how to make a profitable trading strategy more profitable?

HOW TO MAKE A PROFITABLE TRADING STRATEGYMORE PROFITABLE?

TERENCE TAI-LEUNG CHONG*

Hong Kong Institute of Asia-Pacific StudiesThe Chinese University of Hong Kong

Shatin, N. T., Hong Kong

Department of International Economics and TradeNanjing University

Jiangsu, 210044, China*[email protected]

TAU-HING LAM

Department of EconomicsThe 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 boththe 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 SETARrule performs better in the U.S. market, while the MA rule performs better in the Chinese market. Inaddition, we find evidence that a new strategy combining the two rules together is able to createsynergy. An immediate implication of our result is that investors are able to improve the performanceof 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 filterrules fail to outperform the buy-and-hold (B–H) strategy. Brock et al. (1992) show that themoving average (MA) and the trading range break (TRB) rules can beat the B–H rule in theDow Jones index. Bessembinder and Chan (1995) show that technical trading rules areprofitable in the stock markets of Malaysia, Thailand and Taiwan. Hudson et al. (1996) andMills (1997) find that trading rules perform well in the FT30 index. Recently, there has

*Corresponding author.

The Singapore Economic Review, Vol. 58, No. 3 (2013) 1350019 (17 pages)© World Scientific Publishing CompanyDOI: 10.1142/S0217590813500197

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http://dx.doi.org/10.1142/S0217590813500197

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 theU.S. market. Chong et al. (2012) conduct similar analysis for the Chinese markets and findthat 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) andMA(50) rules.1

Our sample consists of totally ten stock market indexes of the U.S. and China, includingthe Dow Jones Industrial Average (DJIA), the NASDAQ Composite Index, the New YorkStock 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 ShanghaiComposite 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 ShenzhenStock 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 areavailable exclusively for local citizens and institutions. They are quoted in RMB and cannotbe traded by foreigners. The B shares could only be traded by foreign investors before 2001.Since February 2001, local investors can also trade the B shares via legal foreign currencyaccounts. The SHC index was launched on July 15, 1991. It consists of all stocks (A sharesand B shares) listed on the Shanghai Stock Exchange. The base day for the SHC index isDecember 19, 1990 and the base value is 100. The SZC index began on April 3, 1991, witha base price of 100. It is a market-capitalization weighted index of stocks in the ShenzhenStock 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 ofdevelopment, market efficiency, institutional setting and the variety of stocks listed. As theU.S. and China’s stock markets are respectively the largest developed and emerging stockmarkets in the world, the result obtained in this paper has important implications on theprofitability of similar rules in other markets.

To mitigate our exposure to data-mining bias, our sample includes ten different stock-market indexes. It is found that the SETAR(200) rule yields substantial returns in four majorU.S. and two Chinese B-share indexes. TheMA(50) rule, on the other hand, is more profitablein the Chinese market. We demonstrate, in almost all cases, that synergy can be achieved bycombining the MA and SETAR trading rules. An immediate implication is that investors canimprove 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 boot-strap analysis. Section 5 explores the synergy of a combined rule and concludes the paper.

1Other studies on combining technical trading rules include Fang and Xu (2003) and Lento (2009).

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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 exten-sions 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-indexreturns:

ΔYt ¼ ðα0 þ α1ΔYt�1ÞI½ΔYt�d ‚ γ� þ ðβ0 þ β1ΔYt�1ÞI½ΔYt�d < γ� þ "t, ð1Þwhere Yt denotes the natural log value of the stock index at day t, γ represents the thresholdvalue, d is the lag length and I½A� is an indicator function that equals 1 if condition A issatisfied. We employ the recursive rolling technique to obtain the SETAR one-step-aheadforecast.

The SETAR trading strategy is as follows:

Buy if Δ Y wtþ1 > 0, ð2Þ

Sell if Δ Y wtþ1 < 0, ð3Þ

where w stands for the length of the observation window and Δ Y wtþ1 refers to the predicted

return that is based upon information from the most recent w observations. In short, if thepredicted price of the next trading day is higher than the price of today, we long the index,otherwise we short it.

2.2. Moving average (MA)

The MA rule is the most widely investigated trading rule. A w-day MA is defined as:

MAtðwÞ ¼Pw

t¼1 Pt

w, ð4Þ

where Pt is the stock price at day t and w represents the bandwidth of the window. The MArule is also studied because of its popularity in the literature (Brock et al., 1992). The ideabehind computing MAs is to smooth out volatile series. When the stock price penetrates itsMA, a trend is considered to be initiated. In our case, let Yt ¼ Pt. According to the MArule, buy and sell signals are generated by the crossing of price and its MA, i.e.,

Buy if Yt >MAtðwÞ, ð5ÞSell if Yt <MAtðwÞ: ð6Þ

Therefore, if the price is higher than the MA, we long the index. Otherwise, we hold a shortposition.

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 willfocus on the SETAR(200) and MA(50) rules.

How to Make a Profitable Trading Strategy More Profitable?

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2.3. Test statistic

On each trading day, a trading signal will be generated and a position will be taken. Thedaily conditional mean and variance of buy (sell) returns can be respectively written as

�bðsÞ ¼1

NbðsÞ

XNt¼1

ΔYtþ1IbðsÞt , ð7Þ

and

�2bðsÞ ¼

1NbðsÞ

XNt¼1

ðΔYtþ1 � �bðsÞÞ2I bðsÞt , ð8Þ

where �bðsÞ is the mean return of the buy (sell) period, �2bðsÞ refers to the conditional

variance of the buy (sell) signals, NbðsÞ represents the number of buy (sell) days, N is thenumber of observations of the sample, ΔYtþ1 is the one-day return and I bðsÞt is an indicatorfunction which equals one if a buy (sell) signal is generated at time t, and equals zerootherwise. The null and alternative hypotheses are respectively

H0 : �bðsÞ ¼ �, ð9ÞH1 : �bðsÞ 6¼ �: ð10Þ

Following Brock et al. (1992), the t-ratio for the mean buy (sell) return is given as follows:

tbðsÞ ¼�bðsÞ � �

ð � 2

NbðsÞþ � 2

N Þ1=2, ð11Þ

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

H0 : �b � �s ¼ 0, ð12ÞH1 : �b � �s 6¼ 0 ð13Þ

and the t-statistic can be expressed as follows:

tðb�sÞ ¼�b � �s

ð� 2

Nbþ �2

NsÞ1=2 : ð14Þ

3. Data and Results

3.1. Data

Our data are obtained from DataStream. For comparison purposes, we use the same sampleperiod as Chong and Lam (2010) and Chong et al. (2012). The sample includes ten stockmarket 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 obser-vations 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),


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3616 daily observations of the SHB (Feb 1992 to Dec 2005), 3913 daily observations ofthe SHC (Jan 1991 to Dec 2005), 3455 daily observations of the SZA (Oct 1992 to Dec2005), 3455 daily observations of the SZB (Oct 1992 to Dec 2005) and 3848 dailyobservations of the SZC (Apr 1991 to Dec 2005). Table 1 reports the summary statistics ofthe daily return of the aforementioned indexes. Note that the returns are leptokurtic andskewed. For the Chinese stock market, the high standard deviation indicates its emergingnature.

A significant serial correlation in stock returns is a sufficient condition for the existenceof trading rule profits. The autocorrelations and the Ljung–Box Q statistics are reported inTable 1. Nine out of the ten indexes have the first-order autocorrelation larger than twicethe Bartlett asymptotic standard error band. All Ljung–Box Q statistics at the fifth lag arestatistically significant at the 1% level.

3.2. Results

Table 2 reports the estimation results of the SETAR model. The reason for choosing thefirst-order SETAR model is due to its simplicity and predictability. Note that most of theestimated coefficients are significant, suggesting that the first-order model is sufficient todescribe the dynamics of the return series.

Tables 3 and 4 report the performance of the two trading rules. Columns 2 and 3 of thetables 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 condi-tional 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. Thenumbers in parentheses are the t-ratios for the hypotheses that the buy (sell) mean isdifferent 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-statisticsfor the buy–sell return are significant. For S&P500, the SETAR trading rule produces asignificant buy–sell return of 0.1164%.

For China, except for SHA, where both the SETAR(200) and the MA(50) rules cannotproduce significant buy–sell returns, the performance of the two rules is good in all otherChinese indexes. For example, all the buy–sell differences are positive and significantlydifferent from zero in a statistical sense for SHB. The SETAR(200) rule, in particular,yields an extremely high buy–sell return of 0.4205%. For the SHC index, the MA(50) ruleyields a buy–sell return of 0.2125%. 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–sellreturn of 0.1872%. For the SZB index, all the buy–sell spreads are significantly positive.The SETAR(200) rule produces a buy–sell return of 0.3525%. For SZC, both the SETAR(200) rule and the MA(50) rule are profitable. The MA(50) rule generates a high buy–sellreturn of 0.2129%. 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 performsbetter in A-share and composite indexes.


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Table1.

Sum

maryStatisticsforDaily

Returns-FullSam

ple

DJIA

NASDAQ

NYSE

S&P500

SHA

SHB

SHC

SZA

SZB

SZC

Obs.

14348

8808

10435

10697

3651

3615

3912

3454

3454

3847

Mean

0.0003

0.0004

0.0003

0.0003

0.0004

�0.0002

0.0006

0.0000

0.0001

0.0003

Std.

0.0090

0.0120

0.0088

0.0096

0.0283

0.0214

0.0263

0.0230

0.0216

0.0237

Skew

�1.6988**

�0.3120*

�1.4806**

�1.2869**

6.1692**

0.3939**

6.2791**

1.0565**

0.3751**

0.9361**

Kurtosis

51.768**

10.787**

37.231**

36.263**

145.97**

5.9399**

155.89**

19.617**

8.0799**

18.861**

JBstat

1609043**

42843**

606495**

589048**

3264553**

5408**

3986644**

56023**

9476**

57584**

�(1)

0.0661

a0.1016

a0.1150

a0.0530

a0.0444

a0.1629

a0.0480

a0.0129

0.1494

a0.0324

a

�(2)

�0.0308a

�0.0118

�0.0117

�0.0190

0.0423

a0.0054

0.0457

a0.0340

0.0332

0.0253

�(3)

�0.0089

0.0131

�0.0093

�0.0188

0.0466

a0.0444

a0.0465

a0.0238

0.0894

a0.0459

a

�(4)

�0.0131

0.0352

a�0

.0149

�0.0212a

0.0315

0.0179

0.0314

0.0711

a0.0822

a0.0557

a

�(5)

0.0163

0.0025

0.0175

0.0092

0.0281

0.0029

0.0264

0.0115

0.0209

0.0283

Bar

std.

0.0084

0.0107

0.0098

0.0097

0.0166

0.0166

0.0160

0.0170

0.0170

0.0161

Q(5)

83.866**

104.57**

145.87**

43.384**

28.156**

104.47**

32.268**

24.474**

133.45**

29.627**

Notes:R

eturns

arecalculated

asthelogdifference

ofthestockindexlevel.“JB

stat”representstheJarque–Beratestforno

rmality.�

(i)istheestim

ated

autocorrelation

atlagi.Q(5)istheLjung–Box

Qstatisticsatlag5.

“Bar

std.”refersto

theBartlettasym

ptoticstandard

errorband

forautocorrelations.A

utocorrelatio

nsgreaterthan

twicetheBartlettasym

ptotic

standard

errorband

aremarkedwith

a .Num

bers

markedwith

*(**)aresignificantat

the5%

(1%)level.


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Table2.

Parameter

Estim

ates

fortheSETA

RModels

SETA

Rparameter

estim

ates

ΔYt¼

ðα0þα 1

ΔYt�

1ÞI½

ΔYt�

d‚γ

�þðβ

0þβ 1

ΔYt�

1ÞI½

ΔYt�

d<γ�þ" t

DJIA

NASDAQ

NYSE

S&P500

SHA

SHB

SHC

SZA

SZB

SZC

α 00.000241

0.000270

0.000725

0.000216

0.000129

�0.000262

0.000314

0.004623

�0.000072

0.000087

(3.0969)**

(2.0697)*

(2.3159)*

(2.2285)

(0.2707)

(�0.72558)

(0.7309)

(4.6316)**

(�0.1901)

(0.2228)

α 10.094023

0.182570

�0.057540

0.079776

0.085265

0.193829

0.088146

�0.104440

0.197521

0.093601

(9.0841)**

(14.5109)**

(�2.4457)*

(6.5100)**

(4.8061)**

(9.7471)**

(5.1417)**

(�3.1853)**

(10.9347)**

(5.3435)**

β 0�0

.006099

0.000665

0.000178

�0.004856

0.004485

�0.022817

0.004071

�0.000755

0.000286

0.004637

(�9.3727)**

(1.3782)

(1.9949)

(�6.2922)**

(2.1502)

(�5.1379)**

(2.1669)

(�1.7851)

(0.2195)

(2.7338)*

β 1�0

.221305

�0.086906

0.150467

�0.176640

�0.204829

�0.298339

�0.195622

0.050695

�0.136618

�0.275861

(�7.9521)**

(�4.5301)**

(14.1397)**

(�5.5182)**

(�4.6607)**

(�3.6059)**

(�4.6111)**

(2.5606)*

(�3.0552)**

(�7.0912)**

γ�0

.013417

�0.015642

0.011406

�0.013815

�0.032851

�0.032084

�0.029832

0.015075

�0.022859

�0.031380

d1

35

12

12

45

2P-value

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

Notes:The

SETA

Rmod

elsareestim

ated

byOLS.Weselect

thethresholdandthelagthat

jointly

give

thesm

allestresidu

alsum

ofsquares.

ΔY

tisthecontinuo

usly

compo

undedreturn

ondayt,disthelagleng

thandγisthethresholdvalue.Num

bers

inparenthesesaret-statisticstestingwhether

estim

ates

arestatistically

different

from

zero.“P-value”isthe50

0-simulationbo

otstrapp

edp-valuetestingthenu

llhy

pothesisof

nothresholdeffect.T

hebo

otstrapprocedureiscond

uctedin

accordance

with

Hansen(199

7)un

dertheassumptionof

homoscedastic

errors.Num

bers

markedwith

*(**

)aresign

ificantat

the5%

(1%)level.


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Table3.

EmpiricalResultsfortheSETA

R(200)Rule

Data

N(Buy)

N(Sell)

�(Buy)

�(Sell)

Buy

Sell

Buy

>0

Sell>

0Buy

–Sell

DJIA

9121

4973

0.008185

0.010392

0.000744

�0.000616

0.52593

0.48824

0.001360

(3.9635)**

(�5.9017)**

(8.5432)**

NASDAQ

5352

3202

0.010296

0.014387

0.001398

�0.001423

0.61117

0.47314

0.002822

(5.0144)**

(�7.0540)**

(10.4519)**

NYSE

6537

3646

0.007681

0.010488

0.000871

�0.000790

0.53297

0.49589

0.001661

(4.2509)**

(�6.2384)**

(9.0837)**

S&P500

6817

3616

0.008602

0.011402

0.000656

�0.000509

0.48306

0.52848

0.001164

(2.6945)*

(�4.0716)**

(5.8577)**

SHA

1567

1832

0.022226

0.024556

0.000359

�0.000016

0.48054

0.51856

0.000375

(0.2740)

(�0.2623)

(0.4641)

SHB

1328

2037

0.023281

0.019287

0.002466

�0.001739

0.47515

0.47128

0.004205

(3.7258)**

(�2.8167)**

(5.6651)**

SHC

1753

1906

0.029665

0.024597

0.001006

�0.000179

0.49914

0.51994

0.001185

(0.7790)

(�0.7455)

(1.3199)

SZA

1229

1976

0.024495

0.020125

0.001044

�0.000591

0.50610

0.51721

0.001636

(1.3672)

(�1.0066)

(2.0552)*

SZB

1457

1747

0.023896

0.019647

0.002119

�0.001405

0.46946

0.49685

0.003525

(2.8061)*

(�2.4685)*

(4.5683)**

SZC

1556

2041

0.023473

0.021886

0.001125

�0.000430

0.49293

0.52131

0.001555

(1.2800)

(�1.0818)

(2.0448)*

Notes:“N(Buy

)”and“N(Sell)”arethenu

mberof

buyandsellsign

als.“�(buy

)”and“�(sell)”arethestandard

deviations

ofbuyandsellperiod

s.“Buy

>0”

and“Sell>

0”arethefractio

nsof

buyandsellreturnsgreaterthan

zero.Num

bers

inparenthesesaret-ratio

stestingthesign

ificance

ofthemeanbuyreturn

from

theun

cond

ition

almean,

themeansellreturn

from

theun

cond

ition

almeanandthebuy–

sellspread

from

zero.N

umbersmarkedwith

*(**

)aresign

ificant

atthe5%

(1%)level.


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Table4.

EmpiricalResultsfortheMA(50)

Trading

Rule

Data

N(Buy)

N(Sell)

�(Buy)

�(Sell)

Buy

Sell

Buy

>0

Sell>

0Buy

–Sell

DJIA

8630

5469

0.00755

0.01095

0.000393

0.000056

0.50927

0.52258

0.000338

(1.0575)

(�1.4400)

(2.1635)*

NASDAQ

5231

3328

0.00954

0.01520

0.000985

�0.000671

0.59128

0.51082

0.001657

(3.0322)**

(�4.1060)**

(6.1828)**

NYSE

6276

3910

0.00735

0.01082

0.000421

0.000041

0.51291

0.53529

0.000380

(1.0313)

(�1.4025)

(2.1074)*

S&P500

6296

4152

0.00806

0.01166

0.000296

0.000181

0.47173

0.57009

0.000115

(0.2931)

(�0.3932)

(0.5947)

SHA

1586

1816

0.02347

0.02352

0.000783

�0.000387

0.49748

0.50165

0.001171

(0.8688)

(�0.8053)

(1.4496)

SHB

1493

1873

0.02236

0.01980

0.001826

�0.001596

0.48426

0.46236

0.003422

(2.9043)**

(�2.5076)*

(4.6865)**

SHC

1767

1896

0.02898

0.02526

0.001490

�0.000636

0.51952

0.49578

0.002125

(1.3957)

(�1.3389)

(2.3681)*

SZA

1454

1751

0.02220

0.02163

0.001058

�0.000813

0.51444

0.50771

0.001872

(1.4708)

(�1.3094)

(2.4075)*

SZB

1541

1664

0.02402

0.01929

0.001762

�0.001258

0.47761

0.49099

0.003020

(2.3299)*

(�2.2048)*

(3.9273)**

SZC

1678

1920

0.02291

0.02225

0.001380

�0.000748

0.51251

0.50417

0.002129

(1.6964)

(�1.5598)

(2.8197)**

Notes:“N(Buy


mberof

buyandsellsign

als.

“�(buy


deviations

ofbuyandsellperiod

s.“Buy

>0”

and“Sell>

0”arethefractio

nsof


zero.N

umbersinparenthesesaret-ratio

stestingthesign

ificance

ofthemean

buyreturn

from

theun

cond

ition

almean,

themeansellreturn

from

theun

cond

ition

almeanandthebuy–

sellspread

from

zero.N

umbersmarkedwith

*(**

)aresign

ificantat

the5%

(1%)level.


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4. Bootstrap Analysis

The significance of the trading-rule returns is also evaluated using the bootstrapped dis-tributions generated from different null models. The bootstrap is conducted as follows:First, residuals of models under the null hypothesis are drawn with replacement to generateartificial returns and prices. The trading rules are then applied to the simulated series. Themeans, standard deviations and t-statistics of the trading rule returns are recorded. Theprocedure 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 thebootstrapped p-value. We first bootstrap the random-walk model with drift:

ΔYt ¼ constantþ "t: ð15ÞThe random-walk specification is consistent with the Efficient Market Hypothesis

(EMH) that stock prices are not predictable. Apart from the random-walk model, we alsobootstrap the generalized autoregressive conditional heteroskedasticity in mean (GARCH-M) model defined as follows:

ΔYt ¼ �0 þ �1"t�1 þ �2ht þ "t, ð16Þht ¼ η0 þ η1"

2t�1

þ η2ht�1, ð17Þ"t ¼

ffiffiffiffih

t

qzt, ð18Þ

where zt � Nð0, 1Þ and ht refers to the conditional variance, which is conditionally nor-mally distributed.

The GARCH-M specification is also consistent with the EMH, where higher ex anteexpected returns are associated with higher conditional volatility. Therefore, the results ofthe GARCH-M simulations allow us to distinguish whether trading-rule returns are due totime 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 η1 and η2 estimates are significant. In

addition, eight series have a positive �2 estimate, implying that a higher expected return isrequired to compensate for the increasing risk.

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., ourconclusions 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-sta-tistics of the SETAR rule is greater than those obtained from the four actual indexes. Forthe MA(50) rule in Table 7, the p-values are also very small for the four U.S. indexes. As aresult, 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


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Table5.

Parameter

Estim

ates

fortheGARCH-M

Model

ΔYt¼

� 0þ� 1" t�1

þ� 2h tþ" t

h t¼

η 0þη 1"2 t�

1þη 2h t

�1" t

¼ffiffiffiffi h tpz t

z t�

Nð0,1Þ

DJIA

NASDAQ

NYSE

S&P500

SHA

SHB

SHC

SZA

SZB

SZC

� 00.000145

0.000484

0.000149

0.000059

�0.000333

�0.001577

0.000087

�0.000213

�0.001640

�0.000282

(1.5003)

(4.0643)**

(1.2216)

(0.3280)

(�1.2402)

(�3.7811)**

(0.3526)

(�0.6136)

(�3.7984)**

(�3.3333)**

� 10.105519

0.235868

0.144617

0.072865

�0.059600

0.152303

�0.014600

0.010405

0.146165

0.015169

(11.486)**

(21.243)**

(13.858)**

(6.9667)**

(�3.1095)**

(8.0817)**

(�0.8146)

(0.5478)

(7.4383)**

(0.8315)

� 24.905549

2.768925

5.613399

5.003145

�0.311700

3.061745

�0.450200

1.220034

3.454249

1.208765

(3.4521)**

(2.1517)

(3.0404)**

(2.1662)

(�0.9829)

(2.6189)*

(�1.3631)

(1.5540)

(2.8534)*

(2.5372)*

η 00.000001

0.000001

0.000001

0.000002

0.000020

0.000022

0.000014

0.000002

0.000047

0.000005

(7.3325)**

(7.8557)**

(6.7780)**

(4.7614)**

(7.0097)**

(7.3578)**

(6.4135)**

(3.4846)**

(9.4552)**

(4.1430)**

η 10.065409

0.115950

0.074359

0.036530

0.318100

0.196695

0.297200

0.071103

0.267521

0.085945

(17.310)**

(15.318)**

(14.139)**

(7.9108)**

(12.469)**

(12.115)**

(11.948)**

(11.364)**

(10.773)**

(12.285)**

η 20.927098

0.875968

0.913356

0.939658

0.738400

0.768936

0.760400

0.931735

0.635973

0.913266

(216.94)**

(115.67)**

(144.65)**

(98.850)**

(39.113)**

(44.494)**

(40.245)**

(167.41)**

(22.942)**

(120.31)**

Notes:T

heGARCH-M

mod

elisestim

ated

usingthemaxim

umlik

elihoo

dmetho

d.ΔY

tisthecontinuo

usly

compo

undedreturn

andh t

isthecond

ition

alvariance.T

henu

mbers


stestingwhether

estim

ates

arestatistically

differentfrom

zero.Num

bers

markedwith

*(**

)aresign

ificantat

the5%

(1%)level.


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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 condi-tional standard deviation of the buy (sell) returns.

For the Chinese market, the results are also consistent with the conventional t-test. ForSHA, the bootstrapped p-values in the “Buy–Sell” column are higher than 5%, implyingthe 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 theMA(50) rule in the SHC index, and by the SETAR(200) and the MA(50) rules in the SZAand 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 variationsin A-share and Composite indexes. However, the p-values in the columns of “�(buy)” and

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 0.0000 1.0000 0.0000 1.0000 0.0120 1.0000 0.0000 0.0000Fra>NASDAQ 0.0000 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 0.0000Fra>NYSE 0.0000 1.0000 0.0000 1.0000 0.0040 1.0000 0.0000 0.0000Fra> S&P500 0.0000 1.0000 0.0000 1.0000 0.0040 1.0000 0.0000 0.0000Fra> SHA 0.4720 0.9880 0.2700 0.7220 0.8160 0.7060 0.3120 0.2860Fra> SHB 0.0220 0.0040 0.0560 0.5640 0.9980 0.8120 0.0860 0.0860Fra> SZC 0.0340 0.5500 0.0140 0.9040 0.9180 0.9700 0.0200 0.0140Fra> SZA 0.0220 0.1380 0.0060 0.8600 0.9860 0.9760 0.0160 0.0120Fra> SZB 0.0000 0.0040 0.0000 1.0000 0.9900 1.0000 0.0000 0.0000Fra> SZC 0.0340 0.5500 0.0140 0.9040 0.9180 0.9700 0.0200 0.0140

Notes: The random-walk series are generated using the scrambled returns. The rows marked with “Fra>” referto the fraction of simulated means, standard deviations and t-statistics that are larger than those from the actualseries.

Table 7. Random-walk Bootstrap Simulation Tests for 500 Replications: MA(50)


Fra>DJIA 0.0920 1.0000 0.0120 0.9720 0.0000 0.9940 0.0100 0.0100Fra>NASDAQ 0.0000 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 0.0000Fra>NYSE 0.0780 1.0000 0.0300 0.9400 0.0000 0.9840 0.0140 0.0200Fra> S&P500 0.3400 1.0000 0.2580 0.2860 0.0000 0.7520 0.2520 0.2520Fra> SHA 0.2580 0.9400 0.0680 0.8920 0.9220 0.9260 0.1000 0.0680Fra> SHB 0.0000 0.0940 0.0000 1.0000 0.9940 1.0000 0.0000 0.0000Fra> SHC 0.0680 0.2160 0.0080 0.9760 0.4640 0.9860 0.0040 0.0080Fra> SZA 0.0280 0.6520 0.0060 0.9380 0.8500 0.9960 0.0060 0.0060Fra> SZB 0.0000 0.0040 0.0000 1.0000 1.0000 1.0000 0.0000 0.0000Fra> SZC 0.0160 0.6880 0.0020 0.9740 0.8720 0.9980 0.0020 0.0020

Notes: The random-walk series are generated using the scrambled returns. The rows marked with “Fra>” referto the fraction of simulated means, standard deviations and t-statistics that are larger than those from the actualseries.


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“�(sell)” in B-share indexes demonstrate that the simulated standard deviations of buysignals are smaller than those derived from the actual series, while the simulated standarddeviations of sell signals are higher than those generated from the actual series.

4.2. GARCH-M model

Tables 8 and 9 report the results of GARCH-M bootstrap simulations for the two tradingrules.

For the U.S. market, the small p-values obtained from the SETAR(200) rule in thecolumns of “Buy–Sell” and the “t-stat(Buy–Sell)” imply that the rule yields a substantialrisk-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-Msimulations cannot replicate the conditional volatility.

Table 9. GARCH-M Bootstrap Simulation Tests for 500 Replications: MA(50)


Fra>DJIA 0.4340 1.0000 0.3840 0.6960 0.0000 0.7180 0.3160 0.3260Fra>NASDAQ 0.0740 1.0000 0.0060 0.9980 0.0000 1.0000 0.0000 0.0000Fra>NYSE 0.4380 1.0000 0.5060 0.5400 0.0000 0.6000 0.4360 0.4440Fra> S&P500 0.6780 1.0000 0.7360 0.3140 0.0000 0.2920 0.7140 0.7180Fra> SHA 0.1000 0.9600 0.0060 0.9680 0.9400 0.9940 0.0100 0.0060Fra> SHB 0.0020 0.0980 0.0020 0.9600 0.9880 0.9980 0.0020 0.0020Fra> SHC 0.0140 0.2360 0.0040 0.9900 0.4700 0.9980 0.0020 0.0040Fra> SZA 0.0460 0.6760 0.0100 0.9280 0.8540 0.9880 0.0160 0.0100Fra> SZA 0.0460 0.6760 0.0100 0.9280 0.8540 0.9880 0.0160 0.0100Fra> SZC 0.0320 0.6780 0.0120 0.9460 0.8640 0.9920 0.0120 0.0100

Notes: The GARCH-M series are generated using estimated parameters and scrambled residuals. The rowsmarked with “Fra>” refer to the fraction of simulated means, standard deviations and t-statistics that are largerthan those from the actual series.

Table 8. GARCH-M Bootstrap Simulation Tests for 500 Replications: SETAR(200)


Fra>DJIA 0.0140 1.0000 0.0000 1.0000 0.0020 1.0000 0.0000 0.0000Fra>NASDAQ 0.9360 1.0000 0.9100 0.4380 0.0000 0.3580 0.8220 0.8200Fra>NYSE 0.1760 1.0000 0.1420 0.9960 0.0020 0.9820 0.0320 0.0380Fra> S&P500 0.1760 1.0000 0.1420 0.9960 0.0020 0.9820 0.0320 0.0380Fra> SHA 0.7260 0.9940 0.5860 0.4880 0.8620 0.3740 0.6320 0.6120Fra> SHB 0.0220 0.0040 0.0560 0.5640 0.9980 0.8120 0.0860 0.0860Fra> SHC 0.1940 0.1400 0.0500 0.8780 0.5720 0.8740 0.0800 0.0860Fra> SZA 0.0320 0.1320 0.0080 0.8840 0.9900 0.9660 0.0160 0.0140Fra> SZB 0.1040 0.0140 0.1280 0.5740 0.9860 0.7580 0.1800 0.1720Fra> SZC 0.0460 0.5100 0.0080 0.8560 0.9180 0.9600 0.0260 0.0180

Notes: The GARCH-M series are generated using estimated parameters and scrambled residuals. The rowsmarked with “Fra> ” refer to the fraction of simulated means, standard deviations and t-statistics that are largerthan those from the actual series.


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For the Chinese market, except for the case of the SETAR(200) rule in SZA and SZCindexes, all the simulated buy–sell returns and their t-ratios are generally higher than thosefrom the original series. For the MA(50) rule, the p-values in the columns of the buy–sellmean and the buy–sell t-statistic are essentially zero for all indexes. The results for thestandard deviations are analogous to those in the random-walk simulations, suggesting thatthe GARCH-M model can successfully replicate the return volatility of the two tradingrules in A-share and composite indexes. Therefore, our bootstrap results show that the tworules 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 explorethe synergy of combining these two profitable trading rules. Combining the trading rulesgenerally reduces the risk and generates fewer noisy trading signals as compared to a singlerule. We define a combined strategy as follows:

Buy if SETAR : Δ Y 200tþ1 > 0 and MA : Yt > MAtð50Þ, ð19Þ

Sell if SETAR : Δ Y 200tþ1 < 0 and MA : Yt < MAtð50Þ: 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 higherbuy–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 0.4205%, while the buy–sell return of the MA(50) rule is 0.3422%. Bothare considered high returns but are still dominated by the combined-strategy return of0.6083%. For the Shenzhen B-share market, the buy–sell return is 0.3525% for the SETAR(200) rule alone, 0.302% for the MA(50) rule alone, but 0.5317% for the combinedstrategy. For the Shenzhen B market, the return buy–sell is 0.3525% for the SETAR(200)rule alone, 0.302% for the MA(50) rule alone, but 0.5317% for the combined strategy.Even for the S&P500 case where the combined strategy does not dominate the two in-dividual strategies, the return difference is not noticeable. The buy–sell return is 0.1164%for the SETAR(200) rule alone, 0.0115% for the MA(50) rule alone, but 0.1157% for thecombined strategy. The combined strategy outperforms the SETAR(200) and MA(50)strategies 90% of the time. Our results provide empirical evidence that a combined tradingstrategy dominates pure trading strategies. A possible explanation is that a trade will not betriggered by the combined rule unless both SETAR and MA conditions are satisfied, socombining trading rules help to reduce the number of false signal and to increase profits.2

2There should be an optimal number of individual rules to be included in the combined strategy. A problem with combiningmany profitable strategies is that it is difficult for all conditions to hold in order to trigger a trade. The more conditions weimpose the more difficult for us to observe a trading signal. In the extreme, there may not be trading signal in all in the entiresample period and we cannot compute the profits of the combined rule. Therefore, combining as many profitable trading rulesas possible is certainly not the way to maximize the combined profits.


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Table10.EmpiricalResultsfortheCom

binedTrading

Strategy:

SETA

R(200)þ

MA(50)

Data

N(Buy)

N(Sell)

�(Buy)

�(Sell)

Buy

Sell

Buy

>0

Sell>

0Buy

–Sell

DJIA

6448

2793

0.00742

0.01196

0.000648

�0.000817

0.74395

0.08879

0.001464

a

(2.8356)**

(�5.7715)**

(7.1589)**

NASDAQ

3815

1788

0.00889

0.01658

0.001488

�0.002259

0.85740

0.05649

0.003746

a

(4.8687)**

(�8.2775)**

(10.817)**

NYSE

4668

2038

0.00693

0.01187

0.000702

�0.001104

0.74636

0.09814

0.001806

a

(2.7311)**

(�6.4198)**

(7.6857)**

S&P500

4742

2063

0.00794

0.01315

0.000489

�0.000669

0.69443

0.17353

0.001157

(1.4071)

(�3.9490)**

(4.5414)**

SHA

893

1140

0.02371

0.02534

0.000627

�0.000618

0.84323

0.22632

0.001245

a

(0.5253)

(�0.9708)

(1.1855)

SHB

800

1344

0.02400

0.01876

0.003505

�0.002577

0.78875

0.19866

0.006083

a

(4.3252)**

(�3.6848)**

(6.4723)**

SHC

1023

1165

0.03277

0.02567

0.002058

�0.000745

0.85533

0.21459

0.002803

a

(1.7354)

(�1.2457)

(2.4102)*

SZA

755

1277

0.02191

0.02417

0.001792

�0.001061

0.82384

0.25294

0.002853

a

(1.9770)*

(�1.5175)

(2.8360)**

SZB

905

1112

0.02502

0.01794

0.003534

�0.001782

0.75580

0.21942

0.005317

a

(4.0839)**

(�2.6074)**

(5.4609)**

SZC

935

1298

0.02402

0.02207

0.001504

�0.001377

0.82032

0.24730

0.002881

a

(1.5152)

(�2.2206)*

(2.9729)**

Notes:“N(Buy


mberof

buyandsellsign

als.“�(buy


deviations

ofbuyandsellperiod

s.“Buy

>0”

and“Sell>

0”arethefractio

nsof


zero.N

umbers


stestingthesign

ificance

ofthemeanbuyreturn

from

theun

cond

ition

almean,

themeansellreturn

from

theun

cond

ition

almeanandthebuy–

sellspread

from

zero.Num

bers

markedwith

*(**

)are

sign

ificantat

the5%

(1%)level.Buy–sellreturnsmarkedwith

agive

ahigh

erbuy–

sellspread

than

theSETA

R(200

)andMA

(50)

rules.


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An immediate implication of our finding is that investors are able to improve the perfor-mance of their portfolios by combining the existing profitable trading rules. Note that thecombined rule has synergy in both China’s and the U.S. market, thus our result applies toboth 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 isbased on trading rules that are already profitable. If a market is very efficient, and notrading rules can beat it, we do not suggest that a profitable trading rule can be constructedby combining unprofitable rules to an efficient market inefficient.

Acknowledgment

We would like to thank Carrella Ernesto and Lumpkin Mcspadden for their able researchassistance.

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 andEconometrics, 7(3), Article 4.

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