Modelling Volatility Cycles: The MF2‐GARCH Model

Con ad, Ch is ian; Engle, Robe F.
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Modelling Vola ili y Cycles: The MF2‐GARCH Model
Jou nal o Applied Econome ics
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John Wiley & Sons
Sugges ed Ci a ion: Con ad, Ch is ian; Engle, Robe F. (2025) : Modelling Vola ili y Cycles: The MF2‐
GARCH Model, Jou nal o Applied Econome ics, ISSN 1099-1255, Wiley, Hoboken, NJ, Vol. 40, Iss. 4,
pp. 438-454,
h ps://doi.o g/10.1002/jae.3118
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Jou nal o Applied Econome ics
RESEARCH ARTICLE OPEN ACCESS
Modelling Vola ili y Cycles: The MF2-GARCH Model
Ch is ian Con ad1,2,3,4 | Robe F. Engle5
1Al ed-Webe -Ins i u e, Heidelbe g Uni e si y, Heidelbe g, Ge many | 2HEiKA - Heidelbe g Ka ls uhe S a egic Pa ne ship, Heidelbe g Uni e si y, Ka ls uhe
Ins i u e o Technology, Ka ls uhe, Ge many | 3KOF Swiss Economic Ins i u e, Zu ich, Swi ze land | 4ZEW Mannheim, Mannheim, Ge many | 5S e n School
o Business, New Yo k Uni e si y, New Yo k, USA
Co espondence: Ch is ian Con ad (ch is ian.con [email protected] g.de)
Recei ed: 6 Ma ch 2023 | Re ised: 22 Oc obe 2024 | Accep ed: 15 Decembe 2024
Funding: This s udy was unded by he Ge man Fede al Minis y o Educa ion and Resea ch (BMBF) and he Baden-Wü embe g Minis y o Science as pa
o Ge many’s Excellence S a egy (ExU 10.2.31).
Keywo ds: long- and sho - e m ola ili y | long- e m o ecas ing | mixed equency da a | ola ili y componen models | ola ili y o ecas ing
ABSTRACT
We p opose a no el mul iplica i e ac o mul i- equency GARCH (MF2-GARCH) model, which exploi s he empi ical ac ha
hedaily s anda dized o ecas e o so one-componen GARCHmodels a ep edic ablebya mo inga e ageo pas s anda dized
o ecas e o s. In con as o o he mul iplica i e componen GARCH models, he MF2-GARCH ea u es s a iona y e u ns, and
long- e m ola ili y o ecas s a e mean- e e ing. When applied o he S&P 500, he new componen model signi ican ly ou pe -
o ms heone-componen GJR-GARCH, heGARCH-MIDAS-RV,and helog-HARmodelinlong- e mou -o -sample o ecas ing.
We illus a e he MF2-GARCH’s scalabili y by applying he new model o mo e han 2100 indi idual s ocks in he Vola ili y Lab
a NYU S e n.
1|In oduc ion
The e is s ong empi ical e idence ha he condi ional a i-
ance o s ock e u ns consis s o se e al componen s. Ea ly e i-
dence o ola ili y componen s was p o ided in, o example,
Ding and G ange 1996 and Engle and Lee 1999. Mo e ecen
e idence can be ound in Ch is o e sen e al. 2008, Kim and
Nelson 2013, Do ion 2016, and Con ad and Kleen 2020, among
o he s. While he GARCH models o Ding and G ange 1996
and Engle and Lee 1999 ha e addi i e ola ili y compo-
nen s, mo e ecen GARCH- ype models decompose he con-
di ional a iance in o mul iplica i e sho - and long- e m com-
ponen s. Fo example, in he Spline-GARCH model o Engle
and Rangel 2008 and he mul iplica i e ime- a ying GARCH
(MTV-GARCH) o Amado and Te äs i a 2013 and Amado
and Te äs i a 2017, he long- e m ola ili y componen is a
de e minis ic unc ion o calenda ime. In con as , in he
GARCH-MIDASo Engle,Ghysels,andSohn2013, helong- e m
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© 2025 The Au ho (s). Jou nal o Applied Econome ics published by John Wiley & Sons L d.
componen depends ei he on a olling window ealized
a iance (hence o h GARCH-MIDAS-RV, see also Wang and
Ghysels 2015) o on low- equency mac oeconomic o inan-
cial a iables(hence o hGARCH-MIDAS-X,seealsoAsgha ian,
Hou, and Ja ed 2013, and Con ad and Loch 2015). Those mul i-
plica i e ola ili y models a e based on he idea ha e u ns ol-
low a s a iona y GARCH p ocess once di ided by he long- e m
ola ili y componen . Howe e , he e is no consensus ye on he
mos sui able app oach o modeling he long- e m componen .
We p opose a no el speci ica ion o he long- e m ola ili y
componen in mul iplica i e GARCH models. The speci ica ion
is mo i a ed by a new empi ical ac ha we documen o
he ola ili y o ecas e o s o one-componen GARCH mod-
els: While he daily s anda dized o ecas e o s a e essen-
ially unp edic able based on pas daily s anda dized o ecas
e o s, a olling window mo ing a e age o he pas daily s an-
da dized o ecas e o s does ha e p edic i e powe . This is
Jou nal o Applied Econome ics, 2025; 40:438–454
h ps://doi.o g/10.1002/jae.3118
438
because one-componen models end o unde p edic o o e -
p edic ola ili y o ex ended ime pe iods. While o e p edic ion
ypically happens du ing economic expansions, unde p edic ion
ma e ializesdu ingeconomic ecessionsando he c isespe iods.
The new model combines a sho - e m GJR-GARCH (see
Glos en, Jaganna han, and Runkle 1993) componen wi h a
long- e m componen speci ied as a mul iplica i e e o model
(MEM) o he pas o ecas e o s o he GARCH componen .
Tha is, he long- e m componen exploi s he p edic abili y in
he a e aged s anda dized o ecas e o s o he sho - e m com-
ponen . In ui i ely, he long- e m componen scales he GARCH
componen ’s ola ili y o ecas up/down i he sho - e m com-
ponen ’s o ecas s ha e unde es ima ed/o e es ima ed ola ili y
in he ecen pas , ha is, he model is lea ning om pas o e-
cas e o s. Because he long- e m componen can ei he e ol e
a he same equency as he sho - e m componen o a a lowe
equency, he new speci ica ion belongs o he class o mixed
equency da a sampling (MIDAS) models pionee ed by Ghysels,
San a-Cla a,andValkano 2004.We e e o hep oposedspeci i-
ca ionasMul iplica i eFac o Mul i-F equencyGARCH.As he
“MF” appea s wice, we abb e ia e he model as MF2-GARCH.
The p ope ies o he MF2-GARCH model clea ly dis inguish
i om p e ious speci ica ions. Fi s , while in o he mul iplica-
i e models, he e is ypically no eedback om he sho - e m
o he long- e m componen (e.g., in he Spline-GARCH), he
MF2-GARCH explici ly speci ies he long- e m componen as a
unc ion o he sho - e m componen ’s pas o ecas e o s. Sec-
ond, because he ac ual economic d i e s o long- e m ola ili y
a e unknown and may a y o e ime, i is challenging o co -
ec lyspeci y helong- e mcomponen in heGARCH-MIDAS-X
in eal ime.Ou speci ica iona oids hisp oblemandisbasedon
a simple MEM equa ion o he long- e m componen . In e es -
ingly, he MF2-GARCH can be ew i en as a GARCH-MIDAS-X
wi h an explana o y a iable “gene a ed wi hin he model.”
Thi d, because he MF2-GARCH is dynamically comple e, ha
is, i ully speci ies he dynamics o he condi ional a iance, i is
s aigh o wa d o cons uc mul is ep ahead ola ili y o ecas s.
We ob ain he ollowing heo e ical esul s o he MF2-GARCH:
Fi s , we de i e he uncondi ional a iance o he daily
e u ns. While he uncondi ional a iance is ime- a ying in
he Spline-GARCH and in ini e in he GARCH-MIDAS-RV o
Wang and Ghysels 2015, e u ns a e co a iance s a iona y in
he MF2-GARCH. Due o he eedback be ween he sho - and
long- e m componen s, he uncondi ional a iance depends no
only on he model pa ame e s bu also on he ou h momen o
he inno a ion. Second, we ob ain he news impac cu e (NIC).
TheNICillus a es ha he esponsi eness onewschanges wi h
hele elo ola ili ywhichisano he essen ial ea u e ha dis in-
guishes heMF2-GARCH omo he modelsin heGARCH am-
ily. Speci ically, condi ional ola ili y is mo e esponsi e o news
du ing low ola ili y pe iods han du ing high ola ili y pe iods.
Thi d,wede i eexp essions o mul is epahead o ecas so con-
di ional ola ili y. Ou esul s show ha he o ecas s a e much
mo e lexible han o ecas s om he nes ed GARCH model. The
o ecas s e lec hecu en s anceo hecondi ional a ianceand
hep e ailing ola ili y egime.In hesho e m, hecondi ional
ola ili y o ecas will app oach he o ecas o he long- e m
componen be o e i con e ges o he uncondi ional ola ili y in
he long un. Fo ecas s om he MF2-GARCH also di e om
o ecas s o s anda d Spline-GARCH o GARCH-MIDAS mod-
els. The o ecas s o he la e models a e ypically assumed o
con e ge o he cu en le el o he long- e m componen . Thus,
he o ecas s om hese models do no ea u e mean e e sion
in he long un.Fo h,we discuss he quasi-maximum likelihood
es ima ion o he MF2-GARCH and p o ide a Mon e Ca lo sim-
ula ion showing ha s anda d asymp o ic esul s lead o alid
in e ence. Finally, we p o ide an analysis o he deg ee o mis-
speci ica ion o he nes ed one-componen GJR-GARCH and he
GARCH-MIDAS-RV when he ue da a-gene a ing p ocess is an
MF2-GARCH. As discussed in Pa on 2020, in he p esence o
es ima ion e o and depending on he employed loss unc ion,
a pa simoniously misspeci ied model migh domina e he ue
bu mo e complex model in e ms o o ecas pe o mance. We
show by simula ions ha o easonable pa ame e alues o he
MF2-GARCH, hedeg eeo misspeci ica iono heGJR-GARCH
and he GARCH-MIDAS-RV is so se e e ha he MF2-GARCH
ou pe o ms bo h models when e alua ed by he squa ed e o
(SE) and he QLIKE loss.
Ou empi ical esul s s ongly suppo he MF2-GARCH. Fi s ,
we es ima e he MF2-GARCH o he S&P 500 and 2142 US
and in e na ional equi ies in he Vola ili y Labo a o y (V-Lab) a
NYUS e n.1Ou in-sample esul sshow ha heMF2-GARCHis
clea ly p e e ed o he nes ed one-componen GJR-GARCH, he
GARCH-MIDAS-RV and he Spline-GARCH. Fo he S&P 500,
we show ha he MF2-GARCH’s es ima ed long- e m compo-
nen isclosely ela ed onewsabou hemac oeconomyandmon-
e a y policy, pa icula ly news abou in la ion and in e es a es.
Thus, we p o ide u he e idence o he close link be ween
economic condi ions and long- e m ola ili y (see, e.g., Engle,
Ghysels, and Sohn 2013, and Con ad and Loch 2015). We also
illus a e ha he MF2-GARCH’s ola ili y o ecas s, which ea-
u e cyclical beha io , a e much mo e lexible han he o ecas s
o he compe i o models. In con as o he (o e ly) smoo h
long- e m componen o he Spline-GARCH, he MF2-GARCH’s
long- e mcomponen adjus sin esponse osho -li edpe iodso
ma ke u moil. Ne e heless, compa ed o he one-componen
GJR-GARCH, he MF2-GARCH’s o ecas s a oid o e es ima ing
ola ili y a e a sho -li ed su ge in ola ili y due o he low pe -
sis ence in i s sho - e m componen .
While mos o he li e a u e on ola ili y o ecas ing ocuses
on sho - e m (e.g., 1-day ahead) p edic ion ho izons, Ch is o -
e sen and Diebold 2000, Engle 2009b, and Ghysels e al. 2019,
among o he s, ha e highligh ed ha in many a eas o inance
long- e m isk o ecas s a e he ele an inpu s. This leads o
he ques ion o how a ahead in o he u u e we can o e-
cas ola ili y. Hence, in e alua ing he ou -o -sample o ecas
pe o mance o he MF2-GARCH, ou ocus is on medium-
and long- e m o ecas ho izons o up o 8 mon hs. We es
whe he he MF2-GARCH, which is designed o cap u e ola il-
i y cycles, leads o be e long- e m p edic ions han he nes ed
GJR-GARCH, he GARCH-MIDAS-RV, he Spline-GARCH, and
Co si and Reno 2012’s log-HAR wi h le e age. Fo he S&P
500, i u ns ou ha he MF2-GARCH ou pe o ms all com-
pe i o models when he o ecas ho izon is beyond 2 mon hs.
The MF2-GARCH’s o ecas pe o mance is pa icula ly s ong
du ing pe iods o high ola ili y, whe e i domina es he com-
pe i o models a all o ecas ho izons. Fo a c oss-sec ion o 20
439
equi ies, ou -o -sample esul s om he V-Lab con i m ha he
MF2-GARCH s ongly ou pe o ms he compe i o models.
The pape is o ganized as ollows. In Sec ion 2, we show ha
he ola ili y o ecas e o s o he GJR-GARCH a e p edic able.
We in oduce he MF2-GARCH and discuss i s p ope ies in
Sec ion 3. The empi ical esul s a e p esen ed in Sec ion 4and
Sec ion5concludes.Fu he modelde ails,p oo s,andaddi ional
ables and igu es can be ound in he Suppo ing In o ma ion.
2|A New Empi ical Fac o Vola ili y Fo ecas
E o s
In his sec ion, we p o ide e idence o a new empi ical ac o
ola ili y o ecas e o s: Rolling window mo ing a e ages o he
s anda dized o ecas e o s o one-componen GARCH models
beha e coun e -cyclically and ha e p edic i e powe o u u e
s anda dized o ecas e o s.
We deno e he log- e u n on day 𝑡by 𝑟𝑡. The condi ional he -
e oskedas ici y in daily s ock e u ns is commonly modeled as a
GARCH p ocess. Daily s ock e u ns a e w i en as
𝑟𝑡=√
ℎ𝑡𝜁𝑡,(1)
whe e 
ℎ𝑡deno es hecondi ional a ianceand he𝜁𝑡a eassumed
obe 𝑖.𝑖.𝑑. wi h mean and a ianceequal o ze oand one, espec-
i ely.Fo illus a ion,we es ima e a GJR-GARCH(1,1)speci ica-
ion o a long ime se ies o daily S&P 500 log- e u ns co e ing
Janua y 1971 o June 2023.2We ob ain he ollowing esul :

ℎ𝑡=0.018
(0.003)+(0.022
(0.006)+0.116
(0.014)𝟏{𝑟𝑡−1<0})𝑟2
𝑡−1+0.905
(0.008)
ℎ𝑡−1(2)
whe e he numbe s in pa en heses a e Bolle sle –Woold idge
obus s anda de o sand𝟏{𝑟𝑡−1<0}equalsonei 𝑟𝑡−1<0,andze o
else. As expec ed, he condi ional a iance is highly pe sis en
and he e is s ong e idence o asymme y.
Se e al es s a is ics ha e been p oposed o check a GARCH
speci ica ion’s adequacy. Fo example, Engle and Ng 1993 and
Halunga and O me 2009 p opose Lag ange Mul iplie (LM) es s
o he null hypo hesis ha a (GJR-)GARCH(1,1) is co ec ly
speci ied.
An al e na i e app oach is o check whe he Equa ion (1) is mis-
speci ied in he sense ha 𝜁𝑡=√𝜏𝑡𝑍𝑡, whe e he 𝑍𝑡a e 𝑖.𝑖.𝑑.
and 𝜏𝑡 ep esen s an omi ed mul iplica i e long- e m ola ili y
componen .Thelong- e mcomponen e ol esei he a hesame
equencyas hedaily e u nso a alowe (e.g.,mon hlyo qua -
e ly) equency. The daily e u ns, 𝑟𝑡, can be ei he s a iona y
o nons a iona y. Fo example, in he Spline-GARCH model 𝜏𝑡
e ol es a he daily equency and—because he long- e m com-
ponen is a de e minis ic unc ion o ime— he daily e u ns
ha e a ime- a ying uncondi ional second momen . In ei he
case, he scaled e u ns, 𝑟𝑡∕√𝜏𝑡, a e assumed o ollow a s a ion-
a y GARCH p ocess. LM es s o an omi ed 𝜏𝑡componen ha e
beenp oposedinLundbe ghandTe äs i a2002andAmadoand
Te äs i a 2017 o daily long- e m componen s. The LM es o
Con ad and Schienle 2020 allows o explana o y a iables in he
long- e m componen and ei he a daily o lowe equency 𝜏𝑡.
The es s o Lundbe gh and Te äs i a 2002 and Con ad and
Schienle 2020 exploi ha he squa ed s anda dized e o s, 𝜁2
𝑡=
𝑟2
𝑡∕
ℎ𝑡,a e𝑖.𝑖.𝑑. unde he null hypo hesis o a cons an long- e m
componen . The Con ad and Schienle 2020 LM es checks
whe he 𝜁2
𝑡is p edic able by 𝑥𝑡−1,𝑥
𝑡−2,…,𝑥𝑡−𝐾, whe e 𝑥𝑡is a
p edic o a iable ha canbeexogenouso “gene a edwi hin he
model.”3Unde he null hypo hesis, heLM es is 𝜒2dis ibu ed
wi h 𝐾deg ees o eedom.
We p opose o use he 𝑚-days olling window a e age o he
squa ed s anda dized e o s,

𝑉(𝑚)
𝑡−1=1
𝑚
𝑚
∑
𝑗=1
𝑟2
𝑡−𝑗

ℎ𝑡−𝑗
,(3)
as hep edic o a iable.Unde henullhypo hesis,𝑟2
𝑡isacondi-
ionally unbiased p oxy o he ue bu unobse able condi ional
a iance and 
ℎ𝑡is a one-s ep-ahead o ecas o he same quan-
i y. I he GARCH model is co ec ly speci ied, he s anda dized
ola ili y o ecas e o s, 𝑟2
𝑡∕
ℎ𝑡, ha e an expec ed alue o one
and a a iance o wo i 𝜁𝑡is Gaussian. Hence, we hink o 
𝑉(𝑚)
𝑡
as a measu e o he local bias o he GARCH condi ional a i-
ance. Fo 𝑚=1, we ob ain 
𝑉(1)
𝑡−1=𝜁2
𝑡−1, which is he p edic o
a iable in he “ARCH nes ed in GARCH” es o Lundbe gh and
Te äs i a 2002.
Figu e 1shows 
𝑉(𝑚)
𝑡based on he condi ional a iances om
he GJR-GARCH in Equa ion (2) o 𝑚∈{1,15,25,45}. In he
uppe igh and bo h lowe panels, i is isible ha , as expec ed,

𝑉(𝑚)
𝑡 luc ua es a ound he alue o one. Howe e , as he wo
lowe panels show, he e a e ex ended pe iods du ing which he
one-componen GARCH model unde es ima es o o e es ima es
ola ili y. Tha is, o 𝑚=25 and 𝑚=45, he e olu ion o 
𝑉(𝑚)
𝑡−1is
inline wi h alocal bias o he GJR-GARCHcondi ional a iance.
Thelocalbiasappea s obecoun e -cyclical:Theone-componen
GARCH model ends o o e es ima e ola ili y du ing expan-
sions and o unde es ima e i du ing ecessions. Fo 𝑚=1,
𝑉(𝑚)
𝑡
is oo noisy o e eal his bias. The e a ealso some spikes in 
𝑉(𝑚)
𝑡.
Thesespikesoccu due oex ao dina ye en swi hunexpec edly
high ola ili y. Fo example, he wo la ges spikes a e due o he
s ock ma ke c ashes on Oc obe 19, 1987 (“Black Monday”) and
Oc obe 13, 1989 (“Mini-C ash”).ThespikeinMa ch 2020is due
o he eme gence o he Co id-19 pandemic and he spike on
Sep embe 14, 2022 due o he elease o highe - han-expec ed
in la ion numbe s.
In Panel B o Table 1, we o mally es whe he 
𝑉(𝑚)
𝑡−1has p edic-
i e powe o 𝜁2
𝑡using he Con ad and Schienle 2020 LM es
wi h 𝐾=1. Fo 𝑚∈{1,5,15}, he null hypo hesis o a cons an
long- e m componen is no ejec ed. When he ue long- e m
componen smoo hly a ies o e ime, his is o be expec ed
because o small 𝑚, 
𝑉(𝑚)
𝑡−1is oo noisy o ha e explana o y powe
o 𝜁2
𝑡. In con as , o 𝑚∈{25,35,45,55}, we s ongly ejec he
nullhypo hesis.Thus, he LM es p o ides e idence o an omi -
ed long- e m componen and sugges s ha 
𝑉(𝑚)
𝑡−1is sui able o
modeling he dynamics o he long- e m componen when 𝑚is
app op ia ely chosen.
Ma hema ical Me hods in he Applied Sciences, 2025
440
FIGURE 1 |AGJR-GARCH(1,1)ises ima ed o dailyS&P500 e u nda a o heJanua y1971 oJune2023pe iod.The igu eshows 
𝑉(𝑚)
𝑡 o 𝑚=1
(uppe le panel), 𝑚=15 (uppe igh panel), 𝑚=25 (lowe le panel), and 𝑚=45 (lowe igh panel). G ay shaded a eas ep esen NBER ecession
pe iods.
TABLE 1 |Summa y s a is ics S&P 500 and LM es .
Mean SD Skewness Ku osis Min Max AC(1)
Panel A: Summa y s a is ics
𝑟𝑡0.03 1.09 −1.00 27.11 −22.93 10.71 −0.02
𝑅𝑉𝑡0.99 2.94 18.01 486.49 0.02 101.29 0.63
Panel B: LM es : Explana o y a iable 
𝑉(𝑚)
𝑡
𝑚1 5 15 25 35 45 55
𝑝- alue 0.930 0.960 0.450 0.030 0.001 0.001 0.010
No e: Panel A shows summa y s a is ics o he daily e u ns, 𝑟𝑡, and he daily ealized a iances, 𝑅𝑉𝑡, o he S&P 500. The columns p esen he mean, he s anda d de ia ion
(sd), skewness, ku osis, he minimum (min) and maximum (max) as well as he i s -o de au oco ela ioncoe icien (AC(1)). Daily e u ns o he S&P 500 co e he
pe iod Janua y 1971 o June 2023. Realized a iancesa e o he pe iod Janua y 2010 o June 2023. Panel B shows he esul s o he Con ad and Schienle 2020 LM es o an
omi ed long- e m componen unde he null hypo hesis o a one-componen GJR-GARCH. We se 𝐾=1. The able shows he 𝑝- alues o he es o di e en choices o 𝑚.
Impo an ly, he beha io o he s anda dized ola ili y o e-
cas e o s is no speci ic o he one-componen GJR-GARCH.
We also es ima ed EGARCH (Nelson 1991), FIGARCH (Baillie,
Bolle sle , and Mikkelsen 1996), and Realized GARCH (Hansen,
Huang, and Shek 2012) models and ob ained e y simila
esul s. Fo example, he co ela ion be ween he 
𝑉(45)
𝑡−1o he
GJR-GARCH and he 
𝑉(45)
𝑡−1o he EGARCH, FIGARCH, and he
Realized GARCH is 0.92, 0.82, and 0.77, espec i ely. Figu e A.1
in he Suppo ing In o ma ion plo s 
𝑉(45)
𝑡−1 o all ou models and
con i ms ha he e is s ong co-mo emen . Fu he mo e, ou
indingsdo no only hold o he S&P 500 bu also o o he in e -
na ionals ockindices.Fo illus a ion,Figu eA.2in heSuppo -
ing In o ma ion eplica es Figu e 1 o he FTSE 100.4
In summa y, he e idence sugges s ha one-componen GARCH
models a e misspeci ied and ha he misspeci ica ion is
de ec able when using sui able mo ing a e ages o pas s an-
da dized o ecas e o s o p edic he cu en s anda dized
o ecas e o .
3|The MF2-GARCH Model
This sec ion in oduces he MF2-GARCH model. In he main
speci ica ion, he sho - and he long- e m componen s e ol e
a a daily equency. Fo his speci ica ion, we de i e he uncon-
di ional a iance o e u ns, he NIC and mul is ep ahead o e-
cas s. In Sec ion 3.2, we sugges se e al di ec ions in which he
MF2-GARCH can be ex ended. In pa icula , we in oduce a
pa ame iza ion ha allows o mul iple equencies, ha is, he
sho - e m componen e ol es a he daily equency, while he
long- e mcomponen a iesa alowe equency.Fu he de ails
441

on he MF2-GARCH a e p o ided in he Suppo ing In o ma-
ion: Sec ion A.1 discusses quasi-maximum likelihood es ima-
ion and p o ides a Mon e Ca lo simula ion. The deg ee o mis-
speci ica ion o he nes ed one-componen GJR-GARCH and he
GARCH-MIDAS-RV when he ue model is an MF2-GARCH is
analyzed in Sec ion A.2 in he Suppo ing In o ma ion. A com-
pa ison o he MF2-GARCH wi h o he componen models is
p o ided in Sec ion A.3 in he Suppo ing In o ma ion.
In gene al and as mo i a ed in Sec ion 2, daily log- e u ns a e
de ined as 𝑟𝑡=𝜎𝑡𝑍𝑡=√ℎ𝑡𝜏𝑡𝑍𝑡. We deno e he in o ma ion se
on day 𝑡by 𝑡.𝜎2
𝑡deno es he condi ional a iance and he sho -
and long- e m ola ili y componen s a e gi en by ℎ𝑡and 𝜏𝑡.We
make he ollowing assump ion abou he inno a ions 𝑍𝑡.
Assump ion 1. Le 𝑍𝑡bei.i.d.Thedensi yo 𝑍𝑡issymme ic
wi h E[𝑍𝑡]=0 and E[𝑍2
𝑡]=1. Fu he , 𝑍2
𝑡has a nondegene a e
dis ibu ion and 𝜅=E[𝑍4
𝑡]<∞.
The assump ion ha he densi y o 𝑍𝑡is symme ic is commonly
made o GJR-GARCH models because i allows o a s aigh -
o wa dcompu a iono hemul is epaheadcondi ional a iance
o ecas (see,e.g.,Zi o 2009).Also,LingandMcAlee 2002make
hisassump ionwhende i ingcondi ions o hes a iona i yand
he exis ence o he ou h momen o he GJR-GARCH. Impo -
an ly, as shown in Alexande , Laza , and S anescu (2021), he
symme y o he densi y o 𝑍𝑡does no p eclude ha he 𝑠-s ep
ahead agg ega ed e u ns exhibi skewness. The assump ion ha
𝜅=E[𝑍4
𝑡]<∞is a necessa y condi ion o ensu ing he ini e-
nesso he uncondi ional a ianceo he e u nsand o he exis-
ence o he a iance o he sco e o he likelihood unc ion.
3.1 |Daily Sho - and Long-Te m Componen s
Wespeci y hesho - e m ola ili y componen asauni a iance
GJR-GARCH(1,1)
ℎ𝑡=(1−𝜙)+(𝛼+𝛾𝟏{𝑟𝑡−1<0})𝑟2
𝑡−1
𝜏𝑡−1+𝛽ℎ𝑡−1,(4)
whe e 𝜙=𝛼+𝛾∕2+𝛽. No e ha he d i ing a iable in
Equa ion(4)is𝑟2
𝑡−1∕𝜏𝑡−1.Thisdis inguishesℎ𝑡 om hedailycon-
di ional a iance, 
ℎ𝑡, in Equa ion (2). We make he ollowing
assump ion abou he pa ame e s o he sho - e m componen :
Assump ion 2. The pa ame e s o he sho - e m
GJR-GARCH componen sa is y he condi ions 𝛼>0,𝛼+𝛾>
0,𝛽>0 and 𝜙=𝛼+𝛾∕2+𝛽<1.
I Assump ions 1and 2hold, hen 𝑟𝑡∕√𝜏𝑡=√ℎ𝑡𝑍𝑡 ollows a
co a iance s a iona y GJR-GARCH(1,1) wi h E[ℎ𝑡𝑍2
𝑡]=E[ℎ𝑡]=
1. I he GJR-GARCH(1,1) ully cap u es he condi ional he -
e oskedas ici y, hen 𝜏𝑡is equal o a cons an and he mul iplica-
i e model educes o a one-componen GJR-GARCH o he
daily e u ns.
Following Engle 2009a, we e e o 𝑟𝑡∕√ℎ𝑡as “deGARCHed
e u ns” and de ine 𝑉𝑡=𝑟2
𝑡∕ℎ𝑡=𝜏𝑡𝑍2
𝑡as he squa ed
deGARCHed e u ns. I he GARCH componen ully cap-
u es he condi ional he e oskedas ici y, hen by Assump ion 1
he 𝑉𝑡a e 𝑖.𝑖.𝑑. Howe e , i 𝜏𝑡is ime- a ying and pe sis en ,
hen 𝑉𝑡is au oco ela ed. Because 𝑉𝑡is a nonnega i e a iable,
we speci y he long- e m componen as a MEM equa ion o he
condi ional expec a ion o 𝑉𝑡:
𝜏𝑡=𝜆0+𝜆1𝑉(𝑚)
𝑡−1+𝜆2𝜏𝑡−1,(5)
whe e
𝑉(𝑚)
𝑡−1=1
𝑚
𝑚
∑
𝑗=1𝑉𝑡−𝑗=1
𝑚
𝑚
∑
𝑗=1
𝑟2
𝑡−𝑗
ℎ𝑡−𝑗.(6)
Recall ha we can hink o he squa ed deGARCHed e u ns as
s anda dized ola ili y o ecas e o s. Hence, 𝑉(𝑚)
𝑡−1isa olling
window measu e o he local bias o he sho - e m compo-
nen ’s condi ional a iance o e he p e ious 𝑚days. No e ha
Equa ion (5) can be w i en is a MEM(1,𝑚) wi h he es ic-
ion ha he 𝑚“ARCH” coe icien s a e gi en by 𝜆1∕𝑚. The
MEM(1,𝑚) is co a iance s a iona y i he sum o he ARCH and
GARCH coe icien s is less han one. By cons uc ion, his is sa -
is ied i 𝜆1+𝜆2<1.
Assump ion 3. Thepa ame e so helong- e m componen
sa is y he condi ions 𝜆0>0,𝜆
1>0,𝜆
2>0 and 𝜆1+𝜆2<1.
Unde Assump ions 1and 3, i holds ha 𝑉𝑡=𝜏𝑡𝑍2
𝑡is a co a i-
ance s a iona y MEM wi h E[𝑉𝑡|𝑡−1]=𝜏𝑡and E[𝑉𝑡]=𝜆0∕(1−
𝜆1−𝜆2). We e e o he pa ame iza ion gi en by Equa ions (4)
and (5) as MF2-GARCH- w-𝑚, whe e “ w-𝑚” s ands o olling
window o leng h 𝑚.
3.1.1 |Uncondi ional Va iance o Daily Re u ns
Nex , we de i e he uncondi ional a iance o he daily e u ns.
Fi s ,no e ha heuncondi ionalmeanand a iancea egi enby
E[𝑟𝑡]=0 and Va [𝑟𝑡]=E[𝑟2
𝑡]=E[𝜏𝑡ℎ𝑡]. The ollowing heo em
p o ides an exp ession o Va [𝑟𝑡].
Theo em 1. Le Assump ions 1–3be sa is ied. I he
MF2-GARCH- w-𝑚p ocess, (𝑟𝑡)𝑡∈ℤ, is co a iance s a iona y,
hen
Γ𝑚=(𝜆11
𝑚𝜙𝜅+𝜆2𝜙)+𝜆1𝜙𝜅1
𝑚
𝑚
∑
𝑗=2𝜙𝑗−1<1(7)
wi h 𝜙𝜅=(𝛼+𝛾∕2)𝜅+𝛽. Theuncondi ional a iance o hedaily
e u ns is gi en by
Va [𝑟𝑡]=(𝜆0+𝜆0
1−𝜆1−𝜆2(1−𝜙)(𝜆1+𝜆2)+Δ
𝑚)∕(1−Γ
𝑚),
(8)
whe e
Δ𝑚=(1−𝜙)𝜆1𝜙𝜆0
1−𝜆1−𝜆2(𝑚−1
𝑚+1
𝑚
𝑚
∑
𝑗=2
𝑗−2
∑
𝑘=1𝜙𝑘).
In he p oo o Theo em 1, we show ha Γ𝑚<1 is sa is-
ied i he e u ns a e co a iance s a iona y. Fo example, o
𝑚=1, he condi ion educes oΓ1=𝜆1[(𝛼+𝛾∕2)𝜅+𝛽]+𝜆2[𝛼+
𝛾∕2+𝛽]<1. This illus a es ha e en i he condi ions which
ensu e ha 𝑟𝑡∕√𝜏𝑡and 𝑉𝑡=𝑟2
𝑡∕ℎ𝑡a e indi idually co a iance
Ma hema ical Me hods in he Applied Sciences, 2025
442
s a iona y (i.e., Assump ions 1–3) a e sa is ied, he condi ion in
Equa ion (7) may be iola ed i 𝜅is su icien ly la ge.
Ingene al, he a ianceo hedaily e u nswilldependonall he
model pa ame e s and he inno a ion’s ou h momen , 𝜅. The
ac ha he uncondi ional a iance depends on 𝜅dis inguishes
he MF2-GARCH- w-𝑚 om s anda d GARCH models and is
due o he co ela ion be ween 𝜏𝑡and ℎ𝑡. The uncondi ional a i-
ance o he e u ns inc eases in 𝜅and dec eases as 𝑚inc eases.5
Theo em 1 e eals ha he MF2-GARCH is undamen ally di -
e en om he Spline-GARCH and he GARCH-MIDAS-RV. In
he Spline-GARCH, he uncondi ional a iance o he e u ns is
ime- a ying, and Va [𝑟𝑡]=∞in he GARCH-MIDAS-RV (see
Wang and Ghysels 2015).
When imposing he es ic ions 𝑚=1 and 𝛾=0, we ob ain a
model ha can be conside ed a “mul iplica i e e sion” o he
addi i e componen model o Engle and Lee 1999.Fo 𝑚=1,
bo h componen s a e symme ic, and we impose he es ic ion
𝛼+𝛽<𝜆
1+𝜆2<1 o ensu e ha 𝜏𝑡is he long- un componen .
In he ollowing co olla y, we de i e he condi ion o he co a i-
ance s a iona i y o he daily e u ns when 𝑚=1 and 𝛾=0 and
s a e hei uncondi ional a iance.
Co olla y 1. Le Assump ions 1–3besa is ied, 𝑚=1,and 𝛾=
0.Thenecessa yandsu icien condi ion o heco a iances a ion-
a y o he MF2-GARCH- w-1p ocess is Γ1=𝜆1(𝛼𝜅 +𝛽)+𝜆2(𝛼+
𝛽)<1. The uncondi ional a iance o he e u ns is gi en by
Va [𝑟𝑡]=𝜆0+𝜆0
1−𝜆1−𝜆2(1−𝛼−𝛽)(𝜆1+𝜆2)
1−[𝜆1(𝛼𝜅 +𝛽)+𝜆2(𝛼+𝛽)] .(9)
I he sho - e m componen is cons an (i.e., 𝛼=𝛽=0), he
exp ession in Equa ion (9) educes o Va [𝑟𝑡]=E[𝜏𝑡]=𝜆0∕(1−
𝜆1−𝜆2). I he long- e m componen is cons an (i.e., 𝜆1=
𝜆2=0), he exp ession in Equa ion (9) educes o Va [𝑟𝑡]=
E[𝜆0ℎ𝑡]=𝜆0. In he la e case, he MF2-GARCH- w-1 educes
o a GARCH(1,1).
Asexpec ed,empi icallywe ind ha 𝑉(1)
𝑡−1=𝑟2
𝑡−1∕ℎ𝑡−1is oonoisy
o se e as a p oxy o he local bias. In Sec ion 4.2.1, we show
ha he op imal choice o 𝑚is a ound 63 o he S&P 500, which
co esponds o a qua e ly mo ing a e age.
3.1.2 |News Impac Cu e
Following Engle and Ng 1993, we use he NIC o illus a e how
he condi ional ola ili y is upda ed in esponse o new in o -
ma ion. Fo a GJR-GARCH(1,1) wi h condi ional a iance 
ℎ𝑡=
𝛼0+(𝛼+𝛾1{𝑟𝑡−1<0})𝑟2
𝑡−1+𝛽
ℎ𝑡−1, he NIC is de ined as
𝑁𝐼𝐶𝐺𝐽𝑅
𝑡+1=
ℎ𝑡+1(𝑟𝑡|
ℎ𝑡)=𝐴𝐺𝐽𝑅
𝑡+(𝛼+𝛾1{𝑟𝑡<0})𝑟2
𝑡,
whe e 𝐴𝐺𝐽𝑅
𝑡=𝛼0+𝛽
ℎ𝑡. Tha is, he NIC is a unc ion o oday’s
e u n and 𝐴𝐺𝐽𝑅
𝑡is known condi ional on 𝑡−1.I 𝛾>0, nega-
i e news, 𝑟𝑡<0, ha e a s onge e ec on ola ili y han posi i e
news. Howe e , he size o he e ec does no depend on he cu -
en le el o ola ili y.
The NIC o he MF2-GARCH- w-𝑚consis s o h ee e ms:
𝑁𝐼𝐶𝑀𝐹
𝑡+1=𝜎2
𝑡+1(𝑟𝑡|𝜏𝑡,ℎ𝑡)=𝐴𝑀𝐹
𝑡+𝐵𝑀𝐹
𝑡
+(𝜆1𝛽1
𝑚+𝜆2(𝛼+𝛾1{𝑟𝑡<0}))𝑟2
𝑡,(10)
whe e
𝐴𝑀𝐹
𝑡=𝜆0(1−𝜙)+𝜆0𝛽ℎ𝑡+𝜆1(1−𝜙)1
𝑚
𝑚−1
∑
𝑗=1
𝑟2
𝑡−𝑗
ℎ𝑡−𝑗
+𝜆1𝛽ℎ𝑡1
𝑚
𝑚−1
∑
𝑗=1
𝑟2
𝑡−𝑗
ℎ𝑡−𝑗
+𝜆2(1−𝜙)𝜏𝑡+𝜆2𝛽𝜎2
𝑡
and
𝐵𝑀𝐹
𝑡=𝜆0(𝛼+𝛾1{𝑟𝑡<0})𝑟2
𝑡
𝜏𝑡
+𝜆1(1−𝜙)1
𝑚
𝑟2
𝑡
ℎ𝑡
+𝜆1(𝛼+𝛾1{𝑟𝑡<0})1
𝑚
𝑟4
𝑡
𝜎2
𝑡
+𝜆1(𝛼+𝛾1{𝑟𝑡<0})𝑟2
𝑡
𝜏𝑡
1
𝑚
𝑚−1
∑
𝑗=1
𝑟2
𝑡−𝑗
ℎ𝑡−𝑗.
The in e cep , 𝐴𝑀𝐹
𝑡, is known condi ional on 𝑡−1. The las e m
dependson 𝑟2
𝑡andmodel pa ame e sand, hence, issimila o he
e m(𝛼+𝛾1{𝑟𝑡<0})𝑟2
𝑡in heGJR-GARCH.Howe e , he e m𝐵𝑀𝐹
𝑡
shows ha he ma ginal e ec o 𝑟2
𝑡also depends on he cu en
condi ional a iance, 𝜎2
𝑡, as well as ℎ𝑡and 𝜏𝑡indi idually.
Figu e2shows he(s anda dized)NICo heMF2-GARCH- w-𝑚
wi h𝑚=63(le panel)and𝑚=21( igh panel).TheNICisplo -
ed as a unc ion o he e u n, 𝑟𝑡, and o 𝜏𝑡=1, 𝜏𝑡=1.5, and
𝜏𝑡=0.5. The pa ame e s a e chosen as in Figu e A.3 in he Sup-
po ing In o ma ion. The sho - e m componen is ixed a i s
uncondi ional expec a ion (i.e., ℎ𝑡=1). No e ha pa ame e s o
helong- e m componen a echosensuch ha E[𝜏𝑡]=1.Tha is,
hesolidblackNIC ep esen sasi ua ioninwhichbo h hesho -
andlong- e mcomponen a ea hei uncondi ionalexpec a ion.
Due o he asymme y e m in he sho - e m componen , he
impac o nega i e e u ns is s onge han he impac o pos-
i i e e u ns. The o he wo lines show ha he e ec o new
in o ma ion becomes s onge /weake when long- e m ola ili y
isbelow/abo ei sexpec a ion.Tha is, hecondi ional ola ili yis
mo esensi i e onews du ing alow- ola ili ype iod han du ing
a high- ola ili y pe iod. This is easonable because a la ge alue
o |𝑟𝑡|is o be expec ed du ing u bulen imes bu less so du ing
anquil imes. In line wi h he obse a ion ha he a iance o
he e u ns dec eases as 𝑚inc eases, he news impac is sligh ly
weake o 𝑚=63 han o 𝑚=21.
3.1.3 |Fo ecas ing Vola ili y
Because he MF2-GARCH- w-𝑚model is dynamically comple e,
we can analy ically de i e ola ili y o ecas s o any desi ed
ho izon. In he ollowing, we assume ha a esea che has
obse ed e u ns up o day 𝑡. Based on he in o ma ion se 𝑡,
she in ends o compu e a ola ili y o ecas o day 𝑡+𝑠. Fi s ,
ecall ha he 𝑠-s ep ahead o ecas o he sho - e m componen
can be compu ed as E[ℎ𝑡+𝑠|𝑡]=1+𝜙𝑠−1(ℎ𝑡+1−1),𝑠≥2 (see,
e.g., Zi o 2009). The o ecas s o he long- e m componen a e
sligh ly mo e in ol ed and a e p esen ed in Appendix A.4 o he
443
FIGURE 2 |The igu eshows heNIC o anMF2-GARCH- w-𝑚modelwi h𝑚=63(le panel)and𝑚=21( igh panel)andpa ame e sasinFigu e
A.3 in he Suppo ing In o ma ion. We ix ℎ𝑡=1 and assume ha he sho - e m componen co ec ly p edic s ola ili y on days 𝑡−1 o𝑡−𝑚−1, ha
is, we se 𝑟2
𝑡−𝑗∕ℎ𝑡−𝑗=1 o 𝑗=1,…,𝑚−1. The NICs a e plo ed as a unc ion o he e u n, 𝑟𝑡, and o 𝜏𝑡∈{0.5,1,1.5}. The NICs a e s anda dized
such ha news impac is ze o o 𝑟𝑡=0 and p esen ed as annualized ola ili ies, ha is, we plo √252(𝜎2
𝑡+1(𝑟𝑡|𝜏𝑡,ℎ𝑡=1)−𝜎2
𝑡+1(𝑟𝑡=0|𝜏𝑡,ℎ𝑡=1)).
Suppo ing In o ma ion. Using hese esul s, E[𝜎2
𝑡+𝑠|𝑡]can be
compu ed as ollows:
Theo em 2. Le Assump ions 1–3and he cons ain in
Equa ion (7)be sa is ied. Then, in he MF2-GARCH- w-𝑚 he
o ecas o he condi ional a iance on day 𝑡+𝑠, 𝑠 ≥1, can be
compu ed as ollows: Fi s , E[𝜎2
𝑡+1|𝑡]=ℎ𝑡+1𝜏𝑡+1. Second, o 𝑠=
2,…,𝑚 he o ecas s can be ecu si ely calcula ed as
E[𝜎2
𝑡+𝑠|𝑡]=(1−𝜙)E[𝜏𝑡+𝑠|𝑡]+𝜆0𝜙E[ℎ𝑡+𝑠−1|𝑡]
+(𝜆11
𝑚𝜙𝜅+𝜆2𝜙)E[𝜎2
𝑡+𝑠−1|𝑡]
+𝜆1𝜙E[ℎ𝑡+𝑠−1|𝑡]1
𝑚
𝑚
∑
𝑗=𝑠
𝑟2
𝑡+𝑠−𝑗
ℎ𝑡+𝑠−𝑗
+(1−𝜙)𝜆1𝜙1
𝑚
𝑠−1
∑
𝑗=2E[𝜏𝑡+𝑠−𝑗|𝑡](1+𝑗−2
∑
𝑘=1𝜙𝑘)
+𝜆1𝜙𝜅𝜙1
𝑚
𝑠−1
∑
𝑗=2𝜙𝑗−2E[𝜎2
𝑡+𝑠−𝑗|𝑡].
(11)
Thi d, o 𝑠>𝑚, he ollowing ecu sion applies:
E[𝜎2
𝑡+𝑠|𝑡]=(1−𝜙)E[𝜏𝑡+𝑠|𝑡]+𝜆0𝜙E[ℎ𝑡+𝑠−1|𝑡]
+(𝜆11
𝑚𝜙𝜅+𝜆2𝜙)E[𝜎2
𝑡+𝑠−1|𝑡]
+(1−𝜙)𝜆1𝜙1
𝑚
𝑚
∑
𝑗=2E[𝜏𝑡+𝑠−𝑗|𝑡](1+𝑗−2
∑
𝑘=1𝜙𝑘)
+𝜆1𝜙𝜅𝜙1
𝑚
𝑚
∑
𝑗=2𝜙𝑗−2E[𝜎2
𝑡+𝑠−𝑗|𝑡].
(12)
We will illus a e he beha io o he ola ili y o ecas s in
Sec ion 4.2.2.
3.2 |Ex ensions o he MF2-GARCH- w
3.2.1 |Modi ica ions o he Daily Long-Te m
Componen
MF2-GARCH wi h be a-weigh s: In Equa ion (6), we assume
ha 𝑉(𝑚)
𝑡−1is based on he a e age o he las 𝑚s anda dized o e-
cas e o s. Ins ead o imposing equal weigh s, we can ake a
weigh ed a e age o he o m
𝑉(𝑚)
𝑡−1=𝑚
∑
𝑗=1𝑤𝑗(𝜔)𝑉𝑡−𝑗=𝑚
∑
𝑗=1𝑤𝑗(𝜔)𝑟2
𝑡−𝑗
ℎ𝑡−𝑗.(13)
Following a common choice in he MIDAS li e a u e (see
Ghysels, San a-Cla a, and Valkano 2006, and Ghysels, Sinko,
and Valkano 2007), we pa simoniously model he weigh s
𝑤𝑗(𝜔)acco ding o a es ic ed be a-weigh ing scheme: 𝑤𝑗(𝜔)=
((1−𝑗∕(𝑚+1))𝜔−1)∕(∑𝑚
𝑘=1(1−𝑗∕(𝑚+1))𝜔−1). By cons uc ion,
heweigh ssum oone.Inaddi ion,weimpose hecons ain ha
he weigh s a e noninc easing (𝜔≥1). Fo 𝜔>1, he weigh s
decline om he i s lag.Fo 𝜔=1, heweigh sa egi enby1∕𝑚,
and hence, we ob ain he model wi h olling window 𝑉(𝑚)
𝑡.We
will e e o his pa ame iza ion as MF2-GARCH-bw-𝑚, whe e
“bw-𝑚” s ands o be a-weigh s o leng h 𝑚.
Realized ola ili y MEM: When 𝑚is small, he a e age s an-
da dized o ecas e o o he sho - e m componen , 𝑉(𝑚)
𝑡=
1∕𝑚∑𝑚
𝑗=1𝑟2
𝑡−𝑗∕ℎ𝑡−𝑗,canbeanoisyp oxy o helocalbias.Ins ead,
i we obse e daily ealized a iances, 𝑅𝑉𝑡, we can base 𝜏𝑡on he
ealized measu e: 𝑉(𝑚,𝑅𝑉 )
𝑡=1∕𝑚∑𝑚
𝑗=1𝑅𝑉𝑡−𝑗∕ℎ𝑡−𝑗. Again, we can
apply he MEM speci ica ion om Equa ion (5). Howe e , wi h-
ou u he assump ions, his speci ica ion is no longe dynami-
cally comple e. We lea e his speci ica ion o u u e esea ch.
3.2.2 |Low-F equency Long-Te m Componen
In he MF2-GARCH- w-𝑚, 𝜏𝑡 a ies a he daily equency.
Ins ead, we can speci y he MF2-GARCH so ha he long- e m
componen a ies a a lowe equency. Fo his speci ica ion, we
in oduce a no a ion ha allows o mixed equencies. We dis-
inguish be ween a low- equency pe iod 𝑡and a high- equency
pe iod 𝑖. The high- equency pe iod 𝑖 ep esen s days while 𝑡
Ma hema ical Me hods in he Applied Sciences, 2025
444
migh ep esen a mon hly, qua e ly, o semiannual equency.
We assume ha he e a e 𝑛days wi hin each pe iod 𝑡, ha is, 𝑖=
1,…,𝑛, and ha we obse e 𝑡=1,…,𝑇 low- equency pe i-
ods. Using his no a ion, we deno e he log- e u n on day 𝑖o
pe iod 𝑡by 𝑟𝑖,𝑡 (whe e we use he con en ion ha 𝑟0,𝑡 =𝑟𝑛,𝑡−1).
Simila ly, we deno e he in o ma ion se on day 𝑖in pe iod 𝑡by
𝑖,𝑡 and de ine 𝑡∶= 𝑛,𝑡. No e ha he new no a ion educes o
he no a ion wi h a daily long- e m componen when 𝑛=1. In
he ollowing, we assume ha Assump ion 1holds o he inno-
a ions 𝑍𝑖,𝑡.
Using he no a ion o mul iple equencies, we w i e he
sho - e m ola ili y componen as
ℎ𝑖,𝑡 =(1−𝜙)+(𝛼+𝛾1{𝑟𝑖−1,𝑡<0})𝑟2
𝑖−1,𝑡
𝜏𝑡
+𝛽ℎ𝑖−1,𝑡 (14)
o 𝑖=2,…,𝑛and wi h ℎ0,𝑡 =ℎ𝑛,𝑡−1. Thus, o 𝑖=1, we ob ain
ℎ1,𝑡 =(1−𝜙)+(𝛼+𝛾1{𝑟𝑛,𝑡−1<0})𝑟2
𝑛,𝑡−1∕𝜏𝑡−1+𝛽ℎ𝑛,𝑡−1.Asbe o e,we
assume ha Assump ion 2holds. We close he model by de in-
inga MEM speci ica ion a he lowe equency. Bya e aging he
squa ed deGARCHed e u ns wi hin low- equency pe iod 𝑡,we
ob ain
𝑉𝑡=1
𝑛
𝑛
∑
𝑖=1
𝑟2
𝑖,𝑡
ℎ𝑖,𝑡
=𝜏𝑡1
𝑛
𝑛
∑
𝑖=1𝑍2
𝑖,𝑡 =𝜏𝑡𝑍𝑡,(15)
whe e 𝑍𝑡=𝑛−1∑𝑛
𝑖=1𝑍2
𝑖,𝑡 wi h E[𝑍𝑡]=1 and Va [𝑍𝑡]=(𝜅−
1)∕𝑛. Again, i ollows om Assump ion 1 ha he 𝑍𝑡a e 𝑖.𝑖.𝑑.
This sugges s he speci ica ion
𝜏𝑡=𝜆0+𝜆1𝑉𝑡−1+𝜆2𝜏𝑡−1.(16)
No e ha he low- equency componen 𝜏𝑡s ill measu es ola il-
i y in daily uni s. We will e e o his pa ame iza ion o he
long- e mcomponen as MF2-GARCH-l -𝑛,whe e“l ”s ands o
low- equency and 𝑛 e e s o he choice o he low- equency
pe iod.6Fo example, when se ing 𝑛=21 o 𝑛=63, he
long- e m componen a ies a he mon hly o qua e ly e-
quency. I Assump ions 1and 3hold, hen 𝑉𝑡=𝜏𝑡𝑍𝑡is a
co a iances a iona yMEM(1,1)wi hE[𝑉𝑡|𝑡−1]=𝜏𝑡andE[𝑉𝑡]=
𝜆0∕(1−𝜆1−𝜆2). A d awback o he low- equency upda ing o
helong- e mcomponen is ha i in oducesadiscon inui yin o
he daily condi ional a iances. Thus, he daily e u ns a e no
longe co a iance s a iona y.
4|Empi ical Applica ion
4.1 |S ock Ma ke Da a
Weuse daily e u nda a o he S&P500 s a ingin Janua y1971
and ending in June 2023. Using he no a ion o 𝑛=1, daily log
e u ns a e compu ed as 𝑟𝑡=100(log(𝑃𝑡)−log(𝑃𝑡−1)), whe e 𝑃𝑡is
he close p ice on day 𝑡. We employ ealized a iances based on
in aday da a p o ided by Tick Da a o e alua e he o ecas pe -
o mance. Daily ealized a iances, 𝑅𝑉𝑡, a e de ined as he sum
o he squa ed i e-minu e in aday log- e u ns on day 𝑡plus he
squa edo e nigh log- e u n(seeBolle sle e al.2018).Wecom-
pu e ealized a iances o he pe iod Janua y 2000 o June 2023.
Table 1shows summa y s a is ics o he daily e u ns and eal-
ized a iances.Theannualized daily e u nsha easamplemean
o 7.38%. The mean o he annualized daily ealized ola ili y is
15.81%du ing hepe iod Janua y2010 oJune2023,which is he
pe iod ha is used o he ou -o -sample o ecas e alua ion. In
addi ion, in Sec ion 4.2.3 we use e u n da a om he V-Lab o
2142 US and in e na ional equi ies.
4.2 |MF2-GARCH in Ac ion
4.2.1 |Applica ion o S&P 500
We i s apply he MF2-GARCH model o he daily log e u ns
o he S&P 500. We mainly ocus on models wi h a daily
long- e m componen and es ima e he MF2-GARCH- w-𝑚and
he MF2-GARCH-bw-𝑚 o he en i e sample pe iod om Jan-
ua y 1971 o June 2023.
Choice o 𝒎:We i s es ima e bo h models o alues o 𝑚
up o 160 and de e mine he op imal 𝑚as he one ha mini-
mizes he BIC.7Fo bo h models, he uppe le panel o Figu e 3
shows he BIC as a unc ion o 𝑚. The lowes alue o he
BIC ma e ializes o 𝑚=63 o bo h models. Fo his alue o
𝑚, he MF2-GARCH- w-𝑚is clea ly p e e ed ela i e o he
MF2-GARCH-bw-𝑚. To in es iga e whe he his pa e n holds
mo egene ally,we ees ima ebo h models o h eesubsamples:
Janua y1971-Decembe 2009(uppe igh panel),Janua y1980-
Decembe 2009(lowe le panel),and Janua y1980 oJune 2023
(lowe igh panel). In all h ee panels, he op imal choice o 𝑚
is a ound 63 and he MF2-GARCH- w-𝑚is he p e e ed model.
O e all, he subsample analysis shows ha he op imal choice o
𝑚is e y s able o he S&P 500 and ha equal weigh s a e p e-
e ed o be a-weigh s.
Pa ame e es ima es (sample pe iod Janua y 1971 o June
2023): The i s wo ows o Table 2(labeled as “𝜏𝑡cons .”)
show he pa ame e es ima es o he nes ed one-componen
GJR-GARCH. The pa ame e es ima es o 𝛼, 𝛾, and 𝛽 ake ypi-
cal alues and indica e s ong pe sis ence in he GARCH compo-
nen (𝛼+𝛾∕2+𝛽=0.982).Nex , he abledisplays hepa ame e
es ima es o he MF2-GARCH- w-𝑚model. Fi s , o he op i-
mal window leng h, ha is, 𝑚=63, he es ima es o he pa am-
e e s in he long- e m componen , 𝜆1and 𝜆2, a e bo h signi i-
can . As expec ed, he pe sis ence in he long- e m componen
(0.982) is much s onge han he pe sis ence in he sho - e m
componen (0.924). Also, due o in oducing a ime- a ying
long- e m componen , he sho - e m GJR-componen is much
less pe sis en han he one-componen GJR-GARCH. The BIC
clea ly a o s he wo-componen MF2-GARCH- w-63 o e he
one-componen model. To illus a e he consequences o choos-
ing𝑚 oo small and oola ge, we p esen pa ame e es ima es o
a mon hly (𝑚=21) and semiannual (𝑚=126) a e aging o pas
o ecas e o s.Fo 𝑚=21, he pe sis ence in he long- e mcom-
ponen inc eases o 0.995. P esumably, his is because inc eas-
ing 𝜆2smoo hes he long- e m componen and, he eby, coun-
e ac s he e ec o dec easing 𝑚.Fo 𝑚=126, he es ima es o
helong- e mcomponen ’spa ame e sa ealmos hesameas o
𝑚=63, bu he s anda d e o s inc ease conside ably.
445
GJR-GARCH, he Spline-GARCH, he GARCH-MIDAS-RV and
he log-HAR in ou -o -sample o ecas pe o mance.
In gene al, he MF2-GARCH model will bene i applica ions
ha equi e long- e m o ecas s o inancial ola ili y, such as
long- un alue-a - isk p edic ions o measu emen o sys emic
isk. Fo example, Con ad, Schoelkop , and Tush e a 2024 show
ha he MF2-GARCH allows o compu e “ ola ili y news” sepa-
a ely o he sho - e m and he long- e m componen s. Assum-
ing a posi i e ela ion be ween expec ed e u ns and he con-
di ional a iance o e u ns in a GARCH-in-Mean ype model,
unexpec ed e u ns can be decomposed in o cash low and dis-
coun a e news. In his amewo k, discoun a e news is mainly
d i enbynews o heMF2-GARCH’slong- e mcomponen .This
is because only news o he long- e m componen is pe sis en
enough o gene a esizable a ia ionin discoun a es.F om his,
i ollows ha he MF2-GARCH’s long- e m ola ili y compo-
nen is a s ong p edic o o he s eng h o he ins an aneous
esponse o he s ock ma ke o su p ises in mac oeconomic
announcemen s (see Con ad, Schoelkop , and Tush e a 2024).
I will also be in e es ing o employ he long- e m componen
in applica ions ha equi e low- equency es ima es o ola ili y,
o example, when analyzing he link be ween inancial ola il-
i yand inancialc ises(see, o ins ance,Danielsson,Valenzuela,
and Ze 2018).
Acknowledgmen s
We a e g a e ul o he co-edi o , E ic Ghysels, and wo anonymous e -
e ees o hei commen s, which g ea ly imp o ed ou pape . We would
like o hank Jö g B ei ung, Ch is ian B ownlees, Rob Capellini, Zeno
Ende s, Jean-Da id Fe manian, Ch is ian F ancq, Ch is ian Gou ie oux,
Onno Kleen, Robinson K use-Beche , Enno Mammen, Anne Opschoo ,
La a Schadwinkel, Julius Schölkop , Timo Te äs i a, and Jean Michel
Zakoïan as well as semina and con e ence pa icipan s a CREST
(Janua y 2020), he ES Wo ld Cong ess (2020), he 13 h Annual SoFiE
Con e ence (2021), and he 11 h ECB Con e ence on Fo ecas ing Tech-
niques (2021) o hei eedback on ea lie e sions o he pape . Funding
by he Ge man Fede al Minis y o Educa ion and Resea ch (BMBF) and
he Baden-Wü embe g Minis y o Science as pa o Ge many’s Excel-
lence S a egy (ExU 10.2.31) is g a e ully acknowledged. Open Access
unding enabled and o ganized by P ojek DEAL.
Da a A ailabili y S a emen
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Open Resea ch Badges
This a icle has been awa ded Open Da a Badge o mak-
ing publicly a ailable he digi ally-sha eable da a neces-
sa y o ep oduce he epo ed esul s. Da a is a ailable a
h ps://doi.o g/10.15456/jae.2025013.1232487362.
Endno es
1In V-Lab, he MF2-GARCH is es ima ed o mo e han 18,000 asse s
om di e en asse classes on a weekly basis. See: h ps:// lab.s e n.
nyu.edu/docs/ ola ili y/MF2-GARCH.
2Theda awillbein oducedanddiscussedinmo ede ailinSec ion4.1.
See also Panel A o Table 1.
3Fo de ails, see he discussion below Assump ion 6 in Con ad and
Schienle 2020, as well as Rema k 7 in he Supplemen a y Appendix
o hei pape .
4We ob ained simila esul s o he DAX and he Hang Seng Index
(HSI). In addi ion, we ound e idence o conside able co-mo emen
ins anda dized ola ili y o ecas e o sin e na ionally(seealsoEngle
and Campos-Ma ins 2023).
5Fo a g aphical illus a ion, Figu e A.3 in he Suppo ing In o ma ion
plo s he annualized uncondi ional ola ili y as a unc ion o 𝑚and
o 𝜅∈{3,5,7}. The model pa ame e s a e chosen as 𝛼=0.02,𝛾=
0.10,𝛽=0.8,𝜆
0=0.01,𝜆
1=0.05, and 𝜆2=0.94.
6We ea 𝑛as a ixed numbe ha is no “ oo la ge.” I 𝑛→∞, hen
𝑍𝑡con e ges o one in p obabili y and, hence, an iden i ica ion issue
a ises due o he linea dependence o 𝑉𝑡and 𝜏𝑡.
7As he choice o 𝑚does no a ec he numbe o pa ame e s, we could
also de e mine he op imal alue based on he likelihood unc ion. We
p e e he BIC because his allows o a meaning ul compa ison wi h
he nes ed one-componen GJR-GARCH.
8Wecheckedwhe he he eiss illp edic abili yin he ola ili y o ecas
e o so heMF2-GARCH- w-63.We oundnoe idence o au oco e-
la ion in he daily squa ed s anda dized esiduals, 
𝑍2
𝑡,whena e aged
a lowe equencies.In addi ion, heempi icaldensi yo 
𝑍𝑡isclose o
being symme ic.
9Tha is, he 𝑠-s ep ahead o ecas is gi en by: Va [𝑟𝑡]+
(𝛼𝐺𝐴 +𝛾𝐺𝐴∕2+𝛽𝐺𝐴)𝑠−1(ℎ𝑡+1𝜏𝑡+1−Va [𝑟𝑡]),whe e𝛼𝐺𝐴,𝛾
𝐺𝐴 and
𝛽𝐺𝐴 a e he pa ame e s o he GJR-GARCH and Va [𝑟𝑡]is he
uncondi ional a iance o he MF2-GARCH.
10 The inc ease in ola ili y was d i en by he Eu opean so e eign deb
c isis and a downg ade o he U.S.’s c edi a ing by S anda d & Poo ’s.
11 The spike in ola ili y was associa ed wi h ea s ha he Fede al
Rese e migh aise in e es a es.
12 We only include s ocks o which he MF2-GARCH pa ame e es i-
ma es sa is y Assump ions 2and 3.
13 We also conside ed log-HAR models wi h ( he log o ) qua e ly and
semiannual a e ages o ealized a iances as addi ional explana o y
a iables. Howe e , hose speci ica ions did no lead o an imp o ed
o ecas pe o mance ela i e o he baseline model. In addi ion, we
conside ed he log-HAR wi hou le e age and he pu e HAR model.
Again, bo h speci ica ions did no lead o imp o emen s in o ecas
pe o mance.
14 AsdiscussedinPa on2020, he ankingo modelsimpliedby heMSE
andQLIKE can di e due omodel misspeci ica iono pa ame e es i-
ma ion e o . See also Sec ion A.2 o he Suppo ing In o ma ion.
15 In line wi h hei esul , we ind ha he MCS includes essen ially all
models when we do no con ol o his pe iod o ins abili y. In his
se ing, he assump ion ha he loss di e ences a e s a iona y, which
unde lies he MCS p ocedu e (see Hansen, Lunde, and Nason 2011,
Assump ion 2), is likely o be iola ed.
16 V-Lab does no p oduce o ecas s o he HAR model o he
GARCH-MIDAS.
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