Fuzzy adaptive decision-making for boundedly rational traders in speculative stock markets

European Journal of Operational Research 202 (2010) 285–293 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor Interfaces with Other Disciplines Fuzzy adaptive decision-making for boundedly rational traders in speculative stock markets Stelios D. Bekiros *,1 European University Institute, Via Delle Fontanelle 10, 50014 San Domenico di Fiesole (FI), Italy a r t i c l e i n f o Article history: Received 10 December 2008 Accepted 16 April 2009 Available online 3 May 2009 Keywords: Fuzzy sets Bounded rationality Technical trading Bubbles Direction-of-change forecasting a b s t r a c t The development of new models that would enhance predictability for time series with dynamic timevarying, nonlinear features is a major challenge for speculators. Boundedly rational investors called ‘‘chartists” use advanced heuristics and rules-of-thumb to make profit by trading, or even hedge against potential market risks. This paper introduces a hybrid neurofuzzy system for decision-making and trading under uncertainty. The efficiency of a technical trading strategy based on the neurofuzzy model is investigated, in order to predict the direction of the market for 10 of the most prominent stock indices of U.S.A, Europe and Southeast Asia. It is demonstrated via an extensive empirical analysis that the neurofuzzy model allows technical analysts to earn significantly higher returns by providing valid information for a potential turning point on the next trading day. The total profit of the proposed neurofuzzy model, including transaction costs, is consistently superior to a recurrent neural network and a Buy & Hold strategy for all indices, particularly for the highly speculative, emerging Southeast Asian markets. Optimal prediction is based on the dynamic update and adaptive calibration of the heuristic fuzzy learning rules, which reflect the psychological and behavioral patterns of the traders. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction Ever since the introduction of the Efficient Markets Hypothesis, fully rational agents were considered the driving forces of markets, which in turn operated in a way to aggregate and process the beliefs and demands of traders reflecting all available information (Fama, 1970; Fama, 1991). But the empirical evidence from financial markets was not in full accordance with the Efficient Markets Hypothesis. The alternative heterogeneous agents’ behavioral model is based on relaxing strict rational agent assumptions and introducing market frictions. Simon (1957) claimed that, boundedly rational agents using simple rules-of-thumb, provides a more accurate and realistic description of human behavior than perfect rationality with optimal decision rules. La Porta et al. (1997); Cenci et al. (1996) and Shiller (2002) argue that stock prices predictability reflects the psychological factors and fashions or fads of irrational investors in a speculative market. Similar results are reported in more recent studies of Madura and Richie (2004) and Sturm (2003). Overall, the study of bounded rationality and the rapid growth of the new field of behavioural economics over the past * Tel.: +39 055 4685 698. E-mail address: Stelios.Bekiros@eui.eu 1 The author acknowledges financial support from the Max Weber Programme at the European University Institute (EUI) where he was a Fellow (2008–2009). 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.04.015 two decades led to the award of the Nobel Prize to V. Smith and D. Kahneman in 2002. The irrational market behavior has also been emphasized by Shleifer and Summers (1990) and Black (1986) in their exposition of noise traders who act on the basis of imperfect information and consequently cause prices to deviate from their equilibrium values. In general, there are two types of agents in heterogeneous agent models: ‘‘fundamentalists”, who base their expectations upon dividends, earnings, growth or even macroeconomic factors, and ‘‘chartists” (noise traders and technical analysts) who instead base their trading strategies upon historical patterns and heuristics and try to extrapolate trends in future asset prices (Brock and Hommes, 1998; Hommes, 2005). The present study focuses on the latter. Specifically, the predictive return sign ability of trading rules that rely on a simple switching strategy is investigated: positive predicted returns are executed as long positions and negative returns as short positions. A similar strategy has been employed, with considerable success, by a number of other researchers (Gençay, 1998b; Gençay, 1998a; Fernández-Rodriguez et al., 2000) etc. In general terms they find that the returns from the switching strategy are higher than those from the passive one for annual returns, even when transaction costs are high. They also find that the asset return predictability is increased during volatile periods. The buy and sell signals are produced from technical trading strategies that incorporate various linear or non-linear econometric models. 286 S.D. Bekiros / European Journal of Operational Research 202 (2010) 285–293 2. Nonlinear modeling and forecasting with neural networks and fuzzy inference systems The major challenge for ‘‘chartists” is the development of new models, or the modification of existing methods, that would enhance forecasting ability particularly for time series with dynamic time variant patterns. Conventional time series analysis, based on stationary stochastic processes does not always perform satisfactorily on economic and financial time series (Harvey, 1989). The reason is that economic data are not generally described by simple linear structural models, white noise or even random walks. The most commonly used techniques for financial forecasting are Regression methods and Autoregressive Moving Average (ARMA) models (Box and Jenkins, 1970). These methods have been used extensively in the past, but they often fail to give an accurate forecast for some series because of their nonlinear structures and some other inherent limitations. Even though ARCH/GARCH models (Bollerslev, 1986) deal with non-constant variance, still some series cannot be explained or predicted satisfactorily, due to inherent chaotic or noise patterns, fat tails, or other nonlinear components. Extensive research in the area of nonlinear modeling has shown that neural networks enhance financial forecasting, mainly because they perform advanced mathematical and statistical processes such as nonlinear interpolation and function approximation. Neural networks are parallel computational models comprising input and output vectors as well as processing units (neurons) interconnected by adaptive connection strengths (weights), trained to store the ‘‘knowledge” of the network. Adya and Collopy (1998) demonstrated the advanced predictive ability of neural networks for time series forecasting. White (1989) and Kuan and White (1994), suggested that the relationship between neural networks and conventional statistical approaches for time series forecasting is complementary. Additionally, the function approximation properties of neural networks have been thoroughly investigated by many authors. The results in Cybenko (1989); Funahashi (1989); Hornik (1991); Hornik et al. (1989); Gallant and White (1992) and Hecht-Nielsen (1989) demonstrated that feedforward networks with sufficiently many hidden units and properly adjusted parameters can approximate any function to any desired degree of accuracy. Poddig (1993) applied a feedforward neural network to predict the exchange rates between American Dollar and Deutsche Mark, and compared results to regression analysis. Other examples using neural networks in stock and currency markets include Gençay (1998); Green and Pearson (1994); Rawani (1993); Weigend (1991); Yao et al. (1996) and Zhang (1994). However, conventional time series analysis techniques as well as neural networks incorporate in terms of input variables, only quantitative factors, such as stock returns, indices and other financial or economic magnitudes. A number of qualitative factors, e.g., macroeconomical or political effects as well as traders psychology may seriously influence the market trend, thus it is important to capture this inherent knowledge. Fuzzy logic has been implemented initially in the area of control systems and decision theory and recently in economic applications with highly promising results. It provides a means of decisionmaking and learning under uncertainty. Specifically, in a fuzzy system numeric variables (inputs and outputs) are translated into fuzzy linguistic terms representing beliefs, e.g. ‘‘low” and ‘‘high”. Each term is described by a membership function, which estimates the ‘‘degree” to which a variable belongs to a fuzzy set. Finally, fuzzy inference rules represented in IF–THEN statements are specified to associate the fuzzy input to the output fuzzy set. The specification of the rules could comprise an efficient mechanism of incorporating expectations and beliefs. In general, Fuzzy systems are widely applied in fields like classification, decision support, process simulation and control systems, exactly because are effective means of modeling human expert knowledge, experience, intuition, etc., (Slowinski, 1993; Slowinski and Stefanowski, 1994; Sugeno, 1988; Kosko, 1992; Klir and Yuan, 1995; Jamshidi et al., 1997; Mamdami, 1974; Mamdami, 1977). Financial and marketing applications have also been reported (Hashemi et al., 1998; Altrock, 1997). One important advantage of fuzzy inference systems is their linguistic interpretability. When implementing fuzzy systems, the focus is paid on modeling fuzziness and linguistic vagueness using membership functions. The fuzzy system approach has been applied to different forecasting problems whereby the operator’s expert knowledge is used for prediction (Kaneko and Takaomi, 1996; Al-Shammari and Shaout, 1998). Although the fuzzy logicbased forecasting shows promising results, the process to construct a fuzzy logic system is subjective and depends on some ad-hoc assumptions. The learning rules derived in this way may not always yield the best forecast, and the choice of membership functions depends on trial and error. Neural networks’ learning ability can be utilized to adjust and fine-tune the fuzzy membership functions. The combination of both techniques results in a hybrid neurofuzzy model which incorporates the learning ability of the neural network and the functionality of the fuzzy expert system. In a neurofuzzy system the basic concept is the derivation of various parameters of a fuzzy inference system by means of adaptive training methods obtained from neural networks (Buckley and Hayashi, 1994; Nishina and Hagiwara, 1997). Recent applications of neurofuzzy models for the prediction of financial prices and volatility can be found in the works of Pantazopoulos et al. (1998); Jalili-Kharaajoo (2004) and Cheng et al. (2007). The present study advances the literature that has utilized separately neural networks or fuzzy logic systems in financial forecasting applications, by presenting a hybrid neurofuzzy approach that leads to superior predictions upon the direction-of-change of the market. The purpose of this paper is to illustrate this concretely through an investigation of the relative direction-of-change predictability of the proposed neurofuzzy trading model compared to other well-established nonlinear models. Finally, this study also provides a significant advancement of an earlier one by Bekiros and Georgoutsos (2007). The remainder of this paper is organized as follows: Section 3 describes how the neurofuzzy model for ‘‘heuristic trading” is constructed. In Section 4 the other forecasting models used in this study are described. Finally, the empirical results are shown in Sections 5 and 6 provides concluding remarks. 3. Decision-making under uncertainty: a hybrid neurofuzzy inference model The neurofuzzy architecture consists of the input, the rule layer and the output layer. In the input fuzzy layer all the input variables are translated into fuzzy linguistic terms. Each term is described by fuzzy membership functions. The type of membership functions is configured in this layer, whereas the parameters of these functions are processed and optimized via neural network training. Fuzzy learning, represented in IF–THEN statements, is specified to associate input and output variables of a system, which in this case is a heterogeneous financial market, while modeling psychology patterns and intuition of the agent. Consequently, the IF–THEN rules’ set-up provides a very realistic model of the decision-making process under which rule-of-thumb traders operate. The rules modeled in the fuzzy rule layer consist of two parts, the ‘‘IF” part and ‘‘THEN” part. The ‘‘IF” part utilizes an ‘‘AND” association. This operator proposed by Zimmerman and Thole (1978) represents the minimum value among all the validity values of the ‘‘IF” part. The output fuzzy layer incorporates the fuzzy membership S.D. Bekiros / European Journal of Operational Research 202 (2010) 285–293 functions for outputs. Finally, in the defuzzification layer, the output is converted from fuzzy variables back into crisp values. A specific architecture where the fuzzy inference layer uses linear dependences of each rule on the system’s input variables, whereby no defuzzification process is required, was introduced by Sugeno (1985). The more general 1st order Sugeno fuzzy model has rules of the form: IF x1 is A AND x2 is B THEN z ¼ w þ /  x1 þ h  x2 ð1Þ where A and B are fuzzy sets while /; h, and w are all constants. Because of the linear dependence of each rule on the system’s input variables the Sugeno system is suited for modeling nonlinear systems by interpolating multiple linear models. In order to forecast the upward and downward trends of the financial market variables a two-input, two rule 1st order Sugeno model is used (Fig. 1), where the parameters /; h and w of the nth rule contribute via a first order polynomial zn ¼ wn þ /n  x1 þ hn  x2 . This model comprises two parameter sets, namely the membership function parameters and the polynomial parameters (/; h, w). In the proposed architecture two membership functions are used for each input corresponding to two regimes, namely ‘‘low” and ‘‘high”. The hybrid training process uses a Levenberg–Marquardt neural backpropagation algorithm (Hagan 287 and Menhaj, 1994) to optimize the membership parameters and a least squares-type algorithm to solve for the polynomial parameters. The polynomial parameters are updated first using a least squares-type algorithm and the membership parameters are then updated by backpropagating the errors. Finally, in order to solve for the neurofuzzy parameters the squared error objective function E ¼ 12 ðy  yt Þ2 is used where yt the target output and y the system output for N size sample. The proposed model operates in five respective steps represented by Sl;i where l ¼ 1; . . . ; 5 the index of each step, i the ith node of step Sl;i and j the number of inputs. In the first step the grades l of the membership functions of each input j are generated as S1;i : lF i ðxj Þ, while in the second step the Q rule weight coefficients are produced S2;i : pi ¼ m j¼1 lF i ðxj Þ. The third step normalizes the rule weight coefficients S3;i : pi ¼ p1pþip2 . Next in the fourth step the rule outputs are calculated as follows: S4;i : yi ¼ pi  zi ¼ pi  ð/i  x1 þ hi  x2 þ wi Þ ð2Þ Finally, in the fifth step all the inputs from the previous step are aggregated producing the output of the system as a piecewise linear interpolating function, dynamically calibrated by the input-dependent normalized weights: S5;i : y ¼ X yi i  1  ð/1  x1 þ h1  x2 þ w1 Þ þ p  2  ð/2  x1 þ h2  x2 þ w2 Þ ð3Þ ¼p The last equation can be reformulated in the following matrix format:  1  x2 p 1 p  2  x1 p  2  x2 p  2   ½/1 h1 w1 /2 h2 w2 T  1  x1 p y ¼ ½p ¼XP ð4Þ The solution for the weight vector P to the above equation, if the X matrix was invertible and considering that the firing strengths are known, could be P ¼ X1  Y. Since this is not usually applicable, other regression methods are used such as lower triangular or more robust orthogonal decompositions. In this study Singular Value Decomposition method (SVD) (Golub and Reinsch, 1971; Golub and Van Loan, 1989; Horn and Johnson, 1991) is used. The SVD method has the advantage of using principal components to remove unimportant information related to white or heteroscedastic noise and thereby lessens the chance of overfitting. The X matrix is decomposed into a diagonal matrix D that contains the singular values, a matrix U of principal components, and an orthogonal normal matrix of right singular values V. The weight matrix is finally solved for using: P ¼ V  D1  UT  Y ð5Þ For the fuzzification of the input variables, symmetric triangular membership functions are used, in order to optimize the neurofuzzy training performance (Ishibuchi et al., 1995). The triangular function contains two parameters, the ai ‘‘peak” and the bi ‘‘support” parameter, as follows: j j ij bi lF i ðxj Þ ¼ 1  bi =2 ; if xj  ai 6 2 ( Fig. 1. Fuzzy adaptive decision-making model. x a 0; ð6Þ else The update rule of the gradient descent algorithm for the ‘‘peak” g @E parameter is given as ai;tþ1 ¼ ai;t  a  , where p the training samp @ai ple size and ga the learning rate (e.g. determines the change of the ai values and eventually the convergence of the square error function). A similar rule applies for the ‘‘support” parameter. After chain partial derivation, the error derivatives are analyzed as @E @E @y @yi @ pi @ lF i ¼    . The derived partial derivatives are  @ai @y @yi @ pi @ lF i @ai 288 S.D. Bekiros / European Journal of Operational Research 202 (2010) 285–293 @E @y @yi ðzi  yÞ @ pi pi ¼ ¼ 1, ¼ Pn and . The ¼ y  yt ¼ e, @ pi @ lF i ðxj Þ lF i ðxj Þ @y @yi i¼1 pi ‘‘peak” and ‘‘support” parameter partial derivatives are
@ lF i 2  sign xj  ai bi or zero otherwise, and ¼ if xj  ai 6 bi
@ai 2 @ lF i 1  lF i xj ¼ . Substituting into the chain rule equation the @bi bi
@E 2e  pi  ðzi  yÞ  sign xj  ai @E
Pn result is ¼ and ¼ @ai @bi bi  lF i xj  i¼1 pi h i
e  pi  ðzi  yÞ  1  lF i xj
Pn , where sign(arg) takes the value of 1 if bi  lF i xj  i¼1 pi the argument is positive and 0 otherwise. Finally, the update rule for the ‘‘peak” parameter is provided in the following recursive equation: ai;tþ1 " # 2e  pi  ðzi  yÞ  signðxj  ai Þ P ¼ ai;t   p bi  lF i ðxj Þ  ni¼1 pi ga ð7Þ whereas for the ‘‘support” parameter: bi;tþ1 ¼ bi;t   gb p  e  pi  ðzi  yÞ  ½1  lF i ðxj Þ b i  lF i ðxj Þ  Xn p i¼1 i  ð8Þ The presented model simulates in a way the adaptive decisionmaking process of the trader, which comprises two passes. In the ‘‘forward” pass the polynomial parameters are calculated using the SVD method, while the membership parameters remain fixed. Next, the outputs are produced using the previously calculated polynomial parameters and in the ‘‘reverse” pass the errors are backpropagated within the layers to determine the membership parameter updates. 4. A ‘‘memory-dependent neural network model The neurofuzzy model is compared against a neural network model and a Buy & Hold strategy in order to examine its relative predictability and profitability performance. In general, a single hidden layer feedforward network with sufficiently hidden units and properly adjusted parameters can theoretically approximate any function to any desired degree of accuracy. Despite the importance of selecting the optimum number of hidden neurons, there is no explicit formula for that matter. The geometric pyramid rule proposed by Masters (1993) considers neurons for a three-layer network with n inputs and m outputs. Katz (1992) indicates that an optimal number of hidden neurons can be found between one-half to three times the number of inputs, whereas Ersoy (1990) proposes doubling the number of neurons until the network’s RMSE performance deteriorates. The output of a neural network is produced via the application of a transfer function. The functionality is to modulate the output space as well as prevent outputs from reaching very large values which can ‘‘block” training. Levich and Thomas (1993) and Kao and Ma (1992) found that hyperbolic sigmoid and tan-sigmoid transfer functions are appropriate for financial markets data because they are nonlinear and continuously differentiable which are desirable properties for network learning. Learning typically occurs through training, where the training algorithm iteratively adjusts the connection weights. Common practice is to divide the sample into three distinct sets called the training, validation and testing (out-of-sample) sets; the training set is the largest and is used by the neural network to learn the patterns presented in the data, the validation set is used to evaluate the generalization ability in order to avoid overfitting and the training set should consist of the most recent observa- tions that are processed for testing predictability. The validation error starts decreasing until the network begins to overfit the data and the error will then begins to rise. The weights are calculated at the minimum value of the validation error. Specifically, if Xt ¼ ðx1;t ; . . . ; xp;t Þ is the input of a single layer feedforward network with q hidden units, the output is given by: " yt ¼ S b0 þ q X bi G ai0 þ i¼1 p X j¼1 aij xj;t !# ¼ f ðxt ; zÞ ð9Þ where i = 1, . . . , q and j = 1, . . . , p. Consider z ¼ ðb0 ; . . . ; bq ; a11 ; . . . ; aij ; . . . ; aqp ÞT as the weight vector and S; G transfer functions. The solution of the network considers estimation of the unknown vector z with a sample of data values. A recursive estimation methodology, which is called backpropagation is used to estimate the weight vector, as follows: ztþ1 ¼ zt þ grf ðxt ; zt Þ  ½yt  f ðxt ; zt Þ ð10Þ where rf ðxt ; zÞ is the gradient vector with respect to z and g the learning rate. The learning rate controls the size of the change of the weight vector on the t-th iteration. The z vector update is achieved via the minimization of the mean square error function. An alternative approach is the Bayesian updating (Foresee and Hagan, 1997), where the weights and biases of the network are assumed to be random variables with specified distributions. A major disadvantage of this method is that it generally takes longer to converge than backpropagation. Whilst feedforward neural networks appear to have no memory since the output at any time instant depends entirely on the inputs and the weights at that instant, recurrent neural networks exhibit characteristics simulating short-term memory. In this study, Elman recurrent neural networks (Elman, 1990) have been utilized. In Elman networks with a single hidden layer the lagged outputs of the hidden neurons are fed back into the hidden neurons themselves. If Xt ¼ ðx1;t ; . . . ; xp;t Þ is the input with q hidden units and t the time index, the output of the network is given by: yt ¼ G½b0 þ q X bi  g i;t  þ et ð11Þ i¼1   P P where g i;t ¼ G ai0 þ pj¼1 aij xj;t þ qh¼1 dih g h;t1 with G the hyperbolic tangent sigmoid transfer function and
T the weight z ¼ b0 ; . . . ; bq ; a11 ; . . . aij . . . ; aqp ; d11 ; . . . dih . . . ; dqq vector. 5. Empirical results The performance of the models is examined using logarithmic returns of the most prominent indices of U.S.A, Europe and Southeast Asia, covering developed and emerging markets with different capitalization and trading practices. Specifically for the United States, Standard & Poor’s 500 and New York Stock Exchange index are considered while for Europe, FTSE100 (UK) and CAC40 (France). In case of Southeast Asia, KLCI Composite (Malaysia), Stock Exchange Weighted (Taiwan), HangSeng (Hong Kong), Jakarta Stock Exchange Composite (Indonesia), Straits Times (New) (Singapore) and SET 100 Basic Industries (Thailand). The sample spans between January 1, 1990 to March 2, 2001 (2915 observations). This sample contains diverse regimes and several ‘‘extreme” events including the Asian crisis and the rise and fall of the tech-market bubble, which makes the analysis for technical traders particularly interesting for trend forecasting. Furthermore, it provides an empirical benchmark also applicable to other turbulent periods such as the financial crisis of 2007–2009 which lead to global recession and was caused by the credit insolvency of investment institutions and high oil prices. 289 S.D. Bekiros / European Journal of Operational Research 202 (2010) 285–293 The daily returns are calculated as rt ¼ logðP t Þ  logðPt1 Þ, where Pt denotes the daily index price. The predictive performance of the models is examined in the period May 5, 1997 to March 2, 2001 (1000 observations) with the use of a 1-day rolling window. To enhance robustness in the results, the out-of-sample period is segmented into three expanding sub-periods, namely PI: May 5, 1997 to April 2, 1999 (500 obs.), PII: May 5, 1997 to March 17, 2000 (750 obs.) and PIII: May 5, 1997 to March 2, 2001 (1000 obs.), the latter covering the entire backtesting period. The inputs xj of the neurofuzzy model correspond to the returns r t of the previous p days while the output y is the forecasted oneday-ahead return ^r t . The inputs in the recurrent neural network correspond to the daily returns over the previous p days, following Gençay (1998) and Fernadez-Rodriguez et al. (2000). For the test period the models utilize a rolling window of all previous observations as a training sample and produce forecasts for each day within the corresponding period. The validation sample for each sub-period is the 30% of the training set, and is used to evaluate the generalization ability and avoid overfitting. The training set consists of the most recent observations that are processed in each sub-period. The training and validation samples utilize a moving window of all previous observations in order to produce forecasts for each day within each backtesting sub-period. The process is repeated in each of the expanding sub-periods. The neurofuzzy model (symbolized as NF) corresponds to a specification with two lags of the returns (j ¼ 2Þ. The procedure for the selection of the lags involved the estimation of piecewise autoregressive models and the calculation of the Ljung-Box statistics for the first 10 lags of the series. Significant autocorrelations of up to the second lag of the return series were identified. Additionally, the Akaike and Schwarz Information Criteria (AIC, SIC) that were estimated for the first six lags provided the minimum value at the second lag. Sensitivity and RMSE analyses for different number of lags were conducted on all indices but the results were not found to be qualitatively different from those presented henceforth. In case of the recurrent neural network (RNN) the best forecasting ability was derived empirically by a topology which incorporated 10 neurons g in the hidden layer and an output layer with a single neuron y. This empirical result follows Katz (1992) and Ersoy (1990). In order to account for the use of nonlinear models instead of linear, a test for the presence of nonlinear dependence in the series is conducted. To that end, the well-known BDS test statistic was used, which under the null of i.i.d. is given by (Brock et al., 1991): W m;T ðeÞ ¼ T 1=2 ½C m;T ðeÞ  C m 1;T ðeÞ=rm;T ðeÞ ð12Þ C m;T ðeÞ is the correlation integral from m dimensional vectors that are within a distance e from each other, when the total sample is T and rm;T ðeÞ is the standard deviation of C m;T ðeÞ. Under the null hypothesis, W m;T ðeÞ, has a limiting standard normal distribution. The BDS test has been applied on: (a) the original data, (b) the Table 1 BDS test. Index Correlation dim. m=2 Dim. distance e¼1 e ¼ 1:5 m=3 e¼1 e ¼ 1:5 Malaysia KLCI Composite Raw data AFR NLSSR 17.91 17.74 12.82 16.03 15.71 13.79 21.99 21.80 16.55 19.65 19.44 18.03 Taiwan Stock Exchange Weighted Raw data AFR NLSSR 11.91 12.06 11.60 13.87 13.75 12.31 16.42 16.51 15.33 18.76 18.65 16.56 Hong Kong HangSeng Raw data AFR NLSSR 10.10 10.19 10.03 11.45 11.73 14.16 13.18 13.35 11.07 14.15 14.51 14.47 Indonesia Jakarta Stock Exchange Composite Raw data AFR NLSSR 23.10 22.01 16.33 21.64 21.40 13.55 27.31 26.74 17.89 24.32 24.66 15.07 Singapore Straits Times (New) Raw data AFR NLSSR 16.43 16.52 11.07 17.18 16.94 7.15 20.12 20.86 13.96 20.71 21.22 9.72 Thailand SET 100 Basic Industries Raw data AFR NLSSR 15.84 15.55 10.10 15.53 15.39 6.20 19.19 19.29 13.11 18.81 18.91 9.62 US SP500 Raw data AFR NLSSR 6.06 6.07 5.20 6.60 6.64 4.35 9.60 9.57 8.16 10.18 10.24 6.67 US NYSE Raw data AFR NLSSR 6.11 6.46 6.17 6.81 7.03 5.34 9.49 9.73 8.32 9.98 10.25 6.90 UK FTSE100 Raw data AFR NLSSR 5.29 5.22 5.47 6.36 6.26 6.17 8.10 8.09 8.52 9.36 9.35 7.85 France CAC40 Raw data AFR NLSSR 4.23 4.30 7.53 5.98 6.07 7.31 5.22 5.19 8.19 7.22 7.24 8.83 Notes: – Raw data = daily index returns, AFR = residuals from an autoregressive filter AR(2), NLSSR = natural logarithm of the squared standardized residuals from AR(2)–GARCH-M (1,1) model. – – m = dimension, e ¼ number of standard deviations of the data. Significance at the 1% level corresponds to the critical value 2.58. 290 S.D. Bekiros / European Journal of Operational Research 202 (2010) 285–293 residuals from an autoregressive filter AR(2) (based on the selected return lags), in order to ensure that the null is not rejected due to linear dependence, and (c) the natural logarithm of the squared standardized residuals from a AR(2)–GARCH-in-mean (1,1) model in order to ensure that rejection of the null is not due to conditional heteroscedasticity (De Lima, 1996). In all cases the null of i.i.d. at the 1% marginal significance level could be rejected and the evidence seemed to suggest that a genuine non-linear dependence is present in the data (Table 1). The trading rule works as follows; at the end of each trading day the models are being re-estimated over a rolling sample with a length equal to the training period. When the output of a model is greater than 0 this is used as a buy signal and a value less than 0 as a sell signal. The total return, when transaction costs are not considered, is estimated as: R¼ nþTþ1 X s  rt ð13Þ nþ1 where T indicates the out-of-sample horizon, rt is the realized return and ^st is the recommended position which takes the value of (1) for a short and (+1) for a long position (e.g. Gençay, 1998b; Jasic and Wood, 2004). In order to evaluate the forecast accuracy of the models, the percentage of correct predictions or correctly predicted signs was calculated as Sign Rate ¼ Th, where h is the number of correct predictions. One other comparative profitability measure is also considered: the Sharpe ratio (SR). The SR is the proportion of the mean return of the trading strategy over its standard deviation, lR SR ¼ rRT . The higher the SR is the higher the return and the lower T the volatility. Finally, as a measure of the predictability the Henriksson–Merton (HM) statistic (Henriksson and Merton, 1991) was employed. According to the test the number of correct forecasts has a hypergeometric distribution, asymptotically distributed as Nð0; 1Þ, under the null hypothesis of no market-timing ability. The empirical results of the comparative implementation of all models are reported in Tables 2–4. Considering total returns, a trading rule with the NF model dominates the RNN and the Buy & Hold (B&H) strategy consistently for all indices in all periods PI, PII, PII. Specifically, the total returns for the trading strategy based on the NF model ranges from the lowest 46.0% (NYSE in PI) to the highest 453.7% (SET 100 in PIII) and thus outperforms impressively the RNN and B&H strategy, the first ranging from 43.2% (SP500 in PII) to 112.4% (HangSeng in PIII) and the latter from 70.8% (SET100 in PIII) to 85.8% (CAC40 in PII). The same applies with the inclusion of transaction costs, which are estimated as 0.05% for each one-way trade, following Hsu and Kuan (2005) and Fama and Blume (1966). Again, the NF trading rule remains significantly profitable and by far better compared to that of the other models. The fact that the NF model outperforms RNN and the B&H strategy is also depicted in the proportion of correctly predicted signs, which is higher compared to the aforementioned models, although the impressive profitability of the NF model may be compromised some times with the marginal improvement of the sign rate (in some cases it is not much higher than 50%). Yet, it is due to the substantial improvement of the quantitative importance of the correctly forecasted signs. The HM test provides a further validation of the statistical significance of the sign rate in case of the NF rule with values such as 3.872 (Straits Times – Singapore in PI), 3.457 (CAC40 in PII) or even 4.525 (KLCI in PIII) at the one-sided 1% level. Instead, the RNN strategy achieves it’s highest only twice in case of FTSE100 (PI and PIII) at 1% level, yet with many non-significant or even negative values. Additionally, the SR (annualized) measuring the profitability per unit of risk over the investigated market period, is much Table 2 Out-of-sample performance of the trading models (sub-period PI: 5/5/1997–2/4/1999). Index Model Total return B&H return RMSE Sign rate HM test Malaysia KLCI Composite NF RNN 156.0 (145.5) 44.4 (32.6) 70.7 0.048 0.038 Sharpe ratio 1.550 0.443 0.536 0.470 3.479* 0.479 Taiwan Stock Exchange Weighted NF RNN 54.0 (45.6) 40.3 (52.2) 13.4 0.017 0.021 1.044 0.775 0.484 0.456 1.698** 1.177 Hong Kong HangSeng NF RNN 94.8 (83.8) 66.6 (54.3) 19.1 0.041 0.032 1.123 0.791 0.456 0.486 1.804** 0.938 Indonesia Jakarta Stock Exchange Composite NF RNN 256.0 (247.0) 9.7 (1.95) 50.4 0.032 0.034 3.067 0.111 0.524 0.478 3.108* 0.105 Singapore Straits Times (New) NF RNN 138.6 (127.7) 10.7 (22.7) 18.9 0.023 0.027 2.055 0.158 0.540 0.490 3.872* 0.346 Thailand SET 100 Basic Industries NF RNN 359.6 (348.6) 24.8 (37.7) 47.8 0.037 0.045 3.131 0.206 0.522 0.474 2.900* 0.116 US SP500 NF RNN 55.8 (47.6) 7.5 (19.4) 44.3 0.012 0.015 1.360 0.190 0.502 0.462 0.143 1.440*** US NYSE NF RNN 46.0 (35.3) 9.4 (21.7) 33.8 0.011 0.014 1.123 0.269 0.508 0.448 0.929 2.481* UK FTSE100 NF RNN 82.7 (73.4) 27.6 (15.4) 35.1 0.012 0.015 2.166 0.712 0.530 0.518 2.355* 2.178** France CAC40 NF RNN 80.1 (71.7) 21.6 (9.8) 45.9 0.016 0.020 1.629 0.443 0.542 0.484 3.150* 0.428 Notes: – – – NF = neurofuzzy model. RNN = recurrent neural network. HT test = Henriksson and Merton (1981) test, asymptotically distributed as Nð0; 1Þ. In parenthesis total return after transaction costs (0.05% average fixed cost for each one-way trade). The sign rate measures the proportion of correctly predicted signs. The Sharpe ratio is defined as the ratio of the mean return of the strategy over its standard deviation (it has been annualized by multiplying it with the squared root of 250). * ** *** Indicate significance at the one-sided 1% levels. Indicate significance at the one-sided 5% levels. Indicate significance at the one-sided 10% levels. 291 S.D. Bekiros / European Journal of Operational Research 202 (2010) 285–293 Table 3 Out-of-sample performance of the trading models (sub-period PII: 5/5/1997–17/3/2000). Index Model Total return B&H return RMSE Sharpe ratio Sign rate HM test Malaysia KLCI Composite NF RNN 206.7 (192.4) 85.5 (68.0) 15.8 0.040 0.033 1.581 0.648 0.532 0.479 3.845* 0.155 Taiwan Stock Exchange Weighted HangSeng 81.1 (68.5) 32.4 (50.4) 142.5 (125.1) 47.9 (29.5) 5.8 Hong Kong NF RNN NF RNN 0.016 0.021 0.035 0.029 1.059 0.427 1.249 0.411 0.491 0.463 0.469 0.476 2.411* 0.663 1.039 0.088 Indonesia Jakarta Stock Exchange Composite NF RNN 327.6 (314.0) 11.5 (29.3) 9.9 0.029 0.032 2.783 0.095 0.508 0.477 2.010** 0.053 Singapore Straits Times (New) NF RNN 178 (160.9) 17.8 (36.2) 11.5 0.021 0.025 1.945 0.190 0.535 0.497 4.083* 0.859 Thailand SET 100 Basic Industries NF RNN 439(422.6) 2.3 (16.8) 31.1 0.035 0.044 2.688 0.016 0.517 0.480 3.659* 0.512 US SP500 NF RNN 71.3 (58.6) 43.2 (61.6) 56.7 0.012 0.012 1.059 0.743 0.495 0.444 0.632 3.125* US NYSE NF RNN 56.3 (40.3) 36.8 (55.3) 38.3 0.011 0.014 0.964 0.712 0.501 0.444 0.759 3.226* UK FTSE100 NF RNN 86 (71.9) 15.4 (3.2) 38.6 0.012 0.015 1.518 0.269 0.519 0.499 France CAC40 NF RNN 106.5 (92.0) 3.6 (14.4) 85.8 0.015 0.019 1.550 0.047 0.548 0.472 3.457* 0.784 24.3 1.627*** 1.154 Notation as in Table 2. Table 4 Out-of-sample performance of the trading models (Total backtesting period PIII: 5/5/1997–2/3/2001) Index Model Total return B&H return RMSE Sign rate HM test Malaysia KLCI Composite NF RNN 266.1 (248.7) 58.8 (35.0) 44.3 0.036 0.030 Sharpe ratio 1.708 0.379 0.534 0.473 4.525* 0.002 Taiwan Stock Exchange Weighted NF RNN 96.4 (79.7) 13.1 (37.3) 40.8 0.019 0.023 0.838 0.111 0.499 0.461 3.398* 1.166 Hong Kong HangSeng NF RNN 214.1 (190.4) 112.4 (87.2) 4.1 0.031 0.027 1.502 0.791 0.478 0.487 0.313 1.006 Indonesia Jakarta Stock Exchange Composite NF RNN 366.5 (348.0) 27 (51.5) 42.6 0.026 0.029 2.561 0.190 0.494 0.466 1.819** 0.763 Singapore Straits Times (New) NF RNN 186.8 (163.5) 11.5 (36.5) 2.1 0.019 0.023 1.613 0.095 0.522 0.502 3.557* 1.447*** Thailand SET 100 Basic Industries NF RNN 453.7(432.0) 23.2 (48.1) 70.8 0.032 0.040 2.277 0.111 0.512 0.471 2.995* 0.177 US SP500 NF RNN 64.3 (47.5) 11.7 (36.4) 39.6 0.012 0.016 0.822 0.142 0.482 0.470 1.072 0.996 US NYSE NF RNN 76.8 (55.2) 10.2 (35.5) 37.4 0.011 0.013 1.138 0.158 0.504 0.469 1.477*** 1.350*** UK FTSE100 NF RNN 77.7 (58.6) 48.7 (23.8) 27.4 0.012 0.014 1.059 0.664 0.512 0.515 1.597*** 2.621* France CAC40 NF RNN 84.1 (75.8) 21.9 (2.2) 68.3 0.014 0.018 0.791 0.237 0.520 0.486 2.027** 0.193 Notation as in Table 2. higher than for the RNN in all indices and sub-periods examined. The fact that B&H strategy outperforms the RNN model in some cases is not in accordance with previous results derived by Fernández et al. (2000) as well as with the conclusions reached by Christoffersen and Diebold (2003). In that, it is noticeable in this study that for some indices and periods the nonlinear RNN model employing an ‘‘active” trading strategy compared to the ‘‘static” B&H, provides with worse even negative (loss) results, e.g. 43.2% for the SP500 in PII, or 11.5% for the SET100 in PIII. However, the B&H strategy never outperforms that of the NF model, with the latter producing remarkably higher profitability results when compared, in all periods and for all indices. It is also worth noticing that the profitability of the NF model is significantly higher in the Asian compared to the US and European markets. These particular emerging markets as they are small and fragmented, illiquid, shallow and characterized by many extreme events provide the ‘‘ideal speculative environment” for noise traders and chartists who act on the basis of imperfect information and on behavioral and psychological patterns (Shleifer and Summers, 1990; Black, 1986). In this context, the NF model proves to be optimal in capturing the adaptive decision-making process of the specific class of market agents. Overall, the predictive ability of the NF model is significantly higher compared to the other models. A plausible explanation is 292 S.D. Bekiros / European Journal of Operational Research 202 (2010) 285–293 that a B&H strategy would be the best for the stock indices in the extreme case with no turning points in the testing period. However, when there are many turning points during a period and the more turning points occur, the better the NF model will be in prediction performance. The dynamic ad-hoc modification of the IF–THEN rules indicating a knowledge emergence mechanism, but most importantly the adaptive ‘‘calibration” of the membership function parameters to match the input regime, comprise the majors factors that lead to precise and prompt identification of market turning points and thus to better classification and prediction results. 6. Conclusions This study introduces a neurofuzzy system for decision-making and trading under uncertainty. The development of new models that enhance predictability for time series with dynamic timevarying, nonlinear features is a major challenge for boundedly rational investors called ‘‘chartists”, who eventually use advanced heuristics and rules-of-thumb to make profit by trading, or even hedge against potential market risks. The present paper expands the literature that has utilized separately neural networks or fuzzy logic systems, nonlinear econometric models or Buy & Hold strategies in the evaluation of the return sign forecasting ability of trading rules by presenting a hybrid neurofuzzy model. The results suggest that with the inclusion of transaction costs, the performance of the proposed neurofuzzy model in terms of market direction-of-change for 10 US, European and Southeast Asian indices, is consistently superior to the recurrent neural network as well as a Buy & Hold strategy for all indices. The examined stock indices are Standard & Poor’s 500, New York Stock Exchange, FTSE100 (UK), CAC40 (France), KLCI Composite (Malaysia), Stock Exchange Weighted (Taiwan), HangSeng (Hong Kong), Jakarta Stock Exchange Composite (Indonesia), Straits Times (New) (Singapore) and SET 100 Basic Industries (Thailand). The neurofuzzy model produced a substantial improvement of the profitability per unit of risk over the investigated market period, as it provided valid information for a potential turning point on the next trading day. Specifically, these results seem to indicate that the neurofuzzy model has been optimally ‘‘trained” to correctly relate changes with the ‘‘sign” of the market one day ahead. The comparison between the trading models in terms of sign prediction demonstrated that the proposed model functioning as a dynamically adjusted piecewise linear interpolator compared to the static nonlinear neural predictor or the naive Buy &Hold strategy, leads to more precise identification of market turning points, while the dynamic change of the inference rules and its parameters allows for adaptive knowledge emergence and optimal regime recognition. 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