Real‐Time Forecasting Using Mixed‐Frequency VARs With Time‐Varying Parameters

Hein ich, Ma kus; Rei , Magnus
A icle — Published Ve sion
Real‐Time Fo ecas ing Using Mixed‐F equency VARs Wi h
Time‐Va ying Pa ame e s
Jou nal o Fo ecas ing
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John Wiley & Sons
Sugges ed Ci a ion: Hein ich, Ma kus; Rei , Magnus (2025) : Real‐Time Fo ecas ing Using Mixed‐
F equency VARs Wi h Time‐Va ying Pa ame e s, Jou nal o Fo ecas ing, ISSN 1099-131X, Wiley,
Hoboken, NJ, Vol. 44, Iss. 7, pp. 2055-2066,
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Jou nal o Fo ecas ing, 2025; 44:2055–2066
h ps://doi.o g/10.1002/ o .3276
2055
Jou nal o Fo ecas ing
RESEARCH ARTICLE OPEN ACCESS
Real- Time Fo ecas ing Using Mixed- F equency VARs Wi h
Time- Va ying Pa ame e s
Ma kusHein ich1 | MagnusRei 2
1S ad we ke Kiel, Uni e si y o Kiel, Kiel, Ge many | 2Deu sche Bundesbank, CESi o, F ank u ,Ge many
Co espondence: Magnus Rei (magnus. ei @bundesbank.de)
Recei ed: 9 Ma ch 2023 | Re ised: 30 Oc obe 2024 | Accep ed: 21 Ma ch 2025
Keywo ds: Bayesian me hods| o ecas ing| mixed- equency models| nowcas ing| ime- a ying pa ame e s
ABSTRACT
This pape p o ides a de ailed assessmen o he eal- ime o ecas accu acy o a wide ange o ec o au o eg essi e models ha
allow o bo h s uc u al change and indica o s sampled a di e en equencies. We ex end he li e a u e by e alua ing a mixed-
equency ime- a ying pa ame e ec o au o eg essi e model wi h s ochas ic ola ili y. Mon e Ca lo simula ion shows ha he
no el model is well- sui ed o es ima e missing mon hly obse a ions in an en i onmen ha is subjec o pa ame e ins abili y.
In a eal- ime o ecas exe cise, he model deli e s accu a e now- and o ecas s and, on a e age, ou pe o ms i s compe i o s.
Pa icula ly, in la ion and unemploymen a e o ecas s a e mo e p ecise.
JEL Classi ica ion: C11, C53, C55, E32
1 | In oduc ion
Mac oeconomis s and, in pa icula , mac oeconomic o e-
cas e s ace wo majo challenges. Fi s , he e a e s uc u al
changes wi hin an economy. Second, in eal ime, o ecas e s
need o p ocess unbalanced da ase s due o indica o s sampled
a di e en equencies and indica o - speci ic publica ion lags.
Conce ning s uc u al change, i is commonly ound ha pa ic-
ula ly modeling ime- a ying ola ili ies enhances VAR- based
in e ence and es ima ion, while luc ua ions in he VAR coe -
icien s a e equen ly conside ed o be less i al ( o example,
Chan and Eisens a 2017; D’Agos ino e al.2013). Since he onse
o he G ea Recession, which p obably caused impo an s uc-
u al shi s, modeling ime- a ying links be ween a iables
has a ac ed anew in e es . Conce ning unbalanced da ase s,
li e a u e s esses he me i s o mixed- equency app oaches in
compu ing p ecise now- and o ecas s, and acking he cu en
s a e o he economy in eal ime ( o ins ance, Kuzin e al.2011;
Scho heide and Song2015; Gö z and Hauzenbe ge 2021).
Howe e , e idence ega ding he o ecas pe o mance o mod-
els allowing o s uc u al shi s in a mixed- equency se ing is
a he spa se. This s udy aims a illing his gap by p o iding a
de ailed assessmen o he eal- ime o ecas accu acy o a bun-
dle mixed- equency models ha allow o s uc u al change.
To his end, we es ima e nonlinea and linea VARs wi h and
wi hou mixed- equencies, including a ully- ledged model
inco po a ing ime- a ying pa ame e s, s ochas ic ola ili ies,
and mixed- equencies—a MF- TVP- SV- VAR. This analysis en-
ables us o ace ou he ela i e impac o he models' mixed-
equency pa and he ime- a ia ion in he models' coe icien s
on he o ecas accu acy. Ou compa ison elies on eal- ime
ou - o - sample now- and o ecas accu acy o bo h poin and den-
si y o ecas s conce ning ou key US mac oeconomic a iables:
GDP, indus ial p oduc ion, CPI, and he unemploymen a e.
O e all, ou o ecas compa ison p o ides wo majo indings.
Fi s , modeling s uc u al change and in aqua e ly dynam-
ics is bene icial o poin and densi y o ecas s, no ably o now
and sho - e m o ecas s. In pa icula , he accu acy o in la-
ion and unemploymen a e o ecas s can be subs an ially in-
c eased. Second, he MF- TVP- SV- VAR deli e s compe i i e
poin and densi y o ecas s—on a e age o e all a iables i
ou pe o ms each compe i o . Ou esul s mo eo e sugges
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.
© 2025 The Au ho (s). Jou nal o Fo ecas ing published by John Wiley & Sons L d.
2056 Jou nal o Fo ecas ing, 2025
ha he combina ion o mixed- equencies, s ochas ic ola ili y,
and ime- a ying pa ame e s is pa icula ly bene icial o in la-
ion nowcas s compu ed wi h only li le in o ma ion abou he
espec i e qua e s. In hose cases, he MF- TVP- SV p o ides
he la ges gains in o ecas accu acy. Inspec ing he mixed-
equency models' o ecas s du ing he G ea Recession e eals
ha allowing o ime- a ia ion in he VAR coe icien s and s o-
chas ic ola ili y is supe io ela i e o only one o hese speci i-
ca ions o in la ion and he unemploymen a e.
Es ima ion o he models' TVP- SV pa ollows Del Neg o and
P imice i (2015). Howe e , we es ima e he hype pa ame e s
ha ela e o he amoun o ime- a ia ion in he pa ame e s
ollowing Ami - Ahmadi e al.(2020).1 Es ima ion o he mod-
els' MF pa is based on Ma iano and Mu asawa (2010) and
Scho heide and Song(2015).
2 | Da a and Fo ecas Se up
We use a da ase consis ing o ou mac oeconomic indica o s,
h ee o which a e sampled a mon hly equency and one is
obse ed qua e ly. The qua e ly se ies is US eal GDP; he
mon hly se ies a e indus ial p oduc ion (IP), he consume
p ice index (CPI), and he unemploymen a e. GDP, IP, and CPI
en e he models in log i s di e ences imes 100. The unem-
ploymen a e emains un ans o med. Fo he VARs es ima ed
on qua e ly equency, he mon hly indica o s en e he mod-
els as qua e ly a e ages. We ob ain eal- ime da a om he
A chi al FRED (ALFRED) da abase. The sample uns om
Janua y 1960 un il Sep embe 2017. The i s 8 yea s a e used
o speci y p io s such ha he es ima ion s a s in Janua y 1968.
We assess he p edic ions ega ding he in aqua e ly in low o
in o ma ion using h ee in o ma ion se s. We assume ha he
o ecas s a e gene a ed a ound he middle o each mon h, when
he cu en indica o eleases a e a ailable. The i s in o ma ion
se (I1) ela es o he i s mon h o each qua e such ha he
o ecas e has in o ma ion up o he middle o Janua y, Ap il,
July, o Oc obe . In hese mon hs, he esea che has obse a-
ions on IP g ow h, in la ion, and unemploymen un il he end
o he espec i e p e ious qua e and a i s and p elimina y
es ima e o GDP e e ing o he p e ious qua e . Conce ning
he second ( hi d) in o ma ion se I2 (I3), one ( wo) addi ional
in o ma ion o he mon hly indica o s and he i s (second)
GDP e ision a e a ailable. As he qua e ly VARs canno cope
wi h he mon hly da a low, we es ima e hem in each ecu sion
based on he balanced in o ma ion se I1, which accoun s o
new in o ma ion only in e ms o da a e isions.
We e alua e ou o ecas s o da a in ages using an expanding
window om Janua y 1990 un il Sep embe 2017. The p edic-
ions a e e alua ed based on qua e ly a e ages. In he case o IP
and CPI, his implies ha we use he p edic ed mon hly g ow h
a es o econs uc he mon hly le els o he se ies, which in
u n a e used o compu e he qua e ly g ow h a es. Fo GDP
g ow h, we use he agg ega ion ule gi en by (5) o ob ain qua -
e ly GDP g ow h. To abs ac om benchma k e isions, we
e alua e GDP g ow h o ecas s based on he second a ailable
es ima e, ha is he o ecas o pe iod
+h
is e alua ed wi h
he ealiza ion aken om he in age published in
+h+2
.
As he emaining a iables a e e ised only a ely and sligh ly,
we e alua e he o ecas based on he la es in age. The maxi-
mum o ecas ho izon
hmax
is se o 4 qua e s. Thus, he mixed-
equency models gene a e o ecas s o
hm=1, …, 12
mon hs.
Fo ecas s o ho izons la ge han one a e ob ained i e a i ely.
3 | Models
We e alua e he o ecas pe o mance o mixed- equency, s o-
chas ic ola ili y, and ime- a ying pa ame e VARs, as well as
he o ecas pe o mance o combina ions o hese ea u es. Fo
he s ochas ic ola ili y models, we use andom walk s ochas ic
ola ili y, which is a pa simonious and compe i i e speci ica ion
(Cla k and Ra azzolo2015).2 Th oughou he pape ,
n=nq+nm
,
whe e
n,nq
, and
nm
deno e he numbe o o al, qua e ly, and
mon hly a iables, espec i ely. Finally,
p
deno es he lag o de .
3.1 | A VAR Wi h Time- Va ying Pa ame e s
and S ochas ic Vola ili y
Fo he VAR wi h ime- a ying pa ame e s and s ochas ic ola-
ili y (TVP- SV- VAR), we assume ha he ec o o a iables (y )
sa is ies
whe e
B0,
is a
n×1
ec o o ime- a ying in e cep s,
Bi,
a e
ma ices o ime- a ying coe icien s and
Ω
is he ime- a ying
n×n
a iance- co a iance ma ix. Le
Ω
=
A
Σ
A�
, whe e he
diagonal elemen s o
Σ
a e he s ochas ic ola ili ies and he
lowe - iangula elemen s o
A
a e he con empo aneous e-
la ions be ween he a iables. Mo eo e , le
𝜎
be he ec o o
he diagonal elemen s o
Σ ,a
he ec o o lowe - iangula el-
emen s s acked by ows o
A
, and
𝛽
he ec o o s acked VAR
coe icien s. We assume ha
whe e
Q
=diag
(
q2
𝛽1,…,q2
𝛽k𝛽
)
.3 We ob ain an SV- VAR by exclud-
ing ime- a ia ion in he VAR coe icien s, and a TVP- VAR by
excluding ime- a ia ion in he a iance- co a iance ma ix.
3.2 | A Mixed- F equency VAR
Es ima ion o he mixed- equency VAR (MF- VAR) ollows he
Bayesian s a e- space app oach o Scho heide and Song(2015).
Le
y
=[y�
q, ,y�
m, ]�
, whe e
ym,
collec s he mon hly a iables and
yq,
deno es he qua e ly a iables a mon hly equency. As
he qua e ly a iables a e obse ed only in he las mon h o
each qua e ,
yq,
con ains missing obse a ions o he i s and
second mon h o each qua e . We cons uc he measu emen
(1)
y =B0, +
p
∑
i=1
Bi, y −i+𝜀 ,𝜀 ∼N(0, Ω
)
(2)
log
𝜎
=log𝜎
−1
+e
,e
=(e
1,
,…,e
n,
)
�
∼N(0, Ψ
)
(3)
a
=a
−
1+
𝜐
,
𝜐
=(
𝜐
�
1, ,…,
𝜐
�
n, )�∼N(0, Φ)
(4)
𝛽 =𝛽 −1+𝜒 ,𝜒 ∼N(0, Q)
2057
equa ion by assuming ha qua e ly GDP g ow h can be disag-
g ega ed in o unobse ed mon hly GDP g ow h (Ma iano and
Mu asawa2003):
Combining he unobse ed wi h he obse ed mon hly a iables
in

y
=[
y�
q, ,y�
m, ]�
, we de ine he s a e ec o by
z
=[
y�
,…,
y�
−p+1]�
and w i e he measu emen equa ion as:
Assuming ha GDP g ow h is o de ed i s in he model,
H
is
gi en by:
The missing obse a ions in
z
a e eplaced by es ima ed s a es
using simula ion smoo hing wi h a ime- a ying dimension
o he s a e- space sys em (Du bin and Koopman2001). The
ansi ion equa ion o he MF- VAR in s a e- space o m is
gi en by:
whe e
𝜇
and
F
con ain he in e cep s and AR- coe icien s, e-
spec i ely.
S
is a
pn ×pn
a iance- co a iance ma ix whe e
he i s
n×n
elemen s equal
Ω
and all emaining en ies a e
ze o. We ob ain he MF- SV- VAR by se ing he i s
n×n
ele-
men s o
S
o
Ω
using he laws o mo ion in (2) and (3). The
MF- TVP- VAR is ob ained by allowing
F
o a y o e ime
acco ding o (4). Including bo h speci ica ions leads o he
MF- TVP- SV- VAR.
3.3 | Es ima ion P ocedu e and P io Speci ica ion
We es ima e he models using Bayesian es ima ion echniques.
The mixed- equency models a e es ima ed wi h 4 lags; he
qua e ly models a e es ima ed wi h 2 lags.4 P io speci ica-
ions ollow Cla k (2011), Del Neg o and P imice i (2015),
and Scho heide and Song (2015). Addi ionally, o he hy-
pe pa ame e s ha ela e o he amoun o ime- a ia ion
in
𝛽 ,ai
, and
log𝜎i
, we ollow Ami - Ahmadi e al.(2020) by
implemen ing ano he laye o p io s o hose hype pa am-
e e s. A de ailed desc ip ion is p o ided in he suppo ing
in o ma ion.
3.4 | Mon e Ca lo E idence
In his sec ion, we accoun o he no el y o he MF- TVP- SV-
VAR and conduc a Mon e Ca lo simula ion o assess he ini e
sample p ope ies o he model. We ocus on a VAR(4) wi hou
in e cep e m and conside ou da a gene a ing p ocesses
(DGPs)—cons an coe icien s (DGP I), a single b eak in bo h
he a iance- co a iance ma ix and he coe icien s (DGP II),
d i ing coe icien s and a iances (DGP III), and cons an co-
e icien s wi h ou lie obse a ions (DGP IV).5 We use he e-
spec i e VAR p ocess along wi h he disagg ega ion cons ain
(5) o mimic a da ase consis ing o wo mon hly and one qua -
e ly se ies. We assess accu acy o he MF- TVP- SV- VAR by
compu ing he RMSEs o he es ima ed missing obse a ions
agains he ac ual simula ed alues and compa e he esul -
ing igu es wi h hose de i ed om a (cons an - coe icien )
MF- VAR.
Table 1 shows ha he MF- TVP- SV- VAR p o ides mo e ac-
cu a e es ima es o DGPs II and III. In pa icula , o he
d i ing- coe icien case (DGP III), he model s ongly ou pe -
o ms he MF- VAR. Fo DGP I, he MF- VAR p o ides, as ex-
pec ed, he bes i . In e es ingly, he MF- TVP- SV- VAR seems
o be inadequa e o DGP IV. Inspec ing he es ima ed coe i-
cien s and a iances e eals ha he MF- TVP- SV- VAR in e -
p e s he ou lie - induced hikes in ola ili y o be pe sis en ,
which leads o compa ably poo es ima es o he VAR's co a-
iance ma ix.6
In sum, hese esul s sugges ha he MF- TVP- SV- VAR is
well- equipped o es ima e missing mon hly obse a ion in ou
da ase , which is likely subjec o some kind o pa ame e ins a-
bili y (see, o example, Chan and Eisens a 2017, and he online
Appendix).
4 | Resul s
Fo he poin o ecas s, which we e alua e in e ms o oo
mean squa ed e o s (RMSE), we i s assess he models'
nowcas accu acy. A e wa ds, we e alua e he accu acy o
he poin o ecas s and p edic i e densi ies wi h espec o
he subsequen qua e s. We e alua e he p edic i e densi ies
using he con inuous anked p obabili y sco e (CRPS).7 We
p o ide esul s o he en i e sample (1995Q1–2017Q4) and o
a sho e sample pe iod (2008Q1–2017Q4) o assess whe he a
possible s uc u al b eak a ound he G ea Recession a ec s
he o ecas pe o mance.
(5)
Δ
3logYq, =yq, =
1
3

yq, +
2
3

yq, −1+
yq, −2+
2
3

yq, −3+
1
3

yq, −
4
(6)
y =H z
(7)
H
=
[
H�
1, H�
2,
]�
(8)
H
1, =
[
1∕30
1×n−12∕30
1×n−110
1×n−12∕30
1×n−11∕30
1×n−101×(p−4)n
],
(9)
H
2, =
[
0n−1×1In−10n−1×pn
]
(10)
z =𝜇+Fz −1+𝜐 ,𝜐 ∼N(0, S)
TABLE 1 | RMSEs o es ima ed missing obse a ions om Mon e
Ca lo simula ion.
DGP I DGP II DGP III DGP IV
MF- VAR 0.96 0.82 1.09 1.02
MF- TVP- SV-
VAR
1.04 0.78 0.74 1.32
No e: DGP I: VAR wi h cons an coe icien s, DGP II: VAR wi h single b eak
in coe icien s and a iances, DGP III: VAR wi h d i ing coe icien s and
a iances, DGP IV: VAR wi h ou lie obse a ions. RMSEs a e based on 10
simula ions pe DGP. Fo each DGP, we se
T=500
.
2058 Jou nal o Fo ecas ing, 2025
TABLE 2 | Real- ime nowcas RMSEs.
1990–2017 2008–2017
Model I1 I2 I3 I1 I2 I3
GDP g ow h
MF- TVP- SV- VAR 0.90 0.86 0.88 0.82 0.80 0.83
MF- SV- VAR 0.89 0.82 0.81
∗
0.81 0.76 0.75
MF- TVP- VAR 0.85
∗∗
0.78 0.79
∗
0.78
∗
0.69
∗
0.72
MF- VAR 0.92 0.86 0.87 0.84 0.79 0.82
Q- TVP- SV- VAR 0.99 1.01 0.99 1.00 1.01 0.98
Q- TVP- VAR 0.98 0.99 0.99 0.97 0.98 0.98
Q- VAR 1.03 1.03
∗
1.03
∗
1.01 1.02 1.02
Q- SV- VAR 0.59 0.59 0.59 0.66 0.66 0.66
IP g ow h
MF- TVP- SV- VAR 0.90 0.80 0.59
∗∗
0.84 0.68 0.52
∗
MF- SV- VAR 0.86 0.78 0.59
∗∗
0.82 0.70 0.52
∗
MF- TVP- VAR 0.86 0.78 0.58
∗∗
0.83 0.70 0.51
∗
MF- VAR 0.87 0.79 0.59
∗∗
0.84
∗∗
0.71 0.53
∗
Q- TVP- SV- VAR 0.99 0.99 0.99 0.93 0.94 0.94
Q- TVP- VAR 1.03 1.04 1.04 1.03 1.03 1.03
Q- VAR 1.06
∗∗∗
1.06
∗∗∗
1.07
∗∗∗
1.05
∗∗
1.06
∗∗
1.06
∗∗
Q- SV- VAR 1.13 1.10 1.09 1.34 1.34 1.34
In la ion
MF- TVP- SV- VAR 0.77
∗
0.48
∗∗
0.28
∗∗∗
0.75
∗
0.46
∗∗
0.26
∗∗
MF- SV- VAR 0.86 0.52
∗∗
0.29
∗∗∗
0.84 0.49
∗∗
0.27
∗∗
MF- TVP- VAR 0.82
∗
0.50
∗∗
0.28
∗∗∗
0.81 0.48
∗∗
0.27
∗∗
MF- VAR 0.90 0.53
∗∗
0.29
∗∗∗
0.87 0.50
∗∗
0.27
∗∗
Q- TVP- SV- VAR 0.91
∗∗
0.91
∗∗
0.91
∗∗
0.90
∗∗
0.90
∗∗
0.91
∗∗
Q- TVP- VAR 0.96
∗∗∗
0.96
∗∗∗
0.96
∗∗∗
0.96
∗∗∗
0.96
∗∗∗
0.96
∗∗∗
Q- VAR 1.04
∗
1.03
∗
1.04
∗
1.03 1.03 1.03
Q- SV- VAR 0.59 0.59 0.59 0.75 0.75 0.75
Unemploymen a e
MF- TVP- SV- VAR 0.80
∗∗
0.57
∗∗∗
0.37
∗∗∗
0.74
∗∗
0.52
∗∗
0.33
∗∗∗
MF- SV- VAR 0.80
∗∗
0.61
∗∗∗
0.38
∗∗∗
0.76
∗∗
0.55
∗∗
0.33
∗∗∗
MF- TVP- VAR 1.00 0.69
∗∗∗
0.37
∗∗∗
1.02 0.67
∗∗
0.33
∗∗∗
MF- VAR 0.79
∗∗∗
0.61
∗∗∗
0.38
∗∗∗
0.74
∗∗
0.56
∗∗
0.33
∗∗∗
Q- TVP- SV- VAR 0.94 0.93 0.95 0.92 0.92 0.93
Q- TVP- VAR 1.01 1.02 1.02 1.01 1.03 1.03
Q- VAR 1.03 1.03
∗
1.04
∗
1.02 1.03 1.03
Q- SV- VAR 0.27 0.26 0.25 0.30 0.30 0.30
No e: RMSEs a e in absolu e e ms o he benchma k model (bo om ow o each panel) and as a ios o he emaining models. Ra ios below uni y ma k ha he
model ou pe o ms he benchma k. Bold igu es indica e he bes esul o he a iable and in o ma ion se .
∗
,
∗∗
, and
∗∗∗
deno e signi icance a he 10%, 5%, and 1%
le el, espec i ely, acco ding o he Diebold–Ma iano- es wi h Newey–Wes s anda d e o s.

2059
4.1 | Nowcas E alua ion
Table2 p esen s he esul s o he nowcas exe cise aking
in o accoun he in o ma ion se s I1 o I3. I p o ides h ee
main akeaways. Fi s , he mixed- equency models ou pe -
o m he qua e ly models. On a e age, o e all in o ma ion
se s and a iables, he bes nowcas pe o mance is ob ained
by he MF- TVP- SV- VAR and he MF- SV- VAR, which im-
p o e on he benchma k (Q- SV- VAR) by oughly 30%. Second,
mos o he ime, he nonlinea MF- models ou pe o m he
linea MF- VAR, indica ing ha —apa om using mon hly
in o ma ion—pa ame e ins abili y is bene icial also in a
mixed- equency se ing. Thi d, he MF- models' ela i e pe -
o mance imp o es wi h mo e in o ma ion a ailable, showing
ha he models a e able o e icien ly p ocess he sequen ial
da a eleases.
Fo GDP g ow h, only MF- models signi ican ly ou pe o m
he benchma k. The bes pe o mance o bo h samples is ob-
ained by he MF- TVP- VAR. This esul sugges s ha , om
a nowcas ing pe spec i e, i is mo e impo an o accoun
o changes in ou pu g ow h dynamics han o accoun o
he decline in ou pu g ow h ola ili y. Mo eo e , he mixed-
equency VARs' ela i e pe o mance imp o es in he 2008–
2017 sample, while he qua e ly models pe o mance emains
a he unchanged. Thus, i appea s ha modeling he in a-
qua e ly low o in o ma ion has gained impo ance. In he
case o IP g ow h, accoun ing pa ame e ins abili y does no
yield mo e p ecise o ecas s. In ac , he mixed- equency
models p o ide in bo h samples i ually iden ical RMSEs.
Agains he backd op o he s ong d op o IP g ow h ola ili y
(see Appendix D o he suppo ing in o ma ion), his esul s is
a he su p ising.
Fo in la ion, he MF- TVP- SV- VAR deli e s he bes pe o -
mance; i imp o es on he benchma k by, on a e age, 50%.
Rega ding bo h samples, he esul s indica e ha no ably wi h
li le in o ma ion abou he cu en qua e he MF- TVP- SV-
VAR p o ides la ge gains in o ecas accu acy ela i e o he
compe ing models. The la e is pa icula ly ele an because
expe o ecas a e usually published in he second mon h o a
qua e . Wi h mo e in o ma ion a ailable, howe e , he di e -
ences owa ds he emaining MF- models anish. Fo he unem-
ploymen a e, he MF- TVP- SV- VAR deli e s he bes nowcas s
on a e age ac oss in o ma ion se s. Howe e , i appea s ha
non- linea i y is no as impo an as o he emaining a iables
as he di e ences o he MF- VAR a e negligible. The la e is
maybe no su p ising gi en ha he luc ua ions in ola ili y
o he unemploymen a e a e less p onounced compa ed wi h
he emaining a iables (see Appendix D o he suppo ing
in o ma ion).
In sum, he nowcas exe cise p o ides s ong e idence in a o
o non- linea models. Fo GDP and IP g ow h, allowing o
ime- a ying pa ame e s wi hou s ochas ic ola ili y s ongly
imp o es o ecas accu acy. Fo in la ion and he unemploy-
men a e he bes o ecas pe o mance is ob ained by models
ha accoun o bo h ime- a ying pa ame e s and s ochas ic
ola ili y.
4.2 | Fo ecas E alua ion
The esul s in Table3 show ha mixed- equency VARs p o ide
compe i i e o ecas s e en o highe ho izons and o bo h sam-
ples.8 In he case o he unemploymen a e, modeling wi hin-
qua e dynamics is pa icula ly bene icial—a each ho izon
e en he wo s pe o ming mixed- equency VAR ou pe o ms
he bes pe o ming qua e ly VAR. Mo eo e , he esul s e-
eal ha he models' o ecas pe o mance subs an ially di e s
ac oss a iables. The bes ela i e pe o mance, o e all a i-
ables and ho izons, is deli e ed by he MF- SV- VAR and he MF-
TVP- SV- VAR; he co esponding RMSEs a e oughly 10% lowe
han hose o he benchma k.
Fo GDP g ow h, he esul s p o ide h ee key indings. Fi s ,
only he Q- TVP- SV- VAR signi ican ly imp o es on he bench-
ma k ega ding bo h samples; o he sho e sample ( igh
panel), he gains a e e en mo e p onounced. Second, he TVP
models ela i e pe o mance s ongly imp o es in he sho e
sample, sugges ing ha ime- a ying in he au o eg essi e coe -
icien s has gained impo ance since he G ea Recession. Thi d,
also he mixed- equency models pe o mance s ongly im-
p o es in he sho e sample (albei in mos cases insigni ican )
wi h he MF- TVP- SV- VAR p o iding he bes pe o mance. Fo
IP g ow h, he benchma k is ha d o bea ( o bo h samples); no
model signi ican ly imp o es on he benchma k.
Fo in la ion and he unemploymen a e, he MF- TVP- SV-
VAR deli e s he bes pe o mance on a e age o e all ho izon,
sugges ing ha ime- a ia ion in each coe icien is c ucial o
hese a iables. Mo eo e , o he unemploymen a e, he MF-
models consis en ly ou pe o m he benchma k, while he qua -
e ly models ail o do. Hence, he esul s p o ide e idence ha
bo h in aqua e ly dynamics and ime- a ia ion in he VAR-
coe icien s a e pa icula ly impo an o hese a iables.
Thus, ou esul s con i m he indings om p e ious s ud-
ies based on qua e ly models (see, among o he s, D’Agos ino
e al.2013; Ba ne e al.2014) by use o mixed- equency mod-
els. In sum, he esul s indica e ha he gains in accu acy due o
a ia ions in he VAR- coe icien s a e smalle han he gains in-
duced by s ochas ic ola ili y. Howe e , using models wi h bo h
ea u es p o ides he mos accu a e o ecas s on a e age o e
all a iables. Finally, he esul s p o ide e idence ha modeling
wi hin- qua e dynamics is bene icial also ega ding sho - e m
o ecas s.
4.3 | P edic i e Densi y E alua ion
The esul s o he CRPS a e displayed in Table4. O e all, he
esul s poin o he use ulness o wi hin- qua e in o ma ion
in deli e ing well calib a ed p edic i e densi ies; he mixed-
equency models p o ide be e esul s on a e age o e all
a iables and ho izons han hei qua e ly coun e pa s. The
MF- TVP- SV- VAR p o ides he bes pe o mance wi h a CRPS
educ ion o abou 10% (on a e age o e all a iables and ho i-
zons). This emphasizes he impo ance o pa ame e ins abili y
o gene a ing accu a e p edic i e densi ies.
2060 Jou nal o Fo ecas ing, 2025
TABLE 3 | Real- ime o ecas RMSEs.
1990–2017 2008–2017
Model h=1 h=2 h=3 h=1 h=2 h=3
GDP g ow h
MF- TVP- SV- VAR 0.98 0.96 0.97 0.94 0.91
∗∗
0.89
∗∗∗
MF- SV- VAR 0.99 1.00 0.98 0.99 0.99 0.97
∗
MF- TVP- VAR 0.99 1.00 0.98 0.96 0.95 0.90
∗∗
MF- VAR 1.05 1.05 0.99 1.02 1.00 0.95
∗∗
Q- TVP- SV- VAR 0.98 0.93
∗∗∗
0.94
∗∗
0.94
∗∗∗
0.88 0.86
∗∗∗
Q- TVP- VAR 0.99 0.93
∗∗∗
0.94
∗∗
0.98
∗∗
0.87
∗∗∗
0.86
∗∗∗
Q- VAR 1.04
∗∗∗
1.00 1.00 1.01 0.98 1.00
Q- SV- VAR 0.61 0.61 0.63 0.69 0.68 0.69
IP g ow h
MF- TVP- SV- VAR 1.04 1.05
∗
1.03 1.00 0.99 0.99
MF- SV- VAR 1.00 1.02 0.99 1.01 1.02 0.99
MF- TVP- VAR 1.05 1.04
∗∗
1.03 1.03 1.02 1.01
MF- VAR 1.06 1.09
∗∗∗
1.04
∗∗
1.05 1.06
∗
1.03
Q- TVP- SV- VAR 1.03 1.01 1.01 0.99 0.98 1.00
Q- TVP- VAR 1.02 1.00 1.00 1.02 0.98 0.98
Q- VAR 1.07
∗∗∗
1.05
∗∗∗
1.03
∗∗∗
1.05
∗∗∗
1.04
∗∗
1.03
∗∗
Q- SV- VAR 1.23 1.26 1.30 1.46 1.48 1.52
In la ion
MF- TVP- SV- VAR 0.81
∗∗∗
0.83
∗∗∗
0.85
∗∗∗
0.81
∗∗∗
0.84
∗∗∗
0.89
∗∗∗
MF- SV- VAR 0.96 1.00 1.01 0.96 0.98 1.00
MF- TVP- VAR 0.91
∗∗
0.91
∗∗∗
0.89
∗∗∗
0.92
∗
0.94
∗
0.94
∗∗
MF- VAR 1.06 1.16
∗∗∗
1.31
∗∗∗
1.03 1.05 1.11
∗∗∗
Q- TVP- SV- VAR 0.89
∗∗∗
0.87
∗∗∗
0.87
∗∗∗
0.89
∗∗∗
0.88
∗∗∗
0.90
∗∗∗
Q- TVP- VAR 0.93
∗∗∗
0.93
∗∗∗
0.91
∗∗∗
0.94
∗∗∗
0.95
∗∗∗
0.95
∗∗∗
Q- VAR 1.07
∗∗∗
1.12
∗∗∗
1.18
∗∗∗
1.04
∗∗∗
1.07
∗∗∗
1.10
∗∗∗
Q- SV- VAR 0.65 0.65 0.62 0.81 0.79 0.73
Unemploymen a e
MF- TVP- SV- VAR 0.81
∗∗∗
0.88
∗∗
0.92 0.78
∗∗∗
0.84
∗∗
0.87
∗∗
MF- SV- VAR 0.84
∗∗∗
0.91
∗
0.94 0.81
∗∗
0.88
∗
0.91
MF- TVP- VAR 0.94 1.00 1.02 0.92 0.97 0.98
MF- VAR 0.84
∗∗∗
0.91 0.95 0.80
∗∗
0.87
∗∗
0.91
Q- TVP- SV- VAR 0.97 1.00 1.01 0.94
∗
0.97 0.98
Q- TVP- VAR 1.03
∗∗
1.04
∗∗∗
1.04
∗∗∗
1.02
∗∗
1.03
∗∗
1.04
∗∗∗
Q- VAR 1.03
∗∗∗
1.02
∗∗∗
1.02
∗∗∗
1.03
∗∗
1.02
∗∗
1.01
Q- SV- VAR 0.45 0.65 0.85 0.54 0.80 1.06
No e: See Table2 o a desc ip ion.
2061
TABLE 4 | Real- ime o ecas CRPS.
1990–2017 2008–2017
Model h=1 h=2 h=3 h=1 h=2 h=3
GDP g ow h
MF- TVP- SV- VAR 0.99 0.97 0.99 0.93 0.91 0.90
MF- SV- VAR 0.99 1.01 0.99 0.99 1.01 0.98
MF- TVP- VAR 1.06
∗
1.12
∗∗∗
1.12
∗∗∗
1.00 1.06 1.04
MF- VAR 1.07
∗∗
1.08
∗∗
1.03 1.04 1.03 0.97
Q- TVP- SV- VAR 0.99 0.97 0.99 0.95 0.92 0.93
Q- TVP- VAR 1.04 1.02 1.02 1.01 0.93 0.93
Q- VAR 1.07
∗∗∗
1.05
∗∗∗
1.05
∗∗∗
1.03 1.02 1.03
Q- SV- VAR 0.33 0.33 0.34 0.36 0.36 0.37
IP g ow h
MF- TVP- SV- VAR 1.03 1.06
∗
1.07
∗
0.95 0.98 1.03
MF- SV- VAR 1.00 1.03 1.02 1.01 1.03 1.02
MF- TVP- VAR 1.02 1.03 1.02 0.98 0.98 0.99
MF- VAR 1.09
∗∗
1.11
∗∗∗
1.07
∗∗
1.05 1.05 1.02
Q- TVP- SV- VAR 1.06
∗∗∗
1.03 1.03 1.00 1.00 1.01
Q- TVP- VAR 1.06
∗∗∗
1.04 1.04 1.04 1.00 0.99
Q- VAR 1.10
∗∗∗
1.06
∗∗∗
1.05
∗∗∗
1.05 1.02 1.02
Q- SV- VAR 0.64 0.66 0.69 0.73 0.76 0.78
In la ion
MF- TVP- SV- VAR 0.81
∗∗∗
0.83
∗∗∗
0.81
∗∗∗
0.81
∗∗∗
0.85
∗∗∗
0.86
∗∗∗
MF- SV- VAR 0.97 1.06 1.08
∗∗∗
0.95 1.03 1.06
MF- TVP- VAR 0.88
∗∗
0.88
∗∗∗
0.81
∗∗∗
0.90
∗
0.91
∗
0.86
∗∗∗
MF- VAR 1.06 1.20
∗∗∗
1.32
∗∗∗
0.99 1.04 1.07
Q- TVP- SV- VAR 0.88
∗∗∗
0.85
∗∗∗
0.83
∗∗∗
0.89
∗∗∗
0.87
∗∗∗
0.86
∗∗∗
Q- TVP- VAR 0.93
∗∗∗
0.91
∗∗∗
0.87
∗∗∗
0.92
∗∗∗
0.91
∗∗∗
0.88
∗∗∗
Q- VAR 1.09
∗∗∗
1.12
∗∗∗
1.15
∗∗∗
1.04 1.05 1.04
Q- SV- VAR 0.32 0.34 0.34 0.42 0.41 0.40
Unemploymen a e
MF- TVP- SV- VAR 0.79
∗∗∗
0.85
∗∗
0.89
∗
0.75
∗∗∗
0.79
∗∗∗
0.83
∗∗
MF- SV- VAR 0.84
∗∗∗
0.89
∗∗
0.93
∗
0.80
∗∗∗
0.85
∗∗
0.89
∗∗
MF- TVP- VAR 1.02 1.11
∗∗
1.16
∗∗
1.02 1.09 1.14
∗
MF- VAR 0.84
∗∗∗
0.91
∗∗
0.94 0.80
∗∗∗
0.85
∗∗
0.88
∗
Q- TVP- SV- VAR 0.96 0.99 1.01 0.92
∗∗
0.95 0.97
Q- TVP- VAR 1.03
∗
1.04
∗∗
1.05
∗∗
1.02 1.03 1.04
Q- VAR 1.03
∗∗
1.01 1.00 1.01 1.00 0.98
Q- SV- VAR 0.23 0.33 0.43 0.27 0.40 0.54
No e: Sco es a e in absolu e e ms o he benchma k (bo om ow o each panel) and as a ios o he emaining models. Ra ios below uni y show ha he model
ou pe o ms he benchma k. Bold igu es ma k he bes esul o he a iable and ho izon.
∗
,
∗∗
, and
∗∗∗
deno e signi icance a he 10%, 5%, and 1% le el, espec i ely,
acco ding o a - es on he a e age di e ence in sco es ela i e o he benchma k model wi h Newey–Wes s anda d e o s.
2062 Jou nal o Fo ecas ing, 2025
Fo GDP g ow h densi y o ecas s, no model signi ican ly im-
p o es on he benchma k. Only he MF- TVP- SV- VAR and Q-
TVP- SV- VAR p o ide an (insigni ican ) imp o emen . Thus,
no only he mean o he p edic i e dis ibu ion is sligh ly
mo e p ecise han hose o he benchma k, bu he en i e dis-
ibu ion. As o poin o ecas s, IP g ow h densi y o ecas
om he benchma k a e supe io compa ed wi h he emain-
ing models.
Rega ding in la ion, he esul s indica e wo ou comes. Fi s ,
he Q- TVP- SV- VAR and he MF- TVP- SV- VAR deli e he la ges
imp o emen s on he benchma k. Hence, as o poin o ecas s,
i is impo an o model ime- a ia ion in bo h he pa ame e s
and he esidual a iances o ob ain p ecise p edic i e densi ies.
Second, including ime- a ia ion in he pa ame e s plays a i al
ole; bo h he MF- TVP- VAR and he Q- TVP- VAR s ongly im-
p o e on he benchma k.
Fo he unemploymen a e, he esul s suppo hose om
he poin o ecas s e alua ion; he MF- TVP- SV- VAR deli e s
he bes pe o mance, imp o ing on he benchma k by up o
20%. Imposing pa ame e ins abili y only in he VAR coe i-
cien s seems o de e io a e he accu acy o he p edic i e den-
si ies. In ac , he MF- VAR and he Q- VAR ou pe o m he
TVP VARs. One eason o his esul s migh be ha he TVP
VARs—pa icula ly he MF- TVP- VAR—exagge a e he ac ual
ime- a ia ion in he VAR coe icien s. Mo eo e and in con-
as o in la ion, each mixed- equency model imp o es bo h
on he benchma k and on i s qua e ly coun e pa . Thus, o
gene a ing p ecise p edic i e densi ies o he unemploymen
a e, i is c ucial o include in aqua e ly in o ma ion and s o-
chas ic ola ili y.
In summa y, he esul s o he p edic i e densi y e alua ion
suppo he indings om he poin o ecas e alua ion. Using
mixed- equency models is bene icial o e all a iables and
ho izons. I signi ican ly imp o es esul s o in la ion and
he unemploymen a e. In addi ion, we con i m he impo -
ance o s ochas ic ola ili y in densi y o ecas ing by use o
mixed- equency VARs. We p o ide e idence ha combining
FIGURE 1 | In la ion o ecas s du ing he G ea Recession. Rows e e o mixed- equency models; columns e e o he o ecas o igins. Ac ual
alues ( ed line) and o ecas s: mean (black line) and 60% and 90% bands om he p edic i e dis ibu ions.