Table 4

(a) See Table 2a for construction of the raw return series. The method for obtaining the filtered returns, R, i, is described in the main text.

(b) See Table 2b for the construction of the raw return series. The method for obtaining the filtered returns, R,j, is described in the main text.

economic factors such as technology and productivity . These distinct sources of volatility persistence could simultaneously influence the return series, resulting in a mixture distribution with different implications for the character of the short- and

The distinct short-run volatility patterns induced by regularly scheduled macroeconomic announcements have been analyzed by Ederington and Lee (1993).

long-run dynamics. A promising first attempt at modeling this interaction between the volatility processes at different time resolutions within a unified framework have been suggested by Muller et al. (1995). In their so-called heterogeneous ARCH, or HARCH, model the volatility at the highest frequency is determined by the sum of numerous ARCH type processes defined over courser time intervals, where each of these components in turn may be linked to the actions of different types of traders with varying time horizons 43.

5.2. Standardized foreign exchange returns

The conjectures underlying a components type formulation of the volatility process are further reinforced by our analysis of the standardized 5-minute returns; /?ли = /?, /(сг,5Лл). If our model provides a good approximation to the data generating process, then this series should display little ARCH effects at daily and lower frequencies, and the intraday ARCH effects should diminish. Consistent with this prediction, the absolute return autocorrelations at the lowest intraday frequencies have been reduced markedly. This is also manifest in the lower curves in Fig. 7a, which depict the correlograms for \Rtn\. Apart from small spikes associated with remaining stochastic periodicity at the daily frequency, the correlations for the absolute returns are generally close to zero beyond the two day lag. Thus, the daily GARCH(1, 1) volatility estimates appear to provide quite satisfactory estimates for the interday volatility dynamics 44. At the same time, Fig. 7a is also indicative of important short-run dynamics that necessarily are unaccounted for by the daily GARCH(1, 1) volatility estimates. This again lends support to our conjecture of distinct short-run, or intraday, components in the fundamental return volatility generating process. The MA(1)-GARCH(1, 1) estimates for the standardized returns in Table 5a reinforce this interpretation by exhibiting a sharp decline in a(/t) + /3(t) as the return horizon increases from five minutes to one hour. In fact, beyond the one hour sampling frequency, the volatility clustering is sufficiently weak that the GARCHO, l) specification breaks down, and only ARCH(l) or homoskedastic MA(1) models are estimated.

5.3. Filtered equity returns

We now turn to the corresponding findings for the S&P 500 returns. In interpreting the results, it is important to recognize that the estimated intraday periodicity now involves interaction terms between the daily volatility level and

Stationary conditions for this new class of time series models are developed in Darorogna et al. (1995).

44 Of course, this apparent lack of any significant long-run correlations in the standardized returns may be due to the relatively short sample of only one year. With a longer span of data the GARCHO, 1) model will most likely fail to capture all the low frequency dynamics (see e.g. Baillie et al., 1996).

(a) See Table 2a for construction of the raw return series. The method for standardizing the returns, R,j, is described in the main text.

(b) See Table 2b for construction of the raw return series. The standardized returns, R, h are generated as described in the main text.

the Fourier functional form, so that not only the level but also the shape of the volatility pattern varies with a,. Thus, our stylized deterministic periodic model discussed in Section 3 is not strictly valid in this context i.e. generally s, пФ s7 for t Ф т. Counter to the results for the DM-$ returns, this time-varying volatility component may weaken the autocorrelations for the raw absolute returns as, effectively, additional noise is injected into the returns process. Fig. 7b seem to indicate that this is indeed the case, as the correlogram for the filtered absolute