1. Introduction

It is widely documented that return volatility varies systematically over the

trading day and that this pattern is highly correlated with the intraday variation of

trading volume and bid-ask spreads. Indeed, these strikingly regular patterns of

market activity measures have provided the impetus for much theoretical work. On the other hand, the dynamics of the intraday return volatility process is mostly

ignored in the empirical market microstructure literature. This is quite surprising

given the notion that news arrivals and the resolution of their informational impact

are intimately related to the dynamics of the return volatility process 1. We

conjecture that the intraday return dynamics is neglected primarily because

standard time series models of volatility have proven inadequate when applied to

high frequency returns data. In fact, previous results reported in the literature are

often contradictory and generally defy theoretical predictions. Consequently, there

is no well established paradigm for intraday volatility modeling, and at present its

inclusion in market microstructure research is tenuous.

In this paper we demonstrate that the difficulties encountered by standard

volatility models arise largely from the aforementioned systematic patterns in

average volatility across the trading day. We further show how practical estimation

and extraction of the intraday periodic component of return volatility is both

feasible and indispensable for meaningful intraday dynamic analysis. Particular

attention is paid to the differing impact of the periodic pattern on the dynamic

return features at the various intraday frequencies. To illustrate the range of

applicability of the developed procedures, the analysis is conducted in parallel for

two different asset classes traded under widely different market structures, namely

the over-the-counter foreign exchange interbank market and an organized exchange

for futures equity index contracts. Moreover, to bring out the distinct

character of the intraday returns process, the findings are contrasted to the

corresponding features of interdaily returns series for the identical assets.

The empirical evidence on the properties of average intraday stock returns dates

back to, at least, Wood et al. (1985) and Harris (1986a) who document the

existence of a distinct U-shaped pattern in return volatility over the trading day i.e.

volatility is high at the open and close of trading and low in the middle of the day.

The existence of equally pronounced intraday patterns in foreign exchange markets

has been demonstrated by Miiller et al. (1990) and Baillie and Bollerslev

(1991) 2

Meanwhile, a separate time series oriented literature has modeled the dynamics

of the intraday return volatility directly, building on the ARCH methodology of

1 For example, theoretical work stress issues such as the process of price discovery, the optimal

timing of trades designed to limit price impact, the differing price response to public versus private

information, the clustering of discretionary liquidity trading and the associated increase in market depth

when private information is short-lived and the particular market dynamics associated with periodic

market openings and closures.

2 Empirical work continues to refine and classify the regularities of high frequency returns in this

dimension. Recent studies include Barclay et al. (1990) and Harvey and Huang (1991) on return

variances over trading versus non-trading periods, Lockwood and Linn (1990) on overnight and

intraday return volatility and Ederington and Lee (1993) on the impact of macroeconomic announcements

on inter- and intraday return volatility.

Direct comparison of these intraday volatility studies is complicated by the

different sampling frequencies employed. Nonetheless, as noted by Ghose and

Kroner (1994) and Guillaume (1994), the results regarding the implied degree of

volatility persistence appear puzzling and in stark conflict with the aggregation

results for ARCH models developed by Nelson (1990, 1992), Drost and Nijman

(1993) and Drost and Werker (1996). One potential explanation is that these

theoretical predictions about the relationship between parameter estimates at

different sampling frequencies do not generally apply in the face of strong intraday

periodicity, a fact that has gone largely unnoticed. The most comprehensive prior

attempt at direct modeling of this intraday heteroskedastic pattern in returns is

provided by a series of papers by the research group at Olsen and Associates on

the foreign exchange market e.g. Miiller et al. (1990, 1993) and Dacorogna et al.

(1993). They apply time invariant polynomial approximations to the activity in the

distinct geographical regions of the market over the 24-hour trading cycle 6

Although this might be a reasonable assumption for the foreign exchange market,

we propose an alternative and more general methodology that allows the shape of

the periodic pattern to also depend on the current overall level of return volatility.

This feature makes the procedure readily applicable to the analysis of high

frequency financial data in general and turns out to be essential for our investigation

of the stock market. While our approach accounts for the pronounced intraday

patterns, we explicitly do not make any attempts to correct for the lower frequency

interdaily patterns that also exist e.g. day-of-the-week and holiday effects which are most certainly present in both of the data sets analyzed here. These inter-daily

features are clearly less significant and not critical for the high frequency analysis

pursued here. Yet, in analyses of longer run phenomena, accounting for these

effects may be equally important and could in principle be incorporated along the

same lines.

The remainder of the paper is organized as follows. Section 2 describes our

data and summarizes the intraday average return patterns. Section 3 contains an

analysis of the correlation structure of both raw and absolute 5-minute returns, as

well as a comparison to the corresponding properties of the two daily time series.

The impact of periodic heteroskedasticity on the 5-minute correlations is strong,

while the evidence of standard conditional heteroskedasticity, although evident at

the daily level, appears weak at many intraday frequencies. This motivates our

simple model of intraday returns that renders formal assessments of the relation

between the intra- and interdaily correlation patterns feasible. Section 4 investigates

the properties of temporally aggregated intraday returns. Estimates of the

degree of volatility persistence at the various sampling frequencies are contrasted

to the theoretical aggregation results. Our estimation strategy for characterizing the

intraday periodicity is presented in Section 5. A relatively simple model that

allows for a direct interaction between the level of the daily volatility and the

shape of the intradaily pattern provides a close fit to the average intradaily

volatility patterns for both return series, with the interaction effect being less

significant for the foreign exchange market. The corresponding time series properties

of the filtered returns obtained by extracting the estimated volatility patterns

from the raw series is also explored. Estimation results for these returns are much

more in line with the theoretical predictions. Moreover, this analysis strongly

suggests that several distinct component processes affect the volatility dynamics.

This finding may help shed new light on the long-memory feature in low

frequency return volatility documented by a number of recent studies. Section 6

contains concluding remarks. Details regarding the construction of the 5-minute

foreign exchange and equity returns employed throughout and the flexible nonparametric

procedure used in the estimation of the intraday periodicity are

contained in the appendices.

It is widely documented that return volatility varies systematically over the

trading day and that this pattern is highly correlated with the intraday variation of

trading volume and bid-ask spreads. Indeed, these strikingly regular patterns of

market activity measures have provided the impetus for much theoretical work. On the other hand, the dynamics of the intraday return volatility process is mostly

ignored in the empirical market microstructure literature. This is quite surprising

given the notion that news arrivals and the resolution of their informational impact

are intimately related to the dynamics of the return volatility process 1. We

conjecture that the intraday return dynamics is neglected primarily because

standard time series models of volatility have proven inadequate when applied to

high frequency returns data. In fact, previous results reported in the literature are

often contradictory and generally defy theoretical predictions. Consequently, there

is no well established paradigm for intraday volatility modeling, and at present its

inclusion in market microstructure research is tenuous.

In this paper we demonstrate that the difficulties encountered by standard

volatility models arise largely from the aforementioned systematic patterns in

average volatility across the trading day. We further show how practical estimation

and extraction of the intraday periodic component of return volatility is both

feasible and indispensable for meaningful intraday dynamic analysis. Particular

attention is paid to the differing impact of the periodic pattern on the dynamic

return features at the various intraday frequencies. To illustrate the range of

applicability of the developed procedures, the analysis is conducted in parallel for

two different asset classes traded under widely different market structures, namely

the over-the-counter foreign exchange interbank market and an organized exchange

for futures equity index contracts. Moreover, to bring out the distinct

character of the intraday returns process, the findings are contrasted to the

corresponding features of interdaily returns series for the identical assets.

The empirical evidence on the properties of average intraday stock returns dates

back to, at least, Wood et al. (1985) and Harris (1986a) who document the

existence of a distinct U-shaped pattern in return volatility over the trading day i.e.

volatility is high at the open and close of trading and low in the middle of the day.

The existence of equally pronounced intraday patterns in foreign exchange markets

has been demonstrated by Miiller et al. (1990) and Baillie and Bollerslev

(1991) 2

Meanwhile, a separate time series oriented literature has modeled the dynamics

of the intraday return volatility directly, building on the ARCH methodology of

1 For example, theoretical work stress issues such as the process of price discovery, the optimal

timing of trades designed to limit price impact, the differing price response to public versus private

information, the clustering of discretionary liquidity trading and the associated increase in market depth

when private information is short-lived and the particular market dynamics associated with periodic

market openings and closures.

2 Empirical work continues to refine and classify the regularities of high frequency returns in this

dimension. Recent studies include Barclay et al. (1990) and Harvey and Huang (1991) on return

variances over trading versus non-trading periods, Lockwood and Linn (1990) on overnight and

intraday return volatility and Ederington and Lee (1993) on the impact of macroeconomic announcements

on inter- and intraday return volatility.

Direct comparison of these intraday volatility studies is complicated by the

different sampling frequencies employed. Nonetheless, as noted by Ghose and

Kroner (1994) and Guillaume (1994), the results regarding the implied degree of

volatility persistence appear puzzling and in stark conflict with the aggregation

results for ARCH models developed by Nelson (1990, 1992), Drost and Nijman

(1993) and Drost and Werker (1996). One potential explanation is that these

theoretical predictions about the relationship between parameter estimates at

different sampling frequencies do not generally apply in the face of strong intraday

periodicity, a fact that has gone largely unnoticed. The most comprehensive prior

attempt at direct modeling of this intraday heteroskedastic pattern in returns is

provided by a series of papers by the research group at Olsen and Associates on

the foreign exchange market e.g. Miiller et al. (1990, 1993) and Dacorogna et al.

(1993). They apply time invariant polynomial approximations to the activity in the

distinct geographical regions of the market over the 24-hour trading cycle 6

Although this might be a reasonable assumption for the foreign exchange market,

we propose an alternative and more general methodology that allows the shape of

the periodic pattern to also depend on the current overall level of return volatility.

This feature makes the procedure readily applicable to the analysis of high

frequency financial data in general and turns out to be essential for our investigation

of the stock market. While our approach accounts for the pronounced intraday

patterns, we explicitly do not make any attempts to correct for the lower frequency

interdaily patterns that also exist e.g. day-of-the-week and holiday effects which are most certainly present in both of the data sets analyzed here. These inter-daily

features are clearly less significant and not critical for the high frequency analysis

pursued here. Yet, in analyses of longer run phenomena, accounting for these

effects may be equally important and could in principle be incorporated along the

same lines.

The remainder of the paper is organized as follows. Section 2 describes our

data and summarizes the intraday average return patterns. Section 3 contains an

analysis of the correlation structure of both raw and absolute 5-minute returns, as

well as a comparison to the corresponding properties of the two daily time series.

The impact of periodic heteroskedasticity on the 5-minute correlations is strong,

while the evidence of standard conditional heteroskedasticity, although evident at

the daily level, appears weak at many intraday frequencies. This motivates our

simple model of intraday returns that renders formal assessments of the relation

between the intra- and interdaily correlation patterns feasible. Section 4 investigates

the properties of temporally aggregated intraday returns. Estimates of the

degree of volatility persistence at the various sampling frequencies are contrasted

to the theoretical aggregation results. Our estimation strategy for characterizing the

intraday periodicity is presented in Section 5. A relatively simple model that

allows for a direct interaction between the level of the daily volatility and the

shape of the intradaily pattern provides a close fit to the average intradaily

volatility patterns for both return series, with the interaction effect being less

significant for the foreign exchange market. The corresponding time series properties

of the filtered returns obtained by extracting the estimated volatility patterns

from the raw series is also explored. Estimation results for these returns are much

more in line with the theoretical predictions. Moreover, this analysis strongly

suggests that several distinct component processes affect the volatility dynamics.

This finding may help shed new light on the long-memory feature in low

frequency return volatility documented by a number of recent studies. Section 6

contains concluding remarks. Details regarding the construction of the 5-minute

foreign exchange and equity returns employed throughout and the flexible nonparametric

procedure used in the estimation of the intraday periodicity are

contained in the appendices.