5.1. Filtered foreign exchange returns

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To further investigate these issues, define the filtered 5-minute return series;

~,,, = R,,n/gt," 38. If the characterization of the 5-minute return series in Eq. (7) is

perfect and the associated estimation error is negligible, then ignoring the impact

of the weak first order return correlation, the filtered returns should conform more

closely to the theoretical aggregation results for the GARCH(1, 1) model. We

explicitly consider how well this hypothesis holds up, but we also keep in mind that the elimination of the main distorting effects of the intraday periodicity may

bring out new features of the volatility process that were difficult to untangle prior

to filtration.

First, we briefly summarize the main characteristics of the filtered series. While

the mean and the standard deviation of these returns are virtually unchanged from

Table la, both skewness and kurtosis are generally reduced by filtering the returns.

For instance, the 5-minute skewness and kurtosis for /~t.n equal 0.175 and 15.8,

respectively. Interestingly, the evidence for negative return autocorrelations at the

very highest frequencies becomes even more pronounced following the filtration,

as measured by Pl = -0.090 for the 5-minute returns. At the same time the first

order absolute 5-minute return autocorrelations decline slightly to pg = 0.292 39

The correlation structure for the absolute 5-minute filtered returns is further

illustrated in Fig. 7a. The upper curves represent the correlogram for the raw

returns, while the middle curves are for the filtered returns. The dramatic reduction

in the periodic pattern is particularly striking for the longest lags. However, from

the daily peaks in the 5 day correlogram it is clear that some periodicity remains,

suggesting the presence of a stochastic periodic, or market specific, component in

the intraday volatility 4o. Note also that the correlations for ]/~t,n] at the daily

frequencies are always below the correlations for the raw absolute returns, ]Rt.,,].

This is consistent with the predictions from Eq. (5).

More direct evidence is provided by the estimated MA(1)-GARCH(1, l)

models reported in Table 4a. Compared to the results in Table 2a, the volatility

parameters now display a much more coherent pattern across the return frequencies volatility process (see e.g. Baillie et al., 1996; Dacorogna et al., 1993; Ding et al.,

1993). Note again the slightly higher peaks associated with the weekly frequencies.

Although the GARCH(1, 1) estimates for the high frequency filtered returns

defy the theoretical predictions, the results are encouraging in terms of our ability

to recover meaningful intraday volatility dynamics. In particular, by eliminating

the deterministic periodicity we were able to uncover an interesting pattern in the

absolute return correlogram which was largely invisible prior to the periodic

filtering. A detailed investigation of the source of this phenomenon is well beyond the scope of the present paper. However, we conjecture that the following factors

have some impact on the observed correlation patterns. First, there may well be

some positively cyclical correlated components left in the ]/~t,n[ series, thus

inducing spurious short-run dynamics in the return volatility. Second, and more

importantly, the results also point to the potential importance of several distinct

intraday volatility processes governed by e.g. economic announcements, the

release of economic statistics, etc. each of which inherently may be of a less

persistent nature than the volatility caused by changing trends in fundamental

 (a) See Table 2a for construction of the raw return series. The method for obtaining the filtered returns,

Rt.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, ~qt.i, is described in the main text.

economic factors such as technology and productivity 42. 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 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 Miiller 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

To further investigate these issues, define the filtered 5-minute return series;

~,,, = R,,n/gt," 38. If the characterization of the 5-minute return series in Eq. (7) is

perfect and the associated estimation error is negligible, then ignoring the impact

of the weak first order return correlation, the filtered returns should conform more

closely to the theoretical aggregation results for the GARCH(1, 1) model. We

explicitly consider how well this hypothesis holds up, but we also keep in mind that the elimination of the main distorting effects of the intraday periodicity may

bring out new features of the volatility process that were difficult to untangle prior

to filtration.

First, we briefly summarize the main characteristics of the filtered series. While

the mean and the standard deviation of these returns are virtually unchanged from

Table la, both skewness and kurtosis are generally reduced by filtering the returns.

For instance, the 5-minute skewness and kurtosis for /~t.n equal 0.175 and 15.8,

respectively. Interestingly, the evidence for negative return autocorrelations at the

very highest frequencies becomes even more pronounced following the filtration,

as measured by Pl = -0.090 for the 5-minute returns. At the same time the first

order absolute 5-minute return autocorrelations decline slightly to pg = 0.292 39

The correlation structure for the absolute 5-minute filtered returns is further

illustrated in Fig. 7a. The upper curves represent the correlogram for the raw

returns, while the middle curves are for the filtered returns. The dramatic reduction

in the periodic pattern is particularly striking for the longest lags. However, from

the daily peaks in the 5 day correlogram it is clear that some periodicity remains,

suggesting the presence of a stochastic periodic, or market specific, component in

the intraday volatility 4o. Note also that the correlations for ]/~t,n] at the daily

frequencies are always below the correlations for the raw absolute returns, ]Rt.,,].

This is consistent with the predictions from Eq. (5).

More direct evidence is provided by the estimated MA(1)-GARCH(1, l)

models reported in Table 4a. Compared to the results in Table 2a, the volatility

parameters now display a much more coherent pattern across the return frequencies volatility process (see e.g. Baillie et al., 1996; Dacorogna et al., 1993; Ding et al.,

1993). Note again the slightly higher peaks associated with the weekly frequencies.

Although the GARCH(1, 1) estimates for the high frequency filtered returns

defy the theoretical predictions, the results are encouraging in terms of our ability

to recover meaningful intraday volatility dynamics. In particular, by eliminating

the deterministic periodicity we were able to uncover an interesting pattern in the

absolute return correlogram which was largely invisible prior to the periodic

filtering. A detailed investigation of the source of this phenomenon is well beyond the scope of the present paper. However, we conjecture that the following factors

have some impact on the observed correlation patterns. First, there may well be

some positively cyclical correlated components left in the ]/~t,n[ series, thus

inducing spurious short-run dynamics in the return volatility. Second, and more

importantly, the results also point to the potential importance of several distinct

intraday volatility processes governed by e.g. economic announcements, the

release of economic statistics, etc. each of which inherently may be of a less

persistent nature than the volatility caused by changing trends in fundamental

 (a) See Table 2a for construction of the raw return series. The method for obtaining the filtered returns,

Rt.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, ~qt.i, is described in the main text.

economic factors such as technology and productivity 42. 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 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 Miiller 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