in High-Frequency Financial Returns
Yuanhua Feng
Department of Mathematics and Statistics
University of Konstanz
D-78457Konstanz, Germany
Email: Yuanhua.Feng@uni-konstanz.de
Abstract
This paper considers simultaneous modelling of seasonality, slowly changing un-
conditional variance and conditional heteroskedasticity in high-frequency nancial
returns. A new approach, called a seasonal SEMIGARCH model, is proposed to
perform thisbyintroducingmultiplicative seasonal andtrend componentsinto the
GARCH model. A data-driven semiparametricalgorithm is developed forestimat-
ing the model. Asymptotic properties of the proposed estimators are investigated
briey. An approximatesignicance testof seasonalityand theuseofMonte Carlo
condence bounds for the trend are proposed. Practical performance of the pro-
posalisinvestigatedindetailusingsomeGermanstockpricereturns. Theapproach
proposedhere provides ausefulsemiparametric extensionof theGARCHmodel.
Keywords: High-frequency nancial data, nonparametric regression, seasonality in
volatility,semiparametric GARCH model,trend involatility.
1 Introduction
Financialreturnsexhibitconditionalheteroskedasticity (CH).Wellknown approachesfor
modellingtheCHaretheARCH(autoregressiveconditionalheteroskedastic,Engle,1982),
GARCH (generalized ARCH, Bollerslev, 1986) models and their extensions. In spite of
theirconditionalheteroskedasticpropertythe ARCHand GARCHmodels are stationary
with constant unconditional variance and are hence time homoskedastic. In recent years
it is however realized that nancial returns also exhibit time heteroskedasticity (TH) or
hence the process is nolonger stationarybut at most localstationary.
Evidence of TH in nancial time series was reported in the literature together with
dierent approaches for modelling it. Mikosch and Starica (1999) showed that the phe-
nomenon ^
1 +
^
1
1 by a tted GARCH(1, 1) model indicates nonstationarity and
proposed the use of a piecewise GARCH model. Beran and Ocker (2001) tted SEMI-
FAR(semiparametricfractionalautoregressive)modelsproposedbyBeran(1999)tosome
volatility series and found signicanttrend involatility. Hardle et al. (2001) introduced
a time-inhomogeneous stochastic volatility model with time varying coeÆcients. A lo-
cal time-homogeneous model with change points is proposed by Mercurio and Spokoiny
(2002),where the volatility isassumed to beconstant inan unknown localtime interval.
A semiparametric GARCH model with a slowly changing scale function (called SEMI-
GARCH) isproposed by Feng (2002)for simultaneously modelling the CH and TH.
The currentpaper considers modellingof volatility inhigh-frequency nancialreturns.
Now the volatility also exhibits daily periodicity (see e.g. Dacorogna et al., 2001). For
modellingdierent componentsinthe volatilityof high-frequency nancialreturnsanew
approach,calledaseasonalSEMIGARCHmodel,isproposedbyintroducinganadditional
multiplicative seasonalcomponent intothe SEMIGARCH model, which extendsthe tra-
ditionalcomponent model of economic time series to the current context. A data-driven
semiparametric algorithmis developed for estimating the model. Although the focus of
this paperisonapplications, necessary asymptoticproperties ofthe proposed estimators
are investigated briey. An approximate signicance test of seasonality and the use of
MonteCarlo condence boundsfor the trend are proposed. Practical performance of the
proposalisinvestigatedindetailusingsomeGermanstock pricereturns. Itisshown that
the proposalworks wellinpractice. Thisnew approachprovides auseful semiparametric
extensionof the wellknown GARCH model.
The paper is organized as follows. Section 2 introduces the model and proposes the
data-driven semiparametricalgorithm. Asymptoticpropertiesofthe proposed estimators
are investigated in Section3. Section 4 describes the signicance test of seasonality and
the Monte Carlo condence bounds for the trend. Applications and discussion on the
practicalperformance of the proposal are given in Section5. Final remarks in Section6
conclude the paper. Proofs of results are put inthe appendix.
2.1 The model
Assume that the log-returns of a nancialtime series follow the model:
Y
i
=+V 1=2
0 v
1=2
(t
i )S
1=2
i h
1=2
i
i
; (1)
where t
i
= i=n is the re-scaled time which guarantees the availability of consistent esti-
mation,V
0
>0isaconstant, v()>0isasmooth, bounded function, S
i
>0is aperiodic
functionwith periodT and
i :=h
1=2
i
i
is assumed to followa GARCH(p, q) modelwith
h
i
=
0 +
p
X
j=1
j
2
i j +
q
X
k=1
k h
i k
(2)
(Bollerslev, 1986). (t) = v 1=2
(t) is called the scale (or volatility trend) function. Let
= (
0
;
1
;:::;
p
;
1
;:::;
q )
0
. It is assumed that
0
> 0,
1
;:::;
p
;
1
;:::;
q
0 and
P
p
i=1
i +
P
q
j=1
j
< 1, which guarantees the existence of a unique strictly stationary
solutionof(2). WithoutlossofgeneralityletT 1
P
T
i=1 S
i
=1, R
1
0
v(t)dt =1andvar(
i )=
1. The last condition implies
0
=1 P
p
i=1
i P
q
j=1
j
. An equivalent representation
of model(1)and (2) is
Y
i
=+V 1=2
0 v
1=2
(t
i )S
1=2
i
i
: (3)
The momentconditions E(
4
i
)<1and E(
8
i
)<1are required for the derivation of the
asymptoticresultsand thepracticalimplementationof anonparametricestimatorofv()
respectively. Necessary and suÆcient conditions which guarantee the existence of high
ordermomentsofaGARCHprocessmaybefounde.g. inLingandLi(1997),Ling(1999)
and Lingand McAleer (2002). See alsoBollerslev(1986) forresults inthe special case of
aGARCH(1, 1) model.
Model(1)and(2) isa semiparametricextensionof the GARCHmodel, which provides
atoolforsimultaneousmodellingofconditionalheteroskedasticity (h
i
),slowscalechange
(v())and seasonality (S
i
)inhigh-frequency nancialreturns. V
0
quanties the averaged
unconditionalvariance of Y
i
. And the total volatility of Y
i
is hence V 1=2
0 v
1=2
(t
i )S
1=2
i h
1=2
i .
NotethatV 1=2
0
isthe absolutelevelofthestandarddeviation. Allotherthreecomponents
are dened relatively and move around the unit level. The introduction of the trend
i
GARCHmodeltoawideclassofnonstationaryprocess. Ifthelocalvarianceofthereturn
serieschangesovertimeand/orifthereisanonconstantperiodictermintheunconditional
variance, then the use of model (1) and (2) will lead to signicantly theoretical and
practicalimprovements.
Furthermore, dene Z
i
= (Y
i
), X
i
= Z
i
= p
V
0 , R
i
= Z
i
= p
V
0 S
i
and
i
= ( 2
i 1).
Then we have
Z 2
i
=V
0 v(t
i )S
i +V
0 v(t
i )S
i
i
; (4)
X 2
i
=v(t
i )S
i +v(t
i )S
i
i
: (5)
and
R 2
i
=v(t
i
)+v(t
i )
i
: (6)
Model(6)isanonparametricregressionwithdependentandheteroskedasticerrors,which
indicatesthatv()canbeestimatedusingwellknowntechniquesinnonparametricregres-
sion (see e.g. Efromovich,1999 and Feng, 2002).
The assumption that
i
follow a GARCH model is made here for simplicity, which
allows ustouse wellknown theoreticalresults forinvestigatingthe propertiesof the pro-
posedmodelandtoestimatetheconditionalheteroskedasticity usinganexistingGARCH
packet. Thisassumptionishowevernotnecessaryandcanbereplacedbyotherparametric
ornonparametric assumptions.
2.2 Estimation of the model
If S
i
1, equations (1) and (2) reduce to the SEMIGARCH modelintroduced by Feng
(2002). Estimation of the SEMIGARCH model was investigated there in detail. The
transformationfrom(4)to(6)shows that,ifV
0 andS
i
canbeestimatedproperlywithout
pre-estimationof v()and ,then v()and canbe estimatedfromthe seasonaladjusted
data following the proposal of Feng (2002). In this paper a semiparametric estimation
procedure followingthis idea willbe proposed.
Let y
1
;y
2
;:::;y
n
denote the observations. At rst can be estimated by the sample
mean y. And one obtains the centralized observations z^
i
= y
i
y. An estimator of the
0
^
V
0
=n 1
n
X
i=1
^ z 2
i
: (7)
Nowdenex^
i
=z^
i
= p
^
V
0
. Letm=[n=T],where[]denotes theintegerpart. Theseasonal
component can be estimated fromx^ 2
i
as follows
^
S
i
=m 1
m 1
X
j=0
^ x 2
jT+i
; i=1;2;:::;T; (8)
and
^
S
i
=
^
S
i kT
fori>T,wherek =[(i 1)=T]. In thenext sectionitwillbeshown that
the eect ofthe unknown volatilitytrendon
^
S
i
isnegligible. That is,the pre-eliminating
of v() is not necessary for estimating S
i
. Alternatively, v() can also be estimated from
^ x
i
without seasonaladjustment. However, there aretwodisadvantages, ifthis is done: 1.
Oneis faced with abandwidthselectionproblemina modelwith periodicerrors. 2. The
errorinthe nonparametricestimatev()^ willcauseanon-negligiblebias in
^
S
i
. Hence, this
alternative estimation procedure willnot beconsidered here.
After obtaining
^
V
0 and
^
S
i
, dene r^
i
= z^
i
= p
^
V
0
^
S
i
to be the standardized, seasonal
adjusted data. Let K(u) denote a second order kernel function. Following Feng and
Heiler(1998) and Feng (2002),a Nadaraya-Watson kernel estimatorof v(t)is given by
^ v(t)=
P
n
i=1 K(
t
i t
b )^r
2
i
P
n
i=1 K(
t
i t
b )
=:
n
X
i=1 w
i
^ r 2
i
; (9)
wherew
i
=K(
t
i t
b )[
P
n
i=1 K(
t
i t
b )]
1
andbisthebandwidth. Andwedene(t)^ =v^ 1=2
(t).
Itisassumedthatb !0,nb!1asn!1,whichtogetherwithotherregularconditions
ensures the consistency of v()^ or().^
Finally,dene the standardized residualsby
^
i
=z^
i
= q
^
V
0
^ v(t
i )
^
S
i :
Following the idea for estimating the parameters in the SEMIFAR model (Beran, 1999
and Beran and Feng, 2002), it is proposed to estimate using the maximum likelihood
estimator (MLE) of Bollerslev (1986) with
i
there being replaced by ^
i
. That is,
^
is
dened as the maximizer of the (approximate) conditional log-likelihood (apart from a
constant)
L()= 1
n n
X
i=1 l
i
() (10)
l
i ()=
1
2 logh
i 1
2h
i
^ 2
i
: (11)
It is assumed that ^
i
= 0 and h
i
= ^ 2
i
= 1
n P
n
j=1
^ 2
j
for i 0. This will not aect the
asymptotic properties of
^
. For more details about the MLE of see Bollerslev (1986)
and Lingand Li (1997). Forcomputing
^
we propose to use the S+GARCH packet.
Asymptotic properties of y,
^
V
0 and
^
S
i
given in the next section ensure that the data-
driven SEMIGARCH algorithmproposed by Feng (2002) can be directly adapted to es-
timate the seasonal SEMIGARCH model. Such a data-driven algorithm processes as
follows:
1. Estimate by y.
2. EstimateV
0
and S
i by
^
V
0 and
^
S
i
asdened above.
3. Run the SEMIGARCH program (in S-Plus) using the standardized, seasonal ad-
justed data r^
i
. Thenweobtain data-driven estimation of v()and .
Formore details see Feng (2002).
The return between the last observation on one day and the rst observation on the
next day is called the overnight return. When analyzing high-frequency data one is also
faced with the problem of how to deal with the overnight returns, because they are
quite dierent from the intraday returns, i.e. those between observations on the same
day. Here, we propose to carry out the proposed algorithm twice with and without the
overnight returns. By the former approach the eect of the overnight returns may be
estimated, since now the overnight returns are treated as returns in a special phase of
the daily period. However, including the overnight returns will cause underestimation
of the seasonality in intraday returns. The latter approach provides us more detailed
informationabout the daily periodicchange in the volatility of the intraday returns.
3 Asymptotic properties
The practical implementation proposed in the last section based on some asymptotic
properties of the proposed estimators, which willbe discussed in this section briey. For
A1. Model (1) and (2) holds with i.i.d. N(0;1)
i
and strictly stationary
i
such that
E(
4
i
)<1.
A2. Thetrendfunctionv(t)isstrictlypositive,bounded andatleasttwice continuously
dierentiable on[0;1].
A3. S
i
are strictly positive and exactlyperiodicwith periodT.
The condition of E(
4
i
) < 1 is suÆcient for the derivation of the asymptoticproperties
of the proposed estimators. However, for the practicalimplementation, E(
4
i
) has to be
estimated (see Feng, 2002). Now, the existence of nite eighth moment of
i
is required.
The asymptoticproperties of yare given inthe following proposition.
Proposition 1 . Under assumptions A1 to A3 we have E(y) = and p
n(y ) D
!
N(0;V
0
), where D
! denotes convergence in distribution.
TheproofofProposition1isgivenintheappendix. Noteinparticularthattheasymptotic
variance of y does not depend on v() and S
i
. We see y is unbiased and has the same
asymptoticvarianceasthatofthesamplemeanofani.i.d. serieswithvarianceV
0
,because
i
are uncorrelated and v(),S
i and
i
are all standardized.
The asymptoticproperties of
^
V
0
are given by
Theorem 1. Under assumptions A1 to A3 we have
i) E(
^
V
0 V
0
)=O(n 1
), var(
^
V
0 )
:
=n 1
2
V
0 .
ii) p
n (
^
V
0 V
0 )
D
!N(0;
2
V0 ),
where
2
V
0
=(nT) 1
V 2
0 Z
1
0 v
2
(t)dt (
T 1
X
j=0 a
j b
j )
(12)
a
j
= T
X
i=1 S
i S
i+j +S
i j
2
;j =0;1;:::;T 1; (13)
b
0
= 1
X
k= 1
2
(kT) (14)
and
b
j
=2 1
X
k=0
2(kT +j);j =1;:::;T 1; (15)
where
2(k)
are the autocovariances of the squared GARCH process 2
i .
The proof of Theorem 1 is given in the appendix. If v(t) 1 and S
i
1, model (1)
and (2)reduces tothe GARCHmodel. Now, resultsinTheorem 1reduce toknown limit
theorem on the sample variance of a GARCH process (see e.g. Davis et al. 1999 and
Mikosch and Starica, 2000).
Asymptotic properties of
^
S
i
are given by the following theorem.
Theorem 2. Under assumptions A1 to A3 we have
i) E(
^
S
i S
i
)=O(m 1
)=O(n 1
).
ii) The asymptotic variance of
^
S
i
is givenby
var(
^
S
i )
:
=m 1
c
f S
2
i Z
1
0 v
2
(t)dt=m 1
2
Si
; (16)
where 2
Si
=c
f S
2
i R
1
0 v
2
(t)dt, c
f
= P
1
i= 1
2(iT) and
2(k) are as dened in The-
orem 1.
iii) p
n (
^
S
i S
i )
D
!N(0;T 2
S
i ).
TheproofofTheorem 2isstraightforwardand isomitted. Theorem 2shows inparticular
thatS
i
canbeestimated p
n-consistentlywithoutpre-eliminatingthe volatilitytrendand
the GARCH eect.
Note that y,
^
V
0 and
^
S
i
are all p
n-consistent. Hence, under conditions A1 to A3
and additionalregularconditions innonparametric regression, the asymptoticproperties
of v()^ and as given in Theorems 1 to 3 in Feng (2002) hold for the corresponding
estimatorsproposedinthe lastsection. Thisisthereason,whythedata-drivenalgorithm
for estimating the SEMIGARCHmodel(Feng, 2002) can be directly used for estimating
model(1)and (2) afterseasonal adjustment.
4 Signicance test and condence bounds
4.1 An approximate signicance test of seasonality
Animportantquestionis,whethertheseasonalcomponentinareturnseriesissignicant.
Toanswer this questiona test should becarried out. In the followingwe willpropose an
approximate signicance test of the nullhypotheses
H i
0 :S
i
=1;i=1;2;:::;T:
It isproposed to reject H i
0
atthe level , if
p
m
^
S
i 1
>z
1
=2
^
S
i s
^ c
f Z
^ v 2
(t)dt;
wherez
1
=2
is the N(0;1)-
1
=2-quantileand
^ c
f
= K
X
i= K
^
2(iT)
with an integer K such that K !1 and K=m ! 0 as n !1. This condition ensures
that P
K
i= K
^
2(iT) is consistent.
1
is chosen so that the joint signicance level of the
test is about . Hence
1
should be much smaller than . If the correlation between
^
S
i
, i=1;2;:::;T, is omitted,then anapproximate value of
1
may be obtained from the
relationship(1
1 )
T
=(1 ). One side tests can be carriedout similarly.
4.2 Monte Carlo condence bounds
Anotherquestion is, if there isa volatilitytrend in the data. This meansthat we should
test the null hypothesis H
0
: v(t) 1 or give condence bounds of v()^ under H
0 . In
method (see Feng, 2002 for a similar idea). The use of Monte Carlo condence bounds
in nonparametric regression is also proposed e.g. by Efromovich (1999). Assume that
we have obtained a tted SEMIGARCH modelfromthe standardized, seasonaladjusted
data. The 100(1 )%Monte Carlo condence bounds are obtained asfollows.
1. Generate a time series of lengthn following the estimated GARCHmodel.
2. Fit aSEMIGARCH modeltothe simulateddata using the bandwidth
^
b .
3. Repeatedly carry out steps 1 and 2 untila given number of replications.
4. Find out proper lower and upper bounds so that the number of estimated trends,
which exceed these bounds atsome places,is not larger than 100 %.
The nullhypothesisH
0
:v(t)1 willberejected atlevel ,if v(t)^ obtained fromthe real
data exceeds these simulated condence bounds at some places. The condence level is
asymptotically (1 ), since
^
is consistent. Here, the bandwidth
^
b is used to keep the
estimated trends from dierent replications to be comparable with each other and with
^
v()obtainedfromtherealdata. Thecondenceboundsinthispaperaredeterminedsuch
that the numbers of estimated trends which exceed the lower and the upper bounds are
the same. Note also that for calculating the total number of exceeding estimates those,
which exceedthe lowerand the upperbounds atthesame time,should not becalculated
twice.
5 Applications
In the following we will apply the proposal to the 20 minute stock price returns (log-
returns)offourGermanrms: AllianzAG,BASFAG,HenkelKGaAandLinde AG.The
dataaretheobservationsoftheGermanXetraelectronictradingsystem. Theobservation
periodis from November 28, 1997 to December 30,1999 including524 observation days.
Here T = 24 for all returns and T = 23 for intraday returns only. For the parametric
part aGARCH(1, 1) modelisused. The estimated parameters p
^
V
0 ,
^
b, ^
1 and
^
1
for all
examplesare giveninTable1. Wesee that p
^
V
0
withovernightreturnsislargerthanthat
overnightreturns ismuchlargerthan the volatilityinany otherphase (seeFigures1to5
below). The selected bandwidthsforthe samedata set withorwithoutovernightreturns
are quite similar, because v() is estimated from the seasonal adjusted data. The tted
GARCH models are all highly signicant. The dierence between the tted GARCH
models in cases with or without overnight returns is not clear. Following the existence
condition for nite high order moments of a GARCH(1, 1) model given by Bollerslev
(1986),it can bechecked that allof the tted GARCH modelshave atleast nite eighth
moment. This together with the results given below shows that nancial return series
often have nite high order momentsbut are ingeneral nonstationary (see Mikosch and
Starica, 1999 and Feng, 2002 for related ndings).
Results of one side signicance tests of H i
0 : S
i
= 1 against H i
1 : S
i
> 1 (for
^
S
i
>1) or
H i
1 :S
i
<1 (for
^
S
i
<1), i=1;2;:::;T, are listed in Table 2 where the codes \1", \0" and
\-1"stand forS
i
>1,S
i
=1andS
i
<1respectively. Inthesetests
1
=0:0022wasused
so that 0:05. Here only results in cases without overnight returns are given. The
observation time intervals are: 9:20{9:40, 9:40{10:00, ..., 16:20{16:40 and 16:40{17:00.
For calculating c^
f
, K = [ p
n =T +0:5] = [ p
12052=23+0:5] = 5 is used. These results
show that the seasonality is for all examples signicant. Observing the Monte Carlo
condence bounds for v()^ shown in Figures1 to5, we can see that h
i
, v(t) as well asS
i
are signicantly non-constant for allexamples.
Detailedresults obtainedfollowingthe seasonalSEMIGARCH modelfor theBASF re-
turnseriesareshowninFigures1and2. Thedataforthisserieswithovernightreturnsare
shown inFigure1a. Estimationresultsforthis example are shown inFigures1bthrough
h. The estimated seasonalcomponent in Figure1b shows that the overnight returns are
clearly dierent from those in other phases. The estimated trend v()^ is shown in Fig-
ure 1c together with 95% (long dashes) and 99% (short dashes) Monte Carlo condence
boundscalculatedfrom400 replications. Figure1dshowsthe standardizedresidualsfrom
which the GARCH(1, 1) model was tted. The conditional standard deviations calcu-
lated following this GARCH modelare shown in Figure 1e. Figure 1f displays the total
volatility, i.e. the product of the three components shown in Figures 1b, c and e and
the averagedstandard deviation p
^
V
0
. The zoomedtotal standard deviationsfor the last
ten days are shown inFigure 1g. Figure1h shows the prediction of the volatility for ve
days inthe future, wherethe estimated trendforthe lastobservation, i.e. ^v(1), isused
as the scale function in the recent future, the averaged standard deviation is assumed
to be unchanged and the prediction of the conditional standard deviations is obtained
usingthe S-Plus GARCHfunction predict(see Martin etal.,1996). Wesee, the GARCH
eectinthepredictiondecayveryquicklyandhencetheseasonalcomponentplaysamore
important role.
Figure 2 shows the same results as given in Figure 1 but for the BASF return series
withoutovernightreturns. Wesee, the seasonalityinthis caseis moreregular. Notethat
the dierence between the two values of
^
S
i
obtained with and without overnight returns
is only due to the dierence of
^
V
0
in these two cases. The estimated trends in these two
cases are almost the same. Estimation results for the Allianz, Henkel and Linde return
seriesare shown inFigures3to5respectivelyforcases with(Figuresatod)and without
(Figures e to h) overnight returns, where some details are omitted to save space. From
Figures2 to 5we see that the seasonal component inthe case withoutovernight returns
has a \U" form over one day. That is the volatility near the open and close time is
generally larger than that near the noon. But the change from one phase to another is
not smooth, especiallyby the Henkel and Linde returns. The seasonality ismost regular
by the Allianzreturns.
The tted trend in the considered time period has a \ form. That is the volatility is
largerinthemiddleofthisobservationperiodandsmallatbothends. Thevolatilitytrend
issmallest atthe current end of these series except for the BASF returns. This property
is important for predicting the future volatility, because it shows that the non-seasonal
unconditionalvarianceatthecurrent endismuchsmallerthanthe averagedlevel. Hence
one can obtain more reasonable prediction for future volatility by introducing the trend
functionintothe parametric GARCHmodel.
Figures 1 to 5 also show that all the condence bounds for v()^ are not symmetric.
The distance between the upperbound and the unit level is always larger than that be-
tween the lower bound and the unit level. This means that the estimated trend from
data generatedby a GARCH modelwithouttrendoftenhas somelarger peaks. Further-
more, followingthe asymptoticnormalityof v(t)^ (see Feng, 2002) we can easilycalculate
condence intervals for the trend at a given point t
0
. However, this does not provide
correctcondence bounds for the trend function onthe whole support [0;1]. The length
acorrespondingcondence intervalof the trend ata given point.
The idea behind the seasonalSEMIGARCH modelcan bewell understood,if we com-
pare the ACF's of dierent time series transformed from the same return series. Let
^ x
i
= z^
i
= p
^
V
0 , r^
i
= z^
i
= p
^
V
0
^
S
i
= x^
i
= p
^
S
i
and ^
i
= z^
i
= q
^
V
0
^ v(t
i )
^
S
i
= x^
i
= q
^ v(t
i )
^
S
i
as de-
ned in Section 2. Note that the time series x^
i
should have both trend and seasonality
in volatility as for the return series itself. r^
i
are seasonal adjusted data and should only
have trend in volatility. ^
i
are consistent estimates of
i
and should have neither trend
nor seasonality involatility. Also dene
^
i
=x^
i
= p
^ v(t
i
)tobethe trend adjusteddata for
comparison,which shouldhave seasonality involatility.
The ACF's of j^x
i j, j^r
i j, j
^
i
j as well as j ^
i
j in all cases are displayed in Figures 6 and 7.
Figures 6a to d show these results for the Allianz return series with overnight returns.
The same results for the Allianzreturn series without overnight returns are displayed in
Figures 6e to h. Figures 6i to p, Figures 7a to h and Figures 7i to p show the same
results as those in Figures 6a to h, but for the BASF, Henkel and Linde return series
respectively. Both, the trend and seasonal eects can be seen clearly from the ACF's of
j^x
i
j. The ACF's of j^r
i
j exhibit only trend eect. This means that the seasonality is well
modelledand eliminatedfollowingthe proposed algorithm,inbothcases with orwithout
overnightreturns. Notealsothatthetrendeectbecomesmoreclearaftereliminatingthe
seasonality. The ACF's of j
^
i
j exhibit only seasonaleect asexpected. That is the trend
isproperlyestimatedand eliminatedfromthese series. Notethat allof the ACF's ofj^x
i j,
j^r
i
j and j
^
i
j indicate non-stationarity in these series. Again, we see that the seasonality
inthevolatilityofthe Henkel andLinde returnsisnotsoregularasthat inthe othertwo
returnseries. But the volatility trendin these two series is moreclear.
The ACF's of j ^
i
j displayed in Figures d, h, l and p of Figures 6 and 7 show that
these seriesseem tobestationaryand that thereis clear GARCHeect in the dataafter
eliminatingboth the seasonality and the trend. Stationary CH models, for instance the
GARCHmodelasconsideredinthispaper, canthenbettedtotheseseasonalandtrend
adjusted series. Figures 1dand h show that the persistence level in ^
i
is sometimes very
high. This indicates that a CH model with long-memory property is sometimes more
preferable. However, this is not considered here.
Inthis paperthe estimationof dierent volatilitycomponents inhigh-frequencynancial
returns is investigated. A new approach to perform this is introduced and the data-
driven algorithmproposed byFeng (2002)isadapted toestimatethemodelinthis paper.
Asymptotic results, signicance test of seasonality and Monte Carlo condence bounds
of the trend are investigated. Data examples show that the proposal works wellin prac-
tice. However, there are still some open questions including the development of a joint
signicance test of seasonality and the development of a theoretical signicance test of
the whole trend function. The latter is also an important open question in standard
nonparametric regression. Alsothe problemof themodelselectionis not discussed. This
problem may perhaps be solved by using the AIC or BIC information criteria. Finally,
it is worthwhile to extend the idea in Feng (2002) and in this paper to other GARCH
variants,e.g. FARIMA-GARCHmodel.
Acknowledgements
ThepaperwasnanciallysupportedbytheCenterof FinanceandEconometrics(CoFE),
University of Konstanz, Gernamy. We are very grateful toProf. JanBeran, Department
ofMathematicsandStatistics,UniversityofKonstanz,fortheadvice. Wewouldverylike
tothankProf. WinfriedPohlmeierandMr. NikolausHautsch,DepartmentofEconomics,
University of Konstanz, for providing us the data. Without their help this paper would
not be nished. Our special thanks go to Mr. Erik Luders, CoFE/ZEW, for helpful
discussions and suggestions, which lead to improve the quality of this paper. Finally,
we are grateful to Prof. Wolfgang Hardle, Prof. Olaf Bunke and Dr. Woocheol Kim,
Humboldt University, Berlin, Prof. Vladimir Spokoiny, Weierstra-Institut, Berlin, and
Prof. Rohit Deo, New York Uniniversity and Humboldt University,for useful comments.
Proof of Proposition 1. It is obvious that y is unbiased. Hence we need to check
var(y)and theasymptoticnormalityofy. Notethattheautocovariancesof
i are
(0)=
1and
(k)=0for jkj>0, since
i
isa standardized GARCH process. We have
var(y)=n 2
V
0 n
X
i=1 v(t
i )S
i
: (A.1)
Let m= [n=T] asdened in (8). Note that v(t
i )
:
=v(t
j
) for ji jj< T and observe the
standardizingassumptions onS
i
and v(t). We have
n 1
n
X
i=1 v(t
i )S
i :
= n 1
m 1
X
j=0 T
X
k=1 v(t
jT+k )S
jT+k
:
= n 1
m 1
X
j=0 v(t
jT+1 )
T
X
k=1 S
jT+k
= n 1
T m 1
X
j=0 v(t
jT+1 )
:
= Z
1
0
v(t)dt: (A.2)
Under the assumptions we have R
1
0
v(t)dt=1,that is var(y) :
=n 1
V
0 .
Furthermore, under the assumptions of model (1) and (2) it can be shown that y is
asymptotically normal, if and only if the sample mean of the GARCH process
i is. It
is well known that the sample mean of a GARCH process with nite fourth moment is
asymptoticallynormal(see e.g. Beranand Feng, 2001). Proposition 1is proved. 3
Proof of Theorem 1. Followingthe results of Proposition 1it can be shown that
E(
^
V
0 )=n
1 n
X
i=1 E[Z
2
i
]+O(n 1
); (A.3)
whereZ
i
=Y
i
are as dened in Section2.
n 1
n
X
i=1 E[Z
2
i
] = n 1
V
0 n
X
i=1 v(t
i )S
i var(
i )
= n 1
V
0 n
X
i=1 v(t
i )S
i
= V
0
+O(n 1
): (A.4)
The lastequation isdue to the same argumentused in (A.2). One obtainsE(
^
V
0 V
0 )=
O(n 1
).
Since yisconsistent, we have for the varianceof V
0
var(
^
V
0 )
:
= n 2
var n
X
i=1 Z
2
i
!
:
= n 2
V 2
0 n
X
i=1 n
X
j=1 v(t
i )v(t
j )S
i S
j
2(i j): (A.5)
The autocovariances
2(k) of the squared GARCH process 2
i
decay exponentially (see
e.g. He and Terasvirta, 1999). Hence P
n
j=1 v(t
i )v(t
j )S
i S
j
2(i
j) converges absolutely.
Let h > 0 such that h ! 0 and nh ! 1 as n ! 1. And let N
n
= [nh]. We have
P
n
j=1 v(t
i )v(t
j )S
i S
j
2(i j) :
= P
ji jjN
n v(t
i )v(t
j )S
i S
j
2(i j) and v(t
i )
:
= v(t
j ) for
ji jjN
n
. This analysis leads to
var(
^
V
0 )
:
= n 2
V 2
0 n N
n
X
i=Nn n
X
j=1 v(t
i )v(t
j )S
i S
j
2
(i j)
:
= n 2
V 2
0 n N
n
X
i=Nn X
ji jjNn v(t
i )v(t
j )S
i S
j
2
(i j)
:
= n 2
V 2
0 n N
n
X
i=Nn v
2
(t
i )
X
ji jjNn S
i S
j
2
(i j)
:
= n 2
V 2
0 n N
n
X
i=Nn v
2
(t
i )
1
X
k= 1 S
i S
i k
2
(k): (A.6)
Furthermore, note that P
1
k= 1 S
i S
i k
2(k) is periodic ini with the same period T and
v 2
(t
i )
:
=v 2
(t
j
)for ji jj<T. Let M
1
=[N
n
=T]and M
2
=[(n N
n
)=T]. We have
var(
^
V
0 )
:
= n 2
V 2
0 M
2
X
j=M
1 v
2
(t
jT )
(
T
X
i=1 1
X
k= 1 S
i S
i k
2(k)
)
:
= (nT) 1
V 2
0 Z
1
0 v
2
(t)dt (
T
X
i=1 1
X
k= 1 S
i S
i k
2
(k) )
: (A.7)
Straightforward calculation leads to
(
T
X
i=1 1
X
k= 1 S
i S
i k
2
(k) )
= T 1
X
j=0 a
j b
j
; (A.8)
wherea
j and b
j
, j =0;1;:::;T 1, are as dened in Theorem 1.
Note again that
^
V
0
is asymptotically normally distributed, if and only if the sample
variance of the GARCH process
i
is. The latter result is shown by Davis et al. (1999)
(see alsoMikosch and Starica, 1999 and Feng, 2002). Theorem 1 isproved. 3
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Kuchenho, Heidelberg: Physica-Verlag.
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University.
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Econometric Theory,15, 824{846.
Ling, S. (1999), \On probability properties of a double threshold ARMA conditional
heteroskedasticity model,"Journal of Applied Probability, 36,688{705.
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Ling, S. and McAleer, M. (2002), \Necessary and suÆcient moment conditions for the
GARCH(r,s)andasymmetricpowerGARCH(r,s)models,"EconometricTheory,18,
722{729.
Mercurio, D. and Spokoiny, V. (2002), \Statistical inference for time-inhomogeneous
volatility models," Discussion Paper, SFB 373, Humboldt University.
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process," Annals of Statistics, 28,1427{1451.
Forallreturns Intraday returns only
Allianz BASF Henkel Linde Allianz BASF Henkel Linde
^
b
opt
0.1588 0.1150 0.1196 0.1262 0.1567 0.1113 0.1213 0.1268
p
^
V
0
0.0050 0.0047 0.0063 0.0063 0.0045 0.0042 0.0058 0.0057
^
1
0.0592 0.0773 0.1232 0.0911 0.0631 0.0769 0.1473 0.0991
^
1
0.8988 0.8380 0.7468 0.8542 0.8908 0.8305 0.6935 0.8379
Table 2. Relative volatility strength during one day (intraday returnsonly)
Allianz 9:00 - 13:00 | 1 1 1 1 0 0 -1 -1 -1 -1 -1
13:00 - 17:00 -1 -1 -1 -1 0 -1 -1 -1 1 1 1 1
BASF 9:00 - 13:00 | 1 1 1 1 0 -1 -1 -1 0 -1 0
13:00 - 17:00 -1 -1 -1 -1 0 0 -1 -1 0 1 1 1
Henkel 9:00 - 13:00 | 1 1 0 1 0 0 0 -1 -1 0 -1
13:00 - 17:00 -1 -1 -1 0 -1 -1 -1 -1 1 1 0 1
Linde 9:00 - 13:00 | 1 1 1 1 0 0 -1 -1 0 0 0
13:00 - 17:00 -1 -1 -1 -1 -1 -1 -1 0 1 0 0 1
0 2000 4000 6000 8000 10000 12000
-0.02 0.0 0.02 0.04
(a) BASF 20 minute returns, all
10 20
1.0 1.5 2.0 2.5
(b) The seasonal component (square root)
0 2000 4000 6000 8000 10000 12000
0.8 0.9 1.0 1.1 1.2 1.3
(c) The scale function
0 2000 4000 6000 8000 10000 12000
-5 0 5 10
(d) The standardized residuals
0 2000 4000 6000 8000 10000 12000
1.0 1.5 2.0 2.5 3.0
(e) The conditional standard deviations
0 2000 4000 6000 8000 10000 12000
0.005 0.010 0.015 0.020
(f) The total standard deviations
0 50 100 150 200
0.004 0.008 0.012
(g) Total SD’s for the last 10 days
0 20 40 60 80 100 120
0.004 0.006 0.008 0.010
(h) Prediction of the SD’s for 5 days
Figure 1: Estimationresults for the BASF returns(with overnight returns).
0 2000 4000 6000 8000 10000 12000
-0.02 0.0 0.02 0.04
(a) BASF 20 minute returns, intraday only
10 20
0.8 1.0 1.2 1.4
(b) The seasonal component (square root)
0 2000 4000 6000 8000 10000 12000
0.8 0.9 1.0 1.1 1.2 1.3
(c) The scale function
0 2000 4000 6000 8000 10000 12000
-5 0 5 10
(d) The standardized residuals
0 2000 4000 6000 8000 10000 12000
1.0 1.5 2.0 2.5 3.0
(e) The conditional standard deviations
0 2000 4000 6000 8000 10000 12000
0.002 0.006 0.010
(f) The total standard deviations
0 50 100 150 200
0.003 0.005 0.007
(g) Total SD’s for the last 10 days
0 20 40 60 80 100
0.0030 0.0040 0.0050
(h) Prediction of the SD’s for 5 days
Figure2: Estimationresultsfor the BASF returns (intraday returns only).
0 2000 4000 6000 8000 10000 12000
-0.04 0.0 0.04 0.08
(a) Allianz 20 minute returns, all
10 20
1.0 1.5 2.0 2.5
(b) The seasonal component (square root)
0 2000 4000 6000 8000 10000 12000
0.8 0.9 1.0 1.1 1.2
(c) The scale function
0 2000 4000 6000 8000 10000 12000
0.01 0.02 0.03
(d) The total standard deviations
0 2000 4000 6000 8000 10000 12000
-0.04 -0.02 0.0 0.02
(e) Allianz 20 minute returns, intraday only
10 20
0.8 1.0 1.2 1.4
(f) The seasonal component (square root)
0 2000 4000 6000 8000 10000 12000
0.8 0.9 1.0 1.1 1.2
(g) The scale function
0 2000 4000 6000 8000 10000 12000
0.005 0.010 0.015 0.020
(h) The total standard deviations
Figure 3: Estimation results for the Allianz returns with (a to d) and without (e to h)
overnight returns.
0 2000 4000 6000 8000 10000 12000
-0.06 -0.02 0.02 0.06
(a) Henkel 20 minute returns, all
10 20
0.8 1.0 1.2 1.4 1.6 1.8 2.0
(b) The seasonal component (square root)
0 2000 4000 6000 8000 10000 12000
0.8 1.0 1.2 1.4
(c) The scale function
0 2000 4000 6000 8000 10000 12000
0.005 0.015 0.025
(d) The total standard deviations
0 2000 4000 6000 8000 10000 12000
-0.06 -0.02 0.02 0.06
(e) Henkel 20 minute returns, intraday only
10 20
0.8 0.9 1.0 1.1 1.2 1.3
(f) The seasonal component (square root)
0 2000 4000 6000 8000 10000 12000
0.8 1.0 1.2 1.4
(g) The scale function
0 2000 4000 6000 8000 10000 12000
0.005 0.010 0.015 0.020 0.025
(h) The total standard deviations
Figure 4: Estimation results for the Henkel returns with (a to d) and without (e to h)
overnight returns.
0 2000 4000 6000 8000 10000 12000
-0.05 0.0 0.05
(a) Linde 20 minute returns, all
10 20
1.0 1.5 2.0
(b) The seasonal component (square root)
0 2000 4000 6000 8000 10000 12000
0.8 1.0 1.2 1.4
(c) The scale function
0 2000 4000 6000 8000 10000 12000
0.01 0.02 0.03 0.04
(d) The total standard deviations
0 2000 4000 6000 8000 10000 12000
-0.04 0.0 0.02 0.04
(e) Linde 20 minute returns, intraday only
10 20
0.8 0.9 1.0 1.1 1.2 1.3 1.4
(f) The seasonal component (square root)
0 2000 4000 6000 8000 10000 12000
0.8 1.0 1.2 1.4
(g) The scale function
0 2000 4000 6000 8000 10000 12000
0.005 0.010 0.015 0.020 0.025
(h) The total standard deviations
Figure 5: Estimation results for the Linde returns with (a to d) and without (e to h)
overnight returns.
0 50 100 150 200
-0.05 0.15
(a) Allianz returns (all)
ACF
Lag
0 50 100 150 200
-0.05 0.15
(b) Allianz (all), seasonal adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(c) Allianz (all), trend adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(d) Allianz (all), standardized residuals
ACF
Lag
0 50 100 150 200
-0.05 0.15
(e) Allianz returns (intraday)
ACF
Lag
0 50 100 150 200
-0.05 0.15
(f) Allianz (intraday), seasonal adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(g) Allianz (intraday), trend adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(h) Allianz (intraday), standardized residuals
ACF
Lag
0 50 100 150 200
-0.05 0.15
(i) BASF returns (all)
ACF
Lag
0 50 100 150 200
-0.05 0.15
(j) BASF (all), seasonal adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(k) BASF (all), trend adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(l) BASF (all), standardized residuals
ACF
Lag
0 50 100 150 200
-0.05 0.15
(m) BASF returns (intraday)
ACF
Lag
0 50 100 150 200
-0.05 0.15
(n) BASF (intraday), seasonal adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(o) BASF (intraday), trend adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(p) BASF (intraday), standardized residuals
ACF
Lag
Figure 6: ACF's for dierent transformed time series obtained from the Allianzreturns
(Figures a toh)and the BASF returns(Figures i top).
0 50 100 150 200
-0.05 0.15
(a) Henkel returns (all)
ACF
Lag
0 50 100 150 200
-0.05 0.15
(b) Henkel (all), seasonal adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(c) Henkel (all), trend adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(d) Henkel (all), standardized residuals
ACF
Lag
0 50 100 150 200
-0.05 0.15
(e) Henkel returns (intraday)
ACF
Lag
0 50 100 150 200
-0.05 0.15
(f) Henkel (intraday), seasonal adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(g) Henkel (intraday), trend adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(h) Henkel (intraday), standardized residuals
ACF
Lag
0 50 100 150 200
-0.05 0.15
(i) Linde returns (all)
ACF
Lag
0 50 100 150 200
-0.05 0.15
(j) Linde (all), seasonal adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(k) Linde (all), trend adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(l) Linde (all), standardized residuals
ACF
Lag
0 50 100 150 200
-0.05 0.15
(m) Linde returns (intraday)
ACF
Lag
0 50 100 150 200
-0.05 0.15
(n) Linde (intraday), seasonal adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(o) Linde (intraday), trend adjusted
ACF
Lag
0 50 100 150 200
-0.05 0.15
(p) Linde (intraday), standardized residuals
ACF
Lag
Figure 7: ACF's for dierent transformed time series obtained from the Henkel returns
(Figures a toh)and the Linde returns(Figures ito p).
