short-run deviations and optimal hedge ratio: evidence from stock futures
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Short-run deviations and optimal hedge ratio:evidence from stock futures
Taufiq Choudhry *
School of Management, University of Southampton, Bradford, Bradford BD9 4JL, UK
Received 26 May 2001; accepted 25 February 2002
Abstract
This paper investigates the effects of the long-run relationship between stock cash index and
futures index on the hedging effectiveness of six stock futures markets. Effectiveness of five
different hedging ratios depending on different estimation procedures is investigated. The
unhedged, the traditional hedge and the minimum variance hedge ratios are all constant while
the bivariate GARCH and GARCH-X hedge ratios are time varying. The effectiveness of the
hedge ratio is compared by investigating the total sample and the out-of-sample performance
of the five ratios. The total sample period consists of daily returns from January 1990 to
December 1999. Two out-of-sample periods used are from January 1998 to December 1999 (2
years) and from January 1999 to December 1999 (1 year). Results show that the time-varying
hedge ratio outperforms the constant hedge ratio.
# 2002 Elsevier Science B.V. All rights reserved.
Keywords: G1; G13; G15
Keywords: Hedge ratio; Bivariate GARCH; Variance
1. Introduction
According to Figlewski (1984) and Fortune (1989), the introduction of stock index
futures is the most important financial innovation affecting the quality of
information about future common stock prices and the possibility of unbundling
the market and non-market components of risk and return. Silber (1985) suggests
* Tel.: �/44-1274-234363; fax: �/44-1274-235680
E-mail address: t.choudhry@bradford.ac.uk (T. Choudhry).
J. of Multi. Fin. Manag. 13 (2003) 171�/192
www.elsevier.com/locate/econbase
1042-444X/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved.
PII: S 1 0 4 2 - 4 4 4 X ( 0 2 ) 0 0 0 4 2 - 7
that futures markets provide two main functions: risk transfer and price discovery.
Risks are transferred to those willing to bear them as hedgers reduce their risk by
paying a premium to speculators.1 In other words, stock index futures can be used to
hedge market risk caused by cash (spot) price fluctuations.2 The existence of futures
trading also enhances the ability of investors to form judgements about expected
forward cash prices (Fortune, 1989).
This paper investigates and compares the risk-reducing effectiveness of differentoptimal (minimum risk) hedge ratios for the stock futures of Australia, Germany,
Hong Kong, Japan, South Africa and the United Kingdom. The main contribution
of the paper is to investigate the effects of short-run deviations from the long-run
relationship between cash index and futures index on the optimal hedge ratio. An
optimal hedge ratio is defined as the proportion of a cash position that should be
covered with an opposite position on a futures market.
In this paper, the optimal hedge ratios are estimated by OLS regressions and the
generalized autoregressive conditional heteroscedasticity (GARCH) model. Theoptimal hedge ratio estimated by means of the OLS regressions are constant, while
the GARCH model, taking into consideration the time-varying distribution of the
cash and futures price changes, provides changing hedge ratio.3 Park and Switzer
(1995) claim that if the joint distribution of stock index and futures prices is changing
over time, estimating a constant hedge ratio may not be appropriate. In other words,
the hedge ratios will certainly vary over time as the conditional distribution between
cash and futures prices changes (Baillie and Myers, 1991). This paper is motivated by
the claims of Park and Switzer (1995) and Butterworth and Holmes (1996) whoindicate the lack of research in this field for stock market futures, especially for
markets other than the United States.
2. Optimal hedge ratios
This section describes the optimal hedge ratio, relying heavily on Cecchetti et al.
(1988) and Baillie and Myers (1991). The returns on the portfolio of an investortrying to hedge some proportion of the cash position in a futures market can be
represented by
1 According to Fortune (1989), risk transfer and price discovery also take place in the absence of
futures markets but these two factors are enhanced in the presence of futures markets. This happens
because the costs of futures transactions are considerably less than the cost of cash (spot) transactions.2 According to Figlewski (1984), risk minimization will depend upon the behaviour of the difference
between the futures prices and the cash prices. Thus some measure of risk will be imposed on hedging a
position in stock, the risk being that the change in the futures price over time will not track exactly the
value of the cash position.3 Park and Switzer (1995) also apply the GARCH method to estimate optimal hedge ratios for the
United States stock futures market. Baillie and Myers (1991) and Myers (1991) study the commodities
futures market using the GARCH method. Park and Switzer (1995) provide citations of other studies that
apply the GARCH model.
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192172
rt�rct �bt�1rf
t ; (1)
where rt is the return holding the portfolio of cash and futures position between t�/1
and t , rct the return on holding the cash position for the same period, rf
t the return on
holding the futures position for the same period and bt�1 the hedge ratio. Thevariance of the return on the hedged portfolio is give by
Var(rt=Vt�1)�Var(rct=Vt�1)�b2
t�1Var(rft =Vt�1)�2bt�1Cov(rc
t ; rft =Vt�1); (2)
where Vt�1 is the information available over the last period. As indicated by
Cecchetti et al. (1988), the return on a hedged position will normally be exposed torisk caused by unanticipated changes in the relative price between the position being
hedged and the futures contract. This ‘basic risk’ ensures that no hedge ratio
completely eliminates risk. The hedge ratio that minimizes risk may be obtained by
setting the derivative of Eq. (2) with respect to b equal to zero. The hedge ratio bt�1
can then be expressed as
bt�1��
Cov(rct ; rf
t=Vt�1)
Var(rft=Vt�1)
�: (3)
The value of bt�1 which minimizes the conditional variance of the hedged portfolio
return is the optimal hedge ratio (Baillie and Myers, 1991).4 Commonly, the value of
the hedge ratio is less than unity so that the hedge ratio that minimizes risk in the
absence of basic risk turns out to be dominated by b when basic risk is taken intoconsideration.5
There are three hedge strategies involving constant hedge ratios: the traditional
one-to-one hedge, the beta hedge and the minimum variance hedge. The traditional
hedge strategy involves adopting a futures position that is equal in magnitude but
opposite in sign to the established cash position (Butterworth and Holmes, 1996).
According to this strategy, the cash price and the futures price move closely together
and if the proportionate price changes in one market exactly match proportionate
price changes in the other market, then price risk will be eliminated.6 The traditionalhedge will not minimize risk because the cash and the futures market may not move
perfectly together. The minimum variance hedge ratio takes account of this lack of
perfect correlation and identifies the hedge ratio which minimizes risk. The minimum
variance hedge ratio is estimated as the slope coefficient of the following regression:
4 Similar analysis is provided in Figlewski (1984) and Myers (1991).5 According to Cecchetti et al. (1988), the optimal hedge ratio b can be expressed as rsc/sf, where r is
the correlation between futures price and cash price, sc the cash standard deviation and sf the futures
standard deviation. Thus, if the futures have the same or higher price volatility than the cash, the hedge
ratio can be no greater than the correlation between them, which will be less than unity.6 With the beta hedge strategy, the hedge ratio is calculated as the beta of the cash portfolio based on
the market model (Figlewski, 1984). This strategy takes account of the fact that the cash portfolio to be
hedged may not match the portfolio on which the futures contract is written (Butterworth and Holmes,
1996). Further, according to Figlewski (1984), the beta hedge ratio is only optimal when the position is to
be held until maturity of the futures. This paper does not investigate the beta hedge.
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192 173
rct �a�brf
t �ot; (4)
where rct ; rf
t and b are defined as before and ot is an error term.7 If the joint
distribution of cash and futures prices is changing through time, regression
employing past data will not correctly estimate the current risk minimizing hedgeratio (Cecchetti et al., 1988). Under time-varying distribution, the bivariate GARCH
models may be applied to estimate time-varying hedge ratios. According to Cecchetti
et al., another problem with the regression estimated hedge ratio is that it does not
take into consideration the effect on expected returns. Hedging away the risk must
also hedge away the expected return to bearing that risk. This is the cost of hedging,
and the reward to risk bearing in the futures market. This paper estimates the
traditional, the minimum variance and the time-varying hedge ratios, and compares
their hedging effectiveness for six stock futures market.
3. The bivariate GARCH, GARCH-X models and time-varying hedge ratio
3.1. Bivariate GARCH
As shown by Baillie and Myers (1991) and Bollerslev et al. (1992), weak
dependence of successive asset price changes may be modelled by means of the
GARCH model. According to the GARCH model, the conditional variance of a
time series depends upon the squared residuals of the process (Bollerslev, 1986). TheGARCH model has the advantage of incorporating heteroscedasticity into the
estimation procedure and it also captures the tendency for volatility clustering in
financial and economic data. The GARCH model may be applied both in univariate
and multivariate forms. Simplicity is the main advantage of a univariate GARCH
model, but it utilizes only the information in one market’s own history (Wahab,
1995). The multivariate GARCH model uses information from more than one
market’s history. According to Conrad et al. (1991), multivariate models provide
more precise estimates of the parameters because they utilize information in theentire variance�/covariance matrix of the errors. Further, the generated regressor
problem associated with univariate models is avoided in multivariate models because
it estimates all parameters jointly (Pagan, 1984). According to Engle and Kroner
(1995), multivariate GARCH models are useful in multivariate finance and
economic models, which require the modelling of both variance and covariance.
7 As pointed out by the referee, lead�/lag effect is important in a relationship between stock cash and
stock futures markets. Frino et al. (2000) provides a good discussion of the lead�/lag relationship between
stock cash and futures market. In this paper, Eq. (4) is also estimated with six leads and lags of the returns
in the future market as dependent variables. Results from the lead�/lag tests are not provided in order to
save space but are available on request.
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192174
Multivariate GARCH models allow the variance and covariance to depend on the
information set in a vector ARMA manner (Engle and Kroner, 1995). This, in
turn, leads to the unbiased and more precise estimate of the parameters (Wahab,
1995).
The following bivariate MA(1)-GARCH(p ,q) model may be used to represent the
returns from the stock cash and futures markets:
yt�m�ot�uot�1; (5)
ot=Vt�1�N(0;Ht); (6)
vech Ht�C�Xp
j�1
Aj vech o2t�j�
Xq
j�1
Bj vech Ht�j; (7)
where yt� (rct ; rf
t ) is a (2�/1) vector containing stock returns from the cash (spot) and
futures markets, Ht is a (2�/2) conditional covariance matrix, C is a (3�/1)
parameter vector (constant), Aj and Bj are (3�/3) parameter matrices and vech is thecolumn stacking operator that stacks the lower triangular portion of a symmetric
matrix. The moving average (MA) term uot�1 is included to capture the effect of
non-synchronous trading. According to Susmel and Engle (1994), non-synchronous
trading induces negative serial correlation and the MA term allows for autocorrela-
tion induced by discontinuous trading in the asset (as suggested by Scholes and
Williams, 1977).
Engle and Kroner (1995) and Wahab (1995) state that various restrictions may be
imposed in this parameterization to make estimation easier. A parsimoniousrepresentation can be obtained by imposing a diagonal restriction on the multi-
variate GARCH parameter matrices so that each variance and covariance element
depends only on its own past values and prediction errors (Bollerslev et al., 1988). In
other words, this presentation is obtained by assuming that matrices Aj and Bj
are diagonal. According to Engle and Kroner (1995) and Baillie and Myers (1991),
the stated restriction seems plausible since it implies that each variance and
covariance depends only on its own past values and prediction errors. The following
equations present a diagonal vech bivariate GARCH(1,1) conditional varianceequation(s):
H11;t�C1�A11(o1;t�1)2�B11(H11;t�1); (7a)
H12;t�C2�A22(o1;t�1o2;t�1)�B22(H12;t�1); (7b)
H22;t�C3�A33(o2;t�1)2�B33(H22;t�1): (7c)
In the bivariate GARCH(1,1) system, the diagonal vech parameterization involves
nine conditional variance parameters. To ensure a positive conditional variance, the
values of C , A11, A33, B11 and B33 are restricted to zero or greater. The ARCHprocess in the residuals from the cash equation is shown by the coefficient of
(o1,t�1)2, (A11), while the coefficient of (o2,t�1)2, (A33), represents the ARCH process
in the futures equation residuals. The parameters A22 and B22 represent the
covariance GARCH parameters, which account for the conditional covariance
between cash and futures prices. Significant covariance parameters imply strong
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192 175
interaction between the cash and futures prices.8 As advocated by Baillie and Myers
(1991, p. 116), it is vital to let the conditional covariance be time-dependent, as in the
bivariate GARCH model, rather than be a constant. This ability of the bivariate
GARCH model to have time-dependent conditional variance makes it ideal to
provide a time-variant hedge ratio.
Given the bivariate GARCH model of the cash and the futures stock returns
presented above, the time-varying hedge ratio can be expressed as
bt�H̄12;t
H̄22;t
; (8)
where H̄12; t is the estimated conditional variance between the cash and futures stock
returns and H̄22; t the estimated conditional variance of the futures returns from thebivariate GARCH model. Given that conditional covariance is time-dependent, the
optimal hedge ratio will be time-dependent.
3.2. Bivariate GARCH-X
An extension of the GARCH model linked to an error-correction model of
cointegrated series has been put forward by Lee (1994). This model is known as the
GARCH-X model. Cointegration implies that in a long-run relationship betweentwo or more nonstationary variables, it is required that these variables should not
move too far apart from each other. Such nonstationary variables might drift apart
in the short-run, but in the long-run they are constrained. Thus, cointegration means
that one or more linear combinations of these variables is stationary even though
individually they are not. The error-correction term, which represents the short-run
deviations from the long-run cointegrated relationship, has important predictive
powers for the conditional mean of the cointegrated series (Engle and Yoo, 1987).
According to Lee (1994), if short-run deviations affect the conditional mean, theymay also affect conditional variance, and a significant positive effect may imply that
the further the series deviate from each other in the short-run, the harder they are to
predict. If the error-correction term (short-run deviations) from the cointegrated
relationship between cash price and futures price affects the conditional variance
(and conditional covariance), then conditional heteroscedasticity may be modelled
with a function of the lagged error correction term. If shocks to the system that
propagate on the first and the second moments change the volatility, then it is
reasonable to study the behaviour of conditional variance as a function of short-rundeviations (Lee, 1994). Given that short-run deviations may affect the conditional
variance and conditional covariance, then it will also influence the time-varying
optimal hedge ratio as defined in Eq. (8).
The following bivariate MA(1)-GARCH(p ,q )-X model may be used to represent
the returns from the stock cash and futures markets:
8 Bera and Higgins (1993) and Engle and Kroner (1995) provide detailed analysis of multivariate
GARCH models.
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192176
yt�m�d(zt�1)�ot�uot�1; (9)
ot=Vt�1�N(0;Ht); (10)
vech Ht�C�Xp
j�1
Aj vech(ot�j)2�
Xq
j�1
Bj vech Ht�j
�Xk
j�1
Dj vech(zt�1)2; (11)
where zt is the error-correction term from the cointegration relationship between
stock cash index and futures index.9 The remaining variables are as described earlier.The error-correction term in the returns equation (Eq. (9)) measures the affect of the
short-run deviations on the cash and futures returns. The squared error term in the
conditional variance and covariance equations (Eq. (11)) measure the influences of
the short-run deviations on conditional variance and covariance.
The following equations present a diagonal vech bivariate MA(1)-GARCH(1,1)-X
conditional variance equation(s) with the squared error-correction term (zt) lagged
once:
H11;t�C1�A11(o1;t�1)2�B11(H11;t�1)�D11(zt�1)2; (11a)
H12;t�C2�A22(o1;t�1o2;t�1)�B22(H12;t�1)�D22(zt�1)2; (11b)
H22;t�C3�A33(o2;t�1)2�B33(H22;t�1)�D33(zt�1)2: (11c)
As advocated by Lee (1994, p. 337), the square of the error-correction term (z)
lagged once should be applied in the GARCH(1,1)-X model. The parameters, D11
and D33 indicate the effects of the short-run deviations between the cash and the
futures prices from a long-run cointegrated relationship on the conditional variance
of the residuals of the cash and futures returns, respectively. The parameter D22
shows the effect of the short-run deviations on the conditional covariance between
cash and futures returns. As stated above, if short-run deviations between cash price
and futures price affect the conditional variance of the cash and futures returns, andthe conditional covariance between the two returns, then optimal hedge as defined in
Eq. (8) will also be affected. In other words, if D33 and D22 are significant then H12
(conditional covariance) and H22 (conditional variance of futures returns) are going
to differ from the standard GARCH model H12 and H22. In such a case, the
GARCH-X time-varying hedge ratio will be different from the GARCH time-
varying hedge ratio. If the two time-varying hedge ratios are different, then the
9 The following cointegration relationship is investigated by means of the Engle�/Granger method:
St�h�a6
i��6giFt�i�zt; where St and Ft are log of cash index and futures index, respectively. In this test
also six leads and lags of the future price index are added in order to investigate the lead/lags effect. This
was suggested by the referee.
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192 177
interesting empirical question arises; which one is more effective? All the above
methods of estimating the hedge ratios (including the constant ratios) are applied
and their effectiveness is compared in this paper.
4. The data and the basic statistics
Daily stock returns from the cash (spot) and the futures markets of Australia,
Germany, Hong Kong, Japan, South Africa and the United Kingdom are used in the
empirical tests. All the data range from January 1990 to December 1999 except for
Germany and South Africa where the data are from January 1991 to December
1999. All futures price indices are continuous series.10 The Australian spot stock
index is based on the All Ordinary price index and the futures prices are based on the
All Ordinary futures index. The All Ordinary share index contains 307 Australian
stocks. The cash and futures prices of the German indices are based on the Dax 30
index and the Eurex-Dax futures index. The Dax 30 contains 30 German stocks. The
Hang Seng price index and Hang Seng index futures represent the Hong Kong cash
and futures prices, respectively. The Hang Seng index contains 33 stocks on the
Hong Kong stock exchange. In the case of Japan, the Nikkei 225 price index is used
for the cash price index and Nikkei stock average futures prices are used for the
futures index. The Nikkei 225 contains 225 Japanese stocks. The JSE industrial index
and the Industrial 25 index represent the South African cash and futures prices.
These indices include 25 South African stocks. The United Kingdom cash and
futures prices are represented by the FTSE-100 index and the FTSE-100 futures
index, respectively. The United Kingdom indices include 100 stocks. All data are
obtained from DATASTREAM .
Stock returns are defined as the first difference in the log of price indices (both
cash and futures). Table 1 shows some of the basic statistics of the 12 stock returns.11
As expected, all series are found to have significant and positive kurtosis implying
higher peaks and fatter tails. Most series are also skewed except the cash returns of
Hong Kong and the futures returns of the United Kingdom. Significant mean is
found only for South Africa and the United Kingdom.
10 The continuous series is a perpetual series of futures prices. It starts at the nearest contract month
which forms the first values for the continuous series, until either the contract reaches its expiry date or
until the first business day of the actual contract month. At this point, the next trading contract month is
taken.11 The stochastic structure of the data is checked by means of several unit root tests. Results from the
augmented Dickey�/Fuller test, the Phillips�/Perron test and the KPSS test show all nine logs of price
indices to be nonstationary at levels and stationary after first difference. Thus all cash and futures returns
are stationary in levels. These unit root test results are available on request.
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192178
Table 1
Basic statistics of cash and futures stock returns total period
Australia Germany Hong Kong Japan South Africa United Kingdom
Cash Futures Cash Futures Cash Futures Cash Futures Cash Futures Cash Futures
Mean 0.00025 0.00025 0.00068 0.00068 0.00069 0.00068 �/0.00028 �/0.00028 0.00048** 0.00047*** 0.00040** 0.00040***
Variance 0.00007 0.00011 0.00014 0.00016 0.00029 0.00038 0.00022 0.00022 0.000095 0.00019 0.00008 0.00011
Skewness �/0.263* �/0.215* �/0.599* �/0.505* 0.060 0.448* 0.339* 0.252* �/1.275* �/0.905* 0.084*** 0.022
Kurtosis 4.660* 4.390* 6.270* 7.810* 11.300* 12.470* 4.210* 2.720* 16.210* 19.310* 2.271* 1.604*
Obs: observations.
* Significance at the 1% level.
** Significance at the 5% level.
*** Significance at the 10% level.
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Table 2
Standard bivariate MA(1)-GARCH(1,1) results
Australia Germany Hong Kong Japan South Africa U.K.
m1 0.00032** (2.35) 0.00078* (4.48) 0.0012* (5.39) 0.00022 (1.28) 0.00080* (4.93) 0.00060* (3.61)
u1 0.0776* (4.74) 0.2092* (11.65) 0.1257* (8.48) 0.2658* (17.86) �/0.0936* (�/4.72) 0.1151* (7.47)
m2 0.00032** (2.12) 0.00083* (4.94) 0.00117* (5.35) 0.000192 (1.06) 0.00076* (3.66) 0.00054* (3.61)
u2 0.2151* (13.87) 0.3267* (19.09) 0.2422* (17.08) 0.2842* (18.79) 0.0812* (4.29) 0.1901* (13.04)
C1 1.70�/10�5* (9.42) 0.47�/10�5* (9.89) 0.01�/10�5* (17.28) 2.08�/10�5* (13.00) 0.81�/10�5* (17.72) 0.27�/10�5* (10.75)
A11 0.1264* (15.31) 0.0723* (13.75) 0.1088* (19.98) 0.1444* (17.19) 0.1740* (19.49) 0.0575* (14.58)
B11 0.6203* (20.13) 0.8951* (130.01) 0.8557* (147.05) 0.7687* (70.69) 0.7120* (62.97) 0.9030* (192.59)
C2 1.87�/10�5* (10.89) 0.47�/10�5* (11.75) 0.98�/10�5* (15.87) 2.16�/10�5* (12.11) 0.81�/10�5* (18.55) 0.30�/10�5* (11.20)
A22 0.1219* (15.21) 0.0611* (13.45) 0.0941* (20.76) 0.1352* (16.37) 0.1320* (19.70) 0.0560* (14.36)
B22 0.6417* (24.97) 0.9046* (147.94) 0.8710* (174.84) 0.7678* (67.12) 0.7600* (88.32) 0.9050* (200.99)
C3 2.42�/10�5* (11.20) 0.68�/10�5* (11.29) 1.04�/10�5* (13.41) 2.33�/10�5* (11.07) 1.44�/10�5* (17.71) 0.03�/10�5* (11.09)
A33 0.1270* (14.78) 0.0638* (11.99) 0.0884* (21.70) 0.1393* (16.08) 0.1230* (21.56) 0.0556* (13.83)
B33 0.6631* (27.57) 0.8965* (117.72) 0.8810* (192.40) 0.7610* (61.05) 0.7780* (86.41) 0.9093* (200.44)
L 24023.02 20187.04 21643.50 22698.18 20676.72 25166.51
/ot=H1=2t /
LB cash 7.92 3.02 7.89 7.22 8.28 9.69
LB future 3.28 8.07 9.05 7.55 5.50 10.39
/o2t =Ht/
LB cash 2.30 0.80 2.66 5.49 5.19 2.30
LB futures 3.49 0.39 5.51 10.55 5.57 7.34
LB: Ljung�/Box statistics for serial correlation of the order 9. t -Statistics in the parentheses; L : log-likelihood function value.
* Significance at the 1% level.
** Significance at the 5% level. ***Significance at the 10% level.
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5. The empirical results
5.1. The bivariate GARCH, cointegration, GARCH-X and OLS results
Table 2 shows the results from the standard bivariate MA(1)-GARCH(1,1) model
for the total period.12 The bivariate MA(1)-GARCH(1,1) results are quite standard.
The ARCH coefficients (A11 and A33) are all positive and significant thus implying
volatility clustering both in the cash returns and the futures returns. The ARCH
coefficients are also less than unity in all cases. The sign and significance of thecovariance parameters (A22 and B22) indicate positive and significant interaction
between the two prices for all six cases. The MA coefficient (u ) is also positive and
significant in all markets except for the South African cash returns. The significant
MA term may be due to non-synchronous trading.13 To assess the general
descriptive validity of the model, a battery of standard specification tests is
employed. Specification adequacy of the first two conditional moments is verified
through the serial correlation test of white noise. These tests employ the Ljung�/Box
Q -statistics on the standardized (normalized) residuals (ot=H1=2t ) and standardized
squared residuals (ot=H2t ): All series are found to be free of serial correlation (at the
5% level). Absence of serial correlation in the standardized squared residuals implies
the lack of need to encompass a higher order ARCH process (Giannopoulos, 1995).
Table 3 shows the cointegration test results between the log of cash price index and
the log of futures price index for the total period.14 Tests are conducted based on the
Engle and Granger (1987) two-step method.15 Description of the Engle and Granger
method is not provided here since it is available in numerous articles and books. In
the cointegration tests, six leads and lags of the futures price index are also added tocheck for possible lead�/lag effect on the relationship and the time-varying hedge
ratio.16 Cointegration between the two price series is found in all six tests. Both the
augmented Dickey�/Fuller tests with and without the trend indicate stationary
12 In a GARCH(p ,q ) model, different combinations of p and q may be applied but, as indicated by
Bollerslev et al. (1992, p. 10), p�/q�/1 is sufficient for most financial and economic series. Bollerslev
(1988) provides a method of selecting the length of p and q in a GARCH model. Tests in this paper were
also conducted with different combinations of p and q with p�/q�/2 being the maximum lag length.
Results based on log-likelihood function and likelihood ratio tests indicate that the best combination is
p�/q�/1. These results are available on request.13 The significant MA term may also be due to different news observed by different investors or the
same news being interpreted differently by investors. This could create a negative serial correlation, as a
result of a process of price adjustment where the price bounces back and forth between centres with
different information.14 As required by any cointegration tests, the stochastic structure of the individual variables has to be
checked. Standard unit root tests such as ADF and KPSS showed that logs of all price indices are
nonstationary in levels but stationary after first difference. Thus the cash and the futures returns are
stationary in levels. These results are available on request.15 Other forms of cointegration tests such as the multivariate Johansen and Juselius (1990) method were
also applied. Results obtained are the same as when using the Engle and Granger method. Johansen tests
results are not presented to save space but are available on request.16 As stated earlier, this was suggested by the referee.
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192 181
Table 3
Cointegration tests between cash price index and futures price index
Con Ft�6 Ft�5 Ft�4 Ft�3 Ft�2 Ft�1 Ft Ft�1 Ft�2 Ft�3 Ft�4 Ft�5 Ft�6 H1 H2 ADF-T ADF-NT
Australia �/0.048 0.021 0.001 �/0.005 0.018 0.004 0.034 0.669 0.130 0.014 0.021 0.011 0.004 0.083 7.22*** 92.67*** �/0.091* (�/8.48) {4} �/0.090* (�/8.48) {4}
Germany �/0.050 0.023 �/0.002 �/0.002 0.020 �/0.021 �/0.006 0.767 0.204 0.019 �/0.005 0.001 �/0.004 0.010 4.80*** 77.25*** �/0.237* (�/9.16) {12} �/0.232* (�/9.08) {12}
HK 0.017 0.023 �/0.010 �/0.016 0.027 0.003 0.017 0.798 0.124 �/0.001 0.000 0.011 �/0.009 0.030 6.37*** 45.29*** �/0.107* (�/8.32) {4} �/0.105* (�/8.29) {4}
Japan 0.228 �/0.009 0.000 0.002 0.004 �/0.005 0.020 0.931 0.037 �/0.009 �/0.001 0.005 �/0.013 0.012 0.94 2.87*** �/0.152* (�/7.86) {8} �/0.134* (�/7.68) {8}
SA �/0.326 �/0.044 �/0.008 �/0.014 �/0.014 0.001 0.021 0.591 0.107 0.045 0.035 0.029 0.023 0.267 1.44 38.73*** �/0.012** (�/3.30) {12} �/0.010** (�/3.10) {12}
UK �/0.076 0.025 �/0.004 0.004 0.009 �/0.008 0.013 0.828 0.077 0.015 0.002 0.009 �/0.005 0.043 5.33*** 34.54*** �/0.055* (�/6.79) {4} �/0.055* (�/6.77) {4}
Number of lags in brackets. H1: null hypothesis that all lead coefficients are equal to zero. H2: null hypothesis that all lag coefficients are equal to zero. ADF-
T: ADF tests with trend. ADF-NT: ADF tests with no trend. t -Statistics in the parentheses.
* Rejection of the null at the 1% level.
** Rejection of the null at the 5% level.
*** Rejection of the unit root at the 5% level or above.
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residuals (error term) in levels from the OLS regression between the two prices.17 In
all tests, cointegration is indicated at the 5% level or above.18 A similar result has
been provided by Ghosh (1993b) for the United States stock futures. Results indicate
a significant effect of the lags in all tests and a significant effect of leads in all cases
except Japan and South Africa. These stationary error terms from the cointegration
test represent the short-run deviations between the cash price and the futures prices.
In the GARCH-X model, these error terms are included in the mean, conditional
variance and covariance equations.
Table 4 shows the bivariate GARCH(1,1)-X model results for the total period.19
These results are similar to the standard GARCH results shown in Table 2. The
ARCH coefficient is significant and less than unity for both the cash and futures
returns in all six tests. Once again the MA coefficient is positive and significant in all
cases except the South African cash returns. All markets again show evidence of
significant interaction between the cash returns and the futures returns. The short-
run deviations between the cash price and future prices have a positive and
significant effect (d1) on the cash returns in all six tests. In the case of the futures
return, the deviations (d2) only significantly influence the German futures return.
The important part of the GARCH-X results is the influence of the short-run
deviations between the cash price and the futures price on the conditional variance
and covariance. The parameters measuring the effects of the short-run deviations on
the conditional variance of cash returns (D11) and futures returns (D33) are mostly
found to be positive and significant. A positive and significant effect of the short-run
deviations on the conditional variance implies that as the deviation between the cash
and future prices gets larger, the volatility of cash and futures returns increases and
prediction becomes more difficult. The short-run deviation effect on conditional
variance is only insignificant in the case of German cash returns. In the case of
German futures returns, the short-run deviations impose a significant negative effect.
The parameter (D22) that measures the effect of the short-run deviations on the
conditional covariance is found to be significant and positive in all cases except for
the United Kingdom. These results clearly show that short-run deviations do indeed
impose a significant effect on the conditional variance of the cash and futures
returns, and also on the conditional covariance between the two returns. The
question to be answered is whether these effects of the short-run deviations also
influence the effectiveness of the time-varying hedge ratio. Once again the
standardized residuals and the standardized squared residuals are found to be free
of serial correlation (at the 5% level). Absence of serial correlation in the
17 The lag length applied in the cointegration ADF test is based on the AIC evidence. The maximum
number of lags applied is 12. Results with the maximum lags required to remove serial correlation are
presented.18 Results from the two smaller samples are similar. These results are not provided in order to save
space but are available on request.19 Once again log-likelihood function and likelihood ratio tests indicate that the best combination is
p�/q�/1.
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192 183
Table 4
Bivariate MA(1)-GARCH(1,1)-X results
Australia Germany Hong Kong Japan South Africa UK
m1 0.00034** (2.40) 0.00113* (5.80) 0.0010* (4.26) 0.00062* (3.13) 0.00064* (3.67) 0.00061* (4.39)
d1 0.085* (4.55) 0.537* (18.372) 0.188* (6.49) 0.259* (6.51) 0.0135* (2.873) 0.0500** (2.14)
u1 0.047* (2.73) 0.055* (2.66) 0.060* (3.64) 0.137* (7.79) �/0.098* (�/4.96) 0.095* (5.71)
m2 0.00032*** (1.91) 0.0012* (5.87) 0.0012* (4.82) 0.00027 (1.32) 0.0007* (3.25) 0.0005* (3.53)
d2 �/0.022 (�/0.96) 0.164* (7.48) �/0.0088 (�/0.29) �/0.021 (�/0.53) 0.0032 (0.006) �/0.0204 (�/0.79)
u2 0.154* (8.53) 0.138* (5.00) 0.158* (9.33) 0.163* (9.43) 0.072* (3.72) 0.162* (9.76)
C1 1.27�/10�5* (8.19) 7.10�/10�6* (9.01) 8.40�/10�6* (10.34) 1.18�/10�5* (10.56) 7.55�/10�6* (15.11) 8.27�/10�6* (13.53)
A11 0.111* (15.11) 0.096* (12.33) 0.107* (19.61) 0.118* (16.17) 0.1550* (15.80) 0.105* (17.00)
B11 0.676* (24.76) 0.845* (78.17) 0.853* (142.72) 0.823* (87.54) 0.7170* (57.84) 0.778* (121.64)
C2 1.41�/10�5* (9.57) 7.51�/10�6* (12.81) 8.27�/10�6* (10.17) 1.20�/10�5* (9.81) 7.03�/10�6* (17.09) 9.45�/10�6* (14.63)
A22 0.107* (14.27) 0.084* (12.53) 0.093* (19.61) 0.108* (14.02) 0.1190* (16.25) 0.104* (17.07)
B22 0.694* (30.39) 0.862* (100.42) 0.868* (167.72) 0.828* (83.27) 0.7660* (78.07) 0.776* (186.05)
C3 1.88�/10�5* (10.16) 1.07�/10�6* (15.96) 8.84�/10�6* (9.60) 1.28�/10�5* (8.88) 1.15�/10�5* (14.79) 1.05�/10�5* (14.51)
A33 0.113* (13.22) 0.087* (12.46) 0.087* (19.71) 0.108* (14.02) 0.1140* (19.36) 0.103* (15.71)
B33 0.707* (32.44) 0.859* (103.91) 0.880* (180.09) 0.827* (75.74) 0.7810* (76.83) 0.789* (151.48)
D11 0.0175** (2.51) 0.0069 (0.635) 0.0247* (7.15) 0.040** (2.24) 0.0009* (5.46) 0.0119*** (1.66)
D22 0.0264** (2.57) �/0.0445* (�/7.78) 0.015* (3.65) 0.034*** (1.76) 0.0025* (7.29) 0.0136 (1.49)
D33 0.0173** (2.44) �/0.0178** (�/2.55) 0.018* (5.04) 0.032*** (1.85) 0.0011* (5.37) 0.0134*** (1.65)
L 24096.55 20458.51 21763.13 22896.82 20715.78 25097.99
/ot=H1=2t /
LB Cash 4.39 1.94 8.21 4.31 6.61 7.43
LB Futures 5.34 1.88 3.66 9.26 4.65 8.93
/o2t =Ht/
LB cash 3.88 0.76 2.43 2.99 5.52 11.41
LB futures 11.07 0.66 3.90 5.30 6.52 10.29
* Significance at the 1% level.
** Significance at the 5% level.
*** Significance at the 10% level.
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standardized squared residuals implies the lack of need to encompass a higher-order
ARCH process.20
Table 5 shows the OLS estimation of Eq. (4) for the total period. It shows the
results from the regression between the cash returns and the futures returns for all six
countries.21 The size of the coefficient on the futures returns shows the constant
minimum variance hedge ratio. In all tests, the coefficient on the future returns is
positive and significantly different from zero at the 1% level. Also, in all cases the
hedge ratio is found to be significantly less than unity by means of the F -test. Thus
results fail to indicate a one-to-one relationship between the cash and the futures
returns. The largest coefficient is found in the case of Japan (0.93) and the lowest for
South Africa (0.5998). These results are quite standard and are found in other studies
also.22
Fig. 1 shows the time-varying hedge ratios from the standard bivariate
GARCH(1,1) and the bivariate GARCH(1,1)-X and the constant minimum variance
hedge ratio for Hong Kong. The top graph shows the standard GARCH time-
varying ratio against the constant ratio and the bottom graph represents the time-
varying ratio from the GARCH-X against the constant ratio. Both time-varying
ratios are clustered around the constant hedge ratio. A close inspection of both time-
varying ratios indicates that they are very similar but not identical. The large dip at
the end is possibly due to the Asian crisis of 1997 and 1998. Graphs of other five
markets also convey the same story and are not provided to save space but are
available on request.
20 Both the GARCH and GARCH-X models were estimated by means of the Berndt et al. (1974)
algorithm.21 As noted earlier in footnote 11, all cash and futures return series are found to be stationary in levels.
Thus their application in OLS regressions is quite safe.22 In the lead�/lags tests, lags had a significant effect in all six tests but leads were only significant in the
cases of Australia and Hong Kong. These results are available on request.
Table 5
OLS regression between cash and futures returns minimum variance hedge ratio
Country Constant Futures returns Diagnostic F -test
Australia 0.000087 (1.083) 0.6529* (86.60) R2�/0.742; DW�/2.43 2118.89**
Germany 0.00016 (1.124) 0.7632* (67.50) R2�/0.660; DW�/2.80 438.72**
Hong Kong 0.00015 (1.086) 0.7885* (110.78) R2�/0.824; DW�/2.70 883.03**
Japan �/0.00001 (�/0.113) 0.9350* (132.27) R2�/0.870; DW�/2.67 97.66**
South Africa 0.00019 (1.741) 0.5998* (74.57) R2�/0.703; DW�/2.04 2475.01**
UK 0.00007 (1.426) 0.8260* (164.59) R2�/0.912; DW�/2.40 1201.71**
DW: Durbin�/Watson. t -Statistics in the parentheses.
* Significant at the 1% level.
** Reject the null at the 5% level or above.
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192 185
Fig. 1. Constant and GARCH (top) and constant and GARCH-X (bottom) time-varying hedge ratio.
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192186
5.2. Total sample hedge ratios comparison result
As indicated by Baillie and Myers (1991) and Park and Switzer (1995), comparison
between the effectiveness of different hedge ratios is made by constructing portfolios
implied by the computed ratios and then comparing the variance of these
constructed portfolios. The portfolios are constructed as (rct �bt�r
ft ); where rc
t is the
cash (spot) returns, rft the futures returns and bt�the estimated optimal hedge ratio.23
The variance of these constructed portfolios is estimated and compared. The change
in variance is calculated as (Varothers�/VarGARCH)/Varothers. Since the main theme of
this paper is to investigate the effectiveness of the time-varying hedge ratio,
comparison of changes in variance is only conducted between the time-varying
ratio portfolios (estimated by means of the GARCH model and the GARCH-X
model) and constant ratio portfolios.
Table 6 shows the variance of the portfolio estimated using the different types of
optimal hedge ratios and the percentage change in the variance of the portfolios
constructed using the time-varying hedge ratio from the bivariate GARCH(1,1)
model, GARCH(1,1)-X model and the three constant hedge ratio portfolios. A
comparison between the standard bivariate GARCH(1,1) and the constant hedge
ratio portfolios indicates that time-varying hedge ratio portfolios have the lowest
variance in only two cases, Australia and Hong Kong. The reduction in the variance
is quite substantial compared with the unhedged portfolio. In the case of Japan, both
the traditional hedge and the minimum variance hedge ratio portfolios outperform
the GARCH time-varying ratio portfolio.
23 In the case of the constant ratio, the time subscript does not exist.
Table 6
Total period (January 1990�/December 1999) portfolio variance and percentage change in the variance
Australia Germany Hong Kong Japan South Africa United Kingdom
Unhedged 0.0000656 0.000142 0.000286 0.000219 0.0000953 0.0000806
Traditional 0.0000307 0.0000574 0.0000671 0.0000294 0.0000581 0.0000103
Minimum 0.0000169 0.0000484 0.0000500 0.0000284 0.0000283 0.0000071
GARCH 0.0000153 0.0000494 0.0000481 0.0000298 0.0000284 0.0000074
GARCH-X 0.0000152 0.0000490 0.0000484 0.0000297 0.0000282 0.0000076
Percentage change in the portfolio variance between GARCH and other methods (excluding GARCH-X)
Unhedged 76.68 65.21 83.18 86.39 70.20 90.82
Traditional 50.16 13.94 28.32 �/1.36 51.12 28.16
Minimum 9.47 �/2.07 3.80 �/4.93 �/0.35 �/4.23
Percentage change in the portfolio variance between GARCH-X and other methods
Unhedged 76.83 65.55 83.31 86.44 70.41 90.57
Traditional 50.49 14.63 38.64 �/1.02 51.46 26.21
Minimum 10.06 �/1.24 3.20 �/3.37 0.35 �/7.04
GARCH 0.65 0.82 �/0.62 0.34 0.70 �/2.70
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192 187
Portfolios created using hedge ratios from the GARCH-X models provide similar
results. Once again in only two cases, Australia and South Africa, the GARCH-X
time-varying ratio provides the lowest variance. As expected, the GARCH-X time-
varying ratio outperforms the unhedged and the traditional hedge in all cases, except
Japan where the traditional hedge does better by a small margin. The minimum
variance hedge ratio does do better in three of the cases, Germany, Japan and the
United Kingdom. The GARCH-X does better than the standard GARCH in allcases, except for Hong Kong and the United Kingdom. In the case of Hong Kong,
the reduction is by less than 1%. For the remaining four countries, the GARCH-X
ratio portfolio produces the lowest variance, though the difference between the two
time-varying ratios is small. There is a substantial drop in the variance between the
constant ratio portfolios and the GARCH-X portfolios especially for Australia,
Hong Kong and South Africa. The inconsistent performance of both GARCH
model ratios may be attributed to the complexity of the GARCH model (Baillie and
Myers, 1991).
5.3. Out-of-sample hedge ratios comparison result
Park and Switzer (1995) and Baillie and Myers (1991) further claim that the more
reliable measure of hedging effectiveness is the hedging performance of different
methods for out-of-sample periods. This paper compares the hedging effectiveness ofthe five different methods during two different out-of-sample time periods. The out-
of-sample periods used are from January 1998 to December 1999 and from January
1999 to December 1999. In order to investigate the out-of-sample hedging
effectiveness of the five hedging methods, the bivariate GARCH, the GARCH-X
models and the minimum variance equations are estimated for the periods January
1990�/December 1997 and January 1990�/December 1998, and then the estimated
parameters are applied to compute the hedge ratios and the portfolios for the two
out-of-sample periods.24 Once again hedging effectiveness is compared by comparingthe variance of these portfolios and the change in the variance.
Table 7 shows the variance of the out-of-sample portfolios and the percentage
change in variance of the portfolios from January 1998 to December 1999.
Comparing the change in the variance between the standard GARCH ratio
portfolios and the constant ratio portfolios indicates the superior performance of
the time-varying hedge ratio. The standard GARCH ratio outperforms all three
constant ratios for all markets. The decline in the variance of the portfolios is again
quite substantial especially for Australia, Hong Kong and the United Kingdom. TheGARCH-X time-varying hedge ratio outperforms all the constant hedge ratios
except in the case of South Africa but fails to provide the lowest variance compared
with the GARCH time-varying ratio in three cases, though the difference between
24 The bivariate GARCH and the minimum variance equation estimations of the period 1990�/1997
and 1990�/1998 are not provided in order to save space but are available on request. These parameters are
similar to the ones estimated for the whole sample period.
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192188
the two time-varying hedge ratio is not much. In the case of Australia, Hong Kong
and Japan, the GARCH-X does better than the standard GARCH. For the
remaining three countries, the standard GARCH outperforms the GARCH-X.
The shorter out-of-sample period of January 1999�/December 1999 results are
presented in Table 8. Once again the standard GARCH ratio does better than the
constant ratios. There is no difference in the minimum variance ratio and the time-
Table 7
Out-of-sample period (January 1998�/December 1999) portfolio variance and percentage change in the
variance
Australia Germany Hong Kong Japan South Africa United Kingdom
Unhedged 0.0000710 0.000257 0.000501 0.000217 0.000202 0.000147
Traditional 0.0000187 0.0000589 0.000147 0.0000273 0.0000768 0.0000106
Minimum 0.0000134 0.0000603 0.000112 0.0000253 0.0000445 0.0000101
GARCH 0.0000121 0.0000539 0.0000964 0.0000251 0.0000433 0.0000088
GARCH-X 0.0000118 0.0000546 0.000096 0.0000250 0.0000450 0.0000091
Percentage change in the portfolio variance between GARCH and other methods (excluding GARCH-X)
Unhedged 82.96 79.03 80.76 88.43 78.56 94.01
Traditional 35.29 8.49 34.42 8.06 43.62 16.98
Minimum 9.70 10.61 13.93 0.79 2.70 12.87
Percentage change in the portfolio variance between GARCH-X and other methods
Unhedged 83.38 78.75 80.84 88.48 77.72 93.81
Traditional 36.90 7.30 34.69 8.42 41.41 14.15
Minimum 11.94 9.45 14.29 1.20 �/1.12 9.90
GARCH 2.48 �/1.30 0.41 0.40 �/3.93 �/3.41
Table 8
Out-of-sample period (January 1999�/December 1999) portfolio variance and percentage change in the
variance
Australia Germany Hong Kong Japan South Africa United Kingdom
Unhedged 0.0000578 0.000186 0.000268 0.000156 0.0000980 0.000123
Traditional 0.0000126 0.0000266 0.000187 0.0000207 0.0000370 0.00000627
Minimum 0.00000990 0.0000300 0.000137 0.0000195 0.0000156 0.00000620
GARCH 0.00000897 0.0000254 0.000103 0.0000195 0.0000156 0.00000531
GARCH-X 0.0000090 0.0000268 0.000104 0.0000197 0.0000154 0.00000560
Percentage change in the portfolio variance between GARCH and other methods (excluding GARCH-X)
Unhedged 84.43 86.34 61.57 87.50 84.80 95.69
Traditional 28.57 4.51 31.34 5.80 57.84 14.52
Minimum 9.09 15.33 24.82 0.00 0.00 14.53
Percentage change in the portfolio variance between GARCH-X and other methods
Unhedged 84.43 85.59 61.19 87.37 84.29 95.45
Traditional 28.57 �/0.75 44.39 4.83 58.38 11.11
Minimum 10.00 11.94 24.09 �/1.03 1.28 9.68
GARCH 0.00 5.51 �/0.97 �/1.03 1.28 �/5.66
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192 189
varying ratio from the GARCH model results for Japan and South Africa. Reducing
the size of the out-of-sample size does not change the GARCH ratio versus the
constant ratio results by much. Once again the decline in the variance is quite large in
some cases. The GARCH-X ratio outperforms the unhedged portfolio in all six tests
but fails against the traditional hedge for Germany and against the minimum
variance hedge for Japan. The traditional hedge and the minimum variance hedge
only outperform the GARCH-X hedge by 1% or less. Comparison between the twotime-varying hedge indicates a better performance by the GARCH-X for Germany
and South Africa. The standard GARCH is just as good as the GARCH-X in the
case of Australia and does better for Hong Kong, Japan and the United Kingdom.
Once again the inconsistent performance of the GARCH time-varying ratios may be
attributed to the complexity of the GARCH model. The out-of-sample results do
provide evidence of superiority of the time-varying hedge ratio of the GARCH
model.
6. Conclusion
It is a well-documented claim in the futures market literature that optimal hedge
ratio should be time-varying and not constant (see Baillie and Myers, 1991; Myers,
1991). This paper investigates the hedging effectiveness of six stock futures. The
paper further studies the effects of short-run deviations from a long-run relationship
between the cash index and futures index on the optimal hedge ratio. The markets
under study are Australia, Germany, Hong Kong, Japan, South Africa and theUnited Kingdom during the period January 1990�/December 1999. Five different
hedging ratios, depending on different estimation procedures, are compared for
hedging effectiveness. Three of these five ratios are constant: the unhedged, the
traditional hedge and the minimum variance hedge. The fourth and the fifth ones,
the standard bivariate GARCH hedge ratio and the bivariate GARCH-X ratio, are
time-varying as they take into consideration the time-varying distribution of the cash
and futures returns. The GARCH-X model further takes into consideration the
effects of the long-run cointegration between the cash price and the futures price.The hedging effectiveness is compared by checking the variance of the portfolios
created using these hedge ratios. The lower the variance of the portfolio, the higher is
the hedging effectiveness of the hedge ratio.
The effectiveness of the hedge ratio is investigated by comparing the total sample
(January 1990�/December 1999) and of out-of-sample performance of the different
hedge ratios for two periods January 1998�/December 1999 (two years) and January
1999�/December 1999 (one year). The parameters of the GARCH model, GARCH-
X model and other models are estimated for the periods January 1990�/December1997 and January 1990�/December 1998, and then hedge ratios are forecast for the
out-of-sample periods January 1998�/December 1999 and January 1999�/December
1999 based upon the estimated parameters.
During the total period, the constant minimum variance hedge ratio does better
than the time-varying GARCH hedge ratio in most cases. In turn the GARCH-X
T. Choudhry / J. of Multi. Fin. Manag. 13 (2003) 171�/192190
does better than the standard GARCH in most of these tests. Using the longer out-
of-sample period (January 1998�/December 1999), both versions of the time-varying
ratios do better than the constant ratios. A comparison between just the two time-
varying GARCH model ratios shows that the standard bivariate GARCH model
performs better than the bivariate GARCH-X model in three cases. The smaller out-
of-sample (January 1999�/December 1999) shows that the standard GARCH model
hedge ratio is again better than the constant ratios in all cases. Considering theGARCH-X, it seems to perform better than the constant ratio in four cases. Once
again the GARCH hedge ratio performance is better than the GARCH-X hedge
ratio in three cases.
These results provide some indication of the superior performance of the time-
varying hedge ratio as compared with the constant ratios. Results also indicate that
taking into consideration the short-run deviations between the cash price and the
futures price may also improve the hedging effectiveness of the time-varying hedge
ratio. Results in this paper advocate further research in this field. Further researchmay be conducted using different frequency of the data, time period, type of futures
markets, etc.
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