2009 Oxford Business & Economics Conference Program
ISBN : 978-0-9742114-1-1
Technical Analysis: A Trading Strategy based on Callable Bull and
Bear Contracts in a Highly Fluctuating Market
TSZ-KIN KWOK
Research Student
Department of Management Sciences
City University of Hong Kong
Tat Chee Avenue, Kowloon, Hong Kong
Tel: 852-98087159
mstkkwok@cityu.edu.hk
ALAN T. K. WAN
Associate Professor
Department of Management Sciences
City University of Hong Kong
Tat Chee Avenue, Kowloon, Hong Kong
Tel: 852- 27887146
msawan@cityu.edu.hk
June 24-26, 2009
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Technical Analysis: A Trading Strategy based on Callable Bull and Bear Contracts in a Highly
Fluctuating Market
ABSTRACT
The Callable Bull and Bear Contracts (CBBCs) play an important role in today’s fluctuating Hong
Kong stock market. Recently, Cheung, Cheung, He and Wan (2007) propose a very encouraging
trading strategy based on the price range and the features of CBBC. In this study, we examine the
proposed trading strategy at the strong bull and bear market period using the higher explanatory
power model: augmented VECM to conduct forecasts. Using data of Hang Seng Index
exchange-traded fund, it is shown that the trading strategy is not always profitable and may suffer
enormous losses. We do not recommend using the proposed trading strategy.
1. INTRODUCTION
Open, High, Low and Close prices play an important role in technical analysis and in
forecasting the mean and volatility of stock prices. Most studies have used only close-to-close stock
prices in determining the mean and volatility; however, Parkinson(1980), and Garman and
Klass(1980) illustrate that, under certain assumptions, the price range which is the difference of the
high and the low is a more efficient volatility estimator than, for example, the traditional daily close
to close price volatility estimator. Further, Beckers (1983) and Wiggins (1993) demonstrate that to
improve the predictive accuracy of the close-to-close prices’ estimation, the extreme-value estimator
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should be employed.
There are several advantages in using the intraday high-low price range for volatility
measurement and forecasting. Andersen and Bollerslev (1998) mention that market microstructure
issues such as non-synchronous trading effects, discrete price observations and bid-ask spreads may
limit the effectiveness of intraday return variances as volatility proxies. Corrado and Truong (2004)
point out that the intraday high-low range volatility provides greater incremental information and can
usefully augment conditional volatility forecasts. In addition, even when high-frequency intra-day
returns are not available, high-low price range data can be obtained easily from financial databases.
Recently, the range estimators have been used to estimate the volatility, for example, Gallant et al.
(1999), Alizadeh et al. (2002), Chou (2005), and Brandt and Jones (2006).
The price ranges not only play an important role in technical analysis, but have also been widely
used by financial markets’ traders. The candlestick charting technique which was introduced by
Japanese rice traders in the 17th century is one of the famous technical analysis techniques which
incorporates bar charts with the price ranges. Recently, Cheung, Cheung, He and Wan (2007)
propose a very encouraging trading strategy based on the price range, and the call price and the
mandatory call event of the Callable Bull/Bear contract (CBBC). By using the CBBCs issued
between June 2006 and April 2007, which is a strong bull period, and data on product related to the
Hong Kong Hang Seng Index, the proposed trading strategy can generate annualized average profit
between 10.64% and 21.09%.
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To explore the idea, we examine the proposed trading strategy at the strong bull and bear market
period with 542 CBBCs in Hong Kong. In addition, Cheung, Cheung, He and Wan (2007) suggest
using the vector error correction model (VECM) and imposing the (1,-1) cointegrating vector
restrictiona to conduct one-period ahead forecasts. The VECM can be written as
p
X t     i X t i  Rt 1   t ,
(1)
i 1
where X t  H t , Lt ' , H t is the daily high, Lt is the daily low, Rt  H t  Lt is the intraday
range,  t is the error term, and  is the first difference operator. However, Cheung (2007) shows
that by considering the three main US stock indexes – the Dow Jones Industrial Index, the NASDAQ
Index, and the S&P 500 Index, from 2 January 1990 to 31 December 2004, the augmented VECMs
incorporating the daily data on opens, highs, lows and closes explain 40-50% of variation in the daily
highs and the daily lows. The augmented VECM is given by
p
q
r
i 1
i 1
i 1
X t     i X t 1i  Rt 1    i Yt i  0 i COt i  t ,
(2)
where Yt i is the vector containing first differencing of the daily open and the daily close, and
COt i  Ct i  Ot i is the interday range. This stimulates our interest to employ the augmented
VECM in generating the high and the low forecasts, so as to obtain the trading signals.
This study also broadens and supplements the paper written by Cheung (2007) in three ways.
First, we adopt the model and use the 1-day ahead forecasts to attest and assure the predictive power
of the model. This extends the model from conceptual base to applicable base. Second, we use the
ARIMA model to cross compare with the VECM and the augmented VECM. This can give a fair
a
The ability of the model to describe stock market data are illustrated by Cheung (2007), and Cheung, Cheung and Wan
(2008)
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view of different models’ forecasting performance. Third, we apply the model in the Hong Kong
Stock market rather than the U.S. Stock market, as each market has their unique and distinguishing
characteristics.
The study is organized as follows: The second section introduces CBBCs in the Hong Kong
market and the trading strategy which is proposed by Cheung, Cheung, He and Wan (2007) in detail.
The third section exhibits the empirical results of the selected models. The fourth section illustrates
the performance of the trading strategy. The final section provides a conclusion of this study.
2. CALLABLE BULL AND BEAR CONTRACTS AND TRADING STRATEGY
The first CBBC in Hong Kong was introduced by the Hong Kong Exchanges and Clearing
Limited (HKEX) in the first half of 2005. The CBBC is a type of structured product and is listed
either as a Bull or a Bear instrument that tracks the performance of underlying asset, such as a share
or an index, without requiring investors to pay for the full price to own the actual asset. Moreover, it
is a single-barrier option contract. In some overseas markets, the listed and the non-listed products
equivalent to the CBBC are named as “Knock Out” or “Stop Loss”, and Contracts for Difference
(CFD) respectively. In Hong Kong, a lifespan of the CBBCs is three months to five years, and
transactions must be settled in cash.
The mandatory call feature is one of the special features of the CBBCs. For a Bull contract, the
call price which is the trigger price must be either equal to or above the strike price. On the other
hand, for a Bear contract, the call price is set either equal to or below the strike price. If the price of
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the underlying asset reaches the call price at any time prior to the expiry date, the mandatory call
event (MCE) will be triggered which means the issuer must call the CBBC and the trading of the
contract will be terminated. The other major feature of the CBBCs is that they move virtually 1:1 to
the underlying asset, so the price movement tends to track the price of the underlying asset closely.
However, when the underlying price is very close to the trigger price, the price movement may not
exactly match with the underlying asset. The HKEX provides more detailed description of these
contracts, including issuers’ information.
Recently, the CBBCs have become a popular instrument in the Hong Kong stock market.
Comparing the turnover of the CBBCs with the equities and warrants, the trading volume of CBBCs
from 2006 quarter 3 to 2008 quarter 1 increased nearly 14 times, while the trading volume of the
equities and the warrants only increased around 2.5 times. Further, starting from the end of 2007
quarter 4, many negative sentiments have surrounded Hong Kong, e.g. sub-prime woes, infinite
delay of “Through-Train”, stagflation of U.S. economy, continued increasing of crude oil’s price…
etc. The Hong Kong Hang Seng Index (HSI) suffered a substantial drop from 31600 points to 21000
points, and the trading volume dropped significantly too. However, the trading volume of the CBBCs
increased dramatically. This indicates the CBBCs play an important role in the fluctuating market.
Cheung, Cheung, He and Wan (2007) suggest an innovative and encouraging trading strategy
based on the CBBC’s characteristics, the detailed steps are shown as follows:
Step 1: Construct 1-day ahead forecasts of daily highs and lows for an
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underlying asset, such as an index or a stock.
Step 2: When the forecasted daily high is higher than the mandatory call
price of a callable bear contract, a buy signal is produced.
Step 3: When the buy signal is observed for m successive day(s), long the
underlying asset.
Step 4: When either the buy signal vanishes for m successive day(s), the
MCE occurs, or the maturity of the CBBC meets; thus, cover the
long position.
Step 5: Repeat Steps 2 to 4, if the position is closed before the MCE and the
maturity of contract.
On the whole, the call feature and the MCE are used to determine the entry and the exit points
of the trading strategy. To generate a sell signal, Steps 2 to 4 should be modified. First, a callable bull
contract is used to generate a sell signal when the forecasted daily low is smaller than the contract’s
call price. Second, Steps 3 and 4 should be amended to accommodate a short-sell of the underlying
asset.
Further, instead of the CBBCs themselves, instruments tracking the performance of the
underlying asset such as the HSI exchange-traded fund and the HSI future are bought or sold. The
advantages and reasons are illustrated in Cheung, Cheung, He and Wan (2007). The researchers also
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demonstrate the performance of using the ETF2833 and the HSI future in the trading strategy is
similar. Thus, the exchange-traded fund (ETF2833) is chosen as the instrument in this study.
In addition, m which is stated above is a predetermined and integral parameter. It is used to
confirm the trading signal. If the value of m is small, too many trades and pseudo signals will be
generated. On the contrary, large m value may forgo many profitable trading opportunities. The value
of m must be chosen scrupulously and therefore, we consider the value of m equals to 1, 2 and 3 in
this study.
In essence, sampling uncertainty should be considered and incorporated in the high ( Hˆ t 1 ) and
the low ( Lˆ t 1 ) estimators. Thus, confidence interval which is used to indicate the reliability of an
estimate is introduced. Hˆ t 1  Hˆ t 1  ˆ t 1 and Lˆt 1  Lˆt 1  lˆt 1 , where ˆ t 1 and lˆt 1 give the
upper range and the lower range of a confidence interval respectively. A more comprehensive
elaboration of the whole trading strategy can be found in Cheung, Cheung, He and Wan (2007).
3. MODELING
The data used in the study is the main Hong Kong index – Hong Kong Hang Seng Index (HSI).
Daily data on opens, highs, lows and closes of the HSI were downloaded from Reuters. All data were
taken natural logarithm. The sample covers from 2nd June 1997 to 30th May 2008, there are totally
2711 observations. The first 10 years’ data from 2nd June 1997 to 31st May 2007 with 2465
observations are used to carry out testing, generate estimation and conduct modeling; while the last
year’s data from 1st June 2007 to 30th May 2008 with 246 observations are used for evaluating the
out-of-sample’s forecasts and executing the proposed trading strategy.
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For the in-sample and the out-of-sample periods, they both encapsulate a strong bull and bear
market period. This characteristic is desirable in two ways: (1) it is good for replicating the
characteristics of the modeling sample; (2) it is an important element for the trading strategy, as one
may earn profit in the bull market, but not in the bear market, and vice versa. It is imperative to
divide the period into two parts: bull market and bear market, which are the first five months from 1st
June 2007 to 31st October 2007 and the last seven months from 1st November 2007 to 30th May 2008
respectively, so as to distinguish where the profits/losses are derived from.
For brevity, we do not report the preliminary analyses including unit root tests and cointegration
tests. The results exhibit the open, the high, the low and the close series are not stationary, and two
cointergrated pairs are found, which are the highs and the lows, and the opens and the closes.
Given that the high and the low series are cointegrated, the dynamic interaction is found.
Therefore, the VECM can be employed to investigate the interactions which can be divided into
short-run and long-run. In order to increment the explanatory power of the VECM, two additional
price series: opens and closes are augmented into the original high-low VECM. Finally, the simplest
model – ARIMA modelb will be used to carry out comparisons. The three models and model
evaluation are presented below.
3.1 VECM
Table 1 (Panel A) presents the result of the VECM (Model 1). The SBC will be employed to
b
Cheung, Cheung and Wan (2008) state that the performance of the forecast based on the random walk model was
consistently worse than the ARIMA model and the VECM. Thus, ARIMA model is considered as the benchmark model.
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select the lag parameter, and if the estimated residuals do not pass the diagnostic test, the lag
parameter will be increased until they could pass. The results are very encouraging that the error
correction term and the range estimator in both the high and the low models are significant, and most
of the parameters’ coefficients are significant as well. The significance of the range estimators
exhibit if the lagged high and the lagged low series deviate from the long-run equilibrium, each of
them would adjust to partially restore the equilibrium relationship.
The negative and the positive signs of the range coefficient in the high and the low models
respectively suggest that the lagged high will decease, while the lagged low will increase. Moreover,
an increase in the today’s range tends to diminish the next day’s high and raise the next day’s low,
and thus, the next day’s range is reduced. The range’s coefficients in the both models are very low.
These indicate a quite long period of time is required to restore back to equilibrium.
The diagnostic test statistics and the corresponding p-value are shown in Table 1 (Panel B). All
of the Q-statistics are insignificant, which illustrate there is no autocorrelation among the residuals;
hence, the lagged terms are adequately chosen. The parameter estimates are quite similar to the
findings conducted by Cheung, Cheung, He and Wan (2007).
3.2 Augmented VECM
The results of the augmented VECM (Model 2) are reported in Table 2. The lagged terms of q
and r are chosen based on the significance of the t-test.
There are some obvious changes in the augmented VECM. Comparing with the original model,
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most of the augmented VECM’s lagged coefficient estimates become more negative, except the
variable  L(-4) in the augmented low model. However, for the intraday range’s estimates, the
changes are very small.
A remarkable improvement is the adjusted R2. The adjusted R2 of the augmented VECMs are
four times larger than the original VECMs. The statistics improve from around 0.07 to around 0.29.
Moreover, the Q-statistics are insignificant in the both models. These results suggest the augmented
VECMs have higher explanatory power.
3.3 ARIMA
The high and the low series are separately modelled in the ARIMA framework, and first
differencing is employed. By examining the correlogram and the partial correlogramc, and using the
SBC to determine the order of p and q, the ARIMA(3,1,2) is selected and determined as the best
model for both the high and the low series.
The parameter estimates are presented in Table 3. All moving average and autoregressive
lagged parameters are significant. The results also indicate the both price series response to the
market information promptly.
3.4 Evaluation
To evaluate the forecasting ability of different models, the in-sample (from 2nd June 1997 to 31st
c
Both the correlograms and the partial correlograms of the high and the low ARIMA models die down quickly within
two standard deviations.
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May 2007) and the out-of-sample (from 1st June 2007 to 30th May 2008) forecasting errors are
considered. The forecast horizon considered is only 1-day, since our trading strategy requires only
the 1-day ahead forecasts. All forecasting models are updated daily. The MAE and MSE are chosen
as the comparison criteria. The results are shown in Table 4.
The results indicate the augmented VECM is the best model during both the in-sample and the
out-of-sample periods, while the worst model is the ARIMA(3,1,2). Noticeably, the results of the
VECM and the augmented VECM are quite similar. Except for the range forecasts in the
out-of-sample period, the forecasting error of each of the Augmented VECM is always slightly
smaller than that of the VECM. As the difference between each comparison criterion is very small, it
is too early and judgmental to make a conclusion. Therefore, a Diebold Mariano (DM) test is
employed to test each pair of forecast series.
Table 5 reports the test statistics of DM test. The results ensure our judgments, and illustrate the
best model in both the in-sample period and the out-of-sample is the augmented VECM, while the
worst model is the ARIMA(3,1,2).
The 1-day ahead forecast is an essential and important component in forming the trading
strategy. Therefore, it would destroy the whole strategy if there are any mistakes. Thus, it is prudent
for us to consider all of the three suggested models in order to prevent a biased and fallacious result,
although the best model is the augmented VECM.
4. TRADING PERFORMANCE AND EVALUATION
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As mentioned in Section 3, the trading period will be divided into two as the characteristics of
this period are very distinctive. Three scenarios are defined to investigate different market situations:
(A) Bull and Bear Market Period from 1st June 2007 to 30th May 2008, (B) Bull Market Period from
1st June 2007 to 31st October 2007, and (C) Bear Market Period from 1st November 2007 to 30th May
2008. During the periods A, B and C, the numbers of available CBBCs are 542, 120 and 468
respectively. These numbers suggest that the issuers are more likely to issue the CBBCs in the
fluctuating and downward trended market.
In period A, there are totally 274 bear and 268 bull contracts. The composition of the bear and
the bull contracts are quite balanced; therefore there is no single contract biased in this period as it
includes both bull and bear markets. On the other hand, there are 72 bear and 47 bull contracts in
period B, and 208 bear and 260 bull contracts in period C. The reason is that the MCE of the bear
contracts are more likely to be triggered in period B. On the contrary, the MCE of the bull contracts
in period C follow the same circumstance. Thus, it is a common practice of issuers to replace the
called contract by issuing same type of new contract. All in all, the different attributes of the above
three periods support us to conduct partitions.
Table 6 reports the performance and profitability of trading the ETF2833 using the three
suggested models with m = 1, 2 and 3. 95% and 99% confidence intervals are considered, and a
one-way 0.5% transaction cost is included in calculating the trading returns. For comparison purpose,
all the returns stated in the table are annualized. Further, the averaged profit/lose is considered to
avoid the big variation of individual return. The overall performance of the trading strategy is not
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very well; thus, traders cannot always earn profits. The discussion of the periods A, B and C are
presented below.
4.1 Period A: Bull and Bear Market Period
The ARIMA model gives the best performance in this period, but the augmented VECM gives
the worst result. Obviously, m=2 gives the highest returns no matter which model is employed.
However, losses are appeared when m=1. Further, when m=3, all models are profitable in the 99% of
confidence interval, while only the ARIMA model is profitable in the 95% confidence interval. It is
hard for us to determine the level of confidence intervals as there is no conspicuous pattern that can
be observed.
Seemingly, traders can make profits by using the trading strategy with m=2; nevertheless, the
percentages of profitable trades do not support the above findings. For m=2 which is the most
profitable one, the percentages of profitable trades ranges from 47.98% to 57.67%. These exhibit
traders only have around 50% of chance to make profits. Hence, traders cannot always have a
favourable position. In conclusion, the trading strategy is not always profitable in the fluctuating
market.
4.2 Period B: Bull Market Period
In this period, the trading strategy gives very encouraging results as the highest annualized
profits is 341.33%. Moreover, the percentages of profitable trades ranging from 61.76% to 91.07%
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upheld the annualized profits and give consistent results. There is only one setting in the augmented
VECM model which is m=2 and confidence interval=95%, however, gives a unpleasant result.
Among the three different settings of m, the case of m=2 garners the best returns which the
average profits range from 285.84% to 341.33% and the minimum percentage of profitable trade is
86.67%. Apparently, the 99% confidence intervals tend to give better results than the 95%
confidence intervals since we can avoid loss. If m=2 and the 99% confidence intervals are
determined, the augment VECM gives the highest profits with higher than 90% of chance to make
profits. In summary, when the market sentiment is bull, the trading strategy is brilliant and every
trader can earn very satisfactory profits.
4.3 Period C: Bear Market Period
In most of the settings, huge amounts of losses ranging from the highest, -116.49%, to the
lowest, -1.46%, are found. The percentages of profitable trades ranged from 33.89% to 48.47% are in
line with the result. Although there is a setting appearing to be profitable, the returns are very small
which is slightly higher than 4%, and the percentage of profitable trades is only 46.73%. These
indicate the profits are not reliable.
Again, m=2 performs the best among those poor results. In addition, there is no difference
between the choices of the 95% confidence interval and the 99% confidence interval. If m=2 is
chosen, the ARIMA model has better performance than the others. Since most of the settings are not
profitable, traders not only cannot make profits by using this trading strategy, but also bear vast
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losses. Thus, the use of this trading strategy is not recommended in the period C.
The results shown in Table 6 are consistent with the findings of Cheung, Cheung, He and Wan
(2007) in the bull market period as the structure of this period is similar to the structure of the
researchers’ trading period. In the bull market period, the trading strategy is very favourable, yet it is
not applicable in the fluctuating and the downward trended periods. These reflect the trading strategy
is not beneficial to traders as no one can exactly predict the market volatility.
There are three main reasons that illustrate the trading strategy is not always applicable and
profitable. First, for the contracts that experienced the MCE in our trading period, the average
number of trading days is 18.54. However, the average of 23 days is found in the researchers’ trading
period. These indicate when the market is fluctuating, the MCE is more likely to be triggered and the
accuracy of the trading signals would be affected by the high market volatility; hence, many false
signals are generated. Second, the profits gain in the bull market may be due to good and favourable
sentiment in the market, but not due to the trading strategy. By using the simple buy and hold trading
strategy to buy the ETF2833 in period B, the annualized profit after adjusting for the one-way
transaction cost is 119.48%. This indicates without using the trading strategy, everyone can earn a
significant amount in the bull market. Therefore, the profits gained by the trading strategy may come
from the upward trend of the market. Third, according to the efficient market hypothesis (Fama,
1970), technical analysis techniques based on the historical underlying prices cannot consistently
earn excess returns (Kirkpatrick and Dahlquist, 2007). The hypothesis supports our findings that one
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can make profit in the bull market period, but cannot consistently gain in the next period.
In short, our empirical findings indicate that we cannot always make profits by using the
trading strategy. Thus, we do not propose using this strategy as no one can actually predict the
market volatility and traders may incur a huge amount of losses
5. CONCLUSIONS
The purpose of this study is to replicate the trading strategy proposed by Cheung, Cheung, He
and Wan (2007) at the period which encapsulates a strong bull and bear market. The trading strategy
is based on the unique characteristics of the Callable Bull/Bear Contract, which is a single-barrier
options contract. To design the entry point(s) and the exit point(s), the information on the call feature
and the mandatory call event are incorporated. Moreover, in order to generate the trading signals, we
not only consider the forecasts from the bivariate model of the daily highs and the daily lows, but
also use the forecasts from the augmented VECM suggested by Cheung (2007) incorporating the
daily data of opens, highs, lows and closes. Moreover, the ARIMA model is employed to cross
compare with the above models.
To determine the best model to conduct the forecasts, comparison criteria such as MSE, MAE,
MAPE and DM test are employed. We find that the best model in both the in-sample period and the
out-of-sample period is the augmented VECM. However, in the prudential view, all of the suggested
models are considered.
Three different scenarios which are the bull and bear market period, the bull market period and
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the bear market period are defined to demonstrate the trading strategy. Our empirical findings
indicate the trading strategy is not profitable in both the bull and bear market period, and the bear
market period. On the other hand, the trading strategy gives a very encouraging result in the bull
market period. However, a significant amount of profit can also be generated by using the simple buy
and hold trading strategy. Thus, the proposed trading strategy is not much superior to the simplest
one. As a result, the trading strategy is not recommended because the average return is not always
positive and traders may experience a huge amount of losses.
While we used three different models to conduct the one-day ahead forecasts, we neither
examine other forecasting methods to perform forecasts nor incorporate more trading information,
for instance, trading volume, into our models. Thus, further investigations on the proposed trading
strategy, and refinement on the forecasting ability are deserved.
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autoregressive range (CARR) model. The Journal of Money, Credit & Banking, 37, pp. 561-582
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2009 Oxford Business & Economics Conference Program
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Corrado, C. & Truong, C. (2004). Forecasting stock index volatility: the incremental information in
the intraday high-low price range, Research Paper Series 127, Quantitative Finance Research
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Kirkpatrick, C.D. & Dahlquist, J.R. (2007). Technical analysis: the complete resource for financial
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Parkinson, M. (1980). The extreme value method for estimating the variance of the rate of return.
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Wiggins, J.B. (1993). Estimating the volatility of S&P 500 futures prices using the extreme-value
method. The Journal of Future Markets, 12, 3, pp.265-273.
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TABLES:
Table 1: High-Low VECM
H
Coefficient
(A) Parameters Estimates
Constant
-0.0011*
 H(-1)
-0.2202***
 H(-2)
-0.1995***
 H(-3)
-0.1722***
 H(-4)
-0.0940**
 L(-1)
-0.3736***
 L(-2)
-0.1436***
 L(-3)
-0.2232***
 L(-4)
-0.0295
HL(.)
-0.0615*
Adjusted
R2
L
T-Statistic
-1.74
-4.45
-3.71
-3.34
-2.27
-7.98
-2.81
-4.59
-0.75
-1.75
-0.0675
Chi-Square
Coefficient
-0.0015**
-0.4448***
-0.2403***
-0.0901*
-0.0795*
-0.2341***
-0.3007***
-0.0097
-0.1405***
-0.0976**
T-Statistic
-2.11
-8.28
-4.12
-1.61
-1.77
-4.60
-5.42
-0.18
-3.28
-2.56
-0.0776
P-value
Chi-Square
P-value
(B) Diagnostic Test
Q-stat(6)
9.41
0.1519
12.46
0.0524
Q-stat(12)
13.88
0.3087
21.08
0.0450
Q-stat(18)
27.38
0.0721
26.86
0.0817
Q-stat(24)
32.36
0.1184
33.73
0.0896
Notes: The Panel A and B report the parameter estimates and the diagnostic test respectively. HL(.)
is the lagged intraday range estimator which is the error correction term imposing the (1,-1)
restriction. Q(n) gives the Q-statistics with n degree of freedom. *, ** and *** indicate the level of
significance at 10%, 5% and 1% respectively.
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Table 2: Augmented VECM
H
Coefficient
(A) Parameters Estimates
Constant
-0.0011**
 H(-1)
-0.7746***
 H(-2)
-0.4088***
 H(-3)
-0.3075***
 H(-4)
-0.1260***
 L(-1)
-0.1431**
 L(-2)
-0.0634
 L(-3)
-0.0843
 L(-4)
-0.0627*
HL(.)
-0.0481
 O(-1)
-0.5009***
 O(-2)
-0.1449*
 O(-3)
-0.1042*
 C(-1)
-0.3331***
 C(-2)
-0.1176
 C(-3)
-0.1817***
OC(-1)
-0.7171***
Adjusted
R2
0.2902
L
T-Statistic
-1.98
-12.48
-5.68
-4.73
-3.43
-2.50
-0.99
-1.42
-1.78
-1.57
-6.12
-1.92
-1.91
-3.97
-1.59
-3.09
-9.13
Coefficient
-0.0015**
-0.1520**
-0.0800
-0.1344*
-0.0393
-0.7914***
-0.4660***
-0.2100***
-0.1120***
-0.1105***
-0.5563***
-0.3019***
-0.1960***
-0.3604***
-0.1784***
-0.2359***
-0.7608***
T-Statistic
-2.37
-2.23
-1.01
-1.89
-0.98
-12.64
-6.64
-3.23
-2.90
-3.29
-6.20
-3.65
-3.28
-3.92
-2.21
-3.66
-8.84
0.2859
Chi-Square
P-value
Chi-Square
P-value
(B) Diagnostic Test
Q-stat(6)
2.24
0.8963
3.31
0.7690
Q-stat(12)
8.09
0.7780
13.37
0.3426
Q-stat(18)
26.72
0.0845
19.42
0.3667
Q-stat(24)
29.66
0.1961
26.82
0.3130
Notes: The Panel A and B report the parameter estimates and the diagnostic test respectively. HL(.)
is the lagged intraday range estimator which is the error correction term imposing the (1,-1)
restriction. OC(.) is the lagged interday range estimator with the (1,-1) restriction. Q(n) gives the
Q-statistics with n degree of freedom. *, ** and *** indicate the level of significance at 10%, 5%
and 1% respectively.
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Table 3: ARIMA model
H
Coefficient
L
T-Statistic
Coefficient
T-Statistic
-0.38
-23.81
-53.36
-11.36
-49.23
-6.44
-0.0001
-1.7490***
-0.9967***
-1.8388***
-1.1618***
-0.0946***
-0.35
-683.33
-345.48
-101.24
-35.83
-4.96
Parameters Estimates
Constant
-0.0001
MA(1)
-0.4140***
MA(2)
-0.9648***
AR(1)
-0.2968***
AR(2)
-0.9183***
AR(3)
-0.1307***
Differencing
1
1
Notes: The table reports the parameter estimates of the ARIMA model. AR(n) refers to n lagged
term(s) in the autoregressive model. MA(n) refers to n lagged term(s) in the moving average model.
*** indicates the level of significance at 1%.
Table 4: Accuracy Measures
High
MAE
MSE
(A) In Sample
ARIMA(3,1,2)
VECM
Augmented VECM
0.01021 0.00023
0.01001 0.00022
0.00864 0.00017
Low
MAE
MSE
0.01112 0.00028
0.01082 0.00026
0.00946 0.00020
Range
MAE
MSE
0.00729
0.00526
0.00525
0.00011
0.00006
0.00006
(B) Out-of-Sample
ARIMA(3,1,2)
0.01462 0.00042
0.01614 0.00048
0.00896 0.00015
VECM
0.01435 0.00041
0.01592 0.00046
0.00633 0.00009
Augmented VECM
0.01258 0.00034
0.01440 0.00040
0.00640 0.00009
Notes: The Panel A and B report the in-sample and the out-of-sample accuracy measures. MAE and
MSE stand for mean absolute error and mean square respectively.
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Table 5: Diebold Mariano Test Statistics
High
Low
Range
C: MAE C: MSE
C: MAE C: MSE
C: MAE
C: MSE
(A) In Sample
Augmented VECM
-11.31 *** -8.54 ***
-10.96 *** -7.50 ***
-16.69 *** -10.40 ***
/ ARIMA(3,1,2)
Augmented VECM
-11.21 *** -8.54 ***
-10.19 *** -7.71 ***
-0.90
-1.80 **
/ VECM
VECM /
-2.95 *** -3.24 ***
-3.30 *** -2.64 **
-16.71 *** -10.27 ***
ARIMA(3,1,2)
(B) Out-of-Sample
Augmented VECM
-3.42 *** -2.14 **
-2.80 *** -2.06 **
-5.97 *** -5.11 ***
/ ARIMA(3,1,2)
Augmented VECM
-3.26 *** -2.22 **
-2.54 *** -1.73 **
1.01
0.91
/ VECM
VECM /
-0.93
-0.37
-0.71
-1.31 *
-6.17 *** -5.27 ***
ARIMA(3,1,2)
Notes: The Panel A and B report the in-sample and the out-of-sample Diebold Mariano test statistics.
‘C: MAE’ represents the Diebold Mariano (DM) test based on mean absolute error (MAE) criterion,
while ‘C: MSE’ represents the DM test based on mean square error (MSE) criterion. A positive test
statistic indicates that the former forecasting model has a larger MSE/MAD than the latter one. **
and *** indicate the level of significance at 5% and 1% respectively.
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Table 6: Performance of Trading using the three models
ARIMA
VECM
Augmented VECM
Avg P&L (Freq profit)
Avg P&L (Freq profit)
Avg P&L (Freq profit)
95%
99%
95%
99%
95%
99%
(A) Bull and Bear Period
m=1
-26.96%
-38.46%
-56.95%
-51.13%
-88.53%
-61.45%
(45.39%)
(44.31%)
(41.67%)
(44.09%)
(38.72%)
(41.51%)
m=2
75.11%
65.67%
45.60%
43.72%
24.09%
24.31%
(57.67%)
(55.12%)
(54.40%)
(53.91%)
(50.00%)
(47.98%)
m=3
12.32%
51.33%
-17.68%
31.35%
-42.83%
4.38%
(47.40%)
(51.72%)
(46.72%)
(49.73%)
(40.59%)
(47.20%)
(B) Bull Period
m=1
154.60%
51.68%
99.12%
39.46%
-19.64%
11.07%
(61.76%)
(63.64%)
(71.43%)
(63.33%)
(72.00%)
(65.00%)
m=2
323.07%
302.70%
316.66%
293.30%
285.84%
341.33%
(86.67%)
(91.07%)
(87.50%)
(87.10%)
(90.20%)
(88.52%)
m=3
160.37%
291.87%
193.86%
277.17%
193.56%
263.01%
(70.00%)
(81.58%)
(67.65%)
(78.26%)
(77.42%)
(77.78%)
(C) Bear Period
m=1
-81.59%
-84.72%
-106.25%
-94.87%
-116.49%
-96.24%
(38.96%)
(39.88%)
(34.85%)
(39.57%)
(33.89%)
(36.28%)
m=2
-10.07%
4.41%
-48.26%
-38.49%
-57.71%
-73.48%
(48.47%)
(46.73%)
(43.15%)
(44.39%)
(39.29%)
(34.71%)
m=3
-10.99%
-1.46%
-59.83%
-20.35%
-75.99%
-58.49%
(42.74%)
(44.30%)
(38.83%)
(42.95%)
(34.62%)
(37.50%)
Notes: The table reports the annualized average profits and loses (net of transaction costs) and the
frequent of profits from trading the ETF2833 by the three different models which are the ARIMA
model, the VECM and the augmented VECM. The Panel A, B and C represent different market
periods which are the bull and bear market period, the bull market period and the bear market period
respectively. ‘m’ corresponds to the number of consecutive day which is used to generate buy/sell
signals. ‘Avg P&L’ refers to the average profits and loses and ‘Freq profit’ refers to the frequent of
profits. The annualized average profits and loses are shown without parentheses, while the frequent
of profits are shown in parentheses under the annualized average profit and lose. 95% and 99%
represent the level of confidence interval.
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