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83

TABLE 8.1 Results of the GARCH( 1,1) estimation in business time.

GARCH( 1,1) parameter estimates for USD-DEM, using the business time scale, for different frequencies. Robust standard errors are given in parentheses. The coefficients with a prime are computed from the (dis)aggregation formulas for the jump hypothesis of Drost and Nijman (1993). The daily interval serves as a reference basis.

 interval «0 «1 «1 +01 10 min 2.15 10" 0.227 0.752 0.979 0.001 0.999 1.000 (0.17 10" (0.0013) (0.0012) 20 min 2.66 10" 0.179 0.816 0.995 0.002 0.997 0.999 (0.15 10" (0.0037) (0.0051) 30 min 2.65 10" 0.143 0.853 0.996 0.003 0.996 0.999 (0.18 10" (0.0062) (0.0101) 1 hr 1.79 10" 0.142 0.784 0.926 0.006 0.992 0.999 (0.42 10" (0.0066) (0.0114) 3.11 10" 0.023 0.970 0.993 0.011 0.986 0.997 (0.13 10" (0.0020) (0.0015) 2.43 10" 0.041 0.941 0.982 0.029 0.962 0.991 (0.22 10" (0.0039) (0.0061) 12 hr 1.07 10" 0.054 0.905 0.959 0.046 0.936 0.982 (0.17 10" (0.0061) (0.0102) 24 hr 1.91 0.068 0.897 0.965 (0.67 10" (0.0095) (0.0153)

coefficients already starts with the 6-hr frequency. The conclusions on temporal aggregation of GARCH are the same. The choice of the time scale has no strong impact on a temporal aggregation study, as long as physical time with its high weight of weekends is avoided.

Detailed results for the two time scales are also listed in Tables 8.1 and 8.2. Table 8.1 presents the results obtained for USD-DEM in the business time scale and Table 8.2 for the same rate but in the #-time scale. The error estimates of the results provide more evidence against the hypothesis of only one GARCH(1,1) process generating the data. Even the theoretically computed coefficients at low frequency, which seem quite close to the estimated coefficients, are often outside the confidence intervals. Only the coefficients for the GARCH process are provided in Tables 8.1 and 8.2, even when an AR(4) term was included in Equation 8.1 for frequencies higher than 2 hr (as discussed at the beginning of Section 8.2.1). We have observed that the inclusion of this autoregressive term in the return equation does not significantly change the values of the GARCH coefficients.

The coefficient estimates are quite similar across different FX rates.3 The hypothesis of only one GARCH(1,1) process is rejected for all the FX rates we tested, not only USD-DEM. The volatility clusters have about the same size-if measured in numbers of time series intervals-for all levels of aggregation. In

3 See Andersen and Bollerslev (1997b) for similar results.

TABLE 8.2 Results of the GARCH(U) estimation in tf-time.

GARCH(U) parameter estimates for USD-DEM, using #-time, for different frequencies. Robust standard errors are given in parentheses. The coefficients with a prime are computed from the (dis)aggregation formulas for the jump hypothesis of Drost and Nijman (1993). The daily interval serves as a reference basis.

 Interval «1 «1 +01 [+0\ 10 min 4.09 10" 0.153 0.839 0.992 0.001 0.999 1.000 (0.27 • 10" (0.0047) (0.0049) 20 min 1.24 • 10" 0.149 0.830 0.979 0.001 0.998 0.999 (0.84 10" (0.0057) (0.0063) 30 min 2.56 • 10" 0.153 0.815 0.968 0.002 0.997 0.999 (0.21 • 10" (0.0077) (0.0091) 1 hr 1.36 • 10" 0.047 0.942 0.988 0.004 0.995 0.999 (0.46 10" (0.0094) (0.0129) 1.65 10" 0.031 0.962 0.993 0.008 0.989 0.997 (0.28 10" (0.0014) (0.0022) 5.93 10" 0.029 0.963 0.992 0.023 0.971 0.994 (0.40 10" (0.0011) (0.0013) 12 hr 1.91 • 10" 0.039 0.948 0.987 0.038 0.949 0.988 (0.45 • 10" (0.0013) (0.0047) 24 hr 8.08- 10" 0.061 0.915 0.975 (2.74 • 10" "7) (0.0119) (0.0155)

other words, the volatility memory seems quite short-lived when measured with high-frequency data and long-lived when measured with data of daily or lower frequency. The information content of the volatility variable o> is not the same for different frequencies. Different volatilities are relevant at different frequencies. We attribute this, along with other authors (Andersen and Bollerslev, 1997a), to the presence of many independent volatility components in the data. This is again the signature of market heterogeneity. The GARCH model does not capture the heterogeneity of traders acting at different time horizons.

This is a plausible explanation of the abnormal results we obtain at high frequencies from the estimation of the GARCH model using a Student-f distribution instead of the normal distribution such as in Baillie and Bollerslev (1989).4 GARCH is misspecified, no matter which form of the conditional distribution of returns is chosen.

To further assess the behavior of the volatility as estimated by GARCH(1,1) processes, we have investigated the temporal stability of the coefficient estimates for several subsamples. Figure 8.3 provides the estimations of the GARCH parameters for USD-DEM at the 2-hr time interval, using r?-time, for subsamples of 6 months, with about 2,190 observations per subsample. As can be seen, the

4 Although the algorithm converges, the sum of the a i and/3i increasingly exceeds 1 as the frequency becomes higher. One also finds excess residual skewness and kurtosis. Since these results are robust to the size of the sample, they cannot be attributed to a larger number of tail observations.

0.00 I I I ~ I I I j .....I I I Ill-1 I I-r-j-ri I -1-1-1

0.5 0.6 0.7 0.8 0.9 1

Beta

FIGURE 8.3 Temporal stability of the GARCH(I,I) coefficients for subperiods of 6 months for the USD-DEM at the 2-hr frequency. The time scale is #-time. In the parameter space, the coefficients are represented by black circles (•) and connected by lines indicating the temporal sequence. Sampling period: 7 years from January 1, 1987, to December 31, 1993.

coefficients are not stable over time. Some of them are in the left half of Figure 8.3, quite far from the others. Moreover, these aberrant coefficients are not directly connected in the temporal sequence. The shifts in coefficient values have an irregular sequence in time, as shown by the lines connecting the points in Figure 8.3. The hypothesis of all parameters being equal across the subsamples can be rejected by using a likelihood-ratio test (see e.g., Hamilton, 1994) with very high significance. This is again a sign of misspecification of the model, but it may also indicate changes in market behavior.

The forecasting quality of GARCH models will be tested in Section 8.4.2. There we shall see that an increasing sample size does not improve the volatility forecasts from GARCH models. The forecasting quality saturates when increasing the sample size after a certain threshold value. The subsamples used for Figure 8.3 are large enough, so the erratic behavior of GARCH coefficients in that figure cannot be attributed to small sample sizes.

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