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61 Power Power FIGURE 5.11 The first 10 lags of the autocorrelation function of \r\p as a function of the power p for USDDEM and USDFRF (first lag on top, 10th at the bottom). The maxima are shown by bullet signs (•). The returns are measured over 30 min (in #time, see Chapter 6). Right above the bottom, at a very low autocorrelation value, there is a horizontal line in both graphs. This is the upper limit of the 95% confidence band of the hypothesis of independent Gaussian observations. themselves is insignificant as it stays mostly inside the confidence band of Figure 5.10. Because the autocorrelation function varies when the absolute returns are raised to a different power, as can be seen in Figure 5.10, we systematically studied the influence of the power p on the autocorrelation function. Some studies on the influence of the power of absolute returns on the autocorrelation have been published (Granger and Ding, 1995; Muller et al, 1998; Bouchaud and Potters, 2000). Granger and Ding (1995) conclude that the exponent p = 1 leads to the highest autocorrelation. Here, as in Muller et al (1998), we report a full analysis of the autocorrelation coefficients as a function of the power p of absolute returns. Figure 5.11 shows how the tails of the distribution influence autocorrelation. Increasing the power of the absolute returns boils down to increasing the relative importance of extreme events in the statistics. In Figure 5.11, we see that the autocorrelation, for the 10 lags considered, decreases when the influence of extreme returns is increased. In other words, extreme events are less correlated with each other than average or small absolute returns. From this study, it seems that the heteroskedasticity is mainly due to the average behavior, not the extreme events. This is represented by a low exponent p smaller than 1; the maximum autocorrelation is for values of p close to onehalf. The results presented in Table 5.3 can well explain a feature shown in Figures 5.10 and 5.11 where the positive autocorrelation of absolute returns is stronger than that of squared returns. The tail index a being almost always below 4, the
TABLE 5.10 Time conversion table. Time conversion table between Greenwich Mean Time (GMT), Europe (MET), USA (EST), and Japan (JPT). The letter D indicates a particular day. Note that GMT and JPT are not subject to daylight saving changes whereas the regions under MET and EST are. GMT MET EST JPT  0:00   1:00   19:00   9:00   3:00   4:00   22:00   12:00   6:00   7:00   1:00   15:00   9:00   10:00   4:00   18:00   12:00   13:00   7:00   21:00   15:00   16:00   10:00   0:00   18:00   19:00   13:00   3:00   21:00   22:00   16:00   6:00 
fourth moment of the distribution is unlikely to converge.24 The denominator of the autocorrelation coefficient of squared returns is only finite if the fourth moment converges, whereas the convergence of the second moment25 suffices to make the denominator of the autocorrelation of absolute returns finite. The second moment is finite if a is larger than 2. Indeed, besides the empirical evidence shown in Figure 5.11, we find that the difference between the autocorrelations of absolute returns and squared returns grows with increasing sample size. This difference computed on 20 years of daily data is much larger than that computed on only 8 years. For a lag of 9 days, we obtain autocorrelations of 0.11 and 0.125 for the absolute returns over 8 and 20 years, respectively, while we obtain 0.072 and 0.038 for the squared returns, showing a strong decrease when going to a larger sample. The same effect as for the daily returns is found for the autocorrelation of 20min returns, where we compared a 9year sample to halfyearly samples. 5.6.2 Seasonal Volatility: Across Markets for OTC Instruments The most direct way to analyze seasonal heteroskedasticity in the form of daily volatility patterns is through our intraday statistics. We construct a uniform time grid with 24 hourly intervals for the statistical analysis of the volatility, the number of ticks, and the bidask spreads. The low trading activity on weekends implies a weekly periodicity of trading activity and is a reason for adding intraweek statistics to the intraday statistics. Both statistics are technically the same, but the intraweek analysis uses a grid of 24 Loretan and Phillips (1994) come to a similar conclusion when examining the tail behavior of daily closing prices for FX rates. 25 For a more formal proof of the existence of the autocorrelation function of stochastic variables obeying fattailed probability distribution, see Davis et al. (1999).
TABLE 5.1 I Average number of ticks. Average number of ticks for each day of the week (including weekends) for the USD against DEM, JPY, CHF, and GBP and XAU (gold) against USD. The sampling period is from January 1, 1987, to December 31,1993. The tick activity has increased over the years except for XAU (gold).       Monday  4888  2111  1773  1764   Tuesday  5344  2438  2043  2031   Wednesday  5328  2460  2033  2022   Thursday  5115  2387  1948  1914   Friday  4495  1955  1633  1670   Saturday       Sunday      
seven intervals from Monday 0:0024:00 to Sunday 0:0024:00 (GMT). With this choice, most of the active periods of the main markets (America, Europe, East Asia) on the same day are included in the same interval. The correspondence of the hours between main markets is shown in Table 5.10. The analysis grids have the advantage of a very simple and clear definition, but they treat business holidays outside the weekends (about 3% of all days) as working days and thus bias the results. The only remedy against that would be a worldwide analysis of holidays, with the open question of how to treat holidays observed in only one part of the world. Another problem comes from the fact that daylight saving time is not observed in Japan while it is in Europe and in the United States. This changes the significance of certain hours of the day in winter and in summer when they are expressed in GMT. An alternative here would be to separate the analysis according to the winter and the summer seasons. An analysis of trading volumes in the daily and weekly grids is impossible as there is no raw data available. The average numbers of ticks, however, give an idea about the worldwide market activity as a function of daytime and weekdays. They are counts of original quotes by representative market makers, though biased by our data supplier. The two bottom graphs of Figure 5.12 improve our knowledge of the intraday and intraweek studies. They show, for example, that even the least active hour, 3:00 to 4:00 GMT (noon break in East Asia), contains about 20 ticks for DEM, a sufficient quantity for a meaningful analysis. The intraweekly results are shown in Table 5.11. The ranking of the FX rates according to the amount of published quotes corresponds fairly well to the ranking obtained by the Bank of International Settlements (BIS) with its survey of the volume of transactions on the FX market (Bank for International Settlements, 1995, 1999). Intraday volatility in terms of mean absolute returns is plotted in the two top histograms of Figure 5.12 for USDDEM. Both histograms indicate distinctly uneven intradayintraweek volatility patterns. The daily maximum of average
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