APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 Published online 19 December 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/asmb.753 The dynamics of the relationship between spot and futures markets under high and low variance regimes Ming-Yuan Leon Li ∗, † Department of Accountancy and Graduate Institute of Finance and Banking, National Cheng Kung University, No. 1, Ta-Hsueh Road, Tainan 701, Taiwan SUMMARY This investigation is one of the first studies to examine the dynamics of the relationship between spot and futures markets using the Markov-switching vector error correction model. Three mature stock markets including the U.S. S&P500, the U.K. FTSE100 and the German DAX 30, and two emerging markets including the Brazil Bovespa and the Hungary BSI, are used to test the model, and the differences between the two sets of markets are examined. The empirical findings of this study are consistent with the following notions. First, after filtering out the high variance regime, the futures price is shown to lead the spot price in the price discovery process, as demonstrated by prior studies; conversely, the spot market is more informationally efficient than the futures market under the high variance condition. Second, the price adjustment process triggered by arbitrage trading between spot and futures markets during a high variance state is greater in scale than that based on a low variance state, and the degree of the co-movement between spot and futures markets is significantly reduced during the high variance state. Third, a crisis condition involved in the high variance state is defined for the two emerging markets, whereas an unusual condition is presented for the three mature markets. Last, the lagged spot–futures price deviations perform as an information variable for the variance-turning process. However, the portion of the variance-switching process accounted for by this signal variable is statistically marginal for the three mature markets selected for this study. Copyright q 2008 John Wiley & Sons, Ltd. Received 5 December 2007; Revised 2 October 2008; Accepted 29 October 2008 KEY WORDS: volatility; lead-lag relation; markov-switching model; stock index futures 1. INTRODUCTION Theoretically, according to the cost-of-carry model, spot and futures prices should converge, and there is no possibility of arbitrage. In practice, when a discrepancy exists between the spot and ∗ Correspondence to: Ming-Yuan Leon Li, Department of Accountancy and Graduate Institute of Finance and Banking, National Cheng Kung University, No. 1, Ta-Hsueh Road, Tainan 701, Taiwan. † E-mail: [email protected] Copyright q 2008 John Wiley & Sons, Ltd. RELATIONSHIP BETWEEN SPOT AND FUTURES MARKETS 697 futures prices, arbitrage behaviors will trigger an adjustment process to this disequilibrium.‡ Prior studies (e.g. [1–10]) invariably use the conventional vector error correction model (VECM) to examine the error correction (EC) process between spot and futures prices. The VECM is briefly presented below. Spot and futures prices are generally found to be not stationary and integrated with an order of one. When spot and futures prices are cointegrated, the lagged spot–futures price deviation should be included in the model, after which the VECM specification is generated. Moreover, this lagged spot–futures price deviation component is known as the EC or mispricing error term. Prior empirical results frequently indicate that when the spot–futures equilibrium is perturbed, it is spot prices that make the greater adjustment in order to re-establish the equilibrium. In other words, the futures prices lead the cash prices in price discovery.§ It must be noted that while numerous studies show that future prices lead to spot prices, certain empirical works indicate the opposite results. In particular, Subrahmanyam [11] and Chan [5] reveal that spot prices tend to lead futures prices and Abhyankar [20] and Silvapulle and Moosa [21] present bi-directional feedback relationship between spot and futures prices. Consequently, the lead–lag relationship between spot and futures prices has posed a longstanding problem in research field. This investigation departs from aforementioned researches in the way the EC term parameters are modeled and proposes a new approach to questions regarding the lead–lad relationship between spot and futures prices. Our underlying assumptions are presented as follows. Plenty of studies, such as Hamilton and Susmel [22], Ramchand and Susmel [23, 24], Li and Lin [25] and Li [26] and among many others, demonstrate the existence of separate high/low volatility regimes in stock markets. Based on this perspective, this investigation hypothesizes that the type of process underlying the convergence toward the spot–futures equilibrium and its speed varies according to market volatility. To account for the dynamics of this interrelationship between spot and futures markets, this investigation designs a Markov-switching VECM (MS-VECM), in which the parameter of the deviation of spot–futures prices changes according to the phase of the volatility regime. Key questions addressed and examined in this investigation include: Is the direction/degree of price adjustment process between spot and futures markets consistent across various market volatility regimes? Moreover, if no such consistency exists, what are the relationships between various volatility regimes and direction/degree of price transmission processes? In addition, we set up the transition probabilities of Markov-switching systems conditional on the absolute value of lagged spot–futures price divergence help to predict the variance-turning process? Last but not least, this study not only adopts three mature markets (U.S., U.K. and Germany) as the research sample but also includes an analysis of two emerging markets (Brazil and Hungary) and examines the differences between the two groups. The comparative analysis is meaningful. ‡ When spot prices are higher than futures prices, spot prices should decrease, and futures prices should increase to return the price relationship to the long-run equilibrium, the opposite being the case when spot prices are lower than futures prices. § Numerous studies have proposed various reasons to explain the role of futures markets in the process of price discovery: Subrahmanyam [11] and Boot and Thakor [12] highlight the advantage of futures contracts implied by index trading mechanisms; Stoll and Whaley [4] and Fleming et al. [13] focus on transaction costs; Chan [5] and Chung [14] deal with price limitations; Stoll and Whaley [4] and Kawaller et al. [2] emphasize leverage effects; Lau and McInish [15] stress tick size; Stephan and Whaley [16] and Chan [5] focus on market liquidity; Fisher [17], Cohan et al. [18], Lo and MacKinlay [19] and Stoll and Whaley [4] indicate the stale price phenomenon in the spot index. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb 698 M.-Y. L. LI Specifically, prices of spot and futures should obey the law of one price; or else arbitrage opportunities would exist. However, this study posits that the spot–futures arbitrage benefits are limited as a result of the impediments for arbitrage trading between spot and futures markets, including transaction costs, transparency and liquidity and so on. Furthermore, this study posits that such arbitrage barriers generally exist in emerging stock markets, such as the Brazil and Hungary markets. A derivative question is: would the state-varying relationship between spot and futures prices demonstrated in this study be consistent with the two types of markets? The remainder of this investigation is organized as follows. Section 2 outlines the underlying models used in this study, including (1) the conventional VECM for the system based on a constant parameter on the EC term and (2) the MS-VECM for the system based on state-varying parameters on the EC term. Subsequently, Section 3 introduces the empirical results and provides economic and financial explanations for them. Finally, Section 4 presents conclusions. 2. MODEL SPECIFICATIONS 2.1. VECM: The system with constant parameters If two data series are non-stationary and share a common stochastic trend, then they can be concluded to be cointegrated. To examine whether spot and futures price series are cointegrated, this study follows Engle and Granger [27] and establishes the following regression: Ft = 0 +1 · St + Z t (1) where Ft and St are the log prices of futures and spot, respectively, which are multiplied by 100 at time t, respectively, and the Z t variable can serve as a measure of the deviation from the equilibrium between spot and futures prices, namely the EC term. If the EC term Z t is a stationary I (0) variable, then spot and futures prices are cointegrated. The prices of spot and futures are generally found to be non-stationary and are integrated with an order of one. As is well known, when prices of spot and futures display cointegration, the EC term should be included in the model and the VECM specification is then given by p Ft = f +f · Z t−1 + i=1 St = s +s · Z t−1 + p i=1 iff ·Ft−i + isf ·Ft−i + q j=1 fsj ·St− j +etf (2) ssj ·St− j +ets (3) q j=1 where denotes the difference operator (such as Ft = Ft − Ft−1 ). Notably, this study sets the EC term, Z t−1 , as (Ft−1 −0 −1 St−1 ), which represents the last period disequilibrium between spot and futures prices. One key feature of the VECM is its consideration of the long-term adjustments to disequilibrium. Briefly, when Z t−1 >0, then Ft should decrease and St should increase to return the price relationship to the long-run equilibrium, the opposite being the case when Z t−1 <0. Therefore, the signs of F and s should be negative and positive, respectively. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb RELATIONSHIP BETWEEN SPOT AND FUTURES MARKETS 699 The covariance matrix of two error terms, e F,t and e S,t in Equations (2) and (3), is presented as follows: f et ∼ BN(0, H ) (4) et = ets where etf and ets are the residuals at time t, BN denotes the bivariate normal distribution and H is a constant 2×2 positive-definite conditional variance–covariance matrix, which is specified in the following equation: (f )2 fs ·f ·s H= (5) fs ·f ·s (s )2 where f and s are the unconditional standard errors of futures and spot returns, respectively; fs is the correlation coefficient between them. Notably, the conventional VECM model suffers from two limitations. First, all the elements in the variance–covariance matrix, namely the H matrix, are constant. Therefore, the hedge ratio derived by the VECM is a constant, too. However, numerous previous studies have pointed out that stock return variances are heterogeneous. Moreover, this work demonstrates a lack of uniformity in the correlation coefficient between futures and spot returns. Second, the conventional VECM assumes the stability of the convergence speed toward the spot–futures equilibrium relationship using constant f and s parameters as in Equations (2) and (3), respectively. 2.2. MS-VECM: The system with state-varying parameters To capture the discrete jump process in stock return volatilities, this work extends the Markovswitching ARCH (SWARCH) model of Hamilton and Susmel [22] to identify the high/low volatility regimes of spot–futures markets at specific points in time and then establishes an MS-VECM framework using state-varying parameters. As with the VECM, this study first regresses Ft on St : Ft = 0 +1 · St + Z t (6) The MS-VECM used in this study is specified as follows: Ft = f +fkt · Z t−1 + St = s +skt · Z t−1 + et |t−1 = f et ets Ht = Copyright q p i=1 p i=1 iff ·Ft−i + isf ·Ft−i + |t−1 ∼ BN(0, Ht ) h ft h f,s t h f,s t h st 2008 John Wiley & Sons, Ltd. q j=1 q j=1 fsj ·St− j +ets (7) ssj ·St− j +ets (8) (9) (10) Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb 700 M.-Y. L. LI where t−1 refers to the information available at time t −1. One key feature of the MS-VECM used in this study is that the variance–covariance matrix, namely Ht , is not only time varying but also state dependent. Specifically, the conditional variance settings of spot and futures returns are specified in the following equations: f )2 m (et−l h ft f f = + t−l 0 gkf t gkf t−l l=1 (11) s )2 m (et−l h st s s = + t−l 0 gkst gkst−l l=1 (12) where kt is the unobservable state variable with possible outcomes of 1, 2, 3, . . . , n, which respects the volatility regime at time t for the spot–futures market. Notably, this investigation extends the one-dimensional SWARCH model of Hamilton and Susmel to establish a two-dimensional system. However, the bivariate framework used in this work is highly intensive in terms of computation time. To keep the number of parameters tractable, this study considers a system involving two volatility regimes: high and low volatility states, namely kt = {1, 2}. Moreover, without incurring a loss of generality, this work normalizes g1f and g2s , the scale coefficient for regime I , to be unity, whereas g2f >1 and g2s >1 in the case of regime II. The bivariate SWARCH model (see Equations (11) and (12)) used in this study employs a conventional ARCH (m) process to capture the conditional variance dynamics of regime I. By contrast, the conditional variances for regime II are g2f andg2s times those of regime I in the equation of futures and spot returns, respectively. Moreover, in the special case with g1f = g2f = 1 and g1s = g2s = 1, the two residual terms in the model follow the fundamental ARCH (m) process. Because this study assumes that the spot–futures market is characterized by two variance regimes, two states are available for modeling the correlation between spot and futures markets and the corresponding covariance is specified as follows: fs f s 1/2 h fs t = kt ×(h t ·h t ) (13) Given the setting of two distinct volatility regimes for the spot–futures market this work sets up a fs system comprising 2-state correlations: fs 1 for the low volatility state and 2 for the high volatility state (see Equation (13)). Furthermore, to capture the nonlinear spot–futures price transmission process across various variance states, this study establishes a setting in which the parameters of the deviation in spot–futures prices vary with the phase of the volatility regime (see Equations (7) and (8)). Specifically, two volatility regimes are defined: (1) regime I, or the low volatility regime, (namely kt = 1), for which the parameters of the EC term, Z t−1 , in the equations for futures and spot returns are f1 and s1 , respectively; while (2) regime II, or the high volatility regime (namely kt = 2), for which the parameters of the EC term, Z t−1 , for the equations for futures and spot returns are f2 and s2 , respectively. As assumed for the MS-VECM adopted in this work, the kt is an unobservable state variable with possible outcomes of 1 and 2. The two states are linked via a first-order Markov process with the following transition probabilities: Copyright q p(kt = 1|kt−1 = 1) = p11 , p(kt = 2|kt−1 = 1) = p12 = 1− p11 p(kt = 2|kt−1 = 2) = p22 , p(kt = 1|kt−1 = 2) = p21 = 1− p22 2008 John Wiley & Sons, Ltd. (14) Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb RELATIONSHIP BETWEEN SPOT AND FUTURES MARKETS 701 where the transition probability p12 yields the probability that regime I is followed by regime II, the opposite being the case for the transition probability p21 , and the transition probabilities p11 and p22 provide the probability that the market state will remain unchanged in the following period. Previous studies have frequently assumed these transition probabilities to remain constant between successive periods. However, studies such as Filardo [28] and Marsh [29] consider the assumption of constant transition probabilities to be too restrictive. Such studies propose models where this assumption is relaxed by making these transition probabilities conditional on observable variables that belong to the available information set. Building on the above discussion, another key feature of this work is to make p11 and p22 conditional on the absolute value of lagged spot–futures price deviations. The idea of time-varying transition probabilities used in this study is described as follows. Theoretically, divergence between spot and futures prices will trigger arbitrage trading between them. This investigation hypothesizes that arbitrage behavior between the spot and futures markets increases the volatility of both markets. Consequently, this study uses the absolute spot–futures price deviation and assigns a greater weight to more recent market momentum to create the following proxy variable to act as a volatility-indicator variable for spot–futures markets: ∗ Z t−1 = 36 ·|z t−1 |+ 26 ·|z t−2 |+ 16 ·|z t−3 | (15) ∗ represents the average value of the absolute spot–futures price deviation over the last where Z t−1 three days and the weighting of the most recent, second most recent and third most recent days follows the pattern 3:2:1.¶ Finally, to ensure that the estimated transition probabilities remain within the defined range, namely 1 > pi j > 0, for i( j) = 1 or 2, this study uses a logistic function defined as follows: ∗ )= p11,t = p(kt = 1|kt−1 = 1, Z t−1 p22,t = ∗ p(kt = 2|kt−1 = 2, Z t−1 )= ∗ ) exp( q0 + q1 · Z t−1 ∗ ) 1+exp( q0 + q1 · Z t−1 ∗ ) exp( p0 + p1 · Z t−1 (16) ∗ ) 1+exp( p0 + p1 · Z t−1 Apparently, a system with constant transition probabilities corresponds to the restriction of q1 = ∗ variable can be inferred p1 = 0. Moreover, the type of variance-tuning news contained in the Z t−1 from the movements in p11,t and p22,t . For example, if p22,t increases and p11,t decreases when ∗ Z t−1 increases, then the transition probabilities from ‘high’ to ‘high’ and from ‘low’ to ‘high’ variance states both rise (i.e. 1− p11,t increases). Regardless of the state of the market at time t −1, the probability of being in the high-volatility state at time t increases. In this case, the information ∗ contained in Z t−1 is high volatility news. More specifically, the high-volatility-news content of ∗ Z t−1 is measured as q1 and p1 , which have negative and positive signs, respectively. Let rt = (Ft , St ) be a (2X1) vector containing the return rates on futures and spot prices. Following Ramchand and Susmel [23, 24], Li and Lin [25], Li [26] and among many others, we ¶ If spot and futures prices differ, arbitrage trading between the spot and futures markets will be triggered regardless of whether the price deviation is positive or negative. Consequently, this study adopts the absolute value of spot–futures price deviation as an indicator of market variances. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb 702 M.-Y. L. LI assume the return rates following the normal distribution and thus calculate the log likelihood of the observed data : L() = T log f (rt |t−1 ; ) (17) t=1 where t−1 refers to the information available at time t −1 and is a vector of population parameters containing the unknown elements of q0 , q1 , p0 , p1 , A , U , f1 , f2 , s1 , s2 ff1 , . . . , ffp , sf ss s s s s fs fs f f f f fs sf ss f s fs 1 , . . . , q , 1 , . . . , p , 1 , . . . , q , 0 , 1 , . . . , m , 0 , 1 , . . . , m , g1 , g2 , g1 , g2 , 1 , 2 . Given s f the restrictions of g1 = 1, g1 = 1, p11,t + p12,t = p21,t + p22,t = 1, 0< p11,t and p22,t <1, the log likelihood can then be maximized numerically with respect to . Furthermore, the regime variables of kt in the model are unobservable; however, one can still use the observed data and the resulting ∧ maximum likelihood estimates to estimate the specific regime probabilities at each point in time.∗∗ Notably, this investigation sets two outcomes for the discrete state variable to represent high and low volatility regimes and the m order of prior-period error squares in the conditional variances (see Equations (11) and (12)). Thus, it is necessary to consider 2(m+1) possible states for each date. For both the VECM and MS-VECM, this work sets the order of auto-regression for the futures and spot returns as unity, namely p = q = 1 (see Equations (2) and (3) for the VECM and Equations (7) and (8) for the VECM, respectively). Furthermore, in the MS-VECM, the number of orders in ARCH is set as two, namely m = 2 (see Equations (11) and (12)).†† Finally, this investigation uses OPTIMUM, a package from GAUSS, in addition to the built-in BFGS algorithm functions, to yield the negative minimum likelihood function.‡‡ Compared with the benchmark model, the VECM with constant parameters, the MS-VECM designed in this study has two key advantages. First, the variance setting in the bivariate SWARCH model is both time and state varying. More specifically, the variance settings in the bivariate SWARCH model (see Equations (11) and (12)) use the conventional ARCH (m) process to characterize the variance dynamics of the low volatility regime, namely regime I. Moreover, the discrete jump process: from g1f (= 1) to g2f (>1) for futures returns, and from g1s (= 1) to g2s (>1) for spot returns, is employed to capture structural change and thus to effectively reduce the severity of the problem of high volatility persistence characteristic of conventional GARCH/ARCH models. It must be noted that our MS-VECM uses the Markov-switching mechanism to control the structural changes in return volatility during the test period and hence effectively mitigate the non-normality problems of the return rates. ∗∗ More specifically, when the information set used for estimation includes signals dated up to time t, the regime probability is a filtering probability. It is also possible to use the overall sample period information set to estimate the state probability at time t. The probability is denoted as a smoothing probability. In contrast, a predicting probability denotes the regime probability for an ex ante estimation, with the information set including signals dated up to period t −1. †† Even with this simple structure involving two lagged ARCH components, there are 24 parameters requiring estimation. A more general structure with higher-order ARCH term could increase the number of parameters to be estimated. Furthermore, similar to Hamilton and Susmel [22] and Li and Lin [25], this study also found that the higher-order ARCH parameter estimates in the SWARCH model generally do not differ significantly from zero. To save space, this study does not report the results of the higher lag order setting. ‡‡ The algorithm of Boyden, Fletcher, Goldfarb, and Shanno (BFGS) can effectively yield the maximum value of the nonlinear likelihood functions, as demonstrated by Luenberger [30]. Additionally, this investigation randomly generates 50 sets of initial values and derives the ML function value for each of them. The mapped converged measure of the greatest ML function value is then used for the parameter estimation. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb RELATIONSHIP BETWEEN SPOT AND FUTURES MARKETS 703 Second, the state-varying system used in this study dispels the unrealistic assumption of a uniform correlation between futures and spot markets inherent in the conventional VECM. The covariance setting in the MS-VECM corresponds to two correlation measures (see Equation (13)); furthermore, the parameters of the EC terms differ according to the volatility state phrase (see Equations (7) and (8)). It must be noted that the analytical techniques used in this study are related to the framework employed by Krolzig et al. [31] and Francis and Owyang [32]. As in this study, these authors establish frameworks that combine a conventional VECM with Markov-switching models to study dynamic transmission processes among certain macroeconomic variables. The analysis in this study differs from theirs with respect to the following points. First, owing to their focus on the dynamic processes that are involved in macroeconomic variables, some population parameters in their models change according to business cycle phases. In contrast, this work identifies the dynamic processes of financial variables, such as spot and futures returns, and creates a nonlinear model from the perspective of a high/low volatility state rather than that of a high/low growth regime. Consequently, this investigation emphasizes interrelationship dynamics between the spot and futures markets across various variance regimes. 3. EMPIRICAL RESULTS AND INTERPRETATION 3.1. Data and preliminary comparative analysis: mature versus emerging The study data comprise the daily stock index futures and spot prices for three mature markets: the U.S. S&P500, the U.K. FTSE100 and the German DAX and two emerging markets: the Brazilian BOVESPA and the Hungarian BSI. The data cover from the period April 3, 1995, through December 12, 2005, and include 2805 daily observations. All the stock prices are stated in dollar terms. Table I summarizes several basic statistics on the logarithmic first difference of the stock index futures and spot prices for all sample markets. Initially, the return mean is close to 0, the skewness coefficient is considerably unequal to zero, and the kurtosis coefficient significantly exceeds three for all cases. However, the two sets of markets differ as follows. First, the absolute values of the minimum/maximum and variance of daily index returns for the two emerging markets significantly exceed those for the three mature markets. Taking the Brazilian and U.S. futures markets as two representative examples, the estimates of minimum/maximum and variance are −19.20%/21.19% and 5.89% (−7.76%/5.75% and 1.33%) for the case of Brazil (U.K.), respectively. This finding is consistent with the notion that the distribution of daily index returns is more/less divergent for the emerging/mature stock markets. Second, regarding the value of the kurtosis coefficient, the values for the two emerging markets clearly exceed those of the three mature markets. Briefly, 10.21 and 21.03 (6.89, 5.82 and 6.45) are the values for the futures markets of Brazil and Hungary (U.S., U.K. and Germany), respectively. Generally, the term used to describe probability distributions with a kurtosis exceeding that of the normal distribution is leptokurtosis. Furthermore, leptokurtosis can also be considered to be a measure of the fatness of the tails of distribution. This finding is consistent with the notion that most markets, particularly poorly developed markets, display more extreme movements than would be predicted by a normal distribution. Because one feature of the nonlinear Markov-switching model is to provide improved description of extreme tail events, this investigation proposes that Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb Copyright q 2008 John Wiley & Sons, Ltd. 0.0325 −0.1259 −7.7621 5.7549 1.3316 6.8903 0.0325 −0.1032 −7.1074 5.5709 1.2151 6.4529 Spot 0.0204 −0.1144 −6.0625 5.9506 1.2974 5.8240 Futures 0.0325 −0.1259 −7.7621 5.7549 1.3316 6.8903 Spot U.K. FTSE100 0.0325 −0.1032 −7.1074 5.5709 1.2151 6.4529 Futures 0.0204 −0.1144 −6.0625 5.9506 1.2974 5.8240 Spot German DAX 30 0.0874 0.0191 −19.1963 21.1942 5.8853 10.2070 Futures 0.0873 0.4042 −17.2292 28.8176 5.0790 17.6112 Spot Brazil Bovespa 0.0985 −0.5751 −19.6778 18.7728 3.8450 21.0275 Futures 0.1008 −0.9189 −18.0340 13.6162 3.0730 16.9172 Spot Hungary BSI Notes: 1. The study data comprise the daily stock index futures and spot prices for three mature markets: the U.S. S&P500, the U.K. FTSE100 and the German DAX and two emerging markets: the Brazilian BOVESPA and the Hungarian BSI. The data cover the period April 3, 1995 through December 12, 2005, and include 2805 daily observations. All the stock prices are stated in dollar terms. 2. The return mean is close to 0, the skewness coefficient is unequal to zero, and the kurtosis coefficient significantly exceeds three for all cases. However, the two sets of markets differ as follows. First, the absolute values of the minimum/maximum and variance of daily index returns for the two emerging markets significantly exceed those for the three mature markets. Taking the Brazilian and U.S. futures markets as two representative examples, the estimates of minimum/maximum and variance are −19.20%/21.19% and 5.89% (−7.76%/5.75% and 1.33%) for the case of Brazil (U.K.), respectively. This finding is consistent with the notion that the distribution of daily index returns is more/less divergent for the emerging/mature stock markets. 3. Regarding the value of kurtosis coefficient, the values for the two emerging markets clearly exceed those of the three mature markets. Briefly, 10.21 and 21.03 (6.89, 5.82 and 6.45) are for the futures markets of Brazil and Hungary (U.S., U.K. and Germany), respectively. Generally, the term used to describe probability distributions with a kurtosis exceeding that of the normal distribution is leptokurtosis. Furthermore, leptokurtosis can also be considered to be a measure of the fatness of the tails of distribution. This finding is consistent with the notion that most markets, particularly poorly developed markets, display more extreme movements than would be predicted by a normal distribution. Because one feature of the nonlinear Markov-switching model is to provide improved description of extreme tail events, this investigation proposes that the higher/lower possibility of rare price movements for the developing/developed markets may be associated with more/less pronounced feasibility of the Markov-switching model. Mean Skewness Minimum value Maximum value Variance Kurtosis Futures U.S. S&P500 Table I. Summary statistics of return rates of futures and spot prices. 704 M.-Y. L. LI Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb RELATIONSHIP BETWEEN SPOT AND FUTURES MARKETS 705 the higher/lower possibility of rare price movements for the developing/developed markets may be associated with more/less pronounced feasibility of the Markov-switching model. Table II lists the unit root and cointergration tests for the stock index futures and spot markets for all markets. The empirical results reveal that both series of futures and spot indices are nonstationary in all cases. However, the logarithmic first difference of stock index including futures and spot is stationary. Additionally, the cointegration test indicates that the EC term, namely Z t , for the stock index futures and spot is stationary, a finding that is consistent for all cases. That is, the cointegration relationship of the price series for the futures and spot markets holds. 3.2. Parameter estimates of the conventional VECM Table III lists the parameter estimates of VECM. As part of its focus on the spot–futures disequilibrium adjustment process, this investigation focuses on discussing two parameters of the EC term, namely f and u in the futures and spot return equations, respectively. This study yields the following results. First, for the U.S. S&P500 market, the f and s estimates are insignificant and significantly positive at the 1% level, respectively. This results shows that when the spot–futures relationship is perturbed, the spot is the market through which the disequilibrium adjustment is made; in other words, the future price leads the spot market in price discovery for the U.S. S&P 500 market, as demonstrated by prior studies. By contrast, the f estimate is significantly positive at the 1% level, but the s estimate is insignificant in the case of the U.K. FTSE100. This result indicates that the spot price provides an information advantage and leads the futures market in price discovery in the case of the U.K. FTSE100. Subsequently, for the German DAX 30 market and the two emerging markets, both the f and s estimates are significant in the 5% level at least, suggesting a bidirectional EC. However, the EC term in the futures return equation is greater in magnitude than that of the spot return equation: −0.3205 versus 0.0585 for the German market, −0.1657 versus 0.0325 for the Brazil market and −0.1499 versus 0.0310 for the Hungarian market. The finding denotes that it is the futures price that makes a greater adjustment in order to reestablish the equilibrium; that is, most of the price discovery takes place at the spot market. To conclude, using conventional VECM with constant parameters to examine the spot–futures price discovery process addresses inconsistent conclusions among various stock markets.§§ 3.3. Asymmetric price adjustment process across various variance states Table IV shows the parameter estimates of the MS-VECM. By considering various market variance states, this work defines a system involving state-varying parameters for the EC term and statevarying correlation measures. The present results show that all the state-varying estimates display significant divergence across various variance states in all cases. To test the null hypothesis for the situation involving the identical EC term parameter and correlation measure, this work first estimates an unrestricted MS-VECM with all state-varying parameters, namely f1 , f2 , s1 , s2 , fs 1, §§ As is well known, in the case of mature and liquid markets, such as the three markets selected in this study, the price adjustment process between spot and futures markets takes place rapidly. Therefore, some studies, such as Herbst et al. [1], Chan et al. [33], Koch [7], Martens et al. [34] and Tse [10], adopt high-frequency data to capture the intraday price discovery process in spot–futures markets. Undeniably, by employing daily stock prices, data with a lower frequency, this study might fail to account for any intraday simultaneity processes in the spot–futures market. However, the present results show that the price disequilibrium between spot and futures markets is pronounced even when investigated on the basis of daily data. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb Copyright q 2008 John Wiley & Sons, Ltd. −0.8717 −0.8903 −14.6333∗ −13.9992∗ −7.5865∗ Spot Spot −1.8953 −1.8926 −26.4927∗ −26.2089∗ −8.2017∗ Futures U.K. FTSE100 Spot −1.7274 −1.7131 −23.7369∗ −23.8241∗ −11.6975∗ Futures German DAX 30 Spot −2.533 −2.488 −13.107∗ −12.991∗ −10.4895∗ Futures Brazil Bovespa Spot −2.975 −3.051 −12.080∗ −11.629∗ −7.6286∗ Futures Hungary BSI Notes: 1. The unit test for the log levels and return rates of futures and spot prices is Dickey and Fuller (1979)’s augmented Dickey–Fuller tests (ADF) for unit roots. Cointegration tests are based on Engle–Granger [27] procedure. 2. This study uses four-order lag lengths for the ADF test. Additionally, the conclusion from the unit root and cointegration tests is robust for the setting with various lag length numbers. 3. The ∗ denotes the significance at the level of 1%. The present results show that both series of futures and spot prices are non-stationary for all cases. However, the logarithmic first difference of stock price including futures and spot is stationary. The cointegration test indicates that the EC term, namely Z t , for the futures and spot prices is stationary for all cases. Namely, the cointegration relationship of the price series for the futures and spot markets holds. 4. The data source is consistent with Table I. Log levels % Returns EC Term Futures U.S. S&P500 Table II. Unit root tests and cointegration tests of futures and spot prices. 706 M.-Y. L. LI Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb Copyright q 2008 John Wiley & Sons, Ltd. 0.9773 (0.0008)∗∗∗ −4189.3936 0.9748 (0.0009)∗∗∗ −4411.4506 0.0202 (0.0184) 0.0167 (0.0338) 0.2524 (0.0530)∗∗∗ −0.2534 (0.0555)∗∗∗ 1.0851 (0.0145)∗∗∗ 0.0671 (0.0580) 1.1377 (0.0152)∗∗∗ 0.1268 (0.0806) 1.1528 (0.0154)∗∗∗ 0.0330 (0.0201) 0.0859 (0.0370)∗∗∗ 0.1931 (0.0735)∗∗∗ −0.2156 (0.0770)∗∗∗ 1.0997 (0.0147)∗∗∗ 0.0196 (0.0191) −0.1118 (0.0354)∗∗∗ −0.0651 (0.0556) 0.0328 (0.0209) −0.0420 (0.0384) −0.1479 (0.0768)∗ U.K. FTSE100 −5998.2639 0.9774 (0.0008)∗∗∗ 0.0360 (0.0155)∗∗ 0.0585 (0.0308)∗ 0.4471 (0.0665)∗∗∗ −0.4513 (0.0661)∗∗∗ 1.5287 (0.0204)∗∗∗ −0.1601 (0.0666)∗∗∗ 1.5422 (0.0206)∗∗∗ 0.0362 (0.0157)∗∗ −0.3205 (0.0237)∗∗∗ 0.1505 (0.0670)∗∗∗ German DAX30 −9804.6929 0.9316 (0.0025)∗∗∗ 0.0798 (0.0204)∗∗∗ 0.0325 (0.0102)∗∗∗ −0.0022 (0.0181) 0.0430 (0.0175)∗∗∗ 2.2470 (0.0300)∗∗∗ 0.2726 (0.0249)∗∗∗ 2.3744 (0.0317)∗∗∗ 0.0712 (0.0283)∗∗∗ −0.1657 (0.0029)∗∗∗ −0.1803 (0.0043)∗∗∗ Brazil Bovespa −10225.294 0.7286 (0.0089)∗∗∗ 0.0938 (0.0308)∗∗∗ 0.0310 (0.0151)∗∗ 0.0590 (0.0168)∗∗∗ 0.0062 (0.0076) 1.7473 (0.0233)∗∗∗ 0.2791 (0.0175)∗∗∗ 1.8833 (0.0251)∗∗∗ 0.0877 (0.0323)∗∗∗ −0.1499 (0.0158)∗∗∗ −0.1815 (0.0212)∗∗∗ Hungary BSI Notes: 1. Please refer to this study’s Equations (2) and (3) for the model specification of the futures and spot return equations. This study sets the lag number order in the VECM to unity, namely p = 1 and q = 1. 2. Please refer to this study’s Equations (4) and (5) for the specification of covariance matrix. 3. The value in the parenthesis denotes the standard error of parameter estimate. 4. The ∗∗∗, ∗∗ and ∗ denote the significance in 1, 2.5 and 5%, respectively. 5. Notably, the conventional VECM model suffers two limitations. First, all the elements in the variance–covariance matrix, namely the H matrix, are constant. Therefore, the hedge ratio derived by the VECM is also a constant. Numerous previous studies have pointed out that stock return variances are heterogeneous. Moreover, this work also demonstrated a lack of uniformity in the correlation coefficient between futures and spot returns. Second, the conventional VECM assumes stability of convergence speed toward the spot–futures equilibrium relationship using constant f and s parameters in Equation (2). 6. The present results show that using the conventional VECM with constant parameters to examine spot–futures price discovery process addresses inconsistent conclusions among various stock markets. 7. Undeniably, by employing daily stock prices, data with a lower frequency, this study might fail to account for any intraday simultaneity processes in the spot–futures market. However, the present results show that the price disequilibrium between spot and futures markets is pronounced even when investigated on the basis of daily data. Log-likelihood fs Correlation s s sf 1 ss 1 s Spot return equation f f ff 1 fs 1 f Futures return equation U.S. S&P500 Table III. Parameter estimates of the VECM for the system with constant parameters. RELATIONSHIP BETWEEN SPOT AND FUTURES MARKETS 707 Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb Copyright q 2008 John Wiley & Sons, Ltd. Log-likelihood fs 1 fs 2 Correlations g2s s s1 s2 sf 1 ss 1 s0 s1 s2 Spot return equation g2f f f1 f2 ff 1 fs 1 f0 f1 f2 Futures return equation q0 p0 q1 p1 Transition probability 0.9809 (0.0012)∗∗∗ −3395.6947 −3735.1549 0.9771 (0.0014)∗∗∗ 0.2154 (0.0472)∗∗∗ −0.1980 (0.0536)∗∗∗ 0.4125 (0.0230)∗∗∗ 0.0963 (0.0173)∗∗∗ 0.0220 (0.0028)∗∗∗ 5.1793 (0.3551)∗∗∗ 0.0444 (0.0151)∗∗∗ 0.0506 (0.0250)∗∗ −0.0313 (0.0391) 0.0949 (0.0175)∗∗∗ 0.0152 (0.0022)∗∗∗ 5.4801 (0.3729)∗∗∗ 0.9802 (0.0012)∗∗∗ 0.9732 (0.0017)∗∗∗ 0.2315 (0.0772)∗∗∗ −0.2484 (0.0800)∗∗∗ 0.5231 (0.0303)∗∗∗ 0.0328 (0.0131)∗∗∗ 0.0960 (0.0243)∗∗∗ 4.4510 (0.3011)∗∗∗ 0.059 (0.0167)∗∗∗ 0.1087 (0.0056)∗∗∗ 0.0261 (0.0441) 0.0287 (0.0114)∗∗∗ 0.0061 (0.0024)∗∗∗ 4.924 (0.3345)∗∗∗ 0.0986 (0.0539)∗ 0.4367 (0.0247)∗∗∗ −0.2224 (0.0328)∗∗∗ −0.0789 (0.0475)∗ −0.1781 (0.0532)∗∗∗ −0.1348 (0.0794)∗ 0.1182 (0.0822) 0.5361 (0.0330)∗∗∗ 0.0370 (0.0156)∗∗∗ −0.0168 (0.0251) 3.0023 (0.1338)∗∗∗ 2.2332 (0.1828)∗∗∗ −0.3512 (0.2353) 0.2585 (0.2389) U.K. FTSE100 0.0507 (0.0171)∗∗∗ 0.0401 (0.0663) 2.8359 (0.1799)∗∗∗ 2.6294 (0.3148)∗∗∗ −0.1588 (0.1921) 0.0835 (0.5011) U.S. S&P500 −5280.0169 0.9761 (0.0014)∗∗∗ 0.9879 (0.0012)∗∗∗ 0.4915 (0.1135)∗∗∗ −0.4855 (0.1138)∗∗∗ 0.8399 (0.0546)∗∗∗ 0.0425 (0.0099)∗∗∗ 0.0267 (0.0069)∗∗∗ 4.4886 (0.3447)∗∗∗ 0.0873 (0.0223)∗∗∗ 0.0425 (0.0130)∗∗∗ 0.1186 (0.1897) 0.0313 (0.0094)∗∗∗ 0.0148 (0.0026)∗∗∗ 4.3881 (0.3387)∗∗∗ −0.2176 (0.1156)∗ 0.8811 (0.0586)∗∗∗ −0.4014 (0.1747)∗∗∗ 0.2159 (0.1152)∗ 0.0790 (0.0230)∗∗∗ −0.0692 (0.0665) 2.2534 (0.2364)∗∗∗ 1.6440 (0.5174)∗∗∗ −0.8481 (0.5647) 0.6904 (1.3667) German DAX30 −8274.7663 0.8866 (0.0101)∗∗∗ 0.9799 (0.0010)∗∗∗ 0.0365 (0.0119)∗∗∗ 0.0073 (0.0082) 2.5115 (0.0977)∗∗∗ 0.1690 (0.0173)∗∗∗ 0.0912 (0.0124)∗∗∗ 3.8483 (0.3370) *** 0.1391 (0.0261)∗∗∗ 0.0616 (0.0225)∗∗∗ −0.1330 (0.0693)∗ 0.1523 (0.0171)∗∗∗ 0.0320 (0.0081)∗∗∗ 4.0690 (0.3455)∗∗∗ 0.1858 (0.0128)∗∗∗ 2.8773 (0.1112)∗∗∗ −0.5749 (0.0588)∗∗∗ −0.1388 (0.0195)∗∗∗ 0.0690 (0.0265)∗∗∗ 0.0250 (0.0238) 3.0684 (0.1704)∗∗∗ −0.0326 (0.0377) −0.6903 (0.1111)∗∗∗ 0.2562 (0.0809)∗∗∗ Brazil Bovespa Table IV. Parameter estimates of MS-VECM for the system with state-varying variances. −8018.6684 0.6137 (0.0151)∗∗∗ 0.9981 (0.0002)∗∗∗ −0.0035 (0.0013)∗∗∗ 0.2136 (0.0170)∗∗∗ 1.1082 (0.0658)∗∗∗ 0.2419 (0.0197)∗∗∗ 0.0913 (0.0211)∗∗∗ 2.3807 (0.1733)∗∗∗ 0.0907 (0.0222)∗∗∗ 0.0371 (0.0123)∗∗∗ 0.0249 (0.0160) 0.2545 (0.0205)∗∗∗ 0.09517 (0.0312)∗∗∗ 3.0399 (0.2211)∗∗∗ 0.2274 (0.0153)∗∗∗ 1.0573 (0.0637)∗∗∗ −0.234 (0.0181)∗∗∗ −0.0196 (0.0069)∗∗∗ 0.0695 (0.0217)∗∗∗ 0.0387 (0.0123)∗∗∗ 2.6352 (0.1943)∗∗∗ 1.9576 (0.1549)∗∗∗ −1.1971 (0.1734)∗∗∗ 0.0503 (0.0652) Hungary BSI 708 M.-Y. L. LI Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb Copyright q 2008 John Wiley & Sons, Ltd. 3.0716 0.5412 12.1321∗∗∗ 2.8832 0.4458 10.8594∗∗∗ U.K. FTSE100 32.1638∗∗∗ 3.4484 0.1846 German DAX30 202.3128∗∗∗ 33.3318∗∗∗ 4.6206 Brazil Bovespa 902.247∗∗∗ 53.4742∗∗∗ 0.29 Hungary BSI using L(H A ). Three restricted models are then estimated by imposing the assumption of a single parameter, including (1) f1 = f2 = f , fs (2) s1 = s2 = s and (3) fs 1 = 2 = . The various restrictions are used to obtain the log-likelihood function of each restricted model, L(H0 ). This function can then be used to calculate a likelihood ratio test, LR = −2[L(H0 )− L(H A )]. In terms of the null hypothesis, each LR statistics exhibits a 2 distribution with one degree of freedom. 3. The ∗∗∗, ∗∗ and ∗ denote the significance in 1, 2.5 and 5%, respectively. The data source is consistent Table I and other notations are consistent with Table III. Notes: 1. Please refer to this study’s Equations (6)–(16) for the model specification of the MS-VECM established by this study. In the MS-VECM, this work sets the order of auto-regression for the futures and spot returns as unity, namely p = q = 1. Furthermore, similarly to Hamilton and Susmel [22] and Li and Lin [25], this study sets the number of orders in ARCH to two, namely m = 2. 2. To test the null hypothesis for the situation involving the identical EC term parameter and correlation measure, this work first estimates an fs unrestricted MS-VECM with all state-varying parameters, namely f1 , f2 , s1 , s2 , fs 1 , 2 , while denoting the model log-likelihood function fs fs 1 = 2 LR statistics f1 = f2 s1 = s2 U.S. S&P500 Table IV. Continued. RELATIONSHIP BETWEEN SPOT AND FUTURES MARKETS 709 Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb 710 M.-Y. L. LI fs 2 , while denoting the model log-likelihood function using L(HA). Three restricted models are then estimated by imposing the assumption of a single parameter, including (1) f1 = f2 = f , (2) fs s1 = s2 = s and (3) fs 1 = 2 = . The various restrictions are used to obtain the log-likelihood function of each restricted model, L(H0 ). This function can then be used to calculate a likelihood ratio, LR = −2[L(H0 )− L(H A )]. In terms of the null hypothesis, each LR value exhibits a 2 distribution with one degree of freedom. Table IV lists the following empirical findings. First, the g2f and g2s estimates markedly exceed unity in all cases. Taking the U.S. S&P500 market as an example, the g2f and g2s estimates are 4.924 and 4.4510 and have standard deviations of 0.3345 and 0.3011 for the futures and spot returns, respectively. This finding is consistent with that of the volatility of regime II being 4.924 times that of regime I for the futures market and 4.4510 times that of regime I for the spot market. Furthermore, the confidence levels of estimates of g2f and g2s do not overlap with unity, namely the values of g1f and g1s are at a level of confidence of 99%, and thus this investigation confidently identifies regime II as a high volatility state and regime I as a low one. Second, the f2 estimate for the high volatility state in the futures return equation is significantly negative in all cases, whereas the f1 estimate for the low volatility state is insignificant for most markets, with the exception of the Hungary BSI market. Conversely, in the spot market, the s1 estimate for the low volatility state is significantly positive for all cases; but the s2 for the low volatility state is insignificant in most markets, with the exception of the Brazil Bovespa market. Notably, the f1 estimate in the Hungary BSI and the s2 estimate in the Brazil Bovespa market, the two exceptions shown above are significant; nonetheless, their signs are inconsistent with the hypothesis.¶ ¶ Taking the U.S. S&P500 market as a representative example, the f1 and f2 estimates in the futures return equation are 0.0401(t-stat. = 0.0663) and −0.1781 (t-stat. = 3.3496), respectively; by contrast, the s1 and s2 estimates in the spot return equation are 0.1087 (t-stat. = 19.4107) and 0.0261 (t-stat. = 0.5918), respectively. These findings show that the spot–futures disequilibrium adjustment process depends primarily on the futures market during a high variance state and on the spot market in the low variance state. In other words, the futures price leads the spot price in price discovery during stable periods; conversely, during volatile periods, the price discovery takes place on the spot market. Furthermore, this finding is robust for all markets. Based on this finding, this study states that pooling data together without considering the impact of variance-switching processes is one of the reasons for the aforementioned inconsistency in the spot–futures price discovery process implied by the conventional VECM. The different types of price discovery processes across various variance states are detailed below. During the low variance state, this study shows that futures markets are more informationally efficient than spot markets. This finding is consistent with earlier studies. However, this study indicates that spot markets serve the price discovery role during the high variance state. ¶ ¶ As hypothesized in this study, when the deviation in spot–futures prices, Z t−1 >0, then Ft should decrease, and St should increase to return the price relationship to the long-run equilibrium, the opposite being the case when Z t−1 <0. Therefore, the sign of f2 and f1 estimates should be negative, whereas the sign of s2 and s1 estimates should be positive. However, the f1 estimate in the Hungary BSI and the s2 estimate in the Brazil Bovespa market are positive and negative, respectively. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb RELATIONSHIP BETWEEN SPOT AND FUTURES MARKETS 711 As is well known, the high variance state in stock markets frequently corresponds to a recession state (see Chen et al. [35], Schwert and Seguin [36], Chen [37] and Hamilton and Lin [38] for related discussion). Moreover, when stock investors face a extremely bear market condition, they prefer to employ an asset reallocation strategy rather than using stock index futures contracts to control the risk of their stock positions; consequently, the spot market, rather than the futures market, serves as a key function. This phenomenon provides one of explanations of why the spot market is more informationally efficient in the price discovery process than the futures market during the high variance state. 3.4. A comparative analysis: high/low variance versus outer/central states While based on a similar framework consisting of state variables, Dwyer et al. [39], Martens et al. [34], Li [40, 41] consider transaction costs and employ the threshold models to define the outer/central regimes; by contrast, this study emphasizes the high/low variance states generated by Markov-switching approaches. A comparative analysis between the two state-varying systems is also examined below. Taking the U.S. S&P500 and the Brazil Bovespa markets as two representative examples, Figures 1 and 2 show the logarithmic futures and spot prices (Ft and St ), spot–futures price deviations (Z t ) and logarithmic first differences of futures and spot prices (Ft and St ), as well as the filtering probability of a high volatility regime ( p(st = 2|t )), for the U.S. S&P500 and the Brazil Bovespa markets, respectively. Clearly, futures and spot price returns appear to be volatile during certain periods, and the Markov-switching system effectively depicts their variance-turning processes. This work identifies the spot–futures market as a high variance state if the corresponding filtering probability exceeds 0.5, that is p(st = 2|t )>0.5; otherwise, it exhibits a low variance state. Furthermore, this study calculates the mean of the absolute spot–futures price deviations, |Z t |, and adopts it as the magnitude of spot–futures price deviations, and the results are listed in Table V. First, as shown in Table V, the observation percentage of a low variance state exceeds that of a high variance in most markets, with the exception of the Hungary BSI market. Second, the magnitude of spot–futures price deviations during high variance periods markedly exceeds that of the low variance state in all cases. Returning to Table IV, our results show that during a high variance state, the spot–futures disequilibrium adjustment process depends primarily on the futures market, accordingly, the f2 estimate is significantly negative; whereas, in a low variance state, the process of spot–futures prices adjustment is mostly triggered by the spot market and the corresponding s1 estimate is significantly positive. However, as shown in Table IV, the f2 estimate is markedly larger in absolute magnitude than the s1 estimate in all cases: 0.1781 versus 0.1087 for the U.S. S&P 500 market; −0.2224 versus 0.0506 for the U.K. FTSE100 market; −0.4014 versus 0.0425 for the German DAX30 market; −0.5749 versus 0.0616 for the Brazil Bovespa market; −0.234 versus 0.0371 for the Hungary BSI market. This finding shows that the scale of the price adjustment occurring in the futures markets during a high variance state is greater than that occurring in the spot markets during a low variance state. The findings presented above show an implicit linkage between the high/low variance states obtained with Markov-switching models and the outer/central regimes generated by the threshold This finding is consistent with the notion that the low volatility state is more persistent than the high volatility state. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb 712 M.-Y. L. LI Futures Spot Futures 7.6 8.0 7.4 6.0 7.2 4.0 7.0 2.0 Spot 0.0 1995/4 1996/4 1997/4 1998/4 1999/4 2000/4 2001/4 2002/4 2003/4 2004/4 2005/4 -2.0 6.8 6.6 -4.0 6.4 -6.0 6.2 -8.0 (a) 6.0 1995/4 1996/4 1997/4 1998/4 1999/4 2000/4 2001/4 2002/4 2003/4 2004/4 2005/4 (c) 1.0 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.21995/4 1996/4 1997/4 1998/4 1999/4 2000/4 2001/4 2002/4 2003/4 2004/4 2005/4 (b) -0.4 -0.6 -0.8 -1.0 -1.2 -1.4 -1.6 -10.0 0.8 0.5 0.3 (d) 0.0 1995/4 1996/4 1997/4 1998/4 1999/4 2000/4 2001/4 2002/4 2003/4 2004/4 2005/4 Figure 1. The logarithmic futures and spot prices, spot–futures price deviation and logarithmic first difference of futures and spot Prices, as well as the filtering probability of a high volatility regime: the case of the U.S. S&P500 market: (a) logarithmic futures and spot prices (Ft and St ); (b) spot–futures price deviations (Z t ); (c) return rates on futures and spot prices (Ft and St ) and (d) filtering probability of high volatility state ( p(st = 2|t )). models of Dwyer et al. [39] and Martens et al. [34]. It must be noted that the asymmetric dynamics involved in the high/low variance states found by this study are unable to be entirely attributed to the usual transaction costs since these are the same for both high and low variance conditions. Furthermore, the state-varying correlations are designed in this study to capture the non-constant co-movements between futures and spot markets across various variance states (see Equation (13)). As shown in Table IV, the correlation between the spot and futures markets for the high variance fs state, namely fs 2 , is lower than that for the low variance state, namely 1 , in all cases and the LR fs statistic for the null hypothesis of fs 2 = 1 is significant at the 1% level for all cases. Moreover, the fs estimate implied by the VECM model with a single correlation measure (see Table III) is lower fs than fs 1 and higher than 2 in all cases. This finding indicates that the conventional VECM, which ignores the nonlinear adjustment process in spot–futures prices across various variance states, underestimates the degree of co-movement between the spot and futures markets in periods of low variance stats and overestimates it during high variance This state-varying correlation phenomenon is detailed below. First, as shown in the findings presented above, the scale of the price adjustment process occurring in the futures markets during a high variance state is higher than that accruing in the spot markets during a low variance state. The process of price adjustment between spot and futures markets is restated below: simultaneous short selling of spot positions and purchasing of futures positions when the mispricing term, Z t−1 , is negative, the opposite being the case when Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb 713 RELATIONSHIP BETWEEN SPOT AND FUTURES MARKETS Futures Spot Futures Spot 40.0 11.0 10.5 30.0 10.0 20.0 9.5 10.0 9.0 0.0 1995/4 1996/4 1997/4 1998/4 1999/4 2000/4 2001/4 2002/4 2003/4 2004/4 2005/4 8.5 8.0 -10.0 7.5 (a) -20.0 7.0 1995/4 1996/4 1997/4 1998/4 1999/4 2000/4 2001/4 2002/4 2003/4 2004/4 2005/4 (c) -30.0 1 12.0 8.0 0.75 4.0 0.5 0.0 1995/4 1996/4 1997/4 1998/4 1999/4 2000/4 2001/4 2002/4 2003/4 2004/4 2005/4 -4.0 0.25 -8.0 (b) (d) -12.0 0 1995/4 1996/4 1997/4 1998/4 1999/4 2000/4 2001/4 2002/4 2003/4 2004/4 2005/4 Figure 2. The logarithmic futures and spot prices, spot–futures price deviation and logarithmic first difference of futures and spot prices, as well as the filtering probability of a high volatility regime: the case of the Brazil Bovespa Market: (a) logarithmic futures and spot prices (Ft and St ); (b) spot–futures price deviations (Z t ); (c) return rates on futures and spot prices (Ft and St ) and (d) filtering probability of high volatility state ( p(st = 2|t )). Z t−1 is positive. Clearly, this price adjustment process implied by the arbitrage behavior between spot and futures markets causes spot and futures prices to tend to move in opposite directions and thus reduces the degree of the co-movements between them. Moreover, during a high variance state, the spot–futures disequilibrium is more influential; that is, market investors are more sensitive to this spot–futures price deviation owing to their tendency toward risk aversion. Consequently, this mental behavior and the higher magnitude of spot–futures price divergences involved in the high variance state speed up the process of spot–futures price adjustment and weaken the correlation between spot and futures markets. 3.5. A comparative analysis: mature versus emerging markets One key feature of this study is to not only adopt three major developed country stock markets (Germany, U.S. and U.K.) as the research sample, but also to include an analysis of two emerging country stock markets (Brazil and Hungary) and to examine the differences between the two groups. This comparative analysis of mature against emerging stock markets is meaningful. As is well known, a key feature of the nonlinear volatility-switching model is its ability to capture rare extremes that occur in the futures–spot relationship, and which strongly influence the dynamic of the relationship between spot and futures markets. Furthermore, this study posits that emerging stock markets generally experience more extreme crisis events than mature stock markets. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb Copyright q 2008 John Wiley & Sons, Ltd. 0.4301 0.3588 0.3819 Mean value of the absolute EC terms, namely |Z t | High variance state Low variance state Whole estimation period 0.4075 0.2983 0.3372 35.61 64.39 100.00 U.K. FTSE100 0.3622 0.2405 0.2939 43.81 56.19 100.00 German DAX 30 1.6238 0.9388 1.0327 13.71 86.29 100.00 Brazil Bovespa 1.8234 0.8971 1.4586 60.62 39.38 100 Hungary BSI Notes: 1. This work identifies the spot–futures market exhibiting a high variance state if the corresponding filtering probability exceeds 0.5, namely p(st = 2|t ) > 0.5; otherwise, a low variance state exists. Furthermore, this study calculates the mean value of the absolute spot–futures price deviations, namely |Z t |, and adopts it as the magnitude of spot–futures price deviations. 2. The observation percentage of a low variance state exceeds that of a high variance in most markets, with one exception being the Hungary BSI market. This finding is consistent with the notion that the low volatility state is more persistent than the high volatility state. Second, the magnitude of spot–futures price deviations during high variance periods markedly exceeds that of the low variance state in all cases. 3. The magnitude of spot–futures price deviations during high variance periods markedly exceeds that of the low variance state in all cases. Moreover, the magnitude of the deviation of spot–futures prices in the case of the two emerging markets considerably exceeds that of the three mature markets, particularly for the high variance state. Specifically, the mean values of the deviation of the spot–futures prices, |Z t |, at the high variance states are 0.4301, 0.4075, 0.3622, 1.6238 and 1.8234 for the U.S. S&P500, the U.K. FTSE100, the German DAX30, the Brazil Bovespa and the Hungary BSI markets, respectively. This investigation thus defines an unusual condition contained in the high variance state for the three mature markets; by contrast, an extreme condition involved in the high variance is identified for the two emerging markets. 32.43 67.57 100.00 Observation percentage (%) High variance state (%) Low variance state (%) Whole estimation period (%) U.S. S&P500 Table V. Comparative analysis across various variance states. 714 M.-Y. L. LI Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb RELATIONSHIP BETWEEN SPOT AND FUTURES MARKETS 715 As shown in Table V, the magnitude of the deviation of spot–futures prices in the case of the two emerging markets considerably exceeds that of the three mature markets, particularly for the high variance state. Specifically, the mean values of the deviation of the spot–futures prices, |Z t |, at the high variance states are 0.4301, 0.4075, 0.3622, 1.6238 and 1.8234 for the U.S. S&P500, the U.K. FTSE100, the German DAX30, the Brazil Bovespa and the Hungary BSI markets, respectively. This investigation thus defines an unusual condition contained in the high variance state for the three mature markets; by contrast, an extreme condition involved in the high variance is identified for the two emerging markets. A further discussion for this phenomenon is detailed below. First, this study denotes that the process of the price adjustment between spot and futures markets occurs very rapidly in mature, liquid markets, as is the case for the three mature stock markets examined in this investigation; meanwhile, for emerging, illiquid markets, such as the Brazil and Hungary markets, the disequilibrium between spot and futures prices takes longer to reduce. This phenomenon explains why the deviation of spot–futures prices in the two emerging markets, particularly in the high variance state, is considerably higher in absolute magnitude than that in the three mature markets. Second, the results presented above show that, during a high variance state, the process of spot– futures prices adjustment mostly depends on futures markets. However, examining the differences between the types of markets, as shown in Table IV, further reveals that the LR statistic for the null hypothesis of f1 = f2 is significant at 10%, a marginal significance, for the three mature markets; whereas it is significant at 1% for the two emerging markets.∗∗∗ This finding is consistent with an unusual condition involved in a high variance state for the mature markets; in contrast, it is characteristic of an extreme condition for the emerging markets, as shown in Table V. Last but not least, the two transition probability parameters: q1 and p1 , as shown in Table IV, have opposite signs for all cases. Specifically, the q1 estimate is negative, and the p1 is positive, as hypothesized by this study. This result indicates that the lagged spot–futures price deviations function as an information variable in the volatility-switching process. However, statistically speaking, the performance of this variance indicator variable is less promising for the three mature markets selected for this study. In particular, using the 5% significance as a criterion, both of the q1 and p1 estimates are insignificant for the three mature markets;††† nonetheless, for the Brazil Bovespa market, both of the q1 and p1 estimates are significant at 1%, and the q1 estimate is significant at 1% for the case of the Hungary BSI market.‡‡‡ The above findings are consistent with the notion that the process of price adjustment between spot and futures markets takes place rapidly in mature, liquid markets, as is the case for the three stock markets examined in this study. Consequently, the portion of the variance-switching process accounted for by the lagged spot–futures price deviation, a signal variable, is statistically marginal. By contrast, the lagged spot–futures price deviation functions as an efficient indicator for the variance-turning process in the emerging markets. ∗∗∗ Notably, using the 5% significance criterion, the LR statistic for the null hypothesis of s1 = s2 is insignificant for all cases. ††† In particular, the q1 estimate is significant at the 10% level, and the p1 estimate insignificant, for two markets: the U.K. FTSE100 and German DAX30, but both estimates are insignificant for the U.S. S&P500 market. ‡‡‡ In the case of Hungary BSI market, although the q1 estimate is significant in the 1% significance, the p1 estimate is insignificant. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb 716 M.-Y. L. LI 4. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS This investigation represents one of the first studies of the application of the MS-VECM to the dynamics of the interrelationship between spot and futures markets. Previous studies dealing with this area have generally focused on the disequilibrium adjustment process between spot and futures markets using a conventional VECM, which suffers from unreliability stemming from its assumption of a uniform convergence process during the whole estimation period. By contrast, this work develops a system in which the parameter of the deviation in spot–futures prices changes according to the phase in which the volatility regime finds itself at a given point in time. The models are tested using three mature stock markets (the U.S. S&P500, the U.K. FTSE100 and the German DAX30) and two emerging markets (the Brazil Bovespa and the Hungary BSI). Based on the empirical findings of this investigation, the following can be concluded. First, the conventional VECM, by failing to take into consideration of the high/low variance state of stock markets, yields inconsistent results regarding the price discovery process of spot–futures markets across the five markets selected for this study. However, after filtering out the high variance regime, the futures price leads the spot price in the price discovery process as demonstrated by prior studies; conversely, the spot market is more informationally efficient than the futures market under the conditions of the high variance state. Second, the price adjustment triggered by arbitrage trading between spot and futures markets in a high variance state is greater in scale than that in a low variance state. Moreover, this phenomenon significantly reduces the degree of the co-movement between spot and futures markets during the high variance state. Third, this study defines a crisis condition involved in the high variance state for the two emerging markets selected by this study, whereas an unusual condition is denoted for the three mature markets. Finally, the lagged spot–futures price deviation functions as an indicator for the variance-turning process. However, the portion of the variance-switching process explained by this signal variable is statistically marginal for the three mature markets; by contrast, in the two emerging markets selected for this study, the signal variable performs well for predicting the volatility-switching process. There are three important caveats to the interpretation of the analytical results of this investigation.§§§ First, this study uses the asset reallocation process involved in a high variance state to explain the asymmetric lead–lag relationship between spot and futures markets across various variance states. However, more in depth discussions dealing with this asset reallocation process and other explanations regarding the asymmetric lead–lag relationship across various variance states in the spot–futures markets are encouraged. Second, this study employs the jump diffusion system controlled by the Markov-switching techniques to design a state-varying parameter approach. It must be noted that certain studies, such as Bollerslev and Engle [42], Diebold [43] and Lamoureux and Lastrapes [44] and among many others, suggest adopting the IGARCH (integrated GARCH) model to control the long memory and high persistence in return volatility. However, Hamilton and Susmel [22], Ramchand and Susmel [23, 24], Li and Lin [25], Li [26] posit that the high persistence in return volatility is caused by structural changes during the estimation period. The comparison analysis between long memory models and state-varying parameter approaches is encouraged. Third, following Brenner and Kroner [9] and Tse [10] and §§§ The author would like to thank an anonymous referee for this suggestion. Copyright q 2008 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2009; 25:696–718 DOI: 10.1002/asmb RELATIONSHIP BETWEEN SPOT AND FUTURES MARKETS 717 among many others, we use a two-step procedure to examine the state-varying co-integration process in spot–futures markets: first, an examination of global co-integration behavior of the time series, and second, an examination of local nonlinear process of the time series. Undeniably, when we run the second step, the EC term is predetermined by using a linear regression model and hence it is exogenous and not taking account of state-varying parameters. Releasing the exogeneity and constant parameter assumptions involved in the EC terms is meaningful for future research. REFERENCES 1. Herbst A, McCormack J, West E. 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