<<

. 2
( 2)



2a and b naturally lend themselves to estimation by the Fourier flexible functional
form introduced by Gallant, 1981, 1982 36. In a related context, Dacorogna et al.
(1993) have proposed estimating the periodicity in the activity in the foreign
exchange market as the sum of three polynomials corresponding to the distinct
geographical locations of the market. Returns measured on their resulting theta-time
scale correspond closely to our filtered returns defined and analyzed below.
However, one advantage of the approach advocated here is that it allows the shape
of the periodic pattern in the market to also depend on the overall level of the
volatility; a feature which turns out to be important for the equity market. Also,
the combination of trigonometric functions and polynomial terms are likely to
result in better approximation properties when estimating regularly recurring
patterns. Furthermore, our approach for estimating st. n utilizes the full time series
dimension of the returns data, as opposed to simply estimating the average pattern
across the trading day. Full details of the approach are provided in Appendix B.
Meanwhile, it is clear from the estimated average intraday periodic patterns
depicted in Fig. 6a and b, that the fitted values, ˜t.n, provide a close approximation
to the overall volatility patterns in both markets. Of course, the usefulness of the
procedure will ultimately depend upon the degree to which it is successful in
identifying the periodic components in a temporal dimension as well. If so, the
approach may serve as the basis for a nonlinear filtering procedure that could
eliminate the periodic components prior to the analysis of any intraday return
volatility dynamics 37


35 This feature is particularly important for the equity returns for which the general U-shaped
volatility pattern is transformed into more of a J-shape on the highest volatility days (see Andersen and
Bollerslev, 1994).
36 This technique has previously been applied to financial return series in a different context by Pagan
and Schwert (1990).
37 This same methodology may also be used directly for prediction of future volatility over different
intraday time intervals. Such intraday volatility prediction may be particularly important in the pricing
and/or continuous re-balancing of hedged intraday options positions. We shall not pursue this issue
any further here, however.
T.G. Andersen, T. Bollerslev / Journal of Empirical Finance 4 (1997) 115-158
142

0.08, , , , , . , , , , , , , . ,



0.07f (a)
[
,˜ 0.06

oo5 [
"˜ 0.04


0.03

0.02

0.01

O.OC i 810 i L i i i i i i i i
40 120 160 200 240 280 320
Five Minute Interval
0 . 1 2 . , . , . , . , . , . , . , . , .




(b)
0.11


E 0.10


0.09

0.08

0.07

˜: 0.06

0.05


0.04 ' ' ;'3'0 . .... .. ;
0 10 2 40 50 60 70 8 90
Five Minute Interval
Fig. 5. Flexible Fourier functional form of intraday average absolute returns, (a) D M - $ , (b) S&P 500.



5.1. Filtered foreign exchange returns


To further investigate these issues, define the filtered 5-minute return series;
˜,,, = R,,n/gt," 38. If the characterization of the 5-minute return series in Eq. (7) is
perfect and the associated estimation error is negligible, then ignoring the impact
of the weak first order return correlation, the filtered returns should conform more
closely to the theoretical aggregation results for the GARCH(1, 1) model. We
explicitly consider how well this hypothesis holds up, but we also keep in mind


38 Alternatively, Rt,. might also be mean adjusted. Since the mean return is practically zero, this is
immaterial.
T.G. Andersen, 72 Bollerslev / Journal of Empirical Finance 4 (1997) 115-158 143


that the elimination of the main distorting effects of the intraday periodicity may
bring out new features of the volatility process that were difficult to untangle prior
to filtration.
First, we briefly summarize the main characteristics of the filtered series. While
the mean and the standard deviation of these returns are virtually unchanged from
Table la, both skewness and kurtosis are generally reduced by filtering the returns.
For instance, the 5-minute skewness and kurtosis for /˜t.n equal 0.175 and 15.8,
respectively. Interestingly, the evidence for negative return autocorrelations at the
very highest frequencies becomes even more pronounced following the filtration,
as measured by Pl = - 0 . 0 9 0 for the 5-minute returns. At the same time the first
order absolute 5-minute return autocorrelations decline slightly to p g = 0.292 39
The correlation structure for the absolute 5-minute filtered returns is further
illustrated in Fig. 7a. The upper curves represent the correlogram for the raw
returns, while the middle curves are for the filtered returns. The dramatic reduction
in the periodic pattern is particularly striking for the longest lags. However, from
the daily peaks in the 5 day correlogram it is clear that some periodicity remains,
suggesting the presence of a stochastic periodic, or market specific, component in
the intraday volatility 4o. Note also that the correlations for ]/˜t,n] at the daily
frequencies are always below the correlations for the raw absolute returns, ]Rt.,,].
This is consistent with the predictions from Eq. (5).
More direct evidence is provided by the estimated M A ( 1 ) - G A R C H ( 1 , l)
models reported in Table 4a. Compared to the results in Table 2a, the volatility
parameters now display a much more coherent pattern across the return frequen-
cies 41. In accord with the theoretical aggregation results the estimates for ct˜k) +
l
tick) increase almost monotonically from the ˜ day to the 2 hour frequency. For
the higher frequencies the theoretical predictions again begin to falter, however,
although less starkly than before. The sum &˜k) +/3˜k˜ remains high, but no longer
increases monotonically and more importantly, &˜k) starts to increase while J3˜k ˜
generally declines. These findings again should be interpreted in light of the
absolute return autocorrelograms in Fig. 7a. The most striking feature is the initial
rapid decay in the autocorrelations, followed by an extremely slow rate of decay
thereafter. This pattern is not consistent with the exponential decay associated with
a GARCH(1, 1) model for the 5-minute returns. Instead, these findings point to a
slow hyperbolic rate of decay in the autocorrelation structure for the absolute
returns, which is consistent with the presence of long-memory features in the


39 See Andersen and Bollerslev for extensive documentation of the summary statistics for both
filtered and standardized returns.
4o The distinction between heat wave, or market specific, volatility as opposed to meteor shower, or
global, volatility clustering was first discussed in Engle et al. (1990). This finding also points to the
potential gains of more general dynamic periodic volatility type modeling (see e.g. Bollerslev and
Ghysels, 1996).
41 To conserve space, we do not report the half life or the mean and median lags.
T. G. Andersen, T. Bollerslee / Journal of Empirical Finance 4 (1997) 115-158
144


(a) Five Days C o r r e l o g r a m
0.35,

0.30 I Filtered R˜.,
Standardized Rtl.a
0.25 I

0.20 f
=
o
0.15
/'˜ ^ J _

0.10 ˜t t' '
i ˜ , , ,
tj
0.05

-0.00

-0.05

-0.10 i i i i i
1 2 3 4
Daily Lag

Forty Days Correlogram
0.35

0.30
Filtered R˜.,
Standsrdlzed R˜..
0.25

0.20

0.15
.-$
it! ,,,jl:,,,
0.10

0.05
Lii.[.}il [.[Li,i,,t
i[J.l
-0.00
'l'"r¢' "˜Ir˜, T,"r" ,p,˜,˜ "T,' 'iI ˜lrV˜ "' IF, I?˜,'˜ ' t ˜
, ˜i'.,, F ,. ,,,,,,,, , ,a',,˜. ,,
I
• , ,

-0.05

--0. I0
0 . .4 . . . 8 1 2' o ' 24 2 32 36 40
' ' ....
Daily Lag

Fig. 7. Absolute returns, (a) D M - $ , (b) S & P 500.



volatility process (see e.g. Baillie et al., 1996; Dacorogna et al., 1993; Ding et al.,
1993). Note again the slightly higher peaks associated with the weekly frequen-
cies.
Although the GARCH(1, 1) estimates for the high frequency filtered returns
defy the theoretical predictions, the results are encouraging in terms of our ability
to recover meaningful intraday volatility dynamics. In particular, by eliminating
the deterministic periodicity we were able to uncover an interesting pattern in the
absolute return correlogram which was largely invisible prior to the periodic
filtering. A detailed investigation of the source of this phenomenon is well beyond
T.G. Andersen, T. Bollerslec / Journal of Empirical Finance 4 (1997) 115-158 145


(b) Five D a y s C o r r e l o g r a m
0.4-0,

0.35 I
i R˜.˜
Filtered
Rt].
Standardized
0.30 I
0.25 ˜-:

:' " " t.
0.20
0.15
,,˜
0.10

0.05

0.00

-0,05 i , i i
4
1 2 3
Daily Lag




Forty Days Correlogram

0.40 . , . , , , . , . , . , . ,


0.35 RL°R˜,a RL.
7˜˜ R--
Fi|t e ˜'ed
Standardized
0.30

0.25
o
.4
•.˜ 0.20

o.15

0.10
0.05 K'Jg • '˜j, Jl;lq'˜d˜.˜12Jl˜9.˜LA.,.˜r.
" ' ' L' ]
˜ l ' ˜ t! " ˜ ,',"','˜"q'*
˜$, ˜ h
" '˜ ,v˜" ˜ . '
0.00
-0.05 i i , i , i i J ˜ i0 , J i J , i i i i
0 4 8 12 16 2 24 28 32 56 40
Daily Lag


Fig. 7 (continued).




the scope of the present paper. However, we conjecture that the following factors
have some impact on the observed correlation patterns. First, there may well be
some positively cyclical correlated components left in the ]/˜t,n[ series, thus
inducing spurious short-run dynamics in the return volatility. Second, and more
importantly, the results also point to the potential importance of several distinct
intraday volatility processes governed by e.g. economic announcements, the
release of economic statistics, etc. each of which inherently may be of a less
persistent nature than the volatility caused by changing trends in fundamental
T. G. Andersen, T. Bollerslev / Journal of Empirical Finance 4 (1997) 115-158
146

Table 4
k T/k a(k ˜ ,G(k) a(k) + fl(k)
(a) MA(1)-GARCH(1, l) models for filtered DM-$ exchange rate returns
l ) k + l,nkRt,i = t£(k) -1- O(k) etk.n- I
R ˜ n ˜- 1 0 0 " E i = ( n - -
'+ e l , (o',k,) e = O˜(k + a(k˜(e[, 1)2 + /3(k)(o-,˜,, ,)2 t = 1, 2 . . . . . 260, n =
) 1, 2 . . . . . 2 8 8 / k
1 74,880 0.176 (0.013) 0.795 (0.016) 0.971
2 37,440 0.167 (0.012) 0.787 (0.015) 0.954
3 24,960 0.172 (0.018) 0.756 (0.025) 0.928
4 18,720 0.171 (0.019) 0.746 (0.029) 0.917
6 12,480 0.135 (0.052) 0.788 (0.096) 0.923
8 9,360 0.064 (0.051 ) 0.904 (0.082) 0.968
9 8,320 0.043 (0.011 ) 0.938 (0.016) 0.981
12 6,240 0.032 (0.011) 0.953 (0.016) 0.985
16 4,680 0.033 (0.010) 0.951 (0.015) 0.984
18 4,160 0.030 (0.009) 0.951 (0.013) 0.981
24 3,120 0.020 (0.007) 0.969 (0.009) 0.989
32 2,340 0.023 (0.006) 0.967 (0.007) 0.990
36 2,080 0.022 (0.007) 0.964 (0.007) 0.986
48 1,560 0.022 (0.008) 0.966 (0.007) 0.987
72 1,040 0.028 (0.013) 0.950 (0.009) 0.978
96 780 0.029 (0.014) 0.951 (0.009) 0.980
144 520 0.040 (0.032) 0.915 (0.016) 0.955

(b) MA(1)-GARCH(I, 1) models for filtered S&P 500 returns
-k -- ˜ __ k
Rt.n = 1 0 0 " ˜ ' i = ( n l ) k + l,nkRt,i -- I'll(k) + O(k)Et,n- 1
+ e2, (o't),) 2 = w(k)+ Ol(k)(e˜,_ i) 2 + fl(k)(cr,!,_ i) 2, t = 1, 2 . . . . . 991, n = 1, 2 . . . . . 8 0 / k
1 79,280 0.096 (0.009) 0.892 (0.011) 0.988
2 39,640 0.086 (0.012) 0.905 (0.014) 0.991
4 19,820 0.088 (0.017) 0.904 (0.019) 0.992
5 15,856 0.071 (0.021) 0.923 (0.022) 0.994
8 9,910 0.058 (0.015) 0.937 (0.016) 0.994
10 7,928 0.058 (0.016) 0.936 (0.017) 0.994
16 4,955 0.109 (0.060) 0.869 (0.074) 0.978
20 3,964 0.084 (0.042) 0.893 (0.054) 0.977
40 1,982 0.099 (0.039) 0.873 (0.053) 0.972

(a) See Table 2a for construction of the raw return series. The method for obtaining the filtered returns,
Rt.i, is described in the main text.
(b) See Table 2b for the construction of the raw return series. The method for obtaining the filtered
returns, ˜qt.i, is described in the main text.



e c o n o m i c f a c t o r s s u c h a s t e c h n o l o g y a n d p r o d u c t i v i t y 42. T h e s e d i s t i n c t s o u r c e s o f
v o l a t i l i t y p e r s i s t e n c e c o u l d s i m u l t a n e o u s l y i n f l u e n c e t h e r e t u r n s e r i e s , r e s u l t i n g in
a mixture distribution with different implications for the character of the short- and



42 The distinct short-run volatility patterns induced by regularly scheduled macroeconomic announce-
ments have been analyzed by Ederington and Lee (1993).
147
T.G. Andersen. T. Bollerslec/ Journal of Empirical Finance 4 (1997) 115-158

long-run dynamics. A promising first attempt at modeling this interaction between
the volatility processes at different time resolutions within a unified framework
have been suggested by Miiller et al. (1995). In their so-called heterogeneous
ARCH, or HARCH, model the volatility at the highest frequency is determined by
the sum of numerous A R C H type processes defined over courser time intervals,
where each of these components in turn may be linked to the actions of different
types of traders with varying time horizons 43

5.2. Standardized foreign exchange returns

The conjectures underlying a components type formulation of the volatility
process are further reinforced by our analysis of the standardized 5-minute returns;
R,. n -Rt.,,/(˜-tgt.n). If our model provides a good approximation to the data
generating process, then this series should display little A R C H effects at daily and
lower frequencies, and the intraday ARCH effects should diminish. Consistent
with this prediction, the absolute return autocorrelations at the lowest intraday
frequencies have been reduced markedly. This is also manifest in the lower curves
in Fig. 7a, which depict the correlograms for I/˜r.,,L Apart from small spikes
associated with remaining stochastic periodicity at the daily frequency, the correla-
tions for the absolute returns are generally close to zero beyond the two day lag.
Thus, the daily GARCH(1, 1) volatility estimates appear to provide quite satisfac-
tory estimates for the interday volatility dynamics 44. At the same time, Fig. 7a is
also indicative of important short-run dynamics that necessarily are unaccounted
for by the daily GARCH(1, 1) volatility estimates. This again lends support to our
conjecture of distinct short-run, or intraday, components in the fundamental return
volatility generating process. The M A ( 1 ) - G A R C H ( 1 , 1) estimates for the stan-
dardized returns in Table 5a reinforce this interpretation by exhibiting a sharp
decline in &(k) +/3(k) as the return horizon increases from five minutes to one
hour. In fact, beyond the one hour sampling frequency, the volatility clustering is
sufficiently weak that the GARCH(1, l) specification breaks down, and only
A R C H ( I ) or homoskedastic MA(1) models are estimated.

5.3. Filtered equity returns

We now turn to the corresponding findings for the S & P 500 returns. In
interpreting the results, it is important to recognize that the estimated intraday
periodicity now involves interaction terms between the daily volatility level and


43 Stationary conditions for this new class of time series models are developed in Darorogna et al.
(1995).
44 Of course, this apparent lack of any significant long-run correlations in the standardized returns
may be due to the relatively short sample of only one year. With a longer span of data the GARCH(I,
I) model will most likely fail to capture all the low frequency dynamics (see e.g. Baillie et al., 1996).
T.G. Andersen, 72 Bollerslet.,/JournalofEmpiricalFinance 4 (1997) 115-158
148

Table 5
k T/k ˜(k˜ ˜(k) a(˜)+ ˜(˜
(a) MA(I)-GARCH(1, 1) models for standardized DM $ exchange rate
RI.,, =- 1 0 0 . ˜ i = O ' I)k+ l.,˜kR˜.i = IX˜k) + O˜k˜e/'.,, I
^k
260, n = 1, 2 . . . . . 2 8 8 / k
Jr- ˜tkn (O'/!n) 2 = ¢..O(k) q- a(k)(˜gk n i ) 2 -{- /˜{/,){{Ttk,, 1 )2, 1 = l, 2 .....
1 74,880 0.182 (0.014) 0.766 (0.021) 0.948
2 37,44(1 0.167 (0.015) 0.760 (0.029) 0.927
3 ' 24,960 0.172 (0.0191 0.706 (0.037) 0.877
4 18,720 0.177 (0.1116) 11.666 (0.034) 0.843
6 12,480 0.173 10.0231 0.603 (0.1/47) 0.776
8 9,360 0.177 (0.028) (1.484 (0.105 ) 0.661
9 8,320 0.123 (0.031/) /).607 (0.143) 0.729
12 6,240 0.158 (0.028) 0.376 (0.071) 0.534
16 4,680 0.184 10.034) -- 0.184
18 4,160 0.089 (0.1126) -- 0.089
24 3,120 0.088 (0.11341 -- 0.088
32 2,340 0.072 (0.031 ) -- 0.072
36 2,080 0.083 (0.1149) -- 0.083
48 1,560 -- -- --
72 1,040 -- -- --
96 780 . . . .
144 520 -- -- --

(b) MA(I)-GARCH(I, l) models for standardized S&P 50(I returns
^k __ ^ __ k
R˜.n = 1 0 0 " ˜ i T ( n I)k+ l,nkRt,i
- tZ(k'˜ + O(k)˜t,a I
+ e˜,, (o-,k,,)" = wc˜) + a˜k)(e,˜,, i) 2 + ˜)(cr, k,,_ i) 2, t = I, 2 . . . . . 991, n = 1, 2 . . . . . 8 0 / k
1 79,280 0.095 10.007) 0.877 (0.012) 0.973
2 39,640 0.107 (0.012) 0.841 (0.022) 0.949
4 19,820 0.127 (0.021 ) 0.764 (0.053) 0.890
5 15,856 0.124 (0.024) 0.765 (0.047) 0.889
8 9,910 0.117 (0.027) 0.727 (0.070) 0.843
10 7,928 0.126 10.(/231 0.681 (0.047) 0.807
16 4,955 0.215 (0.074) 0.512 (0.051 ) 0.727
20 3,964 0.159 (0.096) 0.551 (0.037) 0.710
40 1,982 0.250 (0.160) -- 0.250

(a) See Table 2a for construction of the raw return series. The method for standardizing the returns,
/˜t i is described in the main text.
(b) See Table 2b for construction of the raw return series. The standardized returns, Rt.i, are generated
as described in the main text.



the Fourier functional form, so that not only the level but also the shape of the
volatility pattern varies with o-,. T h u s , o u r s t y l i z e d d e t e r m i n i s t i c p e r i o d i c m o d e l
d i s c u s s e d in S e c t i o n 3 is n o t s t r i c t l y v a l i d in t h i s c o n t e x t i.e. g e n e r a l l y st, n 4: s ....
f o r t 4= r . C o u n t e r to t h e r e s u l t s f o r t h e D M - $ r e t u r n s , t h i s t i m e - v a r y i n g v o l a t i l i t y
c o m p o n e n t m a y w e a k e n t h e a u t o c o r r e l a t i o n s f o r t h e r a w a b s o l u t e r e t u r n s as,
e f f e c t i v e l y , a d d i t i o n a l n o i s e is i n j e c t e d i n t o t h e r e t u r n s p r o c e s s . F i g . 7 b s e e m to
i n d i c a t e t h a t t h i s is i n d e e d t h e c a s e , as t h e c o r r e l o g r a m for the filtered absolute
149
T. G. Andersen. T. Bollerslev / Journal of Empirical Finance 4 (1997) 115 158

returns, IRt,n[, lies substantially above that for the raw series, [R,.,,I. 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(10) and VR A, are all markedly higher than the corresponding statistics for the
raw returns in Table lb 45. This is also in line with the M A ( I ) - G A R C H ( 1 , 1)
estimation results reported in Table 5b. The parameter estimates obtained as we
I
move from the ˜ day to the 40-minute return horizon are again consistent with the
theory. Thereafter, the sum ˜(k) and /3(k) decay slightly, but more importantly ˜(k)
starts to increase. However, all the intraday estimates now consistently p ˜ n t
towards a very high degree of volatility persistence and in all instances c˜(˜)+/3(k )
are higher than the estimates for the raw return series in Table 2b. Note also, that
in line with the findings for the D M - $ 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 D M - $ , the first order autocorrelations for the
standardized absolute returns for the S & P 500, [Rt,,,I, remain highly significant for
^k
I
the lower intraday frequencies, Even at the ?- 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(k ) estimates for the intraday
GARCH(1, 1) models for / ˜1,7l 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 D M - $ 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 D M - $ 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, 1) model may capture a large portion of the day-to-day volatility
clustering, the model's deficiency in dealing with long-memory behavior necessar-
ily becomes more transparent when the time span of the data increases.


45 For instance, pA for the aggregated filtered returns, Ig˜,,I, equal 0.371, 0.379, 0.341, 0.335, 0.305,
0.303, 0.292, 0.290 and 0.292 for k = 1, 2, 4, 5, 8, 10, 16, 20 and 40, respectively.
T.G. Andersen, T. Bollerslel" / Journal of Empirical Finance 4 (1997) 115-158
150


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.


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 require-
ment 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 Ziirich, the 7th World
T.G. Andersen, T. Bollerslev / Journal of Empirical Finance 4 (1997) 115-158 15 l


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


Appendix A. Data description

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

The D M - $ 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 Associ-
ates. 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 e x p { 1 / 3 . [ln(1.6055)+
!n(1.6065)]/2 + 2 / 3 • [ln(1.6050) + ln(1.6055)]/2} = 1.6055. The nth 5-minute
return for day t, Rt, ., 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 MiJller 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 D M - $ 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. R,,,,,
n = 1, 2 . . . . . 288, t = 1, 2 . . . . . 260.

A.2. The standard and poor's 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
152 T.G. Andersen, T. Bollerslet: / Journal o f Empirical Finance 4 (1997) 1 1 5 - 1 5 8


through December 31, 1989. The data specify the time, to the nearest 10 seconds
and the exact price of the S & P 500 futures transaction whenever the price differs
from the previously recorded price 4˜,. The calculation of the returns is based on
the last recorded logarithmic prices for the nearby futures contract over consecu-
tive five minute intervals. The price record covers the full trading day in the
futures market from 8.30 a.m. (central standard time) to 3.15 p.m. Although, the
New York Stock Exchange closes at 3.00 p.m., we retain the last three 5-minute
returns from the futures market in the analysis reported on below. The first return
for the trading day, i.e. from 8:30 to 8:35 a.m., constitutes another unusual time
interval. This period incorporates adjustments to the information accumulated
overnight, and consequently displays a much higher average return variability than
any other 5-minute interval. In effect, this is not a 5-minute return, and we
therefore delete it in the subsequent analysis. Alternatively, it would be possible to
account for this special return interval using dummy variables. However, any such
procedure is invariably ad hoc in nature. Furthermore, informal investigations
reveal little sensitivity to the exact treatment of the overnight returns. We thus
elect to work exclusively with the 5-minute returns. Following Chan et al. (1991),
we also exclude the October 15 through November 13, 1987 time period around
the stock market crash due to the frequent trading suspensions. Outside these four
weeks trading suspensions were rare, but did occur. In these instances the missing
prices were determined by linear interpolation, leading to identical returns over
each of the intermediate intervals. This obviously smoothes the series over the
missing data points which will mitigate the effect of sharp price changes subse-
quent to a trading suspension. Experimentation with exclusion of trading days with
missing observations indicate that the findings pertaining to the degree of volatility
persistence reported on here are virtually unaffected by this interpolation. All in
all, these corrections result in a sample of 991 days, each consisting of 80 intraday
5-minute returns, for a total of 79,280 observations i.e. R,,,,, n -- 1, 2 . . . . . 80,
t = l , 2 . . . . . 991.



Appendix B. Flexible Fourier form modeling of intraday periodic volatility
components

From Eq. (7), define,

xt,,, =- 2 1 o g [ I R , . , , - E(R,,,,)I] log or,2 + log U = log s t,n + log Z t,n •
2 2
-
(A.I)


46 We are grateful to G. Andrew Karolyi for providing us with this 5-minute price series. The same
set of data has also been analyzed from a different perspective in Chan et al. (1991).
T.G. Andersen. T. Bollerslev/ Journal of Empirical Finance4 (1997) 115-158 153

Our modeling approach is then based on a non-linear regression in the intraday
time interval, n, and the daily volatility factor, o-,,
x,,,, = f ( 0 ; o " t, n) + u, .... (A.2)
where the error, ut.,,-= log Z ˜ , , - E(log Z˜,,), is i.i.d, mean zero. In the actual
implementation the non-linear regression function is approximated by the follow-
ing parametric expression,
J [ n //2 D
f( O;cr,, n) =E°'tJi-o [ ˜oj +/x,j˜ +/-'.2j-˜, + Ea,jI,,=,:i
i=1


+i Yl,J c o s - - - - ˜ + 6pj sin N '

where N j - N - I ˜ i _ j , N i = ( N + I)/2 and Nz=-N-I˜i t , x i 2 = ( N + l)(N+
2 ) / 6 are normalizing constants. For J = 0 and D = 0, Eq. (A.3) reduces to the
standard flexible Fourier functional form proposed by Gallant (1981, 1982).
Allowing for J > 1 and thus a possible interaction effect between o-/ and the
shape of the periodic pattern might be important in some markets, however. Each
of the corresponding J flexible Fourier forms are parameterized by a quadratic
component (terms with ix-coefficients) and a number of sinusoids (the 7- and
6-coefficients). Moreover, it may be advantageous to also include time specific
dummies for applications in which some intraday intervals do not fit well within
the overall regular periodic pattern (the A-coefficients).
Practical estimation is most easily accomplished using a two-step procedure.
Firstly, a generated x,, . series, 2t,,,, is obtained by replacing E(R,.,,) with the
sample mean of the returns, ˜',., and crt with the estimates from a daily volatility
model, say 6-t. Substituting ˜t for o-, and treating ˜-,.,, as the dependent variable
in the regression defined by Eqs. (A.2) and (A.3) allow the parameters to be
estimated by ordinary least squares (OLS). Note that from Eq. (3), 6,2 represents
an estimate of M(sZ)o-t 2, so that after substitution for o-t in Eq. (A.2), the term
- l o g M ( s 2) is implicitly included in the constant term in Eq. (A.3), /Zoo. Let
f , , , - f ( 0 ; 6 " , , n) denote the resulting estimate for the right hand side of Eq.
(A.3) 4v. The normalization T - I ˜ , , _ I,N˜,t=I,[T/N]St,n ˜ 1, where [T/N] denotes
the number of trading days in the sample, then suggests the following estimator of
the intraday periodic component for interval n on day t,
T" e x p ( £ , , , / 2 )
^ ˜
st,,, y.˜T/ff]EN= , e x p ( f , . / 2 ) (A.4)

Note that although the periodic modeling procedure is designed for fitting the


47 Given consistent estimates for 6" the resulting parameter estimates will generally be consistent.
t,
However, the use of generated regressors may result in a downward bias in the conventional OLS
standard errors for the parameter estimates (see Pagan, 1984).
T.G. Andersen, T. Bollersler' / Journal of Empirical Finance 4 (1997) 115-158
154

a v e r a g e volatility p a t t e r n a c r o s s the N i n t r a d a y i n t e r v a l s , the s e c o n d - s t a g e e s t i m a -
tion o f Eq. ( A . 3 ) is b a s e d o n a t i m e series r e g r e s s i o n that i n c l u d e all T i n t r a d a y
returns. U t i l i z i n g this a d d i t i o n a l i n f o r m a t i o n in the data r a t h e r t h a n s i m p l y fitting
t h e a v e r a g e i n t r a d a y p a t t e r n , e n h a n c e s the e f f i c i e n c y o f the e s t i m a t i o n .
T h e first step o f o u r p r o c e d u r e i n v o l v e s the d e t e r m i n a t i o n o f the daily volatility
f a c t o r e s t i m a t e s i.e. ˜t. G i v e n the r e l a t i v e s u c c e s s o f the daily M A ( 1 ) - G A R C H ( 1 ,
1) m o d e l s in e x p l a i n i n g the a g g r e g a t i o n r e s u l t s for the i n t e r d a i l y f r e q u e n c i e s in
b o t h m a r k e t s , this a p p e a r s to be a n a t u r a l c h o i c e . Next, the n u m b e r o f i n t e r a c t i o n
t e r m s , J a n d the t r u n c a t i o n lag for the F o u r i e r e x p a n s i o n , P , m u s t b e d e t e r m i n e d .
T h i s is d o n e p r i m a r i l y o n the b a s i s o f p a r s i m o n y i.e. for e a c h o f the r e t u r n series
w e c h o o s e the m o d e l that best m a t c h e s the basic s h a p e o f the p e r i o d i c p a t t e r n w i t h
the f e w e s t n u m b e r o f p a r a m e t e r s . T h e r e s u l t i n g e s t i m a t e s for the D M - $ r e t u r n s
with J=0and P=6are,
n n2
0.72 - 8.39 - - +5.59--
(1.06) (4.14) NI (4.14) N2
2"n'n 27rn 2˜-2n 21r2n
- 2.51 c o s - - - 0.40 s i n - - - 0.38 c o s - - +0.06sin
(--6.15) N (- 1044) N ( 3.71) N (2.70) N
L,, =
27r3n 2rr3n 27r4n 27T4n
+ 0.42COS-- -- 0.09 s i n - - -- 0.02 cos + 0.35 sin
(8.79) N (4.89) N (-0.53) N (20.48) N
27r5n 0.22 27r5n 27r6n 2rr6n
-- 0.12 c o s - - +- sin 0.23 cos +0.01sin
(-5.38) N (13.35) N ( 12.67) N (0.45) N

w h e r e the n u m b e r s in p a r e n t h e s e s i n d i c a t e h e t e r o s k e d a s t i c r o b u s t t-statistics. It is
e v i d e n t f r o m the c o r r e s p o n d i n g fit in Fig. 6a, that this r e p r e s e n t a t i o n p r o v i d e s a n
e x c e l l e n t o v e r a l l c h a r a c t e r i z a t i o n o f the a v e r a g e i n t r a d a i l y p e r i o d i c i t y in t h e
D M - $ m a r k e t . C o n s i s t e n t w i t h Fig. 2a, the b a s i c s h a p e o f the p e r i o d i c p a t t e r n
a p p e a r s i n v a r i a n t to the daily volatility level i.e. J = 0.
In contrast, o u r p r e f e r r e d m o d e l , the S & P 500, r e t u r n s sets J = 1 a n d P = 2,
1,/2
n
1.85 -3.07-- - 2.68 - -
-
(I 62) N˜ (- 2.05) N2
(-3.03)
- 0.16 l,,=d,
f.,, = I n_ de d3
-- 0.62 q- 1 . 1 1 1 , , _
( - 1.83) (2.99)
(0.53)
27rn 2˜-2n
2am 2˜2n
+ 1.18 c o s - - - 0.59 sin + 0.28 cos 0.14 sin
(3.11) N (-6.14) N (294) N (-2.31) N

n n2
- 0.54 1.73 -- + 1.57--
( - 0.9s) N˜ (1.29) N2
(0.95)
- 0.69 l.=d˜
- 0.30 /,, d2
--(O-ol'˜9)In=d'
( - 0.97) (2.02)
2˜-n 2rr2n 2˜r2n
- 0.37 cos 2 ˜ n + 0.12 s i n - - - 0.17 cos 0.03 sin
(1.30) N
L ( - 1.06) N ( - 1.97) N (0.50) N

A l t h o u g h f e w o f the c o e f f i c i e n t s in the e x p a n s i o n c o r r e s p o n d i n g to J = 1 are
i n d i v i d u a l l y s i g n i f i c a n t , l e a v i n g o u t the i n t e r a c t i o n e f f e c t results in a s e e m i n g l y
155
T.G. Andersen, T. Bollerslev / Journal of Empirical Finance 4 (1997) 115-158


i n f e r i o r o v e r a l l fit. A s s e e n in Fig. 2b, the v o l a t i l i t y p r o f i l e for the last f i f t e e n
m i n u t e s o f t r a d i n g ( i n t e r v a l s 78, 79 a n d 80) s h o w s an a b r u p t c h a n g e f r o m the
o v e r a l l s m o o t h i n t r a d a y pattern. T h r e e d u m m y v a r i a b l e s are i n c l u d e d to m i n i m i z e
the d i s t o r t i o n s that m a y o t h e r w i s e arise f r o m this d i s t i n c t p e r i o d i.e. d I = 78,
d 2 = 7 9 a n d d 3 = 80. T h e r e s u l t i n g fit d e p i c t e d in Fig. 6 b a g a i n testifies to the
s u c c e s s o f this r e l a t i v e l y s i m p l e p r o c e d u r e for m o d e l i n g the p e r i o d i c i t y in i n t r a d a y
f i n a n c i a l m a r k e t volatility.


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