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SCHOOL OF FINANCE AND ECONOMICS
UTS:BUSINESS
WORKING PAPER NO. 122 JANUARY, 2003
The Role of Intra-Day and Inter-Day Data Effects in Determining Linear and Nonlinear Granger Causality Between Australian Futures and Cash Index Markets Robert M. Eldridge Maurice Peat Maxwell Stevenson ISSN: 1036-7373 http://www.business.uts.edu.au/finance/
The Role of Intra-day And Inter-day Data Effects In
Determining Linear And Nonlinear Granger Causality Between
Australian Futures And Cash Index Markets
by
Robert M. Eldridge * Maurice Peat **
Maxwell Stevenson **
* Department of Economics and Finance, School of Business, Southern Connecticut State University, New Haven, CT, U.S.A.
** School of Finance and Economics, Faculty of Business, University of Technology Sydney, Sydney, Australia
The Role of Intra-day And Inter-day Data Effects In
Determining The Lead-Lag Relationship Between
Australian Futures And Cash Index Markets
JEL Codes C32,G13,G14
Abstract
In order to explain the incidence of Granger causality between indices from the
futures and the underlying cash market, as reported by numerous empirical studies in
the literature, it is important to account for mean and volatility (second-order)
persistence effects in the data. Further, there is need to control for inter-day and intra-
day effects by imposing an appropriate autocorrelation structure upon each of the
index returns from both markets. Once all these effects are controlled for, then linear
Granger causality ceases to be statistically significant and the associated lead-lag
phenomenon is no longer observable when the information flow between the spot and
futures markets is completed within a five-minute observation interval. Additionally,
nonlinear Granger causality testing indicates no compelling need to account for
nonlinear effects (beyond the second-order moment condition) in order to explain
causality. This result supports the price discovery role of futures markets.
1
1. Introduction Prices in the futures and the cash markets are related contemporaneously according to
the theoretical Cost-Of-Carry (COC) model (Cornell and French, 1983). On the other
hand, numerous empirical studies reported in the literature1 find futures index price
changes can usually lead underlying cash index price changes by up to forty minutes,
or lag them for at most ten minutes.
The identification of a lead-lag relationship and associated linear Granger causality
between futures index returns and returns on the cash index, is important for several
reasons. Firstly, the Efficient Markets Hypothesis (EMH) should apply to both
markets.2 Any persistence of linear Granger causality between the two markets
provides contrary evidence to the EMH. Under all forms of the EMH, a no-arbitrage
condition should apply between assets in both markets. Secondly, the theoretical COC
model specifies a nonlinear representation of the contemporaneous prices from the
futures and the underlying markets. The presence of a lead-lag relationship brings into
question the validity of using a linear bivariate model, the model most often used to
empirically link the two markets. Finally, empirical validation of a non-
contemporaneous relationship between price changes in both markets has implications
for the presumed important economic price discovery function of the futures market3.
Abhyankar (1998) uses high-frequency (five-minute) return data to model the lead-lag
relation between the U.K. index futures and cash markets. His data was filtered for
1 These studies include those of Kawaller, Koch and Koch (1987), Stoll and Whaley (1990), Harris (1989), Wahab and Lashgari (1993), Grunbichler, Longstaff and Schwartz (1994), Tse (1995), Abhyankar (1995), (1998), Shyy, Vijayraghavan and Scott-Quinn (1996), Pizzi, Economopoulos and O’Neill (1998), Min and Najand (1999) and Frino and West(1999). 2 EMH postulates that, in perfect and frictionless markets, no asset can be used to predict price changes in another.
2
mean effects and volatility persistence, his findings confirm those of previous
empirical results that documented a lead of some fifteen minutes in the futures index
returns4 5. His analysis first applies a linear autoregressive (AR) filter to the
underlying index returns to remove autocorrelation in the series. This filtering is
intended to reduce the chance of spurious correlations influencing the direction of
causal effect.6 He finds strong evidence of linear Granger causality in the one
direction, from the futures to the cash market. Linear Granger causality and the
associated lead-lag phenomenon is still present after further filtering for volatility
persistence using an exponential generalised autoregressive conditional
heteroscedasticity (EGARCH) model. Also of interest is his finding of bidirectional
nonlinear Granger causality in the residuals from the bivariate linear model using the
Hiemstra and Jones (1994) variation of the Baek and Brock (1992) test. This result
indicates that, after accounting for linear Granger causality from the futures index
returns to the cash index returns, both returns have nonlinear predictive power for
each other. A conclusion that follows logically is that, if these nonlinearities in the
data are appropriately modelled, neither market should lead nor lag the other.
In this paper, we critique and extend the work of Abhyankar (1998). We examine the
intra-day and inter-day7 autocorrelation structure in the two index return series. The
index series used are five-minute observations of the Share Price Index contract (SPI)
3 Under US commodities law, approval of futures contracts require an economic justification – generally that of price discovery. 4 Returns here are defined as a percent relative price change as opposed to the normal investment definition of return. 5 Evidence of a lead in the cash market index over the futures index is weak. 6 It has been argued that removing autocorrelation helps to mitigate the non-synchronous trading effect, a market imperfection often regarded as contributing to the lead-lag anomaly [Stoll and Whaley (1990)]. 7 The result of moving from the closing price on one day to the opening price of the next, and of moving from each subsequent observation on a day to the corresponding observation on the next.
3
on the Sydney Futures Exchange and the corresponding observations of the
underlying All Ordinaries Index (AOI) as traded on the Australian Stock Exchange8.
McInish and Wood [(1984), (1985)] have established the existence of consistent
patterns in intra-day returns. It is then possible that there exists an inter-day
autocorrelation effect associated with these consistent intra-day patterns. Accordingly,
to mitigate the effect of this autocorrelation, we filter both index return series with an
appropriate autocorrelation structure while, at the same time, accounting for first-
order moment (mean) and second-order moment (volatility persistence) effects. After
this filtering, we find no evidence of linear Granger causality in either direction using
a bivariate vector error correction model (VECM) linking both series. Further, there
is little evidence of nonlinear Granger causality from the residuals of the linear model.
This is in contrast with the results obtained by Abhyankar (1998), and by us when we
replicated his study using his filtering process on our Australian index data. Like
Abhyankar (1998), then we find strong evidence of linear Granger causality, with a
lead of some fifteen minutes by the SPI returns over the AOI returns. Also, when we
don’t account for both higher order moment effects as well as intra-day and inter-day
structure in the data, we find compelling evidence of nonlinear Granger causality
when we test the residuals from the linear vector error correction (VECM) models.
The remainder of the paper is structured as follows. In section 2 we establish the
existence of, and possible explanations for, an empirical lead-lag relationship between
spot and futures index prices in the light of the assumptions underlying the theoretical
COC model. In Section 3 we detail the methodology, with the data being described in
Section 4. In Section 5, we review our empirical results and conclude in Section 6.
8 It is a derivative of the underlying All Ordinaries Index (AOI) that is compiled from stocks traded on the Australian Stock Exchange.
4
2. Relationship Between Prices in Futures and Cash Markets
The Cost-Of-Carry (COC) model is a model used by traders to determine whether
futures contracts are correctly priced. It is a continuous-time representation of the
relationship between the ‘fair value’ of a futures contract and the ‘fair value’ of the
underlying spot index, plus the cost of carrying the spot index for the duration of the
contract. The COC model is represented by the following equation,
))(( tTdrtt eSF −−= (1)
Ft is the index9 futures price at time t, St is the spot index price at time t, r is the
continuously compounded cost of carrying the cash index from t, d is the dividend
yield on the index and T is the expiration date of the futures index contract. T – t is
the time remaining to expiration of the futures contract, while r – d is the time-value
rate of return held in a portfolio which matches the stock index, net of the flow of
dividends from the index. The difference between the futures and the spot prices is
called the basis and represents the net cost of carry.
The COC model assumes that the two markets are perfectly efficient, frictionless and
act as perfect substitutes. Accordingly, profitable arbitrage should not exist because
new information arrives simultaneously to both markets and is reflected immediately
in both futures and spot prices. However, as previously referenced in the
introduction, many empirical studies have established the existence of a lead-lag
relationship between price changes in most futures and spot markets. It has been
argued that persistence in the lead-lag relationship between index futures and spot
index prices can be traced to one or more market imperfections, such as costs of
transactions, liquidity differences between the two markets, asymptotic information
9 Henceforth, the term index will refer to a stock index
5
and non-synchronous trading effects. Other market imperfections likely to have a
major impact on the lead-lag relationship between spot and futures index price
changes are automation of one or the other market, short selling restrictions, different
taxation regimes on futures and stocks, dividend uncertainties and marking to market
imperfections.
3. Methodology
The methodology followed in this paper generally follows Abhyankar (1998),
including the variations to the estimation of the lead-lag relationship employed by
Wahab and Lashgari (1993).
In previous studies, tests used to establish bivariate linear Granger causality and to
explain the lead-lag phenomenon existing between returns in futures and cash market
indices have been constructed from different linear specifications which model the
relationship between the two series. A multivariate regression approach, where
returns from the cash market index are regressed on leads and lags of returns from the
futures market plus an error correction term, has been adopted in the studies of
Fleming, Ostdiek and Whaley (1996) and Abhyankar (1998), among others.
Alternatively, after establishing that the cash and futures market indices are
cointegrated, a bivariate vector-error-correction model (VECM) is used by Wahab
and Lashgari (1993) and Pizzi et al. (1999). Irrespective of what linear specification
is used, strong evidence has been reported that the direction of causality is
unidirectional and supports the price discovery role for the futures market. However,
it should be noted that the lead of the futures over the cash market is more pronounced
the higher the frequency of the data used10.
10 Compare the results using daily returns in Wahab and Lashgari (1993) with those for five-minute returns in Abhyankar (1998).
6
We first test for evidence of linear Granger causality by specifying a linear model that
relates contemporaneous returns of each index to lags of the returns of both
indices. The resultant model is a bivariate vector autoregressive model (VARM).
Using the first-differences of the logarithms of the two index series, this linear model
can be represented algebraically as:
∆lnFt = a(L) ∆lnFt + b(L) ∆lnSt + ε∆lnFt, (2)
∆lnSt = c(L) ∆lnFt + d(L) ∆lnSt + ε∆lnSt, , t = 1, 2, …, T (3)
where ∆lnFt is the first-difference of the logarithm of the index futures price and
∆lnSt corresponds to the same transform of the market index11. a(L) is a polynomial
function in the lag operator whose roots lie outside the unit circle, as does those for
b(L), c(L) and d(L).12 We select the number of lags for the polynomials in the
equation system given by (2) and (3) to be nine, which is the observed number of lags
necessary to capture the longest lag structure between the two indices as previously
reported13. Further, following Wahab and Lashgari (1993), we also recognise that the
index futures and the spot index series are most likely cointegrated and, thus, the
appropriate linear specification is not given by (2) and (3) but rather by a bivariate
vector error correction model (VECM). To determine whether either variable linear
Granger-cause the other, we include an error correction term or cointegrating vector
(CV) on the right-hand side of both equation (2) and (3). The CV is the one period
lagged error of the linear relationship between the levels of both indices. It is
determined after testing for cointegration and estimating the VECM by the method
proposed by Johansen (1988).
11 Both time series being stationary and of length T. 12 The test of whether ∆lnFt strictly Granger-causes ∆lnSt is a test of the null that all the coefficients of c(L) are statistically equivalent to zero. Reversing the direction and considering whether ∆lnSt Granger-causes ∆lnFt , is a test of the null that all the coefficients contained in b(L) are statistically equal to zero. 13 See references to the empirical lead-lag studies given in the introduction to this paper.
7
As well as using the return series for each index, we test for linear Granger causality
using the residuals from three further transformations of each of the return series. The
first of these transformations, an autoregressive moving average (ARMA) model, is
designed to filter the inter-day and intra-day mean effects14. The second
transformation, an exponential generalized autoregressive conditional
heteroscedasticity (EGARCH) model, is employed to remove the effect of volatility
persistence. The ARMA and EGARCH filters were both used by Abhyankar (1998).
However, we could not satisfactorily filter the intra-day and inter-day volatility
persistence in the data using an EGARCH specified model. Accordingly, we applied
an additional transformation that comprised an autoregressive conditional
heteroscedasticity (ARCH) model in which the squared residuals from an AR process
are regressed on their lagged values. The number of lags of the squared residuals is
chosen to account for both intra-day and inter-day second-order moment effects
(volatility persistence) in the data, while the AR process is used to filter the inter-day
and intra-day mean (first-order moment) effects. Both sets of parameters, one set
within the AR structure for the mean and the other as parameters of the lagged
squared residuals terms, are estimated simultaneously as a system.
The presence and direction of linear Granger causality is determined by significance
of the lags of the other variable in each equation of the VECM. The number of
significant lagged transformed index futures coefficients in the VECM adjustment of
equation (3) quantifies the extent of the lead of the futures market over that of the
cash market. Further, the number of significant lagged transformed spot index
coefficients in the error-corrected variant of equation (2) determines the lead in the
other direction.
8
There is no reason to believe that causality should be strictly linear. In fact, it is likely
that linear Granger causality tests will have low power against many types of
nonlinear causality [Brock (1991)]. Evidence of nonlinear structure in stock returns
has been documented by a number of researchers including Scheinkman and LeBaron
(1991), Brock, Hsieh and LeBaron (1991), Hsieh (1991) and Eldridge et al (1993).
Further, using data at observed frequencies that vary from one minute to one hour,
Abhyankar, Copeland and Wong (1997) find evidence of nonlinear structure in U.K.
futures and cash index returns. Dwyer, Locke and Yu (1996) use a variant of the
COC model with nonzero transactions costs to justify a nonlinear relationship
between minute-by-minute S&P 500 futures and cash market indices15. In the light of
this evidence of nonlinear structure within and between the two markets, it follows
that the interaction between the two index series might be more appropriately
modelled by a multivariate nonlinear specification. If the residuals and the squared
residuals from the estimated linear VECM’s (using data filtered for mean effects and
volatility persistence) are devoid of significant autocorrelation effects, then there is no
reason to believe that the linear bivariate model is misspecified. However, if this is
not the case, there is a need to test the residuals for nonlinear Granger causality.
Baek and Brock (1992) proposed a test for multivariate nonlinearity. This test, which
uses the concept of the correlation integral, can be seen as an extension of the BDS
test [Brock, Dechert and Scheinkman (1996)]. A significant BDS statistic implies that
points in m-history space have a higher probability of clustering together than what is
probable with truly random data. The significance of the test statistic associated with
Baek and Brock (1992) determines the existence and the direction of causality
14 This is the transformation advocated by Stoll and Whaley (1990) to account for the effect of non-synchronous trading. 15 They propose a threshold error correction model to capture the nonlinear dynamics between the two markets, and conclude that arbitrage activity in the futures market (i.e. mean reversion in the basis) is a determinant in the convergence of the cash and futures prices.
9
between two sets of vectors. When the leads and lags of one vector are close (in a
probability sense), without the need to have information concerning the probability of
closeness of the lead and lags of the other, then this is evidence of nonlinear non-
Granger causality. Hiemstra and Jones (1994) further modified the Baek and Brock
(1992) test to improve the small sample properties of the test and to relax the
assumption that the series to which the test is applied are independently and
identically distributed (i.i.d.). Before any conclusions can be drawn from the test
regarding the need to consider nonlinearities in the multivariate modelling of the two
index series, the data should be first filtered for first and second-order moment
effects. This ensures that there should be fewer spurious rejections of the null of non-
Granger causality due to the presence of structural breaks and heteroscedasticity in the
data.
Abhyankar (1998) uses an exponential generalised autoregressive conditional
heteroscedasticity (EGARCH) model to filter his high frequency data, and then uses
the residuals from a linear model as input to the Hiemstra and Jones (1994) variant of
the Baek and Brock (1992) test for multivariate nonlinearity. He rejects the null and
finds evidence of significant nonlinear Granger causality in the residuals, concluding
that both returns have nonlinear predictive power for each other after accounting for
the linear lead of the futures market.
In this study, we also filter our data for first and second-order moment effects using
an EGARCH model, but also filter for these effects using the ARCH-type model
discussed previously. Following Abhyankar (1998), the residuals from linear models
10
using these two sets of filtered data become input to the Hiemstra and Jones (1994)
test for multivariate nonlinearity16.
4. Data Description
The data used in this study are five-minute price observations of the four Share Price
Index (SPI) contracts maturing in September and December, 1995, as well as March
and June, 1996, along with the corresponding recorded prices of the All Ordinaries
Index (AOI)17. All observations for the SPI contracts and the AOI prices have been
spliced together to construct a continuous series.
SPI futures contracts began trading on the Sydney Futures Exchange (SFE) in 1983
and, by tracking the movement of the underlying share market, serve as a substitute
for owning a diversified portfolio of shares that form the AOI18. Before automated
trading was introduced in January, 2000, the trading floor of the SFE operated
between 9.50 a.m. to 12.30 p.m. and from 2.00 p.m. to 4.10 p.m. Australian Eastern
Standard Time (AEST). As most trading occurs in the nearest expiry month, we used
the index futures time series based on the near contract, shifting to the next-to-nearest
contract when it had the higher volume traded. Over the study period, trading on the
Australian Stock Exchange (ASX) was fully automated and operated from 10.00 a.m.
(AEST) to the close of trade at 4.00 p.m. on the same day. Opening times for trading
were staggered with all stocks trading by 10.10 a.m.
16 In Appendix B, using notation that closely follows Hiemstra and Jones (1994), we describe this test for two stationary series, Xt and Yt, where Xt = ∆ln(Ft), and Yt = ∆ln(St). 17 Intra-day price observations for both series used in this study have been obtained from the Securities Industry Research Centre of Asia-Pacific (SIRCA), Sydney, Australia. 18 At maturity, the value of the contract in 1995 and 1996 was the actual AOI on the last day of trading, multiplied by 25 Australian dollars.
11
Due to the lunch break between 12.30 p.m. and 2.00 p.m. on the SFE, there is a
problem matching up the index prices from the both markets. We use data from both
exchanges from 10.15 a.m. to 12.30 p.m. to cover each morning trading session,
discarding the first five minutes of trading from 10.10 a.m. to account for possible
anomalous trading after the start-up procedure for the AOI. For a similar reason
pertaining to the start-up of the SPI after lunch, we discard the first five minutes of
trading in both contracts in the afternoon session which runs from 2.00 p.m. to 4.00
p.m. The nearby futures prices and those of the underlying index are matched for
each five-minute interval for each of the four contracts during the last half of 1995
and the first half of 1996. This pairing provides 28 pairs of observations for each
morning and 24 for each afternoon, or 52 observations for each trading day.
The number of observations totalled 3380, 3016, 3224 and 3016 for the September
’95, December ’95, March ’96 and June ’96 contracts, respectively19. Spliced
together, both index series comprise 12,636 observations. Table 1 provides summary
statistics on the five-minute futures and spot data. The higher mean SPI price
indicates that the futures contract trades at a premium to the AOI. This contango
condition, in which the futures price exceeds the price of the underlying asset, is
expected given that the cost-of-carry is more likely to be positive, along with evidence
that futures prices attract a risk premium [see Bessembinder (1993)]. The greater
standard deviation for the SPI futures confirms the findings of Schwert (1990) that the
futures market is more volatile than the underlying cash market. The Trading Cost
Hypothesis [Fleming, Ostdiek and Whaley (1996)] posits that the futures market will
react more quickly to market-wide information shocks due to lower transaction costs
in that market. In times of lower liquidity in the futures market, we would expect
19 This was the total number of observations in each contract after discarding 7 days that had incomplete data for some of the 52 intervals in a day for either the futures or the underlying index.
12
Table 1 Summary Statistics - Five Minute Intraday Observations; July 3, 1995 To June 28, 1996 Variable Mean Median Maximum Minimum Standard
Deviation Skewness* Excess
Kurtosis** SPI Price Level (Ft)
2206.812 2211.500 2364.500 2016.500 75.50431 -0.137982 -0.873667
First-Differences of Logarithm of SPI (∆lnFt )
0.000008 0.000000 0.022386 -0.027832 0.001141 -0.133716 83.49925
AOI Price Level (St)
2190.664 2209.850 2336.600 1993.700 73.08914
-0.225232 -1.080245
First-Differences of Logarithm of AOI (∆lnSt )
0.000009 0.000000 0.021268 -0.020539 0.000778
0.083698 152.2311
* Skewness is calculated as 2
32
31
mmb = , while Kurtosis as 2
2
42 m
m=b , where
n
xxm
n
i
ki
k
∑=
−= 1
)(,
k = 2,3,4; m3 and m4 are the centred third and fourth moments respectively. ** Excess kurtosis is measured by b2 – 3. This gives an indication of departure from normality. A negative excess kurtosis value indicates thinner tails than the normal distribution.
higher intra-day volatility. The distributions of both index prices are skewed to the
left, while the measure of kurtosis (or curvature) point to distributions that have
thinner tails than the normal distribution.
Given that the futures index is a derivative security of the cash index and that both of
them are subject to the same impact from changes in market fundamentals, it comes
as no surprise that these two price series are cointegrated. Using Augmented Dickey-
Fuller tests we concluded that both series were integrated of order one. Further, the
Johansen Maximum Likelihood test [Johansen (1988), Johansen and Juseluis (1990)]
established that the two series are cointegrated20. It follows that an error correction
13
model is the correct linear specification to use in testing for causality [Engle and
Granger (1987)].
With the presence of a unit root in both price series, we created first-differences of the
logarithm of both series in order to induce stationarity. We use these return series in
the remainder of the study.
5. Empirical Results
Recall equation (1) that algebraically links the spot and futures markets in the COC
model:
))(( tTdrtt eSF −−=
Taking logarithms of both sides and re-arranging results in:
tt StTdrF ln))((ln +−−= (4)
If the log of the futures price, Ft, and the log of the cash price, St, both have unit roots,
then the COC model would indicate that they are cointegrated with a long-run linear
relationship between the log of both price series given by equation (4). Theoretically,
the coefficient of the logarithm of the spot index price is one. Cointegration of the
futures and the cash indices implies that a vector error correction model (VECM) is an
appropriate specification for a linear model and can be represented by the following
equation system:
∑ ∑
∑ ∑
= =−−−
= =−−−
+∆+∆+=∆
+∆+∆+=∆
p
i
p
jtjtjititt
p
i
p
jtjtjititt
eFSCVF
eFSCVS
1 122212
1 111111
lnlnln
lnlnln
ϕδγ
ϕδγ (5)
20 Output from all these tests are not reported here but are available from the authors on request.
14
Engle and Granger (1987) show that cointegrated series have a corresponding VECM
[as in equation system (5)] that combines the short-term pricing dynamics with the
long run equilibrium relationship, a feature which differentiates the VECM from
standard causality models. The long run equilibrium relationship is represented by the
lagged cointegrating vector (CVt-1), which is retrieved from the Johansen (1998)
estimation procedure.
In Section 3 we noted that linear Granger causality is tested using the return series for
each index, as well as the filtered return series using transformations which are meant
to remove the effects of mean and volatility persistence from the data. The
correlogram of the index futures return series (∆lnFt ) indicates a lack of significant
autocorrelation 21up until lag 52, which suggests that intra-day effects are not
responsible for autocorrelation in the first-order moment (see Appendix A).
However, inter-day autocorrelation is evident with significant Q-statistics at lags 52,
80 and 104. This autocorrelation structure corresponds to daily effects, with a one-day
plus the next morning effect indicated by significance at lag 80. While previous
studies which use intra-day data also find little evidence of intra-day correlation in the
futures index returns [Chan (1992), Abhyankar (1998)], they seem to overlook the
incidence of autocorrelation at longer lags which correspond to inter-day changes in
the level of the mean. On the other hand, the correlogram for the spot index return
series (∆lnSt ) exhibits strong positive autocorrelation from lag 1, which also includes
the longer inter-day lags (see Appendix A).
The results of our empirical tests for linear and nonlinear Granger causality follow for
the raw return series and their various filtered transformations.
21 Significant autocorrelation is identified from the correlogram by significant Ljung-Box Q statistics.
15
5.1 Linear Granger causality test results
Table 2 reports the results of our linear Granger causality test where the input series
are the raw returns (the first difference of the logarithms) for the futures index, ∆lnFt,
and the corresponding raw returns for the underlying index, ∆lnSt. Equations VECM1
and VECM2 model changes in the logarithm of the spot and futures indices,
respectively. Recall that the lag structure of length nine was chosen to capture the
longest lag structure between the two indices. From VECM1 we observe that the
coefficients on the lagged futures returns correspond to significant t-values for up to
eight lags. This is evidence that changes in the lagged futures returns Granger-cause
current changes in the underlying cash market returns for up to forty minutes.
Further, it provides evidence that the linkage between the futures and cash market
indices is not contemporaneous, as is assumed by the COC model.
The error correction term, CVt-1, is expressed as:
(6) )19649.2()9927.34(
454659.0ln941744.0ln 111
−−−−= −−− ttt SFCV
where ln and are logarithms of the lagged index futures and spot index,
respectively.
1−tF 1ln −tS
As reported in Table 2, a significant coefficient22, γ1, in VECM1 indicates that the
cash index adjusts towards the long-run equilibrium. This can be interpreted as the All
Ordinaries Index (AOI) being responsible for most of the adjustment back to the long-
run equilibrium. This interpretation is consistent with the proposition that past futures
prices cause current cash market prices. The coefficients of changes in the lagged
22 Statistical significance is determined from the t-values.
16
logarithmic spot index in VECM2 are insignificant except for lags 2 and 6, where
they are significant at the 5% level. They provide weak evidence that there may be
Granger causality in the other direction.
Table 2: Vector Error Correction Model (VECM) For Raw Returns Of The Share PriceIndex Future And The All Ordinaries Index Using The Johansen CointegratingVector (CV)
VECM1: ∑ ∑= =
−−− +∆+∆+=∆9
1
9
111111 lnlnln
i jtjtjititt eFSCVS ϕδγ
VECM2: ∑ ∑= =
−−− +∆+∆+=∆9
1
9
122212 lnlnln
i jtjtjititt eFSCVF ϕδγ
VECM1 for ∆lnSt VECM2 for ∆lnFt
Regressor Coefficient t-value Coefficient t-value
CointegratingVector
0.002692 2.15640* -0.003121 -1.67375
∆lnSt-1 -0.121777 -8.46893* 0.012164 0.56637
∆lnSt-2 -0.113637 -7.84407* -0.043305 -2.00141*
∆lnSt-3 -0.087563 -6.02276* -0.015095 -0.69516
∆lnSt-4 -0.053118 -3.64805* 0.027753 1.27618
∆lnSt-5 -0.026237 -1.80139 0.024514 1.12690
∆lnSt-6 -0.020575 -1.41662 0.046526 2.14474*
∆lnSt-7 -0.018652 -1.28937 -0.000974 -0.04508
∆lnSt-8 -0.009270 -0.64754 -0.013804 -0.64560
∆lnSt-9 -0.010170 -0.73660 0.004753 0.23049
∆lnFt-1 0.163506 16.9341* -0.005136 -0.35615
∆lnFt-2 0.109179 10.9812* 0.021087 1.42006
∆lnFt-3 0.072975 7.27114* -0.000620 -0.04137
∆lnFt-4 0.051854 5.14713* -0.016884 -1.12207
∆lnFt-5 0.040022 3.96438* -0.004496 -0.29818
∆lnFt-6 0.027730 2.75169* -0.024506 -1.62817
∆lnFt-7 0.022584 2.24796* 0.000716 0.04775
∆lnFt-8 0.021007 2.10780* 0.010600 0.71212
∆lnFt-9 0.008543 0.88125 -0.004954 -0.34213
* denotes statistical significance at the 5% level
We also note that the statistically insignificant coefficient corresponding to γ2, the
coefficient of CVt-1 in VECM2, indicates a lack of involvement by the index futures
in the long-run adjustment process. Thus, we conclude that information flows from
17
the futures to the cash market, and that this is the direction of the linear Granger
causality.
The correlograms of the residual series from equations VECM1 and VECM223,
suggest a need to account for both intra-day and inter-day mean effects in the data
(see Appendix A). Significant negative autocorrelation was noted at lag 52 (one day’s
trading) and at lag 80 (one and a half day’s trading) for both series, with further
significance at lag 104 (two day’s trading) for the residual series related to the index
futures returns.
We filtered the index and index futures returns series with an autoregressive moving
average (ARMA) model to remove first-order moment (mean) effects. For the index
futures series, our ARMA model contained significant autoregressive (AR) lags at 1
to 28, 52, 104 and 156. The significant moving average (MA) lags were at 28, 52, 104
and 156. Residuals from this model had a correlogram with no significant
autocorrelation (see Appendix A) and formed the transformed (filtered) series,
. Similarly, after filtering the spot index returns series with an ARMA model
with significant AR lags at 1 to 104, 156, 208 and 260 and corresponding MA lags at
multiples of 52 up to 260, the filtered series, , was free of significant
autocorrelation.
'ln tF∆
'ln tS∆
Table 3 contains the estimation results from the VECM, now with the transformed
input series, ∆ and . 'ln tF 'ln tS∆
23 VECM1RES and VECM2RES respectively
18
Table 3: Vector Error Correction Model (VECM) For ARMA Returns Of The Share Price Index Futures And The All Ordinaries Index Using The Johansen Cointegrating Vector (CV)
VECM3: ∑ ∑= =
−−− +∆+∆+=∆9
1
9
13
'1
'11
'1
' lnlnlni j
tjtjititt eFSCVS ϕδγ
VECM4: ∑ ∑= =
−−− +∆+∆+=∆9
1
9
14
'2
'21
'2
' lnlnlni j
tjtjititt eFSCVF ϕδγ
VECM3 for '
ln tS∆ VECM4 for '
ln tF∆
Regressor Coefficient t-value Coefficient t-value
Cointegrating Vector
0.002213 1.81173 -0.003404 -1.80796
'1ln −∆ tS
-0.169268 -12.1555* 0.012329 0.57433
'2ln −∆ tS
-0.107066 -7.57366* -0.056603 -2.59740*
'3ln −∆ tS
-0.068781 -4.85092* -0.025147 -1.15051
'4ln −∆ tS
-0.047810 -3.37135* 0.009646 0.44124
'5ln −∆ tS
-0.032995 -2.32696* 0.013485 0.61694
'6ln −∆ tS
0.017446 -1.23204 0.056353 2.58155*
'7ln −∆ tS
-0.022364 -1.58159 -0.010739 -0.49268
'8ln −∆ tS
-0.017172 -1.22194 -0.023291 -1.07512
'9ln −∆ tS
-0.002467 -0.17994 -0.003636 -0.17201
'1ln −∆ tF
0.137297 15.1542* -0.006067 0.43438
'2ln −∆ tF
0.085252 9.20169* 0.028292 1.98095*
'3ln −∆ tF
0.055457 5.95288* 0.017502 1.21873
'4ln −∆ tF
0.039869 4.27614* -0.002193 -0.15256
'5ln −∆ tF
0.029164 3.12549* -0.005588 -0.38851
'6ln −∆ tF
0.016418 1.76036 -0.027596 -1.91944
'7ln −∆ tF
0.017538 1.88150 0.004492 0.31265
'8ln −∆ tF
0.012952 1.39458 0.009874 0.68967
'9ln −∆ tF
0.003425 0.37500 0.001673 0.11885
* denotes statistical significance at the 5% level
We note from equation VECM3 that the first five coefficients of the transformed,
logarithmic changes of the lagged futures index are significantly different from zero.
19
This indicates Granger causality from the futures market to the cash market but for a
reduced number of lags from when the input series were not mean adjusted. Again,
from equation VECM4, there is only weak evidence that Granger causality runs in the
other direction from the cash market to the futures. After inspecting the correlogram
of the residuals from equations VECM3 and VECM4 we observed that, while the
residuals themselves had been purged of autocorrelation, this was not the case for the
squared residuals (see Appendix A). Their correlogram indicated statistically
significant positive correlation at lags at least equal to 52. Significance in the lagged
square residuals indicates the need to account for the effect of volatility persistence in
the input series to the vector error correction model (VECM).
Abhyankar (1998) filtered his transformed U.K. futures and cash market returns for
volatility persistence using an EGARCH model. We also used EGARCH models to
filter both the index futures and spot returns. For the futures returns, the
corresponding AR structure for the mean had significant lags at 1 to 3, 8, 11, 28,
while the spot returns had significant lags at 1 to 5, 8, 11, 52, 80 and 84.24. The results
of the VECM, when the EGARCH filtered returns ( and ) are used as
input, are given in Table 4.
''ln tF∆ ''ln tS∆
As was the case for the previous transformed series, there is strong evidence of linear
Granger causality, with the direction of causality supporting the notion of price
discovery emanating from the futures market. What is evident from the correlogram
of the residuals and squared residuals from equations VECM5 and VECM6 is that
while no autocorrelation remains in the residuals, significant autocorrelation is still
24 We chose the AR structure for the mean for both sets of returns to specifically account for both intra-day and inter-day effects in the data.
20
present in the squared residuals at lags greater than 52 (see Appendix A). We
concluded that the EGARCH model does not do a satisfactory job at filtering out both
intra-day and inter-day volatility effects.
21
Table 4: Vector Error Correction Model (VECM) For EGARCH Filtered SharePrice Index Futures Returns And The All Ordinaries Index Returns UsingThe Johansen Cointegrating Vector (CV)
VECM5: ∑ ∑= =
− +−∆+−∆+=∆9
1
9
15111
''1
''ln''ln''lni j
tjit ejtFitSCVtS ϕδγ
VECM6: ∑ ∑= =
− +−∆+−∆+=∆9
1
9
16221
''2
''ln''ln''lni j
tjit ejtFitSCVtF ϕδγ
VECM5 for ''
ln tS∆ VECM6 for ''
ln tF∆
Regressor Coefficient t-value Coefficient t-value
CointegratingVector
1.912195 1.18144 -1.578647 -0.95893
''1ln −∆ tS
-0.165088 -13.59010* 0.006265 0.50702
''2ln −∆ tS
-0.074224 -6.02523* -0.016908 -1.34937
''3ln −∆ tS
-0.055082 -4.46298* -0.011410 -0.90895
''4ln −∆ tS
-0.037244 -3.01490* 0.014501 1.15406
''5ln −∆ tS
-0.025943 -2.09896* 0.007325 0.58265
''6ln −∆ tS
-0.010751 -0.87100 0.010467 0.83370
''7ln −∆ tS
-0.010809 -0.87704 -0.003062 -0.24430
''8ln −∆ tS
-0.023497 -1.91276 -0.009789 -0.78343
''9ln −∆ tS
-0.010983 -0.91440 -0.004754 -0.38919
''1ln −∆ tF
0.206605 17.2309* -0.001287 -0.10552
''2ln −∆ tF
0.086344 7.05318* 0.010699 0.85923
''3ln −∆ tF
0.061945 5.04737* 0.011803b 0.94554
''4ln −∆ tF
0.043119 3.51022* -0.024067 -1.92623
''5ln −∆ tF
0.032275 2.62399* 0.004527 0.36188
''6ln −∆ tF
0.018276 1.48796 0.004879 0.39052
''7ln −∆ tF
0.011555 0.94153 0.010718 0.85862
''8ln −∆ tF
0.009551 0.77962 0.004269 0.34255
''9ln −∆ tF
0.007819 0.64602 0.000584 0.04743
• denotes statistical significance at the 5% level
5.2 Nonlinear Granger causality test results
The persistence of the lead-lag effect after filtering our data with both AR and
EGARCH models is consistent with the findings in Abhyankar (1998). He proposes
that it could be nonlinear effects that explain this persistence and, after accounting for
nonlinearity, neither market should lead nor lag the other. Following Abhyankar
(1998), we tested the residuals from both linear VECM’s (reported in Tables 3 and 4)
for nonlinear Granger causality using the Hiemstra and Jones (1994) variant of the
Baek and Brock (1992) test.25 Our findings coincided with those of Abhyankar
(1998), who found evidence of bidirectional nonlinear Granger causality.
Table 5 contains the results for the nonlinear Granger causality test that tests the
residuals of the VECM when the ARMA filtered index returns ( and )
are used as input to the linear model. Following Takens (1983) we select a value ε
such that ε/σ takes on the values of 0.5, 1.0 and 1.5. In essence, ε, scaled by σ, yields
a measure of the number of n–tuples within the vector that are sufficiently close to
each other to provide meaningful information
'ln tF∆ 'ln tS∆
26. The value of the embedding
dimension, m, is set to 1 as suggested by the Monte Carlo experiments of Hiemstra
and Jones (1994). There is strong evidence of bidirectional feedback between the two
transformed index return series.
25 For this test we used the code kindly supplied by the late Craig Hiemstra. 26 Below some critical value of ε, increasing it adds no further n-tuples, and thus no further information. Similarly, above some larger value of ε, all n–tuples are accommodated and increasing ε adds no further information about the data.
22
This result is repeated when the input series to the linear VECM model are the
and series that have been filtered with an EGARCH model, namely '
and . The results of the Hiemstra and Jones (1994) test for this case are found
tF∆
'ttS∆
ln S∆
ln F∆
''t
Table 5 Results For Nonlinear Granger Causality Test Of Residuals From The VECM With Input Series 'ln tF∆ and 'ln tS∆ .
Ho: SPI Do Not Cause AOI Ho: AOI Do Not Cause SPI
σe
1.5 1.0 0.5 1.5 1.0 0.5
Number of Lags
(Lu = Lv)
1 0.0028** (5.307)
0.0082** (8.095)
0.0165** (9.517)
0.0020** (5.136)
0.0053** (7.198)
0.0107** (8.459)
2 0.0038** (5.496)
0.0107** (7.981)
0.0231** (8.830)
0.0021** (4.863)
0.0073** (8.054)
0.0186** (9.434)
3 0.0040** (4.990)
0.0109** (6.982)
0.0254** (7.125)
0.0026** (5.330)
0.0093** (8.564
0.0225** (8.535)
4 0.0042** (4.840)
0.0115** (6.533)
0.0273** (5.945)
0.0025** (4.780)
0.0087** (7.397)
0.0249** (7.144)
5 0.0040** (4.316)
0.0123** (5.985)
0.0273** (4.577)
0.0027** (4.642)
0.0090** (6.735)
0.0282** (6.163)
6 0.0041** (3.926)
0./0127** (5.541)
0.0287** (3.873)
0.0025** (4.139)
0.0088** (6.140)
0.0307** (5.309)
7 0.0037** (3.269)
0.0117** (4.580)
0.0289** (3.181)
0.0024** (3.670)
0.0089** (5.753)
0.0313** (4.259)
8 0.0044** (3.573)
0.0117** (4.085)
0.0340** (2.996)
0.0024** (3.526)
0.0097** (5.819)
0.0406** (4.337)
9 0.0045** (3.348)
0.0125** (3.912)
0.0357** (2.510)
0.0023** (3.199)
0.0101** (5.630)
0.0466** (3.925)
Notes: The first entry in each cell is the statistic given by equation B.8 (Appendix B) and refers to the difference between the conditional probabilities equated in equations B.3 and B.4. It is a standardised test statistic that is asymptotically
distributed as a standard normal variate. Lu and Lv are the number of lags in the residual series used in the test. ** Significance at the 1% level.
in Table 6. While the evidence in favour of not rejecting the hypothesis of nonlinear
Granger causality running in both directions is unequivocal, it does not appear as
strong using the EGARCH filtered input to the linear model as when using the ARMA
filtered data.
23
Hiemstra and Jones (1994) warn that their nonlinear Granger causality test lacks
power when linearities and ARCH effects are still in the data. Also, evidence of
significant autocorrelation in the correlograms of the squared residuals from the
various VECMs is present, irrespective of the transformed data used as input. Highly
significant autocorrelation was found at lags that are multiples of 52, suggesting the
Table 6 Results For Nonlinear Granger Causality Test Of Residuals
From The VECM With Input Series '∆ and'ln tF . ''ln tS∆
Ho: SPI Do Not Cause AOI Ho: AOI Do Not Cause SPI
σe
1.5 1.0 0.5 1.5 1.0 0.5
Number of Lags (Lu = Lv)
1 0.0048** (6.441)
0.0120** (9.605)
0.0231** (11.880)
0.0035** (6.639)
0.0073** (8.439)
0.0094** (7.750)
2 0.0037** (4.177)
0.0103** (6.518)
0.0237** (8.294)
0.0035** (5.719)
0.0090** (8.397)
0.0169** (8.777)
3 0.0062** (5.160)
0.0142** (7.321)
0.0321** (8.100)
0.0046** (6.422)
0.0116** (8.928)
0.0200** (7.406)
4 0.0045** (3.774)
0.0110** (4.995)
0.0277** (5.146)
0.0037** (5.158)
0.0108** (7.649)
0.0211** (5.711)
5 0.0075** (5.494)
0.0156** (6.009)
0.0309** (4.460)
0.0042** (5.186)
0.0110** (6.728)
0.0249** (5.125)
6 0.0056** (3.740)
0.0120** (3.983)
0.0278** (3.143)
0.0029** (3.699)
0.0102** (5.753)
0.0245** (3.994)
7 0.0071** (4.236)
0.0147** (4.359)
0.0413** (3.669)
0.0040** (4.516)
0.0118** (5.826)
0.0180** (2.261)
8 0.0057** (3.150)
0.0115 (3.028)
0.0453** (3.154)
0.0037** (3.881)
0.0124** (5.438)
0.0199 (1.718)
9 0.0075** (3.762)
0.0148** (3.478)
0.0414 (2.320)
0.0045** (4.302)
0.0141** (5.434)
0.0081 (0.457)
Notes: The first entry in each cell is the statistic given by equation B.8 (Appendix B) and refers to the difference between the conditional probabilities equated in equations B.3 and B.4. It is a standardised test statistic that is asymptotically
distributed as a standard normal variate. Lu and Lv are the number of lags in the residual series used in the test. ** Significance at the 1% level.
need for a model to account for volatility persistence in both indices that also takes
into account inter-day as well as intra-day effects. The question remains as to
whether linear or nonlinear Granger causality, along with a persistent lead-lag effect,
remains after appropriately accounting for these inter-day and intra-day first and
second-order moment effects.
24
To answer this question we used a simple ARCH-type model where squared residuals
form an AR process with lags that include orders that are multiples of 52. This AR
process is part of a system where the ARCH model of the squared residuals depends
upon the residuals from an AR process used for removing the mean effects from the
data. In using this model as a filter, we need to simultaneously estimate the
parameters in the AR model that account for the first-order moment effects, plus the
parameters in the ARCH model of the squared residuals. We use this model to filter
both the ∆ and tF tS∆ series to account for first-order moment (mean) and second-
order moment (volatility persistence) effects. If, for example, we were to choose the
series to fit to this model, then the filtered series would be the residuals from
the following system
tS∆ln
27:
LL ++++++= −−−−
2104104
25252
2211
2ttqtqtot εβεβεβεββε
]lnlnln[ln 525211 LL +∆+∆++∆+−∆= −−− tptptott SSSS ααααε (6)
The residuals from this system, once estimated, are called and . '''ln tF∆ '''ln tS∆
Table 7 contains the estimated coefficients and their corresponding t-values resulting
from the estimated VECM with the above residual series as inputs. There is only one
significant coefficient in the VECM at a long lag of on itself. Clearly, these
results refute the existence of linear Granger causality for both transformed index
series. An examination of the correlograms of the residual series, VECM7RES and
VECM8RES, along with their squared residual series, VECM7RESQ and
'''ln tS∆
25
Table 7: Vector Error Correction Model (VECM) For ARCH Filtered Returns of the Share Price Index Futures And The All Ordinaries
Index Using The Johansen Cointegrating Vector (CV)
VECM7: ∑ ∑= =
− +−∆+−∆+=∆9
1
9
17111
'''1
'''ln'''ln'''lni j
tjit ejtFitSCVtS ϕδγ
VECM8: ∑ ∑= =
− +−∆+−∆+=∆9
1
9
18221
'''2
'''ln'''ln'''lni j
tjit ejtFitSCVtF ϕδγ
VECM7 for '''
ln tS∆
VECM8 for '''
ln tF∆
Regressor Coefficient t-value Coefficient t-value
Cointegrating Vector -0.000008 -0.75753 0.000005 0.32315
'''1ln −∆ tS
0.007167 0.44177 -0.023243 -0.86898
'''2ln −∆ tS
-0.000239 -0.01477 -0.013919 -0.52068
'''3ln −∆ tS
-0.010647 -0.65671 -0.007083 -0.26499
'''4ln −∆ tS
-0.007986 -0.49259 -0.008398 -0.31420
'''5ln −∆ tS
0.005095 0.31432 -0.011044 -0.41324
'''6ln −∆ tS
0.000478 0.02952 0.006610 0.24739
'''7ln −∆ tS
-0.016355 1.00928 -0.004654 -0.17419
'''8ln −∆ tS
0.037207 2.29614* -0.003725 -0.13945
'''9ln −∆ tS
0.009390 0.57919 0.007669 0.28694
'''1ln −∆ tF
-0.003219 -0.32705 0.023043 1.42020
'''2ln −∆ tF
0.001739 0.17669 0.020917 1.28939
'''3ln −∆ tF
0.017309 1.75889 0.018121 1.11700
'''4ln −∆ tF
0.011112 1.12904 0.002210 0.13622
'''5ln −∆ tF
0.002053 0.20861 0.004833 0.29786
'''6ln −∆ tF
-0.004575 -0.46492 -0.004101 -0.25282
'''7ln −∆ tF
0.005531 0.56210 0.004357 0.26859
'''8ln −∆ tF
-0.016027 -1.62921 0.008318 0.51290
'''9ln −∆ tF
-0.003491 -0.35486 -0.006224 -0.38373
* denotes statistical significance at the 5% level
VECM8RESQ, reveal a lack of any further autocorrelation in all but the case of the
filtered AOI series at multiples of 52 and greater (see Appendix A).
26
27The specification of an AR structure for the model used to remove the mean effects, prior to simultaneous
Further, we test for nonlinear Granger causality between the residuals from the
VECM model with and as input vectors. Table 8 contains the
results from this test. We find only very weak evidence of nonlinear Granger
causality.
'''ln tS∆ '''ln tF∆
Table 8 Results For Nonlinear Granger Causality Tests Of Residuals From The VECM Model With Input Series and'''ln tF∆ '''ln tS∆
Ho: SPI Do Not Cause AOI Ho: AOI Do Not Cause SPI
σe
1.5 1.0 0.5 1.5 1.0 0.5
Number of Lags
(Lu = Lv)
1 0.0001 (0.740)
0.0007* (2.138)
0.0034** (4.560)
-0.0001** (-6.007)
-0.0001 (-1.55)
0.0009 (1.961)
2 0.0003 (1.118)
0.0011** (2.445)
0.0037** (4.470)
0.0001 (0.597)
0.0001 (0.369)
0.0008 (1.464)
3 -0.0001 (-0.404)
0.0004 (1.101)
0.0011 (1.754)
0.0000 (0.200)
0.0003 (0.896)
0.0011 (1.818)
4 -0.0001 (-0.728)
0.0005 (1.090)
0.0010 (1.357)
0.0000 (-0.224)
0.0001 (0.419)
0.0003 (0.559)
5 0.0000 (-0.159)
0.0004 (0.906)
0.0004 (0.583)
-0.0002 (1.417)
0.0000 (-0.039)
0.0001 (0.214)
6 -0.0001 (-0.217)
0.0004 (0.913)
0.0004 (0.432)
-0.0003 (-1.865)
0.0000 (-0.006)
-0.0004 (-0.575)
7 -0.0001 (-0.455)
0.0004 (0.929)
0.0002 (0.252)
-0.0004* (-2.293)
0.0000 (-0.032)
-0.0002 (-0.201)
8 -0.0004* (-2.001)
-0.0002 (-0.505)
-0.0003 (-0.323)
-0.0005** (-2.679)
-0.0002 (-0.362)
0.0000 (-0.014)
9 -0.0006** (-6.229)
-0.0004 (-0.851)
0.0000 (0.000)
-0.0006** (-3.058)
-0.0003 (0.731)
-0.0004 (-0.435)
Notes: The first entry in each cell is the statistic given by equation B.8 (Appendix B) and refers to the difference between the conditional probabilities equated in equations B.3 and B.4. It is a standardised test statistic that is asymptotically
distributed as a standard normal variate. Lu and Lv are the number of lags in the residual series used in the test. * Significance at the 5% level of significance using a one-sided test. ** Significance at the 1% level.
Filtering of the futures and cash index series for third and higher order moment effects
would be necessary to eliminate remaining significant statistics in Table 8.
27
estimation of all the coefficients, was the same we used to remove the mean effects in our EGARCH model.
Accordingly, after accounting for inter-day and intra-day mean and volatility
persistence effects in the data, we concluded that neither significant linear nor
nonlinear Granger causality exists in either direction between the transformed index
returns of the futures and cash markets.
6. Conclusions
In order to explain the incidence of Granger causality and the associated lead-lag
phenomenon between indices of the futures and cash market as reported in many
empirical studies in the literature, we need to account for mean and volatility
persistence effects in the data. This can be achieved by accounting for inter-day and
intra-day effects using an appropriate autocorrelation structure within each of the
index returns from both markets. Once these autocorrelation and higher order
moment effects have been controlled for, linear Granger causality ceases to be
statistically significant. The lead-lag relation may still persist, however, it will not be
observable when the information flow between the spot and futures markets is
completed within a five-minute observation interval. Further, using the Hiemstra and
Jones (1994) modification of the Baek and Brock (1992) nonlinear Granger causality
test, we find no compelling need to account for nonlinear effects (beyond the second-
order moment condition) in order to explain causality. Accordingly, the EMH
assumption that underlies the COC model seems appropriate.
The results in this paper are conditioned on the fact that we have used Australian
futures and cash market data. The Australian market is closed while most Northern
Hemisphere markets are trading. It remains an interesting question as to whether the
information flows from other international markets that trade synchronously but in
different time zones to that of Australia, have the same impact on inter-day and intra-
28
day data in their neighbouring markets. In addition, of interest is the application of the
methodology used in this paper to larger and deeper markets.
29
7. References Abhyankar, A., 1995, Return and Volatility Dynamics in the FT-SE 100 Stock Index
and Stock Index Futures Markets, Journal of Futures Markets, 15, 457-488.
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Abhyankar, A., L. Copeland, and W. Wong, 1997, Uncovering Nonlinear Structure in Real-Time Stock Market Indices: The S & P 500, the DAX, the Nikkei 225 and the FT-SE 100, Journal of Business and Economic Statistics, 15(1), 1-14.
Baek, E., and W. Brock, 1992, A Nonparametric Test for Independence of a
Multivariate Time Series, Statistica Sinica, 2(1), 137-156. Bessembinder, H., 1993, An Empirical Analysis of Risk Premia in Futures Markets,
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Brock, W., D. Hsieh, and B. LeBaron , 1991, A Test of Nonlinear Dynamics, Chaos,
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Brock, W., W. Dechert, J. Scheinkman, and B. LeBaron, 1996, A Test for
Independence Based on the Correlation Dimension, Econometric Reviews, 15(3), 197-235.
Chan, K, 1992, A Further Analysis of the Lead-Lag Relationship Between the Cash
Market and Stock Index Futures Market, Review of Financial Studies, 5, 123-152.
Cornell, B. and K. French, 1983, The Pricing of Stock Index Futures, Journal of
Futures Markets, 3,1-14. Dwyer, G., P. Locke, and W. Yu, 1996, Index Arbitrage and Nonlinear Dynamics
Between the S & P 500 Futures and Cash, Review of Financial Studies, 9(1), 301-332.
Eldridge, R., C. Bernhardt, and I. Mulvey, 1993, Evidence of Chaos in the S&P500
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Engle, R. and C. Granger, 1987, Co-Integration and Error Correction: Representation, Estimation and Testing, Econometrica, 50, 978-1008.
Fleming, J., B. Ostdiek, and R. Whaley, 1996, Trading Costs and the Relative Rates
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Frino, A. and A. West, 1999, The Lead-Lag Relationship Between Stock Indices and Stock Index Futures Contracts: Further Australian Evidence, Abacus, 35(3), 333-341.
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Harris, L., 1989, S&P 500 Cash Stock Price Volatility, Journal of Finance, 44, 1155-
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32
8. Appendix A
Probabilty values (Q-statistics) for various lags of the correlograms of input series to the different error correction models and the correlograms of their residual and residual squared series.
Lag Series
1 2 3 4 5 6 7 8 9 10 51 52 79 80 104 156
∆St ∆Ft
0.00 0.99
0.000.98
0.000.49
0.000.56
0.000.54
0.000.66
0.000.72
0.00 0.78
0.000.84
0.000.72
0.000.76
0.000.01
0.000.01
0.000.01
0.00 0.04
0.00 0.14
VECM1RES VECM2RES
0.98 0.99
0.99 1.00
0.99 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
0.99 1.00
0.99 0.99
0.99 0.87
0.060.02
0.38 0.08
0.050.02
0.16 0.07
0.22 0.20
'S∆ t '
tF∆
0.62 0.98
0.88 1.00
0.97 1.00
0.99 1.00
0.99 1.00
0.99 1.00
0.99 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 0.99
1.00 1.00
VECM3RES VECM3RESQ
0.99 0.31
1.00 0.57
1.00 0.76
1.00 0.88
1.00 0.94
1.00 0.97
1.00 0.99
1.00 0.99
1.00 0.99
1.00 0.99
1.00 1.00
1.00 0.00
1.00 0.00
1.00 0.00
1.00 0.00
1.00 0.00
VECM4RES VECM4RESQ
0.99 0.99
1.00 0.99
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 0.00
1.00 0.00
1.00 0.00
0.99 0.00
0.99 0.00
VECM5RES VECM5RESQ
0.98 0.95
0.99 0.92
1.00 0.98
1.00 0.99
1.00 0.99
1.00 0.99
1.00 0.99
1.00 0.99
1.00 0.99
1.00 1.00
0.99 1.00
0.99 0.00
0.99 0.00
0.99 0.00
1.00 0.00
1.00 0.00
VECM6RES VECM6RESQ
0.99 0.14
1.00 0.05
1.00 0.03
1.00 0.06
1.00 0.09
1.00 0.14
1.00 0.18
1.00 0.25
1.00 0.31
1.00 0.37
0.99 0.98
0.99 0.00
0.99 0.00
0.99 0.00
0.94 0.00
0.90 0.00
VECM7RES VECM7RESQ
0.99 0.95
1.00 0.99
1.00 0.99
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 0.00
1.00 0.00
1.00 0.00
1.00 0.00
0.98 0.00
VECM8RES VECM8RESQ
1.00 0.97
1.00 0.99
1.00 0.99
1.00 0.99
1.00 0.99
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
31
Appendix B
Let Xmt = a lead vector of length m (B.1a)
= (Xt, Xt+1, … , Xt+m-1), m = 1, 2, …
t = 1,2 …,T – m + 1
= (XLxLx-tX t-Lx, Xt-Lx+1, … , Xt-1), Lx = 1, 2, …
(B.1b)
t = Lx + 1, Lx + 2, …, T
= (YLyLy-tY t-Ly, Yt-Ly+1, … , Yt-1), Ly = 1,2, … (B.1c)
t = Ly +1, Ly + 2, …, T
Given values for m, with Lx, and Ly both assumed ≥ 1 and e > 0, then the probability
measure of Y not strictly Granger-causing X is given by 28
( )( )eXXeXXP
eYYeXXeXXP
LxLxs
LxLxt
ms
mt
LyLys
LyLyt
LxLxs
LxLxt
ms
mt
<−<−=
<−<−<−
−−
−−−− , (B.2)
Non-Granger causality implies that the probability that the closeness of the two lead
vectors, as measured by the supremum norm, is going to be same irrespective of
whether we have information on the closeness of the lagged Y vectors or not. The
probabilities in equation (B.2) above can be re-expressed in terms of their marginal
and joint probabilities. That is,
( )( )
( )( )eXXP
eXXeXXP
eYYeXXP
eYYeXXeXXP
LxLxs
LxLxt
LxLxs
LxLxt
ms
mt
LyLys
LyLyt
LxLxs
LxLxt
LyLys
LyLyt
LxLxs
LxLxt
ms
mt
<−
<−<−
<−<−
<−<−<−
−−
−−
−−−−
−−−−
,
,
,,
= (B.3)
Then, Y is said not to strictly Granger-cause X if
33
( ),,
,,
eXXeXXP
eYYeXXeXXP
LxLxs
LxLxt
mx
mt
LyLyS
LyLyt
LxLxs
LxLxt
mx
mt
<−<−=
<−<−<−
−−
−−−− (B.4)
When the probabilities in (B.3) hold, then (B.4) follows, and a test of Y not strictly
Granger-causing X can be evaluated by the calculation of four correlation integrals in
the following equality:
),(4),(3
),,(2),,(1
eLxCeLxmC
eLyLxCeLyLxmC +
=+ (B.5)
If {xt} and {yt} are realisations of {Xt} and {Yt}, then to describe the evaluation of
C1, C2, C3 and C4, first define I(•) to be the indicator variable such that,
( ) ,1,, =−− exxI LxLxs
LxLxt if the vectors are within e distance of each other,
and zero otherwise. Further, using a property of the supremum norm that
LxLxs
LxLxt xx −− and
( ) ( ),,, eXXPeXXeXXP LxmLxs
LxmLxt
LxLxs
LxLxt
mx
mt <−=<−<− +
−+
−−− (B.6)
we can express the correlation integrals, C1 to C4, as:
( ) ( ) ( )
( ) ( ) (
( )
)
( )
( ) ( )∑∑
∑∑
∑∑
∑∑
<−−
<
+−
+−
<−−−−
<−−
+−
+−
−=
−=+
−=
−=+
st
LxLxs
LxLxt
st
LxmLxs
LxmLxt
st
LyLys
LyLyt
LxLxs
LxLxt
st
LyLys
LyLyt
LxmLxs
LxmLxt
exxInn
neLxC
exxInn
neLxmC
eyyIexxInn
neLyLxC
eyyIexxInn
neLyLxmC
,,)1(
2,,4
,,)1(
2,,3
,,.,,)1(
2,,,2
,,.,,)1(
2,,,1
where t, s = max (Lx, Ly) + 1, … ,T-m+1
n = T + 1 –m – max (Lx, Ly) . (B.7)
28 ||•|| is the supremum norm, a distance measure.
33
The test of the null hypothesis that Y does not Granger-cause X is shown by Hiemstra
and Jones (1994) to be a test of whether the statistic
( )),,,(,0
),,(4),,(3
),,,(2),,,(1
2
~ nLyLxmNasy
neLxCneLxmC
neLyLxCneLyLxmCn
σ
+−
+
(B.8)
Hiemstra and Jones (1994) derive an estimator for σ2(m,Lx,Ly,n) in equation (B.8)
which doesn’t require i.i.d. errors but allows for weak dependence in the error
structure.
33