson and Boyer (998) review he various saisical and marke-based approaches o esimaing he correlaion beween reurns. Ineresingly, our approach enables quanile models wih compleely differen specificaions o be esimaed for differen quaniles. For example, whils he model for he 95h quanile migh be a complex nonlinear funcion of k and σ, he model for he 5h quanile migh be a simple linear funcion. Yar and Chafield (990) noe ha one advanage of a heoreical mehod for esimaing predicion inervals over an empirical procedure is ha he heoreical formulae give insigh ino how he forecas error variance varies wih k. A similar poin can be made for our new proposal. Wih modern compuing power, we feel ha he compuaional inensiy of he approach is no a significan consrain. From a heoreical perspecive, here may be inefficiencies due o he likely correlaion beween dependen variable observaions. I is no clear how o handle his in quanile regression, and i is clearly an area for furher research if he basic appeal of he mehod is acceped.

14 5. A COMPARISON OF EXCHANGE RATE QUANTILE ESTIMATES In his secion, we describe a sudy ha compared our new proposal wih radiional quanile esimaion mehods. Our analysis used 08 daily observaions of he exchange raes for he German deusche mark and he Japanese yen quoed agains he U.S. dollar from 4 July 988 o 5 July 996. We esimaed quaniles of he muliperiod log reurns for holding periods of lengh, 3, 5, 7, 0, and 5 days. Alhough hese periods are chosen arbirarily, for daily reurns, i seems quie reasonable o consider a range of holding periods from one day o hree weeks. For example, Duffie and Pan (997) and Jorion (997) repor ha wo weeks has been proposed by various organisaions as a sandard for value a risk calculaions. We compared he quanile esimaes wih he corresponding acual muliperiod reurns o reveal pos-sample performance. We carried ou his procedure for 000 moving windows, each consising of 04 observaions, o give 000 pos-sample quanile esimaes for each of he holding periods. We compared hree differen quanile esimaors of he s, 5h, 5h, 75h, 95h and 99h quaniles. We chose several quaniles in he ails of he disribuion as esimaion of he ail is ofen considered of greaes imporance. The quanile esimaors ha we used are based on ARCH esimaes of he volailiy. We acknowledge ha beer esimaors may exis, however, we fel ha i was sensible o employ sraighforward and commonly used esimaors in our sudy. In he nex subsecion, we describe he quanile esimaors, and in he subsecion ha follows we presen he resuls. 5.. Quanile Esimaion Mehods We fied an ARMA-GARCH model o an iniial daa se of 04 log reurns using he common pracice of maximising a Gaussian likelihood funcion. We did no find any significan ARMA erms. We concluded ha he mos suiable model was GARCH(,) wih he

15 condiional mean assumed o be a consan. This is consisen wih numerous analyses of exchange rae daa in he lieraure (e.g. Jorion, 995; Xu and Taylor, 995; Andersen and Bollerslev, 997). We produced quanile esimaes based on he GARCH variance forecass and a Gaussian disribuion. A second GARCH quanile esimaor was consruced from he GARCH variance forecass and he empirical disribuion of in-sample sandardised residuals. A sandardised -period reurn is calculaed by dividing he -period residual by he -sep-ahead volailiy forecas, σ. A sandardised muliperiod residual is he k-period residual divided by he muliperiod volailiy forecas, ˆ σ, k. We also esimaed quaniles using he quanile regression neural nework procedure described in he previous secion. The reurns vecor, used as dependen variable, consised of reurns from holding periods of lengh, 3, 5, 7, 0, and 5. We used GARCH(,) -sepahead volailiy forecass, σ, and he lengh of he holding period, k, as explanaory variables. We acknowledge ha if he GARCH model is misspecified, hen he quanile regression neural nework approach will suffer. In view of his, here is a srong argumen for using anoher mehod, such as a moving average, o produce he -sep-ahead forecass for he new approach. However, since GARCH(,) forecass are he mos obvious benchmark o use wih exchange rae daa, we fel ha he simples and leas conroversial opion was o use GARCH(,) -sepahead forecass in our mehod. Furhermore, if we were o use -sep-ahead forecass from anoher mehod in he new approach, and he muliperiod forecasing resuls were found o be noably differen o hose of he GARCH(,) benchmark, we would wonder wheher he difference was due largely o he choice of -sep-ahead forecas used. We applied separae enfold cross-validaion for each of he six quaniles o deermine he opimal number of hidden unis and values of he regularisaion parameers, λ and λ. This led o eiher one or wo hidden unis being used in he neural neworks. This is consisen wih several rules-of-humb for he appropriae number of unis; Kang (99) suggesed n/, and 3

16 Tang and Fishwick (993) proposed n, where n is he number of inpus. We did no reesimae he neural nework weighs for each of he 000 moving windows used in he sudy. Insead, we simply used he firs window of 04 daily observaions. We fel ha if he neural nework quanile model was o be re-esimaed, hen bes pracice would dicae ha he penaly funcion parameers and nework archiecure should also be re-esimaed. This is clearly no pracical in his kind of sudy. For consisency, we also did no re-esimae he GARCH parameers for each moving window. The quanile regression neural nework minimisaion in expression (6) was carried ou in Gauss for UNIX. We used he quasi-newon opimisaion algorihm of Broyden, Flecher, Goldfarb and Shanno (see Luenberger, 984, page 69). 5.. Pos-Sample Resuls As wih volailiy forecasing, he unobservable naure of quaniles implies ha he convenional measures of forecas accuracy, such as MSE, are no direcly applicable. The mos popular measure of quanile esimaor accuracy is he percenage of observaions falling below he quanile esimaor. For an unbiased esimaor of he θh quanile, his will be θ%. This crierion is employed by Alexander and Leigh (997) in a value a risk sudy, and is used exensively for evaluaing quanile esimaors and predicion inervals (e.g. Granger e al., 989; Taylor and Bunn, 999; Taylor, 999b). In his paper, we use his measure as a basis for comparison of he hree esimaors. Table I compares esimaion of he s, 5h, 5h, 75h, 95h and 99h quaniles for he 000 pos-sample German deusche mark reurns a he differen holding periods. The able shows he percenage of pos-sample reurns falling below he quanile esimaors. The aserisks indicae he enries ha are significanly differen from he ideal value a he 5% significance level. The accepance region for he hypohesis es is consruced using a Gaussian disribuion and he sandard error formula for a proporion. We have highlighed in bold he bes measures 4

17 for each quanile a each holding period. Table II compares pos-sample esimaion for he Japanese yen TABLES I & II Table I shows ha for he deusche mark daa, he GARCH volailiy esimaor wih empirical disribuion performs very well for he 5h, 5h 75h and 95h quaniles. The quanile regression mehod maches he GARCH model wih empirical disribuion for he 5h and 95h quaniles, and ouperforms i for he 99h. The GARCH model wih Gaussian disribuion appears o be he overall winner only for he s quanile. Ineresingly, all hree mehods severely underesimae he 5h quaniles. Table II shows ha for he yen daa he quanile regression mehod ouperforms he oher wo mehods for four of he six quaniles. The GARCH model wih empirical disribuion is he mos successful for he 75h quanile and he GARCH model wih Gaussian disribuion performs he bes for he 99h quanile. In order o give some indicaion of he relaive overall performance for he hree mehods a he differen holding periods, we calculaed chi-squared goodness of fi saisics (see Hull and Whie, 998). For each mehod, a each holding period, we calculaed he saisic for he oal number of pos-sample German deusche mark and Japanese yen reurns falling wihin he following seven caegories: below he s quanile esimaor, beween he s and 5h esimaors, beween he 5h and 5h, beween he 5h and 75h, beween he 75h and 95h, beween he 95h and 99h, and above he 99h. Table III shows he resuling chi-squared saisics. The able shows ha he GARCH model wih empirical disribuion performs he wors. The GARCH model wih Gaussian disribuion is beer han he quanile regression approach for four of he seven holding periods, bu overall here is lile o choose beween he wo. Unforunaely, we canno sum he chi-squared saisics across holding periods o give a single summary measure for each of he hree mehods because hese 5

18 saisics are no independen. To provide anoher indicaion of he relaive overall performance, we summed he number of imes ha each mehod ouperformed he oher wo in Tables I and II. These resuls, which are repored in Table IV, confirm our conclusion from Table III ha he quanile regression mehod is very compeiive TABLE III & IV In our iniial empirical work wih he deusche mark reurns, we used jus 436 observaions o esimae he neural nework quanile regression model. However, he resuls for -sep-ahead quanile esimaion were poor so we decided o expand he esimaion daa se o 04. The need for a large esimaion daa se is, perhaps, no surprising since boh neural neworks and empirical approaches o densiy esimaion generally require sizeable amouns of daa. 6. SUMMARY AND CONCLUSIONS We have presened a nonparameric approach o esimaing he condiional densiy of muliperiod reurns. The mehod uses hisorical reurns from a range of differen holding periods and produces quanile models which are funcions of he holding period, k, and he -sep-ahead volailiy forecas, σ, as suggesed by heoreically derived variance expressions. We avoided he need o specify appropriae explanaory variables by using an arificial neural nework o esimae he nonlinear quanile models. Using exchange rae daa, we performed comparaive analysis which gave encouraging resuls. Research invesigaing he usefulness of he mehod for oher exchange raes is currenly in progress. Alhough in his paper we used -sep-ahead volailiy forecass from a GARCH(,) model as inpu o he quanile regression mehod, oher -sep-ahead volailiy forecasing mehods could cerainly be used for exchange raes and for oher applicaions. 6

19 ACKNOWLEDGEMENTS We would like o acknowledge he helpful commens of wo anonymous referees. REFERENCES Alexander, C.O. and Leigh, C.T., On he Covariances Marices Used in Value a Risk Models, Journal of Derivaives, 4 Spring (997), Andersen, T.G. and Bollerslev, T., Answering he Criics: Yes, ARCH Models Do Provide Good Volailiy Forecass, Working Paper 603, Naional Bureau of Economic Research, 997. Baillie, R.T. and Bollerslev, T., The Message in Daily Exchange Raes: A Condiional Variance Tale, Journal of Business and Economic Saisics, 7 (989), Baillie, R.T. and Bollerslev, T., Predicion in Dynamic Models wih Time-Dependen Condiional Variances, Journal of Economerics, 5 (99), 9-4. Bishop, C.M., Neural Neworks for Paern Recogniion, Oxford, Oxford Universiy Press Bollerslev, T., Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 3 (986), Boudoukh, J., Richardson, M. and Whielaw, R.F., Invesigaion of a Class of Volailiy Esimaors, Journal of Derivaives, 4 Spring (997), Burgess, A.N., Robus Financial Modelling by Combining Neural Nework Esimaors of Mean and Median, Proceedings of Applied Decision Technologies, UNICOM Seminars, Brunel Universiy, London, UK, 995. Diebold, F.X., Hickman, A., Inoue, A. and Schuermann, T., Scale Models, Risk, January (998), Donaldson, R.G. and Kamsra, M., Forecas Combining wih Neural Neworks, Journal of Forecasing, 5 (996), Duffie, D., and Pan, J., An Overview of Value a Risk, Journal of Derivaives, 4 Spring (997), Engle, R.F., Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 50 (98), Gibson, M.S. and Boyer, B.H., Evaluaing Forecass of Correlaion Using Opion Pricing, Journal of Derivaives, 5 Winer (998), Granger, C.W.J., Whie, H. and Kamsra, M., Inerval Forecasing: An Analysis Based Upon ARCH-Quanile Esimaors, Journal of Economerics, 40, 87-96,

20 Hansen, B.E., Auoregressive Condiional Densiy Esimaion, Inernaional Economic Review, 35 (994), Hull, J., and Whie, A., Value a Risk When Daily Changes in Marke Variables Are No Normally Disribued, Journal of Derivaives, 5 Spring (998), 9-9. Jorion, P., Predicing Volailiy in he Foreign Exchange Marke, Journal of Finance, 50 (995), Jorion, P., Value a Risk, Illinois: Irwin, 997. Kang, S., An Invesigaion of he Use of Feedforward Neural Neworks for Forecasing, Unpublished Docoral Thesis, Ken Sae Universiy, 99. Koenker, R.W. and Basse, G.W., Quaniles, Economerica, 46 (978), Koenker, R.W. and Basse, G.W., Robus Tess for Heeroscedasiciy Based on Quaniles, Economerica, 50 (98), Kroner, K.F., Kneafsey, K.P. and Claessens, S., Forecasing Volailiy in Commodiy Markes, Journal of Forecasing, 4 (995), Luenberger, D.G., Linear and Nonlinear Programming, Reading, MA: Addison-Wesley, 984. RiskMerics TM Technical Documen, fourh ediion. J.P. Morgan/Reuers, 996. Tang, Z. and P.A. Fishwick, Feedforward Neural Nes as Models for Time Series Forecasing, ORSA Journal on Compuing, 5 (993), Taylor, J.W., A Quanile Approach o Esimaing he Disribuion of Muliperiod Reurns, Journal of Derivaives, 7 Fall (999a), Taylor, J.W., Evaluaing Volailiy and Inerval Forecass, Journal of Forecasing, 8 (999b), -8. Taylor, J.W. and Bunn, D.W., A Quanile Approach o Generaing Predicion Inervals, Managemen Science, 45 (999). Whie, H., Nonparameric Esimaion of Condiional Quaniles Using Neural Neworks, in Arificial Neural Neworks: Approximaion and Learning Theory, A.R. Gallan e al., Cambridge and Oxford, Blackwell (99), Xu, X., and Taylor, S.J., Condiional Volailiy and he Informaional Efficiency of he PHLX Currency Opions Marke, Journal of Banking and Finance, 9 (995), Yar, M., and Chafield, C., Predicion Inervals for he Hol-Winers Forecasing Procedure, Inernaional Journal of Forecasing, 6 (990), Zhang, G., Pauwo, B.E. and Hu, M.Y., Forecasing wih Arificial Neural Neworks: The Sae of he Ar, Inernaional Journal of Forecasing, 4 (998),

21 Table I. Percenage of pos-sample German deusche mark reurns falling below quanile esimaes Holding Period s 5h 5h 75h 95h 99h GARCH(,) GARCH(,) Quanile GARCH(,) GARCH(,) Quanile GARCH(,) GARCH(,) Quanile GARCH(,) GARCH(,) Quanile GARCH(,) GARCH(,) Quanile GARCH(,) GARCH(,) Quanile * * 3.0 * 3. *.3 *.7 * *.3 *.9 *. * *.3 * * * 9. * 7.9 * 8.4 * 6.7 * 6.8 * 7.7 *.3 *.0 * 9.9 * 0.0 * 7.8 * 7.6 * 8.4 * 9. *.0 * 0.5 * 9. * 7. * 7.4 * 7.0 * * * * significan a 5% level bold indicaes bes performing mehod for each quanile a each holding period 9

22 Table II. Percenage of pos-sample Japanese yen reurns falling below quanile esimaes Holding Period s 5h 5h 75h 95h 99h GARCH(,) GARCH(,) Quanile GARCH(,) GARCH(,) Quanile GARCH(,) GARCH(,) Quanile GARCH(,) GARCH(,) Quanile GARCH(,) GARCH(,) Quanile GARCH(,) GARCH(,) Quanile.0 *.5 *.3 *.4 *.8 *. *.9 *.0.5 *.7 *.8 * 3.3 * 3. * 4. * * *.3 * * 6.5 * 6.8 * 7.3 * * 7. * 7.6 * 7.9 * 8.7 * 3. * * 6.8 * *.6 * * * 78. * 79. * 79.4 * * 77.7 * * 77.7 * * * 97. ** * 93.5 * 9.9 * 9.6 * 93. * 93.5 * * 98. * * 97.4 * 96.6 * 97.0 * 97. * 97.5 * * 97.5 * 98.0 * 98. * 98. * * significan a 5% level bold indicaes bes performing mehod for each quanile a each holding period 0

23 Table III. Chi-squared goodness of fi saisics summarising he overall performance of he hree esimaors for he pos-sample German deusche mark and Japanese yen reurns Holding Period GARCH(,) GARCH(,) Quanile 85.9 * 8.6 * 80.4 * 74.9 * 97.0 * 59.6 * 80. * * 00.5 * 74. * 95.4 * 57.3 * 73.7 * 47.3 * 3.9 * 57.6 * 0.3 * 7.3 * 64.8 * 88.5 * * significan a 5% level bold indicaes bes performing mehod for each holding period Table IV. Number of imes an esimaor ouperformed he ohers for he pos-sample German deusche mark and Japanese yen reurns Deusche Mark (Table I summary) Japanese Yen (Table II summary) Toal GARCH(,) GARCH(,) Quanile bold indicaes bes performing mehod