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