small Af = 5 min (q = 288). The stochastic error is proportional to ,/1/288, but the factor u(Afref, m, 2; /,-) with m = 260 leads to another stochastic error component proportional to yj1/260. Both error components together are proportional to V1/288+ 1/260 % VI/137. This is distinctly smaller than the value without bias correction, /1/24 (where the bias makes the error even larger).

So far, the bias discussion has been restricted to realized volatility with an exponent p = 2 in Equation 3.8. When choosing another exponent (such as p = 1, a good choice for many following studies), the bias discussion becomes more complicated. The scaling behavior deviates from Gaussian scaling, as seen in Section 5.5, and data gaps have a stronger influence on the bias than in the case p - 2. For exponents other than 2, a bias correction with a formula such as Equation 7.4 is less successful, and more research is needed. The technique of bias correction is rather new and will be improved by ongoing research. The realized volatility studies of the following sections are older and do not contain any bias correction. However, the choice of very short return intervals (such as 5 min) has been avoided, so the size of the bias is limited.

7.3 CONDITIONAL HETEROSKEDASTICITY 7.3.1 Autocorrelation of Volatility in #-Time

This section analyzes the autocorrelations of returns and realized volatility in physical and j/-time.3 The study utilizes a 20-min frequency instead of an hourly one. We did not take smaller intervals than 20 min in order to avoid a strong bias, as explained in Section 7.2. The autocorrelation function of the USD-DEM is shown in Figure 7.4 for up to 720 lags. The confidence intervals in Figure 7.4 refer to 95% confidence for a Gaussian random process around the sample mean. Because the distributions of returns and volatility are not Gaussian, the confidence intervals are provided as a reference rather than for exact statistical significance.

In Figure 7.4, the autocorrelation function of volatility has a distinct structure, which is far beyond the confidence intervals. For lags of any integer number of days, clear peaks are found. These peaks indicate the daily seasonality. The weekly seasonality is highly visible in the form of high autocorrelation for lags around 1 week and low autocorrelation for lags of about half a week (which frequently means the correlation of working days and weekends). Finally, there is a finer structure with small but visible peaks at integer multiples of 8 hr, corresponding to a frequency three times the daily frequency. Our world market model with three continental markets is confirmed by this observation. Apart from these seasonal peaks there must be a positive component of the autocorrelation that declines with increasing lag. In Figure 7.4, this component cannot be observed as it is overshadowed by seasonality.

The autocorrelations of returns, unlike those of volatility (absolute returns), are close to zero and within the confidence intervals for most of the lags. The

3 Absolute returns are studied here.

. -

1 week

0 100 200 300 400 500 600 700

Lag (in 20-min intervals)

FIGURE 7.4 The autocorrelation function of USD-DEM returns and volatility (absolute returns). The data sampling is in 20-min frequency in physical time for lags up to 10 days. The 95% confidence interval is for a Gaussian random process. The sampling period is from March 3, 1986, to March 3, 1990.

squared returns, instead of absolute returns, may also be used as a proxy for the underlying volatility. Autocorrelations of square returns also exhibit similar seasonality peaks as those of absolute returns, but are less pronounced. It is well known that the theoretical autocorrelation of squared returns is meaningful only if the kurtosis of the return process is finite, which is not guaranteed for currency returns.

A similar autocorrelation analysis is also carried out with the ?-time scale instead of the physical time t, and it is presented in Figure 7.5. There are no large seasonal peaks in the volatility autocorrelations of the ?-time. This is due to the fact that the #-scale is constructed to eliminate the intraday seasonality. The autocorrelation of volatility is significantly positive and declines at an hyperbolic

FIGURE 7.5 The autocorrelation function of the USD-DEM returns and the absolute returns at 20-min data frequency in &-time. The number of lags is up to 10 ? days. The first lag is marked by an empty circle. The exponential decay is shown with a dashed line. The hyperbolically decay fits best to the autocorrelation function of the absolute returns. The figure on the right is the same autocorrelation function for the absolute returns extended to a much larger number of lags with the superimposition of the hyperbolic decay.

rate. This behavior can be explained by the presence of a long memory process in the underlying data-generating process of returns. The rate of decline in the autocorrelation is, however, slower than an exponential decline, which would be expected for a low-order GARCH process, Bollerslev (1986).

The autocorrelation function of volatility (Figure 7.5) is not completely free of seasonalities. A narrow peak can be identified at a lag of 1 week. This peak might be due to the day of the week effects. In our framework, the activity is assumed to be the same for all working days, which may exhibit slight variations across the working days. A small local maximum at a lag of around 1 average business day (one-fifth of a week in #); a small local maximum at a lag of 2 business days and maxima at 3 and 4 business days also exist. A plausible reason for these remaining autocorrelation peaks is a market-dependent persistence of absolute returns. Autocorrelations with a lag of 1 business day compare with the behaviors of the same market participants, whereas autocorrelations with lags of one half or

1 business days compare with the behaviors of different market participants (on opposite sides of the globe). The market-dependent persistence decreases after

2 business days. The predominance of the "meteor shower hypothesis" found by Engle et al. (1990) is confirmed by the fact that the autocorrelation curve in

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