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.

Abhyankar, A., 1998, Linear and Nonlinear Granger Causality: Evidence from the U.K. Stock Index Futures Market, Journal of Futures Markets, 18(5), 519-540.

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,

Journal of Futures Markets, 13, 611-630. Brock, W., 1991, Causality, Chaos, Explanation and Prediction in Economics and

Finance, in: Casti, J., and Karlquist (Eds), Beyond Belief: Randomness, Prediction and Explanation in Science, Boca Raton, FL, CRC Press, 230-279.

Brock, W., D. Hsieh, and B. LeBaron , 1991, A Test of Nonlinear Dynamics, Chaos,

and Instability: Statistical Theory and Economic Evidence (MIT Press, Cambridge, Mass).

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

Cash Index, Advances in Futures and Options Research, 6, 179-192.

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

of Price Discovery in Stock, Futures and Options Markets, Journal of Futures Markets, 16(4), 353-387.

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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.

Grunbichler, A., F. Longstaff, and E. Schwartz, 1994, Electronic Screen Trading and

the Transmission of Information: An Empirical Examination, Journal of Financial Intermediation, 3, 166-187.

Harris, L., 1989, S&P 500 Cash Stock Price Volatility, Journal of Finance, 44, 1155-

1175. Hiemstra, C. and J. Jones, 1994, Testing for Linear and Nonlinear Granger Causality

in the Stock Price-Volume Relation, Journal of Finance, 49(5), 1639-1664. Hsieh, D., 1991, Chaos and Nonlinear Dynamics: Application to Financial Markets,

Journal of Finance, 5, 339-368. Johansen, S., 1988, Statistical Analysis of Cointegration Vectors, Journal of

Economic Dynamics and Control, 12, 231-254. Johansen, S., and K. Juselius, 1990, Maximum Likelihood Estimation and Inference

on Cointegration with Applications to the Demand for Money, Oxford Bulletin of Economics and Statistics, 52, 169-209.

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the S & P 500 Futures and the S & P 500 Index, Journal of Finance, 42(5), 1309-1329.

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Week Effects, Journal of Financial Research, 8(2), 119-126. McInish, T. and R.Wood, 1984, Intertemporal Differences in Movements of Minute-

To-Minute Stock Returns, Financial Review, 19(4), 359-371. Min, J. and N. Najand, 1999, A Further Investigation of the Lead-Lag Relationship

Between the Spot Market and Stock Index Futures: Early Evidence from Korea, Journal of Futures Markets, 19(2), 217-232.

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Relationship Between Stock Index Cash and Futures Markets: A Cointegration Approach, Journal of Futures Markets, 18(3), 297-305.

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Studies, 3, 77-102.

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Shyy, G., V. Vijayraghavan, and B. Scott-Quinn, 1996, A Further Investigation of the Lead-Lag Relationship Between the Cash Market and Stock Index Futures Market with the use of Bid/Ask Quotes: The Case of France, Journal of Futures Markets, 16(4), 405-420.

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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