The predictability of security returns with simple technical trading rules

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1 Journal of Emirical Finance The redictability of security returns with simle technical trading rules Ramazan Gençay Deartment of Economics, UniÕersity of Windsor, 401 Sunset, Windsor, Ont., Canada N9B 3P4 Acceted 9 July 1997 Abstract Technical traders base their analysis on the remise that the atterns in market rices are assumed to recur in the future, and thus, these atterns can be used for redictive uroses. This aer uses the daily Dow Jones Industrial Average Index from 1897 to 1988 to examine the linear and nonlinear redictability of stock market returns with simle technical trading rules. The nonlinear secification of returns are modelled by single layer feedforward networks. The results indicate strong evidence of nonlinear redictability in the stock market returns by using the ast buy and sell signals of the moving average rules. q 1998 Elsevier Science B.V. All rights reserved. JEL classification: C45; C5; C53; G14 Keywords: Market efficiency; Technical trading rules; Feedforward networks 1. Introduction Technical traders base their analysis on the remise that the atterns in market rices are assumed to recur in the future, and thus, these atterns can be used for redictive uroses. The motivation behind the technical analysis is to be able to identify changes in trends at an early stage and to maintain an investment strategy until the weight of the evidence indicates that the trend has reversed. One oular rule for deciding when to buy and sell in a market is the moving average rule. This rule basically involves the calculation of a moving average of the raw rice data. The simlest version of this rule indicates a buy signal whenever the rice climbs r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S

2 348 ( ) R. GençayrJournal of Emirical Finance above its moving average and a sell signal when it dros below. The underlying notion behind this rule is that it rovides a means of determining the general direction or trend of a market by examining the recent history. For instance, an n-eriod moving average is comuted by adding together the n most recent eriods of data, then dividing by n. This average is recalculated each eriod by droing the oldest data and adding the most recent, so the average moves with its data but does not fluctuate as much. The larger the n, the smoother is the moving average rule and measures a longer-term trend. The literature on financial forecasting documented evidence on the redictability of current returns from ast returns as well as the redictability of current returns from other variables such as dividend yields and various term structure variables. This literature also documented significant relationshis between exected returns and fundamental variables such as the rice earnings ratio, the market-to-book ratio and evidence for systematic atterns in stock returns related to various calendar eriods such as the weekend effect, the turn-of-the-month effect, the holiday effect, the January effect and the redictability originating from bid ask bounces. For instance, Lo and MacKinlay Ž find that weekly returns on ortfolios of NYSE stocks groued according to size show ositive autocorrelation. Conrad and Kaul Ž examine the autocorrelations of Wednesday-to- Wednesday returns Ž to mitigate the nonsynchronous trading roblem. for sizegroued ortfolios of stocks that trade on both Wednesdays. Similar to the findings of Lo and MacKinlay Ž they find that weekly returns are ositively autocorrelated. Cutler et al. Ž resent results from many different asset markets generally suorting the hyothesis that returns are ositively correlated at the horizon of several months and negatively correlated at the 3 5 year horizon. Lo and MacKinlay Ž reort ositive serial correlation in weekly returns for indices and ortfolios and negative serial correlation for individual stocks. Chora et al. Ž 199., De Bondt and Thaler Ž 1985., Fama and French Ž and Poterba and Summers Ž find negative serial correlation in returns of individual stocks 1 and various ortfolios over three to ten year intervals. Jegadeesh Ž finds negative serial correlation for lags u to two months and ositive correlation for longer lags. Lehmann Ž and French and Roll Ž reort negative serial correlation at the level of individual securities for weekly and daily returns. Overall, the findings of recent literature confirm the findings of earlier literature that the daily and weekly returns are redictable from ast returns and other economic and financial variables. 1 One interretation of negative correlation in longer horizons is that there can be substantial mean-reversion in stock market rices at longer horizons. On the other hand, the obvious concern is that inferences based on long horizon returns are based on extremely small samle size which makes the mean-reversion hyothesis fragile.

3 ( ) R. GençayrJournal of Emirical Finance The redictability of stock market returns led the researchers to investigate the sources of this redictability. In Brock et al. Ž 199., two of the simlest and most oular trading rules, moving average and the trading range brake rules, are tested through the use of bootstra techniques. They comare the returns conditional on buy Ž sell. signals from the actual Dow Jones Industrial Average Index to returns from simulated series generated from four oular null models: the random walk, the ARŽ. 1, the GARCH-M and the exonential GARCH Ž EGARCH.. Brock et al. Ž 199. find that returns obtained from buy Ž sell. signals are not likely to be generated by these four oular null models. They document that buy signals generate higher returns than sell signals and the returns following buy signals are less volatile than returns on sell signals. In addition, they find that returns following sell signals are negative which is not easily exlained by any of the currently existing equilibrium models. Their findings indicate that the GARCH-M model fails not only in redicting returns, but also in redicting volatility. They also document that the EGARCH model erforms better than the GARCH-M in redicting volatility, although it also fails in matching the volatility during sell eriods. Overall, the findings in Brock et al. Ž 199. show that buy signals consistently generate higher returns than sell signals and the second moments of the distribution of the buy and sell signals behave quite differently because the returns following buy signals are less volatile than returns following sell signals. The asymmetric nature of the returns and the volatility of the Dow series over the eriods of buy and sell signals indicate that the linear conditional mean estimators fail to characterize the temoral dynamics of the security returns and suggest the existence of nonlinearities as the data generation mechanism. This aer investigates the nonlinear redictability of security returns from ast returns as well as from the simlest forms of technical trading rules, namely moving average rules. The single layer feedforward networks are used as the nonlinear conditional mean estimator. To measure the erformance of the nonlinear conditional mean estimator against linear secifications, the oular linear null models such as the simle AR and GARCH-M models are also studied. The evidence of redictability should contain a diligent search for out-of-samle confirmation and subsamle analysis in a reasonably lengthy data set so that the roblems or interretations relating to the samle secific conditions are avoided. This aer rovides a detailed out-of-samle analysis of both the linear and nonlinear conditional mean estimators over 90 years of daily data and in subsamles. The ten most recent observations of each subsamle are ket out for out-of-samle rediction uroses, for a total of 0 observations. The advantage of constructing the forecast horizon from different subsamles is that it enables us to analyze the erformance of the technical trading rules under different market conditions. This is esecially imortant in observing the erformance of these rules in trendy versus sluggish market conditions in which there is no clear trend in either direction. As a measure of erformance the out-of-samle mean square

4 350 ( ) R. GençayrJournal of Emirical Finance rediction error Ž MSPE. is used. The data set is the daily Dow Jones Industrial Average Index from 1897 to The results of this aer indicate that nonlinear conditional mean secifications rovide significant forecast redictions over the linear models. The imrovement in the forecast accuracy is more ronounced in the nonlinear models which use ast buy sell signals, relative to the nonlinear models which use ast returns. The OLS and GARCH-M Ž 1,1. models with ast buy sell signals rovide an average of 1.65% and.95% forecast imrovements over the benchmark model with ast returns. In nonlinear conditional mean secifications, the models with ast returns rovide an average of 4.7% forecast imrovement over the benchmark linear model with ast returns. This forecast imrovement is an average of 10.8% for the nonlinear model with ast buy sell signals. In section two a brief descrition of the data is resented. Estimation techniques are described in section three and emirical results in section four. Conclusions follow thereafter.. Data summary The data series includes the first trading day in 1897 of the Dow Jones Industrial Average Ž DJIA. Index to June 30, 1988, a total of 90 years of daily data. All of the stocks are actively traded and roblems associated with nonsynchronous trading should be of little concern with the DJIA. The data set is studied in subsamle eriods , , and These subsamles are chosen for two reasons. The first subsamle ends with the closure of the stock exchange during World War I. The second includes both the rise of the twenties and the turbulent times of the deression. The third includes the eriod of World War II and ends in June 196, the date at which the Center for Research in Securities Prices Ž CRSP. begins its daily rice series. The last covers the eriod that was extensively researched because of data availability. Secondly, Brock et al. Ž 199. studied the same data set with the same subsamles, which makes our results comarable to theirs. To study the sensitivity of our results to samle variation, each of these four subsamles are further slit into four year subsamles. The volatile eriods are the , , and The and the eriods exhibit the rise of the 190 s and the times of the great deression. The and eriods exhibit an uward trend towards the end of 19 and 1950, resectively. The other remaining eriods do not demonstrate a strong uward or downward trend when the reference oint is the level of the DJIA at the starting date of that articular eriod. The daily returns are calculated as the log differences of the Dow level. All of the suberiods excet one show signs of skewness and all eriods show evidence of kurtosis. The subsamles in the and eriods

5 ( ) R. GençayrJournal of Emirical Finance are highly skewed and have unusually high kurtosis. The source of this letokurtic structure in the eriod originates from the eriod which contains the World War I eriod. All eriods show some evidence of autocorrelation in the first lag. The Ljung-Box-Pierce statistics are also calculated for the first 10 lags and are distributed x Ž 10. under the null of identical and indeendent observations. Twenty-one series out of 6 give strong rejection of the null hyothesis of identical and indeendent observations. 3. Methodology Let t, t s 1,,..., T be the daily Dow series. The return series are n calculated by rts log t y log ty1. Let mt denote the time t value of a n n ny1 moving average rule of length n. mt is calculated by mt s 1rn Ýis0 tyi. The buy and sell signals are calculated by s n 1,n sm n 1 ym n t t t where n1 and n are the short and the long moving averages, resectively. The rules used in this aer are Ž n, n. swž 1, 50., Ž 1, 00.x 1 where n1 and n are in days. To comare the erformance of the test regressions the linear regression rtsaq Ý birtyiqet e t;idž 0, st. 1 with lagged returns is used as the benchmark model. The test regressions are constructed such that each test regression embeds the benchmark model as a secial case. The linear test regression is n 1,n r saq b r q h s t Ý i tyi Ý i tyi qet where e ;IDŽ0, s.. In case of the GARCH-M Ž 1,1. t t rocess the test model is written as n 1,n 1r rtsaq Ýbirtyiq Ýhistyi qg ht qet 3 where e t;n 0, ht and htsd0qd1hty1qde ty1. The test regression for the single layer feedforward network model with d hidden units is written as d n 1,n t 0 Ý ij tyi Ý j ž j Ý ij tyi / t t t js1 r sa q b r q h G a q g s qe e ;ID 0, s where G is the activation function which is chosen to Ž 4. The buy and sell signal is not a discrete indicator but a continuous variable in this aer.

6 35 ( ) R. GençayrJournal of Emirical Finance GŽ x. s ya x 1qe be the logistic function. It has the roerty of being a sigmoidal 3 function. In this aer, the network architecture is a single layer feedforward network. Many authors have investigated the universal aroximation roerties of neural networks ŽGallant and White Ž 1988, 199.; Cybenko Ž 1989.; Funahashi Ž 1989.; Hecht-Nielsen Ž 1989.; Hornik et al. Ž 1989, Using a wide variety of roof strategies, all have demonstrated that under general regularity conditions, a sufficiently comlex single hidden layer feedforward network can aroximate any member of a class of functions to any desired degree of accuracy where the comlexity of a single hidden layer feedforward network is measured by the number of hidden units in the hidden layer. For an excellent survey of the feedforward and recurrent network models, the reader may refer to Kuan and White Ž The out-of-samle forecast erformance of Eqs. Ž. Ž. 4 are measured by the ratio of their mean square rediction error s Ž MSPE. to that of the linear benchmark model in Eq. Ž. 1. To differentiate the redictive ower of the ast returns, the MSPE of the GARCH-M Ž 1,1. model with lagged returns 1r t Ý i tyi t t t Ž t. t 0 1 ty1 ty1 r saq b r qg h qe e ;N 0, h, h sd qd h qd e is also comared to that of the benchmark model in Eq. Ž. 1. In addition, the out-of-samle forecast erformance of the single layer feedforward network model with lagged returns d t 0 Ý j ž j Ý ij tyi/ t t t js1 Ž 5. r sa q b G a q g r qe e ;ID 0, s Ž 6. Ž. is also studied and comared to the benchmark model in Eq. 1. The choice for the number of hidden units in a feedforward network is determined by cross-validation. 4. Out-of-samle evidence For each subsamle the out-of-samle redictive erformance of the benchmark and test models is examined. The forecast horizon is chosen to be 10 days. There 3 w x G is a sigmoidal function if G:R 0, 1, Ga 0 asa y`, Ga 1 asa ` and G is monotonic.

7 ( ) R. GençayrJournal of Emirical Finance are subsamles in equally saced intervals for the entire data set, a total of 0 observations for the out-of-samle forecasts. Each forecast horizon corresonds to either a downward, uward or a non-trending market behavior. Therefore, the results across the subsamles reveal evidence for the erformance of technical trading rules across various market conditions. Out-of-samle forecasts are comletely ex ante by using only the information actually available. Let MSPE t and MPSE b be the mean square rediction errors of the test and benchmark models, resectively. To measure the out-of-samle erformance between the test and benchmark models, the ratio of the mean square rediction errors, MSPE t rmspe b is used. MSPE t rmspe b is less than one if the test model rovides more accurate redictions. Similarly, the ratio is greater than one if the redictions of the test model are less accurate relative to the benchmark model. In this study, a 10% forecast imrovement is adoted as a benchmark measure of erformance. The ratios of mean square rediction errors which rovide 10% or more forecast imrovement are in bold in each table. The MSPE s of the benchmark model ŽEq. Ž 1.., the GARCH-M Ž 1,1. ŽEq. Ž 5.. and the feedforward network models ŽEq. Ž 6.. with ast returns are resented in Table 1. The MSPE s of the benchmark model are reorted in levels. MSPE s of the GARCH-M Ž 1,1. and the feedforward network models are reorted as a ratio to the MSPE s of the benchmark model. All three secifications are estimated with four lags, s 4, of the ast returns. The results in Table 1 indicate that the out-of-samle forecast erformance of the GARCH-M Ž 1,1. model does not outerform the benchmark model. The ratio of the MSPE s of both models are close to the unitary value excet for the eriod. The eriod rovides a 9.3 forecast imrovement over the benchmark model. The findings in Brock et al. Ž 199. show that the linear conditional mean estimators fail to characterize the temoral dynamics of the security returns and suggest the existence of nonlinearities as the data generation mechanism. Here, the single layer feedforward network regression is used as a nonarametric method to model the conditional mean returns of the stock data. The results of feedforward network regression ŽEq. Ž 6.. are resented in the last column of Table 1. Those suberiods which rovide a 10% forecast imrovement are shown in bold. In the majority of the suberiods, the feedforward network model rovides smaller MSPE s in comarison to the benchmark and the GARCH-M Ž 1,1. ŽEq. Ž 5.. models. The largest forecast imrovement occurs at the eriod which is a 1.3 forecast imrovement over the benchmark model. Overall, the results of the feedforward network regression with ast returns indicate forecast imrovement over the benchmark model and the GARCH-M Ž 1,1. ŽEq. Ž 5.. secification. The redictability of the current returns with the ast buy sell signals of the moving average rules are investigated in three different regression models. These test models are OLS ŽEq. Ž.., GARCH-M Ž 1,1. ŽEq. Ž 3.. and feedforward networks ŽEq. Ž 4... The test models are secified such that each test model is nested with the benchmark model. Each of these secifications are estimated for

8 354 ( ) R. GençayrJournal of Emirical Finance Table 1 MSPE s of the models with ast returns Date Obs. Benchmark GARCH-M Ž 1,1. Feedforward r1r1963 9r1r w0.1653x ,7 r1r r1r w0.3117x ,8 3r1r197 31r1r w0.3805x ,8 r1r r1r w0.17x ,8 r1r r1r w0.687x ,7 3r1r r06r w1.176x ,6 3r1r r1r w0.649x ,6 r1r r1r w0.390x ,7 r1r r1r w0.637x ,7 r1r r1r w0.3735x ,7 3r1r r1r w0.3897x ,8 r1r r1r w0.573x ,7 r1r r1r w0.7973x ,7 r1r r1r w0.3700x ,8 r1r193 31r1r w0.363x ,6 3r1r197 31r1r w3.8557x ,6 r1r r1r w1.1855x ,7 r1r r1r w0.3877x ,7 r1r r1r w1.4790x ,7 r1r r1r w1.0559x ,8 3r1r r1r w0.088x ,8 3r1r r1r w1.0887x ,8 Average The linear benchmark model is r s a qý b r qe. The GARCH-M Ž 1,1. t i tyi t model is rts a q 1r Ý birtyiqg ht qe t, e t; N 0, ht and htsd0qd1hty1qde ty1. The feedforward network d Ž. y4 model is rts a0qý js1 bjg a jqýgijrtyi qe t. MSPE s are =10. The MSPE s of the benchmark model ŽEq. Ž 1.. are reorted in levels. The MSPE s of the GARCH-M Ž 1,1. ŽEq. Ž 5.. and the feedforward network models ŽEq. Ž 6.. are reorted as a ratio to the MSPE s of the benchmark model. two different moving average rules. These rules are Ž n, n. swž 1, 50., Ž 1, 00.x 1 where n1 and n are in days. For convenience, these rules are referred to as A' Ž 1, 50. and B' Ž 1, 00. in the following tables. All three secifications are estimated with four lags, s 4, of the buy sell signals. In Table, the OLS regression ŽEq. Ž.. results are resented. For each rule, only the eriod rovides more than 10% forecast imrovement. These are 17.7 and 18.3% for rules A and B, resectively. Also, there is evidence of more redictability for the time eriod between The MSPE s of this time eriod are consistently lower than the benchmark model. The results in Table indicate that the ast buy sell signals in a linear conditional mean secification

9 ( ) R. GençayrJournal of Emirical Finance Table The ratio of the MSPE s of the OLS model with ast buy sell signals to the MSPE s of the benchmark model Date Obs. Rule A Rule B r1r1963 9r1r r1r r1r r1r197 31r1r r1r r1r r1r r1r r1r r06r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r193 31r1r r1r197 31r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r Average The linear benchmark model is rts a qý birtyiqe t. The linear model with buy sell signals is n 1,n r s a qý b r q` h s qe. Rules A and B refer to the Ž n, n.' wž 1, 50., Ž 1, 00.x t i tyi i tyi t 1 where n and n are in days. 1 rovide a slight forecast imrovement in comarison to the benchmark arametric model with ast returns. In Table 3, the results of the GARCH-M Ž 1,1. model ŽEq. Ž 3.. are resented. The GARCH-M Ž 1,1. model makes an imrovement over the OLS model. In general, the MSPE s of the GARCH-M Ž 1,1. model are smaller than the OLS model in Eq. Ž.. The average MSPE s of the GARCH-M Ž 1,1. model for rules A and B are 3. and.7% smaller than the MSPE s of the benchmark model. For the eriod, the GARCH-M Ž 1,1. model rovides 18.9 and 18.3% forecast imrovement over the benchmark model for rules A and B, resectively. In Table 4, the feedforward network regression ŽEq. Ž 4.. results with ast returns and ast buy sell signals are resented. The results indicate that 14 of the suberiods rovide at least 10% forecast imrovement in comarison to the

10 356 ( ) R. GençayrJournal of Emirical Finance Table 3 The ratio of the MSPE s of the GARCH-M Ž 1,1. model with ast buy sell signals to the MSPE s of the benchmark model Date Obs. Rule A Rule B r1r1963 9r1r r1r r1r r1r197 31r1r r1r r1r r1r r1r r1r r06r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r193 31r1r r1r197 31r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r r1r Average The linear benchmark model is r s a qý b r qe. The GARCH-M Ž 1,1. t i tyi t model with buy sell n 1,n 1r signals is r s a qý b r qý h s qg h qe where e ; NŽ 0, h. t i tyi i tyi t t t t and htsd0 q d h qd e. Rules A and B refer to the Ž n, n.' wž 1, 50., Ž 1, 00.x where n and n are in days. 1 ty1 ty1 1 1 benchmark model with ast returns. Esecially for the suberiod, the imrovement in the MSPE of the feedforward network model is as large as 7.9%. The forecast imrovements of the feedforward network regression with the ast buy sell signals are substantial and dominate all other secifications including the feedforward network regression with ast returns. Between the two moving average rules, rule A rovides more accurate out-of-samle redictions relative to rule B. This may be due to the fact that rule B oversmooths the data. The analysis above uses the continuous buy sell signals as regressors. To comare the erformance of the discrete buy sell signals as redictors of the security returns, an indicator variable, d n 1,n t is formed. Let 1 m n 1 ym n n,n Ž t t.)0 d 1 t s ½ Ž 7. 0 otherwise

11 ( ) R. GençayrJournal of Emirical Finance Table 4 The ratio of the MSPE s of the feedforward model with ast buy sell signals to the MSPE s of the benchmark model Date Obs. Rule A Rule B r1r1963 9r1r , ,7 r1r r1r , ,7 3r1r197 31r1r , ,9 r1r r1r , ,7 r1r r1r ,8 0.87,8 3r1r r06r , ,8 3r1r r1r ,7 1.01,8 r1r r1r , ,8 r1r r1r , ,9 r1r r1r , ,8 3r1r r1r , ,8 r1r r1r , ,7 r1r r1r , ,7 r1r r1r , ,7 r1r193 31r1r , ,8 3r1r197 31r1r , ,8 r1r r1r , ,8 r1r r1r ,8 0.83,8 r1r r1r ,8 0.88,8 r1r r1r , ,8 3r1r r1r , ,8 3r1r r1r , ,7 Average The linear benchmark model is rts a qý birtyiqe t. The feedforward network with buy sell signals is r d Ž n 1,n t s a 0 qý b ij r tyi qý js1 h j G a j qý g ij s tyi. qe t. Rules A and B refer to the Žn 1, n.' wž 1, 50., Ž 1, 00.x where n1 and n are in days. The first entry under each rule is the ratio of MSPE s. The second entry is the number of hidden units of the feedforward network model indicated by the cross-validation technique. where n1 and n are the short and long moving averages, resectively. The linear test regression is then written as n 1,n r saq b r q h d t Ý i tyi Ý i tyi qet 8 where e ; IDŽ0, s. t t. The redictive erformance of the linear models with discrete buy sell signals is within one ercent of the redictive erformance of the model in Eq. Ž.. This suggests that the redictive erformance of discrete buy sell signals are similar to that of the continuous buy sell signals. The results of this aer confirm the findings of Brock et al. Ž 199. that the linear conditional mean estimators may not successfully characterize the temoral

12 358 ( ) R. GençayrJournal of Emirical Finance dynamics of the security returns. The results show that the use of the ast buy sell signals in the linear conditional mean estimations rovides only a slight imrovement in the forecast erformances relative to the benchmark model. The main findings of this aer indicate that the nonlinearities in the stock return series lay an imortant role in modelling the conditional mean of the security returns. The forecast gains originate from the utilization of the ast buy sell signals as inuts in the nonarametric conditional mean secifications. In the feedforward network model with ast buy sell signals, the forecast gains over the benchmark model are much more ronounced. 5. Conclusions This aer has used the daily Dow Jones Industrial Average Index from 1897 to 1988 to examine the linear and nonlinear redictability of stock market returns with simle technical trading rules. Evidence of nonlinear redictability in stock market returns is found by using the ast buy and sell signals of the moving average rules. The forecast gains originate from the utilization of the ast buy sell signals as inuts in the feedforward network secifications. The evidence across subsamles indicates that the two moving average rules studied here rovide at least a 10% forecast imrovement in the volative years of the Great Deression and in the trendy years of The erformance of these rules is more moderate in the eriod in which there is no clear trend in either a ositive or negative direction. The technical trading rules used in this aer are very oular and very simle. The results here suggest that it is worthwhile to investigate more elaborate rules and the rofitability of these rules after accounting for transaction costs and brokerage fees. Acknowledgements I thank the editor, the associate editor and two anonymous referees for helful comments which imroved the results in the aer. I also thank Buz Brock, Blake LeBaron, Chung-Ming Kuan, Tung Liu for discussions and Blake LeBaron for roviding the data; Xian Yang for roviding excellent research assistance; and the Natural Sciences and Engineering Research Council of Canada and the Social Sciences and Humanities Research Council of Canada for financial suort. References Brock, W.A., Lakonishok, J., LeBaron, B., 199. Simle technical trading rules and the stochastic roerties of stock returns. Journal of Finance 47,

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