TABLE 8.4 HARCH coefficients for USD-DEM.

Comparison between the coefficients and impacts of the two HARCH processes from a half-hourly USD-DEM series, which is equally spaced in ?-time over 10 years. Instead of the coefficients Cj, the impacts are given. These provide a direct measure of the impacts of the market components on the HARCH variance. The market components are those of Table 8.3 for HARCH and as in Equations 8.28 and 8.30 for EMA-HARCH. The distribution of the random variable e(t) is normal with zero mean and a unit variance.

The result of the optimization procedure is a set of Cj coefficients from which the component impacts are calculated using Equation 8.23 (for HARCH) or Equation 8.33 (for EMA-HARCH). The sum of impacts lj must be below one for stationarity of the process (Equations 8.24 and 8.34). In Table 8.4, the coefficients for both the HARCH and EMA-HARCH are shown with their /-statistics for USD-DEM. They are obtained on exactly the same data set. The log-likelihoods can be compared because both models have the same number of independent coefficients. Clearly, the log-likelihood is improved by going to EMA-HARCH. In both cases, all coefficients are highly significant according to the /-statistics and contribute to the variance equation. The stationarity property is fulfilled in both cases. The HARCH and EMA-HARCH have total impacts of0.8567 and 0.9386, respectively. The impacts of the different components are remarkably similar. Two small differences are worth noticing. The relative importance of the long-term components is slightly higher for EMA-HARCH (37% instead of 35%) and the minimum for the fourth component is more pronounced in EMA-HARCH. The /-statistics are also consistently smaller for EMA-HARCH than for HARCH tyit still highly significant in all cases. The residuals in both formulations still present an excess kurtosis, as was noticed in Muller et al. (1997a) for HARCH.

These results show that we have achieved the goal of redesigning the HARCH process in terms of moving averages. We are able to keep and even improve on the properties of the original HARCH and to considerably reduce the computational

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Market Components Market Components

FIGURE 8.5 Impacts of market components of HARCH processes with components as defined in Table 8.3. Each HARCH model has been made for a particular FX rate by fitting a half-hourly time series equally spaced in i?-time over 7 years. The differences between the impacts, in particular the low values of the fourth component, are highly significant (see the error values of Table 8.4). The values for USD-DEM are those presented in Table 8.4 and they are not fundamentally different from those of other FX rates.

time to optimize the model. The EMA formulation of the process equation reduces this time by a factor of 1000, making the computation of HARCH volatility much more tractable even with limited computational power. In the next section, we will explore the forecasting ability of these models and compare it to a more traditional approach to volatility.

The impacts /, are also plotted in a histogram (Figure 8.5) where it is possible to compare the results for different FX rates. The impact of the fourth component is the weakest among all impacts. This is not only for USD-DEM but also many other rates and also for other sampling periods.10 The fourth component has a

10 When a 7-year sample is split into two parts of 3 1/2 years, the estimated coefficients on both of these subsamples are quite stable.

TABLE 8.5 Results of the EMA-HARCH for the LIFFE Three-Month Euromark.

Results of the EMA-HARCH process estimate for 3-hr ?-time intervals for the different forward rates for the LIFFE Three-Month Euromark. The underlying data are from the LIFFE Three-Month Euromark. Standard errors are given. Instead of the coefficients / (for i > 0), the corresponding impacts are given. Data sample: from April 6, 1992, to December 30, 1997, representing 16,774 observations. The forward rates are labeled according to the market conventions for forward rate agreements, as explained in the text. The 3x6 forward interest rate, for example, applies to the interval starting in 3 months and ending in 6 months.

typical time horizon of around 12 hr-too long for intraday dealers and too short for other traders. This naturally explains the weakness of that component.

When comparing the impacts of Figure 8.5 with the component definition of Table 8.3, we see further interesting features captured by HARCH models. First, the short-term components have, in all cases, the largest impacts. These short-term components model essentially the intraday dealers and the market makers who are known to dominate the market. Second, the similarity in the impacts of the USD-CHF and USD-DEM are plausible as it is well known that the Swiss National Bank policy was tightly tied to the USD-CHF to the USD-DEM rates. The relative weakness of the longer-term components for the GBP-USD is another relevant piece of information that can be gathered from this parametrization and has been confirmed to us by market participants. Since the 1992 crisis, the long-term investors have been reluctant to invest in this market and have been more concentrated on the cross rate GBP-DEM. The relative impact of the fifth and the sixth components are in the same order for USD-CHF and USD-DEM but inverted in the case of both USD-JPY and GBP-USD.

8.3.6 HARCH in Interest Rate Modeling

As described in Chapter 7, we haved performed a lead-lag correlation analysis and established the HARCH effect for forward interest rates implied by interest rate futures, constructed according to Section 2.4.2 (see Figure 7.9). In this section, we use the HARCH parametrization in terms of market components to investigate