Asymmetries, causality and correlation between FTSE100 spot and futures: A DCC-TGARCH-M analysis

International Review of Financial Analysis 24 (2012) 26–37

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International Review of Financial Analysis

Asymmetries, causality and correlation between FTSE100 spot and futures: A DCC-TGARCH-M analysis Juan Tao a,⁎, Christopher J. Green b, 1 a b

Department of Business, Economics and Management, Xi'an Jiaotong-Liverpool University, Suzhou, Jiangsu 215123, PR China Department of Economics, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom

a r t i c l e

i n f o

Article history: Received 24 June 2011 Received in revised form 18 May 2012 Accepted 5 July 2012 Available online 6 August 2012 JEL classification: G13 G14 Keywords: Index futures Causality Conditional correlation DCC-TGARCH-M CCF test

a b s t r a c t We use DCC-TGARCH-M to study asymmetries in the conditional variance in FTSE100 spot and futures returns before and after cost-reducing market microstructure changes on the London Stock Exchange and the London International Financial Futures Exchange. We find bidirectional causality-in-mean and that negative shocks have a larger impact on the conditional variances than positive shocks. There is little evidence of causality-invariance. The results support a risk premium explanation of asymmetric volatility before the microstructure changes; afterwards, there is evidence of a risk premium effect in futures but a momentum effect in spot. Following the microstructure changes, the speed at which the markets absorbed news increased, as did the asymmetric volatility effect of bad news. We also document regular temporary declines in the conditional correlations following contract expiration. This is consistent with the increased uncertainty following expiration, when investors' attention switches to the next near contract, and the no-arbitrage linkage between spot and futures is temporarily reduced. © 2012 Elsevier Inc. All rights reserved.

1. Introduction It is well-established that sharp movements in stock returns tend to be associated with increased conditional stock volatility, and this relationship is often asymmetric: large negative shocks are associated with a greater increase in volatility than large positive shocks (Ederington & Guan, 2010). There are two main explanations for the asymmetry. First is the risk premium (or volatility feedback) theory that argues that sharp return movements create expectations of increased volatility; these require correspondingly higher expected returns and therefore lower prices (Campbell & Hentschel, 1992). Second is the leverage theory that asserts that a negative return shock increases company leverage because of the decline in equity values. Given the volatility of a firm's assets, its equity volatility will necessarily increase (Black, 1976; Christie, 1982). The empirical evidence on these explanations remains inconclusive, although leverage alone appears to be insufficient to account for observed asymmetries (Figlewski & Wang, 2001). Relationships between returns and volatility in index futures markets are less well understood. A risk premium effect could arise in futures markets in a similar way to that in stock markets, with increased

⁎ Corresponding author. Tel.: +86 512 88161709; fax: +86 512 88161999. E-mail addresses: [email protected] (J. Tao), [email protected] (C.J. Green). 1 Tel.: +44 1509 222711; fax: +44 1509 223910. 1057-5219/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.irfa.2012.07.002

volatility rewarded by higher expected returns. But the leverage effect has no direct parallel in futures markets because changes in leveraged positions are limited to an intra-day basis as they are controlled by the daily resettlement (McMillan & Speight, 2003). However, there is a further possibility in futures markets: irrespective of the underlying cause, asymmetries in volatility may spill over from spot to futures, or vice-versa. If volatility spill-overs are combined with asymmetries, a bad news shock in either market may increase volatility and its persistence in both markets. We call this the spill-over hypothesis. Conditional volatility in futures markets has been studied by several authors. Lien and Tse (2002) survey this literature. There is evidence from standard symmetric GARCH-class models that volatility can spill over in either direction between spot and (index) futures (Chan et al., 1991; Racine & Ackert, 2000; Tse, 1999), but this leaves unexplored the issue of volatility asymmetries. Meneu and Torro (2003) find evidence of volatility asymmetries in the Spanish IBEX spot and futures markets: negative spot volatility shocks increase spot volatility but negative futures volatility shocks reduce futures volatility. Meneu and Torro (2003) document these asymmetries using volatility impulse response functions, and then consider the implications for hedging. They do not test the leverage or risk premium theory, nor do they study the relationship between volatility and the underlying futures contracts. Likewise, Chang et al. (2010) find evidence of volatility asymmetries in the spot and futures markets for Asian natural rubber, but do not seek to explain these asymmetries or consider their relationship to the futures contracts. McMillan and Speight, (2003) study conditional

J. Tao, C.J. Green / International Review of Financial Analysis 24 (2012) 26–37

volatility using three years of intra-day FTSE100 futures prices and report similar asymmetries to those commonly found in stock markets. McMillan and Speight (2003) also provide a more direct test of the risk premium effect, but find no evidence in its favour and conclude that the asymmetries must be due to noise traders. However, they do not investigate the spot market, and therefore do not consider whether the asymmetries in futures could be attributed to spill-overs between spot and futures.2 Overall therefore, it seems clear that volatility asymmetries do exist in at least some futures markets, but they are not very well-documented, and their nature and more particularly their causes are not well-understood. In this paper, we study volatility asymmetries in the FTSE100 stock index and index futures market and the linkages between these two markets, using as framework the dynamic conditional correlation model (DCC-GARCH: Engel, 2002). Our research contributes to the literature in several ways, the most important of which are the following. First, we extend Meneu and Torro (2003) by formally testing the risk-premium and spill-over hypotheses using the conditional means and variances. Second, we extend McMillan and Speight (2003) by estimating and testing these hypotheses in a bivariate model of spot and futures. We aim to pin down more precisely whether asymmetric volatility in futures can be explained either by the spill-over or the risk premium hypothesis, or some combination of the two. Third, our data period runs from 28/10/1986 through 30/12/2005 and includes two important cost-reducing market microstructure changes. First was the move from quote-based trading to a hybrid system of dealers alongside an electronic order book by the London Stock Exchange (LSE) on 20/10/1997; second was the move from open outcry to electronic trading by the London International Financial Futures Exchange (LIFFE) on 30/11/1998. We investigate if these cost-reducing changes increased the speed at which the impact of news is absorbed by the markets, and therefore if they resulted in an enhanced asymmetric conditional variance effect following the arrival of bad news. Fourth, the pricing of futures contracts tends to be less efficient around expiration dates (Frino & McKenzie, 2002). As a contract approaches expiration, investors close their positions and switch attention to the next near contract. Volumes in the expiring contract decline and, initially, the pricing of the next near contract involves greater uncertainty. As a result, the arbitrage links between spot and the nearest futures contract either side of an expiration may be temporarily weakened. The implications of this for conditional volatility have not been studied at all. In a further new contribution we investigate the relationship between the time series pattern of conditional volatility and contract expiration. To study these issues, we begin by splitting the data into two samples at the first market microstructure change (on the LSE) on 20/10/1997, but we check this split in several ways as we proceed. We perform preliminary unit root and cointegration tests allowing for the possibility of a break in the data around the market microstructure changes. It transpires that spot and futures prices are cointegrated as we would expect; and this enables us to use a vector error correction model (VECM) to study the conditional means of spot and futures. The use of a VECM in this context is standard (for example, Wahab & Lashgari, 1993). However, we depart from the usual VECM by formulating the model so as to allow for the periodic jumps in the basis at contract expiration without the need to use an arbitrary rollover formula (following Green & Joujon, 2000). We use the estimated VECM to verify formally the validity of the sample split and test the residuals for ARCH. We then extend the VECM using a threshold DCC-GARCH-inMean (DCC-TGARCH-M) specification to model the conditional error variance processes. Previous studies have used a range of GARCH models (including the DCC model) but not all these can easily be nested 2 Bollerslev, Litvinova, and Tauchen (2006) find volatility asymmetries in highfrequency datasets similar to McMillan and Speight (2003), but also do not consider spot-futures linkages.

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in a single general model (Chang et al., 2010). DCC-GARCH has the flexibility of univariate GARCH, with parsimonious parameterisation of the conditional correlations (Engle, 2002); and, as it is a generalisation of the constant conditional correlation (CCC) model, we can test if the estimated DCC model can be simplified to a CCC model. The threshold parameters in the model control for and enable us to test the existence of asymmetric volatility. In addition, the “in-Mean” terms enable a test for the risk premium effect, as suggested by McMillan and Speight (2003). Finally, we set out the cross-correlation function (CCF) approach of Cheung and Ng (1996) to test for causality-in-variance and volatility spill-overs between spot and futures. This is effectively a two-step approach: DCC-TGARCH-M followed by CCF; and it substantially reduces the number of parameters to be estimated in comparison for example with the VARMA method proposed by Ling and McAleer (2002). Our key findings can be summarised as follows. First, we document the existence of significant volatility asymmetries in both the FTSE100 spot index and index futures prices: in general negative volatility news shocks have a significantly larger effect on the conditional variance in each market than do positive shocks. The CCF tests suggest that variance shocks in each market are impounded more-or-less simultaneously in both markets. Second, we find clear evidence of a risk premium effect originating in the spot market in the first sample period. In the second period the evidence is more ambiguous, but is broadly consistent with a risk premium effect in the futures market and a momentum effect in the spot market. Third, the evidence supports the view that the two cost-reducing market microstructure changes were associated with increased efficiency in the response of the markets to news. In the conditional mean model, the error-correction effect shows that returns adjusted more rapidly to movements in the basis in the second period than in the first, with a greater role for price discovery in the futures market. In the DCC-GARCH model there is an enhanced asymmetric variance effect and a substantial decline in the persistence of the conditional correlations from the first to the second sample period, suggesting that news shocks were absorbed more rapidly in the two markets following the market microstructure changes. Finally, the time series of conditional correlations contains more-or-less regular spikes when the correlation declines immediately following contract expiration. This is consistent with the argument that there is greater uncertainty at these times when trading attention switches to the next near contract, and the no-arbitrage linkage between spot and futures is temporarily attenuated. Furthermore, the correlations tend to return to their “normal” pattern more rapidly after the microstructure changes, providing further evidence that pricing efficiency increased between the two sample periods. The rest of the paper is organised as follows. In Section 2, we discuss the properties of the FTSE100 spot and futures data, and outline the key market microstructure changes in London during the sample period. Section 3 reports on the preliminary unit root and cointegration tests. The empirical methods used in the main part of the analysis are set out in Section 4. Section 5 presents the estimation results; while the conditional correlations are discussed in Section 6. Finally, Section 7 contains some concluding remarks.

2. Data Futures and spot prices are abbreviated by Ft and St, with: ft ≡ lnFt; st ≡ lnSt; the basis is defined as bt = ft − st. Daily returns are: Δkft = ft − ft − k; Δkst = st − st − k; where k is the number of days between contiguous observations. Thus, t – k refers to the observation lagged by one working day but k calendar days. In general k = 1, but k > 1 for intervals that cover weekends or holidays.3 rt is the interest 3 Since k varies over time, strictly, we should write kt. To avoid excessive subscripting we generally abbreviate kt as k where no ambiguity will result.

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J. Tao, C.J. Green / International Review of Financial Analysis 24 (2012) 26–37

rate, dt is the index dividend yield, T is the expiration date of a contract and coct = (rt − dt)(T − t) the cost of carry. We use daily closing values for the FTSE100 stock index quoted on the London Stock Exchange (LSE) and daily settlement prices for the futures contracts quoted on the London International Financial Futures Exchange (LIFFE). The data run from 28/10/1986, following “Big Bang” on the LSE when fixed commissions were abolished and trading moved from the floor to a screen-based system, through 30/12/2005, a total of 4832 daily observations. The FTSE100 futures contract and the shares in the underlying FTSE100 index are heavily traded, but less well-studied than the major US contracts; and the possible effect of market microstructure changes has not been considered in this setting. The FTSE100 index was quote-based through 20/10/1997 and hybrid thereafter, with dealers alongside an electronic order system (Tao, 2008: 62). Therefore, the widely documented non-synchronous trading problem for transactions-based indices (Fleming et al., 1996) should be less serious for the FTSE100 because market makers can always update their quotes immediately in response to new information. We use daily data because futures are resettled daily. If a contract is mis-priced at purchase or sale, the mis-pricing is effectively settled at the end of the day when the contract is marked to market. Settlement prices determine daily net gains or losses, margin calls, invoice prices for deliveries and hence the actual cash flows of traders. Resettlement therefore creates a strong argument for using daily data. FTSE100 futures prices were extracted from NYSE Euronext-Liffe (2009). To construct the cost of carry, measures of the interest rate and dividend yield are required. Strictly, the interest rate for any day t is for a loan with maturity (T − t). This requires estimation of a complete term structure of interest rates to match the maturity of each loan with the time-dependent maturity of the futures contract. This is evidently impracticable, and we therefore used the London Discount Market Overnight Rate to represent the interest rate on a loan with maturity (T − t). 4 The dividend yield is that on the FTSE100 index converted to a daily rate. The daily stock index, interest rate and dividend yield were all extracted from Datastream Advance (2009). As most futures contracts, including the FTSE100, are heavily traded only in the last three months before expiration, data for the last three months of each contract were linked through time to create a single futures price history. We spliced data from successive contracts by shifting to the next near contract on each expiration day. This creates a jump in the basis at expiration as a new contract enters the data. A common solution is to smooth the data, for example, by averaging across near and further contracts as delivery approaches, or by splicing the return series at a single point in time. Such methods create an artificial time series that ignores the cost of shifting between contracts because of movements in the basis, and we therefore do not adopt this approach. Instead, we followed Green and Joujon (2000) in utilising real transactions prices and defining a set of time-related variables to model the rollover jumps. See Section 4 below. The arbitrage link between futures and spot is influenced by the costs of trading in the two markets. It is therefore important to perform the empirical analysis over a time period with relatively stable transactions costs. This was used as the principle for splitting the overall dataset. Transactions costs in London that impinge on futures trading have been particularly affected by two discrete microstructure changes. The first occurred at the LSE on 20/10/1997 when the quote-driven dealer system (SEAQ) was supplemented by an order-driven system (SETS) for trading the most liquid FTSE100 securities.5 Under SETS, buy and 4 Given that LDMON is the interest rate for overnight loans, it could be lower or higher than the true rate for a loan maturing at the futures delivery date, depending on the shape of the yield curve. Therefore, the use of LDMON could bias the estimate of the fair futures price. However, since the average maturity of futures contracts used in this study is only 1.5 months, the bias is expected to be small. 5 SEAQ continued in parallel with SETS, with dealers quoting prices over telephone and trading off-exchange. Therefore traders could choose between the order-driven system and the dealership system.

sell orders are automatically reported and matched by the electronic order-book, in principle providing greater transparency and a narrower bid-ask spread. We therefore set 17/10/1997 as the end of the first sample period. The second important microstructure change occurred at LIFFE on 30/11/1998, when open outcry trading was replaced by LIFFE CONNECT, an electronic platform that was expected to reduce costs. It might therefore be preferable to set 20/10/1997–27/11/1998 as a second sample and 30/11/1998–30/12/2005 as a third. However, the central period (1997–1998) is too short to obtain satisfactory empirical results and we therefore tentatively decided to work with just two samples, split at the introduction of SETS on the LSE: 28/10/ 1986–17/10/1997 (2776 observations) and 27/10/1997–30/12/20056 (2051 observations). Although this split was determined on an a priori basis, we tested it in three ways. First, the unit root tests allow for the possibility of an endogenous one-time shift in the intercept of the time series (Section 3). Second, we formally tested the validity of the 20/10/1997 break-date using the estimated model of the conditional mean (Section 4). Third, we re-estimated the model within the shorter period 30/11/1998–30/12/2005 following the introduction of LIFFE CONNECT, and the results were essentially the same as those for the whole of 27/10/1997–30/12/2005; we therefore report results only for 27/10/1997–30/12/2005.

3. Preliminary data analysis: Unit root and cointegration tests Summary statistics suggest that there are differences between the two samples, with the mean returns and basis all declining but a more mixed picture for the measures of variation (Table 1). The observed high kurtosis is suggestive of a time-varying variance. We used Augmented Dickey–Fuller tests (ADF: Dickey & Fuller, 1981) to check for unit roots in each sample. The ADF test tends to be biased in favour of the null of a unit root if there are structural breaks in the series (Brooks, 2002). We therefore also used the Zivot-Andrews test (ZA: Zivot and Andrews, 1992) which allows for an endogenous one-time break in the intercept of the time series process. The null hypothesis of a unit root is that: yt = μ + yt−1 + et. The alternative is that yt is trend-stationary with a one-time break at an unknown time which is to be estimated. The ZA test involves the regression:

yt ¼ μ þ θDU t ðλÞ þ βt þ αyt−1 þ

k X

cj Δyt−j þ et

ð1Þ

j¼1

where: DUt(λ) = 1 if t = Tλ, and 0 otherwise; T is the number of observations; and λ is the estimated value of the break fraction: for a break at TB, λ = TB/T. If the t-statistic for α = 1 exceeds its critical value, the null of a unit root is rejected in favour of stationarity with structural change at the estimated breakpoint. Both tests suggest that futures and spot prices are I(1), and that futures and spot returns, the basis and cost of carry are I(0) (Table 2).7 We look next at cointegration between futures and spot prices. For this purpose we carried out Johansen's maximum eigenvalue and trace tests separately on each sample (Johansen, 1988). We can see that there exists a cointegrating relationship between futures and spot with a single normalized cointegrating vector close to [1, −1] as we would expect (Table 3).

6 The second sub-sample starts from 27/10/1997 to allow for the necessary lags in the error-correction model. 7 We do not report separate tests of the interest rate or dividend yield as they enter the model only indirectly through the cost of carry.

J. Tao, C.J. Green / International Review of Financial Analysis 24 (2012) 26–37 Table 1 Summary statistics, 28/10/1986 – 30/12/2005. Shows summary statistics for daily data with: Δkft = futures return; Δkst = spot return; bt = the basis; rt = interest rate; dt = dividend yield; coct = (rt − dt)(T − t) = the cost of carry. Δkft

Δkst

bt

28/10/1986–17/10/1997 (N = 2776) Mean 0.0006 0.0005 Median 0.0005 0.0007 Standard 0.0100 0.0085 deviation Skewness −0.0660 −0.1730 Kurtosis 3.0536 3.4541 Minimum −0.0656 −0.0639 Maximum 0.0599 0.0544 27/10/1997–30/12/2005 (N = 2051) Mean 0.0001 0.0001 Median 0.0000 0.0004 Standard 0.0126 0.0121 deviation Skewness −0.0682 −0.1481 Kurtosis 2.4760 2.2951 Minimum −0.0608 −0.0589 Maximum 0.0637 0.0590

rt

dt

coct

0.0050 0.0039 0.0068

0.0902 0.0831 0.0342

0.0428 0.0424 0.0052

0.0058 0.0040 0.0056

0.4178 3.0202 −0.0468 0.0339

0.4871 −1.0618 0.0238 0.1800

0.0796 −0.0936 0.0303 0.0576

1.2823 1.0684 −0.0028 0.0274

0.0017 0.0014 0.0041

0.0497 0.0478 0.0116

0.0280 0.0271 0.0053

0.0026 0.0018 0.0026

0.1847 1.6993 −0.0181 0.0252

0.5611 −0.3982 0.0275 0.0752

0.0767 −1.3534 0.0193 0.0424

1.1639 0.9012 −0.0010 0.0125

29

Table 3 Cointegration tests. Johansen (1988) test statistics for cointegration between ft and st: the lag length for the first-differenced variables included in the test (lmax) was selected following Schwert (1989): lmax = int{4(T/100)0.25}. This defines 9 lags for October 1986–October 1997 and 8 lags for October 1997–December 2005. * denotes rejection of the null hypothesis at the 1% level. Trace 1% Number of cointegrating statistic Critical Value vectors

Maximum 1% critical eigenvalue value statistic

Normalized cointegrating vector ft

4. Empirical methodology

76.06* 0.08

18.52 6.63

1.0000 −0.9956

27/10/1997–30/12/2005 None 69.26* 19.94 At most 1 1.89 6.63

67.37* 1.89

18.52 6.63

1.0000 −1.0112

First, we allow for the passage of time by including the time interval (kt) as a regressor. 8 We also correct the basis for trend by including the number of days remaining in the contract (T − t) at each point in time. A linear combination of (T − t + kt) and bt − k can be interpreted as the lagged trend-corrected basis and may be used as the ECT. Second, we model the jump in the basis at expiration of each contract, by introducing a new contract dummy (zt) defined as: 

4.1. Vector error correction model (VECM)

zt ¼

As the cointegrating vector for futures and spot prices is close to [1, −1], it simply defines the basis (ft − st). It follows that a vector error-correction model (VECM) can be used to examine price dynamics between spot and futures, with the lagged basis as the I(0) errorcorrection term (ECT). We write the cost of carry as a stochastic relationship: f t −st −ðrt −dt ÞðT−t Þ ¼ ut or bt −ðrt −dt ÞðT−t Þ ¼ ut

ð2Þ

where ut is I(0). We expect the cost of carry, (rt −dt)(T −t), and therefore the basis to be trend-stationary within any individual contract, with a declining trend governed by the time to expiration. When contracts are spliced together in a single time-series, we get the familiar “sawtooth” pattern in the basis, with a jump each time a new contract enters the data. Following Green and Joujon (2000), we model the deterministic component of these movements by introducing a set of dummy variables.

st

28/10/1986–17/10/1997 None 76.14* 19.94 At most 1 0.08 6.63

T g −t þ kt 0

if T g−1 −t þ kt ¼ 0 otherwise

ð3Þ

We index each contract by g: Tg is the expiration date of contract g. If contract (g − 1) expires at (t − kt), then Tg − 1 − t + kt = 0 and bt − k = ut − k. On the next working day (t), contract g enters the data and bt = (rt − dt)(Tg − t) + ut. Hence the expectation at (t − kt) of the change in the basis at a rollover is: Et−kΔkbt = (rt − dt)(Tg − t). In the absence of a rollover, the expected change in the basis is: − kt(rt − dt). Since kt controls each day's trend change in the basis, zt models the part of the rollover jump not already controlled by kt. Thus, deducting (− kt(rt − dt)) from Et − kΔkbt yields the rollover jump (zt): (rt −dt)(Tg −t+kt). When a new contract enters the data, we have: Et−kΔkbt =(rt −dt)(Tg − t+kt) −kt(rt −dt)=(rt − dt)(Tg −t) as required. The complete VECM for spot and futures returns can therefore be written as:

Δkt st ¼ α 0 þ

M1 X

α 1j Δkt−j st−j þ

j¼1

M2 X

α 2j Δkt−j f t−j þ

j¼1

M3 X

α 3j kt−j

j¼0

M4   X þ α 4 bt−kt þ α 5 T g −t þ kt þ α 6j zt−j þ e1t

ð4aÞ

j¼0

Table 2 Unit root tests. ADF is the t-statistic for the Augmented Dickey–Fuller test. The 1% critical value is −3.43. ZA is the minimum t statistic for the Zivot-Andrews unit root test. The 1% critical value is −5.34. The lag length (lmax) in the tests was selected following Schwert (1989): lmax = int{4(T/100)0.25}. Data: ft = futures price; st = spot price; Δkft = futures return; Δkst =spot return; bt =the basis; coct =(rt −dt)(T−t)=the cost of carry. * denotes rejection of the null hypothesis of a unit root at 1% level. Variable

ft st Δkft Δkst bt coct

28/10/1986 – 17/10/1997

27/10/1997 – 30/12/2005

Lag length (lmax) = 9

Lag length (lmax) = 8

ADF t-statistic

ZA t-statistic

ADF t-statistic

ZA t-statistic

−0.3845 −0.3319 −15.8512* −14.9140* −8.5164* −8.2250*

−3.3700 −3.2313 −19.1950* −18.0516* −9.0105* −10.0813*

−1.3772 −1.3661 −15.3915* −15.2344* −6.9774* −7.0703*

−3.2648 −3.2891 −20.4466* −20.1594* −7.7390* −9.4300*

Δkt f t ¼ β0 þ

N1 X j¼1

β1j Δkt−j st−j þ

N2 X

β2j Δkt−j f t−j þ

j¼1

N3 X

β3j kt−j

j¼0

N4   X þ β4 bt−kt þ β5 T g −t þ kt þ β6j zt−j þ e2t

ð4bÞ

j¼0

Distributed lags in kt and zt are needed to match those of Δkft and Δkst, given the cost of carry relationship. Ceteris paribus, we expect kt − j to have a positive effect on current returns. The effect of zt − j on the basis (β6j − α6j) should be positive, to reflect the jump in the spread between futures and spot at contract rollovers. 8

Racine and Ackert, (2000) also allow for such a day of the week effect.

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J. Tao, C.J. Green / International Review of Financial Analysis 24 (2012) 26–37

4.2. Estimation and testing the VECM Initially, we estimated Eqs. (4a) and (4b) by OLS. To form a model that is parsimonious but adequate, we used the same criterion as for the unit root and Johansen tests to select the initial lag lengths,9 giving 9 lags in 28/10/1986–17/10/1997 and 8 lags in 27/10/1997–30/12/2005. We then simplified by repeatedly testing the joint significance of the four coefficients on the highest lag of Δkft, Δkst, kt, and zt, using F tests and a 1% critical region to control the power of the sequential test procedure. We tested and deleted as necessary the four coefficients jointly as we might expect the lag lengths of Δkft, Δkst, kt, and zt to be comparable, given cost of carry. We also checked the residuals using the Ljung-Box Q test. The final model for the conditional means included 4 lags in each period,10 except that the zt−j (j > 0) were not significant in the spot equation in either period.11 As we would expect the switch to a new contract to be reflected mainly in the futures price, we deleted the zt−j (j >0) from the spot equation. We checked the sample split at 20/10/1997 by estimating Eqs. (4a) and (4b) across the entire data period (28/10/1986–30/12/2005) using 4 lags, and then performing Chow tests at 20/10/1997. Since the change in microstructure at this date is known, Chow tests seem to be both suitable and sufficiently powerful for detecting coefficient instability in this case. The null hypothesis of no break at 20/10/1997 is decisively rejected.12 This result gives some confidence that the introduction of SETS on the LSE did lead to a change in the pricing process affecting spot and futures. We therefore pursue separate estimation on the two samples: 28/10/1986–17/10/1997 and 27/10/1997–30/12/2005. 13 Finally, we tested the residuals of Eqs. (4a) and (4b) for ARCH effects (Table 4 panel A.) Clearly there is substantial evidence of ARCH, and we therefore turn to modelling the conditional variances. 4.3. VECM for the mean and DCC-TGARCH-M for the variance-covariance The VECM diagnostics suggest the existence of a high-order ARCH process; but, as Bollerslev (1986) argued, a low-order GARCH model is a more parsimonious way to parameterise time series which exhibit high-order ARCH diagnostics. We therefore used a DCC-TGARCH-M (1,1) model to estimate the conditional variance and covariance process from Eqs. (4a) and (4b) . We carried out diagnostics on the standardised residuals from this model and found no evidence of ARCH effects in either sample (Table 4 panel B). We are therefore confident that DCC-TGARCH-M (1,1) does provide a satisfactory parameterisation of the conditional variance effects. To obtain the DCC-GARCH model, we assume that the error vector Εt = (e1t,e2t)' from Eqs. (4a) and (4b) is conditionally multivariate normal:   Εt ϕt−kt ∼Nð0; Ht Þ

ð5Þ

Here, ϕt − kt is the information available at time t − k, and Ht is the covariance matrix:  σ H t ¼ Dt Rt Dt ≡ 1t 0 9

0 σ 2t



1 ρ12;t

ρ12;t 1



σ 1t 0

0 σ 2t

 ð6Þ

Schwert (1989) suggests a maximum lag of: lmax = int{4(T/100)0.25}. In the first period, the F test and the Q test gave the same number of lags. In the second period, the F test implied 3 and the Q test 4 lags. We retained 4 lags but, for the final DCC-TGARCH-M estimates we tested whether 3, 4, or 5 lags were acceptable in the mean model. See Table 5 and the discussion in section 5. 11 The results of these F tests are available from the authors on request. 12 The test statistics were F(21,4784) = 2.9598 and 4.2277 for the spot and futures equation respectively. 13 As noted earlier, the VECM was also estimated over 30/11/1998–30/12/2005 and the results were found to be qualitatively identical with those for 27/10/1997–30/12/ 2005. 10

Table 4 ARCH-LM test on residuals. Panel A shows χ2 statistics testing the null hypothesis of no ARCH effects up to order m in the residuals of Eqs. (4a) and (4b). Panel B shows χ2 statistics testing the null hypothesis of no ARCH effects up to order m in the residuals of Eqs. (5)–(10b). χ2(m) is the 1% critical value of the χ2 statistic. All the χ2 in Panel A are highly significant. All the χ2 in Panel B are not significant. Panel A. ARCH-LM test residuals of conditional mean model (Eqs. 4a-4b) Order (m)

1 2 3 4 5 6 7 8 9 10

28/10/1986–17/10/ 1997

27/10/1997–30/12/ 2005

Spot

Futures

Spot

Futures

46.6 148.7 156.2 211.9 212.0 256.0 262.7 264.9 265.1 272.5

28.3 88.8 109.1 131.9 138.1 164.4 165.5 172.2 182.0 183.3

147.3 358.2 446.4 474.0 502.4 531.8 536.8 571.0 571.4 574.5

155.2 340.4 410.3 429.4 466.0 492.5 500.6 555.0 557.0 557.6

χ2(m)

6.63 9.21 11.3 13.3 15.1 16.8 18.5 20.1 21.7 23.2

Panel B. ARCH-LM test on residuals of DCC-TGARCH-M Model (Eqs. (5)–(10b)) Order (m)

1 2 3 4 5 6 7 8 9 10

28/10/1986–17/10/ 1997

27/10/1997–30/12/ 2005

Spot

Futures

Spot

Futures

2.584 2.893 5.023 5.182 7.592 7.796 8.013 8.013 8.436 9.209

0.959 0.970 2.247 2.301 3.013 3.356 3.748 4.073 4.088 4.193

0.031 0.995 1.045 1.432 1.974 2.255 2.307 6.109 11.839 12.960

0.126 0.903 0.995 1.226 1.683 1.702 2.063 8.360 12.615 12.801

χ2(m)

6.63 9.21 11.3 13.3 15.1 16.8 18.5 20.1 21.7 23.2

where σ 21t and σ 22t are the conditional variances of spot and futures returns respectively. Both are assumed to follow a univariate GARCH (1,1) process. Rt is the conditional correlation matrix with ρ12,t being the conditional correlation between spot and futures. The processes of σ 21t, σ 22t and ρ12,t are defined below. To test for asymmetries in the variance process, we followed Glosten et al. (1993) to allow for a different response of conditional volatility to past positive and negative volatility shocks. Zero was used as the threshold value to separate the impacts of past shocks. Thus, we used the TGARCH(1,1) model for each σ 2it (i = 1,2): 2

2

2

2

σ it ¼ ψi0 þ ψi1 ei;t−k þ ψi2 ei;t−k Ii þ ψi3 σ i;t−k

ð7Þ

where Ii (i = 1,2) is the indicator function, with Ii = 1 if ei,t − k b 0 and Ii = 0 otherwise. A significant positive value of ψi2 implies that negative volatility shocks increase the variance more than do positive shocks. The long-run persistence of positive shocks is given by (ψi1+ ψi3) and that of negative shocks by (ψi1 + ψi2+ ψi3). Provided conditional shocks follow a symmetric distribution, the average short-run persistence is (ψi1 +ψi2/2) and the average long-run persistence is (ψi1 +ψi2/2+ ψi3).14 Univariate TGARCH(1,1) models for spot and futures returns are estimated first. The standardized residuals, εit = eit/σit (i = 1,2), are then used to estimate the conditional correlation between the two markets using: 

0

Q t ¼ ð1−θ1 −θ2 ÞQ þ θ1 Ξt−k Ξt−k þ θ2 Q t−k −1

−1

Rt ¼ S t Q t S t

ð8Þ ð9Þ

14 For the second moments to exist, the parameters must satisfy: (ψi1 + ψi2/2 + ψi3 b 1) (Ling & McAleer, 2002).

J. Tao, C.J. Green / International Review of Financial Analysis 24 (2012) 26–37

" Qt ¼

σ 21t

σ 12;t σ 22t

# is the conditional covariance matrix of the stan-

σ 12;t dardized residuals, with conditional variances σ2it (i=1,2), and conditional covariance σ12,t; Q⁎ is the unconditional covariance matrix; and θ1 and θ2 are parameters reflecting news and persistence respectively. In the estimation procedure Q⁎ is replaced by the sample analogue: T X 0 Ξt Ξt , where Ξt =(ε1t,ε2t)' is the standardized residual vector. St T −1 t¼1

is a 2×2 diagonal matrix of the conditional standard deviations:     1 ρ12;t σ 1t 0 is the conditional correlation St ¼ ; and Rt ¼ ρ12;t 1 0 σ 2t matrix, with ρ12,t =σ12,t/σ1tσ2t. DCC-TGARCH enables us to model the combination of volatility asymmetry and conditional correlation. In addition, if return shocks produce higher volatility, the GARCH model entails that the volatility will persist. According to the risk premium argument, informed investors will then require a compensating increase in returns. This implies that conditional volatility will have a direct “CAPM-like” effect on the conditional means,15 which can be captured by the GARCH-M model (Engle et al., 1987). Thus we modify (4a) and (4b) to allow for the possible direct impact of the conditional variances on the mean processes:

Δkt st ¼ α 0 þ

M1 M2 M3 X X X α 1j Δkt−j st−j þ α 2j Δkt−j f t−j þ α 3j kt−j þ α 4 bt−kt



j¼1

þα 5 T g −t þ kt



j¼1

j¼0

M4 2 X X 2 þ α 6j zt−j þ α 7h σ h;t þ e1t j¼0

h¼1

ð10aÞ

Δkt f t ¼ β0 þ

N1 N2 N3 X X X β1j Δkt−j st−j þ β2j Δkt−j f t−j þ β3j kt−j þ β4 bt−kt j¼1

j¼1

j¼0

N4 2   X X 2 þβ5 T g −t þ kt þ β6j zt−j þ β7h σ h;t þ e2t j¼0

h¼1

ð10bÞ According to the risk premium theory, conditional volatility in either market will affect its own conditional mean: α7,1 >0, β7,2 >0. However, we also allow for the possibility of spill-overs between spot and futures with the hypothesis that an increase in the conditional variance of futures will increase the spot return (α7,2 > 0) and correspondingly for the futures return (β7,1 > 0). In summary we estimate Eqs. (10a) and (10b) as the mean model simultaneously with the DCC-TGARCH model for the error process: Eqs. (5)–(9). 4.4. CCF test for causality-in-variance DCC-TGARCH-M tells only about the contemporaneous relationships between volatilities. We study volatility spill-overs between spot and futures using the Cross-Correlation Function (CCF) procedure of Cheung and Ng (1996) to test for causality-in-variance between the two markets. For this, we calculate the squared-standardized residuals from the DCC-TGARCH-M model and estimate their CCF at different lags (hi): CCF(hi). If the CCF(hi) at all leads and lags, are not significantly different from zero then there is no evidence of causality-in-variance (Cheung & Fung, 1997; Cheung & Ng, 1996). pffiffiffi To test for a causal relationship at a specified lag hi, the quantity T CCF ðhi Þ can be compared with the standard normal distribution. CCF test for causality-in-variance shares with DCC-TGARCH-M the advantage that it does not involve simultaneous modelling of both intra- and inter-series conditional 15 This argument is also suggested by McMillan and Speight (2003) and Cheng, JahanParvar, and Rothman (2010).

31

variance dynamics. It therefore substantially reduces the number of parameters to be estimated. 5. Estimation results Maximum likelihood estimates for the DCC-TGARCH-M (1,1) model are reported in Tables 5 and 6. We consider first the conditional mean models (Table 5). For 28/10/1986–17/10/1997 (Panel A), there is evidence of bidirectional causality-in-mean between futures and spot markets, with causality from spot to futures much stronger than the reverse. In general, the spot equation is less well-determined than the futures equation. The coefficients for kt − j (j = 1,…,4) are positive and significant in both equations, indicating that the passage of time has a significant effect on price changes in both markets; the zt−j are also mostly significant and positively signed in the futures equation as expected, suggesting that the price jumps at contract rollovers should not be ignored in explaining spot-futures price dynamics.16 The ECT (the basis) is correctly signed negative in the futures equation and correspondingly positive in the spot equation. However, it is only significant (and larger) in the futures equation, suggesting that divergences from equilibrium in the two markets are corrected primarily in the futures market. In the second sample 27/10/1997–30/12/2005 (Panel B), causality is again bidirectional but tends to run more strongly from spot to futures. The kt−j (j >0) are not significant at conventional levels but the zt−j remain mostly significant and positive. The coefficient on zt is negative in the futures equation, but it is the difference (β60 − α60) that shows the net impact of zt on the basis, and this is positive as we expect. In general, the spot (mean) equation is quite poorly determined with few significant coefficients apart from the ECT, which is positive in the spot equation and also in the futures equation. However, it is the difference (β4 −α4) that shows the net impact of the ECT on next day's basis adjustment, and this is negative as we would expect. Thus, causality-in-mean generally runs more strongly from spot to futures. Stronger causality from futures to spot is a more common finding (for example, Zhong et al., 2004) because of the role that index futures are thought to play in aggregating market information (Chan, 1992). However, other researchers have also found evidence of bi-directional causality: both in the UK (Wahab & Lashgari, 1993) and elsewhere (Chan, 1992). 17 Bi-directional causality could be explained by the fact that the FTSE100 was a quote-based index (with a hybrid market after 20/10/1997), and could therefore reflect information more quickly than the transactions-based futures price. Bidirectional causality is possible if traders use spot and futures markets for different purposes or if there are microstructure and information mechanisms in the two markets that ensure lagged price adjustment even after the direct price impact of information has been fully understood by investors. A further point is that the ECT is stronger in futures in the first period but stronger in spot in the second period. According to Tse (1999) a larger response in futures indicates that spot lead futures in price discovery (and vice-versa). In the first period this is consistent with the causal lead of spot over futures and a greater role for spot in price discovery. In the second period, the larger ECT on spot implies a greater role for price discovery in the futures market, and this is consistent with the switch to more efficient electronic futures pricing on LIFFE CONNECT together with the move from quote-based pricing to a hybrid market for FTSE100 shares on the LSE. Although the mean spot equation is not very well determined especially in the second period, in testing down, we sought to retain smoothness in the lag structure as explained above, and a 4th-order model for the means is supported by likelihood ratio tests as against 3rd or 5th order (LR:4 and LR:5 tests in Table 5). The mean equation 16 The negative sign of zt in the spot equation is further evidence that the net impact of the rollover on the basis is positive. 17 Some recent studies do not report the VECM and this makes it difficult to compare; for example Meneu and Torro (2003).

32

J. Tao, C.J. Green / International Review of Financial Analysis 24 (2012) 26–37 Table 5 (continued) Table 5 (continued)

Table 5 Results of DCC-TGARCH-M(1, 1) Mean Eqs.: (10a) and (10b) Δkt st ¼ α 0 þ

M1 X

α 1j Δkt−j st−j þ

j¼1

þ

M4 X

α 6j zt−j þ

2 X

2 α 7h σ h;t

N1 X

N4 X j¼0

M3 X

  α 3j kt−j þ α 4 bt−kt þ α 5 T g −t þ kt

β 1j Δkt−j st−j þ

N2 X

Variables

þ e1t

β 2j Δkt−j f t−j þ

j¼1

β6j zt−j þ

2 X

N3 X

β 3j kt−j þ β4 bt−kt

  þ β 5 T g −t þ kt

j¼0

2

β 7h σ h;t þ e2t

h¼1

LR:i are likelihood ratio (χ2) tests. LR:z is the test to exclude zt−j j = 1,…,4 from spot; LR:5 is the test to exclude st−5, ft−5, kt−5 from spot and futures and zt−5 from futures; LR:4 is the test to exclude st−4, ft−4, kt−4 from spot and futures and zt−4 from futures; LR:σ2 is the test to exclude σ2s,t and σ2f,t from spot and futures; LR:σ2f is the test to exclude σ2f,t from spot and futures. **Significant at the 1% level; *significant at the 5% level. Panel A 28/10/1986 – 17/10/1997 Spot return Eq. (10a)

Futures return Eq. (10b)

Variables

Coefficients

t-statistics

Coefficients

t-statistics

Intercept Δst−1 Δst−2 Δst−3 Δst−4 Δft−1 Δft−2 Δft−3 Δft−4 kt kt−1 kt−2 kt−3 kt−4 zt zt−1 zt−2 zt−3 zt−4 bt−1 Tg – t + kt σ2s,t

−0.00378 −0.12442 0.05444 0.03214 0.08144 0.17229 −0.04275 −0.02206 −0.06665 2.368E-05 0.00077 0.00062 0.00076 0.00057 −2.440E−05

2.61** 2.68** 1.33 0.71 2.01* 4.08** 1.20 0.57 1.90 0.11 2.97** 2.45* 3.21** 2.70** 2.50*

0.02552 −0.10694 8.67912

1.04 1.71 2.12*

−0.00623 0.33892 0.31337 0.15075 0.15790 −0.25870 −0.31479 −0.14401 −0.14299 0.00023 0.00107 0.00097 0.00105 0.00081 6.944E−05 4.090E−05 1.991E−05 1.459E−05 4.720E−06 −0.08822 −0.06461 16.71149

3.82** 6.98** 6.71** 2.76** 3.38** 5.53** 7.37** 3.02** 3.34** 0.93 3.65** 3.41** 3.90** 3.39** 6.37** 7.61** 2.77** 1.69 0.72 3.03** 0.89 3.63**

Test LR:z; χ2(4) LR:5; χ2(7) LR:σ2; χ2(4)

χ2 value 3.6149

Significance

Test

χ2 value

Significance

LR:4; χ2(7) LR:σ2f; χ2(2)

17.547

0.01**

0.46

10.545

0.16

20.173

0.00046**

0.48582

0.78

Results of DCC-TGARCH-M(1, 1) mean Eqs.: (10a) and (10b) Panel B 27/10/1997–30/12/2005

Variables Intercept Δst−1 Δst−2 Δst−3 Δst−4 Δft−1 Δft−2 Δft−3 Δft−4 kt kt−1 kt−2 kt−3 kt−4

Spot Return Eq. (10a)

Futures Return Eq. (10b)

Coefficients

Coefficients

0.00161 0.00200 −0.18081 −0.05694 −0.07286 −0.04456 0.16161 0.00308 0.08170 0.00028 7.526E−05 −0.00030 0.00039 8.103E−05

Panel B 27/10/1997–30/12/2005

j¼0

h¼1

j¼1

þ

α 2j Δkt−j f t−j þ

j¼1

j¼0

Δkt f t ¼ β0 þ

M2 X

Results of DCC-TGARCH-M(1, 1) mean Eqs.: (10a) and (10b)

t-statistics 1.23 0.02 1.67 0.62 1.00 0.50 1.53 0.04 1.14 1.37 0.30 1.30 1.57 0.34

0.00185 0.64002 0.25524 0.21992 0.03599 −0.67234 −0.26960 −0.26772 −0.02784 0.00027 9.210E−06 −0.00029 0.00038 0.00016

t-statistics 1.37 7.29** 2.36* 2.43* 0.51 7.84** 2.55** 3.09** 0.40 1.30 0.04 1.26 1.46 0.69

zt zt−1 zt−2 zt−3 zt−4 bt−1 Tg – t + kt σ2f,t σ2s,t Test LR:z; χ2(4) LR:5; χ2(7) LR:σ2; χ2(4)

Spot Return Eq. (10a)

Futures Return Eq. (10b)

Coefficients

t-statistics

Coefficients

t-statistics

2.56**

−1.598E−05 3.462E−05 2.322E−05 1.607E−05 1.590E−06 0.11490 −0.49926 45.08835 −54.26113

0.91 6.53** 4.12** 2.99** 0.30 2.40* 8.18** 4.72** 5.04**

−4.703E−05

0.17328 −0.42894 43.46723 −52.48480 χ2 value

3.60** 6.70** 4.06** 4.30** Significance

1.7453

0.78

6.7695

0.45

20.356

0.00042**

Test

χ2 value

Significance

LR:4; χ2(7) LR:σ2f; χ2(2)

26.266

0.00045**

13.609

0.0011**

is a framework for understanding the correlations rather than an end in itself and we therefore decided to retain the insignificant variables instead of deleting any further individual variables, when the resulting lag structure may include implausible spikes. Turning next to the variance equations (Table 6), in the first period (Panel A), there is high and highly significant long-run persistence in the variance in both equations (ψi1 + ψi2/2+ ψi3), a finding which is consistent with other comparable results (Brooks et al., 2002). The parameters (ψi2) which measure the variance asymmetry are positive and significant in both equations. This provides firm evidence that the volatilities in both markets react more strongly to bad news than to good news. This is consistent with the results of Meneu and Torro (2003) for Spain, and of Chang et al. (2010) for the natural rubber market.18 The conditional correlation terms are also highly significant as we would expect, and the persistence in the conditional correlation (θ1 + θ2) is also very high; in other words, and perhaps not surprisingly, news has a strong common effect in the two markets. In the second period (Panel B), there is also high persistence in both conditional variances, with a strong asymmetric effect of news (ψi2); indeed this effect is stronger than in the first period: the asymmetry coefficients (ψi2) are somewhat larger, and the coefficients measuring the symmetric effect (ψi1) are both insignificant. The increase in asymmetry in both markets could be attributable to several factors; but, arguably, the most likely explanation may be found in the market microstructure changes that occurred between the two periods and perhaps to key events in the second period which includes the internet boom and bust, with over-confidence followed by a sustained period of bad news. Thus, the speed at which news was impounded into the markets was increased due to the cost-reducing microstructure changes, and the asymmetric effect in conditional variances caused by bad news was also enhanced. In the second period too there is less persistence in the conditional correlations, suggesting also that the impact of news was absorbed more quickly than before by the markets. Given the observed variance asymmetries and their persistence, we now look back at the mean equations to provide information about the risk premium hypothesis (Table 5). In the first period, only the spot variance (σ 2s,t) is significant and signed positive in both equations, 19 and this is consistent with the risk premium hypothesis. Furthermore, it suggests that risk premium effects arose only from aversion to spot volatility. The net effect on the basis of 18 McMillan and Speight (2003) report similar asymmetric effects in a short-span high-frequency dataset of FTSE100 futures. 19 The insignificant futures variance was deleted from the model.

J. Tao, C.J. Green / International Review of Financial Analysis 24 (2012) 26–37 Table 6 Results of DCC-TGARCH-M(1, 1) conditional variances. 2

2

2

2

Conditionalvarianceequation : σ it ¼ ψi0 þ ψi1 ei;t−k þ ψi2 ei;t−k Ii þ ψi3 σ i;t−k 0

Conditionalcovarianceequation : Q t ¼ ð1−θ1 −θ2 ÞQ þ θ1 Ξt−k Ξt−k þ θ2 Q t−k −1

Conditionalcorrelationequation : Rt ¼ Q t

LR:CCC is a likelihood ratio test for the null that the conditional correlations are time-invariant (CCC vs. DCC). **Significant at the 1% level; *significant at the 5% level. Panel A 28/10/1986–17/10/1997 Spot variance equation

Futures variance equation

Coefficient

Variable

Coefficient

t-statistics

Coefficient

t-statistic

ψi0 ψi1 ψi2 ψi3 (ψi1 + ψi2/2 + ψi3)

Intercept e2i,t−k e2i,t−kIi σ2i,t−k

2.370E−06 0.05987 0.04139 0.88606 0.96663

6.78** 8.72** 4.63** 95.50**

2.430E−06 0.04329 0.04812 0.90705 0.97440

7.06** 8.01** 5.79** 114.45**

Conditional covariance equation

θ1 θ2 θ1 + θ2

Ξt−kΞ′t−k Qt−k

Coefficient

t-statistic

0.08006 0.90500 0.98506

12.39** 110.15**

Test

χ2 value

Significance

LR:CCC; χ2(1)

252.19

0.000

Panel B 27/10/1997–30/12/2005 Spot variance equation

Futures variance equation

Coefficient

Variable

Coefficient

t-statistics

Coefficient

t-statistic

ψi0

Intercept

5.30**

e2i,t −k e2i,t −kIi σ2i,t −k

9.500E− 07 0.00206 0.10162 0.93525 0.98812

5.79**

ψi1 ψi2 ψi3 (ψi1 + ψi2/ 2 + ψi3)

9.700E− 07 0.00062 0.09755 0.93821 0.98761

0.10 10.21** 167.33**

0.34 10.47** 165.53**

Conditional covariance equation

θ1 θ2 θ1 + θ2

Ξt− kΞ′t−k Qt−k

Table 7 CCF test for causality-in-variance. CCF(hi) is the correlation between the standardized residuals ε21,t and ε22,t−k, indicating causality-in-variance from spot to futures for h b 0 and from futures to spot for h > 0. The critical values of the test in this sample are CCF(hi) = 0.04893 (1%) and CCF(hi) = 0.03135 (10%). **Significant at the 1% level; ⁎significant at the 10% level. 28/10/1986–17/10/1997

−1

QtQt

Coefficient

t-statistic

0.18139

9.58**

0.52265 0.70404

9.13**

Test

χ2 value

Significance

LR:CCC; χ2(1)

112.63

0.000

σ 2s,t is also positive (β71 − α71). In the second period in contrast, both variances (σ 2s,t, σ2f,t) are significant: σ2f,t is signed positive and its net effect on the basis is positive; whereas σ2s,t is signed negative as is its net effect on the basis. This would again suggest a risk premium effect, but arising from aversion to futures volatility.20 The negative signs on σ2s,t are not consistent with the risk premium argument21 but could be attributable to a momentum effect (Zhong et al., 2004) by which the decline in share prices associated with a positive volatility shock is expected to persist rather than to be a rational adjustment to the 20 McMillan and Speight (2003) find that the effect of futures volatility on futures is positive but insignificant. 21 Cheng et al. (2010) find a mixture of positive and negative signs in their study of Middle Eastern stock markets.

33

27/10/1997–30/12/2005

Lag (h)

CCF (hi)

Lag (h)

CCF (hi)

Lag (h)

CCF (hi)

Lag (h)

CCF (hi)

0 1 2 3 4 5 6 7 8 9 10

0.79375** −0.02604 0.01305 −0.02514 −0.00345 −0.02668 0.01272 −0.01644 0.00033 0.00747 0.01491

−1 −2 −3 −4 −5 −6 −7 −8 −9 −10

−0.0041 0.00744 −0.02335 0.02635 −0.01679 0.02498 0.01977 −0.00638 −0.01238 0.00607

0 1 2 3 4 5 6 7 8 9 10

0.95804** −0.00670 0.02029 0.00742 −0.01542 0.01592 0.01550 0.01536 0.04420** −0.04779** 0.02378

−1 −2 −3 −4 −5 −6 −7 −8 −9 −10

0.00030 0.02050 −0.00101 −0.00675 0.01695 −0.00065 −0.00469 0.05241** −0.04936** 0.01699

shock. The size of the volatility coefficients is also of interest as they increase considerably between the two samples. This implies a larger impact of volatility on the conditional means and is consistent with an increase in the overall efficiency of the trading systems in absorbing information as we would have expected following the introduction of SETS and LIFFE CONNECT. Next we use the CCF test of causality-in-variance to consider if there are variance spill-overs between the two markets. This could have important implications: for example, the risk premium effect observed in one market might be attributable in part to risks that spilled over from the other market. We computed the CCF for up to 10 leads and lags of the squared standardized residuals of the DCC-TGARCH-M model (Table 7). The contemporaneous cross-correlation CCF(0) is highly significant as we would expect, but in the first period there is no evidence of causality-in-variance. In the second period there is a little evidence of bidirectional causality but only at lags 8 and 9. This could be due to the greater stock market volatility in the second period (Table 1), associated to some extent with the internet boom and bust,22 but it is less easy to see why the causality should be at these specific lags. Our finding differs from Tse (1999), who found S&P futures volatility spilling over into spot, and Meneu and Torro (2003), who found the IBEX spot volatility spilling into futures. Since the variance of spot and futures returns tends to be closely related to the rate of information flow (Chan et al., 1991), these results suggest that information was essentially impounded into FTSE100 spot and futures markets simultaneously in both periods. 6. Conditional correlations We turn next to the dynamic conditional correlations between spot and futures. These are shown together with the two-standard-error bands calculated from the time series in Figs. 1 and 2. Although the two markets were highly correlated in both periods they sometimes lost their close association, as indicated by several obvious spikes in the time plots. This suggests that index arbitrageurs may not always be continuously active, resulting in a periodic loosening of the link between the two markets. Antoniou and Garrett (1993) also found that arbitrage relationships between the two markets may sometimes break down. In the first period, the conditional correlations are generally rising over time, suggesting that the link between the two markets was becoming closer. A logistic trend was fitted to the conditional correlations by non-linear least squares: ρsf,t = γ1/(1 + γ2 exp(−γ3t)); and

22 Using the CCF test, Fujii (2005) found that stock volatility spill-overs across markets increased markedly at times of financial crisis.

34

J. Tao, C.J. Green / International Review of Financial Analysis 24 (2012) 26–37 1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 Conditional Correlation Lower Band (CCC) Upper Band (DCC) Trend

0.65 0.6 11/5/86

11/5/87

11/5/88

11/5/89

11/5/90

11/5/91

11/5/92

Constant Correlation Upper Band (CCC) Lower Band (DCC)

11/5/93

11/5/94

11/5/95

11/5/96

Fig. 1. Conditional correlations: 03/11/1986–17/10/1997.

yielded estimated coefficients of: γ1 = 0.970 (198.3); γ2 = 0.024 (4.67); γ3 = −0.001 (7.94).23 This is also shown in Fig. 1. At the start of our data period investors were probably still learning about the use of the futures market for arbitrage,24 as well as adjusting to the effects of the “Big Bang” on the LSE in 1986. Both these factors would tend to generate an increase over time in the conditional correlations. In the second period, the correlations are higher on average and less variable than in the first period,25 and there is no discernable trend in the data.26 This too is consistent with the argument that the two markets became more closely linked over time, with an apparent step improvement following the microstructure changes of 1997/98. Next, we compared our DCC model to the constant conditional correlation (CCC-)GARCH model (Bollerslev, 1990). This involved re-estimating (10a) and (10b) with a CCC-TGARCH-M(1,1) error structure, that is: assuming, a constant correlation between spot and futures rather than the time-varying correlations of the DCC model. The estimated correlation coefficient in each period and its two-standarderror bands are also shown in Figs. 1 and 2. The data decisively reject the CCC model, as evidenced by likelihood ratio tests comparing DCC and CCC (LR:CCC, Table 6). This gives further confidence that the conditional correlations did indeed change over time. We turn finally to consider in more detail the time series pattern of the conditional correlations plotted in Figs. 3 and 4. These also show contract expiration dates and it can be seen that there is a loose association between expirations and the periodic spikes in the conditional correlations. To explore this further we regressed the conditional correlations on dummy variables corresponding to days in the run-up to expiration and the days following entry of a new contract into the dataset. To get a clear picture we included 30 days either side of expiration. The regression coefficients are plotted in Figs. 5 and 6 together with the two-standarderror bands about zero. These show clearly a sharp drop in the conditional correlation immediately after a new contract enters the dataset, and when in theory the basis is at its maximum in the data; the correlation gradually increases until it is back within the standard error bands by about day 20 in period one and more quickly (day 7) in period two. This pattern cannot be explained by the well-documented tendency for volume to decrease just before expiration (Frino & McKenzie, 2002), as the decline in correlation occurs after the next deferred contract

23

A logistic trend allows for the constraint: ρsf,t ≤ 1. t ratios are shown in parentheses. The FTSE100 futures market was established in 1984. The mean correlation increased from 0.93439 to 0.97422; the standard deviation of the correlation decreased from .0.03844 to 0.013716. It is easily calculated that the difference between the means is significant. 26 Efforts to fit a logistic or linear trend produced negligible t statistics. 24

turns into the near contract and enters the data. When a new contract enters the data, pre-existing open interest is substantial; 27 it declines in the last few days of a contract as positions are closed, and attention switches to the next near contract, but in the last few days of the contract there are no changes in the pattern of conditional correlations comparable to those in the first few days after a new contract enters the data. The pattern we do observe is consistent with the argument that the pricing of the new near-contract involves greater uncertainty and this temporarily reduces the conditional correlation between spot and futures. Likewise, the decrease in the time it takes the correlations to return to their “normal” pattern, as between period one and two, is consistent with the view that pricing efficiency increased between the two periods.

7. Conclusions We find evidence of bidirectional causality-in-mean between FTSE100 spot and futures markets over two samples of data separated by transactions-cost-reducing microstructure reforms (SETS at the LSE and CONNECT at LIFFE). Causality-in-mean runs more strongly from spot to futures in both periods; and this is consistent with the trading process in the two markets, that is, a largely quote-based spot index28 and transactions-based futures price. There is a significant errorcorrection (basis) effect and this is stronger in the second period when the implied adjustment speed is faster than in the first, suggesting that with lower transactions costs, the adjustment process brings prices together more quickly. The variables reflecting contract rollover are also significant and correctly signed underlining the importance of modelling such institutional features. There is strong evidence of asymmetry in the conditional variance in both markets: negative information shocks have a significantly larger effect on the conditional variance than do positive shocks, and this asymmetric effect is stronger in the second period following the costreducing microstructure reforms. In the first period, we find evidence for a risk premium effect in the spot market. In the second period, the evidence is more ambiguous: spot and futures variances are significant in both mean equations and the sign of the futures variance is consistent with a risk premium effect originating in futures but affecting both markets. However, the sign of the spot variance is more consistent with a momentum effect than with the risk premium. There is little

25

27 The original maturity of all contracts in the dataset is 9 months; only the last 3 months are included in the analysis. 28 LSE switched to a hybrid system with the introduction of SETS.

J. Tao, C.J. Green / International Review of Financial Analysis 24 (2012) 26–37

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1.05

1

0.95

0.9

0.85

0.8

0.75

0.7 11/5/97

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Conditional Correlation

Constant Correlation

Lower Band (CCC)

Upper Band (CCC)

Upper Band (DCC)

Lower Band (DCC)

11/5/02

11/5/03

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Fig. 2. Conditional correlations: 03/11/1997–30/12/2005.

1

0.95

0.9

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0.65 Constant Correlation

Conditional Correlation

0.6 11/5/86

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11/5/89

11/5/90

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contract

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Fig. 3. Correlations and contract expiration: 03/11/1986–17/10/1997.

1

0.95

0.9

0.85 Conditional Correlation

0.8 Constant Correlation

0.75 contract

0.7 11/5/97

11/5/98

11/5/99

11/5/00

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11/5/02

11/5/03

Fig. 4. Correlations and contract expiration: 03/11/1997–30/12/2005.

11/5/04

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J. Tao, C.J. Green / International Review of Financial Analysis 24 (2012) 26–37 0.02 0.01

2 standard errors

0 -0.01 -0.02 -0.03

Daily effect on Conditional Correlation (regression coefficient)

-0.04 -0.05 -0.06 -30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30

Days relative to expiration (day 0) Fig. 5. Futures expiration and conditional correlations 28/10/1986–17/10/1997.

0.01 0.005

2 standard errors

0 -0.005 -0.01 -0.015

Daily effect on Conditional Correlation (regression coefficient)

-0.02 -0.025 -0.03 -0.035 -0.04 -30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30

Days relative to expiration (day 0) Fig. 6. Futures expiration and conditional correlations 27/10/1997–30/12/2005.

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