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 Muller 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.

5 Examples of early contributions are Engle et al. (1990) for foreign exchange markets and Hamao et al. (1990) for various national equity index returns.

4 See for example Baillie and Bollerslev (1991) and Chan et al. (1991).

5 This literature is exemplified by Bollerslev and Domowitz (1993), Locke and Sayers (1993), Laux and Ng (1993), Foster and Viswanathan (1995) and Goodhart et al. (1993). This research is partially motivated by an attempt to identify the economic origins of the volatility clustering phenomenon as motivated by the mixture of distributions hypothesis; see for example Clark (1973), Tauchen and Pitts (1983), Harris (1986b, 1987), Gallant et al. (1991), Ross (1989) and Andersen (1994, 1996).

6 One may note that the de-volatilization procedure proposed by Zhou (1992) implicitly adjusts for the intraday periodicity in the adaptive calculation of the volatility increments from tick-by-tick observations. Along similar lines, the notion of time deformation in modeling time varying volatility in financial markets has recently been advocated by Ghysels and Jasiak (1994).

Engle (1982). Most of these studies fall into one of three categories. Firstly, some authors investigate the interrelation between returns in geographically separated financial markets that trade sequentially, with a focus on the transmission of information as measured by the degree of spill-over in the mean returns and/or volatility from one market to the next 3. A second strand of this literature is concerned with the lead-lag relations between two or more markets that trade simultaneously 4. Finally, a third group of papers explores the role of information flow and other microstructure variables as determinants of intraday return volatility 5.

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. Muller 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 non-parametric procedure used in the estimation of the intraday periodicity are contained in the appendices.

2. Intraday return periodicity

Our primary data set consists of 5-minute returns for the Deutschemark-U.S.$ (DM-$) exchange rate from October 1, 1992 through September 30, 1993, comprising 74,880 observations, and the Standard and Poors 500 (S&P 500) composite stock index futures contract from January 2, 1986, through December 31, 1989, consisting of a total of 79,280 observations. A more detailed description of the data sources and the calculation of the 5-minute returns is provided in Appendix A. In addition, we use two daily time series of 3,649 spot DM-$