Forecasting stock indices: a comparison of classification and level estimation models

Transcription

1 Inernaional Journal of Forecasing 16 (2000) locae/ ijforecas Forecasing sock indices: a comparison of classificaion and level esimaion models Mark T. Leung *, Hazem Daouk, An-Sing Chen a, b c a Deparmen of Operaions and Decision Technologies, Kelley School of Business, Indiana Universiy, Bloomingon, IN 47405, USA b Deparmen of Finance, Kelley School of Business, Indiana Universiy, Bloomingon, IN 47405, USA c Deparmen of Finance, Naional Chung Cheng Universiy, Ming-Hsiung, Chia-Yi, 621 Taiwan Absrac Despie abundan research which focuses on esimaing he level of reurn on sock marke index, here is a lack of sudies examining he predicabiliy of he direcion/ sign of sock index movemen. Given he noion ha a predicion wih lile forecas error does no necessarily ranslae ino capial gain, we evaluae he efficacy of several mulivariae classificaion echniques relaive o a group of level esimaion approaches. Specifically, we conduc ime series comparisons beween he wo ypes of models on he basis of forecas performance and invesmen reurn. The esed classificaion models, which predic direcion based on probabiliy, include linear discriminan analysis, logi, probi, and probabilisic neural nework. On he oher hand, he level esimaion counerpars, which forecas he level, are exponenial smoohing, mulivariae ransfer funcion, vecor auoregression wih Kalman filer, and mulilayered feedforward neural nework. Our comparaive sudy also measures he relaive srengh of hese models wih respec o he rading profi generaed by heir forecass. To faciliae more effecive rading, we develop a se of hreshold rading rules driven by he probabiliies esimaed by he classificaion models. Empirical experimenaion suggess ha he classificaion models ouperform he level esimaion models in erms of predicing he direcion of he sock marke movemen and maximizing reurns from invesmen rading. Furher, invesmen reurns are enhanced by he adopion of he hreshold rading rules Elsevier Science B.V. All righs reserved. Keywords: Forecasing; Mulivariae classificaion; Sock index; Trading sraegy 1. Inroducion markes around he world. The increasing diversiy of financial index relaed insrumens, Trading in sock marke indices has gained along wih he economic growh enjoyed in he unprecedened populariy in major financial las few years, has broadened he dimension of global invesmen opporuniy o boh individual and insiuional invesors. There are wo basic *Corresponding auhor. Tel.: ; fax: reasons for he success of hese index rading addresses: mleung@indiana.edu (M.T. Leung), vehicles. Firs, hey provide an effecive means hdaouk@indiana.edu (H. Daouk), for invesors o hedge agains poenial marke finasc@ccunix.ccu.edu.w (A.-S. Chen) risks. Second, hey creae new profi making / 00/ $ see fron maer 2000 Elsevier Science B.V. All righs reserved. PII: S (99)

2 174 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) opporuniies for marke speculaors and arbi- performance of various mulivariae classificarageurs. Therefore, being able o accuraely ion echniques relaive o some economeric forecas sock marke index has profound impli- and arificial inelligence forecasing echniques; caions and significance o researchers and and (3) o develop effecive rading sraegies praciioners alike. guided by he direcional forecass and o es Alhough here exiss a vas number of ari- he relaive performance of hese invesmen cles addressing he predicabiliy of sock mar- schemes. ke reurn as well as he pricing of sock index The remainder of his paper is organized as financial insrumens (e.g. S&P 500 fuures), follows: A lieraure review and he background mos of he proposed models rely on accurae of his sudy are summarized in he nex secion. forecasing of he level (i.e. value) of he In Secion 3, we provide he concepual foundaunderlying sock index or is reurn. In mos ion of he esed mulivariae classificaion cases, he degree of accuracy and he accep- models: parameric linear discriminan analysis, abiliy of cerain forecass are measured by he logi, probi, and he nonparameric probabilisic esimaes deviaions from he observed values. neural nework. In addiion, we review several Depending on he rading sraegies adoped level-based forecasing models (adapive exby invesors, forecasing mehods based on ponenial smoohing, Bayesian vecor auoregminimizing forecas error may no be adequae ression, mulivariae ransfer funcion, and o mee heir objecives. In oher words, rading mulilayered feedforward neural nework) which driven by a cerain forecas wih a small fore- will be esed agains heir classificaion councas error may no be as profiable as rading erpars. Then, we discuss he design of he guided by an accurae predicion of he direcion experimen in Secion 4. This secion also of movemen (or sign of reurn.) Therefore, oulines he proposed index rading sraegies predicing he direcion of change of he sock and how o apply he direcional forecass and marke index and is reurn is also significan in he poserior probabiliies supplied by he classihe developmen of effecive marke rading ficaion models. sraegies. In recen years, here has been a growing number of sudies looking a he direcion or rend of movemens of various kinds of financial 2. Background insrumens (such as Maberly, 1986; Wu & 2.1. Predicabiliy of sock marke reurns Zhang, 1997; O Connor, Remus & Griggs, 1997). However, none of hese sudies provide a Sock prices do no follow random walks is comparaive evaluaion of differen classifica- he ile of a heavily cied paper by Lo and ion echniques regarding heir abiliy o predic MacKinlay (1988). These auhors claim ha he sign of he index reurn. Given his noion, considerable evidence exiss and show ha we examine various forecasing models based sock reurns are o some exen predicable. on mulivariae classificaion echniques and Mos of he research is conduced using daa compare hem wih a number of parameric and from well-esablished sock markes such as he nonparameric models which forecas he level US, Wesern Europe, and Japan. of he reurn. Specifically, he major conribu- For he US, several sudies have examined ions of his sudy are: (1) o demonsrae and he cross-secional relaionship beween sock verify he predicabiliy of sock index direcion reurns and fundamenal variables. Variables using classificaion models; (2) o compare he such as earnings yield, cash flow yield, book o

3 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) marke raio, and size have been found o have Ross (1986) find ha changes in aggregae some power in predicing sock reurns in hese producion, inflaion, he shor-erm ineres cross-secional sudies. Basu (1977), Fama and raes, he slope of erm srucure as measured by French (1992), and Lakonishok, Shleifer and he reurn difference beween long-erm and Vishny (1994) are examples of such cross- shor-erm governmen bonds, and he risk secional sudies. These sudies in general find premium as measured by he reurn difference posiive relaionships beween sock reurns and beween low grade bonds and high grade bonds earnings yield, and beween cash flow yield and are oher macroeconomic facors ha have some book-o-marke raio, and a negaive relaion- power o predic sock reurns. Also, predicship beween sock reurns and size. abiliy is no limied o sock reurns. For For he Japanese sock marke, Jaffe and example, Campbell and Mankiw (1987) and Weserfield (1985) and Kao, Ziemba and Cochrane (1988) find predicabiliy in he GNP, Schwarz (1990) find some evidence of predic- and Huizinga (1987) finds predicabiliy in abiliy in he behavior of daily and inraday exchange raes. paerns in index reurns. Kao and Schallheim Furhermore, Chen e al. (1986) sugges ha (1985) documen size and seasonal anomalies. expeced reurns are a funcion of business Chan, Hamao and Lakonishok (1991) relae condiions in ha he expeced sock marke cross-secional differences in reurns on Japan- premium is negaively relaed o he recen ese socks o he underlying behavior of earn- growh of economic aciviy proxying for he ings yield, size, book o marke raio, and cash healh of he curren economy and posiively flow yield. relaed o he expeced fuure growh of econ- Fundamenal variables are no he only ype omic aciviy and is condiional variance. Chen of cross-secional variables ha conain infor- (1991) sudies he relaion beween changes in maion for predicabiliy. DeBond and Thaler financial invesmen opporuniies and changes (1985, 1987), Chopra, Lakonishok and Rier in he economy. He provides addiional evi- (1992) documen ha a sock s ranking in erms dence ha variables such as he defaul spread, of is performance relaive o he marke can he erm spread, he 1-monh T-bill rae, he conain predicabiliy. Exreme losers have been lagged indusrial producion growh rae, and shown o ouperform he marke over sub- he dividend price raio are imporan deersequen years. minans of fuure sock marke reurns. He In ime-series analysis, Fama and French inerpres he abiliy of hese variables o fore- (1993) idenify hree common risk facors, he cas fuure sock marke reurns in erms of heir overall marke facor, facors relaed o firm size correlaions wih changes in he macroeconomic and book-o-marke equiy which seem o ex- environmen. plain average reurns on socks and bonds. Fama The empirical evidence reviewed above is an and Schwer (1977), Campbell (1987), and illusraion of large body of heoreical work Fama and French (1988a,b) find macroeconom- ha deals wih Arbirage Pricing Models. Ross ic variables such as shor-erm ineres raes, (1976) inroduces he Arbirage Pricing Theory expeced inflaion, dividend yields, yield (APT) and derives a model where sock reurns spreads beween long and shor-erm govern- are a linear funcion of a se of sae variables men bonds, yield spreads beween low grade (facors). The APT does no imply predicabiliy bonds and high grade bonds, lagged sock price- per se, since he facors are assumed o be earnings raios, and lagged reurns have some observed a he same ime reurns are. A sream power o predic sock reurns. Chen, Roll and of lieraure called Condiional Asse Pricing

4 176 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) Models has worked on deriving models ha explores he relaionship beween he direcion uses some available informaion se o explain of inerday and inraday price change. O Connor asses reurns. Ferson and Harvey (1991) show e al. (1997) conduc a laboraory-based experiha predicabiliy in socks reurns are no men and conclude ha individuals show differnecessarily due o marke inefficiency or over- en endencies and behaviors for upward and reacion from irraional invesors, bu raher o downward series. This furher demonsraes he predicabiliy in some aggregae variables ha usefulness of forecasing he direcion of change are par of he informaion se. More specifical- in he price level, ha is, he imporance of ly, hey argue ha sock reurns are predicable being able o classify he fuure reurn as a gain because macro variables (such as ineres raes or a loss. The findings in hese sudies are or consumpion growh) which deermine o a reasonable because an accurae poin esimaion, cerain exen sock reurns, are hemselves as judged by is deviaion from he acual predicable. observaion, may no be a good predicor of he I is a well-esablished pracice in he recen direcion of change in he insrumen s price empirical finance lieraure o accoun for he level. This is a common case for he raders in predicabiliy of sock reurns given he inves- ha he prediced direcion will immediaely ors informaion se by using macroeconomic affec heir decisions on buying or selling he variables ha are public informaion. For exam- insrumen. Finally, in heir sudy on he All ple, an even sudy of insider rading by Ferson Ordinaries Index fuures raded a he Ausralian and Schad (1996) shows ha he omission of Associaed Sock Exchanges, Hodgson and variables like lagged sock reurns and previous Nicholls (1991) sugges conducing an evaluaineres raes could lead o misleading resuls. ion of he economic significance of he direc- Sock reurns predicabiliy given aggregae ion of price changes in fuure research. variables in he invesors informaion se is a well-acceped fac. The quesion ha remains is 2.3. Model inpus how o use he informaion se in an opimal way for forecasing and rading. Sudies relaed o he predicabiliy of reurn and is deerminan facors are ample and one 2.2. Forecasing he direcion of index reurn can easily find such sudies in he lieraure. Hence, for he sake of breviy, we will omi he Mos rading pracices adoped by financial discussion of economic raionale in his paper. analyss rely on accurae predicion of he price Table 1 displays he se of poenial macrolevels of financial insrumens. However, some economic inpu variables which are used by he recen sudies have suggesed ha rading srae- forecasing models analyzed in his paper. This gies guided by forecass on he direcion of he sudy confines he poenial inpu variables o change in price level are more effecive and ineres raes, consumer price index, indusrial may generae higher profis. Wu and Zhang producion, and lagged reurns. These are he (1997) invesigae he predicabiliy of he mos easily available inpu variables ha are direcion of change in he fuure spo exchange observable o a forecaser. Though oher macrorae. In anoher sudy, Aggarwal and Demaskey economic variables can be used as inpus, he (1997) find ha he performance of cross-hedg- general consensus in he lieraure is ha he ing improves significanly if he direcion of majoriy of useful informaion for forecasing is changes in exchange raes can be prediced. subsumed by he ineres raes and lagged Based on he S&P 500 fuures, Maberly (1986) reurns.

5 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) Table 1 Lis of poenial macroeconomic inpu variables and forecased oupu variables Inpu variables ST Shor Term Ineres Rae Firs difference of 3-monh T-bill rae for he US, and firs difference of call money rae for he UK and Japan. LT Long Term Ineres Rae Firs difference of long erm governmen bond rae for he US, firs difference of 20-year governmen bond rae for he UK, and firs difference of long erm governmen bond rae for Japan. R Lagged Index Reurns Lagged erms of he coninuously compounded excess reurns of he broad marke index for he differen counries, respecively. CPI Consumer Price Level Firs difference of consumer price index for he hree counries, respecively. IP Indusrial Producion Level Firs difference of indusrial producion for he hree counries, respecively. Oupu variables R Reurns on Index Coninuously compounded excess reurns of he sock marke index (S&P 500, FTSE 100, and Nikkei 225) esimaed by level esimaion models. C Probabiliy Given he Direcion of Reurn Probabiliies esimaed by he classificaion models ha he excess reurn will be posiive in he nex period. 3. Forecasing index reurns 3.1. Daa The financial and macroeconomic daa se used in his sudy is obained from he TSM daa base and he Ciibase compiled by DSC Daa Services, Inc. and Ciicorp Economic Daabase, respecively. The enire daa se cov- ers he period from January 1967 o December 1995, a oal of 348 monhs of observaions. The daa se is divided ino wo periods: he firs period runs from January 1967 o December 1990 (288 monhs of observaions) while he second period is from January 1991 o December 1995 (60 monhs of observaions.) The firs period, which is assigned o in-sample esimaion, is used o deermine he specificaions of he models and parameers for he forecasing echniques. I also serves he pur- pose of validaing he esimaed models. The second period is reserved for ou-of-sample evaluaion and comparison of performances beween various forecasing models. To es he robusness of he classificaion models, hree globally raded broad marke indices S&P 500 for he US, FTSE 100 for he UK, and Nikkei 225 for Japan, are ex-

6 178 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) amined. The forecased variables are he con- sudy. Alhough hese models are based on inuously compounded 1 monh excess reurns differen saisical echniques, hey share a of hese indices. As i is shown in Eq. (1), he common rai he abiliy o generae he excess reurn on an index is defined as he probabiliy of group membership. In oher coninuously compounded reurn on he price words, hese models are able o esimae he index minus he riskfree ineres rae: probabiliy of an upward (or downward) movemen in he sock index and hus provide a P R 5 ln ]] 2 r (1) recommendaion for rading. S D P where P is he price of he sock index raded a Linear discriminan analysis period and r is he US riskfree (1-monh Discriminan analysis is a mulivariae T-bill) ineres rae in period. Dividends are saisical echnique ha invesigaes he differignored for his sudy. The reason o forecas he ences beween wo or more groups of observa- excess reurns (raher han he index levels or ions wih respec o a se of independen (inpu) he oal reurns) is ha hey provide a measure variables. These independen variables, called of how well our models perform relaive o he discriminaor variables, are used o disinguish minimum reurns gained from deposiing he he characerisics among differen groups. In money in a riskfree accoun. In addiion, our he discriminan analysis, hese discriminaor sudy assumes ha we are in he posiion of an variables are combined o form a se of mahe- American invesor. This leads o our adopion of maical equaions, known as he classificaion US riskfree T bill rae, as opposed o he ineres funcions. There is a classificaion funcion for raes of oher riskfree insrumens issued by each group of observaions. The classificaion foreign governmens, in deriving he excess funcion which yields he highes Z score indi- reurns. This is because an American invesor caes he group membership of he inpu vecor always has a choice of no invesing bu simply o be classified. A he same ime, a probabiliy puing he money ino a US accoun which based on he Z scores can also be calculaed o pays he shor erm riskfree ineres rae. deermine he mos likely group membership. The independen variables for predicing he In his sudy, we conduc Fisher s discrimin- 1 index reurns are all observable on or before he an analysis o classify he direcion of excess las day of he monh preceding he monh o be reurn. Ineresed readers should refer o Hair, forecased. For insance, for he predicion of Anderson, Taham and Black (1995) or oher he index reurn for March 1990, all indepen- saisical exs for a more deailed descripion. den variables mus be observable on or before Afer he process of model selecion and valida- he las day of February Consrucing he ion, he Fisher discriminan analysis yields he daa in his manner ensures ha he esimaion following classificaion funcions based on he of ou-of-sample forecass will be similar o he in-sample daa from January 1967 o December pracice in he real world. Tha is, only observ (he noaions follow he ones explained in able, bu no fuure, daa are used as inpus o Table 1). The forecas is classified as an upward he forecasing models. movemen (a posiive reurn) if Z pos.z neg, ha is, he Z score of he classificaion funcion for 3.2. Forecasing by classificaion models In his secion, we briefly summarize he classificaion models used in his comparaive 1 Linear discriminan analysis in our experimen is performed by he SPSS saisical package.

7 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) posiive reurn is larger han he one for nega- P(Yi5 1uX i) 5 F(Xib ) (8) ive reurn: where he dependen variable Y akes he value US S&P 500 of eiher 0 or 1. The quesion hinges on he SP US Zpos R ST (2) value of he parameer P, he probabiliy ha Y equals one. X is he se of explanaory variables SP US Z R ST (3) and F(?) is a nonlinear funcion of he conneg diional mean. The only difference beween he UK FTSE 100 logi and probi models is he funcional form of FTSE FTSE Z R R F(?). F(?) is he cumulaive densiy funcion pos 22 (CDF) of he logisic disribuion for he logi FTSE FTSE R R24 model, whereas i is he CDF of he normal UK UK LT CPI disribuion if a probi model is used. Readers should check ou Greene (1993) for a more (4) horough descripion of he mehod. Binary choice models are paricularly suied FTSE FTSE Zneg R R22 for our sudy since we wan o predic he FTSE FTSE R R direcion in he movemen of a sock index. The direcion of a movemen is binary in naure (up UK UK LT CPI or down). Therefore, he logi and probi models (5) will permi us o calculae he probabiliy of an up or down move given all he explanaory 2 Japan Nikkei 225 variables in our informaion se. Based on he in-sample daa, he chosen logi model spe- Zpos R R R23 cificaions are as follows: R (6) US S&P 500 SP SP Ĉ5 F( R R22 neg 22 SP SP SP R R R R R (7) US US US Z R R ST ST ST 23 These esimaed funcions are hen used o US US deermine he sign of reurn for each monhly ST ST 25) (9) ou-of-sample period. In our experimen, he discriminan scores Z are also used o compue UK FTSE 100 ˆ ˆ FTSE FTSE C, he poserior probabiliy of group classifica- C5 F( R R22 ion for he sign of reurn in he nex period. FTSE FTSE UK R R ST The mahemaical operaion can be found in UK UK UK Hair e al. (1995) ST ST ST 24 UK UK ST ST Binary choice models: logi and probi UK Binary choice models are appropriae o use ST 27) (10) when rying o model dependen variables ha 2 can ake on only binary values (e.g. 0 or 1). The Logi and probi esimaions are performed by RATS general form of he model is: compuer package.

8 180 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) Japan Nikkei 225 neural nework models are also used o predic he sign of index reurn. Our classificaion Ĉ5 F( R R22 4 nework models are based on he Probabilisic JAP JAP R LT ST Neural Nework (PNN) proposed by Spech JAP (1988, 1990). A few sudies such as Yang LT 23 ) (11) (1999) and Kim (1998) which employ PNN for financial forecasing have shown some promis- The esimaed probi models are: ing resuls. PNN is concepually buil on he Bayesian mehod of classificaion. Theoreical- US S&P 500 ly, given enough informaion, he Bayesian SP SP Ĉ5 G( R R22 mehod can classify a new sample wih he SP SP SP maximum probabiliy of success (Wasserman, R R R , p. 37). A complee descripion of PNN US US US ST ST ST 23 and is mahemaical background can also be US US found in he same ex ST ST 25) (12) As explained in Secion 3.1, our nework raining and validaion scheme involves dividing he firs period daa ino wo segmens. The UK FTSE 100 FTSE FTSE firs segmen is used o rain he nework Ĉ 5 G( R R22 whereas he second segmen is used o de- FTSE FTSE UK R R ST ermine and validae he nework archiecure UK UK and model specificaion. For he sake of breviy, ST ST 23 deails of his procedure are no repored here. UK UK ST ST Enhusiasic readers can obain he informaion from he auhors or refer o Chen and Leung UK UK ST ST 27 ) (13) (1998). The seleced models for ou-of-sample forecasing/ evaluaion have he following funcional Japan Nikkei 225 forms: ˆ US S&P SP ˆ SP US R s LT ST 22 C 5 G( R R JAP JAP C 5 FR, ST (15) JAP LT 23) (14) UK FTSE 100 FTSE ˆ FTSE FTSE FTSE FTSE UK The esimaed probabiliy Cˆ ha we would C 5 G sr, R 22, R 23, R 24, LT d realize a posiive reurn in he nex period is a (16) logisic CDF F(?) of explanaory variables if a logi model is used in forecasing. Likewise, if a probi model is used insead, Cˆ is he probabili- 3 y derived from a normal CDF. Many neural nework applicaions are relaed o financial decision making. Hawley, Johnson and Raina (1990) and Refenes (1995) provide an overview of neural nework Probabilisic neural neworks models used in he fields of finance and invesmen. 4 In addiion o he classificaion models de- Readers who are ineresed in he program codes should scribed in he las wo secions, nonparameric refer o Masers (1995). d 3

9 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) Japan Nikkei 225 firs 228 monhs wihin he in-sample period (from January 1967 o December 1985). The Ĉ 5 H sr d (17) esimaed parameers are hen validaed using where Cˆ is he poserior probabiliy of realizing he remaining 60 monhs (from January 1986 o a posiive reurn in he nex period, and F(?), December 1990) wihin he in-sample period. G(?), and H(?) represen he funcional relaionprovides Since we assume ha hisorical performance ships beween he dependen and independen meaningful informaion o predic he variables esimaed by he PNN. fuure, hese values for b are used in he forecasing in he reserved ou-of-sample period Forecasing by level esimaion models Vecor auoregression wih Kalman Adapive exponenial smoohing filer updaing Makridakis, Wheelwrigh and McGee (1983) The vecor auoregression (VAR) has proven and Maber (1978) described an exension o o be a successful echnique for forecasing radiional exponenial smoohing model, gener- sysems of inerrelaed ime-series variables in ally known as adapive exponenial smoohing. he macroeconomics lieraure. Noaionally, a This approach coninuously evaluaes he per- VAR model wih a lag lengh of p can be formance in he previous period and updaes he represened as: smoohing coefficien. The form of he adapive p 5 exponenial smoohing model is similar o ha Z5O f(s)z2s1 (23) of he simple single exponenial smoohing s51 model: E( 9 ) 5S (24) Rˆ 5 a X 1 (1 2 a )R ˆ 11 (18) where Z is an (n 3 1) vecor of variables where Rˆ is he forecas for period and X is he measured a ime period, f(s) isan (n 3 n) acual observaion made in period, and marix of he coefficiens, p is he lag lengh of E he variables, is an (n 3 1) vecor of random ] a 11 5 UU (19) M disurbances, and S is he variance covariance marix. Deails of he VAR echnique can be E5 be1 (1 2 b )E (20) found in Hamilon (1994). 6 The specificaion of he VAR is deermined M5 bueu 1 (1 2 b )M (21) by accessing he performance of alernaive VAR specificaions in forecasing he las 60 e 5 X 2 R ˆ (22) monhs (from January 1986 o December 1990) of he in-sample period. Based on our exa and b are parameers beween 0 and 1 and u?u perimenal resuls, he following specificaions denoes absolue values. are seleced: Based on he resuls of experimen, he values of b are se o be 0.75, 0.90, and 0.95 for he US S&P 500 SP (25) S&P 500, FTSE, and Nikkei predicion models, Z 5hR j, p 5 3 respecively. These values are deermined by UK FTSE 100 accessing he performance of he models in he FTSE UK Z 5hR, ST j, p 5 2 (26) 5 6 We implemen he adapive exponenial smoohing VAR esimaions are performed by RATS compuer models using Excel spreadshee. package.

10 182 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) Japan Nikkei 225 Japan Nikkei 225 (27) 8 Z 5hR j, p 5 2 MA(3) wih long erm ineres raes (LT) R Kalman filer updaing is hen used o generae ( ) 60 monhly forecass for he reserved ou-of- 1]]]]]]]]]] LT sample period (from January 1991 o December ( L L L ) 1995) using he VAR specificaion chosen. 2 3 Similar o he adapive exponenial smoohing Mulivariae ransfer funcion models, he ransfer funcion models are esi- A mulivariae ransfer funcion model is maed using he firs 228 observaions wihin essenially an ARIMA model wih added ex- in-sample period (from January 1967 o Deogenous variables. I is believed ha he addi- cember 1985). Afer hese esimaed models are ion of exogenous variables can improve he validaed by he las 60 monhs of observaions accuracy of our forecass if hese variables help in he in-sample period, hey are used o generexplaining he sock excess reurns. Ineresed ae ou-of-sample forecass from January 1991 readers should refer o Makridakis e al. (1983) o December In all cases, Indusrial for a deailed explanaion of his echnique. In Producion (IP) and Consumer Price Index our experimen, we follow Box and Jenkins (CPI) did no add predicive value o he model. hree-sage mehod (see Appendix A) aimed a selecing an appropriae model for he purpose Mulilayered feedforward neural of esimaing and forecasing a ime series. The nework 7 seleced models are wrien as follows: Alhough he essenial operaions of neural neworks are he same (i.e. he neworks accep US S&P 500 a se of inpus and, hrough heir processing ARMA(1, 1) wih shor erm ineres raes (ST) unis, produce a corresponding se of oupus), neural neworks can appear in many archiecur- R520.48R al forms. To provide a comparison wih he (20.017) PNN classifier, we es he performance of he 1]]]]]]]]]] 2 3 ST ( L L L ) mulilayered feedforward neural nework (MLFN), commonly known as backprop ne- (28) work. Unlike he PNN which suggess a group classificaion for a given se of inpus, MLFN UK FTSE 100 provides a poin esimae or forecas as he MA(1) wih long erm ineres raes (LT) oupu. Readers can refer o Hassoun (1995) and Zhang, Pauwo and Hu (1998) for an explana- R ( L) 8 The denominaor expression for he lagged LT erm has a 1]]]]]]]]]] 2 3 LT roo inside he uni circle (0.996). This does no violae he ( L L L ) sabiliy condiions for he ransfer funcion. The reason is (29) ha here are no AR erms for R. I suffices for R and LT o be saionary for he model o be sable. Resuls from saionariy ess show ha R and LT are indeed saionary (a 7 Esimaion of mulivariae ransfer funcion models is more complee analysis is available upon reques from he performed by RATS compuer package. auhors). (30)

11 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) ion of neural nework model. Wasserman funcions deduced by he neural neworks. The (1993) also provides a descripion and com- rained neworks are hen applied o he foreparison of he archiecures and mahemaical casing of he index reurns in he ou-of-sample foundaion of PNN and MLFN. period (January 1991 hrough December 1995). Based on he resuls from our sudy, we selec 9 he nework archiecure which leads o consisen and reasonable performance in he validaion 4. Simulaion sudy and rading sraegies sample period from January 1986 o 10 December The seleced model specificaions 4.1. Empirical evaluaion are as follows (he noaions follow hose Each of he forecasing models described in described in Table 1): he las secion is esimaed and validaed by he US S&P 500 in-sample daa. This model esimaion and SP SP SP SP SP US US ˆR 5 F(R, R, R, R, ST, LT ) selecion process is hen followed by an empiri cal evaluaion which is based on he ou-of- (31) sample daa covering 60 monhly observaions from January 1991 o December A his UK FTSE 100 sage, he relaive performance of he models is FTSE ˆR measured by wo primary crieria: FTSE FTSE FTSE FTSE UK UK 5 G(R, R 22, R 23, R 24, ST, LT ) number of correc forecass (his) of he sign (32) of index reurn; and excess reurns obained from index rading. Japan Nikkei 225 JAP A comparison of he performance beween he ˆR 5 H(R, R 22, R 23, R 24, LT ) (33) groups of classificaion and level esimaion where F(?), G(?), and H(?) are he arbirary models can hus be carried ou. The excess reurns are derived from rading sraegies 9 An imperaive issue in designing a nework is deermining which are driven by he forecass made by he he appropriae number of unis in he hidden layer. classificaion and level esimaion models. The Unforunaely, here is no consisen answer o his queslogic of hese rading sraegies and heir resuls ion. Alhough a large hidden layer may provide greaer flexibiliy in funcional mapping, i may also lead o he will be discussed in he nex wo secions. problem of overfiing and hamper he predicive srengh The number of correc forecass of he sign of of he nework model. Neverheless, Salchenberger, Cinar reurn for each forecasing model is repored in and Lash (1992) offered a simple guideline of which he Table 2. The corresponding hi raios are also number of hidden unis should be abou 75% of he given. The average hi raio for he group of all number of inpu unis. This poin was echoed by Jain and four classificaion models is 61.67% whereas Nag (1995). Based on he resuls from our sudy, we selec he nework archiecure which leads o consisen and ha for he group of all four level esimaion reasonable performance in he validaion sample period models is 56.11%. The pooled sandard error is from January 1986 o December The resuling and he z value is 4.28, suggesing he consruc of he nework conains a hidden layer wih en group of classificaion models perform signifihidden unis for boh he US and UK models. For he canly beer han he group of level esimaion Japan model, he nework conains one hidden layer wih six hidden unis. models. In addiion, we also es he null 10 MLFN esimaions are performed by ThinkPro compuer hypohesis of no predicive effeciveness, ha package. is, wheher he hi raio of a group of models is

12 184 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) Table 2 Comparison of he predicive srengh of classificaion and level esimaion models a US S&P 500 UK FTSE 100 Japan Nikkei 225 Number Raio Number Raio Number Raio Classificaion Discriminan analysis * * models Logi * * * Probi * * * Probabilisic neural nework * * * Level esimaion Adapive exponenial smoohing * models Vecor auoregression wih Kalman filer Mulivariae ransfer funcion Mulilayered feedforward neural nework * * a The able repors he number of imes a forecasing model correcly predics he direcion of he index reurn over he 60 ou-of-sample forecas periods from January 1991 hrough December A raio marked wih an aserisk (*) indicaes a 95% significance level based on a one-sided es of H 0: p50.50 agains H a: p The bes classificaion and level esimaion models for each index are in bold. significanly differen from he benchmark of forecasing comparison. This gives a oal of The saisical ess indicae ha he hi monhly periods. In he beginning of each raios for boh groups of classificaion and level monh, an invesor has o decide o purchase he esimaion models are significanly differen index fund and shor he T-bill, or o shor he from 0.5, which jusify he capaciy of hese index fund and purchase he T-bill, depending forecasing models in he predicion of index on he forecas of excess reurn in he nex reurn. Moreover, he resuls implies ha he period. Invesing in his fashion does no require level esimaion models are useful in he fore- iniial capial, making all rading sraegies casing of he sign of reurn alhough hey do comparable on an equal basis. no perform as well as he classificaion models. The rading sraegies beween he classificaion and level esimaion models are slighly 4.2. Simple hreshold rading sraegy differen o ake ino accoun of he unique naure relaed o each ype of forecass. For he Before we presen our rading sraegies, we classificaion models, le Cˆ 11 be he esimaed need o describe he operaional deails of he poserior probabiliy of posiive reurn (upward rading simulaion. The rading simulaion asmovemen) for he period 1 1. On he oher sumes ha, in he beginning of each monhly hand, le Rˆ 11 denoe he forecas of excess period, he invesor makes an asse allocaion reurn o be realized a he end of period 1 1, decision of wheher o shif asses ino T-bills or if he level esimaion models are used. Thus, sock index fund. I should be noed ha he he rading sraegies (i.e. decision rule) can be price of he sock index fund is direcly proporexpressed as follows: ional o he index level. Furher, i is assumed ha he money ha has been invesed in eiher Classificaion models T-bills or sock index fund becomes illiquid and remains locked up in ha securiy unil he If (C ˆ ) hen end of he monh. The horizon of he rading Purchase index fund and shor T-bill simulaion runs from January 1991 o December Else if (C ˆ 11#0.5) hen 1995, he same ou-of-sample period used in he Shor index fund and purchase T-bill

13 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) Level esimaion models A comparison of he average reurns over ˆ each ype of models also shows ha rading If (R 11.0) hen sraegies relied on classificaion scheme are Purchase index fund and shor T-bill ˆ more profiable han hose driven by level Else if (R 11#0) hen esimaes alhough he relaive performance Shor index fund and purchase T-bill varies from index o index. Among all hree indices, Nikkei rading realizes he mos benefi Using hese rading sraegies, we can compue from using classificaion forecasing insead of he excess reurn over all 60 ou-of-sample level esimaion wih respec o he average periods (from January 1991 hrough December reurn (increases from o 82.57%.) How- 1995) for each forecasing model. Table 3 ever, he larges gain in reurn (from o abulaes he reurns from rading he S&P 500, 66.56%) is found in he rading of FTSE 100 FTSE 100, and Nikkei 225 indices. The average when he bes model from each caegory is reurns over each ype of forecasing models compared. These observaions seem o indicae (classificaion versus level esimaion) are also ha he Japanese and Briish markes may be presened. The resuls show ha he reurn less efficien han he US marke. based on rading guided by classificaion models The las column in Table 3 indicaes he is 54.28%, compared o a reurn of 36.31% for average of reurns across all hree indices for level esimaion models. The difference of each forecasing model. Similar o he hi raes, 17.97% is approximaely equal o an annualized he reurns generaed by neural nework models 3.59% of excess reurn. For comparison pur- are beer han he oher models wihin he same pose, we also compue he excess reurn for a groups. Probabilisic Neural Nework (PNN), pure index rading sraegy, i.e, always buy he which yields a oal excess reurn of 64.10%, is index. This invesmen scheme yields a oal of he bes performer among he forecasing 13.67% over he 60 ou-of-sample monhly models evaluaed in his sudy. This observaion periods, which is lower han he reurns ob- suggess some meri of evaluaing he PNN a a ained from he wo ypes of forecasing models. more vigorous level. Table 3 Excess reurn from index rading guided by he classificaion and level esimaion forecass a US UK Japan Average for S&P 500 FTSE 100 Nikkei 225 Each Model Classificaion Discriminan analysis models Logi Probi Probabilisic neural nework Average for four classificaion models Grand average 54.3 Level esimaion Adapive exponenial smoohing models Vecor auoregression wih Kalman filer Mulivariae ransfer funcion Mulilayered feedforward neural nework Average for four level esimaion models Grand average 36.3 a The able repors he excess reurn (%) from index rading over all 60 ou-of-sample monhly periods from January 1991 hrough December The bes classificaion and level esimaion models for each caegory are in bold.

14 186 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) Muliple hreshold rading sraegies becomes more confiden abou a negaive reurn whencˆ 11 decreases from 0.5. For he muliple The rading sraegies discussed above in- hreshold rading sraegies, he course of radvolve a single hreshold which deermines he ing is no deermined by a single hreshold a course of rading. In oher words, if he prob- 0.5 bu wo hresholds away from his midpoin. abiliy forecas Cˆ 11 made by a classificaion Le g, where 0.5#g #1, be he coefficien of model is greaer han he hreshold of 0.5, he reliabiliy an invesor would like o mainain in likelihood of realizing a posiive reurn is higher his rading. I can be viewed as he degree of an and an invesor should buy he index fund o invesor s confidence on he predicive srengh capure he appreciaion. Oherwise, a negaive of he forecasing models or he risk level he is more likely and an invesor should buy he would like o ake. The smaller he value of g, T-bills o earn he ineres. The same logic holds he less confidence or risk an invesor exhibis. for he rading sraegy driven by a level esi- Thus, he muliple hreshold rading can be mae Rˆ 11 excep he hreshold o rigger he summarized as: rading is zero. Since he resuls show ha direcional forecass generally ouperforms he If (C ˆ /g ) hen level esimaes, we aemp o enhance he Purchase index fund and shor T-bill rading sraegy by beer uilizing he infor- Else if (C ˆ 11, 1 2 (0.5/g )) hen maion (i.e. C ˆ 11) supplied by he classificaion Shor index fund and purchase T-bill models. The poserior classificaion prob- Else abiliies, which can be viewed as indicaors of Do nohing (no rading) forecas reliabiliy, are compared agains muliple levels of hreshold. Inuiively, he more This rading scheme consiss of he wo courses Ĉ11 is larger han 0.5, he more he classifica- of acion used in simple hreshold rading and ion model is confiden abou a posiive reurn he opion of no rading. (0.5/g ) and 12(0.5/ in he nex period. Conversely, he model g ) are essenially he hresholds o rigger he Table 4 Difference of excess reurns beween he simple and muliple hreshold rading sraegies a various levels of reliabiliy Coefficien of Difference Coefficien of Difference Coefficien of Difference reliabiliy (%) reliabiliy (%) reliabiliy (%) a Difference of excess reurns is compued as he excess reurn per rade obained from simple hreshold rading subracing he excess reurn per rade obained from muliple hreshold rading over he 60 ou-of-sample monhly periods. Coefficien of reliabiliy is used by an invesor o align he confidence level of forecas wih his risk level in muliple hreshold rading. a

15 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) buying and selling of index fund, respecively. wih he value of g and i is no fair o compare Any probabiliy higher or lower han hese across he reurns wih differen number of hresholds suggess a high degree of confidence rades. Fig. 1 plos he difference in excess wih respec o he invesor s desired reliabiliy reurns versus he coefficien of reliabiliy g. levels. If (0.5/g ) # C ˆ 11 # 1 2 (0.5/g ), he The choice of g by he rader is an empirical forecasing model is no cerain abou is classirange issue. I is reasonable o hink ha he opimal ficaion and hus an invesor can avoid poenial of g should no be in exreme bu raher loss by saying ou of he marke, i.e. making no lies inside he domain of definiion (i.e. no on rade. he boundary). If g is very close o 1 hen he Using hese muliple hreshold rading sraeidenical muliple hreshold rading rule is essenially gies, we generae he excess reurn for he o he simple hreshold sraegy. This ou-of-sample period. Resuls in Table 4 show will neuralize any advanage of he muliple he difference of excess reurns beween he hreshold rule over he simple hreshold rule. In simple and muliple rading sraegies for varirule oher words, he capaciy of muliple hreshold ous coefficiens of reliabiliy g. The differences o screen ou unreliable forecass is elimi- are expressed in excess reurn per rade, insead naed. If g is very close o 0.5 hen he of overall reurn in he enire ou-of-sample probabiliy forecas Cˆ 11 needs o be exremely period. I is because he number of rades varies high or low (close o 0 or 1) in order o rigger a Fig. 1. Difference of excess reurns beween he simple and muliple hreshold rading sraegies versus coefficien of reliabiliy.

16 188 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) rade ransacion. The consequence is ha here will seldom be a rade and herefore no much profi. We hypohesize ha he muliple hres- hold rule will provide he rader wih an advan- age over he single hreshold rule for inermediae values of g. Fig. 1 offers suppor o his conjecure. For high values of g (above 0.8), he difference in excess reurns is very close o 0 (acually slighly negaive). For his range of g, he hresholds are oo close o 1 o provide he muliple hreshold rading rule an advanage over he single hreshold counerpar. On he oher hand, for low values of g (less han 0.66), he rading hreshold is oo close o 0.5 which lead o a oal absence of rading in many cases (his explains why he graph is cu off a g of 0.67). For inermediae values of g (beween 0.67 and 0.75), he muliple hreshold rule provides subsanially beer reurns relaive o he single hreshold rule. In his range, he difference in excess reurns is around 0.7% per rade. This reurn is on a per-rade basis, and hus i represens an excess reurn of 8.4% on an annualized basis. I is because our simulaion assumes ha a rade can ake place only a he beginning of a monh and a rader can make up o 12 rades per year. The range over which g should be chosen is, as saed earlier, an empirical issue. Based on he daa applied o his sudy, i is recommended a rader o choose a g a around 0.7. This corresponds o he upper and lower hresholds of 0.7 and 0.3, respecively. Also, we expec he recommended range of value o hold in general for differen asses. A probabiliy above 0.7 (below 0.3) implies ha he forecasing mehod is fairly confiden ha he index will in- crease(decrease). Hence, resricing rading o hese cases can reduce he number of cosly misakes. 5. Conclusions We show ha he forecasing performance of a group of classificaion models is superior o ha of a group of level esimaion models. The classificaion models included in he sudy are aimed a forecasing he sign (direcion) of index reurn whereas he level esimaion models ake he convenional approach o esi- mae he value of he reurn. The classificaion models perform beer han heir level esima- ion counerpars in erms of hi rae (number of imes he prediced direcion is correc). More ineresingly, he classificaion models are able o generae higher rading profis han he level esimaion models. This is a clear message for financial forecasers and raders. The message is ha heir focus should be on accuraely predic- ing he direcion of movemen as opposed o minimizing he esimaes deviaions from he acual observed values. Praciioners should seriously consider incorporaing classificaion models ino heir forecasing ki. In pracice, raders inrigued by our resuls could use his- orical reurns from he asse hey specialize in and es if he use of classificaion models could have generaed higher profis han wha hey acually obained from level esimaion models. If his is he case, hen hose raders should a leas consider using he classificaion models in addiion o he more radiional models hey are using. In addiion, we are able o show ha he poserior classificaion probabiliies compued by he classificaion models can be successfully used in developing muliple hreshold rading sraegies ha enhances rading profis. Under his rading scheme, a rade does no ake place unless he forecas is accepable based on he invesor s reliabiliy sandard. Also, he empirical es suggess he range of hresholds for which he sraegy will work bes. Acknowledgemens The auhors would like o acknowledge he helpful commens of he wo anonymous referees.

17 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) Appendix A Chan, L., Hamao, Y., & Lakonishok, J. (1991). Fundamenals and sock reurns in Japan. Journal of Finance 46, In our experimen, we follow Box and Jen Chen, A., & Leung, M. (1998). Dynamic foreign currency kins hree-sage mehod aimed a selecing an rading guided by adapive forecasing. Review of appropriae model for he purpose of esimaing Pacific Basin Financial Markes and Policies 1, 383 and forecasing a ime series: 418. Chen, N. (1991). Financial invesmen opporuniies and A.1. Idenificaion sage he macroeconomy. Journal of Finance 46, Chen, N., Roll, R., & Ross, S. (1986). Economic forces We use he SARIMA procedure in RATS and he sock marke. Journal of Business 59, saisical sofware o deermine plausible Chopra, N., Lakonishok, J., & Rier, J. R. (1992). Measuring abnormal performance: Do socks overreac? models. The SARIMA procedure uses sandard Journal of Financial Economics 31, diagnosics such as auocorrelaion funcion Cochrane, J. H. (1988). How big is he random walk in (ACF), parial auocorrelaion funcion (PACF), GNP? Journal of Poliical Economy 9, and plos of series. DeBond, W. F., & Thaler, R. (1985). Does he sock marke overreac? Journal of Finance 40, DeBond, W. F., & Thaler, R. (1987). Furher evidence on A.2. Esimaion sage invesor overreacion and sock marke seasonaliy. Each of he enaive models is fi and he Journal of Finance 42, various coefficiens are examined. In his sage, Fama, E., & French, K. (1988a). Permanen and emporary componens of sock prices. Journal of Poliical he esimaed models are compared using san- Economy 96, dard crieria such as AIC, SBC, and significance Fama, E., & French, K. (1988b). Dividend yields and of coefficiens. expeced sock reurns. Journal of Financial Economics 22, A.3. Diagnosic checking sage Fama, E., & French, K. (1992). The cross-secion of expeced sock reurns. Journal of Finance 47, 427 SARIMA procedure is used o check if he 465. residuals from he differen models are whie Fama, E., & French, K. (1993). Common risk facors in noise. The procedure uses diagnosics ess such he reurns on socks and bonds. Journal of Financial Economics 33, as ACF, PACF, Ljung-Box Q-saisic for serial Fama, E., & Schwer, W. (1977). Asse reurns and correlaion, and Jarque-Bera normaliy es. inflaion. Journal of Financial Economics 5, Ferson, W., & Harvey, C. (1991). The variaion of economic risk premiums. Journal of Poliical Economy 99, References Ferson, W., & Schad, R. (1996). Measuring fund sraegy and performance in changing economic condiions. Aggarwal, R., & Demaskey, A. (1997). Using derivaives Journal of Finance 51, in major currencies for cross-hedging currency risks in Greene, W. H. (1993). Economeric analysis, 2nd ed, Asian emerging markes. Journal of Fuures Markes Macmillan, New York. 17, Hair, J. F., Anderson, R. E., Taham, R. L., & Black, W. C. Basu, S. (1977). The invesmen performance of common (1995). Mulivariae daa analysis, 4h ed, Prenice Hall, socks in relaion o heir price-earnings raios: A es of Englewood Cliffs, NJ. he efficien marke hypohesis. Journal of Finance 32, Hamilon, J. (1994). Time series analysis, 2nd ed, Prince on Universiy Press, New Jersey. Campbell, J. (1987). Sock reurns and he erm srucure. Hassoun, M. H. (1995). Fundamenals of arificial neural Journal of Financial Economics 18, neworks, MIT Press, Cambridge, MA. Campbell, J. Y., & Mankiw, N. G. (1987). Permanen and Hawley, D., Johnson, J., & Raina, D. (1990). Arificial ransiory componens in macroeconomic flucuaions. neural sysems: A new ool for financial decision-mak- American Economic Review 77, ing. Financial Analyss Journal 23,

18 190 M.T. Leung e al. / Inernaional Journal of Forecasing 16 (2000) Hodgson, A., & Nicholls, D. (1991). The impac of index Wasserman, P. (1993). Advanced mehods in neural comfuures markes on Ausralian share marke volailiy. puing, Van Nosrand Reinhold, New York. Journal of Business Finance and Accouning 18, 267 Wu, Y., & Zhang, H. (1997). Forward premiums as 280. unbiased predicors of fuure currency depreciaion: A Huizinga, J. (1987). An empirical invesigaion of he long non-parameric analysis. Journal of Inernaional run behavior of real exchange raes. Carnegie Rocheser Money and Finance 16, Conference Series on Public Policy 27, Yang, Z. (1999). Probabilisic neural neworks in bank- Jaffe, J., & Weserfield, R. (1985). Paerns in Japanese rupcy predicion. Journal of Business Research 44, common sock reurns: Day of he week and urn of he year effecs. Journal of Financial and Quaniaive Zhang, G., Pauwo, B., & Hu, M. (1998). Forecasing wih Analysis 20, arificial neural neworks: The sae of he ar. Inerna- Kao, K., & Schallheim, J. (1985). Seasonal and size ional Journal of Forecasing 14, anomalies in he Japanese sock marke. Journal of Financial and Quaniaive Analysis 20, Kao, K., Ziemba, W., & Schwarz, S. (1990). Day of he Biographies: Mark T. LEUNG is Assisan Professor of week effecs in Japanese socks. In: Elon, E., & Managemen Science a he Universiy of Texas. He Grubber, M. (Eds.), Japanese capial markes, Harper received his BSc and MBA degrees from he Universiy of and Row, New York. California, and Maser of Business and PhD in operaions Kim, S. (1998). Graded forecasing using an array of managemen from Indiana Universiy. His research inerbipolar predicions: Applicaion of probabilisic neural ess are in financial forecasing and modeling, applicaions neworks o a sock marke index. Inernaional Journal of AI echniques, scheduling and opimizaion, and planof Forecasing 14, ning and conrol of producion sysems. He has published Lakonishok, J., Shleifer, A., & Vishny, R. W. (1994). in Decision Sciences, Inernaional Review of Financial Conrarian invesmen, exrapolaion, and risk. Journal Analysis, Review of Pacific Basin Financial Markes and of Finance 49, Policies, Compuers and Operaions Research, and a Lo, A. W., & MacKinlay, A. C. (1988). Sock marke variey of conference proceedings. prices do no follow random walks: Evidence from a simple specificaion es. Review of Financial Sudies 1, Hazem DAOUK is a PhD candidae in Finance, Kelley School of Business, Indiana Universiy. He received a Maberly, E. D. (1986). The informaional conen of he DESCF from ICS Paris, France, and an MBA from he inerday price change wih respec o sock index Universiy of Maryland. His research ineress include fuures. Journal of Fuures Markes 6, volailiy of asse prices, inernaional finance, and finan- Maber, V. A. (1978). Forecas modificaion based upon cial economerics. He has published in he Journal of residual analysis: A case sudy of check volume esima- Financial Economics and Compuers and Operaions ion. Decision Sciences 9, Research. Makridakis, S., Wheelwrigh, S. C., & McGee, V. E. (1983). Forecasing: mehods and applicaions, 2nd ed, An-Sing CHEN is Professor of Finance a Naional Chung Wiley, New York. Cheng Universiy, Taiwan. Dr Chen received his BSc in O Connor, M., Remus, W., & Griggs, K. (1997). Going business from Ken Sae Universiy, MBA in finance and up-going down: How good are people a forecasing PhD in Business Economics from Indiana Universiy. His rends and changes in rends? Journal of Forecasing areas of ineres are forecasing and modeling, opion and 16, derivaives, inernaional finance, and applied econome- Refenes, A. P. (1995). Neural neworks in he capial rics. He has published aricles in a variey of journals, markes, Wiley, New York. including Inernaional Journal of Finance, Journal of Ross, S. (1976). The arbirage heory of capial asse Invesing, Inernaional Review of Financial Analysis, pricing. Journal of Economic Theory 13, Inernaional Review of Economics and Finance, Review Spech, D. (1988). Probabilisic neural neworks for classiof Pacific Basin Financial Markes and Policies and ficaion, mapping, or associaive memory. In: IEEE Compuers and Operaions Research. Inernaional Conference on Neural Neworks. Spech, D. (1990). Probabilisic neural neworks. Neural Neworks 3,