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71

TABLE 6.3 Quality test of the #-scale.

Test for the quality of the #-scale as calculated in Equation 6.21. This ratio illustrates the reduction of intraweekly volatility fluctuations when using the #-scale.

USD-DEM

USD-JPY

GBP-USD

USD-CHF

XAU-USD

Volatility ratio 0.28

0.29

0.25

0.29

0.25

see that the quiet periods during the weekends are in the upper graph in physical time. They give the sense of periodicity. In the lower graph, where hourly returns are computed in #-hours, the seasonality is removed and the picture resembles much more those made with weekly or daily data (omitting weekends). Another remarkable feature of these graphs is the number of large movements. During this period, the USD-DEM experienced price changes as high as 1.5% in an hour.

6.3.3 A Ratio Test for the #-Scale Quality

There are various ways to measure the quality of a #-time scale. Because the goal of such a scale is to remove the daily and weekly seasonality of volatility, it is natural to test the extent to which this has been achieved. Here we define a quantitative test that allows discrimination between various possible business time scales.

The absolute returns on an intraweekly sample as described in Section 6.2.3 are first computed on the physical time scale. We define the size of the weekly fluctuations of mean volatility:

Fv(l)

(6.20)

where i is the index of the hourly interval in the statistical week and N = 168 the total number of these intervals. Absolute returns are observed and averaged over m weeks with index j, for each hour of the statistical week. The fluctuations, which are large when analyzed in physical time t, should be strongly reduced when analyzed in #-time. For analyzing the fluctuations in #-time, the sampling over one full week is again divided into 168 intervals. Instead of being equally spaced in physical time, they are now equally spaced in #-time. This condition can be formally written as #(f,+i) - i?(r/) = 1 hr, where the hour is now measured on the #-scale. The sequence that fulfills this condition is computed by numerical inversion of the ft(t) function on one week. The volatiliy ratio is defined by

F~v(V)/Fv<t) (6.21)

where Fv(#) and Fvet) measure the deseasonalized and raw volatility fluctuations. This ratio measures the quality of the extent to which the & scale successfully eliminates the seasonal fluctuations of the volatility.

In Table 6.3, the resulting ratio is between 0.25 and 0.29 for all rates indicating the quality of the #-scale. For a perfect ?-scale, the measure tends to zero, and



for physical time, the measure is one. Any other &-scale derivation can also be measured the same way, the one with the lowest ratio being the best intraday deseasonalization method. In the next chapter, we will utilize the #-scale in analyzing the autocorrelation function of absolute returns.

6.4 FILTERING INTRADAY SEASONALITIES WITH WAVELETS

The previous sections show that the practical estimation and extraction of the intraday periodic component of the return volatility is feasible. The literature also demonstrated that such extraction of the seasonal volatility component is indispensable for meaningful intraday studies. Earlier studies have shown that strong intraday seasonalities may induce distortions in the estimation of volatility models and are also the dominant source for the underlying misspecifications as studied in (Guillaume etal., 1994; Andersen and Bollerslev, 1997b). Besides, Section 7.3 reveals how such a periodic component pulls the calculated autocorrelations down, giving the impression that there is no persistence other than particular periodicities.

To illustrate the impact of seasonalities, Gencay et al. (2001a) consider the following AR(1) process with a periodic component:

yt=a + (3y,-\+J23-OSi+ t = l...T (6.22)

where Sit = sin(f) + mi, a = 0.0, y0 = 1.0, = 0.99, and T = 1000. Periodic components are Pi = 3, = 4, P3 = 5, and P4 = 6 so that the process has 3, 4, 5, and 6 period stochastic seasonality. The random variables et and vit are identically and independently distributed disturbance terms with zero mean. The signal-to-noise ratio, r], in each seasonal component is set to 0.30.

Figure 6.8 presents the autocorrelation of the simulated AR(1) process with and without the periodic components. The autocorrelation of the AR(1) process without seasonality (excluding 3.0S,-r from the simulated process) starts from a value of 0.95 and decays hyperbolically as expected. However, the autocorrelation of the AR(1) process with the seasonality indicates the existence of a periodic component. The underlying persistence of the AR(1) process in the absence of the seasonality component is entirely obscured by these periodic components. An obvious route is to filter out the underlying seasonalities from the data. A simple method for extracting intraday seasonality that is free of model selection parameters is proposed by Gencay et al. (2001a). The proposed method is based on a wavelet6 multiscaling approach which decomposes the data into its low and high-frequency components through the application of a nondecimated discrete wavelet transform. In Figure 6.8, the solid line is the autocorrelation of the nonseasonal AR( 1) dynamics and the dotted line is the autocorrelation of the deseasonalized series with the method proposed in Gencay et al. (2001a). As

6 An introduction to wavelets can be found in a book by Gencay et al. (2001b).



0 10 20 30 40 50 60 70 80 90 100

Lags

FIGURE 6.8 Sample autocorrelations for the simulated AR(I) process (straight line), AR( I) plus seasonality process (dot-dashed line), and wavelet transformation of the AR( I) plus seasonality process (dotted straight line).

Figure 6.8 demonstrates, wavelet methodology successfully uncovers the nonsea-sonal dynamics without imposing any spurious persistence into the filtered series.

With this method, Gencay et al. (2001a) study two currencies, namely the 5-min Deutschemark - U.S. Dollar (USD-DEM) and Japanese Yen - U.S. Dollar (USD-JPY) price series for the period from October 1, 1992, to September 29, 1993. This data set is also known as the HFDF-I data set. Figure 6.9 presents autocorrelations of the 5-min absolute return series. This shows that the intradaily absolute returns exhibit strong intraday seasonalities. This phenomenon is well-known and reported extensively in the literature; (see for example, Dacorogna etal, 1993; Andersen and Bollerslev, 1997a).

For a long memory process (see Hosking, 1996), the autocovariance function at lag satisfies y(k) ~ Xk~a where X is the scaling parameter and e [0, 1]. A leading example is the fractionally integrated process for which a = 1 - 2d and d is the order of fractional integration, fn Andersen and Bollerslev (1997a), the fractional order of integration is estimated as d - 0.36 for the same USD-DEM series utilized in this example. Andersen et al. (2001) calculate six d estimates from various volatility measures for the USD-DEM and USD-JPY series. These six d estimates vary from 0.346 to 0.448. In this example, the fractional integration parameter is set d = 0.4 to represent the average of these six estimates. Figure 6.10 presents the autocorrelograms of the filtered 5-min absolute returns along with the estimated autocorrelogram of a long memory process with d - 0.4. These



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