High frequency analysis of lead-lag relationships between European financial markets; a theoretical and a practical view

(1)

High frequency analysis of lead-lag relationships

between European financial markets;

a theoretical and a practical view

Alrik Krol June 2007

(2)

Abstract

In this thesis, I investigate if previous found lead-lag relations between returns on portfolios with high trading volume and returns on portfolios with low trading volume also appear between short-term intraday returns on European indices. Furthermore, I investigate if the stock market trader can profit from these relations. Relations are

(3)

Contents

1. Introduction 4

2. Literature review 6

2.1 EMH and lead-lag relations 6

2.2 High frequency data 11

2.3 Trading costs 12

3. Data description 14

4. Methodology 21

4.1 Correlation structure in returns 21

4.2 VAR model 22 4.3 Opening effect 23 5. Empirical Results 25 5.1 Cross-autocorrelation matrices 25 5.2 Lead-lag relations 31 5.3 Opening results 39 6. Trading Strategies 40

6.1.1 Lead-lag trading strategy 40

6.1.2 Trading strategy opening 41

6.2.1 Profitability lead-lag trading strategy 43 6.2.2 Profitability opening strategy 45

7. Conclusion 49

(4)

1. Introduction

Patterns in average stock returns that cannot be explained by the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) are considered anomalies (Fama and French, 2006). Some of the more prominent anomalies are the ‘January Effect,’ the ‘Weekend Effect,’ and the ‘Small Firm Effect’ (Frankfurter and McGoun, 2001). This thesis deals with the anomaly of time series return predictability emerging from lagged correlations between securities (‘cross-autocorrelation’).

The predictability of stock market returns may contradict the efficient market hypothesis (EMH) tested by the CAPM model (Frankfurter and McGoun, 2001). The EMH states that stock market prices follow a random walk. This may imply that predictability of returns, arising from cross-autocorrelations patterns, provide evidence against the EMH. However, Fama (1970) notes three conditions for capital market efficiency: (1) There are no transaction costs in trading securities; (2) all available information is costlessly

available to all market participants and (3) all agree on the implications of current information for the current price and distributions of future prices of each security.

Lo and MacKinlay (1990) found cross-autocorrelation effects between portfolios; they found an asymmetric structure in which returns on portfolios consisting of firms with high market capitalization (large stocks) lead returns on portfolios consisting of firms with low market capitalization (small stocks), but not vice versa. Subsequent studies of Badrinath et al. (1995), Chordia and Swaminathan (2000) and Gebka (2002) attribute this phenomenon to differences in speed of adjustment to common information between high and low trading volume portfolios; information is faster incorporated in high trading volume portfolios than information is incorporated in low trading volume portfolios.

(5)

Trading volume will be defined as the turnover ratio. The lead-lag phenomenon will be investigated from the viewpoint of the stock market trader; a much shorter time interval than earlier studies will be used by using high-frequency data. Cross-autocorrelation effects are investigated between the Euro Stoxx 50 (SX5e), the German Xetra Dax (DAX), the Swiss Market Index (SMI), and the Dutch AEX Index (AEX) for 1, 5, 10 and 30-second intraday time intervals. Furthermore, I will make a distinction between a theoretical and a practical view; I will not only look if the lead-lag effects are significant (theoretical), but also if the stock market trader can profit from their possible appearance (practical).

I hypothesize that there are short-term lead-lag effects between the indices of different European countries due to the fact that information is faster incorporated by traders in indices with a high trading volume than in indices with a low trading volume.

Furthermore, I hypothesize that the trader can profit from these effects due to informational advantages of high-frequency data.

I calculate cross-autocorrelation matrices and estimate a vector autoregressive (VAR) model to find potential lead-lag effects. Furthermore, I develop a trading strategy which may profit from lead-lag relations. Only some support is presented for volume lead-lag relations. Although the high trading DAX future lead all low trading index futures, the low trading SMI future lead the high trading SX5e and AEX futures. The trading strategies which exploit the found lead-lag relations appear to be profitable.

The rest of the thesis is organized as follows: I will give a literature overview in section 2; section 3 will comprise a description of the data; section 4 will consist of the

(6)

2. Literature overview

2.1 EMH and lead-lag relations

In 1900 Bachelier published his thesis ‘Theory of Speculation.’ The importance of his thesis, in which Bachelier asserted that stocks follow random walks, went unnoticed. Decennia later, after new publications of Cootner (1962) and Fama (1965), the random walk hypothesis would be subject to much debate under financial economists. The random walk theory states that successive price changes of a stock are completely

independent; this implies that the series of price changes are not predictable. The random walk theory is an assumption used in many financial models and received broader

support from many empirical studies after the publications of Bachelier, Cootner and Fama.

The random walk hypothesis is an important underlying assumption of the EMH. An efficient market is defined by Fama (1965) as “a market where there are large numbers of rational, profit-maximizers actively competing, with each trying to predict future market values of individual securities, and where important current information is almost freely available to all participants.” Fama continues “in an efficient market, competition among the many intelligent participants leads to a situation where, at any point in time, actual prices of individual securities already reflect the effects of information based both on events that have already occurred and on events which, as of now, the market expects to take place in the future. In other words, in an efficient market at any point in time the actual price of a security will be a good estimate of its intrinsic value.”

(7)

Fama (1970) mentions the practical criterion of the stock market trader. According to Fama, the random walk model is valid for the trader “as long as knowledge of the past behavior of the series of price changes cannot be used to increase expected gains. More specifically, the independence assumption is an adequate description of reality as long as the actual degree of dependence in the series of price changes is not sufficient to allow the past history of the series to be used to predict the future in a way which makes expected profits greater than they would be under a naïve buy-and hold model.”

Fama (1970) notes that it is easy to determine sufficient conditions for capital market efficiency. According to Fama “a market in which (1) are no transaction costs in trading securities, (2) all available information is costlessly available to all market participants and (3) all agree on the implications of current information for the current price and distributions of future prices of each security. In such a market, the current price of a security obviously “fully reflects” all available information.” However, Fama states that these conditions are sufficient for market efficiency, but not necessary: “…transactions costs, information that is not freely available to all investors, and disagreement among investors about the implications of given information are not necessarily sources of market inefficiency, they are potential sources.” So, according to Fama sources of market inefficiency may be the payment of transaction costs or brokerage fees but also the use of high-frequency data, which is not freely available data.

(8)

Lo and MacKinlay (1990a) questioned to what extent the data of De Bondt and Thaler is consistent with stock market overreaction and distilled this into an empirical question: “Are return reversals responsible for the predictability in stock returns?” Lo and MacKinlay state: “A more specific consequence of overreaction is the profitability of a contrarian portfolio strategy, a strategy that exploits negative serial dependence in asset returns in particular.” And follow with: “It is the apparent profitability of several contrarian strategies that has led many to conclude that stock markets do indeed

overreact.” In their article, Lo and MacKinley question the reverse implication, namely, if the apparent profitability of many contrarian strategies necessarily implies stock market overreaction. This counterintuitive result is, according to the authors, a consequence of positive cross-autocovariances across securities, from which contrarian portfolio

strategies benefit. If for example, a high return for security A today implies a high return for security B tomorrow, then a contrarian strategy will be profitable even if each

security’s returns are unpredictable using past returns of that security alone. If this is the case it is not required that stock markets overreact. Lo and MacKinley use weekly equally-weighted and value-weighted CRSP NYSE-AMEX stock-return indexes from 1962 till 1987. Over half of the expected profits they examine may be attributed to cross-autocorrelation and not to negative cross-autocorrelation of individual securities. Furthermore, using cross-autocorrelation matrices they find that the returns of portfolios of larger stocks generally tend to lead the returns on portfolios of smaller stocks.

(9)

portfolios consisting of firms followed by many analysts lead returns on portfolios of firms that are followed by fewer analysts.

Furthermore, Badrinath et al. (1995) provide empirical evidence for the transmission of information between equity securities. They present evidence that the past returns on a stock held by informed institutional traders will be positively correlated with the contemporaneous returns on stocks held by non-institutional uninformed traders. Badrinath calculates cross-autocorrelations matrices and estimates a VAR test to find possible lead-lag relations. Furthermore, he controls for market size. By composing portfolios, using all firms on the NYSE and AMEX for which complete daily and monthly return series are available for the period 1980-1988, Badrinath shows that the returns on portfolios of stocks with higher levels of institutional ownership lead the returns on portfolios of stocks with lower levels of institutional ownership.

Boudoukh et al. (1994) argue that lead-lag effects are a spurious phenomenon; portfolio cross-autocorrelations should vanish after controlling for non-synchronous trading and portfolios own positive auto-correlations. Boudoukh et al. suggest to use an AR(1) model and conclude that cross-autocorrelation effects might persist even if the lagged returns of large firms have no predictive power at all beyond that contained in small firms lagged returns due to non-synchronous trading and portfolios own positive auto-correlations.

However, Chordia and Swaminathan (2000) explored a relation between asymmetric cross-autocorrelation and different levels of trading volume. Daily and weekly returns on high volume portfolio lead returns on low volume portfolios after controlling for size. They calculated cross-autocorrelations matrices and estimated a VAR model.

(10)

Moreover, Gebka (2002) shows that both volume- and size- related cross-autocorrelation patterns exist on the Warsaw stock exchange. Using cross-autocorrelation matrices and the VAR model, he shows that the slower adjustment of low volume and small size portfolios to common information happen independently from each other. Furthermore, Gebka found that portfolio cross-autocorrelations also persist after adjusting for

portfolios own positive autocorrelations, as found by Boudoukh et al.

Table 1 summarizes the articles mentioned, mentions their used methods and summarizes the conclusions.

Table1: Previous research: Methods and conclusions

Who Method Conclusion

Lo and MacKinlay Cross-autocorrelation matrices

Returns on large portfolios lead returns on small portfolios

Brennan et al. Granger causality regressions

Returns on portfolios consisting of firms followed by many analysts lead returns on portfolios of firms that are followed by fewer analysts

Badrinath et al. Cross-autocorrelation matrices, VAR model

Returns on portfolios of stocks with higher levels of institutional ownership lead the returns on portfolios of stocks with lower levels of institutional ownership Boudoukh et al. AR(1) model Lead-lag effects are a spurious phenomenon, portfolio

cross-autocorrelations should vanish after controlling for non-synchronous trading and portfolios own auto-correlations

Chordia and Swaminathan

Cross-autocorrelation matrices, VAR model

Daily and weekly returns on high volume portfolio lead returns on low volume portfolios after controlling for size, own autocorrelation and non-synchronous trading

Gebka Cross-autocorrelation

matrices, VAR model

Both volume- and size- related cross-autocorrelation patterns exist after controlling for own autocorrelations and non-synchronous trading

(11)

also appear on a short-term intraday basis I first need to explore the special characteristics of high-frequency data.

2.2 High frequency data

With the enormous advances in processing technology and data acquisition the study of high (or ultra) frequency data has become more and more feasible. High frequency data is defined to be a full record of transactions and their associated characteristics (also called marks) (Engle, 2000). High frequency data is required to analyze information flows between markets on short term intraday intervals. The salient feature of such high-frequency data is that they are fundamentally irregularly spaced (Engle, 2000). For some research questions the differences in time interval is not of real importance and there can be relied on estimating models in transaction time; for instance, microstructure issues like the informational content of traded volumes for (future) prices (Karpoff, 1987) or the relation between pricing and clustering of transactions (Easley and O’Hara, 1992). From an econometric perspective such hypotheses require an analysis of the marks in high frequency data (Herwartz, 2006). However, for the analysis of information flows between financial markets regularly spaced time intervals are of utmost importance (De Jong and Nijman, 1997). The usual approach to tackle the problem of irregular time intervals is to convert the data to fixed time intervals. Although such a conversion of data comes at the cost of loosing trading information, it appears inevitable in order to overcome statistical issues (Herwartz, 2006).

High frequency data is for an important part composed of market microstructure noise (AitSahalia et al., 2005). Several studies have focused on how transaction prices suffer from microstructure biases:

(12)

Furthermore, non-synchronous trading biases are addressed by Herwartz (2006) and Lo and MacKinley (1990a); transaction prices are based on the last observation obtained in a particular time interval of fixed length. The last transaction price of a period, however, can be a price in the beginning or end of that particular period. Moreover, if the intervals are small, some intervals may contain no observation at all. Lo and MacKinley show how this may result in spurious own autocorrelations and spurious cross-autocorrelations between portfolios returns; for example, consider stocks A and B. Suppose that the returns of stock A and B are independent but stock A trades less frequently than stock B. If news affecting the stock markets arrives in a particular fixed time period, it is more likely that stock B will reflect this information than stock A, simply because stock A may not trade in the particular period when the news arrives. Furthermore, Lo and MacKinley state that this may have as implication that cross-autocorrelation between portfolios is asymmetric and is due solely to the assumption that securities in different portfolios have different probabilities of non-synchronous trading.

2.3 Trading costs

If anomalies in stock markets are discovered an important question is if these anomalies can be exploited into real profits and if they are not just ‘paper-profits.’ Trading costs play a crucial role in determining whether these profits are economically viable.

There exist implicit, explicit and opportunity trading costs (Harris, 2003); implicit costs are intangible costs that traders pay when they trade (Patnaik et al., 2004). These costs consist of the bid-ask spread and the price impact of trades. The bid-ask spread is the difference between the best bid and best offer for a particular security. This implies that you have to buy and sell against different prices in the same market. For instance,

Korajczyk and Sadka (2004) consider half of the bid-ask spread as cost. The price impact of trades can be, for example, a trader who has to pay more than the prevailing bid price in the market, because of the size of his order.

(13)

(Patnaik et al., 2004). Some of these costs are proportional to the size of the trade involved. However, some of these costs can in some cases be avoided. Individuals and companies only have to pay brokerage fees and capital gain taxes by online brokers (www.interactivebrokers.com). Moreover, depending on the country individuals live in, they can be exempted from capital gain taxes. This is, for example, the case in the Netherlands.

Opportunity costs of the trade are foregone possibilities, because of commitment of money or securities to that particular trade. The money cannot be invested elsewhere, which may lead to missed returns. Furthermore, trading costs can differ between agents in the market. For example, active traders may receive discount and professional investors pay lower brokerage fees than retail investors (NSE, (2002, pp. 135)).

(14)

3. Data description

The data is provided by the databases of the International Marketmakers Combination (IMC). The data contains high frequency data of SX5e, DAX, SMI, and AEX futures for the period 20 March 2006 till 16 March 2007. For robustness reasons, the period will be divided in 4 quarters: 20 March 2006 till 16 June; 19 June till 15 September; 18

September till 15 December; and 15 December till 16 March 2007.

Table 2 summarizes the characteristics of the investigated indices; the number of stocks, market capitalization and opening times of the indices are summarized. The market capitalization is a weighted average of the index for the period 20 March 2006 till 16 March 2007. Furthermore, Appendix 1 gives the stocks of which the indices comprise and their influence in the index.

Table 2: Market characteristics: Number of stocks, market capitalization and opening times of the SX5e, DAX, SMI and AEX futures

Number of Stocks Market Cap (€) Futures Opened

SX5e 50 2.490 bln 08:00-22:00 DAX 30 810 bln 08:00-22:00 SMI 26 730 bln 09:00-17:27 AEX 241 495 bln 09:00-17:302 1 25 as of 2 March 2007 2 opens at 08:00 as of 22 January 2007

Following Chordia and Swaminathan (2000) and others, I use turnover ratios as a proxy for volume. The turnover ratio (TR) is calculated for each quarter. The TR is the average volume of the most recent future divided by the average open interest of that future. Table 3 shows the average volume, open interest and TR for each quarter. DAX, SX5e, AEX and SMI futures are used in this research, because of their different turnover ratios.

(15)

future contract (Chordia and Swaminathan (2000). Therefore, I expect that the DAX adjusts quickest to new information, followed by the SX5e, AEX and SMI.

av. volume av. open interest turnover ratio av. volume av. open interest turnover ratio

SX5e 901404 1836161 0.49 741948 2020255 0.37

DAX 174901 242478 0.72 136981 198642 0.69

AEX 38712 115158 0.34 33166 114670 0.29

SMI 39986 335141 0.12 35321 266935 0.13

av. volume av. open interest turnover ratio av. volume av. open interest turnover ratio

SX5e 808743 2087266 0.39 954458 2106098 0.45

DAX 146303 232201 0.63 171182 249127 0.69

AEX 37912 135978 0.28 42546 130151 0.33

SMI 40782 286184 0.14 50203 303322 0.17

Table 3: Volume, open interest and turnover ratios of the SX5e, DAX, AEX and SMI futures

This table shows the average volume, average open interest and turover ratio for each quarter. The turnover ratio is defined as average volume / average open intereset.

December-March September-December

March-June June-September

Expected dividend payments of the underlying stocks of the indices will not influence intraday lead-lag relations, because the dividend payments will be incorporated in the futures price. However, interest rates shifts can have a different impact on futures with a different maturity date. Such an unlikely event has a negligible impact on this study; the AEX future is the only index future with different maturity dates. Furthermore, only 1 of thousands of returns will be affected by such an event.

Table 4 gives an overview of the different expiration dates of the different futures. For liquidity reasons the future with the earliest maturity date will be used.

Table 4: Futures expiration schedule (06-07)

Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar

SX5e 16 15 15 16

DAX 16 15 15 16

SMI 16 15 15 16

(16)

I only use the data when SX5e, DAX, AEX and SMI futures were all open for trading. Normally, this is between 09:00 and 17:27 each day. However, there are some data omissions; table 5 shows the periods which are not taken into account. The second column shows the time of the particular incomplete day which is taken into account. I assume that the data omissions do not have any influence on my research, because it is only a fraction of the total data and the data omissions arisen on a completely random manner.

Table 5: Closed days and data omissions with explanations

Dates Trading time Explanation

14-4/ 17-4/ 1-5/ 25-12/ 26-12/ 1-1

- Public Holiday (all indices closed)

25-5/ 5-6/ 1-8/ 2-1 - SMI closed

16-6/ 15-9/ 15-12/ 16-3 - Expiration (SMI is not trading )

21-4/ 19-5/ 21-7/ 18-8/ 20-10/ 17-11 / 19-1/ 16-2

09:00-16:00 Expiration date AEX future (which

expires at 16:00) 18-6 09:00-15:06 Data omission 24-6 10:40-17:27 Data omission 14-9 09:40-17:27 Data omission 29-12 09:00-14:00 Data omission 8-2 09:00-14:50 Data omission 7-2/ 14-3 - Data omission

(17)

quarter future mean median max min std dev N skewness kurtosis JB P

Mrt-Jun DAX -7.32E-07 0 0.017 -0.028 8.87E-06 1170114 -0.109 10.6 2.80E+06 0.00

SX5e -7.06E-07 0 0.017 -0.014 1.32E-05 1170114 -0.004 7.7 1.08E+06 0.00

AEX -1.28E-06 0 0.497 -0.494 2.26E-05 1170114 0.234 8311 3.37E+12 0.00

SMI -1.16E-06 0 0.032 -0.032 6.50E-06 1170114 -0.381 57.1 1.43E+08 0.00

Jun-Sep DAX 4.77E-07 0 0.029 -0.019 8.51E-06 1139067 0.062 12.5 4.31E+06 0.00

SX5e 5.16E-07 0 0.032 -0.013 1.30E-05 1139067 0.024 8.2 1.29E+06 0.00

AEX 1.01E-06 0 0.255 -0.251 1.89E-05 1139067 0.176 2527 3.02E+11 0.00

SMI 5.15E-07 0 0.019 -0.022 6.13E-06 1139067 0.015 31.9 3.97E+07 0.00

Sep-Dec DAX 6.06E-07 0 0.02 -0.026 6.61E-06 1182548 0.007 16.2 8.60E+06 0.00

SX5e 3.72E-07 0 0.02 -0.015 1.14E-05 1182548 0.004 8.1 1.28E+06 0.00

AEX 2.06E-07 0 0.171 -0.169 1.51E-05 1182548 0.103 2276 2.55E+11 0.00

SMI 3.39E-09 0 0.016 -0.017 5.28E-06 1182548 -0.003 25.4 2.48E+07 0.00

Dec-Mrt DAX -1.14E-07 0 0.019 -0.031 6.92E-06 1020872 -0.32 24.1 1.89E+07 0.00

SX5e -6.12E-07 0 0.024 -0.024 1.18E-05 1020872 -0.014 8.3 1.21E+06 0.00

AEX -5.01E-07 0 0.234 -0.236 1.78E-05 1020872 -0.038 2365 2.37E+11 0.00

SMI -1.04E-07 0 0.033 -0.04 6.08E-06 1020872 -0.383 73.8 2.13E+08 0.00

Mrt-Jun DAX -2.72E-06 0 0.031 -0.026 1.44E-05 356656 -0.182 11.7 1.14E+06 0.00

SX5e -2.80E-06 0 0.030 -0.029 1.91E-05 356656 -0.033 6.6 1.88E+05 0.00

AEX -5.00E-06 0 0.457 -0.455 3.04E-05 356656 0.135 3286 1.60E+11 0.00

SMI -3.60E-06 0 0.019 -0.028 1.06E-05 356656 -0.193 17 2.91E+06 0.00

Jun-Sep DAX 1.25E-06 0 0.038 -0.024 1.32E-05 376902 0.081 13.7 1.79E+06 0.00

SX5e 1.44E-06 0 0.031 -0.020 1.82E-05 376902 0.008 5.9 1.36E+05 0.00

AEX 2.92E-06 0 0.195 -0.195 2.70E-05 376902 0.002 811 1.03E+10 0.00

SMI 1.58E-06 0 0.017 -0.016 9.90E-06 376902 0.031 11.9 1.25E+06 0.00

Sep-Dec DAX 1.88E-06 0 0.039 -0.019 1.02E-05 387122 0.195 20.4 4.91E+06 0.00

SX5e 1.16E-06 0 0.027 -0.024 1.60E-05 387122 -0.007 6.3 1.77E+05 0.00

AEX 6.55E-07 0 0.171 -0.169 2.18E-05 387122 0.026 918 1.35E+10 0.00

SMI 9.30E-07 0 0.013 -0.013 8.57E-06 387122 -0.045 11.9 1.29E+06 0.00

Dec-Mrt DAX -6.09E-07 0 0.024 -0.031 1.10E-05 344667 -0.347 21.4 4.86E+06 0.00

SX5e -2.16E-06 0 0.036 -0.031 1.68E-05 344667 -0.057 8.6 4.45E+05 0.00

AEX -1.22E-06 0 0.161 -0.166 2.44E-05 344667 -0.203 811 9.39E+09 0.00

SMI -9.64E-07 0 0.032 -0.086 9.75E-06 344667 -1.913 192 5.15E+08 0.00

This table gives an overview of the descriptive statistics of the returns on the DAX, SX5e, AEX and SMI futures calculated from 1-second intraday time intervals for the 9.00-17.27 period. For each quarter and future are the mean, median, maximum, minimum, standard deviation, number of returns (N), skewness, kurtosis and the Jarque-Berra (JB) with accompanying P-value summarized in the various columns.

Table 6: Descriptive statistics of 1-second intraday returns on the DAX, SX5e, AEX and SMI futures

(18)

Mrt-Jun DAX -5.64E-06 0 0.031 -0.047 1.99E-05 178284 -0.298 13.1 7.64E+05 0.00

SX5e -5.63E-06 0 0.040 -0.050 2.47E-05 178284 -0.109 7.7 1.65E+05 0.00

AEX -9.95E-06 0 0.456 -0.455 3.98E-05 178284 0.102 2184 3.53E+10 0.00

SMI -6.91E-06 0 0.029 -0.030 1.45E-05 178284 -0.111 14.1 9.10E+05 0.00

Jun-Sep DAX 3.59E-06 0 0.070 -0.086 1.84E-05 188420 0.184 65.7 3.08E+07 0.00

SX5e 4.08E-06 0 0.068 -0.081 2.33E-05 188420 0.268 28.3 5.03E+06 0.00

AEX 7.02E-06 0 0.180 -0.180 3.40E-05 188420 -0.178 373 1.08E+09 0.00

SMI 4.09E-06 0 0.058 -0.086 1.40E-05 188420 -0.278 119 1.05E+08 0.00

Sep-Dec DAX 3.02E-06 0 0.063 -0.061 1.41E-05 193528 -0.285 98.6 7.37E+07 0.00

SX5e 1.52E-06 0 0.076 -0.074 2.02E-05 193528 -0.130 34.6 8.07E+06 0.00

AEX -2.06E+10 0 0.171 -0.169 2.79E-05 193528 -0.080 483 1.86E+09 0.00

SMI 1.11-6 0 0.062 -0.066 1.22E-05 193528 -0.507 114 9.99E+07 0.00

Dec-Mrt DAX 1.16E-06 0 0.114 -0.047 1.57E-05 172294 4.185 307 6.64E+08 0.00

SX5e -1.40E-06 0 0.152 -0.053 2.18E-05 172294 3.023 193 2.59E+08 0.00

AEX 8.40E-07 0 0.161 -0.136 2.93E-05 172294 2.220 421 1.26E+07 0.00

SMI -1.42E-07 0 0.137 -0.054 1.38E-05 172294 6.249 626 2.78E+09 0.00

Mrt-Jun DAX -2.91E-05 0 0.17 -0.30 4.20E-05 59440 -11.18 920 2.08E+09 0.00

SX5e -2.80E-05 0 0.19 -0.32 4.71E-05 59440 -9.82 859 1.81E+09 0.00

AEX -3.87E-05 0 0.46 -0.45 6.80E-05 59440 -4.82 1066 2.80E+09 0.00

SMI -2.47E-05 0 0.12 -0.20 3.01E-05 59440 -11.74 1013 2.53E+09 0.00

Jun-Sep DAX 1.05E-05 0 0.16 -0.17 3.54E-05 62829 -1.79 336 2.91E+08 0.00

SX5e 1.22E-07 0 0.15 -0.16 3.90E-05 62829 -1.03 205 1.07E+08 0.00

AEX 2.19E-05 0 0.18 -0.20 5.08E-05 62829 -0.68 205 1.07E+08 0.00

SMI 1.56E-05 0 0.13 -0.11 2.69E-05 62829 1.66 298 2.27E+08 0.00

Sep-Dec DAX 1.17E-05 0 0.07 -0.10 2.48E-05 64519 -0.94 135 4.68E+07 0.00

SX5e 8.42E-06 0 0.07 -0.12 3.06E-05 64519 -0.71 91 2.10E+07 0.00

AEX 4.22E-06 0 0.13 -0.13 3.85E-05 64519 -0.30 163 6.89E+07 0.00

SMI 6.67E-06 0 0.07 -0.08 2.14E-05 64519 -0.20 110 3.10E+07 0.00

Dec-Mrt DAX 9.03E-06 0 0.12 -0.10 2.86E-05 57438 1.78 182 7.70E+07 0.00

SX5e 3.37E-06 0 0.15 -0.13 3.54E-05 57438 1.79 170 6.64E+07 0.00

AEX 8.24E-06 0 0.21 -0.19 4.61E-05 57438 2.58 290 1.97E+08 0.00

SMI 6.52E-06 0 0.15 -0.10 2.48E-05 57438 4.12 339 2.70E+08 0.00

This table gives an overview of the descriptive statistics of the returns on the DAX, SX5e, AEX and SMI futures calculated from 30-second intraday time intervals for the 9.00-17.27 period. For each quarter and future are the mean, median, maximum, minimum, standard deviation, number of returns (N), skewness, kurtosis and the Jarque-Berra (JB) with

This table gives an overview of the descriptive statistics of the returns on the DAX, SX5e, AEX and SMI futures calculated from 10-second intraday time intervals for the 9.00-17.27 period. For each quarter and future are the mean, median, maximum, minimum, standard deviation, number of returns (N), skewness, kurtosis and the Jarque-Berra (JB) with accompanying P-value summarized in the various columns.

(19)

Table 6 gives an overview of the descriptive statistics of the returns calculated from 1-second time intervals. The different columns report the specific quarter, index future, mean, median, maximum, minimum, standard deviation, number of returns (N), skewness, kurtosis and Jarque-Berra (JB) with accompanying P-value. Because of the short time interval the mean values for the futures are low. There is virtually no difference between the mean returns of the various futures. Furthermore, the median value for all futures is 0. The relative high maximum and minimum of the AEX futures is remarkable. These outliers are spikes in the AEX future; a temporarily strong up or downside movement which is quickly corrected. Due to the high number of returns (N) I assume that these spikes will have a negligible impact on this study. There are almost no differences between the standard deviations of the futures. In order to test for normality the Jarque-Berra test is used, from which the hypotheses are given by:

normality H :₀ normality non H_a : −

The p-value corresponding to the Jarque-Berra is 0.00, and as a result I reject the null hypothesis at a significance level of 0.01%. I conclude that the residuals are not normally distributed and this could imply that the skewness and kurtosis differ significantly from the normal distribution values of 0 and 3, respectively. The non-normality of the returns can influence my results, because the VAR model assumes normally distributed returns; I will have to be cautious when interpreting the results.

(20)

0.01% significance level, which means I will have to be cautious when interpreting the results of the VAR model.

Table 8 and table 9 show a relative decrease in the spikes of the AEX future for the 10 and 30-second returns. The mean values and standard deviations are still virtually the same for the various futures on each time interval. I also reject the Jarque-Berra

hypothesis of normality for the 10 and 30-second returns, which means that I will have to be cautious when interpreting the results of the VAR model, because the VAR model assumes normally distributed returns.

Table 10 shows the tick values of the different index futures. A tick is the minimum change of a security; in this case the minimum change of the index futures. The minimum change can be different for the various futures and is given on the first row. Furthermore, the second row gives the value of one tick for each index future. The tick values will be used to calculate the exact profit & loss (P&L) balance for the trading strategy in section 6; the difference in ticks will be determined after opening and closing a position in index futures and this number of ticks will be multiplied by their value. The third row shows the average (March 06-March 07) index points for each index and the last row shows the values of a 1 percent change for the futures, which deviates from each other because of differences in index points and tick values.

DAX SX5e AEX SMI

Tick in points 0.5 1 0.05 1

Value of tick € 12.5 € 10 € 10 Sfr 10

Average index points 6244.5 3877 474.75 8355

Value of 1% change € 1,561 € 388 € 948 Sfr 836

Table 10: Tick in points, value of 1 tick, average index points and the value of a 1% change for the different index futures

Tick in points is given on the first row. The second row gives the value of 1 tick. The third row shows the average (March 06-March 07) index points for each index and the last row shows the value for a 1 percent change in each future. These characterisitcs are

(21)

4. Methodology

I follow Chordia and Swaminathan (2000) and Brennan (1993) in their method to find lead-lag effects. First, I calculate the cross-autocorrelation matrices of the returns on the different index futures. I will use this correlation structure to determine the amount of lags I specify in the VAR model, which I use to determine if returns on index futures with a high trading volume lead returns on index futures with a low trading volume.

Returns on the futures of the market indices will be calculated as: (1) R_it =lnP_it −lnP_it₋_s

where P is the observed variable for index i at time t and _it P_it₋_s is the observed variable for index i of the previous period. This will be done for several periods, e.g. s= 1, 5, 10, 30 seconds.

4.1 Cross-autocorrelation structure of returns

The time series {R } is a vector of returns of index i in period t, i =1,2,3,4 and t is the t-_it th observation of the 1, 5, 10 or 30-second returns. Vector 4

ℜ ∈

t

R is given

byR =t [R1t,R2t,R3t,R4t]T, ∀t. As in Gebka (2002), Chordia and Swaminathan (2000)

and Brennan (1993) the different cross-autocorrelation matrices are calculated from this matrix up to 4 lags.

The cross-autocorrelation matrices are examined for correlation coefficients between the different index futures. Significance of correlations is determined at a 5% and 1% level. Using the 1% significance level I specify the amount of lags that I will use in the VAR model. Due to noise (mentioned by AitSahalia et al., 2005) I only take a particular lag into account if the cross-correlation for particular futures are significant at a 1% level for all quarters. I assume that this way of lag determination will not influence my

(22)

4.2 VAR model

Hereafter, a vector autoregression (VAR) model will be specified as in Gebka (2002), Chordia and Swaminathan (2000) and Brennan (1993). A VAR can be efficiently

estimated by running ordinary least squares (OLS) on each equation individually, because the regressors are the same for both regressions. I start by considering two indices. Index B is assumed to lead returns on index A, but not vice versa. The following VAR model can now be estimated:

(1) _t K k k t B k K k k t A k t A a a R b R u R = +

_∑

+

_∑

+ = − = − 1 , 1 , 0 , (2) _t K k k t B k K k k t A k t B c c R d R v R = +

_∑

+

_∑

+ = − = − 1 , 1 , 0 ,

where R_K_,_t₋_i is the return on index future K at time t-i. The terms

∑

= − K k k t A kR a 1 , and

∑

= − K k k t B kR d 1

, take the own autocorrelations of the futures into account.

The VAR test is used to answer the question if returns on index futures with a high trading volume can predict returns on index futures with a low trading volume better than returns on index futures with a low trading volume can predict returns on index futures with a high trading volume.

If, in regression (1), lagged returns on index future B have predictive power for the current returns on index future A, controlling for the own autocorrelations of index future A, returns of index future B are said to lead returns on index future A. I examine, in equation (1), if the sum of the coefficients belonging to the return on index future B is greater than zero. This test determines if there exist cross-autocorrelations independent of autocorrelations.

(23)

Next, I test if lagged returns on index future B can better predict returns on index future A than lagged returns on index future A can predict returns on index future B. As in Chorida and Swaminathan (2000) I calculate the asymptotic Z-statistic:

V c b z K k k K k k

∑

= = − = 1 1 were

∑

= K k k b 1

is the sum of coefficients corresponding with the high trading index future

and

∑

= K k k d 1

is the sum of coefficients corresponding with the low trading index future. V

is the estimated variance.

This test corresponds with the following hypothesis:

∑

= K k k b H 1 0 :

∑

= = K k k c 1

∑

= K k k a b H 1 :

∑

= > K k k c 1

This test is performed to determine if returns of index B lead returns on index A. It tests any asymmetry in cross-autocorrelation between high trading volume and low trading volume indices.

4.3 Opening effect

Furthermore, I study a possible opening effect between the DAX and SX5e futures. Futures on both indices start trading at 08.00 and closes at 22.00. Between the close and opening (‘overnight period’) of the futures, there is a limited amount of information that influences both futures differently. However, there can be overnight events with a

(24)

However, companies with the greatest influence in the DAX also appear in the SX5e and the SX5e comprises of 50 stocks (against 30 in the DAX), which diminishes the influence of 1 company on the total return of the SX5e (see Appendix 1). So, it is unlikely that overnight releases of information influences DAX and SX5e futures significantly

different or will influence my conclusions. Moreover, even if such an event takes place it will only affect 1 of 248 trading days. Another overnight event that can influence both index futures differently is an unexpected change in dividend policy of a company with a significant influence in the SX5e. Such a change will affect the SX5e and DAX futures differently, because the DAX is a total return index (adjusted for dividends) and the SX5e is not, which can result in a different percentage opening price. However, there were no announcements of large unexpected changes in dividend policy, so this influence is negligible.

So, the SX5e and DAX futures should open with the same relative price change at 08.00. If both indices open differently, I expect a price reversal in the 8.00-8.01 period; I expect that the relative return on the lagging index future moves towards the relative return on the leading index future. I will calculate the difference between the relative opening returns on the SX5e and DAX futures and I will calculate the difference between the relative returns on the SX5e and DAX futures after the first minute of the new trading day. Thereafter, I will perform a regression analysis over these differences:

(3) R_DAX_,₈_.₀₁ −R_SX₅_e_,₈_.₀₁ =α₀ +β(R_DAX_,_opening −R_SX₅_e_,_opening)+ε

were R_DAX_,₈_.₀₁−R_SX₅_e_,₈_.₀₁ is the difference in relative returns between the DAX and SX5e futures for the 8.00-8.01 period and R_DAX_,_opening −R_SX₅_e_,_opening is the difference in the relative opening between the DAX and SX5e futures.

(25)

5. Empirical Results

This section presents the empirical results; section 5.1 shows the cross-autocorrelation matrices for the 1, 5, 10 and 30-second period returns on the DAX, SX5e, AEX and SMI index futures and determines the amount of lags used in the VAR model. Section 5.2 reports the results of the VAR model and lead-lag relations between the index futures. Section 5.3 reports the results of the opening effect.

5.1 Cross-autocorrelation matrices

The cross-autocorrelation matrices for the 1, 5, 10 and 30-second period returns on the DAX, SX5e, AEX and SMI futures are reported in tables 11, 12, 13 and 14. Table 11 shows the cross- autocorrelations from 1-second intraday returns on the DAX, SX5e, AEX and SMI futures for the 9.00-17.27 daily period. The cross-autocorrelation matrices are given for the four quarters: The first column shows cross-autocorrelation matrices for the March-June quarter; the second column denotes cross-autocorrelation matrices for the June-September quarter; and the third and fourth column report these correlation matrices for the September-December and December-March quarters, respectively. Furthermore, the first row with matrices shows the cross- and autocorrelations without lags. The second row matrices gives the correlations at 1 lag; the third row reports matrices with two lags of the 1-second period; the fourth row shows the cross-autocorrelation matrices at the third lag and the final row denotes the cross-autocorrelation with four lags of a 1-second period. * denotes significance at a 5% level and ** denotes significance on a 1% level.

As expected, table 11 shows negative first order autocorrelations significant at a 1% level. This can be explained by the bid-ask bounce, mentioned by Patnaik et al. (2004). Furthermore, table 11 shows no positive first order autocorrelations, so a potential lead-lag effect on a 1-second intraday basis is not a consequence of positive autocorrelations, as Boudoukh et al. (1994) argues for longer time periods. Although further

(26)

Table 11: Cross-autocorrelation matrices from 1-second intraday returns on the DAX, SX5e, AEX and SMI futures

DAXt SXt AEXt SMIt DAXt SXt AEXt SMIt DAXt SXt AEXt SMIt DAXt SXt AEXt SMIt

DAXt 1.000 0.111** 0.075** 0.063** 1.000 0.111** 0.085** 0.080** 1.000 0.098** 0.082** 0.079** 1.000 0.163** 0.113** 0.111** SXt 0.111** 1.000 0.054** 0.038** 0.111** 1.000 0.057** 0.057** 0.098** 1.000 0.053** 0.047** 0.163** 1.000 0.067** 0.064** AEXt 0.075** 0.054** 1.000 0.033** 0.085** 0.057** 1.000 0.042** 0.082** 0.053** 1.000 0.041** 0.113** 0.067** 1.000 0.056** SMIt 0.063** 0.038** 0.033** 1.000 0.080** 0.057** 0.042** 1.000 0.079** 0.047** 0.041** 1.000 0.111** 0.064** 0.056** 1.000 DAXt-1 -0.147** 0.080** 0.059** 0.068** -0.150** 0.066** 0.063** 0.083** -0.151** 0.051** 0.058** 0.083** -0.105** 0.072** 0.069** 0.100** SXt-1 0.084** -0.193** 0.040** 0.045** 0.066** -0.188** 0.038** 0.050** 0.059** -0.188** 0.031** 0.048** 0.066** -0.187** 0.040** 0.054** AEXt-1 0.034** 0.022** -0.151** 0.028** 0.035** 0.018** -0.180** 0.036** 0.038** 0.019** -0.169** 0.039** 0.034** 0.023** -0.179** 0.040** SMIt-1 0.024** 0.020** 0.018** -0.085** 0.029** 0.022** 0.023** -0.075** 0.028** 0.018** 0.021** -0.072** 0.033** 0.026** 0.025** -0.086** DAXt-2 0.010** 0.050** 0.044** 0.053** -0.008** 0.037** 0.036** 0.062** -0.012** 0.030** 0.032** 0.053** -0.009** 0.037** 0.039** 0.066** SXt-2 0.034** -0.058** 0.026** 0.034** 0.027** -0.071** 0.023** 0.036** 0.025** -0.079** 0.019** 0.030** 0.018** -0.071** 0.020** 0.036** AEXt-2 0.011** 0.014** -0.106** 0.019** 0.013** 0.013** -0.054** 0.022** 0.013** 0.007** -0.054** 0.024** 0.009** 0.014** -0.085** 0.023** SMIt-2 0.008** 0.013** 0.010** -0.033** 0.013** 0.011** 0.011** -0.026** 0.012** 0.007** 0.010** -0.023** 0.007** 0.013** 0.011** -0.022** DAXt-3 0.005** 0.031** 0.030** 0.041** -0.003** 0.025** 0.021** 0.041** -0.012** 0.020** 0.020** 0.038** -0.011** 0.020** 0.017** 0.038** SXt-3 0.012** -0.027** 0.018** 0.026** 0.011** -0.037** 0.012** 0.025** 0.007** -0.043** 0.009** 0.020** 0.002* -0.037** 0.008** 0.021** AEXt-3 0.003** 0.007** -0.044** 0.016** 0.004** 0.008** -0.048** 0.017** 0.004** 0.007** -0.078** 0.015** 0.001 0.006** -0.055** 0.015** SMIt-3 0.004** 0.010** 0.007** -0.015** 0.004** 0.009** 0.008** -0.011** 0.000 0.007** 0.006** -0.013** 0.001 0.007** 0.003** -0.016** DAXt-4 -0.006** 0.020** 0.016** 0.030** -0.006** 0.015** 0.012** 0.030** -0.012** 0.011** 0.007** 0.027** -0.014** 0.009** 0.004** 0.023** SXt-4 0.000 -0.017** 0.008** 0.017** 0.003** -0.023** 0.005** 0.016** 0.000 -0.028** 0.003** 0.012** -0.004** -0.025** 0.002* 0.014** AEXt-4 -0.001 0.007** -0.048** 0.012** 0.004** 0.006** -0.018** 0.012** -0.001 0.004** -0.038** 0.011** -0.002* 0.002 -0.025** 0.009** SMIt-4 0.002* 0.004** 0.005** -0.010** 0.002 0.007** 0.002** -0.009** 0.002* 0.004** 0.000 -0.011** -0.002* 0.006** 0.002* -0.010**

This table shows cross-autocorrelation matrices from 1-second intraday returns on the DAX, SX5e, AEX and SMI futures for the 9.00-17.27 period. Correlations of four quarters are shown above: The first column with matrices reports cross-autocorrelation matrices for the quarter March-June; the second column with matrices denotes cross-autocorrelations for June-September; the third column shows the correlations for September-December and the fourth column for December-March. Furthermore, the matrices on the first row show

cross-correlations without lags. The matrices on the second row report cross-autocorrelation matrices at one lag of 1-second; the third row gives matrices at two lags of 1-second. The fourth row shows the cross-autocorrelation matrices at the third lag and the fifth row gives the cross-autocorrelations at four lags of a 1-second period. * shows significance at a 5% level and ** shows significance at a 1% level.

September-December December-March

(27)

Table 12: Cross-autocorrelation matrices from 5-second intraday returns of the DAX, SX5e, AEX and SMI futures

DAXt 1.000 0.428** 0.292** 0.245** 1.000 0.370** 0.276** 0.283** 1.000 0.324** 0.264** 0.272** 1.000 0.412** 0.309** 0.324** SXt 0.428** 1.000 0.233** 0.191** 0.370** 1.000 0.195** 0.211** 0.324** 1.000 0.174** 0.181** 0.412** 1.000 0.218** 0.225** AEXt 0.292** 0.233** 1.000 0.146** 0.276** 0.195** 1.000 0.156** 0.264** 0.174** 1.000 0.152** 0.309** 0.218** 1.000 0.185** SMIt 0.245** 0.191** 0.146** 1.000 0.283** 0.211** 0.156** 1.000 0.272** 0.181** 0.152** 1.000 0.324** 0.225** 0.185** 1.000 DAXt-1 -0.047** 0.130** 0.114** 0.166** -0.071** 0.107** 0.077** 0.168** -0.097** 0.079** 0.063** 0.140** -0.076** 0.087** 0.061** 0.135** SXt-1 0.056** -0.165** 0.073** 0.114** 0.048** -0.200** 0.046** 0.107** 0.033** -0.231** 0.034** 0.080** 0.024** -0.207** 0.036** 0.081** AEXt-1 0.027** 0.045** -0.221** 0.079** 0.024** 0.040** -0.213** 0.075** 0.017** 0.028** -0.238** 0.064** 0.015** 0.034** -0.258** 0.058** SMIt-1 0.019** 0.037** 0.030** -0.068** 0.020** 0.034** 0.026** -0.050** 0.013** 0.028** 0.019** -0.059** 0.010** 0.036** 0.020** -0.070** DAXt-2 0.009** 0.029** 0.011** 0.044** -0.015** 0.019** 0.007** 0.051** -0.018** 0.021** 0.005** 0.038** -0.011** 0.021** 0.008** 0.024** SXt-2 0.019** -0.018** 0.009** 0.028** 0.002 -0.044** 0.005** 0.033** 0.003* -0.049** 0.004* 0.022** 0.005** -0.038** 0.003 0.017** AEXt-2 0.016** 0.017** -0.064** 0.017** 0.003* 0.005** -0.078** 0.023** 0.004** 0.006** -0.063** 0.017** 0.003 0.011** -0.053** 0.015** SMIt-2 0.016** 0.013** 0.007** -0.014** 0.004* 0.007** 0.000 -0.014** 0.006** 0.013** 0.004** -0.017** 0.011** 0.016** 0.009** -0.017** DAXt-3 0.006** 0.018** 0.004** 0.013** -0.008** 0.005** -0.003 0.018** -0.007** 0.008** 0.006** 0.015** 0.008** 0.010** 0.010** 0.014** SXt-3 0.006** -0.008** 0.001 0.009** 0.002 -0.021** -0.005** 0.009** 0.003 -0.019** 0.002 0.009** 0.009** -0.010** 0.010** 0.011** AEXt-3 0.006** 0.008** -0.024** 0.006** 0.004* 0.003 -0.034** 0.009** -0.001 0.006** -0.037** 0.005** 0.007** 0.007** -0.015** 0.003 SMIt-3 0.004* 0.009** 0.002 -0.012** 0.001 0.002 -0.001 -0.013** 0.004** 0.004** 0.005** -0.009** 0.012** 0.009** 0.011** -0.005** DAXt-4 0.001 0.005** 0.004* 0.000 -0.007** -0.003 -0.002 0.004* -0.006** 0.001 0.001 0.005** 0.003 0.011** 0.012** 0.012** SXt-4 0.004* -0.006** 0.003* 0.002 0.000 -0.007** 0.003 0.004* 0.000 -0.011** 0.004* 0.004* 0.006** -0.001 0.011** 0.012** AEXt-4 0.003 0.003 -0.013** 0.004* 0.000 0.001 -0.021** 0.003* 0.001 0.000 -0.016** 0.004* 0.005** 0.003 -0.004* 0.011** SMIt-4 0.004* 0.004* 0.005** -0.001 0.001 -0.001 0.002 -0.008** 0.001 0.000 0.002 -0.012** 0.005** 0.008** 0.012** 0.001

This table shows cross-autocorrelation matrices from 5-second intraday returns on the DAX, SX5e, AEX and SMI futures for the 9.00-17.27 period. Correlations of four quarters are shown above: The first column with matrices reports cross-autocorrelation matrices for the quarter March-June; the second column with matrices denotes cross-autocorrelations for June-September; the third column shows the correlations for September-December and the fourth column for December-March. Furthermore, the matrices on the first row show cross-correlations without lags. The matrices on the second row report autocorrelation matrices at one lag of 5-seconds; the third row gives matrices at two lags of 5-seconds. The fourth row shows the

cross-autocorrelation matrices at the third lag and the fifth row gives the cross-cross-autocorrelations at four lags of a 5-second period. * shows significance at a 5% level and ** shows significance at a 1% level.

(28)

DAXt 1.000 0.584** 0.402** 0.360** 1.000 0.535** 0.398** 0.428** 1.000 0.479** 0.381** 0.410** 1.000 0.580** 0.473** 0.476** SXt 0.584** 1.000 0.349** 0.304** 0.535** 1.000 0.317** 0.345** 0.479** 1.000 0.287** 0.305** 0.580** 1.000 0.383** 0.380** AEXt 0.402** 0.349** 1.000 0.226** 0.398** 0.317** 1.000 0.263** 0.381** 0.287** 1.000 0.248** 0.473** 0.383** 1.000 0.326** SMIt 0.360** 0.304** 0.226** 1.000 0.428** 0.345** 0.263** 1.000 0.410** 0.305** 0.248** 1.000 0.476** 0.380** 0.326** 1.000 DAXt-1 -0.076** 0.087** 0.061** 0.135** -0.060** 0.082** 0.049** 0.138** -0.070** 0.073** 0.040** 0.119** -0.047** 0.071** 0.050** 0.099** SXt-1 0.024** -0.207** 0.036** 0.081** 0.029** -0.187** 0.032** 0.100** 0.022** -0.220** 0.021** 0.073** 0.017** -0.178** 0.029** 0.067** AEXt-1 0.015** 0.034** -0.258** 0.058** 0.018** 0.032** -0.245** 0.068** 0.018** 0.027** -0.270** 0.060** 0.010** 0.034** -0.231** 0.055** SMIt-1 0.010** 0.036** 0.020** -0.070** 0.013** 0.028** 0.014** -0.051** 0.017** 0.031** 0.017** -0.047** 0.018** 0.040** 0.024** -0.048** DAXt-2 0.004 0.020** 0.009** 0.012** -0.015** -0.001 -0.002 0.015** -0.016** 0.005* 0.004 0.013** 0.007** 0.023** 0.026** 0.025** SXt-2 0.011** -0.011** 0.008** 0.012** 0.001 -0.021** 0.002 0.01** 0.002 -0.030** 0.006* 0.013** 0.013** -0.006** 0.025** 0.024** AEXt-2 0.006* 0.008** -0.041** 0.009** 0.002 0.004 -0.063** 0.009** -0.002 0.006** -0.041** 0.005* 0.012** 0.017** -0.009** 0.017** SMIt-2 0.006** 0.013** 0.008** -0.008** 0.001 0.001 0.000 -0.017** 0.002 0.006** 0.004 -0.020** 0.016** 0.018** 0.027** -0.002 DAXt-3 0.005* 0.008** 0.005* 0.016** 0.000 -0.002 0.001 0.013** -0.002 0.004 0.009** 0.006** 0.009** 0.011** 0.021** 0.030** SXt-3 0.003 0.001 0.004 0.015** -0.001 -0.019** -0.002 0.008** 0.000 -0.012** 0.009** 0.003 0.011** 0.005* 0.018** 0.025** AEXt-3 0.008** 0.008** -0.001 0.010** -0.001 -0.004 -0.018** 0.006* 0.000 -0.002 -0.009** 0.003 0.006** 0.008** 0.001 0.021 SMIt-3 0.008** 0.009** 0.008** 0.004 0.000 -0.006* 0.000 -0.006** 0.007** 0.004 0.010** -0.005* 0.011** 0.012** 0.015** 0.015** DAXt-4 -0.005* 0.005 0.006* 0.019** 0.003 0.003 0.001 0.012** 0.003 0.000 0.009** 0.017** 0.000 0.006* 0.011** 0.018** SXt-4 0.001 -0.001 0.010** 0.016** 0.006** 0.003 0.006** 0.010** 0.004* -0.003 0.007** 0.010** -0.003 -0.003 0.007** 0.011** AEXt-4 0.001 0.004 0.011** 0.013** 0.003 0.003 0.004 0.006** 0.009** 0.005* -0.006* 0.013** 0.001 0.003 0.000 0.011** SMIt-4 0.000 0.004 0.001 0.012** 0.004 0.010** 0.006** 0.007 0.008** 0.007** 0.016** 0.001 0.001 0.006* 0.009** 0.007**

This table shows cross-autocorrelation matrices from 10-second intraday returns on the DAX, SX5e, AEX and SMI futures for the 9.00-17.27 period. Correlations of four quarters are shown above: The first column with matrices reports cross-autocorrelation matrices for the quarter March-June; the second column with matrices denotes cross-autocorrelations for June-September; the third column shows the correlations for September-December and the fourth column for December-March. Furthermore, the matrices on the first row show cross-correlations without lags. The matrices on the second row report autocorrelation matrices at one lag of 10-seconds; the third row gives matrices at two lags of 10-seconds. The fourth row shows the cross-autocorrelation matrices at the third lag and the fifth row gives the cross-cross-autocorrelations at four lags of a 10-second period. * shows significance at a 5% level and ** shows significance at a 1% level.

(29)

DAXt 1.000 0.851** 0.661** 0.661** 1.000 0.802** 0.656** 0.672** 1.000 0.724** 0.619** 0.594** 1.000 0.798** 0.678** 0.642** SXt 0.851** 1.000 0.646** 0.632** 0.802** 1.000 0.611** 0.621** 0.724** 1.000 0.553** 0.520** 0.798** 1.000 0.641** 0.595** AEXt 0.661** 0.646** 1.000 0.512** 0.656** 0.611** 1.000 0.509** 0.619** 0.553** 1.000 0.454** 0.678** 0.641** 1.000 0.524** SMIt 0.661** 0.632** 0.512** 1.000 0.672** 0.621** 0.509** 1.000 0.594** 0.520** 0.454** 1.000 0.642** 0.595** 0.524** 1.000 DAXt-1 -0.005 0.040** 0.031** 0.052** -0.020** 0.020** 0.025** 0.066** -0.038** 0.038** 0.046** 0.066** -0.007 0.054** 0.067** 0.082** SXt-1 0.014** -0.050** 0.025** 0.040** 0.018** -0.102** 0.021** 0.056** 0.011** -0.139** 0.043** 0.049** 0.017** -0.074** 0.059** 0.067** AEXt-1 0.017** 0.025** -0.186** 0.032** 0.010* 0.013** -0.185** 0.047** 0.012** 0.026** -0.172** 0.049** 0.015** 0.038** -0.122** 0.064** SMIt-1 0.013** 0.023** 0.015** -0.013** 0.012** 0.010** 0.017** -0.023** 0.021** 0.029** 0.043** -0.038** 0.028** 0.048** 0.059** 0.004 DAXt-2 -0.002 0.000 0.007 0.031** -0.017** -0.021** -0.003 0.005 0.013** 0.011** 0.031** 0.018** -0.010* 0.001 0.005 0.007 SXt-2 0.004 -0.001 0.010* 0.034** -0.013** -0.021** 0.000 0.008* 0.011** 0.003 0.025** 0.019** -0.010* -0.002 0.003 0.009* AEXt-2 0.000 0.000 -0.002 0.027** -0.007 -0.014** -0.004 0.006 0.009* 0.007 0.023** 0.010** -0.011* -0.003 -0.013** -0.002 SMIt-2 -0.007 -0.006 0.001 0.014** -0.001 -0.011** 0.004 0.002 0.010** 0.010* 0.021** 0.000 -0.006 0.002 0.004 -0.005 DAXt-3 0.002 0.004 0.005 0.025** 0.000 -0.003 0.009* 0.000 0.000 0.003 0.017** 0.006 -0.022** -0.015** -0.010* -0.009* SXt-3 0.001 0.002 0.004 0.025** -0.003 -0.011** 0.005 -0.003 0.004 0.001 0.017** 0.003 -0.029** -0.026** -0.014** -0.015** AEXt-3 -0.001 0.004 0.002 0.019** 0.000 -0.003 -0.001 -0.002 0.002 0.002 0.001 0.004 -0.022** -0.018** -0.010* -0.010* SMIt-3 0.003 0.006 0.008* 0.019** 0.005 0.005 0.014** -0.009* -0.006 -0.003 0.013** -0.003 -0.018** -0.013** -0.008 -0.021** DAXt-4 -0.010* -0.013** -0.003 0.015** 0.009* 0.007 0.011** 0.002 -0.008 -0.010** 0.009* 0.001 -0.005 -0.007 -0.003 0.008 SXt-4 -0.009* -0.012** 0.001 0.015** 0.009* 0.009* 0.012** 0.001 -0.009* -0.01* 0.007 0.003 0.001 -0.002 0.000 0.011** AEXt-4 -0.001 -0.009* 0.001 0.017** 0.004 0.002 0.003 0.000 -0.006 -0.008 0.017** 0.002 -0.007 -0.009* -0.005 -0.004 SMIt-4 -0.001 -0.005 0.003 0.015** 0.002 0.001 0.006 0.001 0.001 -0.005 0.010* 0.001 -0.010** -0.012** -0.011** 0.006

This table shows cross-autocorrelation matrices from 30-second intraday returns on the DAX, SX5e, AEX and SMI futures for the 9.00-17.27 period. Correlations of four quarters are shown above: The first column with matrices reports cross-autocorrelation matrices for the quarter March-June; the second column with matrices denotes cross-autocorrelations for June-September; the third column shows the correlations for September-December and the fourth column for December-March. Furthermore, the matrices on the first row show cross-correlations without lags. The matrices on the second row report autocorrelation matrices at one lag of 30-seconds; the third row gives matrices at two lags of 30-seconds. The fourth row shows the cross-autocorrelation matrices at the third lag and the fifth row gives the cross-cross-autocorrelations at four lags of a 30-second period. * shows significance at a 5% level and ** shows significance at a 1% level.

(30)

volume index (SMI) is striking. For example, the correlation coefficient of

the 1-period lagged DAX with the SMI at no lags is for all periods about three times greater than the cross-autocorrelation coefficient of the lagged SMI with the DAX.

In almost all cases, the high trading volume index has higher cross-correlations with the low trading volume index than the lower does with the higher. This is the case in all quarters. I should be careful with drawing conclusions, however, because the cross-autocorrelations are not adjusted for own cross-autocorrelations, which are the correlations in the own returns of a security. Moreover, the autocorrelations of the SMI future seem significantly lower, at lag 1, than those of the other futures. Furthermore, the

non-synchronous trading bias may result in spurious cross-autocorrelations as explained by Lo and MacKinley (1990b).

All correlations up until two lags are significant at a 1% significant level for the 1-second returns. Although there are still significant correlations for the third and fourth lag, these cross-correlations are lower than for the first two lags. Some cross-autocorrelations between the index futures are significant at a 1% significance level for all quarters, so these lags are still taken into account when estimating the VAR model. However, as Gebka (2002) and Brennan et al. (1993) noted, it is unlikely that the fourth lag will make a difference in the conclusions, because of the relative low impact of this lag.

(31)

1% level. Not a single cross-autocorrelation is significant for all periods at the fourth lag at a 1% significance level and I will therefore include three lags of 5-second returns into my VAR model.

Table 13 denotes cross-autocorrelations matrices for 10-second returns. Again, this table is composed in the same manner as the previous cross-autocorrelation tables. Cross-autocorrelation coefficients increased at 0 and 1 lags compared with the 1 and 5-second returns. Furthermore, own autocorrelations of the index futures look stable compared to the 5-second returns. There is still an asymmetric cross-autocorrelation structure; the high trading volume index future has higher cross-correlations with the low trading volume index future than the lower does with the higher. Just as for the 5-second returns there is no cross-autocorrelation coefficient at the fourth lag that is significant for all quarters on a 1% level and I will therefore include only three lags of the 10-second returns into my VAR model.

Table 14 gives the cross-autocorrelations from 30-second intraday returns. Just as in the previous periods, the relative high trading volume indices have higher cross-correlations with the relative low trading volume indices than vice versa. There is also pattern of asymmetry and an indication of lead-lag relations for returns on a 30-second intraday basis. For the 30-second periods I will take only one lag of 30-second returns into account, because at a 1% significance level further lags did not have significant cross-autocorrelations between index futures for all quarters.

Section 5.1 showed asymmetric cross-autocorrelation matrices; high trading volume index future has higher cross-correlations with low trading volume index future than the lower does with the higher. However, the non-synchronous trading bias may result in spurious cross-autocorrelations as explained by Lo and MacKinley (1990b).

5.2 Lead-lag relations

(32)

and SMI futures. Table 15 shows the results generated by the VAR model. Results are summarized for the quarters: March June 06; June September 06; September 06-December 06 and 06-December 06-March 07. The dependent/independent column shows the dependent index future first, followed by the independent index future. L1 refers to a₁ or

1

c and H1 denotes b₁ or d₁ in equation (1) and equation (2). Similarly, Low refers to

∑

= K k k a 1 or

_∑

= K k k c 1

and High refers to

_∑

= K k k b 1 or

_∑

= K k k d 1

. Column six denotes the

adjustedR , which is the adjusted coefficient of determination. 2 Z( A) is the asymptotic

Z-statistic belonging to the null hypothesis

∑

= K k k b H 1 0 :

∑

= = K k k c 1

in each bivariate VAR,

where the alternative hypothesis is

∑

= K k k a b H 1 :

∑

= > K k k c 1

. K is equal to 4, meaning that four

lags of 1-second returns are taken into account. Finally, the symbols * and ** denote significance at the 5% and 1% levels, respectively.

Starting for the March–June quarter, the results indicate that lagged returns on the high volume trading DAX future do indeed have predictive value for the low volume trading AEX, SX5e and SMI futures. This is the case for all quarters. The sum of coefficients belonging to lagged returns on the DAX is positive and significant at the 1 percent level. Furthermore, the results suggest predictability beyond lag 1 for all quarters: the

(33)

dependent/ independent

L1 Low H1 High L1 Low H1 High

SX5e/DAX -0.274** -0.668** 0.223** 0.706** 0.092 153.8** -0.263** -0.667** 0.199** 0.624** 0.086 138.2** DAX/SX5e 0.089** 0.214** -0.178** -0.278** 0.041 0.074** 0.187** -0.180** -0.303** 0.038 AEX/DAX -0.215** -0.630** 0.247** 0.870** 0.076 229.2** -0.231** -0.541** 0.230** 0.699** 0.067 200.8** DAX/AEX 0.021** 0.044** -0.155** -0.188* 0.025 0.028** 0.065** -0.164** -0.246** 0.028 SMI/DAX -0.109** -0.230** 0.065** 0.223** 0.027 43.7** -0.106** -0.224** 0.080** 0.256** 0.031 44.8** DAX/SMI 0.050** 0.094** -0.152** -0.178** 0.023 0.070** 0.140** -0.162** -0.240** 0.027 AEX/SX5e -0.203** -0.588** 0.132** 0.454** 0.065 79.6** -0.217** -0.493** 0.104** 0.325** 0.060 85.3** SX52/AEX 0.032** 0.098** -0.228** 0.486** 0.057 0.035** 0.112** -0.229** -0.534** 0.056 SMI/SX5e -0.099** -0.203** 0.034** 0.116** 0.018 -42.4 -0.090** -0.181** 0.036** 0.119** 0.017 -61.5 SX5e/SMI 0.086** 0.236** -0.225** -0.475** 0.055 0.108** 0.293** -0.229** -0.535** 0.060 SMI/AEX -0.093** -0.184** 0.012** 0.040** 0.013 -101.5 -0.084** -0.356** 0.017** 0.053** 0.012 -93.0 AEX/SMI 0.123** 0.361** -0.192** -0.551** 0.056 0.129** 0.332** -0.211** -0.471** 0.051 dependent/ independent

SX5e/DAX -0.224** -0.601** 0.234** 0.692** 0.070 208.7** -0.274** -0.703** 0.251** 0.728** 0.090 207.5** DAX/SX5e 0.029** 0.066** -0.166** -0.274** 0.030 0.064** 0.141** -0.134** -0.252** 0.024 AEX/DAX -0.255** -0.660** 0.185** 0.583** 0.081 146.0** -0.253** -0.669** 0.301** 0.877** 0.085 293.1** DAX/AEX 0.058** 0.145** -0.176** -0.308** 0.036 0.022** 0.048** -0.116** -0.197** 0.015 SMI/DAX -0.100** -0.212** 0.059** 0.263** 0.027 63.5** -0.126** -0.262** 0.114*** 0.322** 0.038 88.2** DAX/SMI 0.062** 0.121** -0.163** -0.266** 0.028 0.061** 0.106** -0.115** -0.196** 0.015 AEX/SX5e -0.232** -0.567** 0.082** 0.263** 0.063 48.4** -0.234** -0.607** 0.122** 0.370** 0.070 63.5** SX52/AEX 0.037** 0.110** -0.210** -0.550** 0.059 0.041** 0.124** -0.230** -0.542** 0.061 SMI/SX5e -0.081** 0.158** 0.019** 0.057** 0.011 -73.6 -0.104** -0.204** 0.042** 0.131** 0.020 -55.7 SX5e/SMI 0.118** 0.305** -0.205** -0.536** 0.056 0.113** 0.298** -0.229** -0.541** 0.061 SMI/AEX -0.084** -0.167** 0.032** 0.100** 0.014 -54.3 -0.097** -0.184** 0.021** 0.061** 0.014 -109.0 AEX/SMI 0.096** 0.251** -0.231** -0.566** 0.064 0.152** 0.388** -0.226** -0.582** 0.064

Table 15: Vector autoregressions for the 1-second returns on the DAX, SX5e, AEX and SMI futures

The following vector autoregressive model is estimated using 1-second intraday returns for the quarters March 06-June 06, June 06-September 06, September 06-December 06 and December 06-March 07:

The dependent/ independent column shows first the dependent future and than the independent future. Low refers to or and High refers to or Similarly, L1 refers to or and H1 denotes or is the adjusted coefficient of determination. is the asymptotic Z-statistic belonging to the null hypothesis

in each bivariate VAR. The alternative hypothesis is K = 4 for the 1-second returns. * and ** denote significant at the 5% and 1% levels, respectively.

March - June

September-December December -March

(34)

SX5e/DAX -0.375** -0.853** 0.417** 0.987** 0.118 244.1** -0.373** -0.738** 0.374** 0.762** 0.120 146.2** DAX/SX5e 0.095** 0.215** -0.109** -0.218** 0.014 0.084** 0.159** -0.123** -0.226** 0.016 AEX/DAX -0.379** -0.903** 0.508** 1.155** 0.135 279.6** -0.327** -0.667** 0.373** 0.728** 0.105 149.6** DAX/AEX 0.028** 0.072** -0.068** -0.101** 0.006 0.030** 0.059** -0.093** -0.153** 0.009 SMI/DAX -0.141** -0.251** 0.152** 0.264** 0.050 48.5** -0.139** -0.248** 0.164** 0.294** 0.052 56.0** DAX/SMI 0.048** 0.096** -0.057** -0.067** 0.004 0.072** 0.117** -0.090** -0.151** 0.008 AEX/SX5e -0.340** -0.815** 0.305** 0.786** 0.110 125.2** -0.293** -0.596** 0.198** 0.419** 0.087 63.3** SX52/AEX 0.084** 0.226** -0.221** -0.426** 0.048 0.079** 0.166** -0.255** -0.460** 0.063 SMI/SX5e -0.112** -0.212** 0.085** 0.170** 0.029 -45.4 -0.098** -0.182** 0.082** 0.134** 0.026 -72.4 SX5e/SMI 0.161** 0.340** -0.199** -0.347** 0.039 0.200** 0.363** -0.255** -0.465** 0.063 SMI/AEX -0.093** -0.170** 0.039** 0.086** 0.017 -113.4 -0.075** -0.134** 0.038** 0.076** 0.014 -79.4 AEX/SMI 0.259** 0.593** -0.282** -0.620** 0.080 0.238** 0.431** -0.272** -0.546** 0.077 dependent/ independent

SX5e/DAX -0.380** -0.742** 0.358** 0.762** 0.124 191.8** -0.389** -0.748** 0.408** 0.848** 0.119 209.5** DAX/SX5e 0.063** 0.126** -0.138** -0.235** 0.018 0.061** 0.122** -0.120* -0.193** 0.012 AEX/DAX -0.349* -0.692** 0.365** 0.744** 0.113 207.1** -0.393** -0.746** 0.434** 0.882** 0.129 239.9** DAX/AEX 0.029** 0.057** -0.118** -0.186** 0.014 0.026** 0.051** -0.096** -0.134** 0.009 SMI/DAX -0.130** -0.224** 0.158** 0.281** 0.040 52.5** -0.150** -0.246** 0.169** 0.282** 0.040 55.0** DAX/SMI 0.065** 0.115** -0.117** -0.187** 0.013 0.054** 0.108** -0.094** -0.136** 0.008 AEX/SX5e -0.313** -0.612** 0.160** 0.359** 0.095 40.5** 0.351** -0.657** 0.212** 0.469** 0.109 49.9** SX52/AEX 0.079** 0.178** -0.292** -0.528** 0.080 0.087** 0.199** -0.270** -0.471** 0.068 SMI/SX5e -0.090** -0.159** 0.062** 0.126** 0.018 -76.2 -0.109** -0.185** 0.072** 0.139** 0.020 -78.7 SX5e/SMI 0.187** 0.367** -0.201** -0.529** 0.080 0.195** 0.388** -0.266** -0.464** 0.066 SMI/AEX -0.080** -0.134** 0.036** 0.071** 0.012 -75.4 -0.094** -0.152** 0.037** 0.073** 0.014 -96.5 AEX/SMI 0.206** 0.408** -0.299** -0.578** 0.088 0.245** 0.505** -0.328** -0.598** 0.098

March - June

(35)

SX5e/DAX -0.388** -0.737** 0.440** 0.835** 0.071 131.4** -0.432** -0.826** 0.425** 0.809** 0.093 128.4** DAX/SX5e 0.102** 0.178** -0.098** -0.175** 0.006 0.098** 0.193** -0.123** -0.231** 0.009 AEX/DAX -0.454** -0.864** 0.553** 1.052** 0.123 177.2** -0.404** -796** 0.424** 0.837** 0.112 140.2** DAX/AEX 0.031** 0.058** -0.037** -0.061** 0.002 0.036** 0.069** -0.076** -0.127** 0.005 SMI/DAX -0.133** -0.184** 0.140** 0.198** 0.035 25.7** -0.159** -0.264** 0.161** 0.260** 0.041 36.0** DAX/SMI 0.039** 0.072** -0.024** -0.025** 0.001 0.058** 0.099** -0.080** -0.127** 0.004 AEX/SX5e -0.417** -0.803** 0.406** 0.821** 0.108 95.2** -0.360** -0.714** 0.294** 0.632** 0.097 90.0** SX52/AEX 0.101** 0.204** -0.172** -0.299** 0.023 0.108** 0.219** -0.237** -0.415** 0.043 SMI/SX5e -0.106** -0.158** 0.083** 0.135** 0.018 -25.4 -0.012** -0.102** 0.095** 0.171** 0.024 -41.1 SX5e/SMI 0.153** 0.272** -0.161** -0.238** 0.021 0.208** 0.355** -0.256** -0.432** 0.043 SMI/AEX -0.080** -0.113** 0.037** 0.068** 0.012 -115.6 -0.088** -0.151** 0.045** 0.086** 0.014 -97.9 AEX/SMI 0.301** 0.585** -0.323** -0.557** 0.075 0.282** 0.545** -0.333** -0.633** 0.085 dependent/ independent

SX5e/DAX -0.439** -0.848** 0.445** 0.888** 0.116 179.5** -0.426** -0.776** 0.469** 0.895** 0.078 147.7** DAX/SX5e 0.076** 0.148** -0.122** -0.222** 0.010 0.067** 0.136** -0.096** -0.168** 0.004 AEX/DAX -0.432** -0.813** 0.448** 0.891** 0.129 152.4** -0.384** -0.670** 0.477** 0.909** 0.082 170.1** DAX/AEX 0.035** 0.064** -0.092** -0.148** 0.007 0.031** 0.059** -0.066** -0.097** 0.003 SMI/DAX -0.132** -0.222** 0.155** 0.244** 0.030 45.7** -0.140** -0.207** 0.148** 0.254** 0.027 32.1** DAX/SMI 0.073** 0.123** -0.098** -0.155** 0.007 0.067** 0.126** -0.074** -0.104** 0.003 AEX/SX5e -0.380** -0.711** 0.252** 0.560** 0.110 64.2** -0.328** -0.589** 0.296** 0.630** 0.068 64.1** SX52/AEX 0.114** 0.246** -0.282** -0.497** 0.064 0.139** 0.279** -0.247** -0.417** 0.039 SMI/SX5e -0.091** -0.167** 0.072** 0.135** 0.015 -61.2 -0.105** -0.163** 0.078** 0.159** 0.017 -60.8 SX5e/SMI 0.225** 0.404** -0.296** -0.509** 0.066 0.239** 0.431** -0.256** -0.399** 0.040 SMI/AEX -0.075** -0.129** 0.042** 0.078** 0.010 -102.8 -0.088** -0.126** 0.047** 0.098** 0.012 -104.8 AEX/SMI 0.278** 0.538** -0.358** -0.636** 0.101 0.298** 0.601** -0.311** -0.504** 0.061

March - June