4.3. Interpretation of the GARCH results .for different return frequencies

This section summarizes the evidence from fitting standard GARCH models to

the return series at different frequencies. Particular emphasis is placed on the type

of distortions that may be induced by the strong periodic intraday patterns which

are ignored in these models. There are a couple of indirect ways to gauge the

effect. First, there are theoretical predictions about the relation between the

parameters at various frequencies. If these are most obviously violated at the

particular frequencies where the intraday periodicity are expected to assert the

maximal impact, this is therefore consistent with the periodic pattern being a

dominant source of misspecification for these models. Second, to the extent that

the periodic pattern is a strictly deterministic intraday phenomenon as suggested in

Section 3, the distortions should be absent from models estimated at daily or

multiple-day frequencies. Consequently, if the theoretical aggregation results work

satisfactorily at the multiple-day frequencies but break down intradaily then this is

further evidence of a significant impact of the periodic pattern on the dynamic

properties of the intraday volatility process. We also relate our findings to the prior

estimates reported from intraday volatility modeling. The comparison shows that

our results are fully consistent with the diverse set of estimates reported in the

literature once we control for the different return frequencies employed in the

studies. Finally, the explicit incorporation of the cyclical pattern in Section 5

verifies that most of the distortions attributable to the intraday volatility cycle may 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 12½ trading days and mean lags of around 8-9 days. However, the

1 I corresponding measures collapse at the intermediate 5-1g hour frequencies

(k = 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 / +/3ck ) < 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=-~r=(t_l)k+l,tkR.r, t= 1, 2 . . . . .

[3,649/k], k = 1, 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 3o

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 GARCH(1, 1) 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 GARCH(1, 1) models at the lower interdaily frequencies should be related to

the daily parameters via the simple formula a~+/3~k)=(Ce~l)+ fl~l/) k. 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

30 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.

3t 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(l) term for the daily DM-$

GARCH(I, 1) model is only -0.034 with an asymptotic standard error of 0.018.

 (a) The returns are based on 3,649 daily quotes tor 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 k 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 poor's 500 composite index

from January 2, 1953 through December 31, 1990. The length of the return intervals equals k 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 Boilerslev

(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 GARCH(1, 1) model (&(z88~ = 0.105 and

/3(2881 = 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(~)+/3(k ~ converging to unity and a(k ) converging towards zero as the

length of the sampling interval, k, decreases. In fact, Nelson (1990, 1992)

T. G. Andersen, T. Bollerslev / Journal of Empirical Finance 4 (1997) 115-158 139

establishes general conditions under which GARCH(1, 1) models, even if misspecified

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

^ ^ 1 1 measured by a(~) +/3(k ), for the models based on ~-17 hourly returns (k = 6-18).

For the 5-15 minute returns (k < 3) the sum of the estimates for ~(k) and /3(k) is

again near unity, but the relative size of the coefficients does not conform to the

theoretical predictions, as &(k) 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 GARCH(1, 1) estimates for a(k ) + /3(k ) close to one but, as in

Table 2a, &(k) 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 (O't,k,) 2 on the lagged squared innovation, (etc.h- )2, relative to the overall volatility level,

(~r,! n_ 1) 2, hence explaining the relatively large estimates for c~k ) at the shortest

1 1 return intervals. For the intermediate 7-1 ~ hour return models the change in the

average volatility between sampling intervals will typically appear much more

abrupt, resulting in significantly lower persistence measures. Beyond the 2-hour

intervals the periodic pattern is averaged over a substantial part of the 24-hour

trading day, and the intraday exchange rate estimates are generally closer to the

implications obtained from daily models.

The results for the S&P 500 equity returns tell a similar story. The interdaily

estimates in Table 3b are again broadly consistent with the a priori predictions

based on the daily GARCH(I, 1) model 33. Although the volatility persistence is

higher than for the foreign exchange returns, &~k) +/3~k) again displays a general

smooth decline and the explicit persistence measures are fairly stable across the

different return horizons. The discrepancy between the half lives, mean lags and

median lags implied by the intradaily and interdaily returns are even stronger than

for the foreign exchange rate data, however 34. Moreover, the pattern in the

intraday estimates for a<~) +/3~k ) reported in Table 2b is again erratic, reaching

lows at the ½-day (k = 40) and 20-25 minute (k = 4, 5) return horizons, and highs

at the 40-50 minute (k = 8, 10) and 5-minute (k-- 1) horizons. We conclude that

the daily GARCH models conform closely to the theoretical predictions, but the

strong intraday periodic patterns in volatility render the intradaily estimates highly

variable and generally hard to interpret.

This section summarizes the evidence from fitting standard GARCH models to

the return series at different frequencies. Particular emphasis is placed on the type

of distortions that may be induced by the strong periodic intraday patterns which

are ignored in these models. There are a couple of indirect ways to gauge the

effect. First, there are theoretical predictions about the relation between the

parameters at various frequencies. If these are most obviously violated at the

particular frequencies where the intraday periodicity are expected to assert the

maximal impact, this is therefore consistent with the periodic pattern being a

dominant source of misspecification for these models. Second, to the extent that

the periodic pattern is a strictly deterministic intraday phenomenon as suggested in

Section 3, the distortions should be absent from models estimated at daily or

multiple-day frequencies. Consequently, if the theoretical aggregation results work

satisfactorily at the multiple-day frequencies but break down intradaily then this is

further evidence of a significant impact of the periodic pattern on the dynamic

properties of the intraday volatility process. We also relate our findings to the prior

estimates reported from intraday volatility modeling. The comparison shows that

our results are fully consistent with the diverse set of estimates reported in the

literature once we control for the different return frequencies employed in the

studies. Finally, the explicit incorporation of the cyclical pattern in Section 5

verifies that most of the distortions attributable to the intraday volatility cycle may 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 12½ trading days and mean lags of around 8-9 days. However, the

1 I corresponding measures collapse at the intermediate 5-1g hour frequencies

(k = 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 / +/3ck ) < 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=-~r=(t_l)k+l,tkR.r, t= 1, 2 . . . . .

[3,649/k], k = 1, 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 3o

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 GARCH(1, 1) 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 GARCH(1, 1) models at the lower interdaily frequencies should be related to

the daily parameters via the simple formula a~+/3~k)=(Ce~l)+ fl~l/) k. 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

30 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.

3t 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(l) term for the daily DM-$

GARCH(I, 1) model is only -0.034 with an asymptotic standard error of 0.018.

 (a) The returns are based on 3,649 daily quotes tor 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 k 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 poor's 500 composite index

from January 2, 1953 through December 31, 1990. The length of the return intervals equals k 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 Boilerslev

(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 GARCH(1, 1) model (&(z88~ = 0.105 and

/3(2881 = 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(~)+/3(k ~ converging to unity and a(k ) converging towards zero as the

length of the sampling interval, k, decreases. In fact, Nelson (1990, 1992)

T. G. Andersen, T. Bollerslev / Journal of Empirical Finance 4 (1997) 115-158 139

establishes general conditions under which GARCH(1, 1) models, even if misspecified

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

^ ^ 1 1 measured by a(~) +/3(k ), for the models based on ~-17 hourly returns (k = 6-18).

For the 5-15 minute returns (k < 3) the sum of the estimates for ~(k) and /3(k) is

again near unity, but the relative size of the coefficients does not conform to the

theoretical predictions, as &(k) 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 GARCH(1, 1) estimates for a(k ) + /3(k ) close to one but, as in

Table 2a, &(k) 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 (O't,k,) 2 on the lagged squared innovation, (etc.h- )2, relative to the overall volatility level,

(~r,! n_ 1) 2, hence explaining the relatively large estimates for c~k ) at the shortest

1 1 return intervals. For the intermediate 7-1 ~ hour return models the change in the

average volatility between sampling intervals will typically appear much more

abrupt, resulting in significantly lower persistence measures. Beyond the 2-hour

intervals the periodic pattern is averaged over a substantial part of the 24-hour

trading day, and the intraday exchange rate estimates are generally closer to the

implications obtained from daily models.

The results for the S&P 500 equity returns tell a similar story. The interdaily

estimates in Table 3b are again broadly consistent with the a priori predictions

based on the daily GARCH(I, 1) model 33. Although the volatility persistence is

higher than for the foreign exchange returns, &~k) +/3~k) again displays a general

smooth decline and the explicit persistence measures are fairly stable across the

different return horizons. The discrepancy between the half lives, mean lags and

median lags implied by the intradaily and interdaily returns are even stronger than

for the foreign exchange rate data, however 34. Moreover, the pattern in the

intraday estimates for a<~) +/3~k ) reported in Table 2b is again erratic, reaching

lows at the ½-day (k = 40) and 20-25 minute (k = 4, 5) return horizons, and highs

at the 40-50 minute (k = 8, 10) and 5-minute (k-- 1) horizons. We conclude that

the daily GARCH models conform closely to the theoretical predictions, but the

strong intraday periodic patterns in volatility render the intradaily estimates highly

variable and generally hard to interpret.