The True Cross-Correlation and Lead-Lag Relationship between Index Futures and Spot with Missing Observations

The True Cross-Correlation and Lead-Lag Relationship between Index Futures and Spot with Missing Observations Shih-Ju Chan, Lecturer of Kao-Yuan University, Taiwan Ching-Chung Lin, Associate professor of Kao-Yuan University, Taiwan Chao-Hsien Lin, Assistant professor of Tainan Woman s College of Art & Technology, Taiwan ABSTRACT This study applies the methodology proposed by De Jong and Nijman (1997) to estimate the true crosscorrelations in the case of missing observations or nonsynchronous trading problem and, hence, to explore the lead-lag relationship between index futures and spot contracts. By examining all four index futures contracts available on the Taiwan Futures Exchange (TAIFEX), this article investigates whether the characteristics of futures contract can guarantee its capability of price discovery. Empirical results show that, although these index futures contracts share the same trading mechanism, their interaction patterns with their respective underlying spots are different. INTRODUCTION The topic concerning the interaction between the index spot and futures markets has attracted the intensive attention of academics and practitioners. Numerous studies have investigated the lead-lag relationship between those two markets. The general conclusion is that index futures tends to lead index spot and, therefore, price discovery is well-believed to be one of most important functions of a futures contract. Some early notable examples include Kawaller et al. (1987), Stoll and Whaley (1990), and Chan (1992), among others. Recent works concerning international markets include studies by Abhyankar (1995) on the U.K. FT-SE 100, Shyy et al. (1996) on the France CAC, Booth et al. (1999) on the German DAX, Iihara et al. (1996) on the Japanese NSA, and Roope and Zurbruegg (2002) on the Taiwan TAIEX. Many articles suggest that the factors of transaction costs (e.g., Fleming et al., 1996), liquidity or trading activity (e.g., Stephan and Whaley, 1990), and short-sale constraint (e.g., Diamond and Verrecchia, 1987) may affect the lead-lag relationship between financial markets. The fact that futures markets tend to have lower transaction costs and less short-sale regulation, providing higher liquidity and financial leverage, and reflect market-wide information supports the previous conclusion of price discovery. While most past researchers have only examined the lead-lag relationship between one futures contract and its underlying spot, there are some issues that need to be considered further. First of all, when there are many index futures contracts of different underlying assets trading on a futures exchange, these futures contracts certainly share the same trading mechanism mentioned in the previous paragraph. However, should we expect that the associations between index futures and spot for all contracts will be the same? In other words, after only observing the lead-lag relationship between one specific futures contract and its underlying spot asset, can we conclude that all the futures contracts traded on a same exchange have the similar lead-lag patterns? If we can observe different lead-lag patterns, it means that the characteristics of the futures contract cannot guarantee its leading capability against its underlying spot. Other factors must exist to dominate the explanations previously mentioned in the previous paragraph. The second motivation of this study is that, thanks to the development of computer technology, financial researchers can get more information to examine the behavior of financial markets. Now, the 10-, 5-, and even 1-minute intraday data are increasingly popular. However, as the time interval becomes shorter,

the problems of nonsynchronous trading and missing observations may occur when selecting the prices of infrequent-trading futures and these problems may cause improper statistical inference. Lo and MacKinlay (1990) show that nonsynchronous trading problem may result serial correlation in observed returns, even though the true returns are serially uncorrelated. Based on the above considerations, this article examines all four index futures contracts available on the Taiwan Futures Exchange (TAIFEX). The first phenomenon trigging our interest to compare the price discovery for these four index futures contracts is that the trading volumes of their underlying indices are much different. The specifications of these index futures contracts will be shown in the next section. By applying a method proposed by De Jong and Nijman (1997), this article is able to estimate the true cross-correlations and investigate the lead-lag relationship when some observations of futures prices are missing. If these four index futures consistently revealed information faster than their respective underlying indices, we may confidently conclude that, in Taiwan financial market, futures contract can always play the major role in the procedure of price discovery. If not, that futures dominates spot should be interpreted cautiously. This remainder of this paper is organized as follows. Section 2 briefly introduces and compares Taiwan index futures contracts. Section 3 is the methodology of true cross-correlation estimation when some observations are missing. Section 4 describes the data and Section 5 contains the empirical results and discusses the findings. Finally, Section 6 is the conclusion. TAIWAN INDEX FUTURES The first Taiwan index futures - Taiwan Stock Exchange (TSE) Capitalization Weighted Stock Index (TAIEX) Futures (TX) - was introduced on July 21, 1998. One year later, the TAIFEX launched two more contracts - TSE Electronic Sector Index Futures (TE) and TSE Banking and Insurance Sector Index Futures (TF). To encourage the participation of investors with fewer funds, the TAIFEX introduced the Mini-TAIEX Futures (MTX) on April 9, 2001. The MTX requires only one-quarter of the contract size of that of TAIEX futures. Table 1 shows the comparisons of these index futures contracts. Table 1. Comparisons of Four Taiwan Index Futures Symbol TX MTX b TE TF Launched Date July 21, 1998 April 9, 2001 July 21, 1999 July 21, 1999 Underlying Index TAIEX a TAIEX TSE c Electronic Sector Index TSE Banking and Insurance Sector Index Contract Size NT$200 Index NT$50 Index NT$4,000 Index NT$1,000 Index Minimum Price One index point One index point 0.05 index point 0.2 index point Fluctuation (NT$200) (NT$50) (NT$200) (NT$200) Futures Index b 5,073 5,083 264.98 673.5

Trading Volume of Index Futures b 16,561 3,521 3,871 2,060 Daily Trading Volume (billions) of Index NT$110.06 NT$110.06 Spot d Market Value (trillions) of Index NT$10.3 NT$10.3 Spot d NT$92.1 (83.73%) NT$6.77 (65.7%) NT$ 6.7 (6.34%) NT$1.75 (17%) a: TAIEX represents Taiwan Stock Exchange Capitalization Weighted Stock Index. b: MTX is Mini-TX futures whose contract size is only one quarter of that of TX futures. c: TSE represents Taiwan Stock Exchange. d: Daily average figures from Oct. 2, 2001 to Mar. 29, 2002. The figures in parenthesis are the ratios of trading volumes (market values) of electronic and banking and insurance sectors to that of the Taiwan stock market. The average contract sizes of TX, TE, and TF index futures during Oct. 2, 2001 to Mar. 29, 2002 were around $0.7 to $1 million New Taiwan dollars (NT), while that of MTX futures contract was only around NT$0.25 million. The TX index futures is undoubtedly the most popular contract in the Taiwan index futures market. Its average daily trading volume during October 2, 2001 to March 29, 2002 reached 16,561 contracts, while those of the MTX, TE, and TF index futures were only 3,521, 3,871, and 2,060, respectively. Because the Taiwanese futures market is still growing, those figures of trading volume are higher than those of average volume in 2001 (11,659, 2,334, 2,807, and 1,596, respectively). The trading volumes of the underlying index spots are also significantly different. At the end of 2001, the market values of the financial sector and the electronic sector are around NT$1.75 and NT$6.77 trillions, respectively. These figures represent 17% and 65.7% of the total market value of Taiwan stock market, respectively. However, the financial sector representing a market value of 17% of Taiwan listed companies contributes only 6.34% of daily trading volume of Taiwan market, while those figures of the electronic sector are 65.7% and 87.73%. Therefore, after considering the factor of market size, the trading behavior of the TF index spot is much inactive than those of other index spots. This fact of less active TF futures and spot markets triggers this study to compare the overall information transmission of the Taiwan index futures market, not just investigate a single contract. METHODOLOGY The first step to examine the lead-lag relationship between index futures and spot is to get the intra-day price series. While those indices can be calculated every minute, the transaction interval between the prices of index futures varies. To obtain the fixed interval prices of index futures, the conventional way is to use the last transaction of each intra-day interval to serve as the index futures price of that interval. However, when the interval is short and trading is not active, some intervals may have no transaction and, hence, the problem of nonsynchronous trading or thin trading problem happens, in which the prices of securities are

mistakenly assumed to be sampled simultaneously. A nonsynchronous problem can create spurious autocorrelation and cross-correlation, as Cohen et al. (1983), and Lo and MacKinlay (1990) have demonstrated. To deal with missing observations and to estimate the true covariances and correlations between return series, in which the price data are irregularly spaced in time, De Jong and Nijman (1997) generalize the setup of Cohen et al. (1983) and creatively transform the true correlation computation process into a form of regression equation. Afterwards, they also provide an empirical application to the lead-lag relationship between the S&P 500 index spot and futures. The basic idea of the De Jong-Nijman method is to find an expression for the expectation of cross-products of price changes or returns. Let S t and F t denote the observed logarithm prices of the index spot and index futures at time t. When prices are irregularly observed or there are some missing observations, the differences between two observed prices at time t i+1 and t i can be written as sums of the unobserved price changes or returns: obs t i 1 obs t i 1 obs ti 1 tti 1 P P P P (1) t i t where P = S or F and the price changes Pt at time t i +1, t i +2 and t i+1 are unobserved true returns. Thus, the cross-product of price changes on index futures and spot markets will be: ti1 y ( S S )( F F ) S F. (2) ij ti1 ti t j1 t j tti 1 t j1 t st j 1 t Conditional on the observed transaction time, the expectation of cross-product y ij is a linear combination of the cross-covariances γ k of the returns: ij ti1 tti 1 t j1 t st j 1 s ti1 t j1 E( y ) E( S F ) (3) tti 1 st j 1 ts where Cov S, F ) E( S F ). h ( t th t th Let x ij (h) denote the number of times that γ h appears in the right-hand side of equation (3). As De Jong and Nijman (1997) derive in their paper, x ij (h) can be computed using the following equation: xij ( h) max( 0, min( ti 1, t j1 h) max( ti, t j h)). (4) Because x ij (h) is the function of transaction time only, E(y ij ) can be rewritten as follows: E H ( yij xij ) hh where it is assumed that the horizon of non-zero covariances is limited to H. x ( h) (5) ij h When considering equation (5) as the regression relationship between observed y ij and x ij (h), the unknown true covariances γ h can be estimated. Equation (5) can be transformed into a vector regression equation:

' ij ij ij N1 N(2H 1) (2H1) 1 N1 y x e. (6) where N is the sample size of available cross-products. Because the covariances of higher order than H are assumed to be zero and the relationships between overnight returns are excluded, N will be much less than the product of the numbers of the index spot and futures prices. Of course, if the omitted covariances (i.e., of order > H) are not equal to zero, this regression will suffer from an omitted variables bias as in a conventional regression model. Equation (6) can be estimated by ordinary least squares (OLS). The OLS estimator is a consistent estimator under the assumption that covariances of higher order than H are all zero. After the estimation of covariances, the computation of cross correlations will be the same as the usual way. The cross correlations are given by ˆ ˆ h h (7) S F ˆ0 ˆ0 While the OLS parameter of γ is consistent, the error terms e ij in equation (6) are likely to be heteroskedastic and autocorrelated (HAC) and, hence, the OLS standard errors will be inconsistent and lead to an invalid inference of t-test. Therefore, we use the method proposed by Newey and West (1987) to compute the covariance matrix of error terms, which is consistent in the presence of heteroskedasticity and autocorrelation (HAC) of unknown form. DATA The sample period of this study is from February 20, 2002 to March 29, 2002. Although there are only 27 trading days during the study period, the original sample size is over 7000 because of 1-minute intraday data. Furthermore, before employing the method proposed by De Jong and Nijman (1997), these price data have to be transformed to a new data set with a sample size of over 80,000 to run a regression model. The data of the TAIFEX index futures and the TSE TAIEX index spot were downloaded from the websites of these exchanges. This article uses the 1-minute intraday data and, therefore, the number of 1-minute intervals during the sample period of 27 trading days is 7290. The price observations of three underlying indices are easy to get and have no any missing observation because they are quoted every minute. However, there are some intervals without transactions for the prices of index futures contracts. To apply De Jong and Nijmans approach to estimate the true cross-correlations, instead of using the price of the last interval to be that of the no- transaction interval as the conventional way in past studies, this article treats the prices of these no-transaction intervals as missing observations. Therefore, not like the conventional way that both the sample numbers of index futures and spot prices are equal, the sample sizes of infrequent-trading futures contracts will be less than 7290. As shown in Table 2, the numbers of no-transaction are 0, 68,179, and 1718, respectively. Therefore, the sample sizes of four index futures contracts are 7290, 7222, 7111, and 5572, respectively. Because the trading behavior of TF futures is less active, the percentage of missing observation of TF futures is as high as 23.57%.

Table 2. Sample Size and Number of No-Transaction TX MTX TE TF # of no-transaction 0 68 179 1,718 Sample size 7,290 7,222 7,111 5,572 % of no-transaction a 0% 0.93% 2.46% 23.57% a: Percentage = (number of no-transaction) 7290 EMPIRICAL RESULTS Following the De Jong-Nijman approach, we first estimate the autocorrelations of spot and futures returns. The estimated autocorrelations of spot and futures returns are presented in Tables 3 and 4, respectively. The correlations of higher order than 10 are assumed to be zero in the estimation. Table 3. Autocorrelations of Spot Lag TX&MTX TE TF 0 0.0010 0.0014 0.0012 1-0.042-0.133 * -0.174 * 2 0.103 * 0.080 * 0.025 3 0.028 0.010 0.016 4 0.024 0.021 0.006 5 0.009 0.012 0.003 6-0.018-0.020-0.011 7-0.019-0.019 0.028 8-0.032-0.029-0.020 9-0.033 * -0.024 0.004 10-0.010-0.005-0.006 a: The numbers of observations for three index spots are all 7290. The percentages of missing observations are all 0%. b: The TX and MTX futures have the same underlying index. c: Figures for Lag 0 are variances of spot prices. The others are correlations. * Significance at the 0.001 level.

Table 4. Autocorrelations of Futures Lag TX MTX TE TF 0 0.0010 0.0010 0.0012 0.0011 1 0.046 * -0.021-0.0611 * -0.447 * 2-0.046-0.034-0.005-0.073 3-0.012 0.000-0.036-0.007 4-0.005-0.035 0.004-0.052 5-0.008 0.022-0.005-0.019 6 0.001-0.024-0.010-0.029 7-0.007-0.038-0.011 0.009 8 0.010 0.011 0.012-0.027 9-0.006 0.008 0.006 0.018 10 0.018-0.005 0.006-0.004 a: Figures for Lag 0 are variances of futures prices. The others are correlations. * Significance at the 0.001 level. The autocorrelations of spot returns of Lag 1 and Lag 2 in Table 3 are all negative and positive, respectively, and four of them are significant at the 0.001 level. Compared with the estimates in De Jong and Nijman (1997), the significant lags and the autocorrelations are smaller. Besides, if we horizontally observe the coefficients of Lags 1 and 2 in Table 3, it shows that, as the trading volume of the index spot decreases (from TX and MTX to TF), the autocorrelations increase. That is, the prices of the underlying index spot of TX and MTX disseminate information faster than other indices. As shown in Table 2, the percentage of missing observation of the TF futures is as high as 23.57%. By employing the De Jong-Nijman approach, we can avoid the possible spurious estimations of autocorrelations and cross-correlations resulting from the conventional way to define the price series of futures. In Table 4, there is no significant serial correlation of higher order than 2. Not only are the numbers of significant lag of futures contracts shorter than those of their respective underlying indices, but their first-order autocorrelations are also smaller, with the exception of TF futures. This evidence suggests that, compared with their respective underlying indices, the futures contracts of TX, MTX, and TE can reflect the relevant information faster. However, while the significant first-order autocorrelation of the TF spot is only -0.174, that of the TF futures is -0.447. This provides preliminary evidence that the interaction between futures and spot in the TF system might be different from those of other systems. By analyzing the cross-correlations between futures and spot for four systems presented in Table 5, we now examine the lead-lag relationship for those four systems. In Table 5, a positive (negative) Lag represents futures (spot) leads spot (futures). We can observe that, as the trading values of underlying indices decrease from TX to TF, the lead-lag structures gradually change in the same direction. In the TX system, futures leads the index spot up to 4 minutes, while there is only one significant lag correlation. Although the MTX futures have the same underlying index as the TX, the trading volume of MTX is lower than that of TX because of a shorter history. Therefore, the significant lead coefficients are the first three and the first two lag coefficients become significant. The lead capability of index futures is weakening and

this trend is more obviously in the TE and TF systems. In the TE system, futures and spot lead each other by as much as two minutes; however, the only significant lead coefficient of the TE futures happens at Lag 2. In the TF system, the lead behavior of futures disappears and the index spot dominates the price discovery procedure. The figures in Table 6 are F-statistics that test that the first five lead or lag correlations are jointly zero. The F-statistics of TE and TF systems are getting smaller and that of the lead correlations of TF is not significant. The joint hypothesis tests again confirm the conclusion that, while the TX and MTX index futures are faster in updating prices and disseminate more information than their respective underlying indices, the TF index futures does not lead the TF index spot. In summary, the interaction patterns of these four Taiwanese futures-spot systems are different. While the TX and the MTX index futures transmit information faster than their underlying spots, the TF index futures does not dominate the procedure of price discovery. Although all Taiwanese index futures contracts have lower transaction costs and less short-sale regulation, providing higher financial leverage, and reflect market-wide information when compared with their respective underlying indices, these characteristics of futures contract does not guarantee a dominant position of information transmission in Taiwan market. To our best knowledge, no model has been proposed to explain how and why this situation happens. Considering the investor structure in Taiwanese markets, one possible reason is that, when Taiwanese investors take less interest in the TF index spot, the liquidity of futures is affected and, hence, the interaction between the TF index futures and spot is different from those of other systems. Table 5. Cross-correlations between Futures and Spot Lag TX MTX TE TF -5-0.029-0.026-0.026-0.002-4 -0.018-0.017-0.015-0.008-3 -0.029-0.027-0.013-0.004-2 0.003 0.002 * 0.040 * 0.037-1 0.196 * 0.182 * 0.139 * 0.119 * 0 0.465 0.432 0.444 0.196 * 1 0.087 * 0.081 * 0.046 0.024 2 0.087 * 0.081 * 0.075 * 0.039 3 0.069 * 0.064 * 0.034 0.000 4 0.051 * 0.047 0.031-0.006 5 0.022 0.020 0.014 0.034 a: The cross-correlations are defined as the covariances, estimated by eq.(6), divided by the estimated standard deviations of futures and spot returns. b: Positive (negative) lag represents futures (spot) leads spot (futures). c: The correlations of higher order than 10 are assumed to be zero in this study. However, because the figures of higher order than 5 are all insignificant, they are not presented in this table for the reason of space. * Significance at the 0.1% level.

Table 6. F-statistics of Joint Hypothesis TX MTX TE TF Lag 29.52 * 33.18 * 16.30 * 6.15 * Lead 20.75 * 20.83 * 11.14 * 2.81 The figures in this table are the F-statistics for the joint hypothesis that all the covariances of horizon 1 to 5 (lead) or horizon 1 to 5 (lag) are equal to zero. * Significance at the 0.001 level. CONCLUSION The general conclusion concerning the lead-lag relationship between index futures and spot markets is that index futures dominates its underlying spot in price discovery. However, if only observing the lead-lag relationship between one specific futures contract and its underlying spot asset, can we conclude that all the futures contracts traded on a same exchange have the similar lead-lag patterns? Besides, the conventional way to create the price series of futures may cause the problem of nonsynchronous trading problem and, hence, lead to a spurious cross-correlation. By using 1-minute intraday data and employing the methodology proposed by De Jong and Nijman (1997) to estimate the true cross-correlations in the case of missing observations or nonsynchronous trading problem, this article investigates the lead-lag relationships between the index futures and spot prices of all the available index futures contracts on the TAIFEX. Observing the relatively infrequent trading activity of the TF system motivates us to compare the overall information transmission of the Taiwan index futures market. Empirical results indicate that, while the TX and the MTX index futures disseminate information faster than their respective underlying indices, the TF index futures does not dominate the TF index spot. That is, being a futures contract does not guarantee a major role in the procedure of price discovery. Why does this happen in Taiwan market? One possible reason is that, when Taiwanese investors take less interest in the TF index spot, it affects the liquidity of the TF futures and, hence, changes the interaction between the TF index futures and spot.

REFERENCES Abhyankar, A. H. (1995). Return and Volatility Dynamics in the FT-SE 100 Stock Index and Stock Index Futures Markets. The Journal of Futures Markets, 15: 457-488. Booth, G. G., R. W. So, and Y. Tse. (1999). Price Discovery in the German Equity Index Derivatives Markets. The Journal of Futures Markets, 19: 619-643. Chan, K. (1992). A Further Analysis of the Lead-lag Relationship between the Cash Market and Stock Index Futures Market. Review of Financial Studies, 5: 123-152. Cohen, K., G. Hawawimi, S. Maier, R. Schwartz, and D. Whitcomb. (1983). Friction in the Trading Process and the Estimation of Systematic Risk. Journal of Financial Economics, 12: 263-278. Diamond, D. W. and R. E. Verrecchia. (1987). Constraints on Short-Selling and Asset Price Adjustment to Private Information. Journal of Financial Economics, 18: 277-311. Fleming, J., B. Ostdiek, and R. E. Whaley. (1996). Trading Costs and the Relative Rates of Price Discovery in Stock, Futures, and Option Markets. The Journal of Futures Markets, 16: 353-387. De Jong, F. and T. Nijman. (1997). High Frequency Analysis of Lead-lag Relationships between Financial Markets. Journal of Empirical Finance, 4: 259-277. Iihara, Y., K. Kato, and T. Tokunaga. (1996). Intraday Return Dynamics between The Cash and The Futures Markets in Japan. The Journal of Futures Markets, 16: 147-162. Kawaller, I. G., P. D. Koch, and T. W. Koch. (1987). The Temporal Price Relationship between S&P 500 Futures and the S&P Index. Journal of Finance, 42: 1309-1329. Lo, A. and A. C. MacKinlay. (1990). An Econometric Analysis of Infrequent Trading. Journal of Econometrics, 45: 181-211. Newey, W. and K. West. (1987). A Simple Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica, 55: 703-708. Roope, M. and R. Zurbruegg. (2002). The Intra-day Price discovery Process between the Singapore Exchange and Taiwan Futures Exchange, The Journal of Futures Markets, 22: 219-240. Stephan, J. A. and R. E. Whaley. (1990). Intraday Price Changes and Trading Volume Relations in the Stock and Stock Option Markets, Journal of Finance, 45: 191-220. Shyy, G., V. Vijayraghavan, and B. Scott-Quinn. (1996). A Further Investigation of the Lead-lag Relationship between the Cash Market and Stock Index Futures Market with the Use of Bid/Ask Quotes: the Case of France. The Journal of Futures Markets, 16: 405-420. Stoll, H. R. and R. E. Whaley. (1990). The Dynamics of Stock Index and Stock Index Futures Returns. Journal of Financial and Quantitative Analysis, 25: 441-468. Subrahmanyam, A. (1991). A Theory of Trading in Stock Index Futures. Review of Financial Studies, 4: 17-51.