Aggregation of Forecasts, Data and Model
Meng-Feng Yena
a
Kai-Li Wangb
Ming-Yuan Lia
Department of Accountancy and Graduate Institute of Finance, National Cheng Kung University, TAIWAN
b
Department of Finance, Tunghai University, TAIWAN
Abstract
This paper introduces three GARCH-based forecast approaches for volatility: Procedures AF
(aggregation of forecasts), AD (aggregation of data), and AM (aggregation of model). The first one refers
to the aggregation of forecasts, a method inspired by Andersen et al.’s (1999) approach. In particular, it
involves summing the volatility forecasts by a strong GARCH(1,1) model for all sub-intervals to provide
the volatility forecasts for the aggregated original intervals. Procedure AD contrasts Procedure AF by
estimating the strong GARCH(1,1) model for the aggregated original intervals, a traditional method in the
literature on GARCH forecasting. In addition, we adopt Drost and Nijman’s (1993) weak GARCH
specification and calculate the parameters of the weak GARCH(1,1) model for the original intervals,
using the ML estimates for the strong GARCH(1,1) model estimated upon the sub-intervals. This weak
GARCH(1,1) model is then used to generate approximates of the volatility for the original intervals,
which constitutes Procedure AM. Via Monte Carlo simulations, we compare the forecast performances of
these three approaches against ‘clean’ data with only the GARCH effect. Moreover, we explore the same
issue in the context of periodicities along with the GARCH effect. The simulation results tend to suggest
that Procedure AF dominates its two competitors. This conclusion leads us to the exploration of whether
accommodating the effect of periodicities further enhances the performance of Procedure AF. To achieve
this goal, we replace the standard GARCH(1,1) model in the framework of Procedure AF by Andersen
and Bollerslev’s (1997) intraday-periodic-component GARCH(1,1), Bollerslev and Ghysels’ (1996)
periodic GARCH(1,1) and our IPC-PGARCH(1,1) models. Our empirical study suggests that the standard
GARCH(1,1) model remains the best volatility predictor under the scheme of Procedure AF.
Keywords: strong- and weak-GARCH, temporal aggregation, IPC-GARCH, PGARCH, IPC-PGARCH
JEL classifications: C15, C52, C53

Corresponding Author: kaiwang@thu.edu.tw; Address: 181, Sec.3, Taichung-Kan Rd., Taichung,
Taiwan, R.O.C.; TEL: +886-4-23590121 Ext.3588; Fax: +886-4-23506835.
Aggregation of Forecasts, Data and Model
1. Introduction
Given the success of the GARCH model pioneered by Engle (1982) and Bollerslev (1986), in the past two
decades has the literature on volatility modelling and forecasting witnessed a huge number of variants of
the GARCH model. However, progress made in improving the accuracy of volatility forecasts has been
marginal relative to the standard GARCH(1,1) specification. Given that extensions to the standard
GARCH(1,1) model do not obviously improve the quality of out-of-sample volatility forecasts, a different
stream of work in modelling financial return volatility has recently emerged and contributed to
high-frequency finance. In particular, it is found that high-frequency intraday data not only provide a
better measure of the unobservable, realised ex-post volatility of daily, or longer-term intervals; such data
also help improve the accuracy of out-of-sample volatility forecasts for these inter-daily intervals.
Andersen and Bollerslev (hereafter AB) (1998a) document that the traditional expedient of
approximating the true conditional innovation variance by the squares of return innovations is the reason
that the out-of-sample volatility forecast performances of the GARCH type of models are poor relative to
their in-sample counterparts. Although squared return innovations are an unbiased estimator of the true
conditional innovation variance, they contain too much idiosyncratic noise relative to the information
they provide about the true conditional variance. To resolve this problem of noise in the traditional proxy
of conditional innovation variance, AB (1998a) find that sampling the underlying data more frequently
and summing the squared return innovations of consecutive sub-intervals within each original interval
will provide a more accurate measure of the true conditional variance of the data sampled at the original
frequency. Given this improved measure of the true conditional innovation variance, in particular, AB
(1998a) highlight that the standard GARCH(1,1) model generates out-of-sample volatility forecasts much
more accurate than the literature suggests. Such an improvement in the accuracy of out-of-sample
volatility forecasts tends to be more prominent as the number of sub-intervals within each original
interval increases. Given AB’s (1998a) finding above, using more frequently sampled data to provide a
better proxy of the true conditional innovation variance should become a standard procedure for the
evaluation of volatility forecasts. Following AB (1998a), Andersen et al. (1999) study whether summing
up forecasts by the standard GARCH (1,1) model for the high-frequency intraday intervals helps forecast
longer-term inter-daily volatility. They find that the above approach indeed provides more accurate daily,
weekly, and monthly volatility forecasts, both theoretically and empirically. In particular, based on the
framework of weak GARCH processes1 and the temporal aggregation theory for a weak GARCH(1,1)
1
A weak GARCH(1,1) process is different from the traditional GARCH(1,1) model by Bollerslev (1986) in that the
former needs not assume any probability distribution for the driving disturbances and it specifies the best linear
1
process proposed by Drost and Nijman (1993, hereafter DN), the simulation results in Andersen et al.
(1999) suggest that summing the volatility forecasts from the calculated weak GARCH(1,1) models for
different intraday sampling frequencies provides more accurate volatility forecasts for daily and longer
intervals. The extent to which the inter-daily volatility forecasts are improved is more evident as the
sampling frequency for the intraday weak GARCH(1,1) model increases. However, when the sampling
frequency for the intraday weak GARCH(1,1) model is an hour or less, Andersen et al. find that empirical
forecasts start to deviate from what their simulation results suggest. Adding up volatility forecasts from
these very high-frequency intraday weak GARCH(1,1) models fails to provide more accurate daily and
longer-term volatility forecasts. In particular, this method performs even worse than estimating the daily
GARCH(1,1) model and generating daily volatility forecasts. Andersen et al. impute the breakdown of
their method to the stylised characteristics often observed at very high frequencies, e.g. intraday volatility
patterns, routine macroeconomic news release effects, discrete price quotes, genuine jumps in the price
path, and multiple volatility components.
However, the parameters of the weak GARCH(1,1) model for these intraday intervals in Andersen
et al. do not reflect at all the stylised characteristics since they are naively calculated via DN’s
aggregation formulae from the parameter estimates of the strong GARCH(1,1) model for the daily
observations, which are free of many of these stylised characteristics. The failure of Andersen et al.’s
method motivates us to investigate whether Andersen et al.’s method is still valid if we directly estimate
the strong standard GARCH(1,1) model for these intraday data rather than calculating the intraday weak
GARCH(1,1) model through DN’s aggregation theory, as Andersen et al. have done. One of the purposes
in the paper is therefore to test whether the strong standard GARCH(1,1) model in the context of a given
stylised characteristics, i.e. volatility periodicities in this study, still provide better volatility forecasts in
the Andersen et al.’s sense than the traditional method. Specifically, we estimate the strong standard
GARCH(1,1) filter for sub-intervals and add up the resulting volatility forecasts to constitute forecasts for
the aggregated intervals. For ease of reference, we term this innovative approach as Procedure AF, AF
referring to ‘aggregation of volatility forecasts from the strong standard GARCH(1,1) filter for
sub-intervals.’ The other approach is the typical practice of estimating the strong standard GARCH(1,1)
model directly for the original intervals and generating volatility forecasts. We denote this traditional
approach by Procedure AD, AD meaning ‘aggregation of data’, since each original interval is the
aggregate of all sub-intervals within it, in the sense of log-return. Given DN’s aggregation formulae,
moreover, we could actually introduce the third way to predict the volatility for the original intervals. In
particular, we could calculate the weak GARCH(1,1) model for the original intervals from the parameter
estimates of the strong standard GARCH(1,1) model for the sub-intervals. The calculated weak
projections rather than the conditional variance of the error terms.
2
GARCH(1,1) model is then used to generate forecasts of the best linear projections of the squared return
innovations which serve as approximations to the volatility of the original intervals. We refer to DN
(1993) for more details. This third prediction approach is denoted Procedure AM, AM suggesting the
‘aggregation of model’ since the weak GARCH(1,1) model for the original intervals is calculated
(aggregated) from the parameter estimates of the strong standard GARCH(1,1) model for the
sub-intervals.
As such, we have, in total, three different approaches to forecasting volatility for the original
intervals: Procedures AF, AD and AM. The intuition behind this is of immediate interest and great
importance for the implementation and evaluation of forecasting strategies. If Procedures AF dominates
the alternative approaches, an implication of it suggests that market participators could stick to
high-frequency intraday data when they wish to use the standard GARCH(1,1) model to forecast the
future volatility of their financial asset returns. In contrast, if either procedure AD or AM proves to be the
best predictor, market practitioners may undertake generating relatively low-frequency volatility forecasts
without having to employ the high-frequency data using the strong standard GARCH(1,1) model at a
significant expense of time.
Turning back to Andersen et al.’s study, their intraday weak GARCH(1,1) models certainly might
be mis-specified under the circumstance of the stylised characteristics, which detracts from these models’
ability to forecast the volatility for these ultra-high-frequency intervals. Given the number of stylised
characteristics documented in the intraday data analysis, Andersen and Bollerslev (1997) highlight the
periodic patterns observed in high-frequency intraday return volatility, in both the foreign exchange and
equity markets. Their results indicate that intraday periodicities are the main reason for DN’s (1993)
temporal aggregation theory to break down for intraday intervals. Consistent with Andersen and
Bollerslev (1997), Fang (2000) documents that the hourly volatilities of three foreign exchange rates, i.e.
JPY/USD, JPY/DEM, and DEM/USD, peak during the overlap of the London and New York trading
hours (about 13:00-17:00 GMT). There is a dip in hourly volatility during lunch hours in Asia (3:00-5:00
GMT). They also find a monotonic decline between 20:00 and 24:00 GMT, the gap between the close of
New York and the open of the Tokyo market. See also Baillie and Bollerslev (1991), Zhou (1996), and
Andersen and Bollerslev (1998b) for similar results.
Following these studies comparing the standard GARCH(1,1) model with many other complicated
variants, we focus our interest on whether the standard GARCH(1,1) model betters those variants which
accommodate periodicities (renamed ‘volatility periodicities for sub-intervals’ or just ‘periodicities’ for
ease of reference, hereafter) in the underlying volatility process. Given no prior efforts in this aspect, the
second principle end of this paper is to investigate, via Monte Carlo simulations, whether Andersen et
3
al.’s method is still valid in the context of un-parameterised periodicities observed in the volatility
process of the sub-intervals. This second issue is related to the first: if the standard GARCH(1,1) model
still outperforms the more complicated models specified for periodicities, Andersen et al.’s method above
should be modified by substituting the estimated strong GARCH(1,1) model for the calculated weak
GARCH(1,1) model in the high-frequency intraday intervals. Otherwise, we have to use more
complicated extensions to the strong standard GARCH(1,1) model to capture periodicities before taking
the advantage of Andersen et al.’s idea of summing the high-frequency intraday volatility forecasts to
form lower-frequency inter-daily intervals. To achieve this goal, we will compare the standard
GARCH(1,1) model to some of its complex variants: periodic GARCH (hereafter PGARCH) by
Bollerslev and Ghysels (1996), Andersen and Bollerslev’s intraday-periodic-component GARCH
(hereafter IPC-GARCH) and our innovative IPC-PGARCH models, which characterise periodicities in
volatility for forecasting the daily volatilities of the U.S. dollar/British pound, German mark/U.S. dollar,
Japanese yen/U.S. dollar exchange rate returns, and the hourly volatilities of NASDAQ-traded
Microsoft’s stock returns.
The rest of this paper is organised as follows: Section 2 documents the framework of our Monte
Carlo simulations and explains the data used in our empirical study. Section 3 formulates the three
forecast approaches based upon the standard GARCH(1,1) specification and the three GARCH variants
for periodicities. Section 4 discusses the results of both the simulations and the empirical study. Section 5
concludes and gives implications for future research efforts.
2. DGP Simulation Process and Real Data Descriptions
2.1.1 Monte Carlo Simulation Structure
We start with the introduction of three different parameterisations of the strong standard GARCH(1,1)
model, which constitute our PGARCH(1,1) DGPs. To focus our attention on the part of conditional
innovation variance, we assume that the conditional mean equation for the returns themselves is simply
made up of zero mean random variables with strong GARCH effects. In particular, the model underlying
our DGPs is given by
(i) Conditional mean:
yt
=0+
where
t
t
=
t ht
,
denotes the innovation and
(2.1)
t
the standardised innovation, which follows a standardised t5 or N(0,1)
distribution, and
(ii) Conditional innovation variance:
4
ht =  H + 1H  t21 + 1H ht 1 .
(2.2)
Similar to the simulation framework in BG (1996)2, the basic GARCH(1,1) model is characterised by
Parameterisation 1 in Table 1. Parameterisation 1 is further modified to Parameterisations 2 and 3 in
Table 1 so as to show the shift in intercept ( 
H
) or parameter
1H across the two stages of each
periodic volatility cycle. Note that these parameterisations must satisfy the conditions illustrated in
footnote 3 beneath Table 1. To save space, the derivation details are available upon request.
Parameterisations 1 and 2 in Table 1 are used to mark the change in intercept ( 
H
) of the GARCH(1,1)
model, whereas Parameterisations 1 and 3 in Table 1 are employed to specify the shift in parameter
1H
across the two stages of each volatility cycle.
2.1.2 DGPs 1 to 4 (PGARCH(1,1) Model)
High-Frequency (Sub-Interval) Observations
The models used in the DGPs for sub-intervals are the PGARCH(1,1) specifications, which are given by
DGPs 1 to 4 below.
Conditional mean (zero mean):
yt =  t = t ht
where
t ~
,
(2.3)
i.i.d. N(0,1) or t5, whereas
ht
denotes the conditional variance of the return innovation
t
and
follows a GARCH(1,1) model given by Eqs.(2.4a) through (2.4d) below.
DGP 1: PGARCH(1,1) of two-stage periodicity in intercept
ht  sH( t )  0.15 t21  0.7ht 1 , where
(2.4a)
s(t) = 1 for odd t and 2 for even t, and
1H = 0.05 (Parameterisation 1 in Table 1)
 2H
= 0.01 (Parameterisation 2 in Table 1).
DGP 2: PGARCH(1,1) of two-stage periodicity in alpha
ht  0.05  1Hs ( t ) t21  0.7ht 1 , where
(2.4b)
s(t) = 1 for odd t and 2 for even t, and
11H
= 0.15 (Parameterisation 1 in Table 1)
12H = 0.05 (Parameterisation 3 in Table 1).
2
In BG’s (1996) simulations for a periodic change in parameter
0.4666 and

H
12
= 0.0727, whilst

H
and

H
1
1H of their PGARCH(1,1) model, they let 11H
=
are fixed at, respectively, 0.05 and 0.7.
5
It is of interest to see what would happen to the out-of-sample forecast performances of the three
approaches if the two-stage PGARCH(1,1) model used in DGPs 1 and 2 is replaced by a five-stage
PGARCH(1,1) model. In other words, we increase the number of stages within each cycle from 2 (in
DGPs 1 and 2) to 5 (in DGPs 3 and 4) below:
DGP 3: PGARCH(1,1) of five-stage periodicity in intercept
ht  sH( t )  0.15( t 1 )2  0.7ht 1 , where
(2.4c)
s(t) = [(t-1) mod 5]+1, and
sH( t )
= 0.05{[ sin
0.05( sin  +1),
(2 s(t )  1)

3
H
H
H
]+1}, i.e. 1 = 0.05( sin +1),  2 = 0.05( sin
+1),  3 =
5
5
5
 4H = 0.05( sin
7
9
H
+1),  5 = 0.05( sin
+1).
5
5
DGP 4: PGARCH(1,1) of five-stage periodicity in alpha3
ht  0.05  1Hs ( t ) ( t 1 )2  0.7ht 1 , where
(2.4d)
s(t) = [(t-1) mod 5]+1, and
1Hs( t )
= 0.15{[ sin
0.15( sin  +1),
(2 s(t )  1)

3
H
H
H
]+1}, i.e. 11 = 0.15( sin +1), 12 = 0.15( sin
+1), 13 =
5
5
5
14H = 0.15( sin
7
9
H
+1), 15 = 0.15( sin
+1).
5
5
Low-Frequency (Original-Interval) Observations
Turning to the observations for the original intervals, they are generated in the following way. But, note that to avoid
the aliasing problem, the original interval is set equal to the periodic cycle in length. Under DGPs 1 and 2, for
instance, the number of stages within each periodic cycle is 2, and therefore, the original interval is equivalent to two
sub-intervals in length.
Observations for the (aggregated) original intervals (DGPs 1 to 4)
m
m
i 1
i 1
y   yt   yi  m (  1)    i  m (  1)   
(2.5a)
m
h   ht =  hi  m (  1) ,
(2.5b)
i 1
3
Note that the values of
of
1Hs (t )
and 1
H
 sH(t )
in DGP 3 and of
1Hs (t )
in DGP 4 above are positive for all five stages. Also, the sums
in DGP 4 are within the unit circle for all five stages.
6
where m is equal to 2 for DGPs 1 and 2, and 5 for DGPs 3 and 4. Also, subscripts

and t refer to the
time scale for, respectively, the sub-intervals and the original intervals. Figure 1 illustrates the
relationship between t and
:
Figure 1: Time Scales of both the sub- and the original intervals
Time scale for sub-intervals t:
012…
m
2m…
t = i + m(   1)…for i = 1,2,…,m
In-Sample Period
T=  m
T+20m = m(  +20)
Out-of-Sample Period
Time scale for the original intervals,  :
0
1
…
2…
In-Sample Period

 +20
Out-of-Sample Period
Note: m denotes the number of sub-intervals within each aggregated original interval.
Given the specifications of all four DGPs, Table 2 gives the sample sizes for both sub-intervals and original intervals.
In addition, taking into account the choice of the driving disturbances for the sub-intervals among N(0,1) and t5
doubles the cases that are examined. Table 3 summarises the settings of all these experiments.
2.2. Real Data Descriptions
The literature on modelling periodicities in volatility involves a variety of empirical financial data,
including foreign exchange (hereafter forex) rates, stock market indices, interest rates, and so on.
However, it appears that no studies have been made based on individual equities. In addition to three
series of five-minute forex rate returns for USD/GBP, DEM/USD, and JPY/USD, this paper studies one
series of one-minute individual equity returns for Microsoft’s stock traded in the NASDAQ national
market system. In what follows, we detail these two types of data separately.
2.2.1 The USD/GBP, DEM/USD, and JPY/USD Foreign Exchange Data
Each of these three forex data sets consists of twelve years of five-minute bid and ask quotes from 1
January, 1987 through 31 December, 1998, comprising 1,262,304 observations. All of the 288 five-minute
intervals during the 24-hour daily trading cycle are used. However, since the forex markets are sluggish
during weekends in most countries—except the inter-bank markets in middle-east Asia—quotes for many
of the 5-minute intervals over weekends are not available. We therefore omit those periods from Friday
7
21:00 (GMT) to Sunday 21:00 (GMT), which has become standard practice in modelling forex data.
Please refer to Bollerslev and Dormowitz (1993) for more discussion in this regard. Disregarding the
weekend periods leaves us with a total of 901,440 5-minute observations for each of the three forex time
series. We take the difference of the natural log of the average of the 5-minute bid and ask quotes as our
5-minute return time series, denoted ‘ultra-high-frequency data.’ These 901,440 5-minute returns are then
aggregated into 75,120 hourly, and then into 3,130 daily returns.
Let
yt and yt denote, respectively, series of the hourly forex log returns and log prices for t = 1 to
75,120, whilst
prices for

y and y represent, respectively, series of the aggregated daily returns and log
= 1 to 3,130. In particular,
yt  yt – yt-1 and
y  y – y 1 .
Note that t = 24(  -1) + i, for i =1 to 24. To equalise the number of observations in the returns and prices
of both the hourly and daily samples, we omit the two prices:
yt 0 and y 0 .
Given the length of our forex samples, moreover, it might be that distinguished features appear in
different periods of time. We therefore divide the entire forex time series for each of all three currencies
into twelve one-year-long sub-samples. To achieve this, we have to truncate the three data sets above,
leaving 898,560 5-minute returns, i.e. 74,880 hourly, and 3,120 daily returns. In other words, we have
74,880 five-minute, 6,240 hourly, or 260 daily returns for each of the 12 one-year-long sub-samples. We
refer to these hourly and daily returns as data for our ‘sub- (or high-frequency) intervals’ and ‘original (or
relatively low-frequency) intervals,’ respectively. Moreover, instead of using the squared daily return
innovations, the squared return innovations of the 288 consecutive 5-minute intervals within each of the
3,130 days are aggregated to be a better proxy for the latent daily volatility than the squared daily return
innovations. The sum of these squared 5-minute data will be used to evaluate the daily volatility forecast
performances of our three forecasting procedures.
2.2.2 The Microsoft’s Equity Data
This set of data for Microsoft’s stock traded in the NASDAQ national market system consists of bid and
ask quotes in US dollar for each of the 721,968 one-minute intervals from 1 November, 2000, 14:19
(GMT) through 16 March, 2001, 23:06 (GMT). Since the NASDAQ market is active only during the
cycle from 13:15 (GMT) through 21:00 (GMT) of each weekday, we omit all the quotes outside these
periods, leaving only 167,337 pairs of valid bid and ask quotes. We further discard 3,537 pairs of them
8
and make a more tractable number of 163,8004. The left 163,800 one-minute bid and ask quotes are
transformed into one-minute returns by taking the difference of the natural log of the average of the bid
and ask quotes. We call these one-minute returns our ‘ultra-high-frequency data’ for this equity case. In
parallel to the forex case above, these ultra-high-frequency one-minute returns are aggregated into 32,760
five-minute returns, and then into 2,730 hourly returns.
Let
yt and yt denote, respectively, series of the five-minute equity log returns and log prices for t =
1 to 32,760, whilst
y and y represent, respectively, series of the aggregated hourly equity log
returns and log prices for

= 1 to 2,730. In particular,
yt  yt – yt-1 and
y  y – y 1 .
Note that t = 24(  -1) + i for i = 1 to 5. To equalise the number of observations in the returns and prices
of both the hourly and daily samples, we omit the two prices:
yt 0 and y 0 .
Also, these two data sets are equally divided into ten sub-samples to take account of the possible
different characteristics across the ten sub-samples, each consisting of 3,276 five-minute returns or 273
hourly returns. We refer to these five-minute and hourly returns as data for our ‘sub-(or high-frequency)
intervals’ and ‘original (or relatively low-frequency) intervals,’ respectively. Moreover, the squared
ultra-high-frequency return innovations of the 60 consecutive one-minute intervals within each of the
2,730 hours are aggregated to measure the true hourly volatility.
2.2.3 Statistical Features of Four Time Series
Prior of the application of the PGARCH(1,1), IPC-GARCH(1,1) and IPC-PGARCH(1,1) models to
estimate the potential periodic patterns in the volatility of the sub-intervals, it is useful to take a look at
their basic statistical features. Tables 4 tabulates the basic descriptive statistics of all four time series, i.e.
three series of hourly forex returns and one series of five-minute equity returns. The results show that the
sample means of all return series are not significantly different from zero. In particular, the kurtosis
results indicate that all the empirical distributions have heavier tails than the normal distribution,
motivating us to specify a leptokurtic Student’s t process to better reflect the data characteristics.
Jarque-Bera (J-B) normal distribution test statistics reject a hypothesis of normal distribution at the 1%
level of significance. In addition, the Ljung-Box Q statistics show strong serial correlation in first and
second moments for most series, suggesting the persistence of linear and nonlinear dependences in the
returns of all series. In particular, the first order autocorrelation coefficients — except the one for the
4
We discard the last 3,537 pairs of quotes in order to make the numbers of aggregated five-minute and hourly returns
integers.
9
USD/GBP case—are significant at the 5% level, suggesting the use of an AR model to correct the return
serial correlations. From the perspective of finance theories, a significant, negative, first-order
autocorrelation coefficient for intraday high-frequency forex returns, such as -0.02418 for our JPY/USD
case, might be necessary to reflect the bid-ask bounce arising from the dealers’ market-making activities.
On the contrary, a significant positive first-order autocorrelation coefficient, such as 0.0158 for our
DEM/USD case or 0.05036 for our five-minute Microsoft’s equity returns is somewhat unexpected
because it suggests opportunities of arbitrage. However, these numbers are small and are likely to be
economically insignificant. These descriptive statistics, taken as a whole, suggest that intraday stock forex
and equity returns deviate from conventional Gaussian assumptions and clearly reject an assumption of
independence. These results are consistent with the empirical findings contained in Huisman et al. (1998),
Hsieh (1989) and Wang et al. (2001), lending themselves to the use of a proper AR-GARCH model to
specify the dependence structures existing in the first and second moment sequences.
3. Forecasting Procedures and Model Specifications
We will examine whether cumulative volatility forecasts from the mis-specified standard GARCH(1,1)
filter, fitted to a time series of high-frequency observations (named as Procedure AF), are more accurate
than forecasts either from the same filter fitted to the aggregated low-frequency time series (named as
Procedure AD) or from the implied low-frequency GARCH(1,1) filter (termed as Procedure AM). We
introduce the three volatility forecasting approaches and the three GARCH variants accommodating
periodicities in volatility below.
3.1 Three Forecasting Procedures:
We have in total three different approaches to forecasting volatility for the original intervals: Procedures
AF, AD and AM. The standard GARCH(1,1) model is fitted to both the high-frequency and low-frequency
observations. To simplify our discussion, we take the GDP 3 and 4 as an example, which set the number
of stages to be 5. The forecasting procedures are specified as follows:
3.1.1 Procedure AF: aggregation of forecasts
yt   t  t t ,
where
 t ~ i.i.d. N(0,1) or standardised Student’s t
 t2   H  1H  t21  1H t21 , and
(3.1a)
1
(3.1b)
10
the superscript
H
is used to mark these parameter estimates being for the sub-intervals (or, relatively,
high-frequency intervals). The strong GARCH(1,1) filter in Eqs.(3.1a) and (3.1b) is first estimated for the
sub-intervals. The resulting volatility forecasts are then aggregated in the sense that forecasts of every
five consecutive sub-intervals sum up to one for the original interval. The motive behind this approach
comes from the zero autocorrelation of the observations for the sub-intervals. To be more specific, let
ˆ T2i T
denote the volatility forecast i-steps ahead of the end of the in-sample period for the sub-intervals,
T (=5000). The forecasts for the one-step-ahead case are calculated in the obvious manner by
incrementing the time subscript by one, substituting estimates of model parameters in

H
1
 H , 1H ,
and
, return innovations, and conditional variances into the conditional variance equation. Thus, the
one-step-ahead forecast of the conditional variance is given by
ˆ T2 1 T = ˆ H + ˆ1H  T2 + ˆ1Hˆ T2 .
(3.2a)
By iterative substitution and setting the value of the future return innovations to their expected value of
zero, forecasts of the general i-step-ahead cases, for i = 2 to 100 (the out-of-sample size of the
sub-intervals under DGPs 3 and 4), are given by
ˆ T2i T = (ˆ H )2 + ( ˆ1H + ˆ1H )i-1[ ˆ T2 1 T - (ˆ H )2 ],
where
(ˆ H ) 2 =
(3.2b)
ˆ H
.
1  (ˆ1H  ˆ1H )
Since the focus of interest is on the comparison of the volatility-forecasting performances of these
three approaches for the original intervals, the next step is to bridge the forecasts for sub-intervals and the
original intervals. The relationship is given by
5
5
5
i 1
i 1
i 1
2
2
2
ˆ2
Var (  s ) = E ( 
s ) = ET [(   T i 5( s 1) ) ] =  ET [ T  i  5( s 1) ] =   T i 5( s 1) T
(3.3)
for s = 1 to 20 (the out-of-sample size for the original intervals under DGPs 3 and 4), where  = 980
refers to the in-sample size of the original intervals under DGPs 3 and 4. The third equality in Eq. (3.3)
above is based on the non-autocorrelation of the observations for the sub-intervals, implied by the i.i.d
attribute of the DGP’s in our simulations.5 For ease of reference, we denote the forecasts thus obtained in
Eq. (3.3) by
AF
(ˆ 
) 2 , where the superscript AF indicates that the volatility forecasts are in fact equal to
s
the aggregation of the forecasts for sub-intervals.
5
On the other hand, while the observations for the aggregated original intervals are not independent of one another,
for they fall into the class of weak GARCH processes, they are still uncorrelated with one another, a property of the
weak GARCH processes (combining Eqs.(5) and (7) with r = 1 in Drost and Nijman (1993) justifies this argument).
Empirical financial data, however, are often characterised by first-order autocorrelation, and failure to take into
account this stylised fact in the mean equation would undermine the conditional variance estimates.
11
As market practitioners might be more interested in knowing volatility forecasts of the return
innovations over a longer time horizon, we also calculate the forecasts for horizons of one through five
steps ahead and one through twenty steps ahead as follows. For example, the number of sub-intervals
within each original interval is 5 under DGPs 3 and 4,
5
5
5
2
2
2
Var ( 1_ 5 ) = E [ 
1_ 5 ] = E  [(    s ) ] = ET [(   T i 5( s 1) ) ] =
s 1
s 1 i 1
25
 E [
s 1
T
2
T s
]
25
ˆ
=
s 1
2
T s T
,
(3.4a)
where the subscript Λ+1_5 denotes forecast horizon of one through five steps ahead for the original
intervals, whilst
 1_ 5
refers to the innovation to the mean spanning the same period of time. Note that
the fourth equality in Eq.(3.4a) is based on the foregoing argument that the observations of the
sub-intervals are of zero autocorrelation.
For the forecast horizon of one through twenty steps ahead, similarly,
100
Var ( 1_ 20 ) =  ˆ T2  s T
(3.4b)
s 1
3.1.2 Procedure AD: aggregation of data
For ease of reference, let the superscript L stand for the original intervals (or, relatively, the low-frequency
intervals). Substitute L for the superscript H in Eqs.(3.1a) and (3.1b), and rewrite them as
y        6,
where
(3.5a)
  ~ i.i.d. N(0,1) or standardised Student’s t
2
, and
 2   L  1L 21  1L 21 ,
(3.5b)
Procedure AD involves estimating the strong GARCH(1,1) filter for observations of the original
intervals. The model is specified in Eqs. (3.5a) and (3.5b). This estimated strong GARCH(1,1) filter is
then used to generate the one-step-ahead through the twenty-step-ahead volatility forecasts for the
original intervals, denoted by
2
ˆ 
s
for s = 1 to 20. Similar to the case of sub-intervals, the
one-step-ahead and multi-step-ahead forecasts are given by
2
L
= ˆ
ˆ 
1
2
ˆ 
s
6
L 2
+ ˆ1   +
= (ˆ )
L 2
ˆ1Lˆ 2 , and
+ ( ˆ1 +
L
L 2
2
- (ˆ ) ] for s = 2 to 20,
ˆ1L )i-1[ ˆ 
1
(3.6a)
(3.6b)
However, having also conducted a separate set of experiments allowing for a constant term in Eqs. (3.1a) and (3.5a),
we do not find the constant’s estimate significantly different from zero. Nor are subsequent results different from
those by assuming zero constant term.
12
where (ˆ ) =
L 2
ˆ L
.
1  (ˆ1L  ˆ1L )
Again, the forecast over the period of one through five steps ahead is computed, and under the
assumption of i.i.d. observations7 for the original intervals, it is given by
5
5
5
s 1
s 1
s 1
2
2
2
ˆ2
Var ( 1_ 5 ) = E [ 
1_ 5 ] = E  [(    s ) ] =  E  [  s ] =    s  .
(3.7a)
The third equality in Eq.(3.7a) is based on the property that observations of the original intervals are
uncorrelated with one another. Similarly, the forecast formula for the forecast horizon of one through
twenty steps ahead is given by
20
2
Var ( 1_ 20 ) =  ˆ 
s
(3.7b)
s 1
3.1.3 Procedure AM: aggregation of model
We calculate the parameters of the weak GARCH(1,1) model for the original intervals. The calculated
weak model is given by
 2
=
 DN
+
1DN  21
+
1DN  21 ,
(3.8)
 2 denotes the linear projection of the squared innovations for the original intervals,  2 , on the
2
2
Hilbert space spanned by {1,   1 ,   2 ,…,   1 ,   2 ,…}.
where
Given the parameter estimates for the strong GARCH (1,1) in Eq.(3.1b), the third forecast approach
involves calculating (but not estimating) the weak GARCH(1,1) model through the DN formulae. The
calculated weak GARCH(1,1) model is then used to generate forecasts of the best linear projections of the
squared innovations for the original intervals. In particular, denote the parameters of the calculated weak
GARCH(1,1) model by
 DN , 1DN ,
and
1DN
in Eq.(3.8), where the superscript
these parameters are obtained through the DN formulae. Also, denote by

2

DN
indicates that
the best linear projection
of the squared innovations for the last in-sample original interval,  , on the Hilbert space spanned by {1,
2

2
2
 1 ,  2 ,…,  
1 ,   2 ,…}. In line with Procedure AD, the out-of-sample forecast of the best linear
2
projection of   s is given by
DN
2
=
 
1
+
L 2
2
= ( )
 
s
7
1DN  2
+ ( 1
DN
+
+
1DN  2 , and
L 2
2
1DN )i-1[  
- ( ) ] for s = 2 to 20,
1
(3.9a)
(3.9b)
As the strong GARCH (1,1) parameters for the original intervals are estimated under the assumption of i.i.d.
observations, the out-of-sample volatility forecasts based on these estimates should be calculated under the same
assumption.
13
where ( ) =
 DN
L 2
1  (1DN  1DN )
.
Turning to the forecast horizon of one through five steps ahead, denote the forecast by
P[  1_ 5 | Γ ], where  refers to the Hilbert space spanned by {1,
2
2
  ,  1 ,  2 ,…,  2 ,  
1,
2
 
2 ,…}. Since the observations for the original intervals are uncorrelated with one another, it follows
AB (1998a) that P[  1_ 5 | Γ ] is given by
2
P[  1_ 5 | Γ ]=P[ (
2
5

s 1
5
5
s 1
s 1
2
2
2
 s ) | Γ ]=  P[  s | Γ ]=     s  .
(3.10a)
For a forecast horizon of one through twenty steps ahead, the forecast formula is similarly obtained as
above. That is,
P[  1_ 20 | Γ ]=
2
20

s 1
2
 s 
.
(3.10b)
Note that the types of data used in these three approaches are not consistent. Procedures AF and AM both
stem from the use of the observations for sub-intervals, which are generated by the strong GARCH(1,1)
model with or without periodicities in conditional variance. On the other hand, Procedure AD involves
using the observations for the original intervals, which are not independent of one another, i.e. they fall
into the weak GARCH(1,1) type of process.
3.2 Model Specifications for Volatility Periodicities
Inspired by the periodic patterns observed in the high-frequency intraday return volatility of foreign
exchange and equity markets, AB (1997) introduce the intraday-periodic-component GARCH(1,1) (or
IPC-GARCH(1,1)) model based on Gallant’s (1981, 1982) Fourier flexible functional form, which
explicitly specifies the intraday periodic components of daily returns. This functional form is a
combination of some trigonometric functions of time and a second-order polynomial of time. Using this
model, AB (1997) successfully separate the intraday periodic patterns and the daily characteristics of the
volatility process. They suggest that if the conditional variances of the intraday return series are filtered
by the intraday periodic components and estimated by the strong standard GARCH(1,1) model, the
parameter estimates will be in accord with DN’s (1993) temporal aggregation theory for weak GARCH
processes. On the contrary, unlike the IPC-GARCH(1,1) model involving Fourier analysis and
complicated model inference procedures, a simpler way for modelling a deterministic, or purely repetitive
intraday periodicity, is simply to use a series of dummy variables or a periodic cycle function. The value
of this function varies across the stages of its corresponding periodic cycle. A general representation of
this method is documented by Bollerslev and Ghysels (1996), where both the intercept and the dynamic
parameters are allowed to vary with a periodic cycle function that repeats at a specified frequency. This
model is referred to as the Periodic GARCH(p,q) or PGARCH(p,q) model.
14
Given the possibility of two-layer periodicity in volatility, i.e. one for the sub-intervals and the
other for the original intervals, we propose an innovative IPC-PGARCH model to nest both AB’s (1997)
IPC-GARCH(1,1) and Bollerslev and Ghysels’(1996) PGARCH(1,1) models. In particular, the IPC part
is used to model the volatility periodicity for the sub-intervals, whereas the PGARCH(1,1) part is applied
to tackle the periodic patterns for the original intervals. It is specified as follows:
ym (  1)t  ym (  1)t  ym (  1)t 1
=
y m(  1)t +  m (  1)t
=
y m(  1)t + (
h 1/ 2
) sm (  1)t m (  1)t ,
m
(3.11a)
where t refers to the time index for the sub-intervals, whilst

the time index for the corresponding
original intervals,
m denotes the number of sub-intervals within each original interval,
ym (  1) t denotes the log price of the underlying financial asset for the [ m(  1)  t ]th
sub-interval,
y m(  1)t denotes the sample mean of ym (  1) t for the entire in-sample period of the
sub-intervals,
h  sL( )  1Ls ( ) 21  1Ls ( ) h 1
(3.11b)
 th original interval.
denotes the conditional variance for the
s( ) = [(  -1) modulus S] + 1, S denoting the number of stages of each periodic cycle observed
with the original intervals,
  = y -  = y -(  0L - 1L y 1 )
interval, and
 m (  1)t
denotes the return innovation for the

th
original
y the corresponding return,
denotes the standardised disturbances for sub-intervals, which are assumed to be
independent of the volatility process
standardised central t distribution with
h for the original intervals and follow a
 H degrees of freedom, and
sm (  1)  t denotes the periodic component for the tth sub-interval within the  th original interval
and satisfies
1  m
 sm( 1)t = 1, where  denotes the in-sample size of the
m  1 t 1
original intervals.
Note that, since the original sampling interval for the three series of forex rate returns and the equity
returns are, respectively, daily and hourly, we set the number of stages, i.e. S, to be 5 to nest the
day-of-the-week effect for the forex returns, and 24 to take into account the possible hourly periodic
15
patterns in volatility for the equity returns. However, to avoid the problem of over-parameterisation, we
reduce the number of stages within each periodic cycle to 2 for the equity returns, and restrict the
periodicity to be exhibited only in the intercept. In other words, we restrict the hourly periodic pattern
within each day to manifest itself through a two-stage cycle in the intercept of the GARCH(1,1) model for
the equity case.
In particular, rewrite Eq.(3.11b) as
h  sL( )  1L 21  1L h 1 , where
s( ) = [(  -1) modulus 5] + 1
s ( ) = 1
=2
for the three series of forex returns,
if [(  -1) modulus 24] + 1 = 1, 2, …, 12, and
if [(  -1) modulus 24] + 1 = 13, 14, …, 24
for the series of equity returns.
3.3 Evaluation of the Out-of-Sample Volatility Forecast Performances
We use the regression-based r2 to measure the volatility forecast performances of our three forecast
approaches. Recall the discussion in the introductory section that greater accuracy of measuring the
unobservable true volatility can be achieved by the use of more frequently sampled data, as documented
in AB (1998a). In light of this, we adopt this novel approach to measure the true realised ex-post volatility
for the original intervals by the sum of squared return innovations of all consecutive ultra-high-frequency
intervals within each original interval. For example, the regression models for calculating the
out-of-sample forecast r2 for the four empirical time series are given below.
Out-of-Sample Forecast r2 for USD/GBP, DEM/USD, and JPY/USD Forex Returns
288
 (y
i 1
)  c  a  (ˆ Daily
)2  u ( j 1) k  ( j 1) k 1 ,
( j 1)  k  ( j 1)  k 1
5 min
2
288[  ( j 1) ( k 1)]i
(3.12)
(predicted one-step-ahead daily conditional variances vs. the corresponding cumulative squared
five-minute returns)
where j = 1 to 12 denotes the index for the rolling window under estimation,
k = 1 to 20 denotes the daily time index for the out-of-sample period within each of the twelve
one-year-long sub-samples,
 = 240 denotes the number of daily observations of the in-sample period for each of the twelve
sub-samples,
5min
y288[
 ( j 1)( k 1)]i , for i = 1 to 288, denotes the five-minute return innovation within the
[  ( j  1)  k ]th daily interval, and
16
(ˆ Daily
)2 denotes the predicted one-step-ahead daily volatility made at the
( j 1)  k  ( j 1)  k 1
[  ( j  1)  k ]th daily interval.
Out-of-Sample Forecast r2 for Microsoft’s Equity Returns
60
 (y
i 1
)  c  a  (ˆ Hourly
) 2  u ( j 1) k  ( j 1) k 1 ,
( j 1)  k  ( j 1)  k 1
1-min
2
60[  ( j 1)  ( k 1)] i
(3.13)
(predicted one-step-ahead hourly conditional variances vs. the corresponding cumulative squared
one-minute returns)
where j = 1 to 10 denotes the index for the rolling window under estimation,
k = 1 to 20 denotes the hourly time index for the out-of-sample period within each of the ten
sub-samples,
 = 253 denotes the number of hourly observations of the in-sample period for each of the ten
sub-samples,
1 min
y60[
 ( j 1)( k 1)]i , for i = 1 to 60, denotes the one-minute return innovation within the
[  ( j  1)  k ]th hourly interval, and
(ˆ Hourly
)2 denotes the predicted one-step-ahead hourly volatility made at the
( j 1)  k  ( j 1)  k 1
[  ( j  1)  k ]th hourly interval.
4. Results of Simulations and The Empirical Study
4.1 Simulation results
The simulation is implemented by estimating the strong standard GARCH(1,1) filter for both the
high-frequency (sub-interval) and low-frequency (original interval) observations under no patterns of
volatility periodicity and with patterns of volatility periodicity in the DGPs. We explore three cases: one
step ahead, one through five steps ahead, and one through twenty steps ahead. The strong standard
GARCH(1,1) filter is used to estimate the in-sample, 4960 in-sample observations for sub-intervals
produced by a strong standard GARCH(1,1) DGP, which is purged of periodic variation in its parameters.
The same strong standard GARCH(1,1) filter is then estimated on the aggregated 2480 observations, the
aggregation interval being two sub-intervals in length. These estimations are replicated 10,000 times for
each of Parameterisations 1 through 3 under both N(0,1) and t5 driving disturbances. It is shown that the
estimates are quite close to their true values with both sets of driving disturbances.
17
In addition, to analyze how the volatility periodicity affects the ML parameter estimates of the
mis-specified strong standard GARCH(1,1) models, we devise a range of volatility periodicities through
DGPs 1 to 4. The results indicate that the periodicity in the intercept and alpha parameter of our two-stage
and five-stage PGARCH(1,1) DGPs does not seem to have significant impacts on the model estimation of
the mis-specified strong standard GARCH(1,1) filter for the high-frequency observations. These estimates
justify the usefulness of the ML method in estimating GARCH parameters regardless of the patterns of
volatility periodicity considered in the DGP. To save space, the ML parameter estimates of the strong
standard GARCH(1,1) filter for the sub- and original intervals are not reported here but available upon
request.
4.1.1 Comparing In-Sample Volatility Forecast Performances of the Three Predictors
Having discussed the parameter estimation of the mis-specified strong standard GARCH(1,1) filter for
both AF and AD observations in the context of volatility periodicity, we proceed with the in-sample
volatility forecast of the three predictors. The discussions are separately made in the context of no
periodicity, and periodicity in the intercept or alpha parameter of the GARCH(1,1) DGPs.
No Periodicity in the DGPs
The in-sample volatility forecast performances of the three predictors under the standard no-periodicity
circumstance are assessed by the regression-based r2 as reported in Table 5.1. From the results reported in
Table 5.1, a message is clear: Procedure AF provides the best in-sample volatility forecasts under all three
parameterisations. This is the case no matter which of the two sets of driving disturbances is used and
whether the true low-frequency volatility or the cumulative squared high-frequency innovations are used
to calculate the in-sample r2. The dominance of Procedure AF over its two competitors is prominent if we
look at the right part of each of the two panels of Table 5.1, where the true low-frequency volatility is
used to estimate the values of r2. On the other hand, there is no obvious difference between the
performances of Procedures AM and AD in that the former has a trivial advantage over the latter in terms
of the values of evaluated by the true low-frequency volatility. These results echo the finding of Andersen
et al. (1999) that summing the volatility forecasts by the high-frequency intraday weak GARCH(1,1)
filter helps forecast daily and longer-term volatility.
Volatility Periodicity in the DGPs
Under the two-stage PGARCH(1,1) DGPs (DGPs 1 and 2), the in-sample volatility forecast performances
of all three predictors are reported in Table 5.2. The results in Table 5.2 suggest that periodicity in the
intercept or alpha parameter of the two-stage PGARCH(1,1) DGP for the observations of the
18
sub-intervals does not appear to change the performance rankings of the three predictors in forecasting the
in-sample volatility of the return innovations of the original intervals throughout the four cases. The
values of r2, for the three predictors, are observed to show the same relationship as in the no-periodicity
standard contexts:
2
2
2
> rAM  rAD ,
rAF
though
2
2
is trivially larger than rAD in all four cases. It is worth noting that periodicity does reduce
rAM
the forecast performance of the three predictors.
4.1.2 Comparing the Three Out-of-Sample Volatility Predictors
No Periodicity in the DGPs
In parallel with the previous section, we begin by discussing the standard no-periodicity cases. The
parameterizations of the standard GARCH(1,1) model for the DGPs are given by Table 6. Of primary
interest to us, Procedure AF, based on the strong standard GARCH(1,1) filter estimated for the
sub-intervals, seems to outperform the other two procedures, regardless of the duration of the forecast
horizon and the choice of the driving disturbances investigated. This result echoes Andersen et al.’s (1999)
finding mentioned above. However, we also find, in almost all the cases, that Procedure AM provides
better out-of-sample volatility forecasts than Procedure AD, although the differences are not as
pronounced as those between Procedure AF and Procedure AD. This result indicates that, although the
DN aggregation formulae are restricted to the class of weak GARCH(1,1) processes, the linear
projections of the squared innovations serve as better references to the true conditional variances of the
innovations than the forecasts given by a strong GARCH(1,1) model estimated at the same frequencies.
As one would expect, moreover, the out-of-sample forecast performance of each of the three predictors
tends to decline as the forecast horizon is extended, as reported in Table 6. This is a fundamental feature
of every forecast technique in that the volatility forecasts will approach the sample unconditional
variances if they are made further into the future, and thus more future unknown short-term innovations
will bring out larger forecast errors altogether.
It is important to mention that the use of cumulative squared innovations of the sub-intervals, as
proxies for the realised ex-post volatility of the original intervals, leads to pronounced downward biases
of the r2 from their true values. This is indicative of the need for more accurate proxies for the true
volatility of the original intervals. However, the underestimated r2, calculated against the cumulative
squared innovations of the sub-intervals, gives approximately the same rankings of the out-of-sample
performances of the three predictors as those by the true r2. This can be seen from the shaded areas in the
19
three panels of Table 6, indicating the best predictor of the three competitors8. It is interesting to observe
that regardless of the forecast procedure and the forecast horizon chosen, the value of r2 evaluated by the
realised ex-post volatility of the original intervals tends to be the smallest under Parameterisation 3,
whereas the values of r2 under the other two parameterisations are roughly comparable to each other. The
true
1H
under Parameterisation 3 is set equal to 0.05, a value one third its counterparts under the other
two parameterisations. It is therefore likely that the out-of-sample performances of the three predictors are
subject to the size of alpha of the GARCH(1,1) DGP in that a larger alpha implies better out-of-sample
forecast performances by our three predictors. Intuitively, this result seems reasonable since a larger alpha
parameter in the underlying GARCH(1,1) DGP suggests that the short-term variation in the volatility
process is more discernible from the long-term smooth evolution of the process. The ML method is thus
expected to provide more accurate parameter estimates for the correctly-specified GARCH(1,1) filter than
would otherwise be the case.
Periodicity in the DGPs
Turning to the conditions under which the effect of volatility periodicity exists in the DGPs, it is of
particular interest to know whether the rankings of these three approaches remain unchanged. The
simulation results regarding this part are presented in Table 7. It is found that Procedure AF generates the
most accurate volatility forecasts out of sample under DGPs 1 to 4. This result is robust to the three
forecast horizons. In short, we find that
2
2
2
rAF
> rAM  rAD ,
2
2
although rAM is slightly larger than rAD in both cases of driving disturbances under DGPs 1 through 4.
This result suggests that the superiority of Procedure AF over the other two predictors is robust to the
simple two-stage and five-stage periodicities in the PGARCH(1,1) DGP. It has been shown that
increasing the number of stages within each periodic cycle of a PGARCH(1,1) DGP does not affect the
dominance of Procedure AF over Procedures AD and AM despite seeing that increasing the number of
stages apparently reduces the forecast performances of the three predictors.
Again, it is clear from all three panels in Table 7 that the r2 calculated against the cumulative
squared innovations of the sub-intervals is in general under-estimated under DGPs 1 through 4 for all
8
The only exception is the case under Parameterisation 3 with a t5 DGP for the forecast horizon of 1-20 steps ahead
in Panel C. In this case, the best predictor seems to be Procedure AD in terms of the underestimated r2. However,
the estimated slope coefficients for the regression for all three approaches are insignificant at the 5% level,
rendering the corresponding r2 meaningless. It is thus not appropriate to attribute the best predictor to Procedure
AD in this case.
20
three forecast horizons and for both driving disturbances. It suggests that the number of consecutive
sub-intervals within each original interval might not be enough to give a satisfactory measure of the
realised ex-post volatilities of the original intervals. Increasing the number of sub-intervals within each
original interval might help reduce the measurement error of this volatility proxy. However, the rankings
of the three forecast approaches evaluated by these biased r2 are consistent with those obtained under the
true r2. Notably, such consistency implies that any conclusions made about the relative forecast
performances of the three predictors will not be misleading even when they are made based on a biased
measure of the true volatility.
4.2 Results of the empirical study
For the empirical study, we investigate the one-day-ahead volatility forecasts for the three forex cases and
one-hour-ahead for the individual equity case. To consider the first-order autocorrelation in the data,
however, an AR(1) model is used for the means of the observations of both sub- and original intervals.
The details are omitted here.
The results reached through our Monte Carlo simulations indicate Procedure AF to be the best
volatility predictor both in sample and out of sample amongst the three competitors, all of which are
based on the standard GARCH(1,1) model, in the context of un-parameterised periodicities. To examine
whether these results are supported by empirical data, we next estimate two types of extremely
high-frequency market data, that is, three series of five-minute forex returns ― USD/GBP, DEM/USD,
JPY/USD, and one series of one-minute individual stock returns ― Microsoft’s equity traded through
NASDAQ. In particular, we will apply the three predictors to these empirical data and examine the
robustness of the conclusion from the simulation results regarding the relative forecast performances of
the three volatility predictors.
Following the discussion in Section 3.2, we also replace the strong standard GARCH(1,1)
specification by AB’s (1997) IPC-GARCH(1,1), BG’s (1996) PGARCH(1,1) and our innovative
IPC-PGARCH(1,1) models, separately, under Procedure AF to generate volatility forecasts for the
original intervals. These forecasts will be compared to those by Procedure AF with the strong standard
GARCH(1,1) model to investigate whether accommodating specific volatility features observed with the
sub-intervals improves the volatility forecast performance of Procedure AF for the original intervals.
In total, we have six volatility predictors for the original intervals: Procedure AF (using the strong
standard GARCH(1,1), IPC-GARCH(1,1), PGARCH(1,1) and IPC-PGARCH(1,1) models, respectively),
Procedure AD (using the strong GARCH(1,1) model) and Procedure AM (using the aggregated weak
GARCH(1,1) model).
21
4.2.1 The ML Estimates of the Three Models for Volatility Periodicity as well as the Strong Standard
GARCH(1,1) Model
We apply the ML method to estimate IPC-GARCH(1,1), PGARCH(1,1) and IPC-PGARCH(1,1) models
as well as the strong standard GARCH(1,1) model using the SIMPLEX and BFGS algorithms. It takes a
non-trivial period of time to complete all the estimations and the resulting volatility-forecasting work due
to the difficulty in finding a global maximum of the log-likelihood function value. To save space, the
parameter estimates of the three features-specific models as well as the strong standard GARCH(1,1)
model are omitted here but are available upon request.
4.2.2 Comparison of Volatility Forecast Performances of the Six Predictors: Procedure AF (strong
standard GARCH, PGARCH, IPC-GARCH, IPC-PGARCH), Procedure AD and Procedure AM
In-Sample Volatility Forecast Performances
All three forecast approaches are used to generate one-step-ahead volatility forecasts for the original
intervals for both types of data. We tabulate the r2 for the out-of-sample forecast performances in Table 8.
For the USD/GB, DEM/GBP and JPY/USD forex returns and Microsoft’s equity returns, the strong
standard GARCH(1,1) model under Procedure AF is the best predictor amongst the six competitors,
which is consistent with our simulation results in the context of periodicities exhibited in the underlying
volatility process. It therefore highlights the advantage of using more frequently sampled data, as
pioneered by Andersen et al. (1999), in forecasting relatively longer-term volatilities. The above results
suggest two important points: (i) Using more-frequently sampled data helps forecast longer-term
volatilities. (ii) Allowing for the effects of periodicities in the standard GARCH(1,1) specification does
not further strengthen the advantage of using more-frequently sampled data in the previous point in terms
of in-sample volatility forecasting.
Out-of-Sample Volatility Forecast Performances
The results reported in both panels of Table 9 suggest that the standard GARCH(1,1) specification
generally leads all three features-specific models in that either Procedure AF or AM, both based on the
standard GARCH(1,1) specification, produce most accurate out-of-sample volatility forecasts. In
particular, Procedure AM dominates all its competitors in the USD/GBP case, whereas Procedure AF
based on the standard GARCH(1,1) model keeps its leading place in the cases of DEM/USD forex rate
and Microsoft’s equity. However, the dominance of Procedure AM over Procedure AF is not very
obvious in terms of the forecast r2 in the case of our USD/GBP forex rate returns. In general, Procedure
AF, based upon the strong standard GARCH(1,1) model, is the best volatility predictor amongst the six
candidates employed.
22
This article represents a substantial step forward in incorporating the periodicity effect into
out-of-sample volatility forecasting comparisons. Our results suggest that modelling potential
periodicities does not lead Procedure AF to better volatility forecasts. We find that only the
IPC-GARCH(1,1) model, among the three used, improves the out-of-sample forecast performance of the
strong standard GARCH(1,1) model to an obvious extent with regards to the JPY/USD forex rate. But for
the other three empirical cases, the IPC-GARCH(1,1) is obviously not able to improve the out-of-sample
volatility predictive power of the strong standard GARCH(1,1) model within the framework of Procedure
AF. This result is consistent with the findings of Dimson and Marsh (1990), Poon and Granger (2005),
Pagan and Schwert (1990), Franses and van Dijk (1996) and Chong et al. (1999) that models tailored for
particular features in the volatility process of the data might generate poorer out-of-sample volatility
forecasts than a more general and more parsimonious model, such as the standard GARCH(1,1)
specification. While this result appears to cast doubt on previous efforts in modelling periodicities based
on the strong standard GARCH(1,1) formulation, it might be a piece of good news for market
practitioners who care about the volatility of their portfolios. Practitioners could just stick to the use of the
strong standard GARCH(1,1) specification since ignoring the effect of periodicity in the underlying
volatility process might not give poorer volatility forecasts than otherwise. In terms of time consumption,
moreover, the parameter estimation of the strong standard GARCH(1,1) model can be completed in a few
seconds while it might take up a lot more time to estimate each of its three complicated variants such as
the IPC-GARCH(1,1) model above.
5 Concluding Remarks
The Monte Carlo simulation results in this study suggest that, based on the standard GARCH(1,1)
specification, Procedure AF dominates Procedures AD and AM both in sample and out of sample. This
finding is not only valid in the standard condition when the volatility process is characterised by only the
GARCH effect, but also robust to the circumstance of un-parameterised periodicities in the underlying
volatility process apart from the GARCH effect. The overall performance ranking of Procedures AF, AD,
and AM, which are all based on the standard GARCH(1,1) specification, is given by this statement:
Procedure AF absolutely dominates Procedures AD and AM, whereas Procedure AM performs better than
Procedure AD though to a trivial extent. This result suggests that Andersen et al.’s method is not only
valid in the standard condition purged of any volatility features except the GARCH effect, but also robust
in the context of both the GARCH and periodicity effects. Un-parameterised periodicities in the volatility
process certainly do not detract from the advantage of Procedure AF over the traditional approach,
Procedure AD. This finding appears to solve the problem faced by Andersen et al.’s method, in which a
sampling interval equal to or shorter than hourly would render their method useless in empirical
23
applications. Indicated by its high r2 across the experiments of our simulations, moreover, Procedure AF
which involves summing volatility forecasts for the sub-intervals to form forecasts for the original
intervals proves promising in volatility forecasting using the strong standard GARCH(1,1) model.
To examine the robustness of the conclusion from our simulation results about the relative forecast
performance of the three volatility predictors, we further undertake a set of empirical studies based on
three series of forex rate returns and one series of individual equity returns. It is shown that
un-parameterised periodicities in the conditional variance of the empirical data for the sub-(or
high-frequency) intervals do not subtract from the dominance of Procedure AF over Procedures AD and
AM for forecasting the volatility of the original (or relatively low-frequency) intervals. Our empirical
study echoes the conclusion from the simulation results that Procedure AF generally outperforms
Procedures AD and AM in both in-sample and out-of-sample comparisons.
One of the distinct contributions in this paper is to investigate whether Procedure AF benefits from
modelling periodicities observed with the underlying volatility process. We substitute three
GARCH-variants accommodating periodicity effect for the strong standard GARCH(1,1) model within
the
framework
of
Procedure
AF:
PGARCH(1,1),
IPC-GARCH(1,1),
and
our
innovative
IPC-PGARCH(1,1). Contrary to the empirical findings in the literature which suggest that the in-sample
volatility forecast performances by these three GARCH variants are better than that of the standard
GARCH(1,1) model, we find that these three complex GARCH variants for the periodicity effect do not
outperform the standard GARCH(1,1) model both in sample and out of sample under the Procedure AF
framework. The out-of-sample performance is actually poor for all these three models under Procedure
AF. This result is consistent with the finding of some previous studies in the literature of GARCH
modelling and forecasting: parsimonious GARCH models give better out-of-sample volatility forecasts
than their complicated variants. An implication of this suggests that practitioners could just stick to the
use of the strong standard GARCH(1,1) specification, since ignoring the effect of periodicity in the
underlying volatility process might not give poorer volatility forecasts than otherwise.
The failure of these three GARCH variants for periodicities to provide more accurate forecasts than
the strong standard GARCH(1,1) specification provides a suggestion of more unknown features in the
volatility process of the four empirical time series considered. Leverage effects in volatility and structural
breaks might both be one of the candidates. Actually, the out-of-sample r2 of Procedure AF based on the
strong standard GARCH(1,1) is less than 0.4 for all four time series. This suggests that modelling the
GARCH effects for the sub-intervals in the four empirical cases provides limited support in the sense of
out-of-sample volatility forecasting. For these four empirical time series, therefore, the advantage of
Procedure AF over Procedures AD and AM might be strengthened via other types of models rather than
the GARCH(1,1)-family.
24
Also notably, our simulation results suggest that substituting the cumulative squared return
innovations of the sub-intervals for the true realised ex-post volatility of the original intervals would
under-estimate the forecast performances of all forecasting procedures. The values of r2 for all three
predictors reported in Tables 6 and 7 are greatly understated by the use of such a proxy in all cases. This
result indicates the need for a more accurate proxy for the true volatility in future research. According to
AB (1998a), increasing the number of sub-intervals might help solve this problem. Nevertheless, the
rankings of the three predictors in terms of both their in-sample and out-of-sample forecast performances
are not affected by the use of the proxy. This is good since the realised ex-post volatility is not observable
and researchers must use some kind of proxy, such as the cumulative squared return innovations of the
sub-intervals, for the latent true volatility.
25
Table 1: Parameterisations of the GARCH(1,1) model for the DGPs
Parameterisations
H
1H
 1H
1H +  1H
( H ) 2
 t ~ t5
 t ~ N (0,1)
kH
Notes: 1.
1
2
3
0.05
0.01
0.05
0.15
0.15
0.05
0.7
0.7
0.7
0.85
0.85
0.75
0.3333
0.0667
0.2
9
9
9
3
3
3
kH denotes the unconditional kurtosis of the standardised innovation  t .
2. ( H ) 2 denotes the unconditional variance of
and
 1H . Namely, ( H )2 
 t , implied by  H , 1H ,
H
.
(1  1H  1H )
3. Parameterisations must all satisfy the following conditions
a(1H , 1H , k H , m)  (1H  1H )m  b(1H , 1H , m)
 0.5 , and
a(1H , 1H , k H , m)  [1  (1H  1H ) 2m ]  2b(1H , 1H , m)
0  1H  1H  19, and 1  (1H  1H ) 2  (kH  1)(1H ) 2 .
Table 2: In-sample and out-of-sample size for the sub- and original intervals under all four DGPs
DGP No
No of
sub-intervals
within each
original interval:
m
in-sample size for
original
intervals:

out-of-sample size
for original
intervals:
20
in-sample size
for
sub-intervals:
T= m
out-of-sample
size for
sub-intervals:
20m
2
2480
20
4960
40
2
2480
20
4960
40
5
980
20
4900
100
5
980
20
4900
100
DGP 1:
two-stage periodicity in
the intercept
DGP 2:
two-stage periodicity in
the alpha parameter
DGP 3:
five-stage periodicity in
the intercept
DGP 4:
five-stage periodicity in
the alpha parameter
9
Whereas
H
1H  1H  1 and 1  (1H  1H )2  (kH  1)(1H )2 would also lead to valid k , this is not
feasible in practice as the first condition implies a non-stationary process.
26
Table 3: Specifications of the experiments
Driving
Case No
DGP No
Driving
Case No
DGP No
Disturbances
Disturbances
Case 1
DGP 1
N(0,1)
Case 5
DGP 1
t5
Case 2
DGP 2
N(0,1)
Case 6
DGP 2
t5
Case 3
DGP 3
N(0,1)
Case 7
DGP 3
t5
Case 4
DGP 4
N(0,1)
Case 8
DGP 4
t5
Table 4: Statistical description of the three series of hourly forex rate and the five-minute equity returns
USD/GBP
(hourly
returns)
DEM/USD
(hourly
returns)
JPY/USD
(hourly
returns)
Microsoft’s
equity
(5-minute
returns)
mean (mean=0)
0.00016
(0.796)
-0.00014
(0.813)
-0.00040
(0.594)
-0.00165
(0.342)
Variance
0.02222
0.02196
0.02762
0.09889
Skewness (sk=0)
-0.3511*
(0.000)
0.3987*
(0.000)
-0.3421*
(0.000)
-1.953*
(0.000)
kurtosis (ku=3)
18.0122*
(0.000)
25.5241*
(0.000)
28.9615*
(0.000)
89.0367*
(0.000)
Jarque-Bera (JB=0)
528511.5*
(0.000)
1320729.5*
(0.000)
1402888.2*
(0.000)
10124978*
(0.000)
ˆ1
0.001685
0.01580*
-0.02418*
0.05036*
Ljung-Box Q-statistics:
Q(1)
0.1594
(0.6897)
15.5695*
(0.0001)
29.1887*
(0.0000)
83.1039*
(0.0000)
Q(10)
16.4466
(0.0875)
41.8371*
(0.0000)
47.0119*
(0.0000)
175.6787*
(0.0000)
Q2(1)
452.5978*
(0.0000)
275.6365*
(0.0000)
712.1273*
(0.0000)
16.4368*
(0.0001)
Q2(10)
1192.3287*
(0.0000)
683.8345*
(0.0000)
1038.8004*
(0.0000)
121.8678*
(0.0000)
Notes: 1. p-value in parentheses.
2. * denotes significance at the 5% level.
3. ˆ1 denotes the sample estimate of the first-order autocorrelation coefficient for the returns.
4. Q2( ) denotes the Ljung-Box statistics for the squared return time series.
27
Table 5.1: Average in-sample volatility prediction performances of the three forecasting procedures
2
evaluated by the regression-based r , under the standard GARCH(1,1) DGP specified by
Parameterisations 1 through 3
Panel A: GARCH(1,1)-N(0,1) DGP
r 2 (evaluated by cumulative
Parameterisations of the Strong
standard GARCH(1,1)-N(0,1)
2:  H =0.01, 1H = 0.15, 1H = 0.7
3:  H =0.05, 1H = 0.05, 1H = 0.7
volatility of original intervals)
sub-intervals)
DGP
1:  H =0.05, 1H =0.15, 1H = 0.7
r 2 (evaluated by the true
squared return innovations of
AF
AD
AM
AF
AD
AM
0.25490
0.06206
0.06217
0.99651
0.53848
0.54679
(0.04107)
(0.02763)
(0.02758)
(0.00499)
(0.06057)
(0.05704)
0.25491
0.06206
0.06217
0.99651
0.53850
0.54679
(0.04107)
(0.02763)
(0.02758)
(0.00499)
(0.06055)
(0.05703)
0.12262
0.00473
0.00479
0.95494
0.36904
0.42484
(0.03897)
(0.00338)
(0.00337)
(0.07270)
(0.09846)
(0.04200)
Panel B: GARCH(1,1)- t5 DGP
r 2 (evaluated by cumulative
Parameterisations of the Strong
squared return innovations of
standard GARCH(1,1)-t5 DGP
sub-intervals)
1:  H =0.05, 1H =0.15, 1H = 0.7
2:  H =0.01, 1H = 0.15, 1H = 0.7
3:  H =0.05, 1H = 0.05, 1H = 0.7
r 2 (evaluated by the true
volatility of original intervals)
AF
AD
AM
AF
AD
AM
0.23293
0.06416
0.06407
0.99602
0.66775
0.67283
(0.07966)
(0.04704)
(0.04678)
(0.00654)
(0.09035)
(0.08919)
0.23295
0.06420
0.06412
0.99602
0.66780
0.67288
(0.07969)
(0.04710)
(0.04685)
(0.00654)
(0.09039)
(0.08924)
0.12495
0.00687
0.00683
0.96239
0.56343
0.60869
(0.06797)
(0.00991)
(0.00956)
(0.06264)
(0.12106)
(0.08183)
Notes: 1. The computation of r2 involves regressing both the cumulative squared return innovations of the sub-intervals,
2

i 1
2
i  2(  1)
h (which is obtained by aggregating
, and the true conditional variance of the original intervals,
the true conditional variance of the sub-intervals from the DGP), on the predicted values of volatility from each
2
of the three predictors. In particular, writing

i 1
2
i  2(  1)
= c1 + a1  (ˆ Low _ i ) 2 + u  (predicted values of
innovation variance of original intervals vs. proxy of true innovation variance of original intervals) and,
h  c2  a2  (ˆ Low _ i ) 2  u (predicted values of innovation variance of original intervals vs. true innovation
variance of original intervals) for i = AF, AD and AM, where for Procedure AF: (ˆ Low _ AF ) 2 =
2
ˆ
i 1
with ˆ t =
ˆ 
L
2
1(2)  1
2.
ˆ H  ˆ1H  t21  ˆ1Hˆt21 , for Procedure AD: (ˆ Low _ AD ) 2
 ˆ ˆ  1 , and for Procedure AM: (ˆ 
L
1(2)
2
r 2 refers to the average of r 2
Low _ AM 2
)
=
 = 
2
DN
(2)
= ˆ 2 =
+ 

L
ˆ(2)
DN 2
1(2)  1
i  2(  1)
+
+
DN 2
1(2)
  1 .
across the N = 10000 Monte Carlo replications.
3. Numbers in the parentheses are the Monte Carlo standard deviations for
r2
across the N = 10000
replications.
28
,
Table 5.2: Average in-sample volatility prediction performances, evaluated by the regression-based r2, of
the three forecasting procedures under DGPs 1 and 2
r 2 (evaluated by cumulative
squared return innovations
Case No
1
2
3
4
r 2 (evaluated by true
volatility of original intervals)
(DGP No, Disturbance)
of sub-intervals)
AF
AD
AM
AF
AD
AM
0.267
0.060
0.061
0.997
0.532
0.541
(0.040)
(0.027)
(0.027)
(0.005)
(0.061)
(0.058)
0.201
0.024
0.024
0.872
0.414
0.431
(0.033)
(0.011)
(0.011)
(0.020)
(0.056)
(0.041)
0.244
0.063
0.063
0.996
0.659
0.664
(0.079)
(0.047)
(0.047)
(0.007)
(0.092)
(0.091)
0.196
0.030
0.030
0.871
0.568
0.580
(0.069)
(0.027)
(0.027)
(0.039)
(0.091)
(0.087)
(DGP 1, N(0,1))
(DGP 2, N(0,1))
(DGP 1, t5)
(DGP 2, t5)
Footnotes to Table 5.1 also apply here.
29
Table 6: Out-of-sample forecast performance of the three forecasting procedures under no-periodicity standard GARCH(1,1) DGPs
Panel A: Forecast horizon of 1 step ahead
Cumulative Squared Return Innovations (of sub-intervals) regressed
on Predicted Conditional Variances (of original intervals)
Parameterisations
Driving
in Table 1
Disturbances
Procedure AF
2
rAF
 = 0.05,
1H = 0.15,
 1H = 0.7
H
2:  = 0.01,
1H = 0.15,
 1H = 0.7
H
3:  = 0.05,
1H = 0.05,
 1H = 0.7
1:
â1
(t- ratio)
Procedure AD
2
rAD
â1
(t- ratio)
Procedure AM
2
rAM
â1
(t- ratio)
True Realised Ex-post Volatilities (of original intervals) regressed on
Predicted Conditional Variances (of original intervals)
Procedure AF
2
rAF
â2
(t- ratio)
Procedure AD
2
rAD
â2
(t- ratio)
Procedure AM
2
rAM
â2
(t- ratio)
H
 H = 0.05,
1H = 0.15,
 1H = 0.7
H
2:  = 0.01,
1H = 0.15,
 1H = 0.7
H
3:  = 0.05,
1H = 0.05,
 1H = 0.7
N(0,1)
0.112
N(0,1)
0.112
N(0,1)
0.072
t5
0.047
t5
0.095
t5
0.006
1:
1.004
(35.54**)
1.004
(35.55**)
0.879
(8.50**)
0.342
(22.11**)
0.758
(32.32**)
0.658
(7.91**)
0.062
0.062
0.002
0.042
0.066
0.006
0.931
(25.63**)
0.931
(25.63**)
0.503
(4.34**)
0.437
(20.98**)
0.603
(26.65**)
0.827
(7.81**)
0.065
0.065
0.003
0.043
0.068
0.005
0.982
(26.42**)
0.982
(26.42**)
0.777
(5.53**)
0.453
(21.28**)
0.634
(26.93**)
0.791
(7.43**)
0.915
0.915
0.767
0.950
0.927
0.818
0.987
(327.46**)
0.987
(327.46**)
0.885
(181.39**)
0.843
(434.83**)
0.934
(355.78**)
0.815
(211.66**)
0.512
0.512
0.267
0.621
0.695
0.479
0.924
(102.48**)
0.924
(102.48**)
0.584
(60.39**)
0.916
(127.69**)
0.769
(150.77**)
0.794
(95.83**)
0.540
0.540
0.373
0.662
0.714
0.657
0.973
(108.38**)
0.973
(108.39**)
0.836
(77.09**)
0.968
(139.56**)
0.811
(157.68**)
0.935
(138.50**)
30
Panel B: Forecast horizon across 1-5 steps ahead
Cumulative Squared Return Innovations (of sub-intervals) regressed
on Predicted Conditional Variances (of original intervals)
Parameterisations
Driving
in Table 1
Disturbances
Procedure AF
2
rAF
 H = 0.05,
1H = 0.15,
 1H = 0.7
H
2:  = 0.01,
1H = 0.15,
 1H = 0.7
H
3:  = 0.05,
1H = 0.05,
 1H = 0.7
â1
(t- ratio)
Procedure AD
2
rAD
â1
(t- ratio)
Procedure AM
2
rAM
â1
(t- ratio)
True Realised Ex-post Volatilities (of original intervals) regressed
on Predicted Conditional Variances (of original intervals)
Procedure AF
2
rAF
â2
(t- ratio)
Procedure AD
2
rAD
â2
(t- ratio)
Procedure AM
2
rAM
â2
(t- ratio)
1:
 H = 0.05,
1H = 0.15,
 1H = 0.7
H
2:  = 0.01,
1H = 0.15,
 1H = 0.7
H
3:  = 0.05,
1H = 0.05,
 1H = 0.7
0.983
N(0,1)
0.099
0.920
0.057
(33.21**)
(24.67**)
0.983
N(0,1)
0.099
(6.98**)
(3.33**)
(4.12**)
0.956
0.275
(58.90**)
0.672
0.213
(61.58**)
0.894
0.258
(92.64**)
0.520
0.002
0.956
0.275
(58.90**)
0.972
0.462
(25.29**)
0.330
0.001
0.894
0.258
(92.64**)
0.974
0.060
(24.67**)
0.699
0.005
0.972
0.462
(25.29**)
0.920
0.057
(33.21**)
N(0,1)
0.974
0.060
(61.58**)
0.345
0.058
(51.96**)
0.548
0.090
(24.74**)
(31.38**)
1:
0.541
t5
0.055
t5
0.095
t5
0.004
0.582
0.035
(24.02**)
(18.90**)
0.636
(32.29**)
(4.34**)
(5.02**)
(83.12**)
0.549
0.121
(55.55**)
0.674
0.409
(80.49**)
0.661
0.236
(65.79**)
0.642
0.394
(107.21**)
0.538
0.0025
0.826
0.303
(60.74**)
0.789
0.535
(27.73**)
0.443
0.0019
0.767
0.270
(88.58**)
0.541
0.072
(27.06**)
0.522
0.724
0.441
(19.98**)
0.513
0.068
(5.94**)
0.625
0.039
0.692
0.174
(37.17**)
(45.87**)
31
Panel C: Forecast horizon across 1-20 steps ahead
Cumulative Squared Return Innovations (of sub-intervals) regressed on
Predicted Conditional Variances (of original intervals)
Parameterisations
Driving
in Table 1
Disturbances
Procedure AF
2
rAF
 H = 0.05,
1H = 0.15,
 1H = 0.7
H
2:  = 0.01,
1H = 0.15,
 1H = 0.7
H
3:  = 0.05,
1H = 0.05,
 1H = 0.7
Procedure AD
â1
2
rAD
(t- ratio)
Procedure AM
â1
2
rAM
(t- ratio)
True Realised Ex-post Volatilities (of original intervals) regressed on
Predicted Conditional Variances (of original intervals)
Procedure AF
â1
(t- ratio)
2
rAF
â2
(t- ratio)
Procedure AD
2
rAD
â2
(t- ratio)
Procedure AM
2
rAM
â2
(t- ratio)
1:
 H = 0.05,
1H = 0.15,
 1H = 0.7
H
2:  = 0.01,
1H = 0.15,
 1H = 0.7
H
3:  = 0.05,
1H = 0.05,
 1H = 0.7
0.781
N(0,1)
0.613
0.022
(14.88**)
0.212
N(0,1)
(10.82**)
0.055
0.0005
(0.74)
0.013
(1.07)
0.657
0.045
(20.64**)
0.177
0.0001
(21.59**)
0.567
0.041
(30.25**)
0.109
5.43E-05
(2.19*)
0.084
0.657
0.045
(20.64**)
0.755
0.012
(10.79**)
0.567
0.041
(30.25**)
0.682
0.012
(14.88**)
0.084
(10.82**)
0.613
0.022
0.755
0.012
(10.79*)
0.781
N(0,1)
0.681
0.012
(21.59**)
0.063
0.0027
(11.44**)
0.105
0.0041
(5.22*)
(6.38*)
1:
0.436
t5
0.008
t5
0.011
t5
3.69E-04
0.458
0.484
0.005
(8.74**)
(7.30**)
0.521
0.423
0.177
(1.92)
H
Notes: 1. The linear regression equations for these r2 are given by
(1.97)
5
(1.38)
H
  T2i5( s1), j  c1  a1   f Lows, _j i  u ,H , j and
s 1 i 1
s 1
H
H
s 1
s 1
(22.23**)
0.156
0.011
(15.15**)
0.516
0.047
(21.78**)
0.240
0.022
(19.44**)
0.485
0.045
(25.54**)
0.144
1.90E-04
0.602
0.037
(18.11**)
0.606
0.061
(8.99**)
3.88E-04
0.542
0.032
(24.15**)
0.441
0.008
(8.99**)
0.189
0.055
(7.45**)
0.008
(10.32**)
0.569
0.006
0.196
0.013
(10.17**)
(11.54**)
 h s, j  c2  a2   f Lows, _j i  v ,H , j , where i = AF, AD or AM and H = 1
(one step ahead), 5 (one through five steps ahead), and 20 (one through twenty steps ahead). 2.** denotes significance at the 1% level, and * at the 5% level. 3. Shaded areas denote the most
accurate predictors amongst the three.
32
Table 7 Out-of-sample forecast performances of the three forecasting procedures under DGPs 1 to 4 with periodicity
Panel A: Forecast horizon of 1 step ahead
Cumulative Squared Return Innovations (of sub-intervals) regressed
on Predicted Conditional Variances (of original intervals)
Case
(DGP No,
No
Disturbance)
Procedure AF
2
rAF
â1
(t- ratio)
Procedure AD
2
rAD
â1
(t- ratio)
Procedure AM
2
rAM
â1
(t- ratio)
True Realised Ex-post Volatilities (of original intervals) regressed on
Predicted Conditional Variances (of original intervals)
Procedure AF
2
rAF
â1
(t- ratio)
Procedure AD
2
rAD
â1
(t- ratio)
Procedure AM
2
rAM
â1
(t- ratio)
1
(DGP 1, N(0,1))
0.108
0.100
(34.87**)
0.059
0.927
(25.12**)
0.063
0.964
(25.89**)
0.908
0.989
(313.54**)
0.506
0.926
(101.19**)
0.535
0.962
(107.30**)
2
(DGP 2, N(0,1))
0.048
1.119
(22.53**)
0.019
0.842
(14.01**)
0.022
1.010
(15.01**)
0.845
1.086
(233.27**)
0.374
0.861
(77.37**)
0.410
1.010
(83.31**)
3
(DGP 3, N(0,1))
0.099
0.929
(33.08**)
0.017
0.541
(13.20**)
0.021
0.740
(14.56**)
0.651
0.973
(136.49**)
0.130
0.609
(38.66**)
0.180
0.890
(46.89**)
4
(DGP 4, N(0,1))
0.093
1.033
(32.06**)
0.014
0.468
(11.70**)
0.018
0.703
(13.43**)
0.475
1.066
(95.12**)
0.081
0.525
(29.76**)
0.116
0.822
(36.20**)
5
(DGP 1, t5)
0.082
0.486
(29.93**)
0.085
0.555
(30.43**)
0.083
0.568
(30.05**)
0.947
0.858
(424.19**)
0.668
0.811
(141.68**)
0.703
0.861
(153.72**)
6
(DGP 2, t5)
0.043
0.592
(21.28**)
0.043
0.797
(21.28**)
0.045
0.794
(21.61**)
0.899
1.079
(297.79**)
0.552
1.138
(110.88**)
0.638
1.200
(132.49**)
7
(DGP 3, t5)
0.121
0.765
(37.14**)
0.039
0.587
(20.05**)
0.051
0.691
(23.11**)
0.781
0.900
(188.67**)
0.203
0.624
(50.45**)
0.350
0.841
(73.39**)
8
(DGP 4, t5)
0.161
0.902
(43.78**)
0.024
0.641
(15.56**)
0.042
0.835
(20.82**)
0.653
1.017
(137.22**)
0.078
0.650
(29.02**)
0.162
0.924
(43.99**)
33
Panel B: Forecast horizon of 1-5 steps ahead
Cumulative Squared Return Innovations (of sub-intervals) regressed on
Predicted Conditional Variances (of original intervals)
Case
(DGP No,
No
Disturbance)
Procedure AF
2
rAF
â1
(t- ratio)
Procedure AD
2
rAD
â1
(t- ratio)
Procedure AM
2
rAM
â1
(t- ratio)
True Realised Ex-post Volatilities (of original intervals) regressed on
Predicted Conditional Variances (of original intervals)
Procedure AF
2
rAF
â1
(t- ratio)
Procedure AD
2
rAD
â1
(t- ratio)
Procedure AM
2
rAM
â1
(t- ratio)
1
(DGP 1, N(0,1))
0.097
0.980
(32.81**)
0.056
0.916
(24.24**)
0.058
0.957
(24.89**)
0.455
0.971
(91.42**)
0.252
0.893
(58.03**)
0.270
0.943
(60.85**)
2
(DGP 2, N(0,1))
0.036
1.035
(19.42**)
0.014
0.742
(12.10**)
0.016
0.906
(12.92**)
0.346
1.019
(72.65**)
0.141
0.741
(40.53**)
0.163
0.913
(44.17**)
3
(DGP 3, N(0,1))
0.028
0.743
(17.10**)
0.005
0.336
(6.79**)
0.006
0.553
(8.03**)
0.144
0.802
(41.05**)
0.022
0.352
(15.01**)
0.035
0.618
(19.00**)
4
(DGP 4, N(0,1))
0.028
0.851
(17.07**)
0.004
0.283
(6.26**)
0.006
0.559
(7.77**)
0.115
0.897
(36.12**)
0.016
0.300
(12.79**)
0.026
0.614
(16.49**)
5
(DGP 1, t5)
0.072
0.593
(27.90**)
0.052
0.568
(23.32**)
0.054
0.592
(23.95**)
0.481
0.743
(96.21**)
0.327
0.694
(69.69**)
0.351
0.731
(73.54**)
6
(DGP 2, t5)
0.036
0.695
(19.30**)
0.023
0.706
(15.29**)
0.025
0.752
(16.06**)
0.462
0.924
(92.50**)
0.289
0.933
(63.72**)
0.321
0.997
(68.66**)
7
(DGP 3, t5)
0.012
0.605
(11.22**)
0.003
0.392
(5.51**)
0.005
0.518
(7.05**)
0.098
0.720
(32.94**)
0.028
0.501
(16.82**)
0.043
0.648
(21.21**)
8
(DGP 4, t5)
0.028
0.799
(16.92**)
0.003
0.366
(5.56**)
0.007
0.635
(8.44**)
0.144
0.947
(41.01**)
0.016
0.430
(12.65**)
0.035
0.731
(18.93**)
34
Panel C: Forecast horizon of 1-20 steps ahead
Cumulative Squared Return Innovations (of sub-intervals) regressed on
Predicted Conditional Variances (of original intervals)
Case
(DGP No,
No
Disturbance)
Procedure AF
2
rAF
1
(DGP 1, N(0,1))
0.021
2
(DGP 2, N(0,1))
0.006
3
(DGP 3, N(0,1))
0.002
4
(DGP 4, N(0,1))
0.002
5
(DGP 1, t5)
0.012
6
(DGP 2, t5)
0.006
7
(DGP 3, t5)
0.002
8
(DGP 4, t5)
0.002
â1
(t- ratio)
0.778
(14.75**)
0.587
(7.60**)
0.241
(4.25**)
0.252
(4.18**)
0.478
(10.84**)
0.547
(8.01**)
0.319
(4.01**)
0.326
(3.90**)
Procedure AD
â1
2
rAD
(t- ratio)
0.603
(10.56**)
0.308
(4.33**)
0.088
(2.16*)
0.096
(2.36*)
0.463
(9.58**)
0.522
(7.08**)
0.112
(1.46)
0.040
(0.53)
0.011
0.002
0.0004
0.0006
0.009
0.005
0.0002
2.79E-05
H
Notes: 1. The linear regression equations for these r2 are given by
Procedure AM
2
rAM
0.011
0.002
0.0003
0.0003
0.009
0.004
0.0004
0.0001
5
â1
(t- ratio)
0.671
(10.68**)
0.399
(4.56**)
0.107
(1.68)
0.109
(1.68)
0.475
(9.64**)
0.522
(6.53**)
0.195
(2.12*)
0.116
(1.19)
True Realised Ex-post Volatilities (of original intervals) regressed on
Predicted Conditional Variances (of original intervals)
Procedure AF
2
rAF
0.083
0.041
0.01
0.007
0.074
0.071
0.0097
0.012
H
  T2i5( s1), j  c1  a1   f Lows, _j i  u ,H , j and
s 1 i 1
s 1
â1
(t- ratio)
0.753
(30.02**)
0.540
(20.56**)
0.277
(9.91**)
0.275
(8.43**)
0.588
(28.18**)
0.643
(27.53**)
0.392
(9.92**)
0.415
(10.85**)
Procedure AD
â1
2
rAD
(t- ratio)
0.560
(20.28**)
0.279
(11.39**)
0.075
(3.71**)
0.080
(3.62**)
0.530
(22.92**)
0.537
(21.01**)
0.155
(4.06**)
0.073
(2.10*)
0.040
0.013
0.001
0.001
0.050
0.042
0.0016
0.0004
H
H
s 1
s 1
Procedure AM
2
rAM
0.044
0.015
0.001
0.001
0.054
0.043
0.003
0.001
â1
(t- ratio)
0.649
(21.41**)
0.372
(12.38**)
0.122
(3.87**)
0.124
(3.51**)
0.562
(23.88**)
0.587
(21.15**)
0.257
(5.61**)
0.170
(3.79**)
 h s, j  c2  a2   f Lows, _j i  v ,H , j , where i = AF,
AD or AM and H = 1 (one step ahead), 5 (one through five steps ahead) and 20 (one through twenty steps ahead).
2. ** denotes significance at the 1% level, and * at the 5% level.
3. Shaded areas denote the most accurate predictors amongst the three competing methods.
35
Table 8: In-sample forecast performances of the six predictors
Panel A: Average in-sample forecast r2 for USD/GBP, DEM/USD, and JPY/USD forex rate returns across the 20 rolling windows
Daily Volatility Forecasts vs. Cumulative Squared 5-min Return Innovations
Procedure AD
strong standard
GARCH(1,1)
Procedure AM
aggregated weak
GARCH(1,1)
strong standard
GARCH(1,1)
PGARCH(1,1)
IPCGARCH(1,1)
IPC- PGARCH(1,1)
2
rAD
2
rAM
2
rAF
2
rAF
2
rAF
2
rAF
USD/GBP
0.165
0.192
0.475
0.230
0.164
0.096
DEM/USD
0.161
0.192
0.580
0.394
0.133
0.083
JPY/USD
0.163
0.145
0.484
0.018
0.123
0.051
Data
Procedure AF
Panel B: Average in-sample forecast r2 for Microsoft’s equity returns across the 20 rolling windows
Hourly Volatility Forecasts vs. Cumulative Squared 1-min Return Innovations
Data
Microsoft’s
Equity
Procedure AD
strong standard
GARCH(1,1)
Procedure AM
aggregated weak
GARCH(1,1)
Procedure AF
strong standard
GARCH(1,1)
PGARCH(1,1)
IPCGARCH(1,1)
IPCPGARCH(1,1)
2
rAD
2
rAM
2
rAF
2
rAF
2
rAF
2
rAF
0.003
0.004
0.857
0.336
0.003
0.007
Note: Figures in the shaded areas denote the best predictors amongst the six competitors.
36
Table 9: Volatility forecast r2 for Procedures AF, AD and AM
Panel A: One-day-ahead out-of-sample volatility forecast r2 for USD/GBP, DEM/USD and JPY/USD forex rate returns
One-Day-Ahead Volatility Forecasts vs. Cumulative Squared 5-min Return Innovations
Data
Procedure AD
strong standard
GARCH(1,1)
2
rAD
USD/GBP
0.133
DEM/USD
0.177
JPY/USD
0.208
â
(t- ratio)
0.752**
(5.216)
0.778**
(6.536)
0.754**
(6.436)
Procedure AM
aggregated weak
GARCH(1,1)
â
2
rAM
0.176
0.191
0.192
(t- ratio)
0.822**
(6.147)
0.601**
(6.848)
0.650**
(6.119)
Procedure AF
strong standard
GARCH(1,1)
2
rAF
0.152
0.352
0.065
â
(t- ratio)
0.799**
(5.652)
0.952**
(10.340)
0.426**
(3.324)
PGARCH(1,1)
2
rAF
0.063
0.094
0.010
â
(t- ratio)
-0.207**
(3.455)
0.539**
(4.541)
0.074
(1.285)
IPCGARCH(1,1)
â
2
rAF
0.126
0.091
0.210
(t- ratio)
0.748**
(5.075)
0.595**
(4.452)
0.499**
(6.488)
IPCPGARCH(1,1)
2
rAF
0.023
0.058
0.061
â
(t- ratio)
0.236**
(2.050)
0.478**
(3.499)
0.184**
(3.202)
Panel B: One-hour-ahead out-of-sample volatility forecast r2 for Microsoft’s equity returns
One-Hour-Ahead Volatility Forecasts vs. Cumulative Squared 1-min Return Innovations
Data
Procedure AD
strong standard
GARCH(1,1)
2
rAD
Microsoft’s
Equity
Notes:
0.054
â
(t- ratio)
0.627**
(3.361)
Procedure AM
aggregated weak
GARCH(1,1)
2
rAM
0.055
â
(t- ratio)
0.615**
(3.378)
Procedure AF
strong standard
GARCH(1,1)
2
rAHF
0.082
â
(t- ratio)
0.580**
(4.204)
PGARCH(1,1)
2
rAF
0.077
â
(t- ratio)
0.881**
(4.069)
IPCGARCH(1,1)
2
rAF
0.0001
â
(t- ratio)
0.002
(0.171)
IPC- PGARCH(1,1)
2
rAF
0.052
â
(t- ratio)
0.477**
(3.274)
1.The linear regression equations for these r2 are given by Eqs. (3.12) and (3.13).
2. Figures in the shaded areas denote the best predictors amongst the six competitors.
3. ** denotes significance at the 5% level.
37
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