FIGURE 8.4 Moment conditions for the coefficients c\ and C2 of a HARCH(2) process with a normally distributed et. The straight line on the right represents the boundary for the stationarity and the existence of the second moment. The curves in the middle and on the left represent the boundaries for the existence of the 4th and 6th moments.

8.3.2 HARCH and Market Components

The memory in the volatility is known to be long, as discussed in Section 7.3.2. Therefore, we need a high order of HARCH, a large n, to model the behavior of empirical time series. This implies a high number of coefficients /, which should not be free parameters of HARCH. We need a parsimonious parametrization. In the case of ARCH, some high-order processes can be formulated as low-order GARCH processes (Bollerslev, 1986), but no analogous method is at hand to reduce the number of HARCH parameters.

Our approach of parsimonious parametrization allows for the exploration of the component structure of the market. The coefficients Cj reflect the relative impact of different market components with different relevant time intervals. Therefore, we select m market components corresponding to m free parameters, each associated to some coefficients Cj in a limited range of j. The j ranges are separated by powers of a natural number p, so the typical time interval size of a component differs from that of the neighbor component by a factor of p. All devalues within one component are assumed to be the same:

i(j) - max I

Cj = Ct = Ci(j) log /

e N < - +2 logp

j = 1 ... P"

(8.22)

Only m different coefficients C, have to be estimated to determine the whole set of = pm~] coefficients Cj.

Table 8.3 presents such a component scheme for a time series in #-time with a basic grid of 30 min, p = 4, and 7 components (m = 7). An interval of 30-min in t?-time means only some 7 min during the daily volatility peaks around 14:00 GMT, some 80 min during the Far Eastern lunch break, and even more during weekends and holidays with their very low volatility. Table 8.3 shows the minimum relevant time intervals of a component rather than the total size of the volatility memory. In fact, the memory of the volatility can greatly exceed the indicated interval. The medium-term traders of component 5, for example, are not interested in the volatility of hourly returns, are most interested in volatilities observed over 1 to 3j days, and are also interested in volatilities observed over longer intervals.

The choice of the number of components, m-1, and the factor between the typical time resolutions of the components, p = 4, is reasonable but somewhat arbitrary. The essential results of this chapter do not strongly depend on this choice and can be found also with other m and p values. The model should cover the variety of relevant time resolutions of the market.9 A too small choice of m misses the chance of revealing the component structure; a too large m (with a small p) leads to too many parameters to be estimated and an unrealistically narrow definition of market components.

A quantity more suitable for the intuitive understanding than a coefficient C, is the impact /, of a market component. The expected variance formula, Equation 8.18, strongly suggests a definition of the impact of the j,h coefficient as jc-j. The impact of the ih component is defined as the sum of the impacts of all its coefficients cy. By inserting the coefficient definitions of Equation 8.22, we obtain

/i = c\ = C\

P nf-] i „-2 1 1

/f = £ jej = (p - 1) p~2---I-- Cj for / > 1

j=P-4i

(8.23)

The impact of the long-term components may be considerable even when the coefficients appear to be small. The impact of the fifth component, for example, is /5 = 3O8I6C5, where p is assumed to be 4 as in Table 8.3.

The stationarity condition (Equation 8.19) can now be formulated in terms of the impacts. Their sum is smaller than 1:

< 1 (8.24)

y The choice of Table 8.3 is m = 7. An even higher value, m = 8, has also been tested, leading to a low, insignificant impact of the eighth component and a rejection in a likelihood ratio test. We conclude that the seventh component is the last relevant one on the long-term side.

TABLE 8.3 Definition of the HARCH components.

A HARCH model with seven market components, each with a range of indices j. All coefficients Cj of a component are identical and only seven parameters need to be estimated. The time intervals are the relevant intervals for the volatility perception of the time components (not the total duration of their memory). These basic intervals are given in !?-time and also in physical time. Two columns show the minimum and maximum size of the interval that can occur for a time component, depending also on the daytime. These time component descriptions may contribute to a better understanding of the model.

Range of j

Approximate Shortest range of time interval (at intervals in daily or

!?-time

weekly volatility peaks)in

Longest interval (hut avoiding weekends and

holidays) in

physical time physical time

Description of the time component

30 min 7 min 80 min Short-term, intraday deal-

ers (arbitrage opportunities), market makers

1-2hr

16 min

3 hr

Intraday dealers with few transactions per day

• 16

-8hr

50 min

Dealers with overnight positions and occasional intraday transactions

4 17 - 64 84- 32 hr 4hr

1 day Few traders concerned

(time intervals often beyond local business hours but less than a business day)

5 65 - 256 1A - 5 days 1 day

257 - 1024

days

days

3 days

21 days (weekends always contained)

Medium-term traders, no intraday trading

Long-term traders

7 1025 - 4096 3 - 12 3 weeks 12 weeks Long-term investors, cen-

weeks (weekends tral banks

always contained)