Predictor Preselection for Mixed‐Frequency Dynamic Factor Models: A Simulation Study With an Empirical Application to GDP Nowcasting

F anjic, Domenic; Schweike , Ka s en
A icle — Published Ve sion
P edic o P eselec ion o Mixed‐F equency Dynamic
Fac o Models: A Simula ion S udy Wi h an Empi ical
Applica ion o GDP Nowcas ing
Jou nal o Fo ecas ing
P o ided in Coope a ion wi h:
John Wiley & Sons
Sugges ed Ci a ion: F anjic, Domenic; Schweike , Ka s en (2024) : P edic o P eselec ion o Mixed‐
F equency Dynamic Fac o Models: A Simula ion S udy Wi h an Empi ical Applica ion o GDP
Nowcas ing, Jou nal o Fo ecas ing, ISSN 1099-131X, Wiley, Hoboken, NJ, Vol. 44, Iss. 2, pp. 255-269,
h ps://doi.o g/10.1002/ o .3193
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Jou nal o Fo ecas ing, 2025; 44:255–269
h ps://doi.o g/10.1002/ o .3193
255
Jou nal o Fo ecas ing
RESEARCH ARTICLE OPEN ACCESS
P edic o P eselec ion o Mixed- F equency Dynamic
Fac o Models: A Simula ion S udy Wi h an Empi ical
Applica ion o GDP Nowcas ing
DomenicF anjic | Ka s enSchweike
Co e Facili y Hohenheim and Ins i u e o Economics, Uni e si y o Hohenheim, S u ga , Ge many
Co espondence: Domenic F anjic ( [email protected])
Recei ed: 17 Oc obe 2023 | Re ised: 20 June 2024 | Accep ed: 20 Augus 2024
Funding: The au ho s ecei ed no speci ic unding o his wo k.
Keywo ds: elas ic ne | high- dimensional | so - h esholding | a ge ed p edic o s | a iable selec ion
ABSTRACT
We in es iga e he pe o mance o dynamic ac o model nowcas ing wi h p eselec ed p edic o s in a mixed- equency se ing.
The p edic o s a e selec ed ia he elas ic ne as i is common in he a ge ed p edic o li e a u e. A simula ion s udy and an
applica ion o empi ical da a a e used o e alua e di e en s a egies o a iable selec ion, he in luence o uning pa ame e s,
and o de e mine he op imal way o handle mixed- equency da a. We p opose a no el c oss- alida ion app oach ha connec s
he p eselec ion and nowcas ing s ep. In gene al, we ind ha p eselec ing p o ides mo e accu a e nowcas s compa ed wi h he
benchma k dynamic ac o model using all a iables. Ou newly p oposed c oss- alida ion me hod ou pe o ms he o he spec-
i ica ions in mos cases.
Jel classi ica ion: C32, C53, E37.
1 | In oduc ion
The mixed- equency dynamic ac o model (DFM) has become a
wo kho se model o mac oeconomic o ecas ing and nowcas ing
(Bok e al. 2018; Giannone, Reichlin, and Small2008). Th ough
a combina ion o ac o analysis and Kalman smoo hing (Doz,
Giannone, and Reichlin2011; S ock and Wa son2002a,2002b),
he model can handle big da a se s,1 cons uc ed om mixed-
equency p edic o s, while also exploi ing he o en sho e pub-
lica ion lags o he p edic o a iables. This is especially use ul
when o ecas ing a no ye eleased lowe equency a iable,
o en cu en qua e GDP, wi h he help o pa ially a ailable
qua e ly and mon hly economic indica o s (see, o example,
Bańbu a e al. 2013; Giannone, Reichlin, and Small2008).
By cons uc ion, he amewo k in i es o he use o inc eas-
ingly la ge da a se s o include p edic o s om all sec o s o he
economy. Howe e , conce ns ha e been aised ha he esul ing
ac o s a e less use ul o o ecas ing pa icula ly when he id-
iosync a ic e o s a e c oss- co ela ed (Boi in and Ng 2006).
The e o e, Bai and Ng(2008) p opose o employ a se o a ge ed
p edic o s (TPs) o he ac o analysis. Mo e speci ically, p edic-
o s a e p eselec ed using he elas ic ne (EN) be o e he es ima-
ion o a ac o model and he cons uc ion o a o ecas .2 The
concep is ex ended o a mixed- equency nowcas ing ame-
wo k by Bessec(2013) and Sili e s o s(2017).
The li e a u e p o ides ambiguous esul s on whe he he
TP app oach can subs an ially imp o e he o ecas ing
accu acy (Bulligan, Ma cellino, and Vendi i 2015; Cas le,
Clemen s, and Hend y2013; Eickmeie and Ng2011; Kim and
Swanson 2014, 2018). Pa icula ly, i is unknown how well
he TP app oach pe o ms in mixed- equency da a s uc u es
ha a e common in empi ical applica ions (Bańbu a e  al.
2013). In p e ious s udies, he p eselec ion s ep o he TP ap-
p oach is mos ly implemen ed as o iginally p oposed by Bai
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any medium, p o ided he o iginal wo k is
p ope ly ci ed.
© 2024 The Au ho (s). Jou nal o Fo ecas ing published by John Wiley & Sons L d.
Jou nal o Fo ecas ing, 2025
and Ng(2008), whe e he numbe o a iables aimed o be ex-
ac ed is se o 30 (see, e.g., Bai and Ng2008; Eickmeie and
Ng2011; Kopoin, Mo an, and Pa e2013; Schumache 2010).
This is in con as o he mo e common app oach in he li -
e a u e on penalized eg essions, whe e he uning pa am-
e e s a e c oss- alida ed using a pa o he sample (Has ie,
Tibshi ani, and Wainw igh 2015; Zou and Has ie2005; Zou,
Has ie, and Tibshi ani2007).
Consequen ly, he ollowing issues in e ms o model speci ica-
ion a ise: I is unclea which algo i hm should be used o sol e
he EN p oblem. A ailable choices a e, o example, coo dina e
descen o LARS- EN. Al hough he minimize o he EN ob-
jec i e unc ion is heo e ically unique, he co esponding pa-
ame e ec o , and hence he placemen and numbe o ze os,
is gene ally no unique. Depending on his choice, i mus be
decided whe he he uning pa ame e s a e de e mined based
on (i) c oss- alida ion (CV)3 o (ii) he maximum numbe o
p edic o a iables o be selec ed om he ull panel o p e-
dic o a iables. Then, he e is he addi ional ques ion o how
o handle mixed- equency da a be o e eeding i in o he EN.
The cu en li e a u e does no p o ide a conclusi e s a emen
on he pe o mance o TP nowcas ing. Only ew s udies epo
e idence o TP models o subs an ially imp o e o e bench-
ma k models when applied o di e en eal wo ld da a se s (see,
e.g., Gi a di, Golinelli, and Pappala do 2017; Kopoin, Mo an,
and Pa e2013; Schumache 2010). Ins ead, mos s udies epo
mixed esul s (Bulligan, Ma cellino, and Vendi i2015; Cas le,
Clemen s, and Hend y2013; Eickmeie and Ng2011; Kim and
Swanson2014,2018). A eason o his could lie in sub- op imal
speci ica ion o he p eselec ion s ep. The cu en li e a u e lacks
s udies ha e alua e he in luence o impo an uning pa am-
e e s and guidelines on how hese pa ame e s should be ali-
da ed. Mos o he esea ch is conduc ed by compa ing he TP
pe o mance agains o he models in selec ed empi ical da a se s
wi hou analyzing in de ail how he uning o he p eselec ion
s ep in luences he nowcas ing pe o mance. When u ning o
he issue o handling mixed- equency da a in he TP amewo k,
he li e a u e p o ides e en less guidance o applied esea che s.
To he knowledge o he au ho s, he e is no la ge- scale simu-
la ion s udy ha in es iga es he abo e- men ioned issues ex-
haus i ely. The pape a hand hus aims o con ibu e o he
li e a u e in he ollowing way: (i) I is in es iga ed whe he
any o he TP speci ica ions can imp o e he nowcas accu-
acy o di e en benchma k models in a con olled mixed-
equency se ing, and (ii) we s udy he e ec s ha he choice
o he uning pa ame e s in he p eselec ion s ep has on he
o e all model pe o mance. Pa icula ly, we compa e he pe -
o mance o da a d i en speci ica ions agains selec ing a ixed
numbe o p edic o s. (iii) We p opose a new TP CV s a egy
based on p io nowcas e o s ha seems o pe o m well com-
pa ed wi h he exis ing app oaches. Addi ionally, we assess he
p edic i e accu acy o hose me hods o empi ical da a se s
ha di e in he numbe o p edic o s and spa ial esolu ion.4
Ou indings om he simula ion s udy and he empi ical ap-
plica ion allow o mul iple conclusions. Mos no ably, we ind
ha o each DGP and eal da a se in es iga ed, a leas one TP
speci ica ion can imp o e he nowcas accu acy compa ed wi h
he benchma k models. Selec ing a ixed numbe o p edic o s
pe o ms easonably well, pa icula ly i he da a se con ains
many i ele an a iables. Howe e , we can show ha he TP
models wi h a da a d i en p eselec ion s ep equen ly ou pe -
o m he models wi h a ixed numbe o selec ed p edic o s. The
da a d i en (o c oss- alida ed) models wo k well in misspec-
i ied o noisy sys ems and achie e he bes esul s in he em-
pi ical applica ion. The p oposed TP speci ica ion ha links he
p eselec ing and nowcas ing s eps by c oss- alida ing based on
he nowcas e o yields he mos consis en esul s, ou pe o m-
ing he benchma k DFM in con olled se ings and in ou empi -
ical applica ion.
The emainde o his pape is o ganized as ollows. Sec ion 2
discusses he necessa y basics o DFM nowcas ing, ollowed by
an ou line o he model speci ica ions and hei implemen a ion.
Sec ion3 desc ibes he se - up and epo s he esul s om ou sim-
ula ion s udy. In Sec ion4, he empi ical da a se s a e in oduced
and he ela i e pe o mance o he es ima o s a e e alua ed in a
eal wo ld con ex . Sec ion5 summa izes he indings o his s udy.
2 | Me hodology
2.1 | DFM
In he ollowing, we b ie ly desc ibe he mixed- equency
DFMs es ima ed wi h he wo- s ep p ocedu e p oposed by
Giannone, Reichlin, and Small(2008) and Doz, Giannone, and
Reichlin (2011). Ini ially, we conside a ec o o s a iona y
mon hly a iables
z
={
zh,
}
H
h=1
and a DFM cha ac e ized by he
ollowing equa ions:
Fu he , i allowing o missing da a a he igh end o he ime
index due o di e en publica ion lags, so- called agged edges, i
is assumed ha
He e,
�
={
,
}
R
=1
is he ec o o s a ic ac o s a ime
, and
𝚲
={𝜆h, }
H,R
h=1, =1
is he ac o loadings ma ix.
𝝃
={
𝜉h,
}
H
h=1
is he
idiosync a ic e o ec o a ime
wi h diagonal co a iance ma-
ix
𝚺𝝃
. The ma ix
𝚼
={𝜐
,
q}
R,Q
=1,q=1
has ull ank
Q
, and he ec-
o o p imi i e shocks
𝝐
={𝜖q, }
Q
q=1
is assumed o be a whi e
noise p ocess wi h ze o mean and co a iance ma ix
IQ
.
Fu he mo e, he oo s o he lag polynomial
𝚽P(
𝕃
)
, whe e
𝚽p
={𝜙
,s,p
}
R,R
=1,s=1
, o all
p=1…,P
, a e assumed o lie ou side
he uni ci cle.
The mixed- equency s uc u e o he DFM is implemen ed ac-
co ding o Ma iano and Mu asawa(2003). We assume ha he
a ge a iable, o example, log GDP, is in eg a ed o o de one.
Fu he , we assume ha qua e ly obse a ions a e eco ded
(1)
z =𝚲 +𝝃 ,𝚽
P
(𝕃) =𝚼𝝐
𝔼(𝝃 𝝃�
)=𝚺𝝃=diag(𝜎2
1,𝝃,…,𝜎2
H,𝝃)
𝔼
(𝝃
𝝃�
−s
)=0, ∀ ,s>0, 𝔼(𝝃
(
𝚼𝝐
−w)
�)=0, ∀w,
.
(2)
𝜎
2
h,𝝃=
{
𝜎2
h>0, i zh, is a ailable
∞, o he wise
.
256
in he las mon h o he qua e .
x ,q
deno es a qua e ly a i-
able, and
x − m,m
, o
m=0,1∕3,2∕3
, is he co esponding la en
mon hly a iable a he las , second, and i s mon h o qua e
. Analogously o ou empi ical applica ion, we use log GDP o
ou illus a ion. Since log GDP is a low a iable, he qua e ly
obse a ions
lngdp ,q
a e ela ed o unobse ed mon hly obse -
a ions
lngdp ,m
ia
and i holds o he qua e ly g ow h a es,
Δ3ln gdp ,q
=
ln gdp ,q
−
ln gdp
−
1,q
, ha
Equa ion(4) exp esses he qua e ly g ow h a es as a unc ion
o he mon hly g ow h a es and he eby p o ides us wi h he
agg ega ion weigh s o he mixed- equency DFM.5
As pa o he model selec ion p ocess, we de e mine he num-
be o common ac o s by op imizing he in o ma ion c i e ion
p o ided in Bai and Ng(2002). To ensu e some o m o dimen-
sionali y educ ion, he maximum alue o

R
o be conside ed is
min(⌊0.5N∗
m⌋, 15)
o he simula ed da a and
min(⌊0.5N∗
m⌋, 5)
o he eal wo ld da a, whe e
N∗
m
is he numbe o mon hly TPs
and
⌊
⋅
⌋
is he loo unc ion. The numbe o p imi i e shocks
Q
is chosen o be es ima ed using he p ocedu e in Bai and
Ng(2007). Fo he o de o he VAR p ocesses o he common
ac o , we ollow he con en ion used in Ma colino de Ma os
e al.(2019), whe e a alue o

P
is e ie ed by compu ing he
AIC, BIC, HQC, and inal p edic ion e o and choosing he
mos pa simonious model indica ed by hese measu es.
2.2 | P epa ing Mixed- F equency Da a
o he P eselec ion S ep
Since he EN is no able o handle mixed- equency da a se s
au oma ically, i equi es a p elimina y s ep c ea ing a bal-
anced se o a iables. The TP li e a u e p o ides wo ideas in
his ega d. The i s app oach p oposed by Sili e s o s(2017)
is a skip- sampling o blocking app oach o en used in MIDAS
models. In his case, he mon hly obse a ions o a p edic-
o a iable a e ans o med in o h ee qua e ly a iables,
he eby aligning he equencies o he qua e ly a iable o
in e es and he mon hly p edic o s. Fo example, unemploy-
men a es as a mon hly p edic o a iable is ans o med in
such a way, acking he unemploymen a e a he i s , sec-
ond, and hi d mon h o a qua e .
Skip- sampling he mon hly a iables has some ob ious
d awbacks. The comple e da a ma ix
Xs
is o dimension
((Nq+3Nm)×𝜏)
, whe e
𝜏<T
is he poin in ime o which all
p edic o s a e obse ed. Gi en he numbe o mon hly obse -
a ions,
Nm
, skip- sampling hus inc eases he dimensionali y
o he p oblem conside ably, which can lead o high compu a-
ional cos s. Mo eo e , since he a iables a e highly co ela ed,
inc easing he o e all numbe o a iables also inc eases he
likelihood o including i ele an a iables.
Al e na i ely, Bessec (2013) agg ega es he mon hly a iables
in o pseudo qua e ly indica o s. The agg ega ion app oach
has he ad an age ha he dimensionali y o he p oblem is
no u he inc eased a he cos o losing some in o ma ion
om he highe equency p edic o s. This s udy di e s om
Bessec (2013) in ha he mon hly a iables a e no a e aged
o e a qua e . Ins ead, we ollow he ideas o Ma iano and
Mu asawa(2003) and agg ega e he mon hly a iables using he
geome ic mean o he h ee mon hly a iables o each qua e .6
Hence, he agg ega ion scheme in his s ep is concep ually simi-
la o he one used in he mixed- equency DFM.
2.3 | P eselec ion wi h he EN
The EN was i s in oduced in Zou and Has ie(2005) and ep-
esen s a combina ion o he Ridge and LASSO egula iza ion.
To sol e he EN p oblem, we conside wo popula algo i hms,
namely, LARS- EN (Zou and Has ie2005) and coo dina e de-
scen (F iedman, Has ie, and Tibshi ani2010), and pu special
emphasis on he uning pa ame e s o hese algo i hms.7
Conside he balanced in age
𝜏
. Fo he a iable selec ion s ep,
le he a ge a iable be he qua e ly ime se ies wi h index one
and de ine
y:=x1,q={x1, ,q}𝜏
=1∈X𝜏,q
. Le
X
={xn, }
M,𝜏
n=1, =1
, a
same- equency p edic o ma ix, be e ie ed om he
p edic o s in
𝜏
by one o he me hods desc ibed abo e, whe e
M
is ei he equal o
N−1
o
Nq+3Nm−1
depending on whe he
agg ega ion o skip- sampling is used. The EN p oblem is
de ined as
whe e
𝜷
={𝛽
n
}
M
n=1
is he coe icien ec o . The pa ame e s
l1
and
l2
a e, espec i ely, he LASSO (Tibshi ani1996) and Ridge
(Hoe l and Kenna d1988) uning pa ame e .
I is also common o de ine
𝛼
=
l
1
(l1+l2)
and ecas (5) in o
He e,
𝛼∈[0,1]
is a mixing, o weigh , pa ame e be ween he
wo size cons ain s.
The i s algo i hm is e e ed o as he LARS- EN algo i hm
(E on e  al. 2004; Zou and Has ie 2005) and is mos com-
monly used in he TP li e a u e (see, e.g., Bai and Ng2008;
Bessec2013; Sili e s o s2017). Despi e i s popula i y in he
TP li e a u e, he p ocedu e has a po en ial d awback when
used in a pu ely da a d i en amewo k. E en ually, a some
poin along he solu ion pa h, each a iable mo es in o he ac-
i e se . Fo la ge da a se s, his equi es mo e s eps o calcu-
la e he ull solu ion pa h, and sol ing he p oblem becomes
compu a ionally cos ly. This is o special conce n wi h espec
o skip- sampling, whe e he numbe o a iables in he model
ma ix is d as ically inc eased.
(3)
ln gdp
,q=
1
3
(ln gdp ,m+ln gdp −1∕3,m+ln gdp −2∕3,m)
,
(4)
Δ
3ln gdp ,q=
1
3Δln gdp ,m+
2
3Δln gdp −1∕3,m+Δln gdp −2∕3,
m
+2
3
Δln gdp −1,m+1
3
Δln gdp −4∕3,m.
(5)
min�‖
y−X
�
𝜷
‖2
+l
1‖
𝜷
‖1
+l
2‖
𝜷
‖2�,
(6)
min�‖
y−X�𝜷
‖
2+l
�
𝛼
‖
𝜷
‖
1+
(1
−
𝛼)
2‖
𝜷
‖
2
��.
257
An al e na i e algo i hm, he so- called coo dina e descen , has
been p oposed by F iedman, Has ie, and Tibshi ani(2010). The
p ocedu e has he ad an age ha i is gene ally as e han he
LARS- EN app oach when applied o la ge c oss sec ions. I
migh hus be possible o use coo dina e descen in he TP ame-
wo k o achie e simila o ecas ing e o imp o emen s wi h
less compu a ional cos s compa ed o he LARS- EN algo i hm.
The EN uning pa ame e s need o be de e mined by he e-
sea che and se e al s a egies a e o e ed in he li e a u e.
Fo he coo dina e descen algo i hm, we ollow he de aul
se ings o he glmne package by F iedman, Has ie, and
Tibshi ani (2010). Fo he LARS- EN algo i hm, he
l2
g id is
chosen o be a sequence o 99 loga i hmically spaced alues be-
ween
ln(1.001)
and
ln(10)
, whe e 0 is added o inco po a e he
pu e LASSO i . The maximum numbe o s eps
k
is se o he
numbe o a iables o he gi en da a se .8
Fo CV, an expanding window ime se ies CV p ocedu e acco d-
ing o Hyndman and A hanasopoulos ((2018), Ch. 5.10) is imple-
men ed. Di e en o he classical CV app oach applied o i.i.d.
da a, his speci ic app oach p ese es he ime- se ies s uc u e
o he unde lying obse a ions.
Fo he TP speci ica ion wi h ixed pa ame e s
l2
and
k
, we use
he alues epo ed by Bai and Ng(2008), ha is,
l2=0.25
and
k=30
. This speci ica ion is o en ound in empi ical s udies.9
2.4 | Model Speci ica ions
We begin wi h a desc ip ion o he i s TP speci ica ion am-
ily ha uses in o ma ion con ained in he da a o une he EN
pa ame e s and he e o e guide he a iable selec ion p ocess.
The main in ui ion behind hese da a d i en pa ame e iza ions
is ha he TP models migh be mo e e ec i e i hey we e spe-
ci ically ained o he unde lying da a ins ead o elying on a
ixed numbe o TPs. Fu he mo e, i mus be assumed ha in
a eal wo ld scena io, he se o a iables leading o he smalles
nowcas ing e o migh be ime- a ying. The da a d i en spec-
i ica ions hus p o ide a mo e lexible amewo k in his ega d
when compa ed o he ixed alue app oach.
To cons uc he nowcas a ime
T
, we i s build he balanced
panel wi h a iables in
X
𝜏
,q
and
X𝜏,m
by ei he agg ega ing o
skip- sampling. As abo e,
𝜏<T
is he mos ecen da e o which
he panel
𝝉
is balanced. Second, we use he a iables con ained
in
X𝜏
and he a ge a iable
y𝜏
o sol e he EN p oblem ei he
ia LARS- EN o coo dina e descen . In bo h cases, we alida e
he pa ame e s o he EN ia ime- se ies CV using a one- s ep-
ahead o ecas . Then,

∗
T=
{
X∗
q,T,X∗
m,T
}
is he in age o TPs
cons uc ed om a iables o which he solu ion o he EN
p oblem in he p e ious s ep has indica ed non- ze o coe i-
cien s. Now, we can use
∗
T
o cons uc a nowcas o
yT
using
he mixed- equency DFM. This is done using a epea ed a i-
able selec ion s ep wi h an expanding window p io o each
nowcas .10 No e ha in o al, he e a e ou speci ica ions unde
in es iga ion o his TP speci ica ion amily. Fo simplici y, we
employ he ollowing e minology. Fo each speci ica ion, he
i s pa o i s name is ei he an “A- ” i he mixed- equency
da a is agg ega ed o an “S- ” i i is skip- sampled. The ollowing
pa indica es whe he he EN is sol ed ia coo dina e descend
(“CD- ”) o he LARS- EN algo i hm (“LE- ”). This is ollowed by
“TPN” o a a ge p edic o nowcas .11
In addi ion o he amily o da a d i en models, he ixed alue
p ocedu es a e in es iga ed. Res ic ing he model o always
selec ing a ixed numbe o p edic o s migh lead o imp o e-
men s in e ms o nowcas ing pe o mance by way o a mo e
s able a iable selec ion s ep. The e minology abo e is also ap-
plied o he wo ixed alue speci ica ions. The skip- sampling
app oach is deno ed by “S- F- TPN” and agg ega ion is deno ed
by “A- F- TPN.”
We also conside wo new ypes o speci ica ions ha we de-
no e by nowcas alida ed TP speci ica ions. In hese da a
d i en speci ica ions, he inal alida ion o he se o TPs is
no achie ed by he EN based on an in- sample i bu a he i
is based on he nowcas ing e o o he p e ious qua e . The
main in ui ion is ha he EN e en ually leads o a se o TPs
e ie ed ia a linea p ojec ion o he a ge a iable on o he
p edic o space. Howe e , ac o models a e used in a second
s ep o link he p edic o s o he a iable o in e es , and a se
o p edic o s leading o he lowes nowcas ing e o may no
gene ally coincide wi h he se o p edic o s iden i ied by he
EN. Thus, a na u al ex ension o he models abo e is o use he
EN as a i s pass il e , gi en a g id o alues o he mixing pa-
ame e , gene a ing 100 candida e se s, and hen e alua e hese
se s using he p e ious pe iod's nowcas ing e o . Mo eo e , i
he alida ion is conduc ed as a pseudo- nowcas ing exe cise,
ha is he publica ion lags a e accoun ed o in he nowcas
s ep, his app oach allows o a a iable selec ion s ep ha im-
plici ly accoun s o he elease s uc u e o all a iables. The
nowcas alida ion p ocedu e sligh ly di e s om he p e ious
wo model amilies in he second s ep. He e, we use
X𝜏−1
and
y𝜏−1
o sol e he EN p oblem ia coo dina e descen o a g id
o
𝛼
alues
𝜶
, whe e
l
is c oss alida ed. Le
:={𝛼∈𝜶,𝜏}
be
he se o in ages, whe e each in age co esponds o a alue
in
𝜶
, and consis s o he a iables o which he solu ion o
he EN p oblem epo s non- ze o coe icien s gi en he co e-
sponding
𝛼
alue. We use

o cons uc a nowcas o
y𝜏
o
each unique in age in he se .12 When doing so, we impose
he elease s uc u e o he unde lying da a obse ed a
T
on o
each
𝛼∈𝜶,𝜏
o explici ly accoun o he impo ance o he ime-
liness o he elease o each p edic o . The TPs se is hen ound
as he se esul ing in he smalles RMSFE. The wo e sions
o hese models a e dis inguished by hei handling o mixed-
equency da a and a e deno ed by “A- N- TPN” and “S- N- TPN,”
espec i ely.
The esul s o i e benchma k models a e also epo ed.
Speci ically, we analyze he o ecas ing e o s o an AR(1),
ARMA, and nai e cons an g ow h model. Fu he mo e, he
nowcas esul s o a DFM using all a iables, e e ed o as
c owded DFM o CDFM, is he mos impo an benchma k o
he TP speci ica ions. In he simula ion s udy, we a e able o
compu e nowcas s om a DFM ha only uses he ele an a i-
ables, he so- called o acle DFM (ODFM). This is done o com-
pa e he esul s o using all he a iables wi h hose o using only
he ones s emming om he same p ocess as he a ge a iable
(CDFM s. ODFM).
258 Jou nal o Fo ecas ing, 2025

3 | Simula ion
3.1 | Se up
In ou simula ion s udy, ou di e en DGPs a e in es iga ed.
Inspi ed by he ideas o Boi in and Ng(2006), each DGP is com-
posed o wo di e en models, which a e s uc u ally iden ical,
o example, bo h se ies admi a ac o model o simila dimen-
sions wi h he same dynamic s uc u e bu a e pa ame e ized
di e en ly. The models a e only connec ed by hei idiosync a ic
e o s, which a e all d awn om he same mul i a ia e dis i-
bu ion wi h a non- diagonal co a iance ma ix, ha is, he idio-
sync a ic e o a e co ela ed ac oss models. Addi ionally, each
DGP is conside ed using ei he a ela i ely high o low signal- o-
noise a io (SNR) in he measu emen equa ion. Fo simplici y,
we deno e he da a s emming om he same p ocess as he a ge
a iable by ele an (subsc ip “ e”) while he o he a iables a e
e e ed o as i ele an (subsc ip s “i ”). The main easons o
using his p ocedu e a e he ollowing: Fi s , c oss- co ela ion
o he e o e ms is he main d i e o why a p eselec ion s ep
was ini ially conside ed. The e o e, we also d aw om c oss-
co ela ed e o s in he same p ocess. Second, when using la ge
da a se s, i is di icul o assume ha all a ailable a iables a e
ele an , ha is, s em om a single model. Thus, we sample
om wo di e en models o p o ide a mo e ealis ic scena io.
Las ly, conside ing di e en SNRs is impo an , since he e ec
o an inc ease in he SNR on he o ecas ing pe o mance is am-
biguous. Fo example, i could be ha a lowe SNR inc eases he
need o a iable p eselec ion, since he e ec s o including i el-
e an p edic o s a e s onge han in he case o a high SNR. On
he o he hand, when he SNR is low, he EN migh no be able
o co ec ly iden i y he impo an p edic o s.13, 14
The i s and second DGPs a e a combina ion o ac o models
ha closely esemble he heo e ical unde lying DFM in he
nowcas s ep. The i s DGP (DGPI) is a combina ion o wo
i e- ac o models, whe e hei unde lying s uc u e is aken
om Bai and Ng(2007). The second DGP (DGPII) is chosen
o be a mo e pa simonious single ac o model wi h each ac-
o ollowing a whi e noise p ocess. Le
x , e
={x
n, , e
}
N
e
n=1
and
x ,i
={x
n, ,i
}
N
i
n=1
be he se s o ele an and i ele an a iables
a ime
. The DGPs a e o mally de ined as
and
The mo i a ion o using he abo e DGPs is wo old. Fi s ,
since he DGPs closely esemble he p ocesses in he Giannone,
Reichlin, and Small(2008) amewo k, i can be assumed ha
he nowcas ing s ep es ima es he ue pa ame e iza ion o
he model mo e p ecisely i he a iables s emming om he
ele an p ocesses ha e been iden i ied co ec ly. Howe e , i
can be expec ed ha he a iable selec ion s ep is uns able,
because he ac o d i en dependence s uc u e has o be ap-
p oxima ed linea ly in he p eselec ion s ep. Compa ing DGPI
and DGPII, i migh be possible o in e on he ole o he com-
plexi y o he unde lying DGP on he a iable selec ion and
nowcas ing accu acy.
The hi d DGP conside ed (DGPIII) is concep ually di e en
om he ones abo e. He e, he da a se con ains a iables ha
a e gene a ed om wo VARMA
(1,1)
p ocesses, whe e he e -
o s a e c oss- co ela ed.
Con a y o he DGPs desc ibed abo e, he TP app oach is
expec ed o pe o m compa a i ely well when applied o da a
gene a ed by DGPIII since he model is linea in he p edic-
o s by cons uc ion. Pa icula ly, he a iable selec ion is ex-
pec ed o be mo e s able. The nowcas ing s ep, howe e , migh
c ea e mo e p oblems o he DFM since he model migh no
be app oxima ed well by a small numbe o common ac o s.
The e o e, including i ele an a iables and c oss- co ela ed
e o s migh hea ily in luence he nowcas ing s ep. We ex-
pec ha o he VARMA pa ame e iza ion, he TP app oach
should esul in conside ably mo e p ecise nowcas s compa ed
o he CDFM.15
The las DGP(DGPIV) is a combina ion o wo symme ic DFMs
whe e he measu emen equa ion includes lags o he la en ac-
o s which ollow a VMA p ocess. Again, he s uc u e o he
DFMs was aken om Bai and Ng(2007). Fo mally, he da a a e
gene a ed ia
Bo h p ocesses in DGPIV a e highly pa ame e ized and highly
dynamic DFMs ha di e om he unde lying model o he
Giannone, Reichlin, and Small(2008) amewo k, whe e he
ac o s ha e an au o eg essi e lag s uc u e and he obse ed
a iables a e no dependen on he lags o he ac o s explici ly.
Especially since he nowcas ing amewo k ea s he lags o
he ac o s as addi ional s a ic ac o s, i migh be expec ed
ha , in gene al, he nowcas ing p ocedu e is less p ecise.
Fu he mo e, since he ac o s ollow a VMA p ocess, i is
likely ha
P
is es ima ed o be la ge hus u he inc easing
he dimensionali y o he es ima ed model. The a iable se-
lec ion s ep is assumed o be e y uns able since a linea i o
he highly complex DGP is di icul o achie e. Summa izing,
(DGPI)
x
, e
=𝚲
e
, e
+𝝃
, e
, e
=𝚽
e
−1, e
+𝝂
, e
x ,i =𝚲i ,i +𝝃 ,i ,i =𝚽i −1,i +𝝂 ,i
𝝂
, e =𝚼 e𝝐 , e, and 𝝂 ,i =𝚼i 𝝐 ,i
𝝃 :=(𝝃�
, e,𝝃�
,i )�∼(0N,𝚺𝝃), whe e N=N e +N
i
R=5, Q=3, and P=1.
(DGPII)
x
, e
=𝝀
e
, e
+𝝃
, e
, e
=𝜖
, e
x ,i =𝝀i ,i +𝝃 ,i ,i =𝜖 ,i
𝝃 :=
(
𝝃�
, e,𝝃�
,i
)
�
∼(0N,𝚺𝝃), whe e N=N e +N
i
R=1, Q=0, and P=0.
(DGPIII)
x
, e
=𝚷
e
x
−1, e
+Θ
e
𝝃
−1, e
+𝝃
, e
x ,i =𝚷i x −1,i +Θi 𝝃 −1,i +𝝃 ,i
𝝃 :=
(
𝝃�
, e,𝝃�
,i
)
�
∼(0N,𝚺𝝃), whe e N=N e +N
i
(DGPIV)
x
, e =
(
𝚲0, e +𝚲1, e𝕃+𝚲2, e𝕃
2)
, e +𝝃 , e
, e =Ψ e𝝐 −1, e +𝝐 , e
x ,i =(𝚲0,i +𝚲1,i 𝕃+𝚲2,i 𝕃2) ,i +𝝃 ,i
,i =Ψi 𝝐 −1,i +𝝐 ,i
𝝃 :=
(
𝝃�
, e,𝝃�
,i
)
�
∼(0N,𝚺𝝃), whe e N=N e +N
i
R=6, Q=2, and P=∞.
259
i is expec ed ha he TP app oach, i a all, only leads o small
imp o emen s o he nowcas ing accu acy in his case.
Fo he pa ame e iza ion o he DGPs, we ollow Bai and
Ng (2007) and d aw he ac o loading ma ices and ec o s
𝚲
e ={𝜆n, , e}
N
e
,R
n=1, =1
,
𝚲
i ={𝜆n, ,i }
N
i
,R
n=1, =1
and
𝝀 e
={𝜆
n, e
}
N
e
n=1
,
𝝀i
={𝜆
n,i
}
N
i
n=1
om a s anda d no mal mul i a ia e dis ibu ion.
This is also he case o he ac o loadings co esponding o he
lagged ac o s in DGPIV. Fu he mo e, o all e o ec o s o he
ac o p ocesses i holds ha
𝝐
, e
∼(0
N
e ,I
N
e )
,
𝝐
,i
∼(0
N
i ,I
N
i )
,
𝜖 , e
∼
(0,1)
, and
𝜖 ,i
∼
(0,1)
. Fo DGPI,DGPII, and DGPIV,
he co a iance ma ix o he idiosync a ic e o
𝚺𝝃
, ha ing nonze o
o - diagonal elemen s, is cons uc ed by
𝚺𝝃=INRIN
, whe e
R
is a
andomly d awn co ela ion ma ix using he co ma ix
unc ion o he clus e Gene a ion package by Qiu and
Joe (2020). Fo DGPIII,
IN
is eplaced by
0.005
⋅
IN
o he
cons uc ion o
𝚺𝝃
, such ha he simula ed se ies ha e a
simila magni ude compa ed o hose d awn om he o he DGPs
be o e s anda diza ion. Fo he low SNR case, he a iances a e
mul iplied by a ac o o 5. The s a ing ec o o each ac o , o in
he case o DGPIII o he a iables, is chosen o be ze o, o he
ze o ec o , and a bu n- in pe iod o 200 obse a ions is used.
Fo DGPI and DGPIV, i is chosen ha
𝚽 e
=
𝚽i =
diag(0.2,0.375,0.55,0.725,0.9)
and
Ψ e =Ψ
i =diag(0.2,0.9)
, e-
spec i ely (Bai and Ng2007). Fu he , he ma ices
𝚼 e
and
𝚼i
a e calcula ed ia
𝚼 e =A eS eA e
and
𝚼i =Ai Si Ai
, whe e
A e
and
Ai
a e andom o hono mal ma ices compu ed ia
he ando ho unc ion p o ided by he p acma package by
Bo che s(2021), and
S e
and
Si
a e diagonal ma ices o ank
Q
wi h elemen s d awn om
(0.8,0.12)
. To ensu e s a iona i y
in DGPIII,
𝚷 e =P eD eP e
and
𝚷i =Pi Di Pi
, whe e
P e
and
Pi
a e ma ices wi h andom en ies d awn om
(−0.5,0.5)
,
and
D e
and
Di
a e diagonal wi h andom en ies d awn om
(0.1,0.8)
. To gua an ee in e abili y,
Θ e
and
Θi
a e con-
s uc ed analogously.
Since he s udy aims o simula e a eal wo ld nowcas ing exe -
cise, a qua e o bo h he ele an and i ele an a iables is
agg ega ed ia(4) o esemble qua e ly obse a ions. While
his does no in luence he ex ac ion o he ac o s in he
Giannone, Reichlin, and Small(2008) p ocedu e, i is expec ed
o ha e an impac on he a iable selec ion s ep. Conside ing
he lag s uc u e, only he a ge a iable is lagged by 90 days.
We e ain om imposing a andom lag s uc u e on he p e-
dic o s since his would only in luence he ela i e TP pe o -
mance i he publica ion lags we e longe han he one o he
a ge a iable, which is a ely he case in p ac ice.16
The da a is d awn o
(T,N e,Ni )={(200,150, 50), (200,100, 100),
(200,50, 150)}
o e alua e he TP p ocedu e o di e en com-
bina ions o ele an and i ele an a iables. The expanding
nowcas ing window spans he las 100 obse a ions while he
i s 100 a e used as he ini ial aining se .17
3.2 | Resul s
To e alua e he pe o mance o he di e en TP speci ica-
ions, he RMSFE aken o e he comple e nowcas ing pe iod
is epo ed. The esul s a e ound in Tables1 and 2 whe e he
ela i e pe o mance wi h espec o he CDFM is highligh ed.
We a i e a ou gene al conclusions. Fi s , in mos cases, i
is ound ha he ODFM model ou pe o ms he CDFM model
which is i sel ou pe o med by some o he TP speci ica ions.
This gi es clea e idence ha (i) TP seems o be a iable op ion
o imp o e nowcas accu acy, and (ii) he op imal se o TP is
no always equal o he se o ele an a iables. In o he wo ds,
e en i he ue model speci ica ion was known, a ge ing p e-
dic o s migh s ill imp o e he nowcas ing pe o mance exploi -
ing he dependence s uc u e in a gi en da a se .
Second, he TP p ocedu e achie es he wo s nowcas ing esul s
o
N e =50
and
Ni =150
. Bo h in e ms o absolu e and ela-
i e pe o mance, a clea inc ease in he RMSE is ound o all
TP speci ica ions. While his is o be expec ed, because ha ing
mo e i ele an a iables in he da a se on a e age leads o a
highe likelihood o selec ing mo e i ele an a iables, i clea ly
shows ha he TP p ocedu e should be used wi h ca e. Simply
adding (possibly i ele an ) a iables o he da a se does no
boos he TP pe o mance.
Thi d, using CV me hods seems o ou pe o m choosing a ixed
numbe o p edic o s. While he A- F- TPN p o ides he lowes
RMSFE o 5 ou o 12 DGP con igu a ions in he high SNR se -
ing, i is also highly ola ile esul ing in RMSFE inc eases by
up o 81%. The nowcas alida ed speci ica ions only achie e he
lowes RMSFE in 4 ou o 12 cases. Howe e , he S- N- TPN spec-
i ica ion is e y eliable bea ing he benchma k CDFM in 9 ou
o 12 DGP con igu a ions. In he low SNR se ing, he nowcas
alida ed speci ica ions p o ide he lowes RMSFE in 9 ou o
12 con igu a ions clea ly ou pe o ming he ixed speci ica ions.
This hin s owa ds he impo ance o using he in o ma ion con-
ained in he da a when applying he TP amewo k, ins ead o
elying on ad hoc solu ions, ha is, selec ing he numbe o p e-
dic o s be o ehand.
Finally, dec easing he signal- o- noise a io conside ably wo s-
ens he o e all pe o mance o he TP amewo k. The TP p o-
cedu e o en esul s in an RMSFE imp o emen o only abou
1%. No e, howe e , ha in his se ing he CDFM p o ides e-
sul s ha a e o en only ma ginally be e han he mo e pa si-
monious AR(1), ARMA, and nai e mean benchma ks, i a all.
The e o e, using TP models can s ill be a iable al e na i e o
imp o e he nowcas ing pe o mance o mixed- equency DFMs
in noisie se ings.
Tu ning o he ela i e pe o mance o each speci ica ion in
mo e de ail, we ind ha mos TP speci ica ions a e able o
p o ide lowe RMSFEs han he mos impo an benchma k
model (CDFM). The S- N- TPN speci ica ion is ound o be he
one ha ou pe o ms he CDFM benchma k mos o en, bea ing
he benchma k in 9 ou o 12 cases. Taking he a e age o e he
DGPs, i is ound ha he A- LE- TPN, S- LE- TPN, and S- N- TPN
p o ide a lowe RMSFE han he CDFM. Accoun ing o he
expec edly bad pe o mance in he case o DGPIV (see Table1)
and only aking a e ages o e DGPI – DGPIII, i is ound ha
each speci ica ion esul s in a lowe RMSFE by a leas 4%. The
single bes pe o ming speci ica ion is ound o be he A- N- TPN
260 Jou nal o Fo ecas ing, 2025
TABLE 1 | Rela i e oo mean squa ed e o o 100 a ge ed p edic o s nowcas s o
N=200
a iables gene a ed wi h a high signal- o- noise a io.
N el
=
150, Ni
=
50
N el
=
100, Ni
=
100
N el
=
50, Ni
=
150
Model DGPI DGPII DGPIII DGPIV DGPI DGPII DGPIII DGPIV DGPI DGPII DGPIII DGPIV
Nai e 1.22 1.06 1.37 4.25 1.19 1.06 1.10 4.18 1.22 1.06 0.99 3.86
AR 1.17 1.06 1.37 4.24 1.14 1.06 1.10 4.17 1.16 1.06 1.02 3.85
ARMA 1.17 1.06 1.37 4.24 1.14 1.06 1.10 4.17 1.16 1.06 1.02 3.85
ODFM 1.02 1.00 1.00 0.95 0.98 1.00 0.77 0.97 0.99 1.00 1.00 0.95
CDFM 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
A- CD- TPN 0.90 0.95 0.97 1.77 0.94 1.02 0.94 1.47 0.99 0.99 0.95 1.63
A- LE- TPN 1.00 0.99 1.00 0.99 0.93 0.99 0.77 1.01 0.98 0.98 1.00 1.23
S- CD- TPN 0.97 1.02 0.88 1.37 1.02 1.02 0.89 1.23 0.93 1.00 0.99 1.20
S- LE- TPN 1.02 1.00 0.96 0.98 0.98 0.99 0.78 0.97 1.00 1.00 1.00 0.96
A- F- TPN 0.92 0.81 1.14 1.81 0.86 0.79 0.73 1.23 0.86 0.95 0.98 1.17
S- F- TPN 0.93 0.91 1.09 1.40 0.90 0.87 0.77 1.56 0.94 0.96 1.01 1.03
A- N- TPN 0.94 1.03 0.79 1.21 1.05 1.01 0.65 1.20 0.88 1.00 0.91 1.27
S- N- TPN 0.98 0.99 0.84 0.97 0.94 1.01 0.67 1.10 0.94 0.96 0.95 1.17
No e: The e o measu e is epo ed ela i e o he benchma k CDFM. Bold p in ed alues indica e he lowes achie ed RMSFE. The model ac onyms co espond o he ollowing speci ica ions: AR: An AR(1) model; ARMA: An
ARMA model wi h lag o de de e mined ecu si ely ia he AIC; Nai e: A nai e uncondi ional mean model; ODFM: A DFM using only he ele an a iables; CDFM: A DFM using all a iables; A- CD- TPN (S- CD- TPN): The se o
a ge ed p edic o s is e ie ed ia he EN, which is sol ed using coo dina e descen , wi h EN pa ame e s e ie ed ia CV. The da a is agg ega ed (skip- sampled); A- LE- TPN (S- LE- TPN): The EN is sol ed ia LARS- EN; A- N- TPN
(S- N- TPN): A TP model whe e he se o a ge ed p edic o s is e ie ed using he nowcas ing e o om he p e ious pe iod; A- F- TPN (S- F- TPN): A TP model whe e he se o a ge ed p edic o s is e ie ed ia he EN, whe e he EN
pa ame e s a e se so ha 30 p edic o s a e selec ed. The a iables a e newly selec ed o each nowcas .
261
TABLE 2 | Rela i e oo mean squa ed e o o 100 a ge ed p edic o s nowcas s o
N=200
a iables gene a ed wi h a low signal- o- noise a io.
N el
=
150, Ni
=
50
N el
=
100, Ni
=
100
N el
=
50, Ni
=
150
Model DGPI DGPII DGPIII DGPIV DGPI DGPII DGPIII DGPIV DGPI DGPII DGPIII DGPIV
Nai e 1.01 0.99 1.33 1.22 1.00 0.99 1.15 1.10 0.99 1.00 1.00 1.00
AR 0.98 0.99 1.32 1.21 0.97 0.99 1.16 1.08 0.96 1.00 1.05 0.99
ARMA 0.98 0.99 1.32 1.21 0.97 0.99 1.16 1.08 0.96 1.00 1.05 0.99
ODFM 1.04 1.00 1.00 0.97 0.99 1.00 0.81 0.91 0.98 1.01 0.99 0.92
CDFM 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
A- CD- TPN 1.02 1.03 0.96 1.00 1.01 0.98 0.91 0.98 0.99 1.00 0.99 0.94
A- LE- TPN 0.99 1.00 0.99 0.98 0.99 1.00 0.81 1.04 1.00 1.00 1.00 1.00
S- CD- TPN 1.01 1.01 0.92 1.04 1.01 1.00 0.88 1.00 1.00 1.00 0.99 1.00
S- LE- TPN 1.00 1.01 0.99 1.05 1.00 1.00 0.81 1.03 0.99 1.00 1.00 1.00
A- F- TPN 1.02 1.03 1.09 1.08 0.98 0.98 0.77 0.96 0.97 1.00 0.97 1.00
S- F- TPN 1.01 1.00 1.03 1.11 1.01 0.97 0.80 0.98 0.99 1.01 1.00 0.93
A- N- TPN 1.06 0.99 0.79 1.03 1.05 0.97 0.68 0.96 0.97 1.00 0.93 1.00
S- N- TPN 0.99 0.99 0.77 0.99 1.00 0.98 0.71 0.99 1.01 0.98 0.94 0.96
No e: The e o measu e is epo ed ela i e o he benchma k CDFM. Bold p in ed alues indica e he lowes achie ed RMSFE. The model ac onyms co espond o he ollowing speci ica ions: AR: An AR(1) model; ARMA: An
ARMA model wi h lag o de de e mined ecu si ely ia he AIC; Nai e: A nai e uncondi ional mean model; ODFM: A DFM using only he ele an a iables; CDFM: A DFM using all a iables; A- CD- TPN (S- CD- TPN): The se o
a ge ed p edic o s is e ie ed ia he EN, which is sol ed using coo dina e descen , wi h EN pa ame e s e ie ed ia CV. The da a is agg ega ed (skip- sampled); A- LE- TPN (S- LE- TPN): The EN is sol ed ia LARS- EN; A- N- TPN
(S- N- TPN): A TP model whe e he se o a ge ed p edic o s is e ie ed using he nowcas ing e o om he p e ious pe iod; A- F- TPN (S- F- TPN): A TP model whe e he se o a ge ed p edic o s is e ie ed ia he EN, whe e he EN
pa ame e s a e se so ha 30 p edic o s a e selec ed. The a iables a e newly selec ed o each nowcas .
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