Montréal
Février 2001
Série Scientifique
Scientific Series
2001s-15
Properties of Estimates of Daily
GARCH Parameters Based on
Intra-Day Observations
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Properties of Estimates of Daily GARCH Parameters
Based on Intra-Day Observations
*
John W. Galbraith
†
, Victoria Zinde-Walsh
‡
Résumé / Abstract
Nous considérons les estimés des paramètres des modèles GARCH pour
les rendements financiers journaliers, qui sont obtenus à l'aide des données
intra-jour (haute fréquence) pour estimer la volatilité intra-journalière. Deux bases
potentielles sont evaluées. La première est fondée sur l'aggrégation des estimés
quasi-vraisemblance-maximale, en profitant des résultats de Drost et Nijman
(1993). L'autre utilise la volatilité integrée de Andersen et Bollerslev (1998), et
obtient les coefficients d'un modèle estimé par LAD ou MCO; la première
méthode résiste mieux à la possibilité de non-existence des moments de l'erreur en
estimation de volatilité. En particulier, nous considérons l'estimation par
approximation ARCH, et nous obtenons les paramètres par une méthode liée à
celle de Galbraith et Zinde-Walsh (1997) pour les processus ARMA. Nous offrons
des résultats provenant des simulations sur la performance des méthodes en
échantillons finis, et nous décrivons les atouts relatifs à l'estimation standard de
quasi-VM basée uniquement sur les données journalières.
We consider estimates of the parameters of GARCH models of daily
financial returns, obtained using intra-day (high-frequency) returns data to
estimate the daily conditional volatility.Two potential bases for estimation are
considered. One uses aggregation of high-frequency Quasi- ML estimates, using
aggregation results of Drost and Nijman (1993). The other uses the integrated
volatility of Andersen and Bollerslev (1998), and obtains coefficients from a
model estimated by LAD or OLS, in the former case providing consistency and
asymptotic normality in some cases where moments of the volatility estimation
error may not exist. In particular, we consider estimation in this way of an ARCH
approximation, and obtain GARCH parameters by a method related to that of
Galbraith and Zinde-Walsh (1997) for ARMA processes. We offer some
simulation evidence on small-sample performance, and characterize the gains
relative to standard quasi-ML estimates based on daily data alone.
*
Corresponding Author: John W. Galbraith, Department of Economics, McGill University, 888 Sherbrooke Street
West, Montréal, Qc, Canada H3A 2T7
Tel.: (514) 985-4008 Fax: (514) 985-4039
email: galbraij@cirano.qc.ca
The authors thank the Fonds pour la formation de chercheurs et l'aide à la recherche (Québec) and the Social
Sciences and Humanities Research Council of Canada for financial support of this research, and CIRANO for
research facilities. We are grateful to Nour Meddahi, Tom McCurdy and Éric Renault, and to participants in the
Econometric Society World Congress 2000 and Canadian Econometric Study Group 2000 meeting, for valuable
comments and discussions.
†
McGill University and CIRANO
‡
McGill University
Mots Clés :
GARCH, données haute fréquence, volatilité intégrée, LAD
Keywords:
GARCH, high frequency data, integrated volatility, LAD
GARCH mo dels are widely used for forecast ing and characterizing the condit ional
volatility of ec onomic and ( particularly) nancial time serie s. Since the original cont
ribu-tions of Engle ( 1982)andB ollerslev (1986),t he models haveb e enest imatedby Maximum
Likelihoo d (or quasi-Maximum Likeliho o d, `QML ') metho ds on observations at the
fre-quencyof interest . Int he case of asset ret urns, t hef req uency of interest is often t hedaily
uct uat ion.
Financial dat a are of ten recorde d at frequencies much higher t han the daily. Even
where our interest lies in volatility at the daily fre quenc y, t hese dat a contain inf
orma-tion which may b e used t o improve our est imate s of mo dels at t he daily frequency. Of
course, followingAndersen and Bollerslev (1998), higher-freq uency dat a may also b e used
to estimat e t he daily volatility directly.
The present pap er considers two p ossible strategies for e st imation of daily GARCH
mo de ls which use information ab out higher-freq ue ncy uct uat ions. The rst uses the
known aggregation relations(Drost andNijman, 1993) linkingt he parametersof GARCH
mo de ls of high-frequenc yand corresp onding low-f req uency observations. When such e st
i-mat es are base d on QML estimat es forthe high-f req uency data, however, relat ively
strin-gent conditions are req uired, which may not b e met in (for ex ample ) asset-re turn data.
The second p ot ential strat egy is t o use the observation of Ande rsen and Bolle rslev
(1998) that the volatility of low-f req uency asset returns may b e estimated by t he sum
of squared high-frequency re turns. While t he resulting est imate may b e used direct ly to
characterize the pro ce ss as in A ndersen and Bollerslev or A ndersen et al. ( 1999) , it is
also p ossible to use the seq ue nce of low- (daily -) f requency estimatedvolatilities toobt ain
estimat es of condit ional volatility mo dels such as GARCH mo dels, e xplicit ly allowing f or
estimat ion error in the est imate d daily volatility. The resulting mo dels may b e e st imated
by a variety of te chnique s (inc luding LS) ; by using the Le ast Absolute Deviations (LA D)
estimat or, it is p ossible to obtain consistent and asy mptotically normal estimat es under
quite general condit ions (in part icular, wit hout requiring t he e xist ence of moments of the
returns). We arethen able to obtainestimat es of GA RCH paramete rs using an estimat or
relat ed t o that of Galbraith and Zinde-Walsh (1997)for ARMAmo dels.
In section 2 we describ e the models and estimators to b e conside red and give some
rele vant de nit ions and notat ion. Sect ion 3 provides several asymptot ic result s, while
section 4 pre sents simulation ev idence on t he nite - sample p erf ormance of regression
estimat ors relat ive t o that of standardG ARCH e st imatesbased on the daily obse rvations
alone.
2. GARCH model esti mati onusing hi gher-fre quency dat a
2.1 Pro cesses and not ation
t X t =X 0 + Z t 0 s X s dW s ; wherefW t
gisaBrownianmotionprocessand 2 s
ist heinst antaneouscondit ionalvariance.
Thisisasp e cialc aseofthest ruct ureusedby,e .g.,Nelson(1992),NelsonandFoster(1994).
The pro ce ss is sampled discretely at an interval of time ` (e .g., e ach minute) . We
are inte rested in volatility at a lower-frequency sampling, with sampling interval h` (e .g.,
daily ), so that there are h high-fre quency observations p er low-frequency observat ion.
De neone unit of time as a p erio d of length `:
We inde x t he full set of observat ions by and the lowe r-fre quency obse rvations by
t; so that t = fh;2h;:::;hTg: The size of the sample of low-f req uency obse rvat ions is
therefore T; and of t he full set of high-f req uency observat ions is hT: Following A ndersen
and B ollerslev (1998),estimat e thecondit ional volat ility at t as t he est imate dcondit ional
variance ^ 2 t = ih X j=(i 01)h+1 r 2 j ; (2:1:0) with r 2 j =( x j 0x j01 ) 2 ; x j
indicat ing t he discret ely-sampled obse rvat ions on thepro c ess
X : See A nderse n and Bollerslev onconvergenceof ^ 2 t t o R t t01 2 s ds:
Now consider A RCH and GARCH mo dels at the lower-frequency observations:
2 t =!+ q X i=1 i " 2 t0i ; (2:1:1) 2 t =!+ q X i=1 i " 2 t0 i + p X i01 i 2 t0i ; (2:1:2) where " t = y t 0 t for a process y t
with condit ional mean t
; or in t he drif tless case
" t =x t : So E(" 2 t j t0i ) 2 t
: Mo dels in t he form ( 2.1.1), ( 2.1.2) are directly estimable if,
as in A ndersen and B ollersle v, we have measurements of 2 t
: We return to t his p oint in
Section 2.3 b elow.
Finally,wewillre ferb elowtothede nit ionsofSt rong,Semi-strongandWeakGARCH
given in Drost and N ijman ( 1993) . In strong GARCH , f" t g is such that z t " t = t
IID ( 0;1); se mi-st rong GARCH holds where f" t g is such that E[" t j" t01 ;:::] = 0 and E[" 2 t j" t01 ;:::] = 2 t
; weak GA RCH holds where f" t g is such t hat P[" t j" t01 ;:::] = 0 and P[" 2 t j" t0 1 ;:::] = 2 t ; where P[" 2 t j" t01
;:::] denotes t he b e st linear predictor of " 2 t
given a
const ant and past values of b oth " t
and " 2 t :
Drost and N ij man (1993) showed t hat t ime-aggregated weak GARCH processes lead
to processes of the same class, and gave determinist ic relations b etween the c o ecients
(and thek urtosis) of t he high f requency pro cessand corresp onding time-aggregat ed
(low-frequency) pro cess f or the weak GARCH (1,1) case. As Drost and N ij man not ed, such
relat ions c an in principle b e used to obt ain est imate s of the paramet ers at one frequency
fromthoseatanot he r. Inthissectionweex aminethest rat egyof low-frequenc yestimat ion
based on prior high-frequency e st imates. Time aggregation relations of course di er f or
sto ck and ow variables; here we t reat ows, such as asset returns.
Consider thehigh-frequency GARCH (1,1) pro ce ss
2 =!+ 1 " 2 01 + 1 2 01 ; (2:2:1) if " (h )t = P th j=t(h 01)+1 " j
is t he aggregat ed ow variable, then it s volat ility at the low
frequency follows t he weak GARCH (1,1) pro cess
2 (h )t = 0 + 1 " 2 (h)t + 1 2 (h )t01 ; (2:2:2) with 0 ; 1 ; 1
givenbyt hecorresp ondingformulae( 13-15)for ; ; inDrostandN ijman
(1993), adjusting for not ation. To obt ain c onsist ent estimat ion by QML of the
high-frequencymo del,itwillb e necessarythatt hepro cessissemi-st rongGARCH :t hest andard
Quasi-MLe st imatoroftheG ARCH modelwillin gene ralb e inconsistentinwe akGARCH
mo de ls (as noted by Me ddahi and Re nault 1996, 2000 and Franc q and Z akoan 1998; see
the latt er refe rence f or an example and M-R 2000 fora Mont e Carlo ex ample on samples
of 80000 { 150 000 simulated lowf req uency observat ions).
We will show that the mapping
0 @ 0 1 1 1 A = 0 @ ! 1 1 1 A (2:2:3)
providedby the Drost-Nijman formulae is ac ontinuously di e rentiablemapping; itis also
analy ticover t he region where the paramete rs are de ned.
Thisimpliest hatanyconsist ente st imatorofthehigh-frequencyparameters( !; 1
; 1
)
leads to a consistent estimator of the low-frequency paramet ers ( 0 ; 1 ; 1 ) ; and
simi-larly thatan asympt ot ically Normal estimatorof t he high-f req uency paramet ersresult s in
asymptot ic N ormality of the low-f req uency parameters.
Denot et he vect or 0 @ ! 1 1 1 A
by and,correspondingly, let= 0 @ 0 1 1 1 A : Then ()= :
Now denot e by 2R the region =f(!; 1 ; 1 )2R 3 j! >0; 1 0; 1 0; 1 + 1 <1g;
thatis,t heregion forwhicht heGARCH(1,1)pro c ess is de ned( see,e .g.,Bollerslev1986).
Theorem 1. For any estimat or ^ of such that ( i) ^ p !; (ii) ^ a N(;V()) , the estimat or ^
= (^) is sucht hat for ^ sat isfy ing (i),
^
p !;
and for^satisfying ( ii) ,
^ a N(;V( ^ )) ;
where the asy mptot ic covariance mat rix is V( ^ )= @ @ 0 V() @ 0 @ :
Pro of. It f ollows from (i) and conseq uently also from ( ii) t hat since 2;
P(^ 2 ) ! 1: Consider now the formulae f or
= () ove r in Drost and N ijman
(1993). From(15) of D-N wec an obt ain 1
froma solut ionto aquadraticequation of the
formZ 2 0cZ+1=0; where c=c(!; 1 ; 1 ;) (2:2:4)
is obtained from t he expre ssion in (15) of D-N. For 2 it f ollows that c > 2 and
therefore 1 = c 2 0 2 ( c 2 ) 2 01 3 1 2 is such t hat 0 < 1
< 1: Moreover, it can b e shown that
1 <( 1 + 1 ) h and so 1
obtained from(13) in D-N also lie s b etween0 and 1.
The transformat ion 0 @ ! 1 1 1 A
can b e writ ten as
0 @ ! 1 1 1 A = 0 B B @ h! 10 ( 1+ 1) h 10( 1 + 1 ) ( 1 + 1 ) h 0 c 2 + 2 ( c 2 ) 2 01 31 2 0 c 2 + 2 ( c 2 ) 2 01 3 1 2 1 C C A ;
where c is givenby (2.2.4) ; it is de ned and di erentiablee verywhere in : .
Note t hat ( as follows from Drost and N ij man 1993) , even if 1
= 0; 1
is
non-zero as long as > 0: As h increases, 1 and 1 decline; given 1 and 1 ; condit ional
heteroskedasticity vanishes f or sucient ly large h: There fore, for substant ial condit ional
heteroskedasticity to b e present in t he low-f requency (aggregated) ow pro ce ss, 1
+ 1
strong GARCH high-frequenc y data to obtain e st imates of : Its asymptotic covariance mat rix is V[^ QML ]=[W 0 W] 01 B 0 B[W 0 W] 01 ; where W 0 W = h X =1 T g 2 g 2 0 and B 0 B= T X =1 " 2 2 01 2 g 2 g 2 0 ; with g = @ 2 @ = 0 @ 1 " 2 01 2 01 1 A
: The asy mptot ic variance of t he estimat or ^
based on ow
aggregat ion is then
@ @ 0 [W 0 W] 0 1 B 0 B[W 0 W] 01 @ 0 @ : (2:2:5) If ^ QML
is t he MLE t his reduc es to
@ @ 0 [W 0 W] 01 @ 0 @ : (2:2:6)
Example 1. Let the high-frequency pro cessb e ARCH (1); aggregation thenleads t o a
weakGARCH (1,1)processfort helow-f requencydata. H owever,t heasympt oticcovariance
mat rix for t he estimator ^
is of rank 2 rather t han 3, since t he middle part in (2.2.5) or
(2.2.6) is of dimension 222: This indic ates that there are c ases where ^ is clearly more ecient than ML (or QML
) base d on low-f req uency dat a alone, with covariance matrix
of rank 3.
While estimat ion is feasible by t his metho d, t he requirement s of this strat egy, even
for consist ent est imation, are fairly se ve re. In particular, the p ot ential inc onsist enc y of
QML estimat ion when only weak GA RCH conditions apply means that we must assume
semi-st rong GARCH at the high freq uency if estimation is by QML . This is, however, an
arbitrary sp ec i cat ion; if t he high frequency data are themselves aggregat es of yet higher
frequencies,the semi-st rongconditionsdo notfollow. Whileconsist entest imationof weak
GARCHmo delsisinprinciplep ossible(seeFrancqandZakoan1998) ,theQMLestimat or
do esnot accomplish this.
More generally, estimation based on aggregat ion presumes knowledge of the
mat ors based on the integrated volat ility, which do not presume knowledge of t he
high-frequency pro ce ss b eyond the condit ions necessary for consistency of the daily volatility
estimat e.
2.3 Estimat ion by regression using inte grated volat ility
Asnotedab ove,mo delsoft heform(2.1.1)and( 2.1.2)aredirect lyestimableifwehave
estimat es of t he conditional variance of t he low-fre que ncy observat ions, 2 t
; for e xample
from the dailyint egrat ed volat ility, as in Andersen and Bollerslev . 1
H owever, we will not
followA nderse nand Bollerslevintreating theobse rvationasex act. Instead,weint ro duce
into themo delthemeasurement error arisinginestimat ionof 2 t
fromt hedaily integrated
volatility ( 2.1.0), asp eci cation also employed by Maheuand McCurdy( 2000) . Let
^ 2 t = 2 t +e t ; (2:3:1) prop ertiesof fe t
gf ollow from (2.1.0)and will b e c onsidered b e low.
The ARCH and G ARCH mo delsb ecome
^ 2 t =!+ q X i=1 i " 2 t0i +e t ; (2:3:2) ^ 2 t =!+ q X i =1 i " 2 t0i + p X i=1 i ^ 2 t0i 0 p X i=1 i e t0i +e t : (2:3:3)
Both (2.3.2) and ( 2.3.3) are in principle estimable as regression mo dels.
The mo del
(2.3.3) has an error t erm with an MA (s) f orm; t he coecients of t his moving ave rage
pro c ess are subje ct t o the constraint emb o died in (2.3.3) that t hey are the same ( up
to sign) as t he co ecient s on lagged values of ^ 2 t
: Estimat ion of these mo dels by L S or
QML does however req uire re lat ive ly strong moment conditions t o hold on the regre ssors
and t he volat ility est imation errors fe t
g; not e that this is unlike the standard GARCH
mo de l estimat ed by QML where conditions are usually applie d to the rescaled squared
innovat ions.
Maheuand McCurdy(2000) nd go o dresultsusingt heconst rainedmo del,e st imated
by QML, on fore ign ex change ret urns. Bollen and Inder (1998) estimate a mo del similar
to ( 2.3.3) by st andard QML me tho ds (without accounting for the error auto correlat ion
struct ure), using int ra-day data to obtain est imate s of an unobservable sequence related
todaily volatility. Thisapproachrequiresconsist ency oft he estimat es of theunobservable
1
Since we will b e discussing low-f req uency paramet ers here af ter, we no longer need to
dist inguish f rom high-frequenc y and will omit the `bar' in sy mb ols, refe rring to 2
; ; ;
obtain consistency of the est imator; Bollen and Inder nd goo d results on a sample of S
& P Index futureswith a large numb erof observat ions p e r day.
Herewewillconsiderestimat ionofmo delshavingtheARCHmode l( 2.3.2) ,followedby
computat ionofGARCHparametersfromtheARCH approximat ion. Themet ho d thatwe
will useshould pro duce estimates which are relatively insensitive t o theerror dist ribut ion
and to ex iste nce of moments of t he errors, which is part ic ularly advantageous in nancial
data. This strategy also has the advantageof pro ducing immediat elyan e st imated mo del
whichis direc tlyuseableforforecasting, andofallowingcomputat ionofparametersof any
GARCH(p ;q) mo del from a give n e st imated ARCH represent at ion. Suc ie nt conditions
forconsistentandasymptot ically normal estimat ionare givenin Section3b e low; itis not
necessary thatthe numb er of intra-day obse rvat ions p erday (h) inc rease wit hout b ound.
To obt ain estimat es of GARCHparametersfromtheA RCH represe ntationwepursue
anestimat ionstrategy relatedt othatof Galbrait handZ inde-Walsh(1994,1997) ,inwhich
autore gressive mo delsare used inestimat ionof MA orA RMAmodels. Here, ahigh-order
ARCH mo del is used, and est imate s of GA RCH (p;q) paramet ers are deduced from the
patterns of ARCH coecients. This is another ex ample of what Galbraith and
Zinde-Walsh (2001) refer to as `analy tical indirec t inf erence', in that a me tho d analogous to
indirectinference ( Gourieroux e tal. 1993) isused, butwherethe mappingfrome st imates
of the auxiliary mo del to the model t o b e est imated is obtained analyt ic ally, rather t han
by simulat ion.
Consider rst a c ase of known condit ional variance t
: The G ARCH pro cess (2.1.2)
has a f orm analogous to the ARMA (p;q) ; using standard result s on re presentat ion of an
ARMA (p;q) pro cess in MA form (see, e.g., Fuller 1976) , we can express ( 2.1.2) in the
form 2 t =+ 1 X `=1 ` " 2 t0 ` ; (2:3:4) with 0 =0and 1 = 1 2 = 2 + 1 1 . . . ` = ` + mi n(`;p) X i=1 i `0i ; `q; ` = m in(`;p) X i=1 i `0i ; `>q; (2:3:5)
=(10 (1)) 01 ! = 10 p X i=1 i ! 01 !: (2:3:6)
Giraitis et al. (2000) give general conditions under which the A RCH( 1)
representa-tion is p ossible f or t he strong GARCH (p;q) case ; only the existence of the rst moment
and summability of theco e cients `
(inour notation)are required forthe ex iste nce of a
stric tly st ationary ARCH (1) solution as given in (2.3.4) .
To estimat ethe modelusingatruncat edversion ofthisA RCH( 1) representation,we
use t heestimated low-freq ue ncy c onditionalvarianc e from( 2.1.0) , de ning theestimat ion
error as in ( 2.3.1) and substituting int o ( 2.3.4) toobtain
^ 2 t =+ k X `=1 ` " 2 t0` +e t : (2:3:7)
The truncat ion paramete r k must b e such thatk!1; k=T !0 forc onsist ent estimat ion
of theGARCH mo del.
Themo del( 2.3.7)mayb eestimat edbyLSor,t oobt ainresult srobusttolessrestrictive
condit ions on the volat ility estimation errors, LA D. A sy mptotic prop erties of the L AD
estimat or areconsidered in Sect ion3. Estimat ion pro ceeds by rst obtaining estimat es of
=( 1 ; 2 ;:::; p
)f rom(2.3.5)f or`>q;followed by estimat ionof theq paramet ersof
fromthe rst q re lat ions of ( 2.3.5), and of ! from ( 2.3.6) .
Beginby de ning v( 0)= 2 6 6 4 q +1 q +2 . . . k 3 7 7 5 ; and v(0i)= 2 6 6 4 q+10i q+20i . . . k 0i 3 7 7 5 : (2:3:8)
Next de net he (k0q)2p mat rix V =[v( 01) v ( 02):::v(0p)]=
2 6 6 4 q q01 ::: q0p+1 q +1 q ::: q0p+2 . . . . . . k 01 k 02 ::: k 0p 3 7 7 5 : where r
=0 forr 0: It f ollows from(2.3.5) that v( 0)= 0
V:
The p21ve ctor of estimat es ^ is de ned by ^ =( ^ V 0 ^ V) 01 ^ V 0 ^ v(0) ; (2:3:9)
` `
de nitions ab ove. A n estimat e of canthen b e obtained using the est imate of and the
relat ions ( 2.3.5) : thatis
^ 1 =^ 1 ^ 2 =^ 2 0 ^ 1 ^ 1 . . . ^ q =^ q 0 m in(q ;p) X i =1 ^ i ^ q 0 i : (2:3:10) Finally, ^ ! =(1^ 0 ^ (1))=^ 10 p X i=1 ^ i ! :
The covarianc e matrix of the estimates can b e obtained easily from t he e st imates
of t he representation (2.3.7) and t he Jacobian of the transformation. Le t the parameter
vect or b e =(!; ; ); and let 2
b e the variance of t he noise e t
in (2.3.1) . Then
var()=J 0
( var^) J;
where ^ is the vect or of estimat ed ARCH parameters and J is the Jacobian of the t
rans-formation ( 2.3.9) -( 2.3.10). Comput at ion of the c ovariance matrix of t he LA D parameter
vect or is discussed in sect ion 3.
3. Asymptoti c prop er ti esof the i nt egrat ed vol at il i ty{r egressionesti ma tes
Int hissect ionwediscuss conditionsforconsistentandasy mptoticallyNormal
estima-tion of the integrated volatility{re gression mo del of Section 2 by LAD. Re sults f or QML
estimat ionof t he ARCH mo del were est ablished by Weiss (1986),using theassumption of
nit efourthmoment sof t heunnormalizeddata. Lumsdaine(1996)establishedconsist ency
andasy mptot icN ormalityoftheQMLEf orGARCHmo delsbyimp osingconditionsonthe
re-scaled dat a, z t = " t = t
; including the IID assumption and the existence of high-order
moment s. Lee and Hanse n ( 1994) gene ralized these results tofz t
g which are notI ID, but
simplystrictly stationary and ergodic.
Rec all t hat t he ARCH (k ) {typ e mo del (2.3.7) is a regression mo de l, and consider f or
each T the0 e ldF t
gene rate dbythe regressors f(" 2 t ;" 2 t01 ;:::;" 2 t0 k ); t =k+1;:::;T:g: Denote by W T
the sy mmetric matrix with e le ment s w ij = T 0c ( P T t=1 " 2 t0 i " 2 t0 j ); and le t V T =W 1 2 T :
Assumption 1. The re existssome c such that
w ij =O p (1) ; W 01 T =O p (1); ( T 0 c 2 ) max " 2 t =o p (1):
The assumpt ion is trivially satis e d with c= 1 if " p osse sses a sucient numb e r of
moment s, but t he assumption do esnot require the existence of mome nts. 2
Next c onsider an assumption on t he daily volatility est imation e rror,
e t =^ 2 t 0 2 t : (3:1:1)
The following assumption emb o dies bot h the Error Assumption of Pollard (1991, p.189)
and the addit ional assumption of Pollard's Theorem2 that the realizat ions of t hepro c ess
and the errors ( here, volat ility estimat ion errors) are assumed indep e ndent .
Assumption2. Thevolatilityest imationerrorsfe t
gareI IDwithmedian0andacontinuous
p osit ive density f( :) in the neighb ourho o d of zero. The sequences of errors fe t
g and of
realizat ions of the pro ce ss f" t
g are indep endent .
Theorem 2. Consider the model (2.3.7),
^ 2 t =+ k X `=1 ` " 2 t0` +e t ;
and supp ose t hat A ssumptions 1and 2 are sat is ed. Then if ^= (^;^ 1
;:::;^ k
) is the the
LAD -estimated paramete r vector,
2f(0)V T (^0) D !N( 0;I k +1 ):
Pro of. Follows from Pollard ( 1991, Theorem 2 and Example 1) f or c = 1; can b e
ext ended to any c: A ssumption 1 satis e s t he conditions (ii)-(iv ) of Theorem 2 of Pollard
(1991), and combined wit h Assumption 2 provides all of the condit ions (i)-(iv) for the
asymptot ic dist ribut ion to hold.
4. Simul at i on evi dence
Inthis section we present evidence primarily on t he nit e-sample p erf ormance of the
regression estimat or of Section 2.3 using the daily int egrate d volatility, and for
compar-ison the standard Quasi-ML est imator base d on daily dat a alone . The QML procedure
2
A sanexampleoftheformerp oint ,c onsiderac asewheret heeighthunconditionalmoment
of" t
ex ist s. Then(i)issatis edforc=1bytheWLLN ,T 01 ( P T t=1 " 2 t0i " 2 t0j ) p !E(" 2 t0i " 2 t0j ):
At thesame time, if we re-writ e theA RCH( k ) mo del(2.3.7) as astat ionary AR(k ) mo del
by de ning w t =" 2 t 0 2 t ( not e t hat E(w t j" 2 t ;" 2 t01 ;:::)= 0 and var( w t )<1); we obt ain " 2 t =+ P k `=1 ` " 2 t0 ` +w t
:Followingex ample 2ofPollard(1991)(generalizingto AR(k ) ),
it f ollows t hat max " 2 t = o p (T 1=2
): Of course, A ssumption 1 can also hold in cases where
Theregressionestimator uses( 2.1.0)foradaily volatility estimat e,followedbyestimat ion
of(2.3.7)by OLSorLAD, andt ransf ormat iontoGA RCH paramet erestimat esvia(
2.3.9)-(2.3.10).
Inthe rstse t of simulat ion exercises, low-frequency (daily)dataalone are simulated.
The innovat ions are N ormal, leading t o a st rong GARCH mo del est imated by a true ML
estimat or. These casesaretheref oreasfavourableasp ossiblefortheQMLestimat or. The
low-frequency samplesize T isset atf200;600g and t henumb erof replications is 2000f or
each e xp eriment . Although the rst set of exp eriments uses simulated low-frequency data
only, the parameter values are chosen f or compat ibility with t he lat er exp eriments; the
foursets of paramet er values used result from the aggregation of high-frequency GARCH
pro c esses having paramet ers f!; ; g equal t o (.01, .018, .98), ( .01, .05, .945) , (.01, .08,
.89) and (.01, .10, .85) . The corresp onding low-f req uency parameters are computed from
the aggregation f ormula of Drost and Nijman (1993) for the GARCH ( 1,1) ow case; the
values are ( 6.102, .0555, .8957) , (5.889, .1371, .7450) , (4.442, .0736, .3934), and ( 3.613,
.0540, .2234) .
Inasecondsetofsimulations,f ort hesamesamplesize sandnumb e rofreplications,the
high-fre quency GARCH pro cess is simulated direct ly as in t he rst e xp eriment s, (strong
GARCH, with normal errors) and aggregat ed to form a (we ak GARCH ) low-frequency
(daily) returns process. Est imate s of the daily GARCH parameters on t hese daily data
are obtained by QMLand t he re gression e st imator.
As j ust indicate d, the `t rue' daily parameters are c omput ed from the aggregat ion
formulaofDrostandNijman( 1993) ,forcomparisonwit ht heestimatesf romeachmet ho d.
Not e thatQML applied direct ly to t he dailydata is now te chnic ally inconsiste nt b ec ause
onlyt heweakGARCHconditionscanb eguaranteedtoapply. Forthere gressionest imate s,
the a dailyvolat ility estimate has a noise variance implied by the numb erof obse rvations
p er day, h; which in these examples is set at 25 to keep overall sample sizes manage able.
Thisnumb er,whilemo de st ,iscompatiblewiththesuggestionsofthemorerecentlit erat ure
on integrat ed volatility (A ndersen etal. 1999) which suggests that the optimal frequency
to use in estimat ion of daily volatility is not nece ssarily the highest f requency available.
A small value, representing a relatively noisy estimat e of daily volatility, is in general less
favourable fort he re lat ive p erformance of t he regression est imator.
Because the QML and LAD-ARCH estimators are evaluat ed he re using di erent
in-formation sets{the LAD-ARCH uses the highe r-fre que ncy dat a, while QML do es not{
the outcome of any comparison dep ends up on the information content of t he addit ional
(int ra-day) dat a. The comparison here is made on paramete r values that app ear t o b e
representat ive of empirical out comes, and uses a relatively mode st information content of
intra-day data. Nonetheless, such a comparison can only b e viewed as illustrat ive . In
exp e rime nts of the ty p e p erformed here , using Normal innovat ions, QML or LAD-ARCH
can dominate the other in a nit e sample as t he estimat es of realized volatility b ecome
Results from the rst se t of simulat ions are contained in Figure 1a/b Results for the
second set of simulat ions are rep ort ed in Figure 2a/b and 3a/b. With t he second set of
simulations wealsorep ortaforecast comparisonof1-ste p-ahe adout-of-samplecondit ional
volatility f orecasts f rom each estimation metho d, evaluat ed by RMS fore cast error.
Begin by comparing c orresp onding q uadrant s of Figure 1a wit h those of Figure 2a,
and of 1b with 2b. The QML estimat or's p erf ormance is little di ere nt b etween these
two c ases, despite t he fact t hat the Figure 1 cases are in fact t rue ML estimat ors, and
that t he Figure 2 c ases show QML in a weak GA RCH c ase where the estimator is not
in general c onsist ent. Howeve r t hese re sults, in line wit h those me ntioned by Drost and
Nijman(1993, p. 922), suggestthatthe QMLEmay b e conve rging tovalues quit e closeto
the true parametervalues.
Nonethelessthe nite-sample p e rformance of theQML est imator inthesec ases shows
problems which are of a fairly standard typ e , not sp e ci c to GARCH estimation. The
QML E'sconce ntration ofprobabilitymass neart heb oundarie s of theidenti c ationregion
are typic al of the `pile-up' problems familiar from case s such as ML est imation of MA
parameters; he re the QMLE p erf orms p o orly whe n is near zero. Estimates of are
particularly p o or in t he se e xamples, showing large probability masses in t he vicinity of
0.9 regardless of the true value of ; in each of t he t hree cases where < 0:1: W here
=0:1373; by cont rast, estimates of are conc entrated near t he true value, particularly
at t he larger sample size. Of course, QMLE p erformanc e improves in general f or larger
values of ; as we move away from t he b oundary of t he identi cation region; the values
chosen here, however, are ty pical of t hose app e aring in the empiric al literat ure.
Figure 3 c ontains result s f or t he LAD-ARCH estimat or in corresp onding case s, f or
h= 25: t hat is, using t he relatively low high-frequency information c ontent noted ab ove.
Nonetheless, addit ional information is b eing exploited, and we might exp ect t hat the
es-timator should p e rform re lat ive ly well. This expectationis b orne out in Figure 3a, where
LAD -A RCH estimates of show much b ett er conformity wit h the Normal and smaller
disp ersion, as well as less pile-up nearz ero.
InFigure3b,weseelessdramaticgainsinestimat ionof ;conformitywiththeNormal
isimprovedrealtivetocorresp ondingFigure2acases,butt hereremainssubst antialpile-up
at = 0 and again we see disp ersion of the estimat es across much of the [0;1] interval.
Howe ver, the large spurious p eak in the density near 0.9 is eliminated. Conformity with
the Normal is markedly improved at T = 600: These c ases, emb ody ing Normal errors
in the simulation DGP, do not o er the most favourable circumstances for comparison
of LA D-A RCH; with subst antially skewe d error distributions, we would exp ect to see the
tradit ionaladvantageofL ADest imationinrobustnesstoske wnessandheav y-tailederrors.
Howe ver, the e st imator do es o er clear gainsnonetheless.
5. Conc ludi ng remark s
Est imationof GA RCH models via int egrat ed volatility is feasible evenwith amo dest
numb er of intra-day observations. Such e st imates have a numb er of appare nt advantages
estimat ion, including est imation of quant iles of volatility other thant he me dian. 3
3
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