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

International Journal of Forecasting 16 (2000) 173–190www.elsevier.com/ locate / ijforecast

Forecasting stock indices: a comparison of classification and levelestimation models

a , b c*Mark T. Leung , Hazem Daouk , An-Sing ChenaDepartment of Operations and Decision Technologies, Kelley School of Business, Indiana University, Bloomington, IN 47405, USA

bDepartment of Finance, Kelley School of Business, Indiana University, Bloomington, IN 47405, USAcDepartment of Finance, National Chung Cheng University, Ming-Hsiung, Chia-Yi, 621 Taiwan

Abstract

Despite abundant research which focuses on estimating the level of return on stock market index, there is a lack of studiesexamining the predictability of the direction /sign of stock index movement. Given the notion that a prediction with littleforecast error does not necessarily translate into capital gain, we evaluate the efficacy of several multivariate classificationtechniques relative to a group of level estimation approaches. Specifically, we conduct time series comparisons between thetwo types of models on the basis of forecast performance and investment return. The tested classification models, whichpredict direction based on probability, include linear discriminant analysis, logit, probit, and probabilistic neural network. Onthe other hand, the level estimation counterparts, which forecast the level, are exponential smoothing, multivariate transferfunction, vector autoregression with Kalman filter, and multilayered feedforward neural network. Our comparative study alsomeasures the relative strength of these models with respect to the trading profit generated by their forecasts. To facilitatemore effective trading, we develop a set of threshold trading rules driven by the probabilities estimated by the classificationmodels. Empirical experimentation suggests that the classification models outperform the level estimation models in terms ofpredicting the direction of the stock market movement and maximizing returns from investment trading. Further, investmentreturns are enhanced by the adoption of the threshold trading rules. 2000 International Institute of Forecasters. Publishedby Elsevier Science B.V. All rights reserved.

Keywords: Forecasting; Multivariate classification; Stock index; Trading strategy

markets around the world. The increasing di-1. Introductionversity of financial index related instruments,

Trading in stock market indices has gained along with the economic growth enjoyed in theunprecedented popularity in major financial last few years, has broadened the dimension of

global investment opportunity to both individualand institutional investors. There are two basic

*Corresponding author. Tel.: 11-812-855-8440; fax: reasons for the success of these index trading11-812-855-8679.

vehicles. First, they provide an effective meansE-mail addresses: [email protected] (M.T. Leung),for investors to hedge against potential [email protected] (H. Daouk),

[email protected] (A.-S. Chen) risks. Second, they create new profit making

0169-2070/00/$ – see front matter 2000 International Institute of Forecasters. Published by Elsevier Science B.V. All rights reserved.PI I : S0169-2070( 99 )00048-5

174 M.T. Leung et al. / International Journal of Forecasting 16 (2000) 173 –190

opportunities for market speculators and arbit- performance of various multivariate classifica-rageurs. Therefore, being able to accurately tion techniques relative to some econometricforecast stock market index has profound impli- and artificial intelligence forecasting techniques;cations and significance to researchers and and (3) to develop effective trading strategiespractitioners alike. guided by the directional forecasts and to test

Although there exists a vast number of arti- the relative performance of these investmentcles addressing the predictability of stock mar- schemes.ket return as well as the pricing of stock index The remainder of this paper is organized asfinancial instruments (e.g. S&P 500 futures), follows: A literature review and the backgroundmost of the proposed models rely on accurate of this study are summarized in the next section.forecasting of the level (i.e. value) of the In Section 3, we provide the conceptual founda-underlying stock index or its return. In most tion of the tested multivariate classificationcases, the degree of accuracy and the accep- models: parametric linear discriminant analysis,tability of certain forecasts are measured by the logit, probit, and the nonparametric probabilisticestimates’ deviations from the observed values. neural network. In addition, we review several

Depending on the trading strategies adopted level-based forecasting models (adaptive ex-by investors, forecasting methods based on ponential smoothing, Bayesian vector autoreg-minimizing forecast error may not be adequate ression, multivariate transfer function, andto meet their objectives. In other words, trading multilayered feedforward neural network) whichdriven by a certain forecast with a small fore- will be tested against their classification coun-cast error may not be as profitable as trading terparts. Then, we discuss the design of theguided by an accurate prediction of the direction experiment in Section 4. This section alsoof movement (or sign of return.) Therefore, outlines the proposed index trading strategiespredicting the direction of change of the stock and how to apply the directional forecasts andmarket index and its return is also significant in the posterior probabilities supplied by the classi-the development of effective market trading fication models.strategies.

In recent years, there has been a growingnumber of studies looking at the direction or 2. Backgroundtrend of movements of various kinds of finan-cial instruments (such as Maberly, 1986; Wu & 2.1. Predictability of stock market returnsZhang, 1997; O’Connor, Remus & Griggs,1997). However, none of these studies provide a ‘Stock prices do not follow random walks’ iscomparative evaluation of different classifica- the title of a heavily cited paper by Lo andtion techniques regarding their ability to predict MacKinlay (1988). These authors claim thatthe sign of the index return. Given this notion, considerable evidence exists and show thatwe examine various forecasting models based stock returns are to some extent predictable.on multivariate classification techniques and Most of the research is conducted using datacompare them with a number of parametric and from well-established stock markets such as thenonparametric models which forecast the level US, Western Europe, and Japan.of the return. Specifically, the major contribu- For the US, several studies have examinedtions of this study are: (1) to demonstrate and the cross-sectional relationship between stockverify the predictability of stock index direction returns and fundamental variables. Variablesusing classification models; (2) to compare the such as earnings yield, cash flow yield, book to

M.T. Leung et al. / International Journal of Forecasting 16 (2000) 173 –190 175

market ratio, and size have been found to have Ross (1986) find that changes in aggregatesome power in predicting stock returns in these production, inflation, the short-term interestcross-sectional studies. Basu (1977), Fama and rates, the slope of term structure as measured byFrench (1992), and Lakonishok, Shleifer and the return difference between long-term andVishny (1994) are examples of such cross- short-term government bonds, and the risksectional studies. These studies in general find premium as measured by the return differencepositive relationships between stock returns and between low grade bonds and high grade bondsearnings yield, and between cash flow yield and are other macroeconomic factors that have somebook-to-market ratio, and a negative relation- power to predict stock returns. Also, predic-ship between stock returns and size. tability is not limited to stock returns. For

For the Japanese stock market, Jaffe and example, Campbell and Mankiw (1987) andWesterfield (1985) and Kato, Ziemba and Cochrane (1988) find predictability in the GNP,Schwartz (1990) find some evidence of predic- and Huizinga (1987) finds predictability intability in the behavior of daily and intraday exchange rates.patterns in index returns. Kato and Schallheim Furthermore, Chen et al. (1986) suggest that(1985) document size and seasonal anomalies. expected returns are a function of businessChan, Hamao and Lakonishok (1991) relate conditions in that the expected stock marketcross-sectional differences in returns on Japan- premium is negatively related to the recentese stocks to the underlying behavior of earn- growth of economic activity proxying for theings yield, size, book to market ratio, and cash health of the current economy and positivelyflow yield. related to the expected future growth of econ-

Fundamental variables are not the only type omic activity and its conditional variance. Chenof cross-sectional variables that contain infor- (1991) studies the relation between changes inmation for predictability. DeBondt and Thaler financial investment opportunities and changes(1985, 1987), Chopra, Lakonishok and Ritter in the economy. He provides additional evi-(1992) document that a stock’s ranking in terms dence that variables such as the default spread,of its performance relative to the market can the term spread, the 1-month T-bill rate, thecontain predictability. Extreme losers have been lagged industrial production growth rate, andshown to outperform the market over sub- the dividend–price ratio are important deter-sequent years. minants of future stock market returns. He

In time-series analysis, Fama and French interprets the ability of these variables to fore-(1993) identify three common risk factors, the cast future stock market returns in terms of theiroverall market factor, factors related to firm size correlations with changes in the macroeconomicand book-to-market equity which seem to ex- environment.plain average returns on stocks and bonds. Fama The empirical evidence reviewed above is anand Schwert (1977), Campbell (1987), and illustration of large body of theoretical workFama and French (1988a,b) find macroeconom- that deals with Arbitrage Pricing Models. Rossic variables such as short-term interest rates, (1976) introduces the Arbitrage Pricing Theoryexpected inflation, dividend yields, yield (APT) and derives a model where stock returnsspreads between long and short-term govern- are a linear function of a set of state variablesment bonds, yield spreads between low grade (factors). The APT does not imply predictabilitybonds and high grade bonds, lagged stock price- per se, since the factors are assumed to beearnings ratios, and lagged returns have some observed at the same time returns are. A streampower to predict stock returns. Chen, Roll and of literature called Conditional Asset Pricing


Models has worked on deriving models that explores the relationship between the directionuses some available information set to explain of interday and intraday price change. O’Connorassets returns. Ferson and Harvey (1991) show et al. (1997) conduct a laboratory-based experi-that predictability in stocks returns are not ment and conclude that individuals show differ-necessarily due to market inefficiency or over- ent tendencies and behaviors for upward andreaction from irrational investors, but rather to downward series. This further demonstrates thepredictability in some aggregate variables that usefulness of forecasting the direction of changeare part of the information set. More specifical- in the price level, that is, the importance ofly, they argue that stock returns are predictable being able to classify the future return as a gainbecause macro variables (such as interest rates or a loss. The findings in these studies areor consumption growth) which determine to a reasonable because an accurate point estimation,certain extent stock returns, are themselves as judged by its deviation from the actualpredictable. observation, may not be a good predictor of the

It is a well-established practice in the recent direction of change in the instrument’s priceempirical finance literature to account for the level. This is a common case for the traders inpredictability of stock returns given the inves- that the predicted direction will immediatelytors information set by using macroeconomic affect their decisions on buying or selling thevariables that are public information. For exam- instrument. Finally, in their study on the Allple, an event study of insider trading by Ferson Ordinaries Index futures traded at the Australianand Schadt (1996) shows that the omission of Associated Stock Exchanges, Hodgson andvariables like lagged stock returns and previous Nicholls (1991) suggest conducting an evalua-interest rates could lead to misleading results. tion of the economic significance of the direc-Stock returns predictability given aggregate tion of price changes in future research.variables in the investors information set is awell-accepted fact. The question that remains is 2.3. Model inputshow to use the information set in an optimalway for forecasting and trading. Studies related to the predictability of return

and its determinant factors are ample and onecan easily find such studies in the literature.2.2. Forecasting the direction of index returnHence, for the sake of brevity, we will omit the

Most trading practices adopted by financial discussion of economic rationale in this paper.analysts rely on accurate prediction of the price Table 1 displays the set of potential macro-levels of financial instruments. However, some economic input variables which are used by therecent studies have suggested that trading strate- forecasting models analyzed in this paper. Thisgies guided by forecasts on the direction of the study confines the potential input variables tochange in price level are more effective and interest rates, consumer price index, industrialmay generate higher profits. Wu and Zhang production, and lagged returns. These are the(1997) investigate the predictability of the most easily available input variables that aredirection of change in the future spot exchange observable to a forecaster. Though other macro-rate. In another study, Aggarwal and Demaskey economic variables can be used as inputs, the(1997) find that the performance of cross-hedg- general consensus in the literature is that theing improves significantly if the direction of majority of useful information for forecasting ischanges in exchange rates can be predicted. subsumed by the interest rates and laggedBased on the S&P 500 futures, Maberly (1986) returns.


Table 1List of potential macroeconomic input variables and forecasted output variables

Input variables

ST Short Term Interest RateFirst difference of 3-month T-bill rate for the US, and

first difference of call money rate for the UK and Japan.

LT Long Term Interest RateFirst difference of long term government bond rate for the US,

first difference of 20-year government bond rate for the UK, andfirst difference of long term government bond rate for Japan.

R Lagged Index ReturnsLagged terms of the continuously compounded excess returns of the broad market index for

the different countries, respectively.

CPI Consumer Price LevelFirst difference of consumer price index for the three countries, respectively.

IP Industrial Production LevelFirst difference of industrial production for the three countries, respectively.

Output variables

R Returns on IndexContinuously compounded excess returns of the stock market index (S&P 500, FTSE 100, and

Nikkei 225) estimated by level estimation models.

C Probability Given the Direction of ReturnProbabilities estimated by the classification models that the excess return will be positive in

the next period.

second period is from January 1991 to De-3. Forecasting index returnscember 1995 (60 months of observations.) Thefirst period, which is assigned to in-sample3.1. Dataestimation, is used to determine the specifica-

The financial and macroeconomic data set tions of the models and parameters for theused in this study is obtained from the TSM forecasting techniques. It also serves the pur-data base and the Citibase compiled by DSC pose of validating the estimated models. TheData Services, Inc. and Citicorp Economic second period is reserved for out-of-sampleDatabase, respectively. The entire data set cov- evaluation and comparison of performancesers the period from January 1967 to December between various forecasting models.1995, a total of 348 months of observations. To test the robustness of the classificationThe data set is divided into two periods: the first models, three globally traded broad marketperiod runs from January 1967 to December indices — S&P 500 for the US, FTSE 100 for1990 (288 months of observations) while the the UK, and Nikkei 225 for Japan, are ex-


amined. The forecasted variables are the con- study. Although these models are based ontinuously compounded 1 month excess returns different statistical techniques, they share aof these indices. As it is shown in Eq. (1), the common trait — the ability to generate theexcess return on an index is defined as the probability of group membership. In othercontinuously compounded return on the price words, these models are able to estimate theindex minus the riskfree interest rate: probability of an upward (or downward) move-

ment in the stock index and thus provide aPt recommendation for trading.]]R 5 ln 2 r (1)S Dt t21Pt21

3.2.1. Linear discriminant analysiswhere P is the price of the stock index traded attDiscriminant analysis is a multivariateperiod t and r is the US riskfree (1-montht

statistical technique that investigates the differ-T-bill) interest rate in period t. Dividends areences between two or more groups of observa-ignored for this study. The reason to forecast thetions with respect to a set of independent (input)excess returns (rather than the index levels orvariables. These independent variables, calledthe total returns) is that they provide a measurediscriminator variables, are used to distinguishof how well our models perform relative to thethe characteristics among different groups. Inminimum returns gained from depositing thethe discriminant analysis, these discriminatormoney in a riskfree account. In addition, ourvariables are combined to form a set of mathe-study assumes that we are in the position of anmatical equations, known as the classificationAmerican investor. This leads to our adoption offunctions. There is a classification function forUS riskfree T bill rate, as opposed to the interesteach group of observations. The classificationrates of other riskfree instruments issued byfunction which yields the highest Z score indi-foreign governments, in deriving the excesscates the group membership of the input vectorreturns. This is because an American investorto be classified. At the same time, a probabilityalways has a choice of not investing but simplybased on the Z scores can also be calculated toputting the money into a US account whichdetermine the most likely group membership.pays the short term riskfree interest rate.

In this study, we conduct Fisher’s discrimin-The independent variables for predicting the1ant analysis to classify the direction of excessindex returns are all observable on or before the

return. Interested readers should refer to Hair,last day of the month preceding the month to beAnderson, Tatham and Black (1995) or otherforecasted. For instance, for the prediction ofstatistical texts for a more detailed description.the index return for March 1990, all indepen-After the process of model selection and valida-dent variables must be observable on or beforetion, the Fisher discriminant analysis yields thethe last day of February 1990. Constructing thefollowing classification functions based on thedata in this manner ensures that the estimationin-sample data from January 1967 to Decemberof out-of-sample forecasts will be similar to the1990 (the notations follow the ones explained inpractice in the real world. That is, only observ-Table 1). The forecast is classified as an upwardable, but not future, data are used as inputs tomovement (a positive return) if Z .Z , thatthe forecasting models. pos neg

is, the Z score of the classification function for3.2. Forecasting by classification models

1In this section, we briefly summarize the Linear discriminant analysis in our experiment is per-formed by the SPSS statistical package.classification models used in this comparative


positive return is larger than the one for nega- P(Y 5 1uX ) 5 F(X b ) (8)i i i

tive return:where the dependent variable Y takes the value

US S&P 500 of either 0 or 1. The question hinges on theSP USZ 5 5.334R 2 0.268ST 2 0.730 (2) value of the parameter P, the probability that Ypos t21 t21

equals one. X is the set of explanatory variablesSP US and F(?) is a nonlinear function of the con-Z 5 2 7.080R 1 0.301ST 2 0.749 (3)neg t21 t21

ditional mean. The only difference between thelogit and probit models is the functional form ofUK FTSE 100

FTSE FTSE F(?). F(?) is the cumulative density functionZ 5 5.926R 2 1.875Rpos t21 t22 (CDF) of the logistic distribution for the logitFTSE FTSE

1 1.925R 1 1.769R model, whereas it is the CDF of the normalt23 t24

UK UK distribution if a probit model is used. Readers1 0.129LT 1 2.262CPI 2 1.070t21 t21 should check out Greene (1993) for a more

(4) thorough description of the method.Binary choice models are particularly suited

FTSE FTSEZ 5 2 3.306R 1 2.274R for our study since we want to predict theneg t21 t22

FTSE FTSE direction in the movement of a stock index. The2 0.656R 2 1.009Rt23 t24 direction of a movement is binary in nature (up

UK UK1 0.515LT 1 2.675CPI 2 1.196 or down). Therefore, the logit and probit modelst21 t21

will permit us to calculate the probability of an(5)up or down move given all the explanatory

2variables in our information set . Based on theJapan Nikkei 225in-sample data, the chosen logit model spe-NIK NIK NIKZ 5 9.107R 2 1.194R 1 1.914Rpos t21 t22 t23 cifications are as follows:

NIK1 3.490R 2 0.763 (6) US S&P 500t24

SP SPC 5 F(0.058 1 12.445R 2 6.116Rt t21 t22NIK NIKZ 5 2 7.885R 1 4.533Rneg t21 t22 SP SP SP1 4.686R 2 5.130R 1 5.390Rt23 t24 t25NIK NIK

1 0.053R 1 0.876R 2 0.733 (7)t23 t24 US US US2 0.643ST 1 0.052ST 2 0.152STt21 t22 t23

These estimated functions are then used to US US2 0.270ST 2 0.368ST ) (9)t24 t25determine the sign of return for each monthly

out-of-sample period. In our experiment, theUK FTSE 100discriminant scores Z are also used to compute

FTSE FTSEˆ ˆC , the posterior probability of group classifica- C 5 F(0.135 1 10.607R 2 4.794Rt t t21 t22

tion for the sign of return in the next period. FTSE FTSE UK1 2.183R 1 2.815R 2 0.054STt23 t24 t21The mathematical operation can be found in

UK UK UK1 0.028ST 1 0.078ST 1 0.052STHair et al. (1995). t22 t23 t24

UK UK1 0.009ST 2 0.006STt25 t263.2.2. Binary choice models: logit and probit

UKBinary choice models are appropriate to use 1 0.172ST ) (10)t27

when trying to model dependent variables that2can take on only binary values (e.g. 0 or 1). The Logit and probit estimations are performed by RATScomputer package.general form of the model is:


3Japan Nikkei 225 neural network models are also used to predictthe sign of index return. Our classificationNIK NIKC 5 F(0.291 1 16.815R 2 4.773R 4t t21 t22 network models are based on the Probabilistic

NIK JAP JAP Neural Network (PNN) proposed by Specht1 2.241R 2 0.649LT 2 0.035STt23 t21 t22(1988, 1990). A few studies such as YangJAP

2 1.057LT ) (11)t23 (1999) and Kim (1998) which employ PNN forfinancial forecasting have shown some promis-

The estimated probit models are: ing results. PNN is conceptually built on theBayesian method of classification. Theoretical-

US S&P 500 ly, given enough information, the BayesianSP SPC 5 G(0.039 1 7.557R 2 0.879R method can classify a new sample with thet t21 t22

maximum probability of success (Wasserman,SP SP SP1 3.141R 2 2.976R 1 4.737Rt23 t24 t25 1993, p. 37). A complete description of PNN

US US US and its mathematical background can also be2 0.557ST 1 0.023ST 2 0.085STt21 t22 t23

found in the same text.US US2 0.210ST 2 0.205ST ) (12)t24 t25 As explained in Section 3.1, our network

training and validation scheme involves divid-ing the first period data into two segments. TheUK FTSE 100

t first segment is used to train the networkFTSE FTSEC 5 G(0.154 1 8.624R 2 1.981Rt21 t22 whereas the second segment is used to de-FTSE FTSE UK

1 0.451R 1 4.018R 2 0.031ST termine and validate the network architecturet23 t24 t21

and model specification. For the sake of brevity,UK UK1 0.004ST 1 0.067STt22 t23 details of this procedure are not reported here.

UK UK Enthusiastic readers can obtain the information1 0.083ST 2 0.015STt24 t25from the authors or refer to Chen and LeungUK UK

1 0.016ST 1 0.105ST ) (13)t26 t27 (1998). The selected models for out-of-sampleforecasting /evaluation have the following func-tional forms:Japan Nikkei 225

NIK NIKˆ US S&P 500C 5 G(0.219 1 9.543R 2 5.451Rt t21 t22SP SP USˆNIK JAP JAP C 5 F R , ST (15)s dt t21 t211 0.287R 2 0.419LT 1 0.077STt23 t21 t22

JAP2 0.417LT ) (14)t23 UK FTSE 100

FTSE FTSE FTSE FTSE FTSE UKˆˆ C 5 G R , R , R , R , LTs dThe estimated probability C that we would t t21 t22 t23 t24 t21t

realize a positive return in the next period is a (16)logistic CDF F(?) of explanatory variables if alogit model is used in forecasting. Likewise, if a

ˆprobit model is used instead, C is the probabili-t3ty derived from a normal CDF. Many neural network applications are related to financialdecision making. Hawley, Johnson and Raina (1990) andRefenes (1995) provide an overview of neural network

3.2.3. Probabilistic neural networks models used in the fields of finance and investment.4In addition to the classification models de- Readers who are interested in the program codes shouldrefer to Masters (1995).scribed in the last two sections, nonparametric


Japan Nikkei 225 first 228 months within the in-sample periodNIK (from January 1967 to December 1985). TheNIKC 5 H R (17)s dt t21 estimated parameters are then validated using

ˆ the remaining 60 months (from January 1986 towhere C is the posterior probability of realizingt

December 1990) within the in-sample period.a positive return in the next period, and F(?),Since we assume that historical performanceG(?), and H(?) represent the functional relation-provides meaningful information to predict theships between the dependent and independentfuture, these values for b are used in thevariables estimated by the PNN.forecasting in the reserved out-of-sample period.

3.3. Forecasting by level estimation models3.3.2. Vector autoregression with Kalman

3.3.1. Adaptive exponential smoothing filter updatingMakridakis, Wheelwright and McGee (1983) The vector autoregression (VAR) has proven

and Mabert (1978) described an extension to to be a successful technique for forecastingtraditional exponential smoothing model, gener- systems of interrelated time-series variables inally known as adaptive exponential smoothing. the macroeconomics literature. Notationally, aThis approach continuously evaluates the per- VAR model with a lag length of p can beformance in the previous period and updates the represented as:smoothing coefficient. The form of the adaptive

p5exponential smoothing model is similar to that Z 5O f(s)Z 1 ´ (23)t t2s tof the simple single exponential smoothing s51

model:9E(´ ´ ) 5 S (24)t t

ˆ ˆR 5 a X 1 (1 2 a )R (18)t11 t t t t where Z is an (n 3 1) vector of variablestˆwhere R is the forecast for period t and X is the measured at time period t, f(s) is an (n 3 n)t t

actual observation made in period t, and matrix of the coefficients, p is the lag length ofthe variables, ´ is an (n 3 1) vector of randomtEt

]a 5 (19)U U disturbances, and S is the variance–covariancet11 Mt matrix. Details of the VAR technique can beE 5 be 1 (1 2 b )E (20) found in Hamilton (1994).t t t21

6The specification of the VAR is determinedM 5 b ue u 1 (1 2 b )M (21) by accessing the performance of alternativet t t21

VAR specifications in forecasting the last 60ˆe 5 X 2 R (22) months (from January 1986 to December 1990)t t t

of the in-sample period. Based on our ex-a and b are parameters between 0 and 1 and u?u

perimental results, the following specificationsdenotes absolute values.are selected:Based on the results of experiment, the valuesUS S&P 500of b are set to be 0.75, 0.90, and 0.95 for the (25)SPh jZ 5 R , p 5 3S&P 500, FTSE, and Nikkei prediction models,

respectively. These values are determined by UK FTSE 100(26)FTSE UKaccessing the performance of the models in the Z 5hR , ST j, p 5 2

5 6We implement the adaptive exponential smoothing VAR estimations are performed by RATS computermodels using Excel spreadsheet. package.


Japan Nikkei 225Japan Nikkei 225(27) 8NIK MA(3) with long term interest rates (LT)h jZ 5 R , p 5 2

R 5 ´ 1 0.38´ 1 0.057´ 1 0.12´t t t21 t22 t23Kalman filter updating is then used to generate (20.0062)60 monthly forecasts for the reserved out-of- ]]]]]]]]]]1 LT2 3 t21(1 2 0.40L 1 0.41L 2 1.02L )sample period (from January 1991 to December

(30)1995) using the VAR specification chosen.

Similar to the adaptive exponential smoothing3.3.3. Multivariate transfer function models, the transfer function models are esti-

A multivariate transfer function model is mated using the first 228 observations withinessentially an ARIMA model with added ex- in-sample period (from January 1967 to De-ogenous variables. It is believed that the addi- cember 1985). After these estimated models aretion of exogenous variables can improve the validated by the last 60 months of observationsaccuracy of our forecasts if these variables help in the in-sample period, they are used to gener-explaining the stock excess returns. Interested ate out-of-sample forecasts from January 1991readers should refer to Makridakis et al. (1983) to December 1995. In all cases, Industrialfor a detailed explanation of this technique. In Production (IP) and Consumer Price Indexour experiment, we follow Box and Jenkins’ (CPI) did not add predictive value to the model.three-stage method (see Appendix A) aimed atselecting an appropriate model for the purpose 3.3.4. Multilayered feedforward neuralof estimating and forecasting a time series. The network

7selected models are written as follows: Although the essential operations of neuralnetworks are the same (i.e. the networks accept

US S&P 500 a set of inputs and, through their processingunits, produce a corresponding set of outputs),ARMA(1, 1) with short term interest rates (ST)neural networks can appear in many architectur-R 5 2 0.48R 1 ´ 1 0.66´t t21 t t21 al forms. To provide a comparison with the

(20.017) PNN classifier, we test the performance of the]]]]]]]]]]1 ST2 3 t21 multilayered feedforward neural network(1 2 0.31L 1 0.17L 2 0.64L )

(MLFN), commonly known as ‘backprop’ net-(28) work. Unlike the PNN which suggests a group

classification for a given set of inputs, MLFNprovides a point estimate or forecast as theUK FTSE 100output. Readers can refer to Hassoun (1995) andMA(1) with long term interest rates (LT)Zhang, Patuwo and Hu (1998) for an explana-

R 5 ´ 1 0.48´t t t21

8The denominator expression for the lagged LT term has a(20.0038 2 0.013L)]]]]]]]]]]1 LT root inside the unit circle (0.996). This does not violate the2 3 t21(1 1 0.83L 2 0.37L 2 0.84L ) stability conditions for the transfer function. The reason is

that there are no AR terms for R. It suffices for R and LT(29)to be stationary for the model to be stable. Results fromstationarity tests show that R and LT are indeed stationary (a

7Estimation of multivariate transfer function models is more complete analysis is available upon request from theperformed by RATS computer package. authors).


tion of neural network model. Wasserman functions deduced by the neural networks. The(1993) also provides a description and com- trained networks are then applied to the fore-parison of the architectures and mathematical casting of the index returns in the out-of-samplefoundation of PNN and MLFN. period (January 1991 through December 1995).

Based on the results from our study, we select9the network architecture which leads to con-

4. Simulation study and trading strategiessistent and reasonable performance in the vali-dation sample period from January 1986 to

10 4.1. Empirical evaluationDecember 1990. The selected model specifica-tions are as follows (the notations follow those

Each of the forecasting models described indescribed in Table 1):

the last section is estimated and validated by theUS S&P 500 in-sample data. This model estimation and

SP SP SP SP SP US US selection process is then followed by an empiri-R 5 F(R , R , R , R , ST , LT )t t21 t22 t23 t24 t21 t21cal evaluation which is based on the out-of-

(31) sample data covering 60 monthly observationsfrom January 1991 to December 1995. At this

UK FTSE 100 stage, the relative performance of the models isFTSE

R measured by two primary criteria:t

FTSE FTSE FTSE FTSE UK UK5 G(R , R , R , R , ST , LT )t21 t22 t23 t24 t21 t21 • number of correct forecasts (hits) of the sign

(32) of index return; and• excess returns obtained from index trading.

Japan Nikkei 225NIK A comparison of the performance between theNIK NIK NIK NIK JAPR 5 H(R , R , R , R , LT ) (33)t t21 t22 t23 t24 t21 groups of classification and level estimation

models can thus be carried out. The excesswhere F(?), G(?), and H(?) are the arbitraryreturns are derived from trading strategies

9An imperative issue in designing a network is determining which are driven by the forecasts made by thethe appropriate number of units in the hidden layer. classification and level estimation models. TheUnfortunately, there is no consistent answer to this ques- logic of these trading strategies and their resultstion. Although a large hidden layer may provide greater

will be discussed in the next two sections.flexibility in functional mapping, it may also lead to theThe number of correct forecasts of the sign ofproblem of overfitting and hamper the predictive strength

of the network model. Nevertheless, Salchenberger, Cinar return for each forecasting model is reported inand Lash (1992) offered a simple guideline of which the Table 2. The corresponding hit ratios are alsonumber of hidden units should be about 75% of the given. The average hit ratio for the group of allnumber of input units. This point was echoed by Jain and four classification models is 61.67% whereasNag (1995). Based on the results from our study, we select

that for the group of all four level estimationthe network architecture which leads to consistent andmodels is 56.11%. The pooled standard error isreasonable performance in the validation sample period

from January 1986 to December 1990. The resulting 0.0130 and the z value is 4.28, suggesting theconstruct of the network contains a hidden layer with ten group of classification models perform signifi-hidden units for both the US and UK models. For the cantly better than the group of level estimationJapan model, the network contains one hidden layer with

models. In addition, we also test the nullsix hidden units.10 hypothesis of no predictive effectiveness, thatMLFN estimations are performed by ThinkPro computerpackage. is, whether the hit ratio of a group of models is


Table 2aComparison of the predictive strength of classification and level estimation models

US S&P 500 UK FTSE 100 Japan Nikkei 225

Number Ratio Number Ratio Number Ratio

Classification Discriminant analysis 34 0.57 36 0.60* 41 0.68*models Logit 36 0.60* 36 0.60* 38 0.63*

Probit 36 0.60* 36 0.60* 38 0.63*Probabilistic neural network 38 0.63* 37 0.61* 38 0.63*

Level estimation Adaptive exponential smoothing 29 0.48 33 0.55 38 0.63*models Vector autoregression with Kalman filter 32 0.53 32 0.53 35 0.58

Multivariate transfer function 32 0.53 34 0.56 35 0.58Multilayered feedforward neural network 38 0.63* 30 0.50 36 0.60*

a The table reports the number of times a forecasting model correctly predicts the direction of the index return over the 60out-of-sample forecast periods from January 1991 through December 1995. A ratio marked with an asterisk (*) indicates a95% significance level based on a one-sided test of H : p50.50 against H : p.0.50. The best classification and level0 a

estimation models for each index are in bold.

forecasting comparison. This gives a total of 60significantly different from the benchmark ofmonthly periods. In the beginning of each0.5. The statistical tests indicate that the hitmonth, an investor has to decide to purchase theratios for both groups of classification and levelindex fund and short the T-bill, or to short theestimation models are significantly different

from 0.5, which justify the capacity of these index fund and purchase the T-bill, dependingforecasting models in the prediction of index on the forecast of excess return in the nextreturn. Moreover, the results implies that the period. Investing in this fashion does not requirelevel estimation models are useful in the fore- initial capital, making all trading strategiescasting of the sign of return although they do comparable on an equal basis.not perform as well as the classification models. The trading strategies between the classifica-

tion and level estimation models are slightly4.2. Simple threshold trading strategy different to take into account of the unique

nature related to each type of forecasts. For theBefore we present our trading strategies, we ˆclassification models, let C be the estimatedt11need to describe the operational details of the posterior probability of positive return (upward

trading simulation. The trading simulation as- movement) for the period t 1 1. On the othersumes that, in the beginning of each monthly ˆhand, let R denote the forecast of excesst11period, the investor makes an asset allocation return to be realized at the end of period t 1 1,decision of whether to shift assets into T-bills or if the level estimation models are used. Thus,stock index fund. It should be noted that the

the trading strategies (i.e. decision rule) can beprice of the stock index fund is directly propor-

expressed as follows:tional to the index level. Further, it is assumedthat the money that has been invested in either Classification modelsT-bills or stock index fund becomes illiquid and

ˆremains ‘locked up’ in that security until the If (C .0.5) thent11

end of the month. The horizon of the trading Purchase index fund and short T-billˆsimulation runs from January 1991 to December Else if (C #0.5) thent11

1995, the same out-of-sample period used in the Short index fund and purchase T-bill


Level estimation models A comparison of the average returns overeach type of models also shows that tradingˆIf (R .0) thent11 strategies relied on classification scheme are

Purchase index fund and short T-bill more profitable than those driven by levelˆElse if (R #0) thent11 estimates although the relative performanceShort index fund and purchase T-bill

varies from index to index. Among all threeindices, Nikkei trading realizes the most benefit

Using these trading strategies, we can compute from using classification forecasting instead ofthe excess return over all 60 out-of-sample level estimation with respect to the averageperiods (from January 1991 through December return (increases from 58.43 to 82.57%.) How-1995) for each forecasting model. Table 3 ever, the largest gain in return (from 35.62 totabulates the returns from trading the S&P 500, 66.56%) is found in the trading of FTSE 100FTSE 100, and Nikkei 225 indices. The average when the best model from each category isreturns over each type of forecasting models compared. These observations seem to indicate(classification versus level estimation) are also that the Japanese and British markets may bepresented. The results show that the return less efficient than the US market.based on trading guided by classification models The last column in Table 3 indicates theis 54.28%, compared to a return of 36.31% for average of returns across all three indices forlevel estimation models. The difference of each forecasting model. Similar to the hit rates,17.97% is approximately equal to an annualized the returns generated by neural network models3.59% of excess return. For comparison pur- are better than the other models within the samepose, we also compute the excess return for a groups. Probabilistic Neural Network (PNN),‘pure index’ trading strategy, i.e, always buy the which yields a total excess return of 64.10%, isindex. This investment scheme yields a total of the best performer among the forecasting13.67% over the 60 out-of-sample monthly models evaluated in this study. This observationperiods, which is lower than the returns ob- suggests some merit of evaluating the PNN at atained from the two types of forecasting models. more vigorous level.

Table 3aExcess return from index trading guided by the classification and level estimation forecasts

US UK Japan Average forS&P 500 FTSE 100 Nikkei 225 Each Model

Classification Discriminant analysis 44.1 28.2 86.1 52.8models Logit 40.7 22.9 84.1 49.2

Probit 40.7 28.1 84.1 51.0Probabilistic neural network 49.8 66.6 76.0 64.1Average for four classification models 43.8 36.5 82.6 –Grand average 54.3

Level estimation Adaptive exponential smoothing 28.6 30.0 67.8 42.1models Vector autoregression with Kalman filter 14.2 10.5 49.4 24.7

Multivariate transfer function 25.8 23.5 47.7 32.3Multilayered feedforward neural network 33.7 35.6 68.9 46.1Average for four level estimation models 25.6 24.9 58.4 –Grand average 36.3

a The table reports the excess return (%) from index trading over all 60 out-of-sample monthly periods from January 1991through December 1995. The best classification and level estimation models for each category are in bold.


4.3. Multiple threshold trading strategies becomes more confident about a negative returnˆwhenC decreases from 0.5. For the multiplet11

The trading strategies discussed above in- threshold trading strategies, the course of trad-volve a single threshold which determines the ing is not determined by a single threshold atcourse of trading. In other words, if the prob- 0.5 but two thresholds away from this midpoint.

ˆability forecast C made by a classification Let g, where 0.5#g #1, be the coefficient oft11

model is greater than the threshold of 0.5, the reliability an investor would like to maintain inlikelihood of realizing a positive return is higher his trading. It can be viewed as the degree of anand an investor should buy the index fund to investor’s confidence on the predictive strengthcapture the appreciation. Otherwise, a negative of the forecasting models or the risk level heis more likely and an investor should buy the would like to take. The smaller the value of g,T-bills to earn the interest. The same logic holds the less confidence or risk an investor exhibits.for the trading strategy driven by a level esti- Thus, the multiple threshold trading can be

ˆmate R except the threshold to trigger the summarized as:t11

trading is zero. Since the results show thatˆdirectional forecasts generally outperforms the If (C . 0.5 /g ) thent11

level estimates, we attempt to enhance the Purchase index fund and short T-billˆtrading strategy by better utilizing the infor- Else if (C , 1 2 (0.5 /g )) thent11

ˆmation (i.e. C ) supplied by the classification Short index fund and purchase T-billt11

models. The posterior classification prob- Elseabilities, which can be viewed as indicators of Do nothing (no trading)forecast reliability, are compared against multi-ple levels of threshold. Intuitively, the more This trading scheme consists of the two coursesC is larger than 0.5, the more the classifica- of action used in simple threshold trading andt11

tion model is confident about a positive return the option of no trading. (0.5 /g ) and 12(0.5 /in the next period. Conversely, the model g ) are essentially the thresholds to trigger the

Table 4aDifference of excess returns between the simple and multiple threshold trading strategies at various levels of reliability

Coefficient of Difference Coefficient of Difference Coefficient of Differencereliability (%) reliability (%) reliability (%)

1.00 0.00 0.88 20.14 0.76 0.240.99 20.04 0.87 20.12 0.75 0.710.98 20.05 0.86 20.11 0.74 0.750.97 20.09 0.85 20.11 0.73 0.790.96 20.07 0.84 20.19 0.72 0.540.95 20.05 0.83 20.03 0.71 0.750.94 20.05 0.82 20.07 0.70 0.770.93 20.05 0.81 0.02 0.69 0.740.92 20.08 0.80 0.16 0.68 1.030.91 20.09 0.79 0.12 0.67 1.320.90 20.12 0.78 0.12 – –0.89 20.15 0.77 0.20 – –

a Difference of excess returns is computed as the excess return per trade obtained from simple threshold tradingsubtracting the excess return per trade obtained from multiple threshold trading over the 60 out-of-sample monthly periods.Coefficient of reliability is used by an investor to align the confidence level of forecast with his risk level in multiplethreshold trading.


buying and selling of index fund, respectively. with the value of g and it is not fair to compareacross the returns with different number ofAny probability higher or lower than thesetrades. Fig. 1 plots the difference in excessthresholds suggests a high degree of confidencereturns versus the coefficient of reliability g.with respect to the investor’s desired reliability

ˆ The choice of g by the trader is an empiricallevels. If (0.5 /g ) # C # 1 2 (0.5 /g ), thet11

issue. It is reasonable to think that the optimalforecasting model is not certain about its classi-range of g should not be in extreme but ratherfication and thus an investor can avoid potentiallies inside the domain of definition (i.e. not onloss by staying out of the market, i.e. making nothe boundary). If g is very close to 1 then thetrade.multiple threshold trading rule is essentiallyUsing these multiple threshold trading strate-identical to the simple threshold strategy. Thisgies, we generate the excess return for thewill neutralize any advantage of the multipleout-of-sample period. Results in Table 4 showthreshold rule over the simple threshold rule. Inthe difference of excess returns between theother words, the capacity of multiple thresholdsimple and multiple trading strategies for vari-rule to screen out unreliable forecasts is elimi-ous coefficients of reliability g. The differencesnated. If g is very close to 0.5 then theare expressed in excess return per trade, instead

ˆof overall return in the entire out-of-sample probability forecast C needs to be extremelyt11

period. It is because the number of trades varies high or low (close to 0 or 1) in order to trigger a

Fig. 1. Difference of excess returns between the simple and multiple threshold trading strategies versus coefficient ofreliability.


trade transaction. The consequence is that there a group of classification models is superior tothat of a group of level estimation models. Thewill seldom be a trade and therefore not muchclassification models included in the study areprofit. We hypothesize that the multiple thres-aimed at forecasting the sign (direction) ofhold rule will provide the trader with an advan-index return whereas the level estimationtage over the single threshold rule for inter-models take the conventional approach to esti-mediate values of g. Fig. 1 offers support to thismate the value of the return. The classificationconjecture. For high values of g (above 0.8), themodels perform better than their level estima-difference in excess returns is very close to 0tion counterparts in terms of hit rate (number of(actually slightly negative). For this range of g,times the predicted direction is correct). Morethe thresholds are too close to 1 to provide theinterestingly, the classification models are ablemultiple threshold trading rule an advantageto generate higher trading profits than the levelover the single threshold counterpart. On theestimation models. This is a clear message forother hand, for low values of g (less than 0.66),financial forecasters and traders. The message isthe trading threshold is too close to 0.5 whichthat their focus should be on accurately predict-lead to a total absence of trading in many casesing the direction of movement as opposed to(this explains why the graph is cut off at g ofminimizing the estimates’ deviations from the0.67). For intermediate values of g (betweenactual observed values. Practitioners should0.67 and 0.75), the multiple threshold ruleseriously consider incorporating classificationprovides substantially better returns relative tomodels into their forecasting kit. In practice,the single threshold rule. In this range, thetraders intrigued by our results could use his-difference in excess returns is around 0.7% pertorical returns from the asset they specialize intrade. This return is on a per-trade basis, andand test if the use of classification models couldthus it represents an excess return of 8.4% on anhave generated higher profits than what theyannualized basis. It is because our simulationactually obtained from level estimation models.assumes that a trade can take place only at theIf this is the case, then those traders should at

beginning of a month and a trader can make upleast consider using the classification models in

to 12 trades per year.addition to the more traditional models they are

The range over which g should be chosen is, using.as stated earlier, an empirical issue. Based on In addition, we are able to show that thethe data applied to this study, it is recommended posterior classification probabilities computeda trader to choose a g at around 0.7. This by the classification models can be successfullycorresponds to the upper and lower thresholds used in developing ‘multiple threshold’ tradingof 0.7 and 0.3, respectively. Also, we expect the strategies that enhances trading profits. Underrecommended range of value to hold in general this trading scheme, a trade does not take placefor different assets. A probability above 0.7 unless the forecast is acceptable based on the(below 0.3) implies that the forecasting method investor’s reliability standard. Also, the empiri-is fairly confident that the index will in- cal test suggests the range of thresholds forcrease(decrease). Hence, restricting trading to which the strategy will work best.these cases can reduce the number of costlymistakes.

Acknowledgements

The authors would like to acknowledge the5. Conclusionshelpful comments of the two anonymous ref-

We show that the forecasting performance of erees.


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