Model Specification for the Estimation of the Optimal Hedge Ratio with
Stock Index Futures: an Application to the Italian Derivatives Market
by
Agostino Casillo1
ABSTRACT
Several techniques to estimate the hedge ratio with index futures contracts have been
proposed in the literature. While these techniques hold theoretical appeal, there is no univocal
evidence as to their effectiveness. This research provides an empirical comparison of four
different econometric techniques in the context of hedging the market risk using the FIB 30
index futures contract of the Italian Derivatives Market. Specifically, the OLS regression
model, the bivariate vector autoregressive model (BVAR), the vector error correction model
(VECM) and the multivariate GARCH model are employed. Then the hedging effectiveness
is measured in terms of ex-post and ex-ante variance minimization. It is generally found that
only the GARCH model is able to outperform the OLS model and the improvement is not
really consistent.
KEYWORDS: Optimal hedge ratio; Hedging; Stock Index Futures; Italian Derivatives
Market.
JEL Classification: G10, G13.
1
University of Birmingham and Associazione “Guido Carli”
1. Introduction
To implement a hedge, it is necessary to calculate the corresponding hedge ratio that is
defined by Hull (2003: 78) as “the ratio of the size of the portfolio taken in futures contracts
to the size of the exposure”.
Researchers have distinguished three hedge strategies: the traditional one to one hedge, the
beta hedge, and the minimum variance hedge. All three strategies require determining the
optimal hedge ratio h* but the traditional strategy emphasizes the potential for futures contract
to be used to reduce risk. The strategy is very simple and involves the adoption of a fixed
hedge h=-1 which consists of taking a futures position that is equal in magnitude, but opposite
in sign, to the spot position. If price changes in the futures market exactly match those in the
future markets the adoption of a one to one strategy will be enough to eliminate the price risk.
The beta hedge strategy is quite similar but it is used when the cash portfolio to be hedged
does not exactly match the portfolio underlying the futures contract. For this reason the
optimal hedge ratio is calculated this time as the negative of the beta of the cash portfolio.
Naturally, if the cash portfolio is that which underlies the futures position, the two strategies
will give the same value for the hedge ratio.
However, in practice the prices on the spot and futures markets do not move exactly
together and a hedge ratio derived from the traditional or beta hedge strategy would not
minimize the risk. In particular Peters (1986) has shown that mispricing adds 20% to the
volatility of the futures contract. Since the futures contract is more volatile than the
underlying index the use of a portfolio beta as a sensitivity adjustment would result in a
portfolio being over-hedged.
This imperfect correlation is taken in consideration instead by the minimum variance
hedge ratio. The model has been proposed by Johnson (1960) and Stein (1961) and developed
by Ederington (1979), who employs portfolio theory to identify the hedge ratio (h*) as:
2
h *t −1 = -
σ S , F ,t
(1.3)
σ F2 ,t
where σ2F,t represents the conditional variance of the futures contract and σ2SF,t is the
conditional covariance between the index futures and the spot position on the same index. The
negative sign just means that the hedging of a long stock position requires a short position in
the corresponding futures contract.
In order to show how the above can be derived let St and Ft be the logarithms of the stock
index and index futures prices respectively. The starting point necessary to derive the model is
represented by the two equations that express the actual returns offered from the two assets at
time t, ∆St = St – St-1=Rs,t and ∆Ft = Ft – Ft-1=Rf,t. Hence, the expected return at time t-1 of a
portfolio (P) consisting in a long position in Qs unit of the stock index and Qf units of the
corresponding futures contract, is given by:
Et-1 (Pt) = Qs Et-1(∆St) – Qf Et-1 (∆Ft) (1.4)
where the price changes are assumed to be stochastic, and the spot position Qs is given.
The resulting variance associated to this portfolio is
σ2p,t = Qs2,t −1 σ2S, t + Q 2f ,t −1 σ2F,t – 2Qs,t-1 Qf,t-1 σS,F, t (1.5)
with σ2S, t and σ2F,t representing the conditional variance of the cash and futures positions. If
the goal is to maximize investor’s expected utility we need to define its utility function, that is
the risk-return trade-off, in order to go on. Following Kahl’s formulation, Brooks et al.
(2002: 335) show that if the investor has the following linear utility function,
U ( Et-1 (Pt), σ2p,t) = Et-1 (Pt) - ψ σ2p,t (1.6)
where ψ is the degree of risk aversion, then agent’s maximisation problem can be expressed
as,
max U( Et-1 (Pt), σ2p,t ) = Qs Et-1(∆St) – Qf Et-1 (∆Ft) +
-ψ ( Qs2,t −1 σ2S, t + Q 2f ,t −1 σ2F,t – 2Qs,t-1 Qf,t-1 σS,F, t,) (1.7).
3
Substituting, we can derive the solution that permits to maximize the utility function, that
is the condition for an optimum,
-
Qf
Qs
=
σ sf
δF
2
σ f Qsνσ 2f
(1.8),
that can also be written as
Qf =
σ sf
δF
Q
s
νσ 2f
σ 2f
(1.9).
Assuming that ψ is indefinitely high2 the investor futures position does not depend from
the risk parameter because the ratio
δF
becomes insignificant. In that case the hedge ratio
νσ 2f
is:
h* =
Qf
Qs
=-
σ sf
σ
=- ρ s (1.10),
2
σf
σf
where ρ is the correlation coefficient between S and F. Substituting this expression into the
portfolio variance equation we obtain the variance of return for the minimum risk hedge ratio,
σ 2MIN = σ 2S (1- ρ 2SF ) (1.11)
Therefore, the return on a hedged position will normally be subject to the risk caused by
unanticipated changes in the relative price between the position being hedged and the futures
contract. The formula clearly expresses that only when there is a perfect correlation the risk
can be completely eliminated by hedging.
However, as recognized by Cecchetti et al. (1988) and Castellino (1990) the Minimum
Variance (MV) hedge ratio is in general inconsistent with the mean-variance framework. In
order to be consistent the MV hedge ratio has to implicitly assume that either expected returns
on the futures contract needs to be zero or that investors are infinitely risk averse, as shown
above, which means that they will renounce an infinite amount of expected return in exchange
for an indefinitely small risk reduction. Such an assumption is undoubtedly unrealistic,
2
i.e. ψ→∞
4
nevertheless the minimum variance hedge ratio is useful in order to provide a benchmark for
the hedging performances.
Many studies have been proposed on the empirical estimation of the optimal hedge ratio
employing various techniques. If the spot and futures prices are not cointegrated and the
conditional variance-covariance matrix is time invariant, it has been shown that a constant
optimal hedge ratio can be obtained from the slope coefficient h in the regression:
∆ St = α + h* ∆Ft + εt. (1.12).
Hence, the simple method of the Ordinary Least Squares regression in which the
coefficient estimate for the future price gives the hedge ratio by regressing the spot on the
future price, has been employed in the past by various studies, like Ederington (1979),
Malliaris and Urrutia (1991) and Benet (1992).
However, this method has suffered various criticisms. It has been shown (Pindyck, 1984;
Poterba and Summers, 1986; Bollerslev, 1986; Baillie and De Gennaro, 1990) that stocks
returns typically exhibit time-varying conditional heteroscedasticity and hence the data do not
support the assumption that the variance-covariance matrix of return is constant over time.
Thus, in order to improve the estimation of the hedge ratio, it is necessary to consider the
possible time-varying nature of the second moments. For this reason the more recent literature
(Baillie and Myers, 1991; Myers, 1991; Sephton, 1993; Park and Switzer, 1995) has proposed
the use of hedging strategies based on the GARCH (generalized autoregressive conditional
heteroscedasticity) class of models which allow the conditional variances and covariances
used as inputs to the hedge ratio to be time-varying.
Another issue addressed by a large number of researchers is the important role played by
the theory of cointegration between futures and spot market in determining the optimal hedge
ratio. In fact, the presence of the efficient market hypothesis and the absence of arbitrage
opportunity imply that spot and futures markets are cointegrated and an error correction
representation must exist.
5
Finally, other researchers have proposed more complex techniques and some special case
of the above techniques for the estimation of the OHR. Among these we mention the random
coefficient autoregressive offered by Bera et al. (1997), the Fractional Cointegrated Error
Correction model by Lien and Tse (1999), the Exponentially Weighted Moving Average
Estimator by Harris and Shen (2002), and the asymmetric GARCH by Brooks et al. (2002).
Nevertheless, whether a distinctive superior hedge ratio estimation methodology exists is a
question that is still under debate.
1.2 Research structure
In spite of the copious literature on the subject there is no general consensus about a
technique that could systematically provide a better estimation of the optimal hedge ratio for
stock index futures. Indeed, if the OLS model shows various inadequacies from a statistical
point of view, more complex models have offered conflicting results from an empirical point
of view.
Thus the objective of this study is to offer a contribution to this question applying various
models to the calculation of the hedge ratio of the stock index futures and the underlying cash
index for the MIB 30 market. Moreover this work expands the range of the markets that has
been tested because to the best of our knowledge no study in the international literature has
been made on the FIB 30 stock index futures of the Italian Derivatives Market.
The rest of the paper is organized as follows. In the next section a detailed review of
previous studies specifically conducted on stock index futures is presented. Then section 3
provides an overview of the data sample while section 4 describes the methodology of the
research, analysing in-depth the statistical tests and econometric models that are employed in
the estimation. Finally section 5 presents the results of the statistical tests and of the hedge
ratio estimation and compares the hedging performances of the various models.
6
2. Literature review
The great part of studies on stock index futures hedging still relates to the United States of
America, although more recently various works have been published in relation to the UK,
Australia and Japan. This section aims to outline the main themes of previous works and
identify the most recent trend of research. The review clearly focuses on studies specifically
conducted on stock index futures but works on optimal hedge ratios estimation that refer to
other types of futures contracts are also reported when their results are considered important
in order to understand the development of the research. In fact, since the existing literature is
quite voluminous, a selection is necessary in order to convey the main themes.
2.1 The early studies
The hedging of basis risk is grounded in the mean variance framework of Markowitz
(1952), and is firstly applied to futures hedging by Working (1953), Johnson (1960), and
Stein (1961) but Ederington (1979) properly formalizes these ideas. He argues that a
minimum variance hedge ratio can be defined as the ratio of the covariance between the
futures and spot price to the variance of the future price. Then he shows that the minimum
variance hedge ratio is just the slope coefficient from an OLS regression where spot and
future prices are respectively the dependent and independent variable.
Ederington’s theories have been followed by numerous studies including Hill and
Schneeweis (1981, 1982); Benninga et al. (1983); Figlewski (1984, 1985); Witt et al. (1987);
Myers and Thompson (1989); Castellino (1990); Myers (1991). These studies have tried in to
analyse in particular the question of whether levels, changes, or return models should be used
in the regression approach to optimal hedge ratio estimation. For example, Witt et al. (1987)
7
study the theoretical and practical differences among these three specifications frequently
used to estimate hedge ratios in relation to the hedging in different agricultural commodities.
They find that hedge ratios based on price-level regressions are as statistically significant as
those by the other two procedures, in disagreement with the opinion of some previous studies,
and argue that theoretically the proper hedge ratio estimation technique depends upon the
objective function of the hedger and the type of hedge being considered. Myers and
Thompson (1989) try to offer a generalized approach in order to solve the same issue.
However also in this case none of these regressions seems to provide an absolute answer to
the question and every one of them appear appropriate under special circumstances.
Moreover, their application to the hedging in corn, soybeans and wheat futures markets shows
that the simple regression model is the more appropriate also if its estimates are close to those
obtained with the generalized approach.
2.2 The development of the research
The first researcher to focus on the use of stock index is Figlewski. Since then numerous
works have followed with varying conclusions. Figlewski (1984) investigates hedging
performances for the Standard & Poor’s 500 futures contract related to a portfolio underlying
the five major stock indexes for the period 1982-83. He includes dividend payments in the
return series but finds that this choice does not alter the obtained results because of their
predictable nature. Consequently, following studies have excluded this component from the
calculation of the returns. He confirms that ex-post minimum variance hedge ratios can be
estimated by OLS using historical data and shows that in all cases hedging performances are
worse using the beta hedge ratio instead of the minimum variance hedge ratio. Also Junkus
and Lee (1985) examining hedging effectiveness of three USA stock indexes futures
8
exchanges, confirm that the minimum variance hedge ratio is superior to all the others in
reducing the risk associated to the cash portfolio.
However Ghosh (1993), through empirical calculation based on several stock portfolios
hedged with Standard & Poor’s 500 index futures, finds the hedge ratios obtained from
traditional models to be underestimated because the cointegration relation is ignored. In fact,
the theory of cointegration developed by Engle and Granger (1981) shows that if two series
are cointegrated there must exist an error correction representation that permits to include
both the short-run dynamics and the long-run information. In accordance Lien and Luo (1993)
prefer an error correction model in order to estimate the hedge ratios for stock index markets,
because of the relationship between spot and futures markets. These considerations are
analytically confirmed by Lien (1996) that proves that the hedger makes a mistake if the
hedging decision is based on the hedge ratio obtained from a first difference model that does
not include an error correction term. Also Chou et al.’s work (1996) agrees with the previous
conclusions investigating the hedging behavior of the Nikkei Stock Index futures. This study
estimates and compares the hedge ratios of the conventional and error correction model
revealing that the error correction model is superior in a statistical sense.
At the same time, other studies were supporting the possibility of improving the estimation
of minimum risk variance ratio through the adoption of time-dependent conditional variance
models such as the GARCH framework developed by Engle (1982) and Bollerslev (1986). In
this way Baillie and Myers (1991) and Myers (1991), who examining commodity futures
report improvements in hedging performances over the constant hedge approach by following
a dynamic strategy based on a GARCH model. However Myers (1991) underlines that despite
significant theoretical advantages, in practice the GARCH model performs only slightly better
than a simple constant hedge ratio. For this reason, he asserts that assuming constant optimal
hedge ratios and using linear regression approaches, may be an acceptable approximation.
9
Later on, Kroner and Sultan (1993) find similar results with five currency futures over the
period 1985-1990.
Park and Switzer (1995) are the first to apply dynamic hedging to the stock index futures,
and in particular to the daily data from the Standard and Poor’s 500 Index Futures and the
Toronto 35 Index Futures. They argue that in this case, and also considering transaction costs,
the GARCH model gives an improved hedging strategy in comparison with the traditional
strategy, the OLS hedge, and the OLS with cointegration between spot and futures. However
Lien and Luo (1994) do not agree about the supposed superiority of GARCH to all the other
models in the hedge ratio estimation. In fact in their opinion however GARCH may
characterize the price behavior of financial assets, the cointegration relation, and hence the
error correction term, is the only truly indispensable element in order to optimise the hedging
performances of the obtained ratio. Fackler and McNew (1994) go beyond, deeply criticizing
the negative features of the GARCH model. In fact, in their opinion GARCH methodology is
“a less than ideal technique for estimating time-varying hedge ratios” (1994: 620). This is
because the estimation of GARCH models requires the use of nonlinear optimization
algorithms and the imposition of inequality restrictions on model parameters.
Holmes’ study (1995) is the first to investigate ex-ante hedging effectiveness for the UK in
relation to FTSE-100 contract. Considering the period 1984-1992, the results show the
possibility to achieve a risk reduction of more than 80% in comparison with the unhedged
portfolio. In a following article Holmes (1996) examines ex-post hedging effectiveness for the
same contract as in the earlier study. He shows that in terms of risk reduction a hedge strategy
based on a minimum variance hedge ratio estimated through a OLS slightly outperforms a
strategy based on a minimum variance hedge ratio estimated using more advanced techniques
such as the ECM and the GARCH approach but also by simpler techniques like the beta
hedge and the unhedged portfolio.
10
2.3 Recent issues in the hedge ratio estimation
Recent researches in optimal hedge ratio estimation have proposed the adoption of
increasingly complex estimation techniques. These studies certainly offer an important
theoretical contribution but it is not yet completely clear if these models are able to offer a
substantial improvement from a practical point of view.
Lypny and Powalla (1998) study the hedging effectiveness of a dynamic hedging strategy
based on a GARCH (1,1) covariance structure, combined with an error correction of the mean
returns, for the German stock index DAX futures. They explain that the adoption of this
model gives significant improvements over a simple constant hedge and over a dynamic
hedge with the error correction but without the GARCH (1,1) covariance structure, which is
consistent with the study by Park and Switzer (1995). Also Chakraborty and Barkoulas (1999)
test the GARCH (1,1) model for future contracts of five leading currencies. However, in their
case despite the empirical evidence supports time-varying optimal hedge ratios the dynamic
model significantly outperforms the static model just in one case out of five.
A fractionally integrated error correction is applied by Lien and Tse (1999) for estimating
the hedge ratio. The results from this model are compared with that obtained from the OLS,
VAR, EC and ARFIMA-GARCH approaches using daily data for the Nikkei Stock Average
Index over the period 1989 to 1996. They obtain that the OLS estimation of the minimum
variance hedge ratio provides the worst outcomes for hedging periods longer than 5 days as
compared to the other methods.
Afterward Miffre (2001) proposes a new methodology, the GMM, that tries to estimate
time-dependent minimum variance hedge ratios minimizing hedging rule as a set of
conditional moments restrictions. However, the following tests suggest that the traditional and
the OLS approaches can even result in better hedging performances than the proposed
methodology.
11
Recently, Yang’s study (2001) describes and estimates four different models to calculate
optimal hedge ratios for the All Ordinaries Share Price Index (AOI) and the corresponding
index futures contract (SPI) on the Australian market. In this case the Error Correction model
is found to perform better off than the OLS regression model, the bivariate vector
autoregressive model (BVAR) and the multivariate diagonal Vec GARCH model. This result
is consistent with Ghosh (1993) and Lien’s (1996) works which had proved that the lack of
consideration for the cointegration between spot and futures markets leads to a smaller than
optimal position.
Also Sim and Zurbruegg’s (2001) study agrees on the importance of utilizing an estimation
technique of the hedge ratio that takes into account the cointegrative relationship between spot
and futures markets but it is considered as much as important as considering the time-varying
nature of market risk. Their arguments are based on an empirical application to the FTSE-100
spot and futures contracts that shows the advantages of employing a time-varying hedge ratio
rather than a simpler constant measure.
The paper by Harris and Shen (2002) offers an alternative to traditional approaches to the
OHR. The researchers adopt an OHR robust to leptokurtosis and compare it to the rolling
window and the EWMA approach in relation to the daily data for the FTSE 100 index futures
for the period 1984-2002. The results show that the variance of the new OHR is as much as
70% lower than the variance of the OHRs obtained from the traditional methods, implying a
substantial reduction in the transaction costs.
Another paper by Lien et al. (2002) uses a constant correlation model in relation to ten spot
and futures markets covering currency, commodity and stock index futures. It asserts that also
if conditional heteroscedasticity is a characteristic of many financial series there is no
definitive conclusion about the superior hedging performances of the GARCH model.
Therefore, the performances of the MV hedge ratio from an OLS and a CC-GARCH are
obtained and compared showing that the GARCH strategy is not able to outperform the OLS
12
strategy. Moreover, given the high transaction costs associated with the GARCH strategy the
authors argue that it should not be considered for hedging purposes but just for data
description.
Brooks et al. (2002) focus on the impact of asymmetries on the hedging of stock-index
position through index futures. The outcome of their research, based on 3850 daily
observations of the FTSE-100 stock index and stock index futures contract spanning the
period January 1985 to April 1999, shows that asymmetric models in which positive and
negative prices changes affect volatility forecast differently permit, in general, to improve the
hedging effectiveness.
In the same year Laws and Thompson (2002) compare the hedge ratios obtained through
the Naïve, OLS, GARCH (1,1), Egarch in Mean and EWMA methods. Their study, based on
the FTSE-100 stock index futures, points out that in this case the EWMA method results in
being superior to the others.
However a recent work by Moosa (2003) questions the theory that the choice of the model
used to estimate the hedge ratio can really affect the effectiveness of hedging. His study
analyses this proposition employing four different models: a level model, a first difference
model, a simple error correction and a general error correction model. These models are
compared in relation to two different sets of data. The first is a set of monthly observations on
cash and futures prices of Australian stocks while the second sample consisted of quarterly
observations covering the period 1987 – 2000 on the spot exchange rates of the pound and the
Canadian dollar against the US dollar. The results show that the model specification does not
make any significative difference for hedging effectiveness in both cases. For this reason
Moosa concludes that “Although the theoretical arguments for why model specification does
matter are elegant…what really matters for the success or failure of a hedge is the
correlation between the prices of the unhedged position and the hedging instrument” (2003:
13
19). In substance, a low correlation necessarily implies a poor hedging performance while a
high correlation produces a good hedging performance.
2.4 Summary of the literature review
It is apparent from previous studies that many researchers have proposed the estimation of
time-varying optimal hedge ratios for stock index futures based on GARCH models or
extensions of them (Baillie and Myers, 1991; Myers, 1991; Kroner and Sultan, 1993; Park
and Switzer, 1995; Sim and Zurbruegg, 2001; Brooks, 2002). Moreover, is also generally
recognized that when the presence of cointegration is detected the inclusion of an error
correction term becomes appropriate (Ghosh, 1993; Lien, 1993; Kroner and Sultan, 1993;
Lien and Luo, 1994). More complex alternatives to the traditional GARCH model have also
been offered in recent years (Lien and Tse, 1999; Harris and Shen, 2000; Miffre, 2001;
Brooks, 2002).
However, some authors have pointed out that despite the fact that the more advanced
techniques have the potential to exhibit a better performance there are some drawbacks that
should be considered. First of all, some of these methods can be too difficult to estimate and
can produce significative transaction costs (Lien, 2002). Moreover, more simple methods like
the OLS one can reach similar levels of performance many times and can be more than
satisfactory for hedging purposes (Myers, 1991; Holmes, 1995; Chakrabothy and Barkoulas,
1999; Miffre, 2001).
Conclusively, there is not the evidence necessary in order to suggest the establishment of a
unique optimal hedge ratio.
14
3. Data
3.1 The Italian Derivatives Market
The Italian Derivatives Market (IDEM) is the Italian market of stock derivatives run by
Borsa Italiana Stock Exchange of Milan. The IDEM market was born in November 1994,
when the telematic trading of FIB 30, the futures contract on the MIB 30 index, began.
The MIB 30 index is an arithmetic average, weighted by market capitalization, based on
the 30 most highly capitalized companies listed in Borsa Italiana and accounts for more than
60% of the total market value of the Italian Stock Exchange. The members of the MIB 30
index are subject to revision by Borsa Italiana two times a year, in March and September. The
index is computed using the transaction prices, rather than using the best bid and ask prices
quotation as it happens for the FTSE 100 Index in the U.K, and is updated every minute as
long as the Borsa Italiana is open. It has basis 31.12.2004=10,000 which means that by
common consent it has been fixed that the 31st December 1994 the index had that value.
The FIB 30 futures contract is quoted in the same units as the underlying index, except that
the decimal is rounded to the nearest 0.5. There are four delivery months: March, June,
September, and December. Trading takes place in the three nearest delivery months, although
volume in the far contract is usually very small. Each contract is therefore traded for 9
months. The FIB30 contracts are cash-settled as opposed to the physical delivery of the
underlying. All contracts are marked to market on the last trading day, which is the third
Friday in the delivery month, at which point all the positions are closed. For the FIB 30 the
settlement price on the last trading day is calculated as an average of minute-by-minute
observations between 9.30 am and 10.00 am, when the trading in an expiring contract ceases.
To the best of our knowledge in international literature no study has been conducted about the
15
model specification for the estimation of the OHR for the FIB 30 contract hence it seems
plausible to choose it as the subject of our study.
3.2 Data
The data used in this study has been obtained from Datastream. It includes the MIB 30
index and the corresponding index futures contract FIB 30 daily closing prices on a daily
basis for the period of 28th November 1994 – 10th June 2004. After removing non-trading
days the series consist of 2,489 observations. Only the first 2,459 observations are used in the
test, leaving the last 30 observations, starting from 11th May 2004, for an ex-ante hedge ratio
performance comparison.
The return series for the cash portfolio and futures contract are calculated as the
logarithmic price change:
 PS ,t 
 (3.1),
∆St = RS,t= log 

 PS ,t −1 
and
 PF ,t 
 (3.2),
∆Ft = RF,t= log 

P
F
,
t
−1


where ∆St and ∆Ft are the daily returns on the cash and futures position and PS and PF are the
spot and futures prices at time t.
Both logarithmic series are represented in Figure 1, which shows a strong correlation
between them. Summary statistics of prices changes of the MIB 30 spot and futures index
returns are also provided in Table 1.
16
4. Research methodology
The literature review in the second section of this work has widely shown that there are a
variety of methods available to obtain the hedge ratio. In this study four different models are
described and estimated to calculate optimal hedge ratios. In the following paragraphs we
present and discuss each model while the last paragraph explains the methodology used in
order to compare the resulting hedge ratios.
4.1 The conventional regression model
As mentioned in the first section, the simplest technique to estimate a hedge ratio is
represented by a linear regression of change in spot prices on change in futures price. This
method has been widely applied in the literature.
Witt et al. (1988) report three basic specifications of this model: a price level, a price
change and a percentage change. However, the use of a price level model could be
inappropriate from both a theoretical and a statistical point of view. In fact Hill and
Schneeweis (1981) Brown (1985) and Wilson (1987) argue that theoretically hedgers attempt
to reduce the risk derived from unexpected price changes from the time the hedge begin until
it is lifted. Moreover, Benninga et al. (1984) point out that statistically price differences are
more appropriate because cash and futures prices can be highly correlated if corresponding
trends exist between the two of them. In that case estimating the hedge ratios by using price
levels may result in an autocorrelation in the residuals leading to a violation of the OLS
assumption and inefficient estimates of hedge ratios.
At the same time the use of a price difference equation seems to be preferable to a
percentage change model. In fact the percentage change model could be useful when cash and
17
futures prices do not change linearly with respect to one another, for example in cross hedging
when different commodities are being compared. On the contrary since in this case we are
considering direct hedging a price differences model should be preferred to a percentage
change model because it is more parsimonious and eases hedge ratio interpretation.
Hence defining the actual returns offered from the two assets at time t as ∆St and ∆Ft the
price model can be obtained regressing ∆St on ∆Ft. The linear regression used is the following
∆St = α + β ∆Ft + εt (4.1)
where α is the constant term while εt is the error term from the Ordinary Least Squares (OLS)
model. The slope of the regression β provides an estimate for the minimum variance hedge
ratio h*.
The OLS technique is quite robust and simple to use but in order to be valid and efficient
there are specific assumptions that must be satisfied. In particular, we are going to use the
Breusch-Godfrey test in order to examine whether serial correlation between the error terms is
present. In fact the Classical Linear Regression Model assumes that cov(εi , ε j)=03 and the
presence of autocorrelation could produce inefficient coefficient estimates implying wrong
estimates for the standard errors. The Breusch-Godfrey test is preferable to the DurbinWatson (DW) because it is not subject to the DW restrictions and permits to detect a wider
range of residual autocorrelation forms. The hypothesis that there is no serial correlation
between the current value and the previous value of each variable of the model (S and F) will
also be tested using the Ljung-Box statistic which permits to verify if k of the correlation
coefficients are jointly equal to 0.
Next, we are going to apply the Bera-Jarque test to examine that disturbances are normally
distributed. The Bera-Jarque test is the most commonly applied statistic to this end and
exploits the properties that characterize the skewness and kurtosis of a normal distribution in
order to do that.
3
i.e. the errors are independent with one another.
18
Finally we are going to test if errors do not have a constant variance, that is in other words
the presence of heteroscedasticity. The tests necessary to detect the existence of
heteroscedasticity and ARCH effects are presented later together with the introduction of the
GARCH model.
4.2 The Bivariate Vector Autoregressive Model
Despite the traditional approach being undoubtedly inviting for its easy application, it
suffers from a number of limitations. As earlier noted by Herbst et al. (1993) one weakness of
the above model is represented by the fact that estimation of the minimum variance hedge
ratio could suffer from problems of serial correlation in OLS residuals. Moreover this model
does not take account of the fact that movements in the spot and futures markets may
influence the current price movement.
These problems can be solved through a bivariate Vector autoregressive model (VAR) that
is a systems regression model where there are two variables, each of whose current value
depends on a combination of previous value of both the variables. For this reason, in order to
eliminate the serial correlation and take in consideration independent variables that can
influence the dependent variable we repropose the following model already tested by Yang
(2001) for the Australian market:
k
k
i =1
i =1
k
k
i =1
i =1
∆S t = α s + ∑ β si ∆S t −i + ∑ λ si ∆Ft −i + ε st (4.2a)
∆Ft = α f + ∑ β fi ∆S t −i + ∑ λ fi ∆Ft −i + ε ft (4.2b)
where α is the intercept, βs, βf, λs and λf are positive parameters, εst and εft are independently
identically distributed error terms. The optimal lag length k that permits eliminating the serial
correlation from the system equation is defined using the multivariate versions of the
19
Akaike’s and Schwarz’s Bayesian information criteria. Possible disagreements between the
two criteria are solved through the likelihood ratio test.
If we let σ2( εft) = σff, and cov (εst, εft) = σsf we can easily define the optimal hedge ratio
from the model as:
h*=
σ sf
σ ff
(4.3).
4.3 The Vector Error Correction Model
The presence or the absence of a cointegration relationship between the variables is an
important investigation to conduct since standard inference procedures do not apply to
regression where the dependent variable and the regressors are integrated. In fact Ghosh
(1993), and Kroner and Sultan (1993) have shown that the regression given by an equation
like the standard VAR is misspecified when spot and future index prices are cointegrated
because it ignores the error correction, excluding the impact of last period’s equilibrium error.
Hence Ghosh (1993) and Lien (1996) show that the existence of a cointegration relationship
between spot and future markets will typically produce a downwardly biased hedge ratio if
the error correction term is not considered in the estimated equation.
Following the theory of cointegration developed by Engle and Granger (1987: 235) we can
define as integrated of order d, denoted xt ~ I (d), “a series with no deterministic component
which has a stationary, invertible, ARMA representation after differencing d times”.
If two variables, xt and yt, are both I(1), it is generally true that a linear combination of
them zt such that
zt = x t + α yt (4.4)
will also be I (1). However it is possible that zt ~ I (0) when a special constraint operates on
the long-run components of the two series, in other words if the two series are cointegrated.
20
To formalize these ideas, let vt be a n x 1 vector of variables, then the components of vt are
said to be cointegrated of order (d, b) if: (i) all the components of vt are I(d); (ii) there exist at
least one vector with coefficient α ( ≠ 0) such that zt = α’vt ~ I(d-b), with b>0.
Ordinary least squares (OLS) method is inappropriate if xt and yt are non-stationary
because the standard errors are not consistent. However, Engle and Granger show that if two
series are non-stationary but a linear combination of them is stationary an error correction
representation, not subject to spurious results, that links these variables must exist.
The theoretical relationship that links spot and futures index price is represented by the
cost-of-carry model:
Ft = St e(r-d) (T –t) (4.5)
where Ft is the stock index futures price at time t, St is the stock index price at time t and T
represents the future expiration date, so that (T-t) is the time to maturity of the future contract.
Finally, (r-d) is the net cost of carrying the underlying stock in the index, which is the time
value cost of wealth tied up in the stock index investment, offset by the flow of dividends
from the underlying basket of stocks
Taking logarithms of both sides the relation gives
log Ft = log St + (r-d) (T-t), (4.6)
implying that the long-term relationship between the logs of the spot and futures price should
be equal to one and consequently the basis should be stationary. If, on the contrary, the future
prices could wander without bound an arbitrage opportunity would arise offering the traders
the possibility of a riskless profit such that the relationship would be soon brought back to
equilibrium. Because cointegration is an econometric way of identifying linear combinations
of variables that are bound by some relationship in the long run, spot and futures are supposed
to be cointegrated since they are the price of the same asset at different points in time. Testing
for cointegration and estimating cointegrating vectors is therefore an important issue in the
specification of the model for calculating the hedge ratio.
21
4.3.1 Tests for unit roots
A stationary series can be defined as one “with a constant mean, constant variance and
constant autocovariances for each given lag” (Brooks, 2002: 367).
The implications of stationarity in series are profound since it affects their own properties
and behaviour. In a pioneering work Granger and Newbold (1974) have shown that regressing
non-stationary variables onto each other leads to potentially misleading inferences about the
estimated parameters and the degree of association. In fact the use of non-stationary data can
imply that the regression of one variable on the other could show significant coefficient
estimates and a high value for the R2 measure even when the two variables are not related to
each other. This kind of model is known as spurious regression and in this case the standard
assumption for asymptotic analysis are not valid.
Therefore, the first step involves testing for the order of integration of the individual
variables under investigation. Various methods have been proposed in order to test for
stationarity. They are based on the fact that a non-stationary series is characterized by a unit
root. Indeed, the data-generating process can be written as
(1-L) yt= α +εt (4.7).
Thus the characteristic equation has a single root equal to one, hence the name.
The early work on testing for a unit root in time series is proposed by Dickey and Fuller
(Fuller, 1976; Dickey and Fuller, 1979). The aim of their test, in its basic version, is to
examine the null hypothesis that in the equation
∆y t = α o y t −1 + ε t (4.8)
the value of the parameter α 0 is equal to 0, i.e. the series has a unitary root for the
characteristic equation and hence is not stationary, versus the one-sided alternative hypothesis
that α 0 <0. If spot and futures prices are found to be non-stationary, the test should be
repeated to detect the order of non-stationarity. In that case the null hypothesis will be that the
series has two unit roots against the alternative hypothesis of just one unit root. However not
22
many series in finance are found to have more than a unit root. The test statistics for the
original DF test do not follow a standard distribution and critical values are obtained through
simulation experiments.
The Dickey-Fuller test is based on the assumption that the error term εt is white noise and
hence not autocorrelated. However if this is not the case the test will be oversized, implying
that the portion of times in which a correct null hypothesis is incorrectly rejected will be
higher than the nominal size chosen. For this reason, Dickey and Fuller, Dickey and Said
(1984), Phillips (1987), Phillips and Perron (1988), and others developed modifications of the
Dickey-Fuller test that apply when εt is not white noise. Basically, in that case the solution is
to “augment” the test adding p lags of the dependent variable of the regression (∆y).
Here we apply the following augmented Dickey-Fuller (ADF) test for the presence of unit
root:
p
∆y = a 0 y t −1 + ∑ α i ∆y t −i + ε t (4.9)
i =1
where ∆y t = y t − y t −1 . Lagged differences ensure that any dynamic structure present in the
dependent variable is erased and that εt is a stochastic term that has mean equal to 0, a
constant variance and follows the assumption of absence of autocorrelation. The test is subject
to the same critical values of the Dickey-Fuller test. The number of lags p is chosen in order
to optimise the trade-off between the removal of all the autocorrelation and the increase in the
coefficient standard errors. The optimal number of lags p of the dependent variable is selected
using two information criteria the Akaike information criterion (AIC) and the Schwartz’s
Bayesian information criterion (SBIC), based on a maximum lags of 7.
Really, it has been explained by Kim and Schmidt (1993) that the accuracy of the DickeyFuller test decreases in presence of ARCH effects, implying the rejection of the null
hypothesis of a unit root too often. However the same paper concludes that the effect is not
great and the test can still be considered quite reliable. An alternative test that utilizes non-
23
parametric correction for serial correlation has been proposed by Phillips and Perron (1988).
Their test is quite similar to the ADF one, and suffers from more or less the same limitations,
but includes a correction to the ADF procedure to allow for autocorrelated residuals. Also if
the two tests give the same conclusion often, Leybourne and Newbold (1999) report that it has
been noted that if the process is stationary but with a root close to the non-stationary boundary
these tests can give different outcomes and that is why we consider both of them in order to
have more robust results.
Once the presence of unit roots is detected it is possible to test for cointegration.
Cointegration is analysed by the two-step procedure of Engle and Granger (1987). The Engle
and Granger test is a single equation technique that basically uses the ADF test on the
residuals εt of the regression
∆y t = α∆xt + ε t (4.10)
to see whether they are stationary or not. However since this test is conducted on the residuals
of an estimated model, then the critical values are different from those specified for an ADF
test on series of raw data. Engle and Granger (1987) have tabulated these new values giving
their name to this application.
4.3.2 Model specification
The finding that St and Ft are cointegrated implies that the cointegrating vector must be
incorporated in the VAR obtaining a Vector Error Correction Model. Denoting the error
correction term as E the VECM can be written as:
k
k
i =1
i =1
k
k
i =1
i =1
∆S t = c s + ∑ β si ∆S t −i + ∑ λ si ∆Ft −i − α s Et −1 + ε st
(4.11a)
∆Ft = c f + ∑ β fi ∆S t −i + ∑ λ fi ∆Ft −i + α f Et −1 + ε ft (4.11b)
24
where the coefficients αs and αf can be interpreted as speed of adjustment factors and measure
how quickly each market reacts to the deviation from the long run equilibrium relationship.
Hence if the cointegrating vector between Ft and St were denoted as (1, -β) the error
correction term E is equal to (st – βft). In this way, the error correction term permits correcting
a proportion of last period’s equilibrium error. Therefore if this equilibrium is positive
because the spot rate is too high in relation to the value of the futures price, the error
correction term should push down the spot price back to equilibrium and αs will have a
negative value. At the same time the error correction term should push up ∆Ft in the second
equation, implying a negative sign for αF. A zero coefficient for one error correction term
would imply that however the two markets move following a long run relationship one market
does all the readjustment, in other words it is a market to follow the other. On the contrary, it
is not possible that αs=αf=0 since otherwise the model would be a standard VAR in
differences and the two variables would not be cointegrated.
The constant hedge ratio can again be calculated as shown in equation (4.3).
4.4 The Bivariate GARCH with error correction model
The problem with the models previous offered is that they implicitly assume that the risk in
spot and futures markets is constant over time, implying that the minimum risk hedge ratio
will be the same all the time irrespective of when the hedging is undertaken. However this
assumption clearly contrasts with the reality since as Bollerslev (1990) and Kroner and Sultan
(1991) have shown the availability of new information produces changes in the risk of the
various assets. Hence the risk minimizing hedge ratios should be time varying. Therefore
conventional models cannot produce risk-minimizing hedge ratios, raising important concerns
regarding the risk reduction properties associated with the traditional methods.
25
There are various statistical tests useful in detecting the presence of a non-constant
variance, or heteroscedasticity, in the data. Most of the tests are applied to the ordinary least
square residuals which will mimic the heteroscedasticity of the true disturbances. However to
formulate most of the available tests it is necessary to specify the nature of the
heteroscedasticity. For this reason one of the most popular methodologies is represented by
White’s general test which makes few assumptions about the likely form of the
heteroscedasticity. Moreover, the presence of ARCH effect is tested as well. Actually, the
volatility clustering is a well-known characteristic of many series of asset return. This
particular specification of heteroscedasticity refers to the observation that in many financial
time series the current level of volatility tends to be positively correlated with its level during
the previous periods. Therefore the Lagrange Multiplier test of Engle (1982) is going to be
used to detect the presence of this feature.
If the data sample is characterized by sudden spikes and periods of increased volatility as
usually happens with spot and futures data, the assumption of a constant variance results
inappropriate. The development of the Autoregressive Conditional Heteroscedasticity
(ARCH) model introduced by Engle (1982) and generalized by Bollerslev (1986) and its
enormous extension and derivations in the early 1980’s has proved to be very useful in
providing an econometric method to calculate time-varying hedge ratios based on the
conditional variance and covariance.
ARCH and GARCH models allow for the conditional variance of a series to be dependent
upon lagged squared error residuals and its own lagged values. In this way the models should
afford better predictions of changes in the basis by internalising the temporal variability of the
covariance matrix of spot and futures price changes and by allowing shocks to volatility to
persist. In general GARCH models are preferable to ARCH models since they are more
parsimonious and avoid overfitting leading consequently to a smaller probability to breach the
non-negativity constraints that characterize the conditional variance equation.
26
Unfortunately, there are several futures of the GARCH methodology that may make it a
less than ideal technique for estimating time-varying hedge ratios. From a practical
standpoint, the adoption of a GARCH model requires the use of nonlinear optimization
algorithms and possibly the imposition of inequality restrictions on model parameters.
Moreover, the methodology can be quite taxing and the only evidence that supports the effort
for this kind of model comes from simulation that compare the risk reduction benefits of an
estimated non-constant hedge ratio to those of an estimated constant hedge ratio.
The model tested herein is a multivariate GARCH with error correction that takes account
for the ARCH effects in the residuals of the error correction model. This kind of model has
already been employed in Brooks et al. (2002). The conditional mean equation of the model is
represented by the following Vector Error Correction model:
∆St= c0 + γ0 Et-1 + εst (4.12a)
∆Ft = c1 + γ1Et-1 + εft (4.12b)
Under the assumption εt│Ωt~(0, Ht), and by defining ht as vech(Ht), where vech(.) denotes
the vector-half operator that stacks the lower triangular elements of a N×N matrix into an
[N(N×1)/2]×1 vector , the bivariate VECM (k) GARCH (1,1) may be written as:
 hss ,t 
vec(HT) = ht=  hsf ,t  = c0 + A1 vec(εs,t-1; εf,t-1)+B1ht-1 (4.13)
h ff ,t 


that can be expanded as:
 hss ,t   c ss ,t   a11

 
 
 hsf ,t  = c sf ,t  + a 21
h ff ,t  c ff ,t   a31

 

a12
a 22
a 32
a13   ε s2,t −1  b11 b12


a 23  × ε s ,t −1 , ε f ,t −1  + b21 b22
a33   ε 2f ,t −1  b31 b32
b13   hss ,t −1 


b23  ×  hsf ,t −1 
b33  h ff ,t −1 
where hss and hff represent the conditional variance of the errors (εs,t, εf,t) from the mean
equations, while hsf represents the conditional covariance between spot and futures prices.
However there are 21 parameters in the conditional variance-covariance structure of the above
bivariate GARCH (1,1) vech model to be estimated, subject to the constraint that Ht be
27
positive-definite for all values of εt in the sample. Given the excessively large number of
parameters needed to estimate the model, Bollerslev (1990) has proposed the assumption that
matrix Ai and Bi are diagonal and the correlation between the conditional variances are to be
constant. However this assumption has been found unrealistic for a large number of financial
series by Bera and Roh (1991) who have conducted a test for constant correlation hypothesis.
Hence Engle and Kroner (1995) have proposed the Bollerslev, Engle, Kroner, and Kraft
(BEKK) parameterization:
H t = C 0*'C 0* + A11*' ε t −1ε t' −1 A11* + B11*' H t −1 B11* (4.14)
where,
 c11*
C = *
c 21
*
0
*

α 11* α 12* 
 β 11*
c12
*
*
; B11 =  *
 ; A11 =  *
* 
0
 β 21
α 21 α 22 
β 12* 
.
β 22* 
The BEKK parameterization has the advantages of requiring the estimation of only 11
parameters in the conditional variance-covariance structure and of guaranteeing that Ht is
positive definite.
4.5 Hedging effectiveness
So far four different strategies have been proposed to estimate optimal hedge ratios for our
data sample. Then the performances of the different hedging models previously discussed are
evaluated and formally compared using ex-post and ex-ante forecasting methods.
In order to do that an unhedged portfolio, consisting of a combinations of shares with the
same proportion as the stock index held on the spot market, and the hedged portfolio,
consisting of share price index held on both the spot and futures markets, are constructed.
Hence the mean and variance of the returns and the percentage variance reduction of the
hedged portfolio are calculated and compared in relation to the unhedged portfolio.
28
According to Kroner and Sultan (1993) the variance of the return of the unhedged and
hedged portfolio are simply expressed as:
Var (U) = σ S2 (4.15)
Var (H) = σ S2 + h 2*σ 2f − 2h *σ sf (4.16)
where σS, σF are the standard deviation of the spot and futures price respectively, σSF is their
covariability, and h* is the optimal hedge ratio calculated following the various models.
Then, the hedging effectiveness of the different portfolios is measured as the percentage
reduction in the variance of the hedged portfolio in comparison to the unhedged portfolio.
Hence, the measure of hedging effectiveness is herein defined as the ratio of the variance of
the unhedged portfolio minus the variance of the hedged position, over the variance of the
unhedged position:
τ=
Var (U ) − Var ( H )
(4.17)
Var (U )
However the ultimate test of superiority of a hedging modelling strategy lies in its ability
to achieve the maximum variance portfolio reduction in the out-of-sample. This is what really
matters from an investor’s viewpoint. As a matter of fact the investor needs to fix its hedging
position ex-ante so that he is less concerned with how well the models have done in the past
than with how well they can perform in the future. Moreover satisfactory performances in the
in-sample can be obtained also by inadequate models due to the benefit of hindsight. Hence
the out-of-sample performance represents a better way to evaluate hedging strategies.
The out-of-sample analysis is conducted for the period 11th May 2004 to 10th June 2004.
For the out-of-sample testing period we forecast the hedge obtained from the GARCH model
for the following day by computing the one period forecast of the variance divided by the one
period forecast of the conditional variance. On the other hand, for the other three models that
produce constant hedge ratios, the estimated hedge ratios are used in the out-of-sample period.
29
5. Empirical results
5.1 Estimations from model 1
The first Optimal Hedge Ratio is estimated via the OLS method by running equation (4.1).
It corresponds to the coefficient of the explanatory variable, which is equal to h*= 0.93172.
The estimated coefficients of the regression are provided in table 2.
Table 2 also provides the standard errors and t-ratios associated to each coefficient to show
each coefficient’s relative significance at 5% level. The results show that the intercept is
statistically insignificant, indicating that there is no linear trend in the data generation process.
Since we have shown previously that the estimated hedge ratio may be a biased result if the
time series data do not satisfy the CLRM assumptions some diagnostic tests are conducted in
order to assess if these assumptions hold. Table 3 shows the results of the analysis. The
Ljung-Box test statistics for fifth, tenth, and fifteenth orders of serial autocorrelation as well
as the Breusch-Godfrey Lagrange multiplier test for the presence of serial correlation between
the error term and 25 of its lagged values, are conducted. According to the statistical rule that
if the tests exceed their respective critical value the null hypothesis is rejected, both the tests
suggest the presence of autocorrelation in the residuals. The presence of autocorrelation
implies that the coefficient estimates are inefficient even for large sample size causing the
possibility that standard error estimates are wrong. The presence of serial residuals can also
be interpreted as a sign of a dynamic structure in the dependent variable that the traditional
regression is not able to capture.
Next, the Jarque-Bera test is performed on the return from spot (∆St) and futures (∆Ft)
prices to analyse the normality of the residuals. The test is asymptotically distributed as a Chisquared distribution with two degrees of freedom under the null hypothesis of normality.
30
Hence the high Bera-Jarque statistic proves that residuals do not follow a normal distribution.
This result also implies that the inference we make about the coefficient estimates could be
wrong, although in this case the large sample size does not give us great cause for concern.
Such significant deviation from normality can probably be attributed to the presence of
certain types of heteroscedasticity like volatility clustering. The systematic changing
variability in the residual graph displayed in Figure 2 is a first sign of heteroscedasticity.
However, to detect the presence of heteroscedasticity White’s test statistic and the Lagrange
Multiplier test for ARCH effects are conducted. White’s test statistic follows a Chi-squared
distribution with degrees of freedom equal to the number of slope coefficients, excluding the
constant, in the test regression. The joint null hypothesis is represented by the proposition that
the error terms are homoskedastic and independent related to the regressor. The ARCH test
statistic diagnoses volatility clustering and is asymptotically distributed as a Chi-Squared
distribution under the null hypothesis of no ARCH effects. Hence, since the value of the
White’s test is greater than the critical value obtained from the Chi-squared distribution table,
the presence of heteroscedasticity is found in accordance with previous literature and our
suppositions. The same conclusion can be obtained from the Lagrange Multiplier, which
confirms the presence of ARCH effects in the residuals. The consequences of ignoring
heteroscedasticity are similar to those of ignoring heteroscedasticity that have already been
shown.
In conclusion, the results obtained from the OLS method make us cautious with the
consequent inferences since the aforementioned problems could imply non-informative
standard errors and indicate misspecified regression models.
5.2 Estimations from model 2
31
Since the presence of autocorrelation has been detected, the OLS method could be not
appropriate. This problem is solved by Yang (2001) by introducing the bivariate Vector
Autoregressive (VAR) model shown in equation (4.2a-b).
The optimal number of lags to use is chosen by applying the multivariate version of
Akaike’s information criterion (MAIC) and Schwarz Bayesian information criterion
(MSBIC). Table 4 displays the values of Akaike’s and Schwarz’s information criteria up to 7
lags. It is clear that MAIC is minimised when 4 lags of each variable are used, while SBIC is
minimised when only 2 lags of each variable are used. This is not surprising since the SBIC
embodies a much stiffer penalty term for the loss of degrees of freedom from adding extra
parameters. Given these results, it has been tested whether lags 2-4 for both the equation
could be restricted to zero using the likelihood ratio. The likelihood ratio test for joint
hypothesis is given by
∧
∧


LR = T log ∑ r − log ∑ u  (5.1)


∧
∧
where T is the sample size, while ∑ r and ∑ u is the determinant of the variance-covariance
matrix of the residuals for the restricted and unrestricted model respectively. This test statistic
is asymptotically distributed as a Chi-squared variate with degrees of freedom equal to the
number of restrictions that in this case is equal to 8.
Since the value of the test results are lower than the critical value, the restriction is
accepted by the likelihood ratio4. Hence, in subsequent analysis, a VAR (2) model is
employed. The estimated coefficients of the regression are provided in table 5.
Tests for autocorrelation are now conducted to verify the persistence of this problem. The
two streams of the residuals from equations (4.2a-b) are displayed in table 6. The Ljung-Box
Q-statistic at lag k tests the null hypothesis of absence of autocorrelation up to order k. For
daily data, the lag of 20, corresponding to a period of four weeks, is chosen. However the low
4
In fact the 5% critical value level for 8 restrictions is 15.507 while the test gives a value of 14.386.
32
levels of the autocorrelation coefficients and the high values of the Q-statistic clearly indicate
that the Bivariate VAR has adequately taken into account the serial correlation previously
detected.
5.3 Test for unit root and estimations from model 3
As it has been explained in the methodological section, whether spot and futures prices
contain a unit root will be tested first and whether they are cointegrated will be examined
later.
The results of the Augmented Dickey-Fuller (ADF) and Phillips-Perron tests for unit root
are reported in table 7. The ADF test statistics based on the Akaike’s information criterion
show that none of the level series are stationary process, while for all the difference series the
hypothesis of a unit root is rejected at the 1% level of significance, suggesting the presence of
stationarity. The same null hypothesis is examined by the Phillips-Perron test which confirms
that the logarithm of the prices are integrated I(1) and first differencing is sufficient to induce
stationarity. This conclusion is in accord with the results of many previous researches
regarding the non-stationary nature of the logarithmic price series.
As differencing once produces stationarity, we may conclude that each series is an I(1)
process, which is necessary for testing for cointegration. This is done by means of EngleGranger test that basically applies the ADF test, but with different critical values, to the
residuals of the cointegrating regression5. The relevant results are presented in table 8. The
null hypothesis that residuals contain a unit root is here clearly rejected in favour of the
alternative hypothesis that they are stationary. Therefore spot and future prices result in being
cointegrated and linked by a long run relationship with each another. The importance of these
findings is that the change in spot return series is not just a function of change in futures
5
i.e. the regression of the logged spot and futures prices: log St =α + β log Ft + u.
33
returns, as assumed by the first differences model shown before, but also of lagged
equilibrium errors and lagged value changes in spot and futures prices. Traditional models
ignore this relationship producing a suboptimal hedge ratio.
Incorporating the error correction into the previous VAR model, the error correction model
is estimated. The results are presented in table 9. They show that γs=-1.1 and γf=0.67 which
means that the error correction term is correctly signed in both equations and implies that spot
prices have a much greater speed of adjustment than the spot prices. Moreover the error
correction of futures is highly significant in both the equation and spot and futures past prices
lags seem able to explain the current movements of both the prices. Therefore a bi-directional
causality seems to exist between the two markets. Besides the table shows that the
cointegrating relationship is St-1-(0.985)*Ft-1=c which basically corresponds to the condition
for long-run market efficiency6.
5.4 Estimations from model 4
In order to examine the efficiency of the VAR model, it could be useful to verify the
features of the residuals. We have already seen that the VAR model has adequately
considered the serial correlation examining the streams of the residuals.
On the contrary, the plot of the actual values of the residuals presented in figure 3 shows
that even if the mean seems constant the variance is still changing through the time and the
presence of autoregressive conditional heteroskedastic (ARCH) effects persists. This is also
confirmed by the analysis proposed by McLeod and Li (1983), which examines the sample
autocorrelation functions of the mean equation squared residuals for a significant Q-statistic at
a given lag. The results, which show a high significance for the Q-statistic for each given lag,
are reported in table 12. Since the presence of heteroscedasticity and ARCH effects is
6
i.e. A cointegrating vector [1,1]
34
detected, the assumption of constant variance over time and the estimation of constant hedge
ratios may be inappropriate. The estimation of time-varying variances and covariances and as
a consequence time-varying hedge ratios based on a GARCH model are therefore expected to
give better results.
The bivariate GARCH (1,1) model in the Bollerslev, Engle, Kroner and Kraft (BEKK)
specification is adopted here. The model is given by equations (4.12a\b-4.14) while table 10
displays the results of the estimation. The bivariate GARCH model has been estimated using
a program for EViews, which uses the Marquardt algorithm to compute maximum-likelihood
estimates. The estimated parameters for the mean equations seem to be statistically significant
implying that the GARCH (1,1) error is able to capture the dynamics in the second moments
of the joint distribution of returns. It is also interesting to note that the cross correlation of the
variances P = 0.9717 is high but not perfect, just as we expected.
Table 13 shows that the GARCH model is really able to remove the serial correlation
previously detected since uncorrelatedness in the vector of squared standardized residuals is
now found for each given lag.
5.5 Hedging effectiveness
In this section, we evaluate and compare the hedging performances of the four hedging
models considered in our study.
The hedge ratio for the conventional regression method is obtained as the estimated
coefficient of the future price in the regression of spot on the future price. The ratio between
the variance and the covariance of the residuals is instead used to obtain the optimal hedge
ratios for the bivariate VAR model and the bivariate VEC model. The optimal hedge ratios for
these three models are presented in table 11. It can be seen that the hedge ratio obtained from
the VEC model is slightly greater than those obtained from the OLS and VAR models. This
35
result is consistent with those from Ghosh (1993) and Lien (1996) where it is noted that the
hedge ratio results biased downward in size when the cointegrating relationship is ignored.
The dynamic hedge ratio obtained from the conditional variance and covariance between
spot and futures price in the bivariate GARCH (1,1) with error correction model is plotted in
figure 4. It shows signs of extreme volatility during the sample period. The sample mean of
the hedge ratio is 0.933158 while the series ranges from a minimum of 0.59 to a maximum of
1.15.
Table 14 displays the in sample hedging performances of the various models. The naïve
method is added just as a term of comparison. The results demonstrate that all hedging
strategies permit achieving substantial risk reductions compared to the unhedged position.
Indeed the unhedged portfolio suffers, not surprisingly, of the highest variance in the return.
The naïve hedge, which assumes a unitary hedge ratio, follows with a variance reduction in
relation to the unhedged portfolio equal to the 92,16%. The hedging performances for the
remaining models do not differ very much. The bivariate GARCH (1,1) with error correction
seems to offer performances slightly superior to the OLS, the VAR and the VAR with error
correction model. In fact the variance reduction associated with the Garch model is 92.7451%
against the 92.7442% of the OLS model, the 92.7438% of the VAR model and the 92.7309%
of the VAR with error correction model. Also considering the results under a risk-return trade
off basis the GARCH model is found to outperform the other models providing the greatest
return and the lowest portfolio variance simultaneously.
The out-of-sample comparison conducted for the last thirty observations is shown in Table
15. All models provide lower portfolio returns and variance reduction than in the sample. In
this case the GARCH method continues to achieve the best performances in variance
reduction but no longer provides the highest portfolio return since in this it is outperformed by
the OLS, which offers a greatest portfolio return of about 0.085%.
36
Conclusion
This study has empirically assessed the appropriateness of various hedging models with the
Fib 30 futures contract of the Italian Derivatives Market. It is worth noting that to the best of
our knowledge no study has been previously conducted in international literature on the Fib
30 index futures and the underlying Mib 30 stock index.
At this purpose the hedge ratios obtained from the conventional OLS regression method,
the VAR model, the VEC model, and the bivariate GARCH with error correction model have
been compared under a variance minimization criterion. Only the first 2,465 observations
have been used to estimate the optimal hedge ratio, leaving the last 30 observations for an outof-sample comparison.
The performances of the hedge ratios in the in-sample and out-of-sample forecast have
offered a similar picture. All the models are able to offer a significative reduction in the
portfolio variance in comparison with the unhedged portfolio and the simpler naïve method.
In the in-sample the GARCH model slightly outperforms the other models being the only one
able to capture the time varying nature of the hedge ratio while the OLS offers the second best
performances, that is quite surprising given the presence of heteroscedasticity and the
existence of a cointegrating relationship between spot and futures markets which should
normally imply the inclusion of an error correction term in the model. The VAR model offers
performances similar to the OLS while the VAR with error correction follows, meaning that
in this case the inclusion of an error correction term is not able to assure any superior
performance. The same classification is offered by the out-of-sample comparison also if in
this context the superiority of the GARCH model is a little more appreciable.
Thus, at first sight the necessity to adopt a dynamic framework could seem justified.
However, since the GARCH model does not seem able to get a very meaningful reduction in
the portfolio variance relative to the constant hedge ratios the consideration of the extra
37
computational and rebalancing costs associated with a dynamic position could lead to decide
that the OLS method can be adequate or even preferable to estimate the optimal hedge ratio
for the FIB 30 stock index futures contract.
Of course this cannot be considered a general result and the GARCH model may provide
significantly better performance in applications related to other stock index futures. At the
same time we believe that these considerations should not be underestimated since they are
supported by various previous studies. For example, in a pioneering work Myers (1991: 40)
has already written in relation to commodities hedging that also if it performs better than the
other models in the hedge ratio estimation, “the extra expense and complexity of the GARCH
model do not appear to be warranted”.
However since in this case neither the error
correction model is able to offer better results than the OLS, this study seems to be
completely coherent with the results reached by Holmes (1996) that shows how the OLS
methods provides superior minimum variance hedge for the FTSE 100 contract than the
GARCH or the Error Correction model.
This evidence leads us to agree with the general conclusions presented in a recent research
by Moosa (2003). In this work Moosa has pointed out that what really matters for a hedger is
the correlation between the stock index futures and the underlying index, while the arguments
about more complex model specification often result “elegant” from a theoretical viewpoint
but not always consistent from an empirical one.
In fact in our case all the strategies produce similar performances and the small changes in
variance reduction do not significatively support the existence of a systematically superior
optimal hedge ratio estimation technique. On the contrary each method has some advantages
and disadvantages that investors should consider. Sometimes a method like the OLS could
even result preferable, offering reliable results with less complexity and costs. At the same
time the implications of the misspecification problems that can derive from this model should
be carefully considered since they can invalidate the resulting inferences.
38
Whether similar conclusion will appear from other applications of these estimation
techniques remains a question for future studies. The analysis in this work also suggests
several other avenues for further research like the development of new measures of hedging
effectiveness able to take account of transaction costs and taxes or the need to study in depth
the ability of the GARCH models to capture irregular market fluctuations and other
phenomena observed in asset return series in order to understand the implications for the
hedge ratio estimation.
39
References
Baillie, R. T. and De Gennaro, R. P. (1990), ‘Stock returns and volatility’, Journal of
Financial and Quantitative Analysis, Vol. 25, pp. 203-214.
Benet, B.A. (1992), ‘Hedge Period Length and Ex-Ante Futures Hedging Effectiveness: The
Case of Foreign-Exchange Risk Cross Hedges’, Journal of Futures Markets, Vol. 12, pp.
163-175.
Benninga, S., Eldor, R. and Zilcha, I. (1983), ‘Optimal Hedging in the Futures market under
price uncertainty’, Economic Letters, Vol. 13, pp. 141-145.
Bera, A.K., Garcia, P. and Roh, J.S. (1997), ‘Estimation of Time-Varying Hedge Ratios for
Corn and Soybeans: BGARCH and Random Coefficient Approaches’, Sankhya B., Vol.
59, pp. 346-368.
Bera, A.K. and Roh, J.S. (1991), ‘A Moment test of Consistency of the Correlation in the
Bivariate GARCH Model’, Mimeo, Department of Economics, University of Illinois at
Urbana-Champain.
Bollerslev, T. (1986), ‘Generalized Autoregressive Conditional Heteroscedasticity’, Journal
of Econometrics, Vol. 31, pp. 307-327.
Bollerslev, T. (1990), ‘Modelling the Coherence in Short-Run Nominal Exchange Rates: A
Multivariate Generalized ARCH Approach’, Review of Economics and Statistics, Vol.
72, pp. 498-505.
Brooks, C. (2002), Introductory Econometrics for Finance, Cambridge: Cambridge University
Press.
Brooks, C., Henry, O.T. and Persand, G. (2002), ‘The Effects of Asymmetries on Optimal
Hedge Ratios’, Journal of Business, Vol. 75, No. 2, pp. 333-352.
Brown, S.L. (1985), ‘A Reformulation of the Portfolio Model of Hedging’, American Journal
of Agricultural Economics, Vol. 67, pp. 508-512.
Castellino, M. (1990), ‘Minimum-Variance Hedging with Futures Re-visited’, Journal of
Portfolio Management, Vol. 16, pp. 74-80.
Cecchetti, S.G., Cumby, R.E. and Figlewski, S. (1988), ‘Estimation of the Optimal Futures
Hedge’, The Review of Economics and Statistics, Vol. 70, pp. 623-630.
Chakraborthy, A. and Barkoulas, J.T. (1999), ‘Dynamic Futures Hedging in Currency
Markets’, The European Journal of Finance, Vol. 5, pp. 299-314.
40
Chen, S.S., Lee, C.F. and Shrestha, K. (2001), ‘On A Mean-Generalized Semivariance
Approach to Determining the Hedge Ratio’, Journal of Futures Markets, Vol. 21, No. 6,
581-598.
Chou, W.L., Denis, K.K.F. and Lee, C.F. (1996), ‘Hedging with the Nikkei index futures: The
conventional model versus the error correction model’, Quarterly Review of Economics
and Finance, Vol. 36, pp. 495-505.
De Jong, D.N., De Roon, F. and Veld, C. (1997), ‘Out-of-sample Hedging Effectiveness of
Currency Futures for Alternative Models and Hedging Strategies’, Journal of Futures
Markets, Vol. 17, No. 7, pp. 817-837.
Dickey, D.A. and Fuller, W.A. (1979), ‘Distributions of Estimators for Autoregressive Time
Series with a Unit Root’, Journal of the American Statistical Association, Vol. 74, pp.
427-431.
Dickey, D.A. and Said, S.E (1984), ‘Testing for Unit Roots in Autoregressive-Moving
Average Models of Unknown Order’, Biometrika, Vol. 71, pp. 599-607.
Ederington, L.H. (1979), ‘The Hedging Performances of the New Futures Markets’, Journal
of Finance, Vol. 34, No. 1, pp. 157-170.
Engle, R.F. (1982), ‘Autoregressive Conditional Heteroskedasticity with Estimates of the
Variance of United Kingdom Inflation’, Econometrica, Vol. 50, No. 4, pp. 987-1007.
Engle, R.F. and Granger C.W.J. (1987), ‘Cointegration and Error Correction: Representation,
Estimation and Testing’, Econometrica, Vol. 55, No. 2, pp. 251-276.
Engle, R.F. and Kroner, K.F. (1995), ‘Multivariate Simultaneous Generalized ARCH’,
Econometric Theory, Vol. 11, No. 24, pp. 122-150.
Fabozzi, F.J. and Kipnis, G.M., eds, (1989), The Handbook of Stock Index Futures and
Options, Homewood, Illinois: Dow Jones-Irwin.
Fackler, P.L. and McNew, P.K. (1994), ‘Nonconstant Optimal Hedge Ratio Estimation and
Nested Hypotheses Tests’, Journal of Futures Markets, Vol. 14, No. 5, pp. 619-635.
Fama, E.F. (1965), ‘The Behavior of Stock Market Prices’, Journal of Business, Vol. 38, No.
1, pp. 34-105.
Figlewski, S. (1984), ‘Hedging Performance and Basis Risk in Stock Index Futures’, Journal
of Finance, Vol. 39, pp. 657-669.
Figlewski, S. (1985), ‘Hedging with Stock Index Futures: Theory and Application in a New
Market’, Journal of Futures Markets, Vol. 5, No. 2, pp. 183-189.
Fuller, W.A. (1976), Introduction to Statistical Time Series, New York: John Wiley & Sons.
Ghosh, A. (1993), ‘Hedging with Stock Index Futures: Estimation and Forecasting with Error
Correction Model’, Journal of Futures Markets, Vol. 13, No. 7, pp. 743-752.
41
Granger, C.W.J. (1981), ‘Some Properties of Time Series Data and Their Use in Econometric
Model Specification’, Journal of Econometrics, Vol. 16, pp. 121-130.
Granger, C.W.J. and Newbold P. (1974), ‘Spurious Regressions in Econometrics’, Journal of
Econometrics, Vol. 2, pp. 111-120.
Greene, W.H. (2003), Econometric Analysis, Fifth Edition, Upper Saddle River, New Jersey
Prentice Hall.
Harris, R.D.F. and Shen, J. (2002), ‘Robust estimation of the optimal hedge ratio’, School of
Business and Economics, University of Exeter.
Herbst, A.F., Kare D.D. and Marshall J.F. (1993), ‘A Time Varying Convergence Adjusted,
Minimum Risk Futures Hedge Ratio’, Advances in Futures and Option Research, Vol.
6, pp. 137-155.
Hill, J. and Schneeweis, T. (1981), ‘A Note on the Hedging effectiveness of Foreign Currency
Futures’, The Journal of Financial Research, Vol. 1, pp. 659-664.
Hill, J. and Schneeweis, T. (1982),
‘The Hedging Effectiveness of Foreign Currency
Futures’, The Journal of Financial Research, Vol. 3, pp. 95-104.
Holmes, P. (1995), ‘Ex ante Hedge Ratios and the Hedging Effectiveness of the FTSE-100
Stock Index Futures Contracts’, Applied Economic Letters, Vol. 2, pp. 56-59.
Holmes, P. (1996), ‘Stock Index Futures Hedging: Hedge Ratio Estimation, Duration Effects,
Expiration Effects and Hedge Ratio Stability’, Journal of Business & Accounting, Vol.
23, No. 1, pp. 63-78.
Hull, J.C. (2003), Options, Futures, and Other Derivatives, Fifth edition, Upper Saddle River,
New Jersey: Prentice Hall.
Johnson, L. L. (1960), ‘The Theory of Hedging and Speculation in Commodity Futures’,
Review of Economic Studies, Vol. 27, pp. 139-151.
Junkus, C.J. and Lee, C. (1985), ‘The use of Three Stock Index Futures in Hedging
Decisions’, Journal of Futures Markets, Vol. 5, No. 2, pp. 201-222.
Kim, K. and Schmidt, P. (1993), ‘Unit-Root Tests with Conditional Heteroscedasticity’,
Journal of Econometrics, Vol. 59, No. 3, pp. 287-300.
Kroner, K.F. and Sultan, J. (1991), ‘Exchange Rate Volatility and Time Varying Hedge
Ratios’, Chapter 2, pp. 397-412 in Rhee, S.G. and Chang, R.P., eds, Pacific-Basin
Capital Market Research, Amsterdam: North-Holland.
Kroner, K.F. and Sultan, J. (1993), ‘Time-Varying Distributions and Dynamic Hedging with
Foreign Currency Futures’, Journal of Financial and Quantitative Analysis, Vol. 28, No.
4, pp. 535-551.
42
Laws, J. and Thompson, J. (2002), ‘Hedging Effectiveness of Stock Index Futures’, Liverpool
Business School, Liverpool John Moores University.
Leybourne, S.J. and Newbold P. (1999), ‘The behaviour of Dickey-Fuller and Phillips-Perron
tests under the alternative hypothesis’, Econometrics Journal, Vol. 2, No.1, pp. 92-106.
Lien, D. (1996), ‘The Effect of the Cointegration Relationship on Futures Hedging’, Journal
of Futures Markets, Vol. 16, No. 7, pp. 773-780.
Lien, D. and Luo, X. (1993), ‘Estimating Multiperiod Hedge Ratios in Cointegrated Markets’,
Journal of Futures Markets, Vol. 13, No. 8, pp. 909-920.
Lien, D. and Luo, X. (1994), ‘Multiperiod Hedging in the Presence of Conditional
Heteroskedasticity’, Journal of Futures Markets, Vol. 14, pp. 927-955.
Lien, D. and Tse, Y.K. (1999), ‘Fractional Cointegration and Futures Hedging’, Journal of
Futures Markets, Vol. 19, No. 4, pp. 457-474.
Lien, D., Tse, Y.K. and Tsui, A.K.C. (2002), ‘Evaluating the Hedging performance of the
Constant-Correlation GARCH model’, Applied Financial Economics, Vol.12, pp. 791798.
Lypny, G. and Powalla, M. (1998), ‘The Hedging effectiveness of DAX Futures’, The
European Journal of Finance, Vol. 4, pp. 345-355.
Malliaris, A.G. and Urrutia, J.L. (1991), ‘The Impact of the Lenghts of Estimation Periods
and Hedging Horizons on the Effectiveness of a Hedge: Evidence from Foreign
Currency’, Journal of Futures Markets, Vol. 3, pp. 271:289.
Markowitz, H. (1952), “Portfolio Selection”, Journal of Finance, Vol. 7, pp. 77-91.
McLeod, A.I. and Li, W.K. (1983), ‘Diagnostic Checking ARMA Time Series Models Using
Squared-Residual Autocorrelations’, Journal of Time Series Analysis, Vol. 4, pp. 269273.
Miffre, J. (2001), ‘Efficiency in the pricing of the FTSE100 futures contract’, European
Financial Management, Vol. 7, No. 1, pp. 9 – 22.
Moosa, I.A. (2003), ‘The Sensitivity of the Optimal Hedge Ratio to Model Specification’,
Finance Letters, Vol. 1, pp. 15-20.
Myers, R. J. (1991), ‘Estimating Time-Varying Hedge Ratios on Futures Markets’, Journal of
Futures Markets, Vol. 11, No. 1, pp. 39-53.
Myers, R. J. and Thompson, S.R. (1989), ‘Generalized Optimal Hedge Ratio Estimation’,
American Journal of Agricultural Economics, Vol. 71, pp. 858-867.
Park, T.H. and Switzer, L.N. (1995), ‘Bivariate GARCH Estimation of the Optimal Hedge
Ratios for Stock Index Futures: A Note’, Journal of Futures Markets, Vol.15, No. 1, pp.
61-67.
43
Peters, E. (1986), ‘Hedged Equity Portfolio: Components of Risk and Return’, Advances in
Futures and Option Research, Vol. 1, pp. 75-92.
Phillips, P.C.B. (1987), ‘Time Series Regression with a Unit Root’, Econometrica, Vol. 55,
pp. 277-302.
Phillips, P.C.B. and Perron, P. (1988), ‘Testing for a Unit Root in Time Series Regression’,
Biometrika, Vol. 75, No. 2, pp. 335–346.
Pyndick, R.S. (1984), ‘Risk, Inflation and the Stock Market’, American Economic Review,
Vol. 3, pp. 335-351.
Poterba, J.M. and Summers, L.H. (1986), ‘Mean Reversion in Stock Prices: Evidence and
Implications’, Journal of Financial Economics, Vol. 22, pp. 27-59.
Sephton, P.S. (1993), ‘Hedging wheat and canola at the Winnipeg Commodity Exchange’,
Applied Financial Economics, Vol. 3, pp. 67-72.
Sim, A.B. and Zurbruegg, R. (2001), ‘Optimal Hedge Ratios and Alternative Hedging
Strategies in the Presence of Cointegrated Time-Varying Risks’, The European Journal
of Finance, Vol. 7, No. 3, pp. 269-283.
Stein, J.L. (1961), ‘The simultaneous Determinations of Spot and Futures Prices’, American
Economic Review, Vol. 51, pp. 1012-1025.
Yang, W. (2001), ‘M-GARCH Hedge Ratios and Hedging Effectiveness in Australian Futures
Markets’, School of Finance and Business Economics, Edith Cowan University.
Wilson, W.W. (1987), ‘Price Discovery and Hedging in the Sunflower Market’, Journal of
Futures Markets, Vol. 9, pp. 377-391.
Witt, H. J., Schroeder, T.C. and Hayenga, M. L. (1987), ‘Comparison of Analytical
Approaches for Estimating Hedge Ratios for Agricultural Commodities’, Journal of
Futures Markets, Vol. 7, pp. 135-146.
Working, H. (1953), ‘Futures Trading and Hedging’, American Economic Review, Vol. 43,
pp. 314-343.
44
Table 1: Descriptive Statistics of FIB 30 and MIB 30 Logarithmic Series
Mean
Stand. Dev.
Q(1)
Q(2)
Q(3)
Q(4)
Minimum
Maximum
Kurtosis
Skewness
FIB 30 (log)
MIB 30 (log)
10.17
0.38
9.82
10.21
10.47
10.84
9.48
10.84
-1.08
-0.24
10.17
0.38
9.82
10.21
10.47
10.84
9.48
10.84
-1.08
-0.25
Notes: (a) “Q(.)” is for quartile. (b) “Stand. Dev.” is for standard deviation.
Table 2: Estimates of the conventional regression model
Variables
α
∆Ft
Coefficients
Standard Error
T-statistic
P-value
3.29E-05
8.06E-05
0.408643947
0.6828
0.931722886
0.005232044
178.0800943
0.0000*
Notes: (a)The results are the estimates of equation (4.1): ∆St= α +β∆Ft+εt. (b) ∆St and ∆Ft represent the
differenced logarithm of spot and futures series respectively, at time t. (c) The standard errors, t-ratios
and P-values are presented besides the corresponding coefficients to show each coefficient’s relative
significance. (d) “*” indicates significance at 5% significance level.
Table 3: Results of the diagnostic tests conducted on model 1
Test
Autocorrelation:
Autocorrelation:
Breusch-Godfrey
Ljung Box
∆St
Q(5)
Q(10)
Q(15)
∆Ft
Q(5)
Q(10)
Q(15)
Normality:
Jarque-Bera
Heteroscedasticity: White's test
Volatility Clustering: ARCH effects test
Test
statistic
392.7690
P-value
0.0000
Conclusion
Reject
17.5001
24.4042
36.5674
12.1031
21.6248
36.1463
5011.9630
70.1076
307.3879
0.0036
0.0066
0.0015
0.0334
0.0171
0.0017
0.0000
0.0000
0.0000
Reject
Reject
Reject
Reject
Reject
Reject
Reject
Reject
Reject
Notes: (a) The statistics tests are applied to the estimates of equation 4.1. (b)The conclusions are valid
for the 5% significance level. (c) “Q(.)” indicates the results of the Ljung-Box Q-statistic at each lag.
45
Table 4: Values of the multivariate version of the Akaike’s (AIC) and Schwartz’s
Bayesian (SBIC) information criteria for different number of lags in model 2
Lags
AIC
SBIC
1
2
3
4
5
6
7
-13.8579
-13.8927
-13.8992
-13.9051*
-13.9047
-13.9033
-13.9041
-13.8437
-13.8691*
-13.8661
-13.8626
-13.8527
-13.8419
-13.8332
Note: “*” indicates the minimum for each information criterion.
Table 5: Estimates of the Bivariate VAR(2) model
∆St
Variable Coefficient
∆S(-1)
-0.3523
∆S(-2)
-0.1911
∆F(-1)
0.3421
∆F(-2)
0.2128
0.0003
Constant
Standard
Error
0.0805
0.0798
0.0777
0.0779
0.0003
∆Ft
t-ratio
-4.3740*
-2.3959*
4.3998*
2.7333*
0.8757
Coefficient
0.1666
-0.0145
-0.1756
0.0300
0.0002
Standard
error
0.0836
0.0828
0.0807
0.0808
0.0003
t-ratio
1.9934*
-0.1746
-2.1771*
0.3718
0.0003
Notes: (a) The results are the estimates of equations (4.2a-b). (b) ∆S(.) and ∆F(.) represent the
differenced logarithm of spot and futures price respectively, at each lag: 1,2. (c) “*” indicates the
statistically significant coefficients at 5% significance level.
46
Table 6: The Autocorrelation Function of the Residuals of VAR model
(a)Residuals for the spot equation
Autocorrelation
Partial Correlation
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|*
.|.
.|.
.|.
*|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|*
.|.
.|.
.|.
*|.
.|.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
AC
PAC
Q-Stat
Prob
0.000
0.000
-0.001
-0.001
-0.003
-0.002
0.004
0.000
0.029
0.015
-0.013
-0.022
0.006
0.039
0.058
0.003
-0.023
-0.019
-0.054
0.035
0.000
0.000
-0.001
-0.001
-0.003
-0.002
0.004
0.000
0.029
0.015
-0.013
-0.022
0.006
0.040
0.059
0.002
-0.023
-0.020
-0.055
0.036
8.E-05
0.0004
0.0018
0.0053
0.0277
0.0405
0.0783
0.0785
2.1238
2.6899
3.0811
4.3321
4.4278
8.2459
16.647
16.664
17.949
18.850
26.136
29.185
0.993
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.989
0.988
0.990
0.977
0.986
0.876
0.340
0.408
0.392
0.401
0.126
0.084
AC
PAC
Q-Stat
Prob
0.000
0.000
-0.001
-0.002
-0.003
-0.002
0.005
0.001
0.034
0.011
-0.015
-0.032
-0.001
0.046
0.058
-0.010
-0.029
-0.015
-0.045
0.031
0.000
0.000
-0.001
-0.002
-0.003
-0.002
0.005
0.001
0.034
0.010
-0.015
-0.032
-0.001
0.046
0.058
-0.011
-0.030
-0.016
-0.045
0.033
0.0003
0.0007
0.0031
0.0136
0.0328
0.0462
0.1037
0.1062
3.0417
3.3147
3.8361
6.4253
6.4274
11.575
19.852
20.122
22.139
22.693
27.677
30.024
0.986
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.963
0.973
0.974
0.893
0.929
0.640
0.178
0.215
0.179
0.203
0.090
0.069
(b)Residuals for the futures equation
Autocorrelation
Partial Correlation
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|*
.|.
*|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|.
.|*
.|.
*|.
.|.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Notes: This table shows the plots and the values of the autocorrelation function (AC) and the partial
autocorrelation function (PAC) of the residuals of equations (4.2a-b). The last two columns are the Qstatistics for higher order autocorrelation and the corresponding probability. The null hypothesis is that
there is no autocorrelations at a given order. The probabilities show us that the null hypothesis should
not be rejected.
47
Table 7: Results of the Unit Root tests based on the logarithmic spot and futures prices
ADF
PP
Significance
level
Critical*
Values
TestStatistics
Conclusion
Critical*
Values
TestStatistics
Conclusion
1%
5%
10%
-3.4345
-2.8632
-2.5677
-1.5799
-1.5799
-1.5799
Accept
Accept
Accept
-3.4345
-2.8632
-2.5677
-1.5753
-1.5753
-1.5753
Accept
Accept
Accept
1%
5%
10%
-3.4345
-2.8632
-2.5677
-22.2530
-22.2530
-22.2530
Reject
Reject
Reject
-3.4345
-2.8632
-2.5677
-50.2616
-50.2616
-50.2616
Reject
Reject
Reject
1%
5%
10%
-3.4345
-2.8632
-2.5677
-1.5616
-1.5616
-1.5616
Accept
Accept
Accept
-3.4345
-2.8632
-2.5677
-1.5594
-1.5594
-1.5594
Accept
Accept
Accept
1%
5%
10%
-3.4345
-2.8632
-2.5677
-22.4954
-22.4954
-22.4954
Reject
Reject
Reject
-3.4345
-2.8632
-2.5677
-51.2210
-51.2210
-51.2210
Reject
Reject
Reject
St
∆St
Ft
∆Ft
Notes: (a) ADF is the Augmented Dickey-Fuller statistics while PP is the Phillips-Perron statistics.
(b)For both the test the null hypothesis is represented by the presence of unit roots. (c)It should be
noted that the critical values for the ADF and PP unit root tests differ after the 4th decimal place.
Table 8: Result of the Engle-Granger test for unit root in the residuals of the
cointegrating equation
Significance
level
Critical
Values
TestStatistics
Conclusion
1%
5%
10%
-4.00
-3.37
-3.02
-6.7640
-6.7640
-6.7640
Reject
Reject
Reject
εt
Note: The null hypothesis of the test is represented by the presence of unit roots in the residuals of the
cointegrating equation (4.10).
48
Table 9: Estimates of the Bivariate VAR(2) Error Correction Model
∆S
Variable
Coefficient
-1.53E-05
Constant
∆S(-1)
-0.1330
∆S(-2)
-0.1878
∆F(-1)
-0.5366
∆F(-2)
-0.1448
-1.1034
ECT
Cointegrating Equation
∆St
1
∆Ft
-0.9847
-1.83E-05
Constant
∆F
Standard
Error
0.0003
0.1605
0.0927
0.1589
0.0915
0.2123
t-ratio
-0.0444
-0.8287
-2.0253*
-3.3760*
-1.5818
-5.1974*
0.0048
-
-204.536*
-
Coefficient
-1.72E-05
-0.4606
-0.2676
-0.2901
-0.0552
0.6679
Standard
Error
0.0004
0.1669
0.0965
0.1353
0.0952
0.2209
t-ratio
-4.79E-02
-2.7596*
-2.7739*
-2.1438*
-0.5797
3.0242*
Notes: (a) The results are the estimates of equations (4.11a-b). (b) ∆S(.) and ∆F(.) represent the
differenced logarithm of spot and futures price respectively, at each lag: 1,2. (c) * indicates the
statistically significant coefficients at 5% level.
Table 10: Estimates of the Bivariate GARCH Error Correction Model
Variable
Coefficient
Std. Error
z-statistic
c0
0.000345
0.00025
1.36674
c1
0.000363
0.00025
1.42313
γ1
-0.000414
0.00029
-1.43253
γ2
0.000524
0.00030
1.76431
c11
0.002418
0.00013
18.17710
c12
0.000724
0.00002
31.19459
c22
0.002262
0.00012
19.34161
α11
0.273361
0.00784
34.85457
α12
0.227850
0.00642
35.49065
α21
0.031781
0.00064
49.57972
α22
0.280177
0.00700
40.00768
β11
0.947653
0.00289
327.70695
β12
0.880972
0.00486
181.26996
β21
0.912456
0.00315
289.66857
β22
0.947080
0.00245
386.97607
p
0.97710
Notes: (a) The results are the estimates of equations (4.12a\b-4.14). (b) ρ indicates the correlation
index.
49
Table 11: Optimal hedge ratios for the OLS, bivariate VAR (2) and VEC model
Optimal hedge ratio
OLS
VAR
VECM
0.93172
0.93349
0.94281
Notes: (a) The hedge ratio for the conventional regression method is obtained as the estimated
coefficient of the future price in the regression of spot on the future price. (b) The hedge ratios for the
VAR and VEC models are obtained as the ratio between the variance and the covariance of their
respective residuals.
50
Table 12: The Autocorrelation Function of the Squared Residuals of VEC model
(a)Residuals for the spot equation
Autocorrelation
|*
|**
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
Partial Correlation
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|*
|**
|*
|*
|*
|*
|
|
|*
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
AC
PAC
Q-Stat
Prob
0.170
0.221
0.174
0.157
0.151
0.155
0.149
0.131
0.195
0.146
0.114
0.130
0.128
0.123
0.112
0.082
0.115
0.073
0.084
0.082
0.170
0.198
0.118
0.084
0.074
0.076
0.065
0.040
0.113
0.049
0.002
0.030
0.035
0.028
0.014
-0.017
0.031
-0.018
-0.002
0.010
71.338
191.44
266.22
327.17
383.46
442.78
497.15
539.15
632.94
685.21
717.28
759.33
799.53
836.90
867.94
884.57
917.19
930.47
948.13
964.96
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
AC
PAC
Q-Stat
Prob
0.171
0.218
0.191
0.148
0.146
0.140
0.159
0.156
0.219
0.155
0.113
0.139
0.125
0.117
0.112
0.067
0.116
0.082
0.081
0.075
0.171
0.194
0.137
0.071
0.065
0.060
0.082
0.072
0.135
0.047
-0.011
0.030
0.028
0.020
0.013
-0.039
0.026
-0.010
-0.004
-0.001
71.576
188.22
278.33
332.00
384.14
432.11
494.63
554.46
673.10
732.24
763.66
811.52
849.85
883.71
914.80
925.88
959.38
975.93
992.26
1006.0
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
(b)Residuals for the futures equation
Autocorrelation
Partial Correlation
|*
|**
|*
|*
|*
|*
|*
|*
|**
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|*
|
|
|*
|*
|*
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Notes: This table shows the plots and the values of the autocorrelation function (AC) and the partial
autocorrelation function (PAC) of the squared residuals of equations (4.11a-b). The last two columns
are the Q-statistics for higher order autocorrelation and the corresponding probability. The null
hypothesis is that there is no autocorrelations at a given order. The probabilities show us that the null
hypothesis should be rejected.
51
Table 13: The Autocorrelation Function of the Squared Residuals of GARCH model
(a)Residuals for the spot equation
Autocorrelation
Partial Correlation
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
AC
PAC
Q-Stat
Prob
0.000
-0.002
-0.003
-0.005
-0.004
-0.004
-0.014
-0.011
-0.013
-0.020
-0.018
-0.024
-0.025
-0.029
-0.032
-0.032
-0.025
-0.040
-0.037
-0.042
0.000
-0.002
-0.003
-0.005
-0.004
-0.004
-0.014
-0.011
-0.013
-0.020
-0.018
-0.024
-0.025
-0.030
-0.033
-0.034
-0.028
-0.044
-0.041
-0.048
0.0005
0.0092
0.0335
0.1029
0.1449
0.1808
0.6853
0.9605
1.3886
2.3917
3.1648
4.5234
6.0179
8.0187
10.478
12.964
14.556
18.540
21.835
26.242
0.982
0.995
0.998
0.999
1.000
1.000
0.998
0.998
0.998
0.992
0.988
0.972
0.946
0.888
0.789
0.675
0.627
0.421
0.293
0.158
AC
PAC
Q-Stat
Prob
0.000
-0.002
-0.003
-0.005
-0.004
-0.005
-0.014
-0.010
-0.013
-0.020
-0.018
-0.021
-0.024
-0.028
-0.030
-0.033
-0.026
-0.038
-0.036
-0.040
0.000
-0.002
-0.003
-0.005
-0.004
-0.005
-0.014
-0.010
-0.013
-0.020
-0.018
-0.022
-0.024
-0.029
-0.031
-0.034
-0.028
-0.041
-0.040
-0.045
0.0004
0.0079
0.0377
0.1040
0.1349
0.1851
0.6453
0.8803
1.2684
2.2467
3.0316
4.1471
5.5112
7.4504
9.6491
12.260
13.868
17.474
20.661
24.577
0.984
0.996
0.998
0.999
1.000
1.000
0.999
0.999
0.999
0.994
0.990
0.981
0.962
0.916
0.841
0.726
0.676
0.491
0.356
0.218
(b)Residuals for the futures equation
Autocorrelation
Partial Correlation
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Notes: This table shows the plots and the values of the autocorrelation function (AC) and the partial
autocorrelation function (PAC) of the squared residuals of equations (4.12a\b-4.14). The last two
columns are the Q-statistics for higher order autocorrelation and the corresponding probability. The
null hypothesis is that there is no autocorrelations at a given order. The probabilities show us that the
null hypothesis should be rejected.
52
Table 14: In-sample hedging performances
Model
Return
Variance
Unhedged 0.000269337 0.000220344
% Var. reduction
-
OLS
4.47913E-05 1.59878E-05
0.92744176
VAR
4.43643E-05 1.59885E-05
0.92743838
VEC
4.21196E-05
1.6017E-05
0.92730945
GARCH
4.48357E-05 1.59858E-05
0.92745093
Naive
2.83365E-05 1.70937E-05
0.92242296
Notes: (a) The in-sample analysis is conducted for the period 28th November 1994 – 10th May 2004
(2459 obs.). (b) The return, the variance, and the percentage variance reduction are presented besides
the corresponding portfolio to show each portfolio’s performance. (c) “% Var. reduction” is the
percentage variance reduction in relation to the unhedged portfolio.
Table 15: Out-of-sample hedging performances
Model
Return
Variance
Unhedged 0.001459176 2.12919E-06
% Var. reduction
-
OLS
-0.000412216 3.03013E-05
0.40982191
VAR
-0.000415774 3.03895E-05
0.40810454
VEC
-0.000434482 3.08611E-05
0.39891933
GARCH
-0.000419827 3.00915E-05
0.41390973
Naive
-0.000549352 3.40516E-05
0.33677767
Notes: (a) The out-of-sample analysis is conducted for the period 11th May 2004 to 10th June 2004 (30
obs.). (b) The return, the variance, and the percentage variance reduction are presented besides the
corresponding portfolio to show each portfolio’s performance. (c) % Var. reduction is the percentage
variance reduction in relation to the unhedged portfolio.
53
Figure 1. The logarithm of MIB30 and
FIB30 Series
Figure 2. The plot of the residuals from the
OLS model
11
0.03
10.5
0.02
10
0.01
LN FIB30
2401
2251
2101
1951
1801
1651
1501
1351
1201
901
1051
751
601
-0.02
451
8.5
301
-0.01
1
0
9
151
9.5
-0.03
Residuals from OLS model
LN MIB30
Figure 3. The plot of the residuals from the VAR Error Correction Model
(a)Squared residuals for the spot equat.
(b)Squared residuals for the futures equat.
0.15
0.15
0.1
0.1
0.05
0.05
0
0
-0.05
-0.05
-0.1
-0.1
-0.15
Residual
Residual
Figure 4. The plot of the dynamic OHR obtained from the GARCH model
1.4
1.2
1
0.8
0.6
0.4
0.2
2398
2257
2116
1975
1834
1693
1552
1411
1270
988
1129
847
706
565
424
283
1
142
0
dynamic OHR
54
55