45 For instance, pA for the aggregated filtered returns, Л* , equal 0.371, 0.379, 0.341, 0.335, 0.305, 0.303, 0.292, 0.290 and 0.292 for к = 1, 2, 4, 5, 8, 10, 16, 20 and 40, respectively.

returns, \Rtn\, lies substantially above that for the raw series, \R, \. Interestingly, this does not just occur at the 5-minute sampling frequency; the absolute return autocorrelation across the nine different intraday frequencies, as measured by pA, QA(\0) and VRA, are all markedly higher than the corresponding statistics for the raw returns in Table lb 45. This is also in line with the MA(1)-GARCH(1, 1) estimation results reported in Table 5b. The parameter estimates obtained as we move from the \ day to the 40-minute return horizon are again consistent with the theory. Thereafter, the sum a(k) and fi(k) decay slightly, but more importantly a(k) starts to increase. However, all the intraday estimates now consistently point towards a very high degree of volatility persistence and in all instances a{k) + /3(A) are higher than the estimates for the raw return series in Table 2b. Note also, that in line with the findings for the DM-$ returns, the 5 day correlogram in Fig. 7b for the 5-minute filtered S&P 500 returns still retains a distinct periodic pattern, indicating the presence of even more complicated stochastic volatility components. Nonetheless, the simple filtering procedure again succeeds in eliminating a large proportion of the systematic intraday variation in the absolute returns and in so doing has unveiled a cleaner and starkly different picture of the volatility dynamics.

5.4. Standardized equity returns

In contrast to the results for the DM-$, the first order autocorrelations for the standardized absolute returns for the S&P 500, \Rk j, remain highly significant for the lower intraday frequencies. Even at the j day return frequency, pA = 0.094 exceed the corresponding asymptotic standard error by more than a factor four. This is also confirmed by the much higher a(k) + /3(lt) estimates for the intraday GARCHd, 1) models for Rkn given in Table 5b. Similarly, the correlograms in Fig. 7b for the standardized returns indicate a much higher degree of volatility persistence in the 5-minute S&P 500 returns than was the case for DM-$ returns. In fact, the standardized absolute return correlogram stays mostly positive for the first 22 trading days, or about a month. This indication of more persistent volatility dynamics is likely attributable to the longer time span for the equity data. For example, Dacorogna et al. (1993) find that the absolute standardized return autocorrelations remain positive for one month when using 20-minute DM-$ data over a four year sample. Additionally, from Guillaume (1994) it is evident that our ability to detect significant long-horizon absolute return correlations is intimately linked to the length of the sampling period. Hence, although the interdaily GARCH(1, 0 model may capture a large portion of the day-to-day volatility clustering, the models deficiency in dealing with long-memory behavior necessarily becomes more transparent when the time span of the data increases.

6. Concluding remarks

Our analysis of the intraday volatility patterns in the DM-$ foreign exchange and S&P 500 equity markets documents how traditional time series methods applied to raw high frequency returns may give rise to erroneous inference about the return volatility dynamics. Explicit allowance for the influence of the strong periodicity, as exemplified by our flexible Fourier form, is a necessary requirement for discovery of the salient intraday volatility features. Moreover, adjusting for the pronounced periodic structure appears critical in uncovering the complex link between the short- and long-run return components, which may help to explain the apparent conflict between the long-memory volatility characteristics observed in interday data and the rapid short-run decay associated with news arrivals in intraday data. More directly, however, our findings have immediate and important implications for a large range of issues in the rapidly growing literature using very high frequency financial data. Examples include investigations into the lead-lag relationship among returns and volatility both within and across different markets, the effect of cross listings of securities, the fundamental determinants behind the volatility clustering phenomenon, the development of real time trading and investment strategies and the evaluation of continuous option valuation and hedging decisions. Only future research will reveal the extent of the biases induced into these studies by the neglect of intraday periodic components.

Acknowledgements

We would like to thank Richard T. Bailie, the editor, an anonymous referee, Dominique Guillaume, Robert J. Hodrick, Charles Jones, Stephen J. Taylor, Kenneth F. Wallis, along with seminar participants at the Olsen and Associates Research Institute for Applied Economics, the workshop on Market Micro Structure at the Aarhus School of Business, the Fall 1994 NBER Asset Pricing Meeting at the Wharton School, the HFDF-I Conference in Zurich, the 7th World

We conclude, that in spite of important institutional differences in the markets and the associated intradaily volatility patterns, there is strong indications that the volatility processes for the foreign exchange and the U.S. equity market share several important qualitative dynamic features. Moreover, these characteristics were largely invisible prior to our filtration of the intraday periodic structures in the high frequency return series. At the same time interesting differences between the average volatility level and volatility persistence in the two markets also emerge. These conclusions would be next to impossible to reach from the, at first sight, rather perplexing estimates obtained directly from the raw high frequency returns.

Congress of the Econometric Society in Tokyo, Duke University and the University of California at Santa Barbara for helpful comments.

Appendix A. Data description

A. 1. The Deutschemark-U.S. dollar exchange rate data

The DM-$ exchange rate data consist of all the quotes that appeared on the interbank Reuters network during the October 1, 1992 through September 29, 1993 sample period. The data were collected and provided by Olsen and Associates. Each quote contains a bid and an ask price along with the time to the nearest even second. Approximately 0.36% of the 1,472,241 raw quotes were filtered out using the algorithm described in Dacorogna et al. (1993). During the most active trading hours, an average of five or more valid quotes arrive per minute; see Bollerslev and Domowitz (1993). The exchange rate figure for each 5-minute interval is determined as the interpolated average between the preceding and immediately following quotes weighted linearly by their inverse relative distance to the desired point in time. For instance, suppose that the bid-ask pair at 14.14.56 was 1.6055-1.6065, while the next quote at 14.15.02 was 1.6050-1.6055. The interpolated price at 14.15.00 would then be exp{l/3 [ln( 1.6055) + ln(1.6065)]/2 + 2/3 [ln( 1.6050) + ln(l .6055)]/2} = 1.6055. The nth 5-minute return for day t, Rtll, is then simply defined as the difference between the midpoint of the logarithmic bid and ask at these appropriately spaced time intervals. This definition of the returns has the advantage, that it is symmetric with respect to the denomination of the exchange rate. However, as noted by Muller et al. (1990), the numerical difference from the logarithm of the middle price is negligible. All 288 intervals during the 24-hour daily trading cycle are used. However, in order to avoid confounding the evidence in the correlation analysis conducted below by the decidedly slower trading patterns over weekends, all the returns from Friday 21.00 Greenwich mean time (GMT) through Sunday 21.00 GMT were excluded (see Bollerslev and Domowitz (1993) for a detailed analysis of the quote activity in the DM-$ interbank market and a justification for this weekend definition). Similarly, to preserve the number of returns associated with one week we make no corrections for any worldwide or country specific holidays that occurred during the sample period. All in all, this leaves us with a sample of 260 days, for a total of 74,880 5-minute intraday return observations i.e. Rln, n = 1, 2, 288, t=l,2, 260.

A.2. The standard and poors 500 stock index futures data

The intraday S&P 500 futures data are based on quote capture information provided by the Chicago Mercantile Exchange (CME) from January 2, 1986