8.3 MODELING HETEROGENEOUS VOLATILITIES

In Chapter 7 we showed that there is asymmetry in the interaction between volatilities measured at different frequencies. A coarsely defined volatility predicts a fine volatility better than the other way around. This effect is not present in a simple GARCH model. More complex types of ARCH models have to be developed to account for the heterogeneity found in high-frequency data, such as the HARCH (Heterogeneous Autoregressive Conditional Heteroskcdasticity) model.

The HARCH process proposed in this section has a variance equation based on returns over intervals of different sizes. The empirical behavior of lagged correlation can be reproduced well with this new process. At the same time, HARCH is able to reproduce the long memory of volatility,5 as found in Section 7.3.2, Dacorogna et al. (1993), and Ding et al. (1993). Moreover, the terms of the conditional variance of HARCH reflect the component structure of the market in a natural way.

As with most processes from the ARCH family, HARCH is based on squared returns,6 with their good analytical tractability. Whereas the convergence problem of the fourth moment of the return distribution forced us to define volatility in terms of absolute returns in the correlation analysis of Section 7.4.1, there is no such constraint for the volatility equation of the HARCH process.

8.3.1 The HARCH Model

In this section, we present the HARCH model as it was first presented by Muller et al. (1997a). This should facilitate the understanding of the approach but this initial formulation as the initial ARCH formulation is cumbersome to compute. In the next sections, we shall see a formulation with a much faster and simpler computation and estimation. As in Equation 8.2, the returns r, of a HARCH(n) process are defined with the random variable et, which is identically and independently distributed (i.i.d.) and follows a distribution function with zero expectation and unit variance:7

r, = a, e,

(8.13)

where

> 0 , cn > 0 , cj > 0 for j = 1 ... n - 1 (8.14)

5 The FIGARCH process, Baillie et al. (1996), has been designed to model the long memory but cannot reproduce the lead-lag correlation of Section 7.4.2 as it is still based on returns observed over intervals of constant size.

6 Except for a process class proposed by Ding et al. (1993), which models volatility in terms of different powers of absolute returns

7 Here, we utilize a normal distribution and alternatively explore Student-f distributions.

The equation for the variance of is a linear combination of the squares of aggregated returns. Aggregated returns may extend over some long intervals from a time point in the distant past up to time / - 1. The heterogeneous set of relevant interval sizes leads to the process named HARCH for "Heterogeneous Autoregressive Conditional Heteroskedasticity." The first "H" may also stand for the heterogeneous market if we follow that hypothesis as proposed in Section 7.4.

The HARCH process belongs to the wide ARCH family but differs from all other ARCH-type processes in the unique property of considering the volatilities of returns measured over different interval sizes. The Quadratic ARCH (QARCH) process (see Sentana, 1991) is an exception. Although QARCH was not developed for treating different interval sizes, it can be regarded as a generalized form of HARCH as explained in Section 8.3.3.

The coefficients c\ ...cn should not be regarded as free parameters of the model. The heterogeneous market approach leads to a low number of free model parameters, which determine a much higher number n of coefficients modeling the long memory of volatility.

The explicit formulation of HARCH(2) may help to illustrate the special properties of the HARCH process.8 The variance equation of HARCH(2) can be written in two forms:

The second form of this HARCH(2) process can be identified as an ordinary ARCH(2) process, except for its last term which contains the mixed product r( irr 2- In other ARCH-type processes, the absolute values of returns matter where in HARCH, also their signs matter. Two subsequent returns of the same size and in the same direction will cause a higher contribution to the variance process than two subsequent returns that cancel out each other.

The variance, the unconditional expectation of squared returns, can be derived from Equation 8.13:

The cross products, such as rt-\rt-2 in Equation 8.15, have no influence here as their expectation is zero. A necessary condition of stationarity is constant unconditional variance:

of = CO + C] rf x + C2 ( ,-] + 2)2 =

CO + (Cl + C2) tf x + C2 rf 2 + 2 C2 rr l -2

(8.15)

(8.16)

E(r2) = E(r2 ;) , i > 1

8 Whereas H ARCH( 1) is identical to ARCH( 1).

Inserting this in Equation 8.16, we obtain the variance

E(r2) =

(8.18)

which must be finite and positive

i < 1

(8.19)

This necessary stationarity condition is also a sufficient condition for both the stationarity of the process and the existence of the variance, the second moment. Proving this is not trivial and we do not follow here the path chosen by Engle (1982) and Bollerslev (1986) because the mixed products, such as rt-\r,-2, make the matrix formulation of the problem difficult. The HARCH process can be seen as a Markov chain. Meyn and Tweedie (1993) have obtained some results for the ergodicity and recurrence of Markov chains that can be used for proving the stationarity and the moment condition. The complete proof is given by Dacorogna etal. (1996).

The conditions for the existence and constant unconditional expectation of higher moments can be obtained through steps analogous to Equations 8.16 through 8.19, but the computation becomes increasingly tedious for higher moments and larger n values. The expectation of the 2mh moment is

Equation 8.13 of the variance is inserted in E(cr2m) and all the terms are explicitly computed. Some products of powers of returns have nonzero expectations, leading to an equation system for these expectations and E(a,2m). The equation system has the dimension m for n =2 and higher for larger n values. In the relatively simple case of the fourth moment (m = 2) of HARCH(2) (n = 2), the expectation E(r2r2 j) has to be computed and solved together with the equation for E(rf4). In the standard case of st following a normal distribution, N(0, 1), E(e4) = 3 and the following necessary condition is obtained to keep the fourth moment finite

The sufficiency of this necessary fourth moment condition is also proven in Dacorogna et al. (1996). In Figure 8.4, the second and fourth moment conditions according to Equations 8.19 and 8.21, plus the sixth moment condition following an analogously derived equation, are plotted for a HARCH(2) process. Processes with a finite second and a diverging fourth moment exist in a large part of the ci-C2-plane. This is remarkable because half-hourly FX returns have an empirical distribution with a tail index between 2 and 4, as found in Section 5.4.2 and Dacorogna et al. (2001a).

E(r2m) = E(afm) E(e/m)

(8.20)

3 \c\ + (ci + c2)2l + c2 [1 + 3 (c2 + 6c,c2 + 4c2)] < 1

(8.21)