HOW TO MAKE A PROFITABLE TRADING STRATEGY MORE PROFITABLE?

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1 The Singapore Economic Review, Vol. 58, No. 3 (2013) (17 pages) World Scientific Publishing Company DOI: /S HOW TO MAKE A PROFITABLE TRADING STRATEGY MORE PROFITABLE? TERENCE TAI-LEUNG CHONG * Hong Kong Institute of Asia-Pacific Studies The Chinese University of Hong Kong Shatin, N. T., Hong Kong Department of International Economics and Trade Nanjing University Jiangsu, , China * chong2064@cuhk.edu.hk TAU-HING LAM Department of Economics The Chinese University of Hong Kong Shatin, N. T., Hong Kong Published 28 August 2013 Chong and Lam and Chong et al. show that SETAR(200) and MA(50) outperform other rules in both the U.S. and the Chinese stock market. This paper investigates the synergy of combining SETAR (200) and MA(50) rules in ten U.S. and Chinese stock market indexes. It is found that the SETAR rule performs better in the U.S. market, while the MA rule performs better in the Chinese market. In addition, we find evidence that a new strategy combining the two rules together is able to create synergy. An immediate implication of our result is that investors are able to improve the performance of their portfolios by combining existing profitable trading rules. Keywords: SETAR model; bootstrap; GARCH-M model; combined strategy; market efficiency. JEL Classifications: C22, G10, G12 1. Introduction The performance of technical trading strategies has long been examined in the literature. For example, Fama and Blume (1966) and Jensen and Bennington (1970) show that filter rules fail to outperform the buy-and-hold (B H) strategy. Brock et al. (1992) show that the moving average (MA) and the trading range break (TRB) rules can beat the B H rule in the Dow Jones index. Bessembinder and Chan (1995) show that technical trading rules are profitable in the stock markets of Malaysia, Thailand and Taiwan. Hudson et al. (1996) and Mills (1997) find that trading rules perform well in the FT30 index. Recently, there has * Corresponding author

2 The Singapore Economic Review been growing interest in nonlinear trading rules (Fernández-Rodríguez et al., 2003; Andrada-Félix et al., 2003; Nam et al., 2005; Pérez-Rodríguez et al., 2005). However, most of the aforementioned studies focus on the performance of a given set of trading rules. Chong and Lam (2010) show that SETAR(200) and MA(50) outperform other rules in the U.S. market. Chong et al. (2012) conduct similar analysis for the Chinese markets and find that most rules fail to produce significant returns, except for the SETAR(200) and MA(50) models during the pre-soe reform period. Based on the results of Chong and Lam (2010) and Chong et al. (2012), this paper investigates the synergy of combining SETAR(200) and MA(50) rules. 1 Our sample consists of totally ten stock market indexes of the U.S. and China, including the Dow Jones Industrial Average (DJIA), the NASDAQ Composite Index, the New York Stock Exchange Composite Index (NYSE), the Standard and Poor s 500 Index (S&P500), the Shanghai A-share Index (SHA), the Shanghai B-share Index (SHB), the Shanghai Composite Index (SHC), the Shenzhen A-share Index (SZA), the Shenzhen B-share Index (SZB) and the Shenzhen Composite Index (SZC) in order to draw robust conclusions. Compared to the U.S. market, the Chinese stock market has a much shorter history. There are two stock exchanges in China. The Shanghai Stock Exchange and the Shenzhen Stock Exchanges were launched on November 26, 1990 and April 11, 1991 respectively. Two types of shares are traded, namely, A shares and B shares. Tradable A-shares are available exclusively for local citizens and institutions. They are quoted in RMB and cannot be traded by foreigners. The B shares could only be traded by foreign investors before Since February 2001, local investors can also trade the B shares via legal foreign currency accounts. The SHC index was launched on July 15, It consists of all stocks (A shares and B shares) listed on the Shanghai Stock Exchange. The base day for the SHC index is December 19, 1990 and the base value is 100. The SZC index began on April 3, 1991, with a base price of 100. It is a market-capitalization weighted index of stocks in the Shenzhen Stock Exchange which tracks the daily price movements of all the shares in the exchange. The U.S. and Chinese stock markets are very different in terms of the size, stage of development, market efficiency, institutional setting and the variety of stocks listed. As the U.S. and China s stock markets are respectively the largest developed and emerging stock markets in the world, the result obtained in this paper has important implications on the profitability of similar rules in other markets. To mitigate our exposure to data-mining bias, our sample includes ten different stockmarket indexes. It is found that the SETAR(200) rule yields substantial returns in four major U.S. and two Chinese B-share indexes. The MA(50) rule, on the other hand, is more profitable in the Chinese market. We demonstrate, in almost all cases, that synergy can be achieved by combining the MA and SETAR trading rules. An immediate implication is that investors can improve the performance of their portfolios by combining the existing profitable trading rules. The rest of this paper is organized as follows: Section 2 presents the methodology. Section 3 discusses the data and reports the empirical results. Section 4 conducts a bootstrap analysis. Section 5 explores the synergy of a combined rule and concludes the paper. 1 Other studies on combining technical trading rules include Fang and Xu (2003) and Lento (2009)

3 How to Make a Profitable Trading Strategy More Profitable? 2. Methodology 2.1. Self-exciting threshold autoregressive (SETAR) model The self-exciting threshold autoregressive (SETAR) model was first proposed by Tong (1978) and further elaborated by Tong and Lim (1980) and Tong (1983). Further extensions of the model include Chen and Tsay (1993) and Astatkie et al. (1997). Recently, Chong et al. (2008) apply the model to predict currency crises. Chong and Lam (2010) show that trading rules based on the SETAR model are profitable in the U.S. stock market. In this paper, we consider a simple two-regime first-order SETAR model for stock-index returns: ΔY t ¼ðα 0 þ α 1 ΔY t 1 ÞI½ΔY t d γšþðβ 0 þ β 1 ΔY t 1 ÞI½ΔY t d < γšþ" t, ð1þ where Y t denotes the natural log value of the stock index at day t, γ represents the threshold value, d is the lag length and I½AŠ is an indicator function that equals 1 if condition A is satisfied. We employ the recursive rolling technique to obtain the SETAR one-step-ahead forecast. The SETAR trading strategy is as follows: Buy if Δ ^Y tþ1 w > 0, ð2þ Sell if Δ ^Y tþ1 w < 0, ð3þ where w stands for the length of the observation window and Δ ^Y tþ1 w refers to the predicted return that is based upon information from the most recent w observations. In short, if the predicted price of the next trading day is higher than the price of today, we long the index, otherwise we short it Moving average (MA) The MA rule is the most widely investigated trading rule. A w-day MA is defined as: P wt¼1 P MA t ðwþ ¼ t, ð4þ w where P t is the stock price at day t and w represents the bandwidth of the window. The MA rule is also studied because of its popularity in the literature (Brock et al., 1992). The idea behind computing MAs is to smooth out volatile series. When the stock price penetrates its MA, a trend is considered to be initiated. In our case, let Y t ¼ P t. According to the MA rule, buy and sell signals are generated by the crossing of price and its MA, i.e., Buy if Y t > MA t ðwþ, Sell if Y t < MA t ðwþ: ð5þ ð6þ Therefore, if the price is higher than the MA, we long the index. Otherwise, we hold a short position. Chong and Lam (2010) and Chong et al. (2012) show that SETAR(200) and MA(50) outperform other rules in the U.S. and the Chinese stock markets. In this paper, we will focus on the SETAR(200) and MA(50) rules

4 The Singapore Economic Review 2.3. Test statistic On each trading day, a trading signal will be generated and a position will be taken. The daily conditional mean and variance of buy (sell) returns can be respectively written as and bðsþ ¼ 1 N bðsþ X N 2 bðsþ ¼ 1 N bðsþ X N t¼1 t¼1 ΔY tþ1 I bðsþ t, ð7þ ðδy tþ1 bðs ÞÞ 2 I bðsþ t, ð8þ where bðsþ is the mean return of the buy (sell) period, 2 bðsþ refers to the conditional variance of the buy (sell) signals, N bðsþ represents the number of buy (sell) days, N is the number of observations of the sample, ΔY tþ1 is the one-day return and I bðsþ t is an indicator function which equals one if a buy (sell) signal is generated at time t, and equals zero otherwise. The null and alternative hypotheses are respectively H 0 : bðsþ ¼, ð9þ H 1 : bðsþ 6¼ : ð10þ Following Brock et al. (1992), the t-ratio for the mean buy (sell) return is given as follows: t bðsþ ¼ bðsþ ð 2 N bðsþ þ, ð11þ 2 N Þ1=2 where is the unconditional daily mean and 2 is the unconditional variance. Next, we evaluate the significance of the buy sell spread, which represents the return of an average complete transaction. The null and alternative hypotheses are H 0 : b s ¼ 0, H 1 : b s 6¼ 0 and the t-statistic can be expressed as follows: ð12þ ð13þ t ðb sþ ¼ b s ð 2 N b þ 2 N s Þ : 1=2 ð14þ 3. Data and Results 3.1. Data Our data are obtained from DataStream. For comparison purposes, we use the same sample period as Chong and Lam (2010) and Chong et al. (2012). The sample includes ten stock market indexes, including 14,348 daily observations of the DJIA (Jan 1951 to Dec 2005), 8809 daily observations of the NASDAQ (Feb 1971 to Dec 2005), 10,436 daily observations of the NYSE (Dec 1965 to Dec 2005), 10,698 daily observations of the S&P500 (Dec 1964 to Dec 2005), 3652 daily observations of the SHA (Jan 1992 to Dec 2005),

5 How to Make a Profitable Trading Strategy More Profitable? 3616 daily observations of the SHB (Feb 1992 to Dec 2005), 3913 daily observations of the SHC (Jan 1991 to Dec 2005), 3455 daily observations of the SZA (Oct 1992 to Dec 2005), 3455 daily observations of the SZB (Oct 1992 to Dec 2005) and 3848 daily observations of the SZC (Apr 1991 to Dec 2005). Table 1 reports the summary statistics of the daily return of the aforementioned indexes. Note that the returns are leptokurtic and skewed. For the Chinese stock market, the high standard deviation indicates its emerging nature. A significant serial correlation in stock returns is a sufficient condition for the existence of trading rule profits. The autocorrelations and the Ljung Box Q statistics are reported in Table 1. Nine out of the ten indexes have the first-order autocorrelation larger than twice the Bartlett asymptotic standard error band. All Ljung Box Q statistics at the fifth lag are statistically significant at the 1% level Results Table 2 reports the estimation results of the SETAR model. The reason for choosing the first-order SETAR model is due to its simplicity and predictability. Note that most of the estimated coefficients are significant, suggesting that the first-order model is sufficient to describe the dynamics of the return series. Tables 3 and 4 report the performance of the two trading rules. Columns 2 and 3 of the tables labeled with N(Buy) and B(Sell) show the number of buy and sell signals. Columns 6, 7 and 10 marked with Buy, Sell and Buy Sell show the daily conditional mean for buy, sell and buy sell returns. Columns 8 and 9 marked with Buy > 0 and Sell > 0 are the fraction of buy and sell signals that produce positive returns. The numbers in parentheses are the t-ratios for the hypotheses that the buy (sell) mean is different from the unconditional mean and that the buy sell spread is different from zero. Both trading rules perform reasonably well in the U.S. market. For DJIA, the SETAR (200) rule produces a buy sell return of 0.136%. For NASDAQ and NYSE, the t-statistics for the buy sell return are significant. For S&P500, the SETAR trading rule produces a significant buy sell return of %. For China, except for SHA, where both the SETAR(200) and the MA(50) rules cannot produce significant buy sell returns, the performance of the two rules is good in all other Chinese indexes. For example, all the buy sell differences are positive and significantly different from zero in a statistical sense for SHB. The SETAR(200) rule, in particular, yields an extremely high buy sell return of %. For the SHC index, the MA(50) rule yields a buy sell return of %. For the SZA index, both the SETAR(200) and the MA (50) rules generate a significant buy sell spread. The MA(50) rule produces a buy sell return of %. For the SZB index, all the buy sell spreads are significantly positive. The SETAR(200) rule produces a buy sell return of %. For SZC, both the SETAR (200) rule and the MA(50) rule are profitable. The MA(50) rule generates a high buy sell return of %. Overall, the two trading rules perform well in the Chinese stock market. The SETAR(200) rule performs better in B-share indexes, while the MA(50) rule performs better in A-share and composite indexes

6 The Singapore Economic Review Table 1. Summary Statistics for Daily Returns-Full Sample DJIA NASDAQ NYSE S&P500 SHA SHB SHC SZA SZB SZC Obs Mean Std Skew ** * ** ** ** ** ** ** ** ** Kurtosis ** ** ** ** ** ** ** ** ** ** JB stat ** 42843** ** ** ** 5408** ** 56023** 9476** 57584** (1) a a a a a a a a a (2) a a a (3) a a a a a (4) a a a a a (5) Bar std Q (5) ** ** ** ** ** ** ** ** ** ** Notes: Returns are calculated as the log difference of the stock index level. JB stat represents the Jarque Bera test for normality. (i) is the estimated autocorrelation at lag i. Q(5) is the Ljung Box Q statistics at lag 5. Bar std. refers to the Bartlett asymptotic standard error band for autocorrelations. Autocorrelations greater than twice the Bartlett asymptotic standard error band are marked with a. Numbers marked with *(**) are significant at the 5%(1%) level

7 How to Make a Profitable Trading Strategy More Profitable? Table 2. Parameter Estimates for the SETAR Models SETAR parameter estimates ΔY t ¼ðα 0 þ α 1 ΔY t 1 ÞI½ΔY t d γšþðβ 0 þ β 1 ΔY t 1 ÞI½ΔY t d <γšþ" t DJIA NASDAQ NYSE S&P500 SHA SHB SHC SZA SZB SZC α (3.0969)** (2.0697)* (2.3159)* (2.2285) (0.2707) ( ) (0.7309) (4.6316)** ( ) (0.2228) α (9.0841)** ( )** ( )* (6.5100)** (4.8061)** (9.7471)** (5.1417)** ( )** ( )** (5.3435)** β ( )** (1.3782) (1.9949) ( )** (2.1502) ( )** (2.1669) ( ) (0.2195) (2.7338)* β ( )** ( )** ( )** ( )** ( )** ( )** ( )** (2.5606)* ( )** ( )** γ d P-value Notes: The SETAR models are estimated by OLS. We select the threshold and the lag that jointly give the smallest residual sum of squares. ΔY t is the continuously compounded return on day t, d is the lag length and γ is the threshold value. Numbers in parentheses are t-statistics testing whether estimates are statistically different from zero. P-value is the 500-simulation bootstrapped p-value testing the null hypothesis of no threshold effect. The bootstrap procedure is conducted in accordance with Hansen (1997) under the assumption of homoscedastic errors. Numbers marked with *(**) are significant at the 5%(1%) level

8 The Singapore Economic Review Table 3. Empirical Results for the SETAR(200) Rule Data N(Buy) N(Sell) (Buy) (Sell) Buy Sell Buy > 0 Sell > 0 Buy Sell DJIA (3.9635)** ( )** (8.5432)** NASDAQ (5.0144)** ( )** ( )** NYSE (4.2509)** ( )** (9.0837)** S&P (2.6945)* ( )** (5.8577)** SHA (0.2740) ( ) (0.4641) SHB (3.7258)** ( )** (5.6651)** SHC (0.7790) ( ) (1.3199) SZA (1.3672) ( ) (2.0552)* SZB (2.8061)* ( )* (4.5683)** SZC (1.2800) ( ) (2.0448)* Notes: N(Buy) and N(Sell) are the number of buy and sell signals. (buy) and (sell) are the standard deviations of buy and sell periods. Buy > 0 and Sell > 0 are the fractions of buy and sell returns greater than zero. Numbers in parentheses are t-ratios testing the significance of the mean buy return from the unconditional mean, the mean sell return from the unconditional mean and the buy sell spread from zero. Numbers marked with *(**) are significant at the 5%(1%) level

9 How to Make a Profitable Trading Strategy More Profitable? Table 4. Empirical Results for the MA(50) Trading Rule Data N(Buy) N(Sell) (Buy) (Sell) Buy Sell Buy > 0 Sell > 0 Buy Sell DJIA (1.0575) ( ) (2.1635)* NASDAQ (3.0322)** ( )** (6.1828)** NYSE (1.0313) ( ) (2.1074)* S&P (0.2931) ( ) (0.5947) SHA (0.8688) ( ) (1.4496) SHB (2.9043)** ( )* (4.6865)** SHC (1.3957) ( ) (2.3681)* SZA (1.4708) ( ) (2.4075)* SZB (2.3299)* ( )* (3.9273)** SZC (1.6964) ( ) (2.8197)** Notes: N(Buy) and N(Sell) are the number of buy and sell signals. (buy) and (sell) are the standard deviations of buy and sell periods. Buy > 0 and Sell > 0 are the fractions of buy and sell returns greater than zero. Numbers in parentheses are t-ratios testing the significance of the mean buy return from the unconditional mean, the mean sell return from the unconditional mean and the buy sell spread from zero. Numbers marked with *(**) are significant at the 5%(1%) level

10 The Singapore Economic Review 4. Bootstrap Analysis The significance of the trading-rule returns is also evaluated using the bootstrapped distributions generated from different null models. The bootstrap is conducted as follows: First, residuals of models under the null hypothesis are drawn with replacement to generate artificial returns and prices. The trading rules are then applied to the simulated series. The means, standard deviations and t-statistics of the trading rule returns are recorded. The procedure is repeated for 500 times to provide a good approximation of the estimators. The proportion of the simulated values larger than those from the actual series gives the bootstrapped p-value. We first bootstrap the random-walk model with drift: ΔY t ¼ constant þ " t : The random-walk specification is consistent with the Efficient Market Hypothesis (EMH) that stock prices are not predictable. Apart from the random-walk model, we also bootstrap the generalized autoregressive conditional heteroskedasticity in mean (GARCH- M) model defined as follows: ΔY t ¼ 0 þ 1 " t 1 þ 2 h t þ " t, h t ¼ η 0 þ η 1 " 2 þ η t 1 2h t 1, qffiffiffiffi " t ¼ z t, h t where z t Nð0, 1Þ and h t refers to the conditional variance, which is conditionally normally distributed. The GARCH-M specification is also consistent with the EMH, where higher ex ante expected returns are associated with higher conditional volatility. Therefore, the results of the GARCH-M simulations allow us to distinguish whether trading-rule returns are due to time varying risk-return equilibrium or market inefficiency. Table 5 reports the estimation results of the GARCH-M model. For the conditional variance equation, all the η 1 and η 2 estimates are significant. In addition, eight series have a positive 2 estimate, implying that a higher expected return is required to compensate for the increasing risk. ð15þ ð16þ ð17þ ð18þ 4.1. Random-walk model The random-walk bootstrap results are reported in Tables 6 and 7. The figures reported in the tables are the fractions of simulated values that are larger than those derived from the actual observations. In Table 6, for the case of the U.S., our conclusions are similar to those obtained from the conventional t-test. For the SETAR(200) rule, the p-values are all zeros for buy sell spreads and the simulated buy sell t-statistics, indicating that none of the simulated buy sell spreads and the simulated buy sell t-statistics of the SETAR rule is greater than those obtained from the four actual indexes. For the MA(50) rule in Table 7, the p-values are also very small for the four U.S. indexes. As a result, we conclude that the SETAR(200) and MA(50) rules are profitable in the U.S. market. Observe from the values of standard deviations that the random-walk simulations

11 How to Make a Profitable Trading Strategy More Profitable? Table 5. Parameter Estimates for the GARCH-M Model ΔY t ¼ 0 þ 1 " t 1 þ 2 h t þ " t h t ¼ η 0 þ η 1 " 2 t 1 þ η 2 h t 1 p " t ¼ ffiffiffiffi h t zt z t Nð0, 1Þ DJIA NASDAQ NYSE S&P500 SHA SHB SHC SZA SZB SZC (1.5003) (4.0643)** (1.2216) (0.3280) ( ) ( )** (0.3526) ( ) ( )** ( )** (11.486)** (21.243)** (13.858)** (6.9667)** ( )** (8.0817)** ( ) (0.5478) (7.4383)** (0.8315) (3.4521)** (2.1517) (3.0404)** (2.1662) ( ) (2.6189)* ( ) (1.5540) (2.8534)* (2.5372)* η (7.3325)** (7.8557)** (6.7780)** (4.7614)** (7.0097)** (7.3578)** (6.4135)** (3.4846)** (9.4552)** (4.1430)** η (17.310)** (15.318)** (14.139)** (7.9108)** (12.469)** (12.115)** (11.948)** (11.364)** (10.773)** (12.285)** η (216.94)** (115.67)** (144.65)** (98.850)** (39.113)** (44.494)** (40.245)** (167.41)** (22.942)** (120.31)** Notes: The GARCH-M model is estimated using the maximum likelihood method. ΔY t is the continuously compounded return and h t is the conditional variance. The numbers in parentheses are t-ratios testing whether estimates are statistically different from zero. Numbers marked with *(**) are significant at the 5%(1%) level

12 The Singapore Economic Review Table 6. Random-walk Bootstrap Simulation Tests for 500 Replications: SETAR(200) Result Buy (Buy) t-stat(buy) Sell (Sell) t-stat(sell) Buy Sell t-stat(buy Sell) Fra > DJIA Fra > NASDAQ Fra > NYSE Fra > S&P Fra > SHA Fra > SHB Fra > SZC Fra > SZA Fra > SZB Fra > SZC Notes: The random-walk series are generated using the scrambled returns. The rows marked with Fra> refer to the fraction of simulated means, standard deviations and t-statistics that are larger than those from the actual series. fail to replicate the volatility of the two trading rules. The p-values in the (buy) and (sell) columns demonstrate that the model overestimates (underestimates) the conditional standard deviation of the buy (sell) returns. For the Chinese market, the results are also consistent with the conventional t-test. For SHA, the bootstrapped p-values in the Buy Sell column are higher than 5%, implying the failure of the trading rules. For the two B-share indexes, the p-values are close to zero, indicating the presence of abnormal returns. Lastly, significant returns are obtained by the MA(50) rule in the SHC index, and by the SETAR(200) and the MA(50) rules in the SZA and the SZC indexes. For the conditional standard deviations, the fractions in the columns of (buy) and the (sell) suggest that the random-walk model is able to replicate the conditional variations in A-share and Composite indexes. However, the p-values in the columns of (buy) and Table 7. Random-walk Bootstrap Simulation Tests for 500 Replications: MA(50) Result Buy (Buy) t-stat(buy) Sell (Sell) t-stat(sell) Buy Sell t-stat(buy Sell) Fra > DJIA Fra > NASDAQ Fra > NYSE Fra > S&P Fra > SHA Fra > SHB Fra > SHC Fra > SZA Fra > SZB Fra > SZC Notes: The random-walk series are generated using the scrambled returns. The rows marked with Fra> refer to the fraction of simulated means, standard deviations and t-statistics that are larger than those from the actual series

13 How to Make a Profitable Trading Strategy More Profitable? Table 8. GARCH-M Bootstrap Simulation Tests for 500 Replications: SETAR(200) Result Buy (Buy) t-stat(buy) Sell (Sell) t-stat(sell) Buy Sell t-stat(buy Sell) Fra > DJIA Fra > NASDAQ Fra > NYSE Fra > S&P Fra > SHA Fra > SHB Fra > SHC Fra > SZA Fra > SZB Fra > SZC Notes: The GARCH-M series are generated using estimated parameters and scrambled residuals. The rows marked with Fra> refer to the fraction of simulated means, standard deviations and t-statistics that are larger than those from the actual series. (sell) in B-share indexes demonstrate that the simulated standard deviations of buy signals are smaller than those derived from the actual series, while the simulated standard deviations of sell signals are higher than those generated from the actual series GARCH-M model Tables 8 and 9 report the results of GARCH-M bootstrap simulations for the two trading rules. For the U.S. market, the small p-values obtained from the SETAR(200) rule in the columns of Buy Sell and the t-stat(buy Sell) imply that the rule yields a substantial risk-adjusted return. For the conditional standard deviations, the p-values in the columns of (buy) and the (sell) are 1.00 and 0.00 respectively, implying that the GARCH-M simulations cannot replicate the conditional volatility. Table 9. GARCH-M Bootstrap Simulation Tests for 500 Replications: MA(50) Result Buy (Buy) t-stat(buy) Sell (Sell) t-stat(sell) Buy Sell t-stat(buy Sell) Fra > DJIA Fra > NASDAQ Fra > NYSE Fra > S&P Fra > SHA Fra > SHB Fra > SHC Fra > SZA Fra > SZA Fra > SZC Notes: The GARCH-M series are generated using estimated parameters and scrambled residuals. The rows marked with Fra> refer to the fraction of simulated means, standard deviations and t-statistics that are larger than those from the actual series

14 The Singapore Economic Review For the Chinese market, except for the case of the SETAR(200) rule in SZA and SZC indexes, all the simulated buy sell returns and their t-ratios are generally higher than those from the original series. For the MA(50) rule, the p-values in the columns of the buy sell mean and the buy sell t-statistic are essentially zero for all indexes. The results for the standard deviations are analogous to those in the random-walk simulations, suggesting that the GARCH-M model can successfully replicate the return volatility of the two trading rules in A-share and composite indexes. Therefore, our bootstrap results show that the two rules perform quite well. 5. The Combined Strategy and Concluding Remarks The success of the SETAR(200) and MA(50) trading rules sparks our interest to explore the synergy of combining these two profitable trading rules. Combining the trading rules generally reduces the risk and generates fewer noisy trading signals as compared to a single rule. We define a combined strategy as follows: Buy if SETAR : Δ ^Y tþ1 200 > 0 and MA: Y t > MA t ð50þ, ð19þ Sell if SETAR : Δ ^Y tþ1 200 < 0 and MA: Y t < MA t ð50þ: ð20þ The performance of the new strategy, as seen from Table 10, is encouraging. Significant buy sell returns are obtained in nine out of the ten indexes. In comparison to Tables 3 and 4, the combined strategy results in fewer transactions and yields a higher buy sell return than the two individual rules. It outperforms the individual SETAR(200) and MA(50) strategies in all the six China s indexes and in three out of the four U.S. indexes. For example, for the Shanghai B-share market, the buy sell return of the SETAR (200) rule is %, while the buy sell return of the MA(50) rule is %. Both are considered high returns but are still dominated by the combined-strategy return of %. For the Shenzhen B-share market, the buy sell return is % for the SETAR (200) rule alone, 0.302% for the MA(50) rule alone, but % for the combined strategy. For the Shenzhen B market, the return buy sell is % for the SETAR(200) rule alone, 0.302% for the MA(50) rule alone, but % for the combined strategy. Even for the S&P500 case where the combined strategy does not dominate the two individual strategies, the return difference is not noticeable. The buy sell return is % for the SETAR(200) rule alone, % for the MA(50) rule alone, but % for the combined strategy. The combined strategy outperforms the SETAR(200) and MA(50) strategies 90% of the time. Our results provide empirical evidence that a combined trading strategy dominates pure trading strategies. A possible explanation is that a trade will not be triggered by the combined rule unless both SETAR and MA conditions are satisfied, so combining trading rules help to reduce the number of false signal and to increase profits. 2 2 There should be an optimal number of individual rules to be included in the combined strategy. A problem with combining many profitable strategies is that it is difficult for all conditions to hold in order to trigger a trade. The more conditions we impose the more difficult for us to observe a trading signal. In the extreme, there may not be trading signal in all in the entire sample period and we cannot compute the profits of the combined rule. Therefore, combining as many profitable trading rules as possible is certainly not the way to maximize the combined profits

15 How to Make a Profitable Trading Strategy More Profitable? Table 10. Empirical Results for the Combined Trading Strategy: SETAR(200) þ MA(50) Data N(Buy) N(Sell) (Buy) (Sell) Buy Sell Buy > 0 Sell > 0 Buy Sell DJIA a (2.8356)** ( )** (7.1589)** NASDAQ a (4.8687)** ( )** (10.817)** NYSE a (2.7311)** ( )** (7.6857)** S&P (1.4071) ( )** (4.5414)** SHA a (0.5253) ( ) (1.1855) SHB a (4.3252)** ( )** (6.4723)** SHC a (1.7354) ( ) (2.4102)* SZA a (1.9770)* ( ) (2.8360)** SZB a (4.0839)** ( )** (5.4609)** SZC a (1.5152) ( )* (2.9729)** Notes: N(Buy) and N(Sell) are the number of buy and sell signals. (buy) and (sell) are the standard deviations of buy and sell periods. Buy>0 and Sell>0 are the fractions of buy and sell returns greater than zero. Numbers in parentheses are t-ratios testing the significance of the mean buy return from the unconditional mean, the mean sell return from the unconditional mean and the buy sell spread from zero. Numbers marked with *(**) are significant at the 5%(1%) level. Buy sell returns marked with a give a higher buy sell spread than the SETAR(200) and MA (50) rules

16 The Singapore Economic Review An immediate implication of our finding is that investors are able to improve the performance of their portfolios by combining the existing profitable trading rules. Note that the combined rule has synergy in both China s and the U.S. market, thus our result applies to both developed and emerging stock markets with different degrees of market efficiency. Note also that our result is still consistent with the EMH since our combined strategy is based on trading rules that are already profitable. If a market is very efficient, and no trading rules can beat it, we do not suggest that a profitable trading rule can be constructed by combining unprofitable rules to an efficient market inefficient. Acknowledgment We would like to thank Carrella Ernesto and Lumpkin Mcspadden for their able research assistance. References Andrada-Félix, J, F Fernández-Rodríguez, MD García-Artiles and S Sosvilla-Rivero (2003). An empirical evaluation of non-linear trading rules. Studies in Nonlinear Dynamics and Econometrics, 7(3), Article 4. Astatkie, T, DG Watts and WE Watt (1997). Nested threshold autoregressive (NeTAR) models. International Journal of Forecasting, 13, Bessembinder, H and K Chan (1995). The profitability of technical trading rules in the Asian stock markets. Pacific-Basin Finance Journal, 3, Brock, W, J Lakonishok and B Lebaron (1992). Simple technical trading rules and stochastic properties of stock returns. Journal of Finance, 47, Chen, R and RS Tsay (1993). Functional-coefficient autoregressive models. Journal of the American Statistical Association, 88, Chong, TTL and TH Lam (2010). Predictability of nonlinear trading rules in the U.S. stock market. Quantitative Finance, 10(9), Chong, TTL, Q He and M Hinich (2008). The nonlinear dynamics of foreign reserves and currency crises. Studies in Nonlinear Dynamics and Econometrics, 12(2), Article 2. Chong, TTL, TH Lam and I Yan (2012). Is the Chinese stock market really inefficient? China Economic Review, 23(1), Fama, EF and ME Blume (1966). Filter rules and stock-market trading. The Journal of Business, 39, Fang Y and D Xu (2003). The predictability of asset returns: An approach combining technical analysis and time series forecasts. International Journal of Forecasting, 19, Fernández-Rodríguez, F, S Sosvilla-Rivero and J Andrada-Félix (2003). Technical analysis in foreign exchange markets: Evidence from the EMS. Applied Financial Economics, 13, Hansen, BE (1997). Inference in TAR models. Studies in Nonlinear Dynamics and Econometrics,2, Hudson, R, M Dempsey and K Keasey (1996). A note on the weak form efficiency of capital markets: The application of simple technical trading rules to UK stock prices 1935 to Journal of Banking and Finance, 20, Jensen, MC and GA Bennington (1970). Random walks and technical theories: Some additional evidence. Journal of Finance, 25, Lento, C (2009). Combined signal approach: Evidence from the Asian-Pacific equity markets. Applied Economics Letters, 16(7),

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