and kurtosis K(m)e where

1 - (ft + a)m

a0{m) = m ----- (8.7)

1 - (p + a)

) = ( + ) -(3(m) (8.8)

K(m)e- 3 + (Ke - 3)/m + 6(Ke - 1)

{m - 1 - m(fi + g) + (ft + q)m}{q - pa(p + a)} X m2(l -p -a)2(l -2Ba)

\P<m) \ < 1 is the solution of the quadratic equation

P(m) a(ft,a,Ke,m)(ft+ar- , , )

(8.9)

(8.10)

1 a-H2 ( , , , ){1 + (P +a)2m] -2b(P,a,m)

~ " (m)

with

( , , £, ) = (8.11)

m(\ - P) + 2m(m - 1)-

( £- l)fl -(P + a)2}

[m- 1 - m(P + a) + (P + a)m}{a - Pa(P + a)} \-(fi+ a)2

b(P,a,m)= {a - p<x(P + a)} 7 (8.12)

1 - (P+a)2

These formulas are used to determine the parameters of the aggregated GARCH processes and can also be used for going from low to high-frequency (i.e., for disaggregation).

When exploring temporal aggregation, we have to choose a time scale. Seasonality is not the subject of an aggregation study, but might disturb it. Eliminating seasonalities by using the #-scale presented in Chapter 6 is a natural choice. However, we have additionally tested an alternative time scale which we call a business time scale in the remainder of this section. This business time simply omits the weekend periods from Friday 22:30 GMT to Sunday 22:30 GMT, when markets are virtually closed.

As a third time scale, we have tried physical time. In physical time, weekends cover two-sevenths of the total sample. This causes a complete breakdown of the estimation procedure, yielding very large a\ estimates. Physical time including weekends is simply unusable here. The aforementioned business time is a usable substitute of physical time from which it only differs in its omission of weekends.

0.25 (-

Beta

FIGURE 8.1 Aggregation of the GARCH(U) for estimated coefficients in business time (•) and theoretically derived coefficients (A) using the (Drost and Nijman, 1993) results for USD-DEM, for different aggregation factors (1 = 10 min; 2 = 20 min; 3 = 30 min; 6 = I hr; 12 = 2 hr; 36 = 6 hr; 72 = 12 hr; 144= 24 hr). The labels of the estimated coefficients (•) are printed in bold. The diagonal dotted line represents the stationarity limit for which ai + j6 = 1. Sampling period: 7 years from January I, 1987, to December 31, 1993.

8.2.3 Estimates of GARCH( 1,1) for Various Frequencies

Time series of USD-DEM have been sampled with frequencies between 10 min and 1 day. For each series, the GARCH(1,1) coefficients have been estimated using the procedure of Section 8.2.1. The resulting coefficients a\ and fi\ (see Equation 8.2) are plotted in Figure 8.1 in the form of black circles, which are labeled by the number of 10-min intervals contained in the time series interval. The label "144" thus means daily sampling.

For comparison, the theoretical values of cc\ and fi\ are also plotted as triangles. The reference values at daily frequency (label 144) are estimated from real data, but the values at all other frequencies are computed from these reference values according to Drost and Nijman (1993), as explained in Section 8.2.2.A computation according to Nelson and Foster (1994) yields similar results that are not plotted here.

0.15

0.8 0.9 1

Beta

FIGURE 8.2 Aggregation of the GARCH( I, I) for estimated coefficients in !?-time (•) and theoretically derived coefficients ( ) using the (Drost and Nijman, 1993) results for the USD-DEM for different aggregation factors (I = I0 min; 2 = 20 min; 3 = 30 min; 6 = I hr; 12 = 2 hr; 36 = 6 hr; 72 = 12 hr; 144= 24 hr). The labels of the estimated coefficients (•) are printed in bold. The diagonal dashed line represents the limit for which o>l + f}\ = 1. Sampling period: 7 years from January I, 1987, to December 31, 1993.

Although the coefficient estimates may look quite reasonable for some lower frequencies, the global picture for all frequencies appears quite odd. In particular, the f3i estimates for frequencies higher than 2 hr decrease down to values close to 0.75, whereas the theory, represented by the triangles in Figure 8.1, suggests that [3 \ should tend to one. The triangles are very far from the corresponding black circles. The hypothesis of volatility being generated by only one GARCH(1,1) process is clearly rejected with the high significance of high-frequency data analyzed over 7 years.

The results of Figure 8.1 are computed on the basis of the business time introduced at the end of Section 8.2.2. Figure 8.2 shows the corresponding results based on #-time. The time scale & is in fact a better choice because of its better deseasonalization. However, the results of Figure 8.2 are similar to those of Figure 8.1. The strong deviation between theoretical and empirically estimated