The intraday measures in Table 2a are converted to minutes whereas the interdaily results in Table 3a are given in days. Furthermore, recall that the weekend returns have been excluded from the intraday series. This may induce a distortion in the return dynamics but, again, our informal analysis found this effect to be inconsequential.

31 Note that any serial dependence in the mean will generally increase the order of the implied low frequency weak GARCH model beyond that of the high frequency GARCH( 1,1) model (see Drost and Nijman (1993) for further details). However, the estimate for the MA(1) term for the daily DM-$ GARCHO, 1) model is only -0.034 with an asymptotic standard error of 0.018.

be eliminated. Hence, our findings apply readily to the majority of the prior high frequency studies in the literature, and, in particular, provide an indication of the magnitude of their potential biases due to the neglect of the intraday periodicity in the volatility process.

The MA(1)-GARCH(1, 1) results for the intraday foreign exchange rates are given in Table 2a. The implied persistence measures reveal an alarming degree of irregularity across the different sampling frequencies. For the longer intraday intervals the estimates, converted into minutes, point to half lives around 18,000, or about 12 trading days and mean lags of around 8-9 days. However, the corresponding measures collapse at the intermediate hour frequencies

(£ = 6-18), becoming less than 4 hours, only to resurrect again at the lowest, 5-10 minute, intervals (k = 1,2) where violations of the a(k) + (i(k) < 1 inequality cause the estimated processes to be covariance nonstationary.

These intraday results contrast sharply with the findings for the interdaily DM-$ returns reported in Table 3a i.e. R = ET=(, 1)A.+,JkRT, r=l, 2, [3,649/А:], k=l, 2, 10 where [ ] denotes the integer value. Here, the persistence measures appear quite consistent over the different return intervals, with the half lives and mean lags fluctuating around 20 and 15 days, respectively 30. As for the intraday returns, the median lag is always substantially lower than the mean lag and measured with some imprecision resulting in numerous violations of the inequality governing the lower bound of the statistic, particularly for the smaller sample sizes.

A formal framework for assessment of the parameter estimates obtained at the various sampling frequencies is available from the results on temporal aggregation in ARCH models provided by Nelson (1990, 1992), Drost and Nijman (1993) and Drost and Werker (1996). Specifically, assuming that the GARCHO, 0 model serves as a reasonable approximation to the returns process at the daily frequency, it follows from Drost and Nijman (1993) that the estimates for the corresponding weak GARCHO, 1) models at the lower interdaily frequencies should be related to the daily parameters via the simple formula a(Jt) +/3(lt) = (c*(1) +/3(1))*\ This implies that the estimated half lives, when converted to a common unit of measurement as in our tables, should be stable across the frequencies 31. Our evidence in Table 3a is in line with this prediction and it is also consistent with

Table 3

(a) The returns are based on 3,649 daily quotes for the Deutschemark-U.S. dollar spot exchange rate from March 14, 1979 through September 29, 1993. Weekend and holiday quotes have been excluded. The length of the return intervals equals к days, for a total of [T/k] observations, where [ ] denotes the integer value. See Table 2a for the definition of the half life, mean lag, and median lag. These measures are converted to trading days.

(b) The returns are based on 9,558 daily observations for the Standard and poors 500 composite index from January 2, 1953 through December 31, 1990. The length of the return intervals equals к days, for a total of [T/k] observations, where [ ] denotes the integer value. See Table 2a for the definition of the half life, mean lag, and median lag. These measures are converted to trading days.

earlier evidence for other interdaily exchange rates reported in Baillie and Bollerslev (1989).

The observations above suggest that the results for the intraday exchange rates in Table 2a are indicative of serious model misspecification. For further analysis, we again use the estimates for the daily GARCHO, 1) model (й(288) = 0.105 and /3(288) = 0.873) as a natural benchmark since these are unaffected by the intraday periodicity. The results of Drost and Nijman (1993) and Drost and Werker (1996) now imply that the intraday returns should follow weak GARCH(1, 1) processes with a(k) + converging to unity and a(k) converging towards zero as the length of the sampling interval, k, decreases. In fact, Nelson (1990, 1992)

establishes general conditions under which GARCHO, 1) models, even if misspec-ified at all frequencies, will satisfy the above convergence results and produce consistent estimates for the true volatility process at the highest sampling frequencies. Unfortunately, these predictions do not allow for deterministic effects in the volatility process. Yet, given the estimated standard errors, the 12 hourly through 2 hourly returns (k = 24-144) are roughly in line with the qualitative predictions. Beyond this point the theoretical results are strongly contradicted, however. The most blatant violations are provided by the much lower volatility persistence, as measured by &{k) + (5{k), for the models based on -j-1 \ hourly returns (k = 6-18). For the 5-15 minute returns (k < 3) the sum of the estimates for a(k) and (lik) is again near unity, but the relative size of the coefficients does not conform to the theoretical predictions, as a(jt) is too large.

Our intraday results in Table 2a are not unusual. They mirror the range of estimates previously obtained in the literature over corresponding return frequencies. In particular, Engle et al. (1990) and Hamao et al. (1990) who primarily rely on returns over six hours or longer find evidence of volatility persistence that is consistent with estimates from daily data. In contrast, Baillie and Bollerslev (1991) and Foster and Viswanathan (1995), on using hourly and half-hourly returns, find much lower volatility persistence 32. However, the volatility persistence measures appear to rebound at the higher frequencies e.g. Bollerslev and Domowitz (1993) report 5-minute GARCHO, 1) estimates for aw + P(k) close to one but, as in Table 2a, a(t) seems too large. For the very highest frequencies, Locke and Sayers (1993) find that 1-minute returns generally display little volatility persistence. Conversely, Goodhart et al. (1993) detect very strong persistence in quote-by-quote data, but also find a marked decline in the persistence once information events are taken explicitly into account, illustrating how specific news arrivals may overwhelm the underlying conditional heteroskedasticity at the extremely high frequencies.

Our findings provide strong, albeit indirect, evidence in support of the conjecture that a contributing factor to the systematic variation in volatility estimates across return frequencies is the interaction between the previously well documented interdaily conditional heteroskedasticity and the intraday periodicity. For the highest frequencies the change in the intraday pattern will generally appear smooth between adjacent returns, and thus have little impact on the overall estimated degree of volatility persistence. However, as argued more formally below, the existence of short-lived intraday volatility components (in addition to the intraday periodicity) will tend to increase the dependence of (сгД)2 on the

Interestingly, Laux and Ng (1993) deviate from these studies by finding high persistence in half-hourly data for the CME currency futures. However, the futures market only operates during the most active trading in the U.S. segment of the foreign exchange interbank market and this represents a period of relative stability for the intraday volatility pattern.