Lead-lag relationship and Hedge Effectiveness:
Hang Seng Index, Hang Seng Index Futures and Tracker Fund
Gordon, Y.N. Tang
Department of Finance and Decision Science, Hong Kong Baptist University, Hong
Kong
E-mail: gyntang@hkbu.edu.hk
Fax: +852-3411-5585
Tel.: +852-3144-7563
Karen, H.Y. Wong
Chu Hai College of Higher Education, Hong Kong
E-mail: hywong@chuhai.edu.hk
Fax : +852-2408-8922
Tel. : +852-2408-9813
1
Abstract
In this article, lead-lag relationship and hedging effectiveness among the Hang Seng
Index, Hang Seng Index Futures and Tracker Fund are examined. By applying the
cointegration test, the long-term relationship among the three markets is confirmed.
Results of the cause and effect relationships between the Hang Seng Index, Hang Seng
Index Futures and Tracker Fund show different lead-lag relationships between each
markets. Hang Seng Index is led by both of the Hang Seng Index Futures and Tracker
Fund. Similarly, Tracker Fund is led by other two markets. Only the Hang Seng Index
Futures, surprisingly, is led by the Tracker Fund. The hedge ratio and hedging
effectiveness reveal that the hedging performance of the HSIF is improved after the
advent of the Tracker Fund.
2
1. Introduction
Nowadays, a variety of securities are available for investors. Exchange-traded-fund is one
of popular securities. Being the first Exchanged-traded-fund in Hong Kong, the Tracker
Fund is always being treated as an alternative of Hang Seng Index Futures for hedging.
The investment in Tracker Fund is more and more because investors can take more
hedging positions in the cash market. After the advent of the Tracker Fund, the interaction
among the Hang Seng Index, Hang Seng Index Futures and Tracker Fund is worth to be
investigated. In order to find out the interaction, two issues are mainly examined in this
study: Lead-lag relationship and hedging effectiveness.
The relationship between cash and futures markets has been extensively discussed in
practitioner and academic. Some studies prove that the lead-lag relationship between
two markets is not significant (Ho, Fang and Woo, 1992) and the relationship does not
exist even (Lim, 1992a and 1992b). In this study, we look for whether any lead-lag
effect exists across the long-term relationship among three markets through cointegation
tests. Studies have stated futures markets lead cash markets (Kawaller, Koch and Koch
(1987), Stoll and Whaley (1990), Chan (1992), Shyy, Vijaraghavan and Scott-Quinn
(1996)). Using Error Correction Models, a lead-lag relationship between two markets:
Hang Seng Index and Hang Seng Futures, is examined. Also, the relationships between
Tracker Fund and other two markets are examined.
Hedging can be achieved via different strategies: risk minimization, profit maximization
and using a portfolio theoretical approach to attain a satisfactory risk-return trade-off.
Here, hedge ratio and hedge effectiveness are the main measures of hedging
performance. Hedging strategy is vary, thus, there are numbers of means to measure the
hedge ratio and effectiveness (see Lien and Tse (1999), Holmes (1996), Ghosh (1993),
Myers (1991), Cecchetti, Cumby, and Figlewski (1988)). To see any effect of the
Tracker Fund on Hang Seng Index Futures, we estimate the hedge ratio and
effectiveness of Hang Seng Index Futures after the existence of Tracker Fund.
The remainder of this article is organized into five sections. Next section provides a
discussion of the data; background information of the Hang Seng Index Futures and the
Tracker Fund respectively. Following this, the methodology is presented. The empirical
findings for general statistics, unit-root test, cointegration, error correction model, hedge
ratio and effectiveness are then discussed. The final section gives conclusions.
2. Data and General Statistics
Two test periods are illustrated in this study. Most of tests are based on the test period
from 12 November 1999 when the Tracker Fund (TraHK) listed to 28 June 2002. To
enhance the study in hedging effectiveness of Hang Seng Index Futures (HSIF), the test
period of this part is extended to 2 January 1998. The Hang Seng Index (HSI) data is
supplied by Hang Seng Index Services Limited while other elements (HSIF and TraHK)
are provided by Hong Kong Exchanges and Clearing Limited (HKEx). The first recorded
HSI and the first nearby transaction of HSIF in each 15-minute interval are chosen. This
means that our study has a 15-minute basis. Similarly, we pick the first nearby TraHK
transaction from each 15-minute interval. The prices are grouped as a series with a 15minute basis. Finally, we use three time series, HSI, HSIF and TraHK, with first reported
prices on a 15-minute basis.
3
The three test series, HSI, HSIF and TraHK, are firstly computed in natural logarithm
form and the return series are calculated as below:
RHSI ,t  p HSI ,t  p HSI ,t 1
(1)
RHSIF ,t  p HSIF ,t  p HSIF ,t 1
(2)
RTraHK ,t  pTraHK ,t  pTraHK ,t 1
(3)
where p HSI ,t , p HSIF ,t and pTraHK are natural logarithms of HSI prices, HSIF prices and
TraHK prices at time t, respectively.
HSIF was firstly introduced in 1986. As the HSIF has been the most active futures in
the world, the Hang Seng Index Futures has been widely studied. Cheng, Fung, and
Chan (2000) examined the impact of the 1997 Asian financial crisis on index futures
markets, Fung and Draper (1999) studied the mispricing of the HSIF under short sales
constraints, Fung, Cheng, and Chan (1997) examined the intra-day patterns of the
HSIF, and Ho, Fan, and Woo (1992) investigated the intra-day arbitrage opportunities
and price behavior of the HSIF. In this case,
Table 1 highlights the details of the HSIF.
Since the first Exchange Traded Fund was issued in 1993, ETFs have expanded rapidly
worldwide. In Hong Kong, the TraHK, listed on 12 November 1999, is the first ETF in
the market. Nowadays, investors are available to trade 10 ETFs (includes two ETFs
under the pilot programme) in Hong Kong. The TraHK which is intended to track
closely the performance of the HSI is successfully being the new popular products for
investors. Same as listed stocks, ETFs are listed on and traded on exchanges. Features
of the TraHK are summarized in Source: Hong Kong Exchanges and Clearing Limited
4
Table 2.
Table 1 Features of HSI Futures
Underlying Index
Contracted Price
Hang Seng Index (HSI)
The price in which index points at which HSI Futures
Contract is registered by the Clearing House
Contracted Multiplier
HK$ 50 per index point
Contracted Value
Contracted Price multiplied by Contract Multiplier
Contract Months
Spot Month, the next calendar month, and March,
June, September and December months
Pre-Market Opening Period
09:15AM-09:45AM (Hong Kong Time)
02:00PM-02:30PM (Hong Kong Time)
Trading Hours
09:45AM-12:30PM (Hong Kong Time)
02:30PM-04:15PM (Hong Kong Time)
Last Trading Day
The business day immediately preceding the last
Business Day of the Contract Month
Final Settlement Day
The first business day after the last trading day
Source: Hong Kong Exchanges and Clearing Limited
5
Table 2 Features of Tracker Fund
Underlying Index
Board Lot Size
Pre-Market Opening Period
Trading Hours
Hang Seng Index (HSI)
500 units
09:30AM-10:00AM (Hong Kong Time)
10:00AM-12:30PM (Hong Kong Time)
02:30PM-04:00PM (Hong Kong Time)
Last Trading Day
N/A
Final Settlement Day
N/A
Source: Hong Kong Exchanges and Clearing Limited
Meanwhile, summary statistics are reported in Table 3 to study the general
characteristics of all data series: the natural logarithm and returns of the HSI, HSIF and
TraHK. The summary covers mean, maximum, minimum, standard deviation, skewness,
kurtosis, and Jarque-Bera test statistics. For all variables, the Jarque-Bera results show
that all data series are non-normally distributed.
On average value, the TraHK has the lowest mean in both the natural logarithm and
returns groups. The mean value of log TraHK (2.613828) is much lower than those of the
other two variables (9.515846 of HSI and 9.515658 of HSIF). However, the mean value
of the TraHK return (-0.002153) is almost the same as those of the others (-0.002469 of
HSI and -0.002498 of HSIF). The mean values of all returns are negative.
In addition, the standard deviation of the TraHK returns (0.636190) is the highest; it is
higher than the HSI (0.428533) and even the HSIF (0.373695). The variability of the
TraHK is much higher than the cash and futures markets. Findings are supported by the
Kurtosis results. The Kurtosis value of the TraHK (114.3595) is much higher than the
others (44.704 of HSI and 29.77123 of HSIF). That means that the distribution of the
TraHK is very leptokurtic. The peaked values created a relatively high standard deviation.
Table 3 Summary Statistics of HSI, HSIF and TraHK
LHSIa
9.515846
9.821735
9.092120
0.181041
0.229012
1.738822
861.1700*
LHSIFb
9.515658
9.817656
9.094534
0.180295
-0.226694
1.739369
858.4887*
LTraHKc
9.521583
9.822820
9.110520
0.177378
-0.223303
1.729718
867.2527*
Mean
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Notes:
a
LHSI represents the natural logarithm of the HSI.
b
LHSIF represents the natural logarithm of the HSIF.
c
LTraHK represents the natural logarithm of the TraHK.
d
HSI Return represents the nominal return of the HSI.
e
HSIF Return represents the nominal return of the HSIF.
f
TraHK Return represents the nominal return of the TraHK.
* Significance at the 1% level.
6
HSI
Returnd
-0.002469
4.700082
-8.413757
0.428533
-0.797778
44.70400
833146.4*
HSIF
Returne
-0.002498
4.625533
-7.263620
0.373695
-0.606431
29.77123
343524.5*
TraHK
Returnf
-0.002153
17.43151
-17.43151
0.636190
-0.489627
114.3595
5932245*
3. Methodology
3.1. Unit Root Test
Many studies, including Granger causality and cointegration test, assume a stationary time
series. It is necessary to apply a unit root test to prove the stationarity of our data series.
There are six data series for the three variables, HSI, HSIF and TraHK, in this article.
These series are divided into two groups: natural logarithms and returns. Thus, we
conduct unit root tests on both log and return prices of the HSI, HSIF and TraHK.
A number of methods are used to test for the presence of unit roots in time series: DickeyFuller (DF), Augmented Dickey-Fuller tests (ADF) (Dickey and Fuller, 1979), the
Sargan-Bhargava (1983) CRDW-test and the non-parametric tests described by Philips
and Perron (1987). Compared with other tests, indeed, the Dickey-Fuller approach is the
most popular one and is commonly used to determine whether the time series is stationary
or not. And hence on we use this approach in this study. In order to expand our study, we
demonstrate the DF test also. Both of the DF and ADF studies are based on the same
hypotheses below:
Hypothesis H1:  u  1 against the alternative hypothesis  u  1
The DF test is, in its simplest form, approximate to the yt as follows:
yt   u yt 1   t
(4)
In case yt is highly autocorrelated, which we will test in a later section, the error term (  t )
will also have high autocorrelation and follow an AR (p) process. As the DF test assumes
its  t is ‘white-noise’, it is necessary to demonstrate the ADF test instead. The ADF test
fixes the problem of the DF test by adding a number of lagged first differences to the
dependent variable, to compensate for the autocorrelated omitted variables that would
exist from  t .
In general, the cash and futures markets are in an upward trend and the TraHK should
follow a trend of the HSI, the trend seems definitely exist in data series of HSI, HSIF and
TraHK? However, for the sake of completeness, we examine cases of DF and ADF tests
with and without a trend. Besides, lag level is considered from one to 10 in the ADF test.
In fact, the test will be adjusted if a trend is not applied in the test.
3.2. Correlation and autocorrelation Tests
We then measure the associations between variables through the correlation test and
further demonstrate the autocorrelation test for all data series, to check the degree which
the series relates to its lagged values over time. Under the autocorrelation test, the
maximum lag term is up to 16, that is, the maximum available level in EViews.
7
3.3. Cointegration Test
Two kinds of cointegration test are available in this study: Trace statistics and MaxEigenvalue statistics. The lag intervals from one to five are used. Under the interpretation
of EViews, the interval of lags represents the level of differentiation of the variable.
Hence, we input the number of lags from zero to four instead.
In this article, numbers of vectors in different variables are examined. In terms of three
data series: the HSI, HSIF and TraHK returns, a trivariate model is applied. To implement
the trivariate model, we use the Johansen (1988) cointegration method to examine the
long term relationship between the HSI, HSIF and TraHK.
Out of the Johansen approach, we illustrate the bi-variate cointegration regressions
between series: (1) HSI and HSIF; (2) HSI and TraHK; and (3) HSIF and TraHK. The
results are further applied to both the DF and ADF tests. To simplify the tests, we choose
up to five lags.
3.4. Error Correction Model (ECM)
After the cointegration relationships were proved, an error correction model was then
applied to test the lead-lag relationship of cointegrated data series in three markets. The
ECM forces the model to move together in the long run by combining the cointegration
relationship among the series. Under the application of EViews, we use a VEC model that
is designed for nonstationary series with cointegration relationships. The results show
possible cause and effect relationship amongst the variables.
To perform the trivariate cointegration test, the respective ECM has two error correction
terms. The cointegration equations are:
Z1,t 1  ln HSI t 1  ln TraHK t 1
(5)
Z 2,t 1  ln HSIFt 1  ln TraHKt 1
(6)
where ln HSI t 1 , ln HSIFt 1 and ln TraHK t 1 represent the logarithm of HSI, HSIF and
TraHK, respectively; Z1,t 1 and Z 2 ,t 1 are the error correction terms. And, the ECMs are as
follows:
5
5
5
rHSIt   HSI  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   HSI (7)
i 1
i 1
i 1
5
5
5
i 1
i 1
5
5
5
i 1
i 1
i 1
rHSIFt   HSIF  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   HSIF (8)
i 1
rTraHKt  TraHK  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   TraHK (9)
8
3.5. Hedge Ratio
Assume the hedger aims to minimize variance of value change of the HSI. At time t-1,
the optimal hedge ratio equals the conditional covariance divided by the conditional
variance futures return:
Xt-1 = Cov(RHIS, t, RHSIF, t | It-1) / Var(RHIS, t)Var(RHSIF, t | It-1 )
(10)
where It-1 is the information set at time t – 1.
3.6. Hedge Effectiveness
Developed by Ederington (1979), the hedging effectiveness is determined by the
squared covariance divided by the product of cash and futures returns variances.
Ht = Cov2 (RHSI, t, RHSIF, t | It-1) / Var(RHSI, t | It-1 ) Var(RHSIF, t | It-1 )
(11)
The hedging effectiveness is in essence the same as the R-squared in a regression of
cash returns on futures returns.
4. Empirical Results
4.1. Correlation Tests
Table 4 shows the correlation coefficients amongst all variables. For the natural logarithm,
all series have around 0.99 correlation coefficients. Correlation relationships are very high,
but for the returns, all series are much lower. The correlation coefficient of the HSI and
HSIF returns is 0.78521; the correlations between TraHK and HSI and TraHK and HSIF
are 0.467254 and 0.415094 respectively.
Table 4 Correlation Coefficients between Variables
LHSIa
1.000000
0.999836
0.999272
0.014238
0.008704
LHSIFb
0.999836
1.000000
0.999272
0.016379
0.015212
LTraHKc
0.999272
0.999272
1.000000
0.010599
0.006512
LHSIa
LHSIFb
LTraHKc
HSI Returnd
HSIF Returne
TraHK
Returnf
0.008135
0.010084
0.020223
Notes:
a
LHSI represents the natural logarithm of the HSI.
b
LHSIF represents the natural logarithm of the HSIF.
c
LTraHK represents the natural logarithm of the TraHK.
d
HSI Return represents the nominal return of the HSI.
e
HSIF Return represents the nominal return of the HSIF.
f
TraHK Return represents the nominal return of the TraHK.
* Significance at the 1% level.
9
HSI
Returnd
0.014238
0.016379
0.010599
1.000000
0.785210
HSIF
Returne
0.008704
0.015212
0.006512
0.785210
1.000000
TraHK
Returnf
0.008135
0.010084
0.020223
0.467254
0.415094
0.467254
0.415094
1.000000
4.2. Autocorrelation Tests
Autocorrelation coefficients of all variables are reported in Table 5. Like the correlation
results, the natural logarithm series of HSI, HSIF and TraHK are highly autocorrelated
(around 0.99 with p-value=0.0000). For the returns series, all coefficients are less than 1.
Around five out of 16 lags are marked negative in each series. Comparing the three
returns series, HSI and TraHK are less autocorrelated, and the results are significant with
p-value=0.0000. The HSIF has markedly higher autocorrelation coefficients, and the
results are not significant. The p-values of HSIF returns are from 0.049 to 0.330.
Table 5 Autocorrelation Coefficients of All Variables
Lag
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
LHSIa
1.000
(0.0000)
0.999
(0.0000)
0.999
(0.0000)
0.999
(0.0000)
0.998
(0.0000)
0.998
(0.0000)
0.998
(0.0000)
0.997
(0.0000)
0.997
(0.0000)
0.997
(0.0000)
0.996
(0.0000)
0.996
(0.0000)
0.996
(0.0000)
0.995
(0.0000)
0.995
(0.0000)
0.995
(0.0000)
LHSIFb
1.000
(0.0000)
0.999
(0.0000)
0.999
(0.0000)
0.999
(0.0000)
0.998
(0.0000)
0.998
(0.0000)
0.998
(0.0000)
0.997
(0.0000)
0.997
(0.0000)
0.996
(0.0000)
0.996
(0.0000)
0.996
(0.0000)
0.995
(0.0000)
0.995
(0.0000)
0.995
(0.0000)
0.994
(0.0000)
LTraHKc
0.999
(0.0000)
0.999
(0.0000)
0.999
(0.0000)
0.998
(0.0000)
0.998
(0.0000)
0.998
(0.0000)
0.997
(0.0000)
0.997
(0.0000)
0.997
(0.0000)
0.996
(0.0000)
0.996
(0.0000)
0.996
(0.0000)
0.995
(0.0000)
0.995
(0.0000)
0.995
(0.0000)
0.994
(0.0000)
HSI Returnd
0.095
(0.0000)
-0.013
(0.1486)
0.008
(0.3741)
-0.017
(0.0589)
-0.004
(0.6567)
0.013
(0.1486)
0.009
(0.3173)
-0.006
(0.5050)
0.002
(0.8241)
0.011
(0.2216)
0.005
(0.5785)
0.011
(0.2216)
0.013
(0.1486)
-0.005
(0.5785)
0.001
(0.9115)
0.013
(0.1486)
Notes:
a
LHSI represents the natural logarithm of the HSI.
b
LHSIF represents the natural logarithm of the HSIF.
c
LTraHK represents the natural logarithm of the TraHK.
d
HSI Return represents the nominal return of the HSI.
e
HSIF Return represents the nominal return of the HSIF.
f
TraHK Return represents the nominal return of the TraHK.
* Significance at the 1% level.
10
HSIF Returne
-0.020
(0.0263)
0.011
(0.2216)
0.000
(1.0000)
-0.010
(0.2665)
0.002
(0.8241)
0.013
(0.1486)
-0.016
(0.0755)
-0.006
(0.5050)
0.001
(0.9115)
0.008
(0.3741)
0.006
(0.5050)
0.001
(0.9115)
0.015
(0.0956)
-0.011
(0.2216)
0.002
(0.8241)
0.008
(0.3741)
TraHK Returnf
-0.294
(0.0000)
-0.004
(0.6567)
-0.005
(0.5785)
0.003
(0.7389)
-0.019
(0.0348)
0.015
(0.0956)
0.001
(0.9115)
-0.009
(0.3173)
0.003
(0.7389)
0.000
(1.0000)
0.010
(0.2665)
0.018
(0.0455)
0.000
(1.0000)
-0.022
(0.0140)
0.002
(0.8241)
0.008
(0.3741)
4.3. Unit Root Tests
Table 6 lists the Dickey-Fuller Tests results of all variables. The results of the natural
logarithm series have high p-values and are not significant. However, the results are
highly significant (p-value=0.0001) in the returns series. All of the returns series are
stationary with and without trends.
Table 6 Dickey-Fuller Test
LHSI
LHSIF
LTraHK
HSI Return
HSIF
Return
TraHK
Return
Without
Trend
-0.695205
[-3.4341]
(.8462)
-0.906297
[-3.4341]
(.7869)
-1.676288
[-3.4341]
(.4434)
-97.3657
[-3.4341]
(.0001)
-109.2858
[-3.4341]
(.0001)
-145.0203
[-3.4341]
(.0001)
With Trend
-3.036362
[-3.9645]
(.1223)
-3.263767
[-3.9645]
(.0724)
-4.484422
[-3.9645]
(.0016)
-97.36822
[-3.9645]
(.0001)
-109.2875
[-3.9645]
(.0001)
-145.0199
[-3.9645]
(.0001)
Notes:
Dickey-Fuller (DF) test (Dickey and Fuller, 1979) is used to determine whether the time series is stationary or not. The DF test has
the hypothesis as below:
Hypothesis H2:
 u  1 against the alternative hypothesis  u  1 .
The DF test is in its simplest form to approximate the
In case
yt as following:
yt   u yt 1   t
yt is highly autocorrelated, which we will test in a latter section, the error term (  t ) will also have high autocorrelation and
follow an AR (p) process.
a
[ ] denotes 1% critical value
b
( ) denotes MacKinnon (1996) one-sided p-value
The Augmented Dickey-Fuller Test (ADF) results are shown in Table 7 to Table 9. In
summary, the natural logarithm series are non-stationary while the returns series are
stationary. The AIC and BIC results are the same in both the HSI and HSIF, while the
TraHK results are always different from those of both the HSI and HSIF. The ADF results
of the HSI (see Table 7) and HSIF (see Table 8) report their lowest AIC and BIC values
in ten lags for both with and without trend series. However, the lowest AIC and BIC
values of the TraHK (see Table 9) occur in one lag for both with and without trend series.
As stated above, HIS, HSIF and TraHK returns series are stationary. For the HIS and
HSIF returns, the lowest AIC and BIC values exist in one lag (see Table 10 and Table
11), but for the TraHK, the results are different. Under both with and without trend
assumptions, the lowest AIC are in four lags and the lowest BIC are in two lags (see
Table12).
11
Table 7 Augmented Dickey-Fuller Test: LHSI
Number
of lags
ADF Test
Statistics
1% Critical
Value
Prob.*
AIC
BIC
Without
Trend
1
2
3
4
5
6
7
8
9
10
-0.840194
-0.807309
-0.825398
-0.794567
-0.794561
-0.813705
-0.824829
-0.815252
-0.819713
-0.838686
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
(0.8071)
(0.8166)
(0.8114)
(0.8201)
(0.8201)
(0.8147)
(0.8116)
(0.8143)
(0.8130)
(0.8057)
-8.350005
-8.350306
-8.350178
-8.350297
-8.350037
-8.349998
-8.349793
-8.349660
-8.349408
-8.349313
-8.348084
-8.347746
-8.346977
-8.346456
-8.345555
-8.344875
-8.344029
-8.343255
-8.342362
-8.341626
With Trend
1
2
3
4
5
6
7
8
9
10
-3.237438
-3.175269
-3.191559
-3.164454
-3.169176
-3.211848
-3.229505
-3.194259
-3.202249
-3.207004
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
(0.0772)
(0.0894)
(0.0861)
(0.0917)
(0.0907)
(0.0821)
(0.0787)
(0.0855)
(0.0839)
(0.0830)
-8.350745
-8.351017
-8.350894
-8.351005
-8.350748
-8.350730
-8.350533
-8.350381
-8.350132
-8.350034
-8.348184
-8.347816
-8.347052
-8.346523
-8.345625
-8.344967
-8.344128
-8.343335
-8.342446
-8.341707
LHSI
Notes: Augmented Dickey-Fuller tests (ADF) (Dickey and Fuller, 1979), is used to determine whether the time series
is stationary or not. It has the same hypothesis as the DF test as below:
Hypothesis H3:
 u  1 against the alternative hypothesis  u  1 .
The ADF test fix the problem of the DF test by adding an number of lagged first differences of the dependent variable
to compensate autocorrelated omitted variables that would exist from
Figures in parentheses are MacKinnon (1996) one-sided p-values.
12
t
Table 8 Augmented Dickey-Fuller Test: LHSIF
LHSIF
Number
of lags
ADF Test
Statistics
1% Critical
Value
Prob.*
AIC
BIC
Without
Trend
1
2
3
4
5
6
7
8
9
10
-0.872615
-0.890765
-0.892205
-0.875868
-0.877691
-0.897404
-0.872028
-0.862152
-0.865515
-0.879511
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
(0.7974)
(0.7918)
(0.7913)
(0.7964)
(0.7958)
(0.7897)
(0.7976)
(0.8006)
(0.7995)
(0.7953)
-8.067249
-8.067116
-8.066872
-8.066711
-8.066463
-8.066400
-8.066386
-8.066184
-8.065940
-8.065765
-8.065328
-8.064556
-8.063671
-8.062869
-8.061980
-8.061277
-8.060622
-8.059779
-8.058894
-8.058078
With Trend
1
2
3
4
5
6
7
8
9
10
-3.250798
-3.266717
-3.257966
-3.244600
-3.257531
-3.297858
-3.271527
-3.249940
-3.244202
-3.250056
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
(0.0747)
(0.0719)
(0.0734)
(0.0759)
(0.0735)
(0.0666)
(0.0710)
(0.0749)
(0.0759)
(0.0749)
-8.067989
-8.067861
-8.067611
-8.067446
-8.067206
-8.067163
-8.067140
-8.066927
-8.066678
-8.066503
-8.065428
-8.064660
-8.063769
-8.062964
-8.062083
-8.061400
-8.060736
-8.059882
-8.058992
-8.058175
Notes: Augmented Dickey-Fuller tests (ADF) (Dickey and Fuller, 1979), is used to determine whether the time series is
stationary or not. It has the same hypothesis as the DF test as below:
Hypothesis H4:
 u  1 against the alternative hypothesis  u  1 .
The ADF test fix the problem of the DF test by adding an number of lagged first differences of the dependent variable to
compensate autocorrelated omitted variables that would exist from
t
Figures in parentheses are MacKinnon (1996) one-sided p-values.
13
Table 9 Augmented Dickey-Fuller Test: LTraHK
LTraHK
Number
of lags
ADF Test
Statistics
1% Critical
Value
Prob.*
AIC
BIC
Without
Trend
1
2
3
4
5
6
7
8
9
10
-1.089116
-0.921503
-0.858574
-0.840575
-0.802058
-0.804542
-0.811313
-0.800542
-0.797494
-0.796153
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
(0.7224)
(0.7820)
(0.8016)
(0.8070)
(0.8180)
(0.8173)
(0.8154)
(0.8185)
(0.8193)
(0.8197)
-7.367440
-7.377098
-7.378351
-7.378217
-7.378543
-7.378284
-7.378075
-7.377875
-7.377619
-7.377395
-7.365520
-7.374537
-7.375150
-7.374375
-7.374060
-7.373161
-7.372311
-7.371470
-7.370573
-7.369708
With Trend
1
2
3
4
5
6
7
8
9
10
-3.58965
-3.373291
-3.29982
-3.279552
-3.24156
-3.243322
-3.267852
-3.247938
-3.242207
-3.224640
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
(0.0307)
(0.0550)
(0.0663)
(0.0697)
(0.0764)
(0.0761)
(0.0717)
(0.0752)
(0.0763)
(0.0796)
-7.368337
-7.377903
-7.379129
-7.378986
-7.379300
-7.379042
-7.378847
-7.378637
-7.378378
-7.378143
-7.365777
-7.374702
-7.375287
-7.374504
-7.374177
-7.373279
-7.372443
-7.371592
-7.370692
-7.369815
Notes: Augmented Dickey-Fuller tests (ADF) (Dickey and Fuller, 1979), is used to determine whether the time series is
stationary or not. It has the same hypothesis as the DF test as below:
Hypothesis H5:
 u  1 against the alternative hypothesis  u  1 .
The ADF test fix the problem of the DF test by adding an number of lagged first differences of the dependent variable to
compensate autocorrelated omitted variables that would exist from
t
Figures in parentheses are MacKinnon (1996) one-sided p-values.
14
Table 10 Augmented Dickey-Fuller Test: HSI Return
Number
of lags
ADF Test
Statistics
1% Critical
Value
Prob.*
AIC
BIC
Without
Trend
1
2
3
4
5
6
7
8
9
10
-73.64212
-59.99632
-53.36945
-47.75562
-43.04432
-39.64476
-37.44254
-35.2374
-33.09438
-31.49429
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
0.859917
0.860047
0.859924
0.860184
0.860226
0.860433
0.860564
0.860817
0.860915
0.861160
0.861837
0.862608
0.863125
0.864026
0.864708
0.865556
0.866328
0.867222
0.867961
0.868847
With Trend
1
2
3
4
5
6
7
8
9
10
-73.64617
-60.00143
-53.37575
-47.76284
-43.05246
-39.65359
-37.45172
-35.24717
-33.10429
-31.50473
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
0.860028
0.860160
0.860034
0.860294
0.860335
0.860543
0.860675
0.860928
0.861029
0.861274
0.862588
0.863361
0.863875
0.864776
0.865458
0.866307
0.867080
0.867974
0.868716
0.869603
HSI Return
Notes: Augmented Dickey-Fuller tests (ADF) (Dickey and Fuller, 1979), is used to determine whether the time
series is stationary or not. It has the same hypothesis as the DF test as below:
Hypothesis H6:
 u  1 against the alternative hypothesis  u  1 .
The ADF test fix the problem of the DF test by adding an number of lagged first differences of the dependent
variable to compensate autocorrelated omitted variables that would exist from
Figures in parentheses are MacKinnon (1996) one-sided p-values.
15
t
Table 11 Augmented Dickey-Fuller Test: HSIF Return
HSIF Return
Number
of lags
ADF Test
Statistics
1% Critical
Value
Prob.*
AIC
BIC
Without
Trend
1
2
3
4
5
6
7
8
9
10
-75.6567
-61.76625
-54.04217
-48.16947
-43.36379
-40.83132
-38.40268
-36.08323
-33.90733
-32.12085
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
(.0001)
(.0001)
(.0001)
(.0001)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
1.143119
1.143363
1.143522
1.143771
1.143836
1.143846
1.144047
1.144292
1.144468
1.144664
1.145039
1.145924
1.146723
1.147613
1.148319
1.148970
1.149811
1.150697
1.151515
1.152351
With Trend
1
2
3
4
5
6
7
8
9
10
-75.66004
-61.77066
-54.04765
-48.17588
-43.37104
-40.8394
-38.4113
-36.09227
-33.91666
-32.13086
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
1.143235
1.143480
1.143637
1.143886
1.143951
1.143959
1.144160
1.144406
1.144585
1.144779
1.145795
1.146682
1.147479
1.148368
1.149074
1.149723
1.150565
1.151452
1.152272
1.153107
Notes: Augmented Dickey-Fuller tests (ADF) (Dickey and Fuller, 1979), is used to determine whether the time series is
stationary or not. It has the same hypothesis as the DF test as below:
Hypothesis H7:
 u  1 against the alternative hypothesis  u  1 .
The ADF test fix the problem of the DF test by adding an number of lagged first differences of the dependent variable to
compensate autocorrelated omitted variables that would exist from
t
Figures in parentheses are MacKinnon (1996) one-sided p-values.
16
Table12 Augmented Dickey-Fuller Test: TraHK Return
TraHK
Return
Number
of lags
ADF Test
Statistics
1% Critical
Value
Prob.*
AIC
BIC
Without
Trend
1
2
3
4
5
6
7
8
9
10
-95.1876
-73.97752
-61.55965
-54.68218
-48.62138
-44.05965
-41.03198
-38.38998
-36.18235
-33.90041
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
-3.4341
(.0001)
(.0001)
(.0001)
(.0001)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
(.0000)
1.833142
1.831879
1.832011
1.831679
1.831938
1.832148
1.832347
1.832603
1.832827
1.832962
1.835063
1.834440
1.835212
1.835521
1.836421
1.837272
1.838111
1.839008
1.839873
1.840650
With Trend
1
2
3
4
5
6
7
8
9
10
-95.1902
-73.98198
-61.56544
-54.68929
-48.62942
-44.06872
-41.04177
-38.40047
-36.19327
-33.9117
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
-3.9645
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
(.0001)
1.833254
1.831986
1.832117
1.831782
1.832041
1.832250
1.832449
1.832705
1.832930
1.833068
1.835815
1.835188
1.835959
1.836264
1.837164
1.838014
1.838854
1.839751
1.840617
1.841396
Notes: Augmented Dickey-Fuller tests (ADF) (Dickey and Fuller, 1979), is used to determine whether the time series
is stationary or not. It has the same hypothesis as the DF test as below:
Hypothesis H8:
 u  1 against the alternative hypothesis  u  1 .
The ADF test fix the problem of the DF test by adding an number of lagged first differences of the dependent variable
to compensate autocorrelated omitted variables that would exist from
Figures in parentheses are MacKinnon (1996) one-sided p-values.
17
t
4.4. Cointegration Tests
Cointegration Test results of Trace and Max-Eigen Statistics are shown in Table 14 and
Table 15 respectively. There are ten outputs from five lags in both tests. In each output,
there are several items: the number of the cointegration equation, eigenvalue, 5% critical
value and 1% critical value, and trace statistics or max-eigen statistics. All of the results
show that there are two cointegration equations in the variables of the HSI, HSIF and
TraHK. These cointegration equations are significant at both the 5% and 1% levels. Thus,
we conclude that the HSI, HSIF and TraHK are trivariately cointegrated. Normalized
cointegration equations are provided in Table 16.
4.5. Error Correction Model (ECM)
As we found the cointegration relationships between the HSI, HSIF and TraHK, the
trivariate cointegration test will now be employed. In this part of study, tests are examined
from one to five lags. We believe that more test results can enforce our findings. To gain a
clear understanding of our results, only the tests with five lags are shown in detail.
To perform the trivariate cointegration, we should firstly compute two error correction
terms in the respective error correction models. The values of the error correction terms
are shown in Table 17. Then, the error correction terms are applied in the cointegration
equations. Table 17 (continued) shows three cointegration equations with variables in five
lags. From these equations, the lead-lag structure between the HSI, HSIF and TraHK are
shown.
The test produces interesting findings. HSIF returns certainly led both HSI and TraHK
returns up to five lags. This means that the influence of the HSIF returns on the HSI and
TraHK are last from 15 minutes to 75 minutes. Similar to the HSIF returns, the HSI
returns led TraHK returns from 15 minutes to 75 minutes. For the TraHK returns, the
influence was only available for 15 minutes. The results show that the TraHK returns led
both the HSI and HSIF for 15 minutes. For other lags, the results of ECM are the same as
the results with five lags. All corresponding results are shown in Table 18 to Table 21.
4.6. Pairwise Cointegration Regressions
Compared to the results of trivariate cointegration from the Johansen approach, we
demonstrate bi-variate cointegration regressions. Pairwise regressions are applied in
different groups: HSI and HSIF; HSI and TraHK; HSIF and TraHK. The findings are
statistically significant, and the results are shown in Table 22. For all pairs, the constant
and coefficient of the independent variables are significant at the 1% level. The R-squared
values are very high and reach around 0.99. This shows that the cointegration regressions
are very successful in estimating the corresponding dependent variables.
Furthermore, we apply ADF tests to all of the above pairs. The purpose is to examine the
stationarity of the cointegration regressions, so we only choose up to five lags in both with
and without trend cases. The results of both with and without trend cases are the same: all
of the regressions are stationary. The lowest AIC and BIC are marked in one lag (see
Table 23 - Table 25).
18
4.7. Hedging Performance
Test Period Classification
In this part, we divide our study into three periods as shown in Table 13. Period 1
covers periods before and after the issue of the TraHK. Period 2 covers the date before
the issue of the TraHK and Period 3 only covers the date after the issue of the TraHK. In
this case, Period 1 (all samples) covers a period between the corresponding month
before the issue of the TraHK and the corresponding months after the issue of the
TraHK. There are 22 months of Pre-HK period, but 31 months for Post-TraHK period.
Details of findings are recorded on Table 28 to Table 30.
Table 13 Hedging Performance – Test Period Classification
Period:
Time Intervals:
1: All Period
2nd January, 1998 - 28th June, 2002
2: Pre-Period
2nd January, 1998 - 11th November, 1999.
3: Post-Period
12th November, 1999 - 28th June, 2002
4.7.1. Hedge Ratios
Table 26 presents the mean and variance of the optimal hedge ratios from three different
methods of data classification. Out of all three periods, the average ratio of Post-TraHK is
higher than that of Pre-TraHK. However, the ratio variance of Pre-TraHK is higher than
that of Post-TraHK. The existence of the TraHK increases the mean ratio and decreases
the ratio variance of the HSIF. This indicates that, with the TraHK, HSIF is more
successful in hedging the HSI and its variability of the hedge ratio is even more stable.
4.7.2. Hedge Effectiveness
Table 27, the same as Table 26, provides hedge effectiveness of the HSIF. The mean
effectiveness increases in Post-TraHK. Generally, better hedging instruments mark higher
hedge effectiveness (Ht). Comparing all periods, the effectiveness variance decreases in
Post-TraHK. It seems the hedging effectiveness of the HSIF performs better and is more
stable after the advent of the TraHK.
5. Summary and Conclusion
A trivariate cointegration test has been used to study the cointegration among the HSI,
HSIF and TraHK. The aim of this article is to examine the long-term relationship amongst
these three markets. Several studies indicate that a long-term relationship exists in the
cash and futures markets. However, there are not many studies of the long-term
relationships of HSI, HSIF and other instruments. The TraHK is very similar to both the
HSI and HSIF. It is worthwhile comparing the relationship between the HSI, HSIF and
TraHK.
19
Based on past findings, we noted that there are long-term relationships between the
TraHK and HSI and between the TraHK and HSIF. The ECM was then applied, we
computed two error correction terms in the respective error correction model. From these
tests, the lead-lag structure amongst the HSI, HSIF and TraHK is shown.
The HSIF returns certainly led both the HSI and TraHK returns from 15 minutes to 75
minutes. Similar to the HSIF returns, the HSI returns led the TraHK returns from 15
minutes up to 75 minutes. For the TraHK returns, the influence was only available for 15
minutes. The results showed that the TraHK returns led both the HSI and HSIF for 15
minutes. Both the HSIF and TraHK are assumed to represent investors’ expectations
about the HSI. One explanation of our findings is that the market is ahead of the news
(Hamilton, 1922). In addition, Tvede (2002) stated when the market is ahead of the news,
it is quite simply ahead of the economy. Thus, the movements of both the HSIF and
TraHK led the movement of the HSI by up to 75 minutes.
Since the HSI, HSIF and TraHK are basically referred to the same instrument, they should
eventually restore equilibrium. Investors, however, can use the above patterns of deviation
among the three markets to obtain a possible arbitrage profit. On the other hand, it is
believed that the chance is small and the profit may not be economically significant after
taking into consideration of transaction costs. This situation is likely to be consistent with
an efficient market.
Furthermore, we studied the hedging performance of the HSIF via the optimal hedge
ratio and the hedging effectiveness. After the advent of the TraHK, there were
significant decreases in the mean and variance of both indicators. As the HSI variability
was reduced after the issue of the TraHK, the HSIF is easier to hedge than the HSI.
Thus, the hedge performance of the HSIF is improved.
20
Table 14 Johansen’s Multivariate Cointegration Test (Trace Statistic)
Variables:
HSI, HSIF, TraHK
Lag-Length:
k=0
Unrestricted Cointegration Rank Test
Hypothesized
Trace
5 Percent
1 Percent
No. of CE(s)
Eigenvalue
Statistic
Critical Value
Critical Value
None **
0.236861
5094.992
34.91
41.07
At most 1 **
0.159201
1991.769
19.96
24.6
At most 2
9.65E-05
1.107407
9.24
12.97
*(**) denotes rejection of the hypothesis at the 5%(1%) level
Trace test indicates 2 cointegrating equation(s) at both 5% and 1% levels
Lag-Length:
k=1
Unrestricted Cointegration Rank Test
Hypothesized
Trace
5 Percent
1 Percent
No. of CE(s)
Eigenvalue
Statistic
Critical Value
Critical Value
None **
0.09248
1931.101
34.91
41.07
At most 1 **
0.068622
817.1801
19.96
24.6
At most 2
9.90E-05
1.136122
9.24
12.97
*(**) denotes rejection of the hypothesis at the 5%(1%) level
Trace test indicates 2 cointegrating equation(s) at both 5% and 1% levels
Lag-Length:
k=2
Unrestricted Cointegration Rank Test
Hypothesized
Trace
5 Percent
1 Percent
No. of CE(s)
Eigenvalue
Statistic
Critical Value
Critical Value
None **
0.055657
1151.47
34.91
41.07
At most 1 **
0.042045
494.1742
19.96
24.6
At most 2
9.99E-05
1.147063
9.24
12.97
*(**) denotes rejection of the hypothesis at the 5%(1%) level
Trace test indicates 2 cointegrating equation(s) at both 5% and 1% levels
Lag-Length:
k=3
Unrestricted Cointegration Rank Test
Hypothesized
Trace
5 Percent
1 Percent
No. of CE(s)
Eigenvalue
Statistic
Critical Value
Critical Value
None **
0.0409
809.4251
34.91
41.07
At most 1 **
0.028258
330.1461
19.96
24.6
At most 2
0.000101
1.164114
9.24
12.97
*(**) denotes rejection of the hypothesis at the 5%(1%) level
Trace test indicates 2 cointegrating equation(s) at both 5% and 1% levels
Lag-Length:
k=4
Unrestricted Cointegration Rank Test
Hypothesized
Trace
5 Percent
No. of CE(s)
Eigenvalue
Statistic
Critical Value
None **
0.034886
651.0502
34.91
At most 1 **
0.020901
243.5521
19.96
At most 2
9.97E-05
1.144516
9.24
*(**) denotes rejection of the hypothesis at the 5%(1%) level
Trace test indicates 2 cointegrating equation(s) at both 5% and 1% levels
21
1 Percent
Critical Value
41.07
24.6
12.97
Table 15 Johansen’s Multivariate Cointegration Test (Max-Eigen Statistic)
Variables:
HSI, HSIF, TraHK
Lag-Length:
k=0
Hypothesized
Max-Eigen
5 Percent
1 Percent
No. of CE(s)
Eigenvalue
Statistic
Critical Value
Critical Value
None **
0.236861
3103.224
22
26.81
At most 1 **
0.159201
1990.661
15.67
20.2
At most 2
9.65E-05
1.107407
9.24
12.97
*(**) denotes rejection of the hypothesis at the 5%(1%) level
Max-eigenvalue test indicates 2 cointegrating equation(s) at both 5% and 1% levels
Lag-Length:
k=1
Hypothesized
Max-Eigen
5 Percent
1 Percent
No. of CE(s)
Eigenvalue
Statistic
Critical Value
Critical Value
None **
0.09248
1113.92
22
26.81
At most 1 **
0.068622
816.044
15.67
20.2
At most 2
9.90E-05
1.136122
9.24
12.97
*(**) denotes rejection of the hypothesis at the 5%(1%) level
Max-eigenvalue test indicates 2 cointegrating equation(s) at both 5% and 1% levels
Lag-Length:
k=2
Hypothesized
Max-Eigen
5 Percent
1 Percent
No. of CE(s)
Eigenvalue
Statistic
Critical Value
Critical Value
None **
0.055657
657.2953
22
26.81
At most 1 **
0.042045
493.0272
15.67
20.2
At most 2
9.99E-05
1.147063
9.24
12.97
*(**) denotes rejection of the hypothesis at the 5%(1%) level
Max-eigenvalue test indicates 2 cointegrating equation(s) at both 5% and 1% levels
Lag-Length:
k=3
Hypothesized
Max-Eigen
5 Percent
1 Percent
No. of CE(s)
Eigenvalue
Statistic
Critical Value
Critical Value
None **
0.0409
479.279
22
26.81
At most 1 **
0.028258
328.982
15.67
20.2
At most 2
0.000101
1.164114
9.24
12.97
*(**) denotes rejection of the hypothesis at the 5%(1%) level
Max-eigenvalue test indicates 2 cointegrating equation(s) at both 5% and 1% levels
Lag-Length:
k=4
Hypothesized
Max-Eigen
5 Percent
1 Percent
No. of CE(s)
Eigenvalue
Statistic
Critical Value
Critical Value
None **
0.034886
407.4981
22
26.81
At most 1 **
0.020901
242.4076
15.67
20.2
At most 2
9.97E-05
1.144516
9.24
12.97
*(**) denotes rejection of the hypothesis at the 5%(1%) level
Max-eigenvalue test indicates 2 cointegrating equation(s) at both 5% and 1% levels
22
Table 16 Johansen’s Multivariate Cointegration Test
Variables:
HSI (HSI), HSIF (HSIF), TraHK (TraHK)
Lag-Length: k = 0
2 Cointegrating Equation(s):
Log likelihood
Normalized cointegrating coefficients (std.err. in parentheses)
LN_FUTURES
LN_INDEX
LN_TraHK
1.000000
0.000000
-1.021088
(0.000710)
0.000000
1.000000
-1.017033
(0.000790)
Lag-Length:
k=1
2 Cointegrating Equation(s):
Log likelihood
Normalized cointegrating coefficients (std.err. in parentheses)
LN_FUTURES
LN_INDEX
LN_TraHK
1.000000
0.000000
-1.021038
(0.001060)
0.000000
1.000000
-1.016989
(0.001160)
Lag-Length:
k=2
2 Cointegrating Equation(s):
Log likelihood
Normalized cointegrating coefficients (std.err. in parentheses)
LN_FUTURES
LN_INDEX
LN_TraHK
1.000000
0.000000
-1.020948
(0.001370)
0.000000
1.000000
-1.016946
(0.001490)
Lag-Length:
k=3
2 Cointegrating Equation(s):
Log likelihood
Normalized cointegrating coefficients (std.err. in parentheses)
LN_FUTURES
LN_INDEX
LN_TraHK
1.000000
0.000000
-1.020842
(0.001670)
0.000000
1.000000
-1.016859
(0.001810)
Lag-Length:
k=4
2 Cointegrating Equation(s):
Log likelihood
Normalized cointegrating coefficients (std.err. in parentheses)
LN_FUTURES
LN_INDEX
LN_TraHK
1.000000
0.000000
-1.020762
(0.001930)
0.000000
1.000000
-1.016801
(0.002090)
23
145462.5
C
0.206514
(0.006790)
0.168105
(0.007550)
147313.2
C
0.206030
(0.010050)
0.167679
(0.011040)
147932.8
C
0.205165
(0.013050)
0.167265
(0.014180)
148251.3
C
0.204147
(0.015950)
0.166432
(0.017240)
148397.6
C
0.203382
(0.018380)
0.165876
(0.019880)
Table 17 Error Correction Term (Number of lags: 5)
A.I. Z1,t-1=α1lnHSIt-1-δ1lnTraHK t-1
Z1,t-1
α1
δ1
1
-1.016693
N.A.
-0.00238
[427.261]
A.II. Z2,t-1=α2lnHSIFt-1-δ2lnTraHK t-1
Z2,t-1
α2
δ2
1
-1.020611
N.A.
-0.0022
[463.160]
(Continued on next page)
24
Table 17 Error Correction Term (Number of lags: 5) Error Correction Term (Number of lags: 5) (continued)
a
rHSI
5
5
5
i 1
i 1
i 1
rHSIt   HSI  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   HSI
B.I.
t
b
cHSI ,1
cHSI , 2
cHSI ,3
cHSI , 4
cHSI ,5
-0.065388
0.085783
-0.41266
-0.279377
-0.149212
-0.118763
-0.055404
0.46363
-0.01261
-0.01301
-0.02022
-0.02131
-0.02119
-0.02005
-0.01632
-0.01789
[-3.39576]
[ 25.9173
]
[-5.18499]
[ 6.59181]
[-20.4106]
[-13.1110]
[-7.04177]
5
5
d HSIF,3
d HSIF, 4
d HSIF,5
eTraHK,1
0.025126
eTraHK , 2
eTraHK,3
0.279521
0.163837
0.100964
0.06241
-0.01965
-0.0197
0.01874
0.01551
[4.2258]
[ 8.31579
]
[ 5.38763
]
[ 4.02434
]
[ 2.80301
]
d HSIF,3
d HSIF, 4
d HSIF,5
eTraHK,1
0.023434
0.005819 - 0.006476
-0.01094
-0.01229
-0.00896
0.01264
0.004133
-0.01006
[ 1.25591
]
eTraHK , 2
eTraHK , 4
eTraHK,5
0.005274
0.003237
0.01019
-0.00942
-0.00741
[ 0.40576]
[ 0.56002
]
[ 0.43677
]
eTraHK,3
eTraHK , 4
eTraHK,5
0.002832
0.007457
-0.00905
Rsquared
Log
AIC
likelihood
Schwarz
SC
-8.456169
8.467055
0.120996
48596.73
Rsquared
Log
likelihood
0.004208
46308.57
-8.057361
8.068247
Rsquared
Log
likelihood
AIC
0.341354
44149.53
-7.681058
7.691944
5
rHSIFt   HSIF  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   HSIF
B.II.
i 1
rHSIF
t
i 1
i 1
d HSIF,1
d HSIF, 2
-0.014387
0.020082
0.027638
0.019083
0.025772
0.018802
-0.01992
-0.02184
-0.02398
-0.02405
-0.02288
0.01893
0.01243
-0.0115
[-0.72237]
[0.91965]
15230]
[ 0.79348
]
[ 1.12663
]
[ 0.99320
]
[ 2.14163
]
[ 0.47366
]
[0.52086]
[ 0.24634
]
[ 0.82423
]
d HSIF,5
eTraHK,1
eTraHK , 2
eTraHK,3
eTraHK , 4
eTraHK,5
a
b
c HSI ,1
cHSI , 2
cHSI ,3
cHSI , 4
cHSI ,5
0.071439
-0.055919
0.000425
-0.035851
-0.008593
-0.046563
-0.01539
-0.01589
-0.02468
-0.02601
-0.02587
-0.02448
[ 4.64074]
B.III.
[-5.92207]
d HSIF, 2
d HSIF,1
[-3.52013]
[ 0.01720]
[-1.37832]
[-0.33220]
5
[.90208]
5
AIC
Schwarz
SC
5
rTraHKt  TraHK  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   TraHK
i 1
25
rTraHK t
i 1
i 1
d HSIF,1
d HSIF, 2
d HSIF,3
d HSIF, 4
0.072562
0.57518
4
0.429191
0.28325
0.165379
0.08207
1
-0.68557
-0.540514 - 0.404262
0.260398
-0.14256
-0.02404
-0.02636
-0.02895
0.02761
-0.02285
-0.01321
-0.01483
0.01501
-0.01388
-0.01092
[ 3.01853]
[ 21.8229
]
[ 5.98963
]
[ 3.59185
]
[51.9091]
[36.4500]
[26.9368]
[18.7654]
[13.0554]
a
b
c HSI ,1
cHSI , 2
cHSI ,3
cHSI , 4
cHSI ,5
-0.02652
0.087306
0.147158
0.118566
0.149386
0.104382
-0.01858
-0.01917
-0.02979
-0.0314
-0.03122
-0.02955
[ 1.42731]
[ 4.55336]
[4.94007]
[ 3.77653]
[ 4.78495]
[ 3.53266]
[4.8251]
Notes: T-statistics are in [].
ab Z1,t-1, and Z2,t-1 refer to the error correction terms of trivariate cointegration test. The cointegration equations are as below:
Z1,t-1=α1lnHSIt-1-δ1lnTrahk t-1, Z2,t-1=α2lnHSIFt-1-δ2lnTrahk t-1
cde The ECM are the followings:
5
5
5
i 1
i 1
i 1
rHSIt   HSI  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   HSI
5
5
5
i 1
i 1
i 1
rHSIFt   HSIF  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   HSIF
5
5
5
i 1
i 1
i 1
rTraHKt  TraHK  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   TraHK
0.02903
[ 9.75769
]
Schwarz
SC
Table 18 Error Correction Term (Number of lags: 1)
A.I. Z1,t-1=α1lnHSIt-1-δ1lnTrahk t-1
Z1,t-1 a
α1
1
N.A.
A.II. Z2,t-1=α2lnHSIFt-1-δ2lnTrahk t-1
Z2,t-1 b
α2
1
N.A.
B.I. rHSI
t
δ2
-1.021038
-0.00106
[-967.135]
  HSI  aZ1,t 1  bZ 2,t 1  CHSI rHSI ,t 1  d HSIF rt 1  eTrahk rt 1   HSI c
a
rHSIt
B.II. rHSIF
t
-0.139704
-0.01154
[-12.1103]
b
0.160594
-0.01176
[ 13.6604]
c HSI ,1
-0.221497
-0.01547
[-14.3176]
d HSIF ,1
eTraHK ,1
0.307728
-0.01398
[ 22.0188]
0.024334
-0.00649
[ 3.74727]
R-squared
0.103326
Log likelihood
48501.02
Akaike AIC
-8.449521
Schwarz SC
-8.44632
  HSIF  aZ1,t 1  bZ 2,t 1  CHSI rHSI ,t 1  d HSIF rt 1  eTrahk rt 1   HSIF d
a
26
rHSIFt
B.III. rTraHK
t
0.0623
-0.01395
[ 4.46666]
b
-0.048024
-0.01421
[-3.37864]
c HSI ,1
0.02085
-0.0187
[ 1.11469]
d HSIF ,1
eTraHK ,1
-0.034693
-0.0169
[-2.05315]
0.020216
-0.00785
[ 2.57483]
 TraHK  aZ1,t 1  bZ 2,t 1  CHSI rHSI ,t 1  d HSIF rt 1  eTrahk rt 1   TraHK
a
rTraHKt
δ1
1.016989
-0.00116
[-877.457]
0.021787
-0.01783
[ 1.22167]
Notes: See Notes to Table 17.
b
0.223466
-0.01817
[ 12.2961]
c HSI ,1
0.129935
-0.02392
[ 5.43313]
R-squared
0.00339
Log likelihood
46321.81
Akaike AIC
-8.069834
Schwarz SC
-8.066633
e
d HSIF ,1
eTraHK ,1
0.316868
-0.0216
[ 14.6664]
-0.345798
-0.01004
[-34.4469]
R-squared
0.260294
Log likelihood
43500.75
Akaike AIC
-7.578318
Schwarz SC
-7.575117
Table 19 Error Correction Term (Number of lags: 2)
A.I. Z1,t-1=α1lnHSIt-1-δ1lnTraHK t-1
Z1,t-1
α1
δ1
1
-1.01695
N.A.
-0.00149
[-682.962]
A.II. Z2,t-1=α2lnHSIFt-1-δ2lnTRAHK t-1
Z2,t-1
α2
δ2
1
-1.02095
N.A.
-0.00137
[-744.930]
B.I.
2
2
2
i 1
i 1
i 1
rHSIt   HSI  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT  I   eTraHK ,i rTRAHK t i   HSI
rHSIt
27
B.II. rHSIF
t
a
b
c HSI ,1
cHSI , 2
d HSIF ,1
-0.10132
-0.01192
[-8.50200]
0.122314
-0.01222
[ 10.0116]
-0.34091
-0.01845
[-18.4774]
-0.15715
-0.01599
[-9.82555]
0.402217
-0.01635
[ 24.6058]
0.172945
-0.01509
[ 11.4610]
2
2
2
i 1
i 1
i 1
eTraHK ,1
eTraHK , 2
0.02656
-0.00774
[ 3.42991]
0.014886
-0.00695
[ 2.14305]
R-squared
Log likelihood
Akaike AIC
Schwarz SC
0.114432
48568.16
-8.461432
-8.45631
R-squared
Log likelihood
Akaike AIC
Schwarz SC
0.003604
46318.54
-8.069444
-8.064322
R-squared
Log likelihood
Akaike AIC
Schwarz SC
0.297434
43792.1
-7.629221
-7.624099
  HSIF  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   HSIF
rHSIFt
B.III. rTraHK
t
a
b
c HSI ,1
cHSI , 2
d HSIF ,1
0.066641
-0.0145
[ 4.59674]
-0.05007
-0.01486
[-3.36860]
0.007657
-0.02244
[ 0.34116]
-0.02339
-0.01946
[-1.20193]
-0.02906
-0.01989
[-1.46136]
d HSIF , 2
0.012523
-0.01836
[ 0.68220]
2
2
2
i 1
i 1
i 1
eTraHK ,1
0.026465
-0.00942
[ 2.80937]
eTraHK , 2
0.010976
-0.00845
[ 1.29889]
 TraHK  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   TraHK
a
rTraHKt
d HSIF , 2
b
0.028385
0.157388
-0.01807
-0.01852
[ 1.57111]
[ 8.49742]
Notes: See Notes to Table 17.
c HSI ,1
cHSI , 2
d HSIF ,1
0.105621
-0.02797
[ 3.77610]
0.062758
-0.02425
[ 2.58816]
0.463219
-0.02478
[ 18.6918]
d HSIF , 2
0.229387
-0.02288
[ 10.0270]
eTraHK ,1
-0.50275
-0.01174
[-42.8243]
eTraHK , 2
-0.24601
-0.01053
[-23.3608]
Table 20 Error Correction Term (Number of lags: 3)
A.I. Z1,t-1=a1lnHSIt-1- δ 1lnTraHK t-1
Z1,t-1
a1
1
N.A.
A.II. Z2,t-1=α2lnHSIFt-1-δ2lnTraHK t-1
Z2,t-1
a2
1
N.A.
δ1
-1.016859
-0.00181
[-561.645]
δ2
-1.020842
-0.00167
[-609.657]
3
3
i 1
rHSI
3
rHSIt   HSI  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   HSI
B.I.
t
a
b
-0.083277
0.10273
-0.01219
[-6.82934]
-0.01254
[ 8.19323]
i 1
cHSI ,1
i 1
d HSIF,1
d HSIF, 2
d HSIF,3
eTraHK,1
eTraHK , 2
eTraHK,3
R-squared
Log
likelihood
AIC
Schwarz
SC
-0.077095
0.438163
0.240397
0.103754
0.023233
0.010299
0.001049
0.118109
48587.35
8.464991
-8.457948
-0.01619
[-4.76144]
-0.01716
[ 25.5409]
-0.01805
[ 13.3169]
-0.01537
[ 6.74928]
-0.00836
[ 2.77998]
-0.00862
[ 1.19481]
-0.00718
[ 0.14606]
cHSI , 2
cHSI ,3
-0.381451
-0.231874
-0.01941
[-19.6518]
-0.0195
[-11.8932]
3
3
3
i 1
i 1
i 1
rHSIFt   HSIF  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   HSIF
B.II.
rHSIF
a
b
c HSI ,1
cHSI , 2
cHSI ,3
0.066714
-0.052105
0.008286
-0.023238
0.00786
-0.01486
[ 4.48821]
-0.01528
[-3.40910]
-0.02366
[ 0.35018]
-0.02377
[-0.97779]
-0.01974
[ 0.39822]
d HSIF, 2
d HSIF,3
eTraHK,1
eTraHK , 2
eTraHK,3
R-squared
Log
likelihood
AIC
Schwarz SC
-0.025603
0.018912
0.007254
0.021633
0.003267
-0.009944
0.00373
46314.82
-8.068976
-8.061933
-0.02091
[-1.22434]
-0.022
[ 0.85946]
-0.01874
[ 0.38711]
-0.01019
[ 2.12350]
-0.01051
[ 0.31094]
-0.00875
[-1.13593]
eTraHK,1
eTraHK , 2
eTraHK,3
R-squared
AIC
Schwarz SC
-0.595704
-0.01249
[-47.7045]
-0.395674
-0.01288
[-30.7201]
-0.202707
-0.01073
[-18.8913]
0.320389
-7.66183
-7.654787
d HSIF,1
t
28
B.III.
3
3
3
i 1
i 1
i 1
rTraHKt  TraHK  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   TraHK
a
rTraHK t
0.029326
-0.01822
[ 1.60955]
Notes: See Note to Table 17.
b
c HSI ,1
cHSI , 2
cHSI ,3
0.120357
-0.01873
[ 6.42424]
0.117256
-0.029
[ 4.04291]
0.068983
-0.02913
[ 2.36803]
0.092522
-0.02419
[ 3.82431]
d HSIF,1
0.529278
0.02563
[ 20.6481]
d HSIF, 2
0.351175
-0.02697
[ 13.0194]
d HSIF,3
0.157842
-0.02297
[ 6.87186]
Log
likelihood
43978.41
Table 21 Error Correction Term (Number of lags: 4)
A.I. Z1,t-1=a1lnHSIt-1- δ 1lnTraHK t-1
Z1,t-1
a1
δ1
1
-1.0168
N.A.
-0.00209
[-487.029]
A.II. Z2,t-1=α2lnHSIFt-1-δ2lnTraHK t-1
Z2,t-1
a2
δ2
1
-1.02076
N.A.
-0.00193
[-528.975]
rHSI
4
4
4
i 1
i 1
i 1
rHSIt   HSI  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   HSI
B.I.
t
cHSI ,1
a
b
-0.07447
0.094882
-0.39881
-0.01242
[-5.99754]
-0.01279
[ 7.41698]
-0.01993
[-20.0085]
cHSI , 2
cHSI ,3
cHSI , 4
d HSIF,1
d HSIF, 2
d HSIF,3
d HSIF, 4
eTraHK,1
eTraHK , 2
eTraHK,3
eTraHK , 4
R-squared
Log
likelihoo
d
AIC
Schwarz
SC
-0.25917
-0.12012
-0.0718
0.451524
0.261716
0.138432
0.060303
0.024977
0.012582
0.00428
0.005307
0.119678
48592.86
-8.466165
-8.457201
-0.0207
[-12.5231]
-0.0199
[-6.03488]
-0.01627
[-4.41221]
-0.01762
[ 25.6305]
-0.01911
[ 13.6931]
-0.01856
[ 7.45700]
-0.01548
[ 3.89524]
-0.00874
[ 2.85932]
-0.00952
[ 1.32205]
-0.00912
[ 0.46946]
-0.00733
[ 0.72395]
eTraHK,1
eTraHK , 2
eTraHK,3
eTraHK , 4
R-squared
Log
likelihoo
d
AIC
Schwarz SC
0.003987
46311.78
-8.068626
-8.059662
0.022148
-0.01066
[ 2.07847]
0.003633
-0.01161
[ 0.31296]
-0.009608
-0.01112
[-0.86384]
-0.001414
-0.00894
[-0.15806]
eTraHK,1
eTraHK , 2
eTraHK,3
eTraHK , 4
R-squared
Log
likelihoo
d
AIC
Schwarz SC
0.331317
44067.09
-7.677429
-7.668465
-0.646996
-0.01296
[-49.9298]
-0.477309
-0.01412
[-33.8095]
-0.316309
-0.01353
[-23.3865]
-0.145858
-0.01087
[-13.4127]
4
B.II.
4
4
rHSIFt   HSIF  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   HSIF
i 1
i 1
i 1
a
b
c HSI ,1
cHSI , 2
cHSI ,3
cHSI , 4
0.06757
-0.01515
[ 4.46084]
-0.052456
-0.01561
[-3.36139]
0.006346
-0.02431
[ 0.26099]
-0.026964
-0.02525
[-1.06803]
0.003671
0.02428
[ 0.15119]
-0.026174
-0.01985
[-1.31844]
d HSIF,1
d HSIF, 2
d HSIF,3
d HSIF, 4
rHSIF
t
29
B.III.
4
4
-0.02446
-0.02149
[-1.13819]
0.021543
-0.02332
[ 0.92396]
0.010989
-0.02265
[ 0.48523]
0.012132
-0.01889
[ 0.64239]
4
rTraHKt  TraHK  aZ1,t 1  bZ 2,t 1   cHSI ,i rHSIt 1   d HSIF ,i rHSIFT I   eTraHK ,i rTraHK t i   TraHK
i 1
a
b
0.028127
-0.01842
[ 1.52700]
0.101763
-0.01898
[ 5.36247]
c HSI ,1
cHSI , 2
i 1
i 1
cHSI ,3
cHSI , 4
0.116397
-0.02953
[ 3.94205]
0.069081
-0.02414
[ 2.86153]
d HSIF,1
d HSIF, 2
d HSIF,3
d HSIF, 4
rTraHK t
Notes: See Note to Table 17.
0.132402
-0.02957
[ 4.47791]
0.092793
-0.0307
[ 3.02248]
0.556068
-0.02613
[ 21.2781]
0.40026
-0.02835
[ 14.1169]
0.235782
-0.02754
[ 8.56181]
0.095537
-0.02297
[ 4.15999]
Table 22 Results of Pairwise Cointegration Regression
Dependent
Variable
Independent
Variable
Constant
Coefficient of
Indep. Var.
R-squared
DW
HSI
HSIF
0.040592*
0.995714*
0.999673
0.670633
HSI
TraHK
-0.155458*
1.015705*
0.998540
0.698711
HSIF
TraHK
-0.195316*
1.019911*
0.998542
0.782488
Notes:
*Significant at 1% level.
30
Table 23 Augmented Dickey-Fuller Test: Residuals of HSI_HSIF
HSI_HSIF
Number of
lags
ADF Test
Statistics
1% Critical
Value
Prob.*
AIC
BIC
31
Without
Trend
0
1
2
3
4
5
-48.091020
-31.777170
-25.557850
-22.234260
-20.421540
-18.962500
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
(0.0001)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
-9.196520
-9.298494
-9.329016
-9.341203
-9.345030
-9.347984
-9.195240
-9.296574
-9.326456
-9.338002
-9.341188
-9.343502
With Trend
0
1
2
3
4
5
-48.314680
-31.938210
-25.693360
-22.356490
-20.537430
-19.072290
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
-9.197923
-9.299149
-9.329419
-9.341488
-9.345258
-9.348166
-9.196003
-9.296588
-9.326218
-9.337647
-9.340776
-9.343043
*MacKinnon (1996) one-sided p-values.
Table 24 Augmented Dickey-Fuller Test: Residuals of HSI_TraHK
HSI_TraHK
Number of
lags
ADF Test
Statistics
1% Critical
Value
Prob.*
AIC
BIC
32
Without
Trend
0
1
2
3
4
5
-49.483010
-30.385640
-22.466660
-17.984350
-15.291280
-13.133830
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
(0.0001)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
-7.669581
-7.818823
-7.887815
-7.928563
-7.950790
-7.971363
-7.668301
-7.816903
-7.885254
-7.925362
-7.946948
-7.966881
With Trend
0
1
2
3
4
5
-49.619490
-30.473480
-22.530920
-18.033240
-15.330280
-13.163880
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
-7.670394
-7.819087
-7.887887
-7.928542
-7.950722
-7.971261
-7.668473
-7.816526
-7.884686
-7.924701
-7.946240
-7.966138
*MacKinnon (1996) one-sided p-values.
Table 25 Augmented Dickey-Fuller Test: Residuals of HSIF_TraHK
Number of
lags
ADF Test
Statistics
1% Critical
Value
Prob.*
AIC
BIC
Without
Trend
0
1
2
3
4
5
-52.990220
-32.401000
-23.896330
-19.259560
-16.506220
-14.185000
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
-3.4307
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
-7.575037
-7.717926
-7.785252
-7.821915
-7.841051
-7.860638
-7.573757
-7.716005
-7.782691
-7.818714
-7.837209
-7.856156
With Trend
0
1
2
3
4
5
-53.017670
-32.418180
-23.908360
-19.268100
-16.512490
-14.189070
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
-3.9588
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
(0.0000)
-7.575085
-7.717849
-7.785130
-7.821773
-7.840898
-7.860477
-7.573164
-7.715288
-7.781929
-7.817931
-7.836416
-7.855355
HSIF_TraHK
33
*MacKinnon (1996) one-sided p-values.
Table 26 Optimal Hedge Ratios
All period
Pre-period
Method
Mean
Variance
Mean
Variance
1
(Sum up)
0.544607
0.001984
0.533812
0.004219
2
(Individual)
0.642671
0.010725
0.610579
0.015557
3
(Grouping)
0.614484
0.003863
0.596491
0.007918
Notes:
We assumed the hedger tries to minimize the variance. At time t-1, the optimal
variance futures return. Therefore,
Post-period
Mean
Variance
0.552267
0.000257
0.665446
0.006409
0.674106
0.000260
p-value
(0.0000)
(0.0500)
(0.0000)
hedge ratio equals the conditional covariance divided by the conditional
X t 1  CovRHSI ,t , RHSIF ,t | I t 1  / Var RHSI ,t | I t 1 Var RHSIF ,t | I t 1 
where It-1 is the information set of time t-1.
t-Statistics
13.142
-2.084
-4.420
34
Table 27 Hedge Effectiveness
All period
Mean
Variance
Method
1
2
3
(Sum up)
(Individual)
(Grouping)
0.500609
0.586438
0.545942
0.001436
0.011923
0.012033
Pre-period
Mean
Variance
0.485049
0.588012
0.582596
0.002834
0.017744
0.008930
Post-period
Mean
Variance
0.511648
0.585321
0.601352
0.000151
0.008243
0.000179
t-Statistics
p-value
31.840
0.350
-0.903
(0.0000)
(0.7300)
(0.3770)
Notes:
Ederington (1979) shows that the hedging effectiveness is determined by the squared covariance divided by the product of cash and futures returns
variances which is calculated as follows:
H t  Cov 2 RHSI ,t , RHSIF,t | I t 1 / VarRHSI ,t | I t 1 VarRHSIF,t | I t 1 
35
Table 28 Optimal Hedge Ratios and Hedging Effectiveness (individual month)
36
Order of
Month
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Pre-TraHK
Hedge Ratioa Hedging Effe.b
0.690922
0.659407
0.624674
0.659515
0.627082
0.652873
0.660762
0.647051
0.678531
0.604651
0.725436
0.666980
0.693670
0.680544
0.709414
0.726555
0.575512
0.575901
0.745142
0.754372
0.562914
0.578960
0.614238
0.567576
0.677862
0.656738
0.233225
0.202212
0.295125
0.266066
0.654139
0.666672
0.632563
0.638453
0.534923
0.493364
0.650811
0.590808
0.613463
0.574384
0.665787
0.613197
0.459991
0.566539
Notes: See Note to Table 26 and Table 27.
Post-TraHK
Hedge Ratio Hedging Effe.
0.755175
0.658563
0.645384
0.559476
0.650086
0.569128
0.723978
0.614371
0.635234
0.574072
0.675056
0.710445
0.629144
0.594733
0.770183
0.604095
0.665557
0.545210
0.653055
0.527264
0.620502
0.536070
0.699145
0.632446
0.681125
0.536182
0.599161
0.581482
0.525479
0.353797
0.720962
0.640843
0.666522
0.670406
0.613225
0.494302
0.531127
0.440265
0.517655
0.425251
0.784229
0.649568
0.808829
0.786073
0.643102
0.619444
0.679706
0.588244
0.625479
0.468184
0.669088
0.620993
0.540055
0.525551
0.622546
0.581311
0.671262
0.638730
0.825628
0.700516
0.781134
0.697949
Table 29 Optimal Hedge Ratios and Hedging Effectiveness (All Sample:
Pre & Post Periods)
37
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Pre & PostTraHK
Month
1
-
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Notes: See Note to Table 26 and Table 27.
Hedge Ratio
0.721243
0.673276
0.661259
0.671442
0.668024
0.673754
0.671647
0.676953
0.671363
0.677262
0.668637
0.667071
0.668899
0.561981
0.531447
0.538962
0.548663
0.549092
0.550203
0.551223
0.737421
0.567638
0.568842
0.570010
0.570440
0.571301
0.571059
0.571386
0.571805
0.572909
0.573793
Hedging Effe.
0.659570
0.620538
0.615549
0.618979
0.613120
0.629547
0.631378
0.635387
0.628687
0.636547
0.627669
0.033116
0.625037
0.521649
0.491377
0.499681
0.513265
0.512542
0.512772
0.513335
0.519208
0.524029
0.525548
0.526207
0.525612
0.526458
0.526467
0.526826
0.527298
0.528041
0.528772
Table 30 Optimal Hedge Ratios and Hedging Effectiveness (Pre / Post Period)
38
Order of Month
Pre-TraHK
Post-TraHK
From
To Hedge Ratio Hedging Effe. Hedge Ratio Hedging Effe.
1
1
0.690922
0.659407
0.755175
0.658563
1
2
0.660149
0.660673
0.682261
0.593928
1
3
0.647821
0.658854
0.671410
0.586232
1
4
0.651842
0.657382
0.685989
0.594694
1
5
0.657159
0.647440
0.675722
0.590921
1
6
0.669884
0.651116
0.676197
0.615129
1
7
0.675507
0.658247
0.668559
0.612101
1
8
0.680164
0.667336
0.674167
0.611134
1
9
0.668349
0.657271
0.673702
0.606415
1
10
0.681433
0.673947
0.673021
0.603023
1
11
0.669768
0.664712
0.667347
0.595586
1
12
0.663989
0.654313
0.669927
0.598649
1
13
0.666990
0.655800
0.670763
0.594231
1
14
0.500020
0.480232
0.666706
0.593435
1
15
0.465151
0.443768
0.664168
0.588472
1
16
0.474115
0.454271
0.667570
0.591757
1
17
0.489716
0.472365
0.667646
0.597733
1
18
0.491749
0.473419
0.665926
0.594258
1
19
0.494706
0.475614
0.661835
0.589537
1
20
0.498098
0.478483
0.658638
0.585857
1
21
0.509673
0.487902
0.661736
0.587437
1
22
0.515598
0.484560
0.677114
0.607746
1
23
0.675717
0.608399
1
24
0.675915
0.607878
1
25
0.674870
0.604522
1
26
0.674783
0.604993
1
27
0.671688
0.603146
1
28
0.670908
0.602843
1
29
0.670946
0.603278
1
30
0.672803
0.604450
1
31
0.674069
0.605574
Notes: See Note to Table 26 and Table 27.
References
39
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