Journal of Forecasing J. Forecas. 17, 401±414 (1998) Moving Average Rules, Volume and he Predicabiliy of Securiy Reurns wih Feedforward Neworks RAMAZAN GENCË AY 1 * AND THANASIS STENGOS 2 1 Universiy of Windsor, Canada 2 Universiy of Guelph, Canada ABSTRACT This paper uses he daily Dow Jones Indusrial Average Index from 1963 o 1988 o examine he linear and non-linear predicabiliy of sock marke reurns wih some simple echnical rading rules. Some evidence of nonlinear predicabiliy in sock marke reurns is found by using he pas buy and sell signals of he moving average rules. In addiion, pas informaion on volume improves he forecas accuracy of curren reurns. The echnical rading rules used in his paper are very popular and very simple. The resuls here sugges ha i is worh while o invesigae more elaborae rules and he pro abiliy of hese rules afer accouning for ransacion coss and brokerage fees. # 1998 John Wiley & Sons, Ld. KEY WORDS echnical rading; feedforward neworks INTRODUCTION Technical analyss es hisorical daa o esablish speci c rules for buying and selling securiies wih he objecive of maximizing pro and minimizing risk of loss. Technical rading analysis is based on wo main premises. Firs, he marke's behaviour paerns do no change much over ime, paricularly he long-erm rends. While fuure evens can indeed by very di eren from any pas evens, he marke's way of responding o brand-new uncerainies is usually similar o he way i handled hem in he pas. The paerns in marke prices are assumed o recur in he fuure, and hus, hese paerns can be used for predicive purposes. Second, relevan invesmen informaion may be disribued fairly e cienly, bu i is no disribued perfecly, nor will i ever be. Even if i were, some invesors, hrough superior analysis and insigh, would always have an edge over he majoriy of invesors and would ac rs. Therefore, valuable informaion can be deduced by sudying ransacion aciviy. * Correspondence o: Ramazan GencË ay, Deparmen of Economics, Universiy of Windsor, 401 Sunse, Windsor, Onario, Canada, N9B 3P4. Conrac gran sponsors: Social Sciences and Humaniies Research Council of Canada, Naural Sciences and Engineering Research Council of Canada. CCC 0277±6693/98/050401±14$17.50 # 1998 John Wiley & Sons, Ld.
402 R. GencË ay and Thanasis Sengos Marke analyss use a combinaion of various echnical indicaors o forecas a possible change in a prevailing rend. For insance, he widely used Wall Sree Technical Marke Index (WSTMI) is composed enirely of en echnical indicaors so i ignores fundamenal daa on he economy, corporae earnings and dividends. These en indicaors are compiled ino one number o faciliae he percepion of changes in invesor psychology, marke acion, speculaion, and moneary condiions ha are usually presen a key marke urning poins. This index aemps o idenify inermediae o long-erm marke moves (3±6 monhs or longer), raher han shor swings. I is of value in con rming he coninuaion of a curren rend in providing early warning of a change in he prevailing rend. Colby and Meyers (1988) repor ha WSTMI, for he period of 18 Ocober 1974 o 31 December 1986 forecas he direcion of he Dow Jones Indusrial Average Index (DJIA) 58.5% of he ime 1 week in advance; 62.6% of he ime 5 weeks in advance; 70.4% of he ime 13 weeks in advance; 79.5% of he ime 26 weeks in advance; and 81.6% of he ime 52 weeks in advance. One common componen of many echnical rules is he moving average rule. This rule basically involves he calculaion of a moving average of he raw price daa. The simples version of his rule indicaes a buy signal whenever he price climbs above is moving average and a sell signal when i drops below. The underlying noion behind his rule is ha i provides a means of deermining he general direcion or rend of a marke by examining he recen hisory. For insance, an n-period moving average is compued by adding ogeher he n mos recen periods of daa, hen dividing by n. This average is recalculaed each period by dropping he oldes daa and adding he mos recen, so he average moves wih is daa bu does no ucuae as much. An n-period moving average is smooher han a p-period (where p 5 n) moving average and measures a longer-erm rend. A ypical moving average rule can be wrien as m ˆ 1=n Xn 1 p i iˆ0 1 According o equaion (1) a buy signal is generaed when he curren price level p is above m, (p 7 m ) 4 0; oherwise a sell signal is generaed. The mos popular moving average rule as repored in Brock, Lakonishok and LeBaron (1992) is he 1-200 rule, where he shor period is one day and he long period is 200 days. Oher popular ones are he 1-50, 1-150, 5-200 and he 2-200 rules. There are oher variaions of he simple moving average rule. One is o add an addiional volume indicaor such ha he rule becomes m ˆ 1=n Xn 1 p i iˆ0 v ˆ 1=k Xk 1 vol j jˆ0 2 where vol is he number of shares raded in period and k is he lengh of he volume average rule. Now, no only he moving average for prices bu also he moving average of volume also mus be aken ino accoun o issue a buy or a sell signal. Conrary o echnical rading analysis, he e cien marke hypohesis saes ha securiy prices fully re ec all available informaion. A precondiion for his srong version of he hypohesis is ha informaion and rading coss are always zero. Since informaion and rading coss are posiive, he srong form of he marke e ciency hypohesis is clearly false. A weaker version of
Feedforward Neworks 403 he e ciency hypohesis saes ha prices re ec informaion o he poin where he marginal bene s of acing on informaion do no exceed he marginal coss (Jensen, 1978). Earlier work nds evidence ha daily, weekly and monhly reurns are predicable from pas reurns. For example, Fama (1965) nds ha he rs-order auocorrelaions of daily reurns are posiive for 23 of he 30 Dow Jones Indusrials. Fisher's (1966) resuls sugges ha he auocorrelaions of monhly reurns on diversi ed porfolios are posiive and larger han hose for individual socks. As surveyed in Fama (1970, 1991), he evidence for predicabiliy in earlier work ofen lacks saisical power and he porion of he variance of reurns explained by he variaions in expeced reurns is so small ha he hypohesis of marke e ciency and consan expeced reurns is ypically acceped as a good working model. Unlike he earlier lieraure which focused on he predicabiliy of curren reurns from pas reurns, he recen lieraure has also invesigaed he predicabiliy of curren reurns from oher variables such as dividend yields and various erm srucure variables. This lieraure also documens signi can relaionships beween expeced reurns and fundamenal variables such as he price±earnings raio, he marke-o-book raio and evidence for sysemaic paerns in sock reurns relaed o various calendar periods such as he weekend e ec, he urn-of-he-monh e ec, he holiday e ec and he January e ec. There has also been exensive recen work on he emporal dynamics of securiy reurns. For insance, Lo and MacKinlay (1988) nd ha weekly reurns on porfolios of NYSE socks grouped according o size show posiive auocorrelaion. Conrad and Kaul (1988) examine he auocorrelaions of Wednesday-o-Wednesday reurns (o miigae he nonsychronous rading problem) for size-grouped porfolios of socks ha rade on boh Wednesdays. Similar o he ndings of Lo and MacKinlay (1988) hey nd ha weekly reurns are posiively auocorrelaed. Culer, Poerba and Summers (1991) presen resuls from many di eren asse markes generally supporing he hypohesis ha reurns are posiively correlaed a he horizon of several monhs and negaively correlaed a he 3±5 year horizon. Lo and MacKinlay (1990) repor posiive serial correlaion in weekly reurns for indices and porfolios and negaive serial correlaion for individual socks. Chopra, Lakonishok and Rier (1992), De Bond and Thaler (1985), Fama and French (1986) and Poerba and Summers (1988) nd negaive serial correlaion in reurns of individual socks and various porfolios over hree-o-en-year inervals. Jegadeesh (1990) nds negaive serial correlaion for lags up o wo monhs and posiive correlaion for longer lags. Lehmann (1990) and French and Roll (1986) repor negaive serial correlaion a he level of individual securiies for weekly and daily reurns. Overall, he ndings of recen lieraure con rm he ndings of earlier lieraure ha he daily and weekly reurns are predicable from pas reurns and oher economic and nancial variables. Evidence of he ine ciency of sock marke reurns led he researchers o invesigae he sources of his ine ciency. In Brock, Lakonishok and LeBaron (1992) (BLL hereafer), wo of he simples and mos popular rading rules, moving average and he rading range brake rules, are esed hrough he use of boosrap echniques. They compare he reurns condiional on buy (sell) signals from he acual Dow Jones Indusrial Average (DJIA) Index o reurns from simulaed series generaed from four popular null models. These null models are he random walk, he AR(1), he GARCH-M due o Engle, Lilien and Robins (1987), and he exponenial GARCH (EGARCH) developed by Nelson (1991). They nd ha reurns obained from buy (sell) signals are no likely o be generaed by hese four popular null models. They documen ha buy signals generae higher reurns han sell signals and he reurns following buy signals are less volaile han reurns on sell signals. In addiion, hey nd ha reurns following sell signals are
404 R. GencË ay and Thanasis Sengos negaive which is no easily explained by any of he currenly exising equilibrium models. Their ndings indicae ha he GARCH-M model fails no only in predicing reurns, bu also in predicing volailiy. They also documen ha he EGARCH model performs beer han he GARCH-M in predicing volailiy, alhough i also fails in maching he volailiy during sell periods. The resuls in BLL documen wo imporan sylized facs. The rs is ha buy signals consisenly generae higher reurns han sell signals. The second is ha he second momens of he disribuion of he buy and sell signals behave quie di erenly because he reurns following buy signals are less volaile han reurns following sell signals. The asymmeric naure of he reurns and he volailiy of he Dow series over he periods of buy and sell signals sugges he exisence of nonlineariies as he daa-generaion mechanism. Overall, he ndings of BLL show ha he linear condiional mean esimaors fail o characerize he emporal dynamics of he securiy reurns and sugges he exisence of possible non-lineariies. Blume, Easley and O'Hara (1994) presen a model in which boh pas price and pas volume provide valuable informaion regarding a securiy. Volume conains informaion regarding he qualiy of informaion in pas price movemens; which perhaps should be more useful for smaller, less widely followed rms. Campbell, Grossman and Wang (1993) invesigae he relaionship beween rading volume and serial correlaion in sock reurns by modelling he ineracions beween liquidiy raders and marke makers. In heir model, marke makers require higher expeced reurn o accommodae he exogenous selling pressure of liquidiy raders. Therefore, price changes accompanied by high volume are more likely o be reversed han are price changes accompanied by low volume. Conrad, Hameed and Niden (1994) form a conrarian porfolio sraegy o es for he relaions beween rading volume and subsequen individual securiy reurns. An exensive survey beween price changes and volume is presened in Karpov (1987). This paper uses he wo simple echnical rading indicaors in equaions (1) and (2) o invesigae he predicive power of hese rules in forecasing he curren reurns. The rule in equaion (2) di ers from he rule in equaion (1) by incorporaing addiional informaion on volume. The comparison beween he wo rules, herefore, will reveal he predicive power of he volume in predicing he curren reurns. The es regressions of his paper conain he pas buy and sell signals of he echnical rading rules in equaions (1) and (2) as regressors o forecas he curren reurns. To measure he performance of he regression, benchmark regression models wih pas reurns as regressors are also sudied. The simple AR and GARCH-M(1,1) models are used as he linear condiional mean esimaors. The single layer feedforward neworks are used as he non-linear condiional mean esimaors. As a measure of performance he ou-of-sample mean square predicion error (MSPE) is used. The daa se is he daily Dow Jones Indusrial Average Index from 2 January 1963 o 30 June 1988, a oal of 6409 observaions. he sudy is carried ou in six subsamples. For each subsample he forecas horizon is chosen o be he las one-hird of he daa se. There are wo advanages of consrucing he forecas horizon from four di eren subsamples. The rs is o avoid spurious resuls as a resul of daa-snooping problems or sample-speci c condiions. The second is ha i enables us o analyse he performance of he echnical rading rules under di eren marke condiions. This is paricularly imporan in observing he performance of hese rules in rendy versus sluggish marke condiions in which here is no clear rend in eiher direcion. The resuls of his paper indicae ha here are no forecas improvemens in predicing curren reurns in linear condiional mean speci caions wih pas buy±sell signals relaive o linear
Feedforward Neworks 405 models which use pas reurns as regressors. In non-linear condiional mean speci caions, he models wih pas reurns provide an average of 2.5% forecas improvemen over he benchmark linear model wih pas reurns. This forecas improvemen is as large as 9.0% for he non-linear condiional mean speci caions which uilize pas buy±sell signals as regressors. The addiion of he volume indicaor furher improves he predicive power of he feedforward nework esimaors o an average of 13% over he benchmark model. In he nex secion a brief descripion of he daa is presened. Esimaion echniques are described in he hird secion and empirical resuls in he fourh. Conclusions follow hereafer. DATA DESCRIPTION The daa series includes he rs rading day in 1963 of he Dow Jones Indusrial Average (DJIA) Index o 30 June 1988, a oal of 6409 observaions. All he socks are acively raded and problems associaed wih non-synchronous rading should be of lile concern wih he DJIA. The daa se is sudied in subsample periods 1963±7, 1968±71, 1972±5, 1976±9, 1980±3 and 1984±8. The summary saisics of he daily reurns for all subsamples are presened in Table I. The daily reurns are calculaed as he log di erences of he Dow level. None of he subperiods excep he 1984±8 period show signi can skewness and excess kurosis. The rs en auocorrelaions are also given in he rows labelled r n. The Barle sandard errors from hese series are also repored in Table I. All periods show some evidence of auocorrelaion in he rs lag. The Ljung±Box±Pierce saisics are shown in he las row. These are calculaed Table I. Summary saisics of he log rs di erenced daily DJIA series January 1963±June 1988 Descripion 1963±88 1963±7 1968±71 1972±5 1976±9 1980±83 1984±8 Sample size 6409 1258 982 1008 1009 1011 1136 Mean*100 0.0187 0.0267 0.0019 0.0042 0.0023 0.0418 0.0472 Sd.*100 0.9598 0.5780 0.7503 1.0960 0.7709 0.9775 1.3752 Skewness 2.8059 0.0589 0.4932 0.2091 0.1650 0.3592 5.7253 Kurosis 86.8674 7.1234 6.3526 3.9791 4.1694 4.3425 113.2671 Max 0.0967 0.0440 0.0495 0.0460 0.0436 0.0478 0.0967 Min 0.2563 0.0293 0.0319 0.0357 0.0304 0.0359 0.2563 r 1 0.1036 0.1212 0.2929 0.2118 0.1130 0.0470 0.0126 r 2 0.0390 0.0269 0.0024 0.0531 0.0090 0.0480 0.1051 r 3 0.0083 0.0243 0.0046 0.0099 0.0197 0.0228 0.0172 r 4 0.0231 0.0480 0.0485 0.0260 0.0188 0.0361 0.0466 r 5 0.0247 0.0267 0.0248 0.0627 0.0051 0.0243 0.1015 r 6 0.0098 0.0153 0.0639 0.0360 0.0530 0.0293 0.0058 r 7 0.0065 0.0014 0.0314 0.0072 0.0114 0.0130 0.0234 r 8 0.0029 0.0396 0.1087 0.0054 0.0632 0.0164 0.0151 r 9 0.0133 0.0107 0.0086 0.0487 0.0216 0.0099 0.0230 r 10 0.0133 0.0064 0.0619 0.0048 0.0166 0.0205 0.0131 Barle sd. 0.0125 0.0282 0.0319 0.0315 0.0315 0.0314 0.0297 LBP 89.6 25.3 109.0 57.3 21.8 8.96 29.5 w 2 005 10 18.307 Noes: r 1,...,r 10 are he rs en auocorrelaions of each series. LBP refers o he Ljung±Box±Pierce saisic and i is disribued w 2 (10) under he null hypohesis of idenical and independen disribuion.
406 R. GencË ay and Thanasis Sengos for he rs en lags and are disribued w 2 (10) under he null of idenical and independen observaions. Five series ou of six give srong rejecion of he null hypohesis of idenical and independen observaions. ESTIMATOR TECHNIQUES Le p, ˆ 1,2,..., T be he daily Dow series. The reurn series are calculaed by r ˆ log(p ) 7 log(p 71 ). Le m n and v k denoe he ime value of a price average rule of lengh n and he volume average of lengh k, respecively. m n and v k are calculaed by m n ˆ 1=n Xn 1 p i iˆ0 v k vol j jˆ0 1=k Xk 1 3 The buy and sell signals for he price average rule are calculaed 1 by s n1;n2 ˆ m n1 m n2 4 where n1 andn2 are he shor and he long moving averages, respecively. The rule used in his paper is (n1,n2) ˆ (1,200) where n1 and n2 are in days. This rule is widely used in pracice. The es regressions for he OLS and GARCH-M models of his rule are r ˆ a 0 b i s n1;n2 1 e e ID 0; s 2 5 and r ˆ a 0 b i s n1;n2 1 gh 1=2 e 6 where e N(0,h ) and h ˆ d 0 d 1 h 1 d 2 e 2 1. The indicaor variable for he volume average rule is calculaed by I k1;k2 ˆ 1; vk1 1; v k1 v k2 4 0 v k2 4 0 7 The volume rule used in his paper is (k1,k2) ˆ (1,10) where k1 and k2 are in days. The linear es regression for he echnical rading rule wih volume indicaor is r ˆ a 0 a 1 I k1;k2 1 b i s n1;n2 i e 8 1 The analysis above generaes coninuous buy±sell signals. An alernaive way o consruc he buy±sell signals is o consruc an indicaor funcion given 1 when s n1;n2 4 0 (he shor moving average is above he long) and 1 oherwise. The resuls of his paper are no sensiive o hese alernaive choices of buy±sell signals.
Feedforward Neworks 407 where e ID 0; s 2. In case of he GARCH-M(1,1) process he es model is wrien as r ˆ a 0 a 1 I k1;k2 1 b i s n1;n2 1 gh 1=2 e 9 where e N(0,h ) and h ˆ d 0 d 1 h 1 d 2 e 2 1. There are numerous non-parameric regression echniques available such as exible Fourier forms, non-parameric kernel regression, waveles, spline echniques and ari cial neural neworks. Here, a class of ari cial neural nework models, namely he single-layer feedforward neworks, is used. The jusi caion for his choice is ha he rae of convergence of hese neworks does no depend on he dimensionaliy of he inpu space. Recenly, Hornik e al. (1994) have shown ha single hidden-layer feedforward neworks can approximae unknown funcions and heir derivaives wih error decreasing a raes as fas as d 1=2 and ha he dimension of he inpu space, p, does no a ec he rae of approximaion, bu only he consans of proporionaliy. This is in sharp conras o he properies of he sandard kernel and series approximans. This is an advanage in erms of having desirable esimaors in small samples. The single-layer feedforward nework regression model wih lagged buy and sell signals and wih d hidden unis is wrien as! r ˆ a 0 Xd jˆ1 b j G a 1j g ij s n1;n2 1 e e ID 0; s 2 where G is he known acivaion funcion which is chosen o be he logisic funcion. This choice is common in he ari cial neural neworks lieraure. The es regression model wih he volume indicaor is wrien as! r ˆ a 0 Xd jˆ1 b j G a 1j a 2j I k1;k2 1 g ij s n1;n2 1 e e ID 0; s 2 Many auhors have invesigaed he universal approximaion properies of neural neworks (Gallan and Whie, 1988, 1992; Cybenko, 1989; Funahashi, 1989; Hech-Nielson, 1989; Hornik, Sinchcombe and Whie, 1989, 1990). Using a wide variey of proof sraegies, all have demonsraed ha under general regulariy condiions, a su cienly complex single hidden-layer feedforward nework can approximae any member of a class of funcions o any desired degree of accuracy where he complexiy of a single hidden-layer feedforward nework is measured by he number of hidden unis in he hidden layer. For an excellen survey of he feedforward and recurren nework models, he reader may refer o Kuan and Whie (1994). To compare he performance of he regression models in (5), (6), (8), (9), (10) and (11) he linear regression 10 11 r ˆ a b i r 1 e e ID 0; s 2 12 is used wih he lagged reurns as he benchmark model. The ou-of-sample forecas performance of equaions in (5), (6), (8), (9), (10) and (11) are measured by he raio of heir mean square predicion errors (MSPEs) o ha of he linear benchmark model in equaion (12).
408 R. GencË ay and Thanasis Sengos A number of papers in he lieraure sugges ha condiional heeroscedasiciy may be imporan in he improvemen of he forecas performance of he condiional mean. For his reason, he MSPE of he GARCH-M(1,1) model wih lagged reurns r ˆ a b i r 1 gh 1=2 e e N 0; h h ˆ d 0 d 1 h 1 d 2 e 2 1 13 is compared o ha of he benchmark model in equaion (12). The ou-of-sample forecas performance of he single-layer feedforward nework model wih lagged reurns! r ˆ a 0 Xd b j G a j g ij r 1 e e ID 0; s 2 jˆ1 14 is also compared o ha of he benchmark model in equaion (12). Feedforward nework regression models require a choice for he number of hidden unis in a nework. Le! o ˆ a 0 Xd b j G a j g ij x i 15 jˆ1 where x 7i is eiher pas reurns (equaion (14)) or pas buy±sell signals (equaions (10) and (11)). The cross-validaed performance measure is formally de ned 2 as XT 1 C T d T ˆ1 h i 2 r ^o d T 16 where o^d T ignores informaion from he h observaion and consequenly provides a measure of nework performance superior o average squared error. A compleely auomaic mehod for deermining nework complexiy appropriae for any speci c applicaion is given by choosing he number of hidden unis d à T o be he smalles soluion o he problem min d2nt C T d 17 where N T is some appropriae choice se. Here, we se N T ˆ {1, 2,..., 10}. The number of lags for he pas buy±sell signals in each regression is chosen o be p ˆ 1, 2 or 3 lags. In models wih volume indicaor, he rs lag of he volume indicaor is always used as a regressor. For each onesep-ahead forecas observaion, he feedforward nework regression is re-esimaed and he opimal nework complexiy is deermined according o he cross-validaed performance measure. Accordingly, a di eren model may be indicaed by he cross-validaed performance measure a di eren forecas horizons. A rolling-sample approach is used so ha same number of observaions are used as he in-sample observaions a every one-sep-ahead predicion. The 2 Moody and Uans (1994) also use cross-validaed performance measure wihin he conex of corporae bond raing predicion.
Feedforward Neworks 409 maximum number of hidden unis (N T ˆ 10) and he maximum number of lags (p ˆ 3) in a given feedforward regression is chosen according o he compuaional limiaions. EMPIRICAL RESULTS For each subsample he ou-of-sample predicive performances of he benchmark and es models are examined. For each subsample he forecas horizon is chosen o be he las one-hird of each daa se. There are wo advanages of consrucing he forecas horizon from six di eren subsamples. The rs is o avoid spurious resuls as a resul of daa-snooping problems or samplespeci c condiions. The second is ha i enables us o analyse he performance of he echnical rading rules under di eren marke condiions. This is paricularly imporan in observing he performance of hese rules in rendy versus sluggish marke condiions in which here is no clear rend in eiher direcion. Ou-of-sample forecass are compleely ex ane by using only he informaion acually available. Le MSPE AND MPSE b be he mean square predicion errors of he es and benchmark models, respecively. To measure he ou-of-sample performance beween he es and benchmark models, he raio of he mean square predicion errors, MSPE /MSPE b is used. MSPE / MSPE b is less han one if he es model provides more accurae predicions. Similarly, he raio is greaer han one if he predicions of he es model are less accurae relaive o he benchmark model. Empirical resuls wih pas reurns The MSPEs of he benchmark (equaion (12)), GARCH-M(1,1) and he feedforward nework models wih pas reurns are presened in Table II. The MSPEs of he benchmark model are repored in levels. MSPEs of he GARCH-M(1,1) and he feedforward nework models are repored as a raio o he MSPEs of he benchmark model. All hree speci caions are esimaed for hree lags of he pas reurns. Table II repors ha he ou-of-sample forecas performance of he GARCH-M(1,1) model does no ouperform he benchmark model. The di erence beween he average MSPEs of boh models is less han 10%. The GARCH-M(1,1) model, however, has more accurae average sign predicions. One furher consideraion is o exploi any poenial non-lineariies ha migh exis in he condiional mean which migh add o he forecasing power of he pas reurns. The resuls of he model in equaion (14) wih feedforward nework esimaion are presened in he las wo columns of Table II. In he majoriy of he subperiods, he feedforward nework model provides smaller MSPEs in comparison o he benchmark and he GARCH-M(1,1) models. The resuls are especially suggesive in he fourh and fh periods where he neural nework model performs considerably beer han he oher models in erms of MSPEs and sign predicions. The average forecas improvemen of he feedforward nework model is abou 2.5% and provides more accurae sign predicions han he GARCH(1,1) model. The resuls also do no seem o be sensiive o he choice of lag lengh. Overall, he resuls of he feedforward nework regression wih pas reurns indicae forecas improvemen over he benchmark model and he GARCH-M(1,1) speci caion. Furhermore, boh GARCH-M(1,1) and feedforward nework models provide more accurae average sign predicions relaive o he benchmark parameric model.
410 R. GencË ay and Thanasis Sengos Table II. MSPEs of he models wih pas reurns Lags OLS Sign GARCH-M(1,1) Sign Feedforward Sign 1963±7 Lag 1 [0.4849] 0.466 1.000 0.467 0.984 0.500 Lag 2 [0.4866] 0.465 0.997 0.467 0.981 0.511 Lag 3 [0.4864] 0.462 0.996 0.463 0.983 0.513 1968±71 Lag 1 [0.4290] 0.533 0.995 0.534 0.994 0.541 Lag 2 [0.4263] 0.523 0.996 0.533 0.990 0.542 Lag 3 [0.4256] 0.511 0.998 0.532 0.982 0.543 1972±5 Lag 1 [1.6164] 0.531 0.995 0.533 0.994 0.535 Lag 2 [1.6151] 0.510 0.996 0.521 0.990 0.545 Lag 3 [1.5556] 0.511 0.998 0.518 0.982 0.543 1976±9 Lag 1 [0.6938] 0.551 0.998 0.554 0.971 0.585 Lag 2 [0.6929] 0.567 0.997 0.568 0.967 0.583 Lag 3 [0.6926] 0.563 0.998 0.565 0.973 0.581 1980±83 Lag 1 [1.1522] 0.431 0.996 0.451 0.968 0.533 Lag 2 [1.1599] 0.433 0.995 0.457 0.943 0.554 Lag 3 [1.1618] 0.430 0.999 0.451 0.959 0.553 1984±8 Lag 1 [4.3259] 0.465 1.000 0.467 0.984 0.510 Lag 2 [4.3495] 0.466 0.997 0.467 0.981 0.512 Lag 3 [4.3743] 0.463 0.996 0.465 0.983 0.521 Noes: The numbers in brackes are he MSPEs of he benchmark model in levels. MSPEs of he GARCH-M(1,1) and he feedforward nework models are repored as a raio o he MSPEs of he benchmark model. MSPEs of he benchmark model are 10 74. `Sign' refers o he average sign predicions in he forecas horizon Empirical resuls wih pas buy±sell signals of he moving average rules The predicabiliy of he curren reurns wih he pas buy±sell signals of he moving average rules are invesigaed wih wo di eren moving average rules. These are he (1,200) rule wihou he volume indicaor and he (1,200) rule wih 10-day volume average indicaor. For convenience, we will call hese rules A and B, respecively. The resuls wih rule A are presened in Table III. Boh OLS and he GARCH-M(1,1) speci caions provide sligh improvemens over he benchmark model wih respec o heir average MSPEs. The GARCH-M(1,1) model, however, provides higher average sign predicions relaive o he OLS model. The las wo columns of Table III are devoed o he feedforward nework regression resuls wih pas buy±sell signals. Again he feedforward nework model ouperforms is compeiors in erms of MSPEs and sign predicions, especially in he rs and fh periods. Overall, i provides smaller MSPEs han he GARCH-M(1,1) model and he feedfoward nework model has more accurae sign predicions. Also, he resuls seen insensiive o he choice of lag lengh. Comparing he resuls of Tables II and III we can see ha i is he feedforward nework model ha improves wih he use of moving average rules, whereas he OLS and he GARCH-M(1,1) models do no seem o perform di erenly beween he wo cases. In Table IV, rule B is sudied. The only di erence beween rules A and B is ha rule B accommodaes for he volume indicaor as an addiional regressor. This measures any addiional forecas gain aained from he volume variable. The es models for he linear models are presened in equaions (5), (6), (8) and (9). The non-linear condiional speci caion is given in equaions (10) and (11). In Table IV, he OLS and he GARCH-M(1,1) speci caions aain an
Feedforward Neworks 411 Table III. The raio of he MSPEs of he models wih pas buy±sell signals o he MSPEs of he benchmark model (moving average rule wihou volume indicaor) Lags OLS Sign GARCH-M(1,1) Sign Feedforward Sign 1963±7 Lag 1 0.997 0.467 0.994 0.469 0.911 0.571 Lag 2 0.998 0.467 0.994 0.468 0.913 0.573 Lag 3 0.995 0.465 0.991 0.465 0.911 0.571 1968±71 Lag 1 0.996 0.536 0.990 0.534 0.905 0.581 Lag 2 0.995 0.527 0.991 0.534 0.908 0.582 Lag 3 0.998 0.514 0.992 0.537 0.906 0.587 1972±5 Lag 1 0.994 0.534 0.993 0.538 0.905 0.591 Lag 2 0.993 0.515 0.991 0.522 0.901 0.593 Lag 3 0.991 0.516 0.990 0.520 0.900 0.590 1976±9 Lag 1 0.990 0.556 0.989 0.555 0.909 0.597 Lag 2 0.991 0.564 0.987 0.570 0.907 0.598 Lag 3 0.994 0.568 0.988 0.567 0.904 0.599 1980±83 Lag 1 0.994 0.435 0.993 0.454 0.900 0.600 Lag 2 0.995 0.431 0.995 0.459 0.901 0.601 Lag 3 0.995 0.435 0.996 0.455 0.903 0.602 1984±8 Lag 1 0.994 0.464 0.990 0.469 0.904 0.597 Lag 2 0.993 0.463 0.991 0.468 0.905 0.598 Lag 3 0.995 0.462 0.992 0.467 0.901 0.599 Table IV. The raio of he MSPEs of he models wih pas buy±sell signals o he MSPEs of he benchmark model (moving average rule wih volume indicaor) Lags OLS Sign GARCH-M(1,1) Sign Feedforward Sign 1963±7 Lag 1 0.983 0.483 0.980 0.487 0.876 0.631 Lag 2 0.984 0.485 0.979 0.491 0.875 0.632 Lag 3 0.983 0.487 0.978 0.490 0.873 0.630 1968±71 Lag 1 0.981 0.538 0.981 0.547 0.867 0.621 Lag 2 0.980 0.539 0.980 0.546 0.873 0.620 Lag 3 0.981 0.540 0.980 0.548 0.870 0.619 1972±5 Lag 1 0.983 0.537 0.978 0.545 0.871 0.618 Lag 2 0.984 0.536 0.976 0.550 0.873 0.619 Lag 3 0.981 0.535 0.975 0.551 0.875 0.620 1976±9 Lag 1 0.979 0.567 0.973 0.571 0.881 0.631 Lag 2 0.980 0.569 0.974 0.575 0.882 0.635 Lag 3 0.978 0.570 0.976 0.576 0.880 0.636 1980±83 Lag 1 0.981 0.455 0.972 0.487 0.871 0.641 Lag 2 0.982 0.456 0.971 0.484 0.869 0.638 Lag 3 0.980 0.457 0.970 0.487 0.867 0.635 1984±8 Lag 1 0.981 0.476 0.972 0.495 0.872 0.633 Lag 2 0.980 0.473 0.969 0.496 0.861 0.631 Lag 3 0.981 0.471 0.968 0.498 0.860 0.638
412 R. GencË ay and Thanasis Sengos average of 2% improvemen over he benchmark model. Furhermore, he GARCH-M(1,1) speci caion has more accurae sign predicions in comparison o he OLS model. The neural nework model improves boh on he MSPEs and sign predicions when compared wih he OLS and GARCH-M(1,1) models, especially for he rs, fh and sixh periods. Furhermore, as before, he choice of lag lengh does no seem o maer. In feedforward nework speci caions, rule B aains an average of 12% forecas gain over he benchmark model. This addiional forecas gain is approximaely 50% more han he forecas performance of he feedforward neworks wihou he volume indicaor. Moreover, he feedforward nework models provide an average of 62% correc sign predicions. I is noiceable ha he OLS and GARCH-M(1,1) models also show improvemen wih he volume indicaor over heir previous performance from he comparison of Tables II±IV. CONCLUSIONS This paper has used he daily Dow Jones Indusrial Average Index from January 1963 o June 1988 o examine he linear and non-linear predicabiliy of sock marke reurns wih some simple echnical rading rules which uilize price and volume averaging. In linear condiional mean speci caions, hese rules do no provide forecas gains over he linear benchmark model wih pas reurns. The GARCH-M(1,1) model, however, provides a higher percenage of sign predicions over he OLS model when pas buy±sell signals of he moving average rules and he volume indicaor are used as regressors. In non-linear condiional mean speci caions, he feedforward nework model does improve on he benchmark model. In addiion, he volume indicaor adds addiional forecas accuracy. The echnical rading rules used in his paper are very popular and very simple. The resuls here sugges ha i is worh while o invesigae more elaborae rules and he pro abiliy of hese rules afer accouning for ransacion coss and brokerage fees. ACKNOWLEDGEMENTS We hank he ediors and wo anonymous referees for heir consrucive commens. Boh auhors hank o he Social Sciences and Humaniies Research Council of Canada for suppor. Ramazan GencË ay also hanks he Naural Sciences and Engineering Research Council of Canada for is suppor. REFERENCES Blume, L., Easley, D. and O'Hara, M., `Marke saisics and echnical analysis: The role of volume', Journal of Finance, 49 (1994), 153±181. Brock, W. A., Lakonishok, J. and LeBaron, B., `Simple echnical rading rules and he sochasic properies of sock reurns', Journal of Finance, 47 (1992), 1731±1764. Campbell, J. Y., Grossman, S. J. and Wang, J., `Trading volume and serial correlaion in sock reurns', Quarerly Journal of Economics, 108 (1993), 905±940. Chopra, N., Lakonishok, J. and Rier, J. R., `Performance measuremen mehodology and he quesion of wheher socks overreac', Journal of Financial Economics, 31 (1992), 235±268. Colby, R. W. and Meyers, T. A., `The Encyclopedia of Technical Marke Indicaors', Business One Irwin, (1988).
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414 R. GencË ay and Thanasis Sengos Poerba, J. M. and Summers, L. H., `Mean reversion in sock prices: Evidence and implicaions', Journal of Financial Economics, 22 (1988), 27±59. Auhors' biographies: Ramazan GencË ay is a Professor in he Economics Deparmen a he Universiy of Windsor. His areas of specializaion are non-linear ime series modelling, nancial forecasing and he deecion of chaoic dynamics from observed daa. Some of his publicaions have appeared in he Journal of he American Saisical Associaion, Physica D, Journal of Nonparameric Saisics, Journal of Applied Economerics, Journal of Forecasing and Journal of Empirical Finance. Thanasis Sengos is a professor in he Economics Deparmen a he Universiy of Guelph. His areas of specializaion are nonparameric economerics, non-linear ime series modelling and he defecion of chaoic dynamics from observed daa. Some of his publicaions have appeared in Review of Economic Sudies, Journal of Moneary Economics, Inernaional Economic Review, European Economic Review and Journal of Economerics. Auhors' addresses: Ramazan GencË ay, Deparmen of Economics, Universiy of Windsor, 401 Sunse Avenue, Windsor, Onario, N9B 3P4, Canada. Thanasis Sengos, Deparmen of Economics, Universiy of Guelph, Guelph, Onario, N1G 2W1, Canada.