from the i.i.d. diffusion process make the realized volatility computed with returns measured at very short intervals no longer an unbiased and consistent estimator of the daily volatility. It is thus interesting to go one step further than in the previous section and to model the bias in order to obtain an easy correspondence between volatilities estimated at different frequencies. Recently, Corsi etal. (2001) have investigated and modeled the bias and its effect on measurements of realized volatility.

There are two limitations to the precision of the estimation of realized volatility. The number of return observations in the measurement period is limited and leads to a stochastic error (noise). One can easily see that for long time intervals (a year and more) it becomes difficult to assess the statistical significance of the volatility estimation because there are not more than a handful of independent observations. This number grows and the noise shrinks when the return measurement intervals shrink, but then the bias starts to grow. Until now, the only choice was a clever trade-off between the noise and the bias, which led to typical return intervals of about an hour. Tick frequency and data gaps play a major role. The goal is to define a superior realized volatility, which combines the low noise of short return interval sizes with the low bias of large return intervals. We shall not enter in the details here since it is still a research in progress. Tt suffices to mention that this is a crucial issue for a widespread use of high-frequency data in volatility estimation.

Gencay et al. (2001d) also provide evidence that the scaling behavior breaks for returns measured at higher intervals than 1 day. Figure 5.9 reports their decomposition of the variance on a scale-by-scale basis through the application of a nondecimated discrete wavelet transform.23 This methodology does not assume any distributional form to the returns. The wavelet variance for each absolute return series is plotted on a log-log scale in Figure 5.9. For example the first scale is associated with 20-min changes, the second scale is associated with 2*20=40-min changes, and so on. Each increasing scale represents lower frequencies. The first six scales capture the frequencies 1/128 < / < 1/2; that is, oscillations with a period length of 2 (40 min) to 128 (2560 min). Because there are 72*20 = 1440 min in a day, we conclude that the first six scales are related with intraday dynamics of the sample.

In the seventh scale, an apparent break is observed in the variance for both series which is associated with 64*20 = 1280-min changes. Because there are 1440 min in a day, the seventh scale corresponds to 0.89 day. Therefore, the seventh and higher scales are taken to be related with 1 day and higher dynamics.

For a power law process, v2 ( ) oc tj~a~1 so that an estimate of a is obtained

by regressing log v2(tj) on log jja~x. Figure 5.9 plots the ordinary least squares (OLS) fits of the sample points for two different regions. Estimated slopes for the smallest six scales are -0.48 and -0.59 for USD-DEM and USD-JPY series,

23 A extensive study of wavelet methods within the context of time series analysis and filtering is presented in Gencay et al. (2001b).

1 2 4 8 16 32 64 128 256 512 1024 2048

1 2 4 8 16 32 64 128 256 512 1024 2048

Aggregation Factors (20 min)

FIGURE 5.9 Wavelet variance for 20-min absolute returns of (a) USD-DEM and (b) USD-JPY from December I, 1986, through December I, 1996, on a log-log scale. The circles are the estimated variances for each scale. The straight lines are ordinary least squares (OLS) fits. Each scale is associated with a particular time period. For example the first scale reflects 20-min changes, the second scale reflects 2 * 20 = 40-min changes, the third scale reflects 4 * 20 = 160-min changes, and so on. The seventh scale is 64 * 20 = 1280-min changes. Because there are 1440-min per day, the seventh scale corresponds to approximately one day. The last scale shows approximately 28 days.

respectively. This result implies that a = -0.52 for the USD-DEM series and a - -0.40 for the USD-JPY absolute return series for the first six scales (intraday).

5.6 AUTOCORRELATION AND SEASONALITY

Before closing this chapter, we investigate the autocorrelation and the seasonality of high-frequency data. Are returns and the volatility serially correlated, beyond the negative short-term autocorrelation in Section 5.2.1? Are there periodic patterns, seasonality, in the data? Clearly, we expect to find very little data during weekends and holidays, but what else can be said about different weekdays and daytimes? We answer these questions by using two types of statistical analysis. The autocorrelation function of a stochastic quantity reveals at the same time serially dependence and periodicity. The autocorrelation function signals a periodic pattern by peaking at lags that are integer multiples of the particular period. We call the other type of analysis an intraday-intraweek analysis. It relates quantities to the time of the day (or the week) when they are observed. We thus obtain average quantities evaluated for every hour of a day or of a week.

0.40

0.30 4

i>

............- it 1 1 1 1 1

0.00 10.00 20.00 30.00

Time Lag (Hours)

FIGURE 5.10 Autocorrelations of hourly returns (o), their absolute values (*), and their squares (+) as functions of the time lag, for XAU (gold) against USD. The band about the zero autocorrelation line represents 95% significance of the hypothesis of independent Gaussian observations.

5.6.1 Autocorrelations of Returns and Volatility

A convenient way to discover stylized properties of returns is to conduct an autocorrelation study. The autocorrelation function examines whether there is a linear dependency between the current and past values of a variable:

, Zlxif - r) - (x}][x(t) - (x)]

px(r) - (5.38)

VEMt - r) - (x)]2J2[x(t) - (x)]2

where x(t) can be any time series of a stochastic variable and is the time lag. The autocorrelation function peaks at lags corresponding to the periods of seasonal patterns.

Here we present an analysis of the autocorrelation function p of hourly returns, their absolute values, and their squares over a sample of 3 years from March I, 1986, to March 1, 1989. We see in Figure 5.10 that the last two variables have a significant, strong autocorrelation for small time lags (few hours), which indicates the existence of volatility clusters or patterns. More interesting is the significant peak for time lags of and around 24 hr. This is a strong indication of seasonality with a period of 1 day. The autocorrelation of the returns