Inflation forecasting in turbulent times

E l, Ma in e al.
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In la ion o ecas ing in u bulen imes
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IHS Wo king Pape 56
Sep embe 2024
In la ion Fo ecas ing in
Tu bulen Times
Ma in E l
Ines Fo in
Ja osla a Hlousko a
Sebas ian P. Koch
Robe M. Kuns
Leopold Sögne
Au ho (s)
Ma in E l, Ines Fo in, Ja osla a Hlousko a, Sebas ian P. Koch, Robe M.
Kuns , Leopold Sögne
Edi o (s)
Robe M. Kuns
Ti le
In la ion Fo ecas ing in Tu bulen Times
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In la ion Fo ecas ing in Tu bulen Times
Ma in E l Ines Fo in Ja osla a Hlousko a Sebas ian P. Koch
Robe M. Kuns Leopold S¨ogne
*
Abs ac
Recen ly, many coun ies we e hi by a se ies o mac oeconomic shocks, mos no ably as a con-
sequence o he COVID-19 pandemic and Russia’s in asion in Uk aine, aising in la ion a es
o mul i-decade highs and suspending well-documen ed mac oeconomic ela ionships. To cap-
u e hese ail e en s, we p opose a mixed- equency Bayesian ec o au o eg essi e (BVAR)
model wi h -dis ibu ed inno a ions o wi h s ochas ic ola ili y. While in la ion, indus ial
p oduc ion, oil and gas p ices a e a ailable a mon hly equencies, eal g oss domes ic p oduc
(GDP) is obse ed a a qua e ly equency. Thus, we apply a mixed- equency amewo k us-
ing he o wa d- il e ing-backwa d-sampling algo i hm o gene a e mon hly eal GDP g ow h
a es. We o ecas in la ion in hose eu o a ea coun ies which ex ensi ely impo ene gy om
Russia and he e o e ha e been hea ily exposed o he ecen oil and gas p ice shocks. To
measu e he o ecas pe o mance o ou mixed- equency BVAR model, we compa e hese in-
la ion o ecas s wi h hose gene a ed by a ba e y o compe ing in la ion o ecas ing models.
The p oposed BVAR models domina e he compe i ion o all coun ies in e ms o he log
p edic i e densi y sco e.
Keywo ds: Bayesian VAR, mixed- equency, o wa d- il e ing-backwa d-sampling, in la ion o e-
cas ing
JEL classi ica ion: C5, E3
*
All au ho s: Ins i u e o Ad anced S udies (IHS), Jose s ¨ad e S aße 39, 1080 Vienna, Aus ia. Leopold
S¨ogne has a u he a ilia ion wi h he Vienna G adua e School o Finance (VGSF). The au ho s would like o
hank pa icipan s o he 2023 Mee ing o he Aus ian Economic Associa ion (NOeG), Joshua C.C. Chan, Osca
Fe nandez, Syl ia F ¨uhwi h-Schna e , Helmu Ho e , and wo anonymous e e ees o help ul commen s ha ha e
con ibu ed o imp o ing he pape . The au ho s g a e ully acknowledge suppo om Pe e G iessl, Ch is ine
Lie z, Johannes Neme h, and Yannic P ohaska. Leopold S¨ogne acknowledges suppo by he Cos Ac ion HiTEc
– CA21163.
1
1 In oduc ion
The COVID-19 pandemic, esul ing supply side dis up ions, he quick economic eco e y, and he
ene gy p ice shock ollowing Russia’s in asion o Uk aine had un o eseen consequences on in la ion
dynamics and ha e posed majo challenges o in la ion o ecas ing. E idence has eme ged ha
pa ame e es ima ion in ime se ies models widely used o mac oeconomic o ecas ing has become
mo e di icul due o he COVID-19 shock and i s a e ma h.
Fo eu o a ea in la ion, Bobeica and Ha wig (2023) documen ha pa ame e es ima es o
Bayesian ec o au o eg essions (BVAR) we e s ongly a ec ed. They p opose o use a a - ailed
dis ibu ion o he e o e ms. To imp o e he accu acy o eu o a ea in la ion o ecas s, hey
also ecommend es ima ing la ge models wi h a igh e p io (compa ed o s anda d BVAR
speci ica ions) and including o -model in o ma ion o o ecas s, such as in o ma ion om he
ECB Su ey o P o essional Fo ecas e s (see also K ¨uge e al., 2017; Banbu a e al., 2021).
O he wo k add essing he ecen ail e en s ocuses on he US economy, such as Lenza and
P imice i (2022), Ca ie o e al. (2022) and Scho heide and Song (2021). Mo e speci ically, Lenza
and P imice i (2022) modi y he inno a ion a iance o he pandemic pe iod. They exploi he
ac ha we know he exac iming o he inc ease in he inno a ions’ a iance du ing he COVID-
19 pe iod (Ma ch and subsequen mon hs in 2020). Whe eas his migh be ue o he pandemic,
i is ha de o disen angle he exac iming o he he e ogeneous e ec s o ising ene gy p ices on
in la ion a es in di e en coun ies. Coun ies ha e aced di e ing dependencies on ene gy supply
om Russia, and go e nmen s ha e been implemen ing di e en policies o mi iga e apidly ising
p ices. Fo he pe iod a e May 2020, he au ho s simply assume ha he esidual a iance will
decay a a mon hly a e o 20%.
Ca ie o e al. (2022) sugges allowing o S uden - dis ibu ed inno a ions and ou lie s in a
ec o au o eg essi e (VAR) model wi h s ochas ic ola ili y. Ex eme obse a ions a e iewed
as ou lie s ha a e cha ac e ized by ansi o y inc eases in ola ili y, in which case i may be
desi able o educe hei in luence on model es ima es. Thei model augmen s he s anda d
s ochas ic ola ili y speci ica ion wi h an ou lie s a e. Fo he ea men o a - ailed e o s in
s ochas ic ola ili y, hey use -dis ibu ed inno a ions. An olin-Diaz e al. (2021) also allow o
sho -li ed ou lie s ha do no lead o a pe sis en ise in he s ochas ic ola ili y p ocess in a
dynamic ac o model o nowcas ing US GDP.
Al e na i ely, Scho heide and Song (2021) econside a mixed- equency VAR o gene a e
mac oeconomic o ecas s o he US du ing COVID-19. The ecommenda ion is o exclude ex-
eme obse a ions du ing a ew mon hs o he pandemic o imp o e he o ecas ing pe o mance.
Howe e , his assumes ha he iming o ou lie s is known ex-an e and does no add ess he
subsequen pe iod o unce ain y p ope ly. Fu he mo e, Cla k e al. (2023) apply Bayesian ma-
chine lea ning echniques o accoun o possible non-linea i y. The au ho s demons a e ha
Bayesian eg ession ees ha e s ong o ecas ing p ope ies in bo h he o e all le el and in he
2

ails, espec i ely.
In his pape , we conside a small-scale Bayesian VAR amewo k wi h di e en e o a iance
speci ica ions o o ecas in la ion in u bulen imes o selec ed Eu opean coun ies. Wo king
wi h he mos ecen in la ion da a, ou expe ience is simila o he e idence desc ibed in he
abo e-men ioned li e a u e. Fo ecas ing in la ion using VAR models based on longe his o ical
ime se ies un il he yea 2019 (p e-COVID-19) esul s in a apid decay o in la ion a es down
o a es obse ed be o e he ecen sha p inc ease. By con as , es ima ing VARs (wi h Gaussian
e o s) wi h he ull 2004 o 2023 da a se can esul in non-s able (explosi e) in la ion o ecas s.
Bo h esul s a e implausible and unsa is ac o y.
The e o e, we explo e wo di e en speci ica ions o ola ili y. Fi s , we choose an app oach
o o ecas ing pos -pandemic in la ion ha is closely ela ed o ha o Bobeica and Ha wig
(2023) using -dis ibu ed dis u bances. Howe e , we conside a mon hly ins ead o he qua e ly
equency in o de o use imelie and ine in o ma ion on in la ion dynamics. Thus, employing
(qua e ly) g oss domes ic p oduc (GDP) esul s in a mixed- equency p oblem. Fu he , we
use he gas p ice as an addi ional ene gy a iable. Second, we conside models wi h s ochas ic
ola ili y o cap u e ex eme e en s. I is a well-known ac ha Bayesian VARs wi h ime-
a ying ola ili y o en p o ide be e poin and densi y o ecas s o mac oeconomic a iables
han models wi h homoscedas ic e o s e ms; see, e.g., Cla k (2011) and Cla k and Ra azzolo
(2015). Speci ically, we conside he e o e ms o be gene a ed by a ac o s ochas ic ola ili y
model as p oposed in Kas ne (2019).
We include da a on in la ion, indus ial p oduc ion, and GDP om six eu o a ea coun ies,
namely Aus ia, Belgium, Finland, Ge many, I aly, and Slo akia, which depend s ongly on na -
u al gas impo s om Russia. To cap u e exogenous shocks a ec ing in la ion we do no only
include he oil p ice bu also he gas p ice. We conside mon hly obse a ions om Feb ua y
2004 o Feb ua y 2024, wi h GDP only obse ed a a qua e ly equency, while he o he da a
a e a ailable a mon hly equencies. Ou unde lying econome ic model is a ec o au o eg es-
si e model elying on mon hly a iables. Tha is, also o GDP he unde lying model applies
mon hly g ow h a es (see, o ins ance P oie i and Gio annelli, 2021, o equen is mon hly
GDP es ima es). To pe o m pa ame e es ima ion his a icle ollows a Bayesian app oach. In
wo king wi h S uden - dis ibu ed noise e ms we mainly ollow Bobeica and Ha wig (2023),
bu – in con as o hem – we apply a Minneso a ype p io o he au o eg essi e pa ame e
ma ices. The same p io o he au o eg essi e ma ices is also applied in he case o s ochas ic
ola ili y. In addi ion, we ha e o accoun o mixed sampling equencies. We adap o wa d-
il e ing-backwa d-sampling, as p oposed in F ¨uhwi h-Schna e (1994), o ob ain samples om
he pos e io dis ibu ion o he unobse ed mon hly GDP g ow h a es.
We conduc a comp ehensi e empi ical analysis including he pe iod o sha p in la ion inc eases
and dec eases be ween mid-2021 and ea ly-2024 in he six eu o a ea coun ies, which we conside
he “ u bulen imes” in his pape . We ind ha he p oposed a ian o a mixed- equency
3
Bayesian VAR wi h a - ailed e o s and he al e na i e a ian wi h s ochas ic ola ili y p o ide
be e ou -o -sample poin and densi y o ecas accu acies han a ba e y o popula compe ing
in la ion o ecas ing models. The compe ing models include a uni a ia e e sion o he p oposed
model, a e sion o he Bayesian VAR model wi h only mon hly a iables, dis ega ding GDP, a
uni a ia e au o eg essi e model, and homoscedas ic and he e oscedas ic e sions o an unobse ed
componen s model. To e alua e he in la ion o ecas s we employ adi ional measu es, such as
he mean absolu e e o and he oo mean squa ed e o , as well as log p edic i e densi y sco es.
This a icle is o ganized as ollows: Sec ion 2 b ie ly desc ibes he VAR model. Sec ion 3
in oduces he mixed- equency p oblem. Then, Sec ion 4 desc ibes ou Bayesian app oach, in
pa icula , he p io s. De ails a e p o ided in a sepa a e appendix. Sec ion 5 discusses he
pe o mance o di e en models and hen p esen s o ecas s and an impulse esponse analysis o
six Eu opean coun ies. The las sec ion concludes.
2 The Model
In his a icle we join ly model indus ial p oduc ion, IP , in la ion, In l , he eal g oss domes ic
p oduc , GDP , he gas p ice, pgas, , and he oil p ice, poil, , by using a ec o au o eg essi e
(VAR) model o o de p. We conside da a a a mon hly equency, and index deno es he
ime index. Fo each coun y, we s ack he a iables in o he i e-dimensional column ec o
y = (∆ ln IP , In l ,∆ ln GDP ,∆ ln pgas, ,∆ ln poil, )⊤∈R˜
kwhe e ˜
k= 5. Then we ge 1
y =a+
p
X
j=1
Ajy −j+ε .(1)
In he ollowing we assume ha he g ow h a es o he oil and he gas p ices a e no a ec ed
by ∆ ln IP ,In l , ∆ ln GDP and he e o e se he co esponding elemen s o Aj o ze o (see
also Equa ion (11) in Appendix A). In his a icle he noise e m ε ei he ollows a S uden -
dis ibu ion (Case 1), o is gene a ed by a s ochas ic ola ili y model (Case 2).
Case 1: Following Bobeica and Ha wig (2023), he noise e m ε ollows an iid mul i a ia e
S uden - dis ibu ion wi h mean ze o, co a iance ma ix Σ, whe e 0 <Σ<∞, and νdeg ees
o eedom. F om Bayesian li e a u e (see, e.g., Geweke, 1993; Bobeica and Ha wig, 2023) a
S uden - dis ibu ed noise e m ε wi h νdeg ees o eedom can be ob ained by d awing ε om
1In his a icle we apply he ollowing no a ion: ∆x deno es x −x −1and ∆ ln x abb e ia es ln x −ln x −1
( ha is, g ow h a es a e calcula ed as loga i hmic g ow h a es). Fo ec o s and ma ices we use bold ace. I no
o he wise s a ed, he ec o s conside ed a e column ec o s. 0a×band 1a×bs ands o a×bma ix o ze os and ones
and 0ais used o abb e ia e 0a×1.⊗deno es he K onecke p oduc and In he iden i y ma ix o dimension n×n.
ec(M) ec o izes he ma ix M, while ech(M) ec o izes he lowe iangula pa o a symme ic ma ix M.
N(·,·), IG (·,·), and W(·,·) deno es he mul i a ia e no mal, he in e se Gamma, and he Wisha dis ibu ion,
espec i ely. U(ν, ¯ν) abb e ia es a uni o m dis ibu ion on he in e al [ν, ¯ν]. ∝s ands o p opo ional o.
4
a mul i a ia e no mal wi h mean ze o and co a iance ma ix Σ := λ Σ, whe e λ is sampled
om an in e se Gamma dis ibu ion IG ν
2,ν
2.
Case 2: Al e na i ely, we conside he noise e ms ε o be gene a ed by a ac o s ochas ic
ola ili y model as p oposed in Kas ne (2019). Tha is, o a ˜
kט
k-dimensional ma ix Σ we
assume
Σ =ΛV Λ⊤+ΣU ,(2)
whe e Λis a ˜
k× -loading ma ix, is he numbe o ola ili y ac o s V =diag (exp(h1 ),...,
exp(h )) ∈R × ,ΣU =diag (exp(h +1 ), . . . exp(h +˜
k )∈R˜
kט
k, and each hj ,j= 1,...,˜
k+ ,
ollows a s able i s o de au o eg essi e p ocess wi h no mally dis ibu ed noise e ms. Then,
ε =Σ1/2
η , whe e η ollows a ˜
k-dimensional s anda d no mal dis ibu ion.
The pa ame e ec o θcollec s all he pa ame e s o he VAR conside ed in (1), ha is a,
and he ec o ized pa ame e ma ices Aj,j= 1, . . . , p. Fo -dis ibu ed inno a ions i also
con ains ech (Σ), λ1, . . . , λT, as well as ν, while o he s ochas ic ola ili y model i con ains all
he pa ame e s o he ac o s ochas ic ola ili y models de ined in Kas ne (2019). We choose p
such ha he au oco ela ions o he esiduals a e insigni ican , i.e., p= 4.
The VAR sys em de ined in (1) esul s in he ma ix polynomial a(z) = Ik−A1z−· · ·−Apzp,
z∈C. We assume ha he s abili y condi ion ( he de e minan o a(z)= 0, o all |z| ≤ 1) is me .
Le Ldeno e he lag ope a o . Then, y =a(L)−1ε , ∈Z, p o ides us wi h he unique (weakly)
s a iona y (and causal) solu ion o (1) (see, e.g., Deis le and Sche e , 2018, Theo em 4.4).
In addi ion, we conduc ed a panel VAR analysis. Howe e , he o ecas ing pe o mance u ned
ou o be be e in he coun y-by-coun y se up. Tha is why we ocus in he main ex on he
coun y-speci ic VARs.
3 Da a and mixed- equency
We use indus ial p oduc ion, in la ion and eal g oss domes ic p oduc o he six coun ies
Aus ia, Belgium, Finland, Ge many, I aly, and Slo akia. The selec ed coun ies a e all pa o
he Eu opean Economic and Mone a y Union (EMU) and also depend s ongly on gas impo s
( om Russia).2Thus, hey a e pa icula ly ulne able o gas p ice shocks and na u al candida es
o analyzing oil and gas p ice shocks as po en ial d i e s o (ene gy) in la ion. We do no conside ,
o ins ance, coun ies like Po ugal o Spain ha impo li le o ze o na u al gas om Russia.
The in la ion a es o he Bal ic coun ies may ha e been a ec ed much mo e by he Russian
in asion in Uk aine due o a gene ally b oade economic in e ac ion wi h Russia, Bela us and
Uk aine and a e he e o e also no conside ed. F ance and he Ne he lands a e no in he coun y
2Fo es ima es o he numbe and di e si y o gas supply sou ces, see, o ins ance, he Eu opean Union Agency
o he Coope a ion o Ene gy Regula o s, h ps://aegis.ace .eu opa.eu/ches /da ai ems/214/ iew, las accessed
29.11.2023.
5
lis , as he o me ex ensi ely implemen ed an i-in la iona y measu es while he la e changed i s
me hod o calcula ing in la ion3du ing he cou se o he yea 2023 (June 2023).4
While indus ial p oduc ion, in la ion, and GDP a e ob iously coun y-speci ic, we use in e -
na ional p ice quo a ions o B en oil as well as o TTF gas, he eby implici ly neglec ing mino
di e ences in coun y-speci ic wholesale p ices. Indus ial p oduc ion and GDP a e published
seasonally adjus ed while he ha monized index o consume p ices (HICP) is no . Ins ead o
seasonally adjus ing he HICP and using mon h-o e -mon h pe cen age changes, we op o wo k
wi h p ice changes on a yea -o e -yea basis (annual in la ion), which e ec i ely ac s as some
so o seasonal adjus men . All a iables measu ed in p ices a e denomina ed in Eu o wi h he
excep ion o he oil p ice, which is o iginally measu ed in US Dolla and hen con e ed o Eu o
using he US Dolla /Eu o exchange a e. The in la ion a e is measu ed in pe cen .
Fu he , we apply he ollowing da a ans o ma ions: we calcula e mon h-o e -mon h loga-
i hmic g ow h a es o indus ial p oduc ion, oil and gas p ices, and qua e -o e -qua e g ow h
a es o GDP. Tha is, we ge he ans o med a iables ∆ ln IP , ∆ ln pgas, , ∆ ln poil, obse ed
on a mon hly basis, and ln GDPq−ln GDPq−1, obse ed on a qua e ly basis, whe e q, q + 1, . . .
deno es a qua e ly ime scale. Fo inal es ima ion we conside he pe iod Feb ua y 2004 o
Feb ua y 2024. The s a ing da e o ou sample is de e mined by he a ailabili y o gas p ices.5
The da a, i s sou ces and ans o ma ions a e summa ized in Table 1.
Va iable Abb e ia ion T ans o ma ion Sou ce Da ase o Code
Indus ial p oduc ion IP ∆ ln IP Eu os a s s inp m
HICP in la ion a e In l Eu os a p c hicp man
Real g oss domes ic p oduc GDPq∆ ln GDPqEu os a namq 10 gdp
TTF NL na u al gas u u e pgas, ∆ ln pgas, Re ini i Eikon TRNLTTD
B en oil p ice in Eu o poil, ∆ ln poil, Re ini i Eikon OILBREN/USEURSP
US Dolla /Eu o exchange a e Re ini i Eikon USEURSP
Table 1: Included a iables.
ep esen s mon hly equency, q ep esen s qua e ly equency. No e ha in la ion is calcula ed
as he yea -o e -yea g ow h a e o he Ha monized Index o Consume P ices (HICP). Fo oil
and gas p ices as well as he exchange a e we use mon hly a e ages o daily quo es.
Obse a ional Scheme: Equa ion (1) desc ibes he da a gene a ing p ocess o y on a mon hly
basis. The a iables obse ed a a mon hly equency a e called as a iables, y
, while he
a iable GDP g ow h obse ed a a qua e ly equency is called a slow a iable, ys
, wi h he
sampling a e o he slow a iable being h ee. In he da a desc ibed abo e he g ow h a e
o indus ial p oduc ion, ∆ ln IP , in la ion, In l , he change o he gas p ice, ∆ ln pgas, , and
he change o he oil p ice, ∆ ln poil, , a e obse ed a a mon hly basis and a e he e o e as
a iables. By con as , GDP is obse ed a a qua e ly a e and is a slow a iable. Mon hly eal
3Swi ching om including only new ene gy con ac s o a me hod ha e lec s all (new and exis ing) con ac s.
4See, o ins ance, A menda iz e al. (2023).
5Ou ans o ma ions ensu e s a iona i y, which is con i med by uni oo es s.
6
−10
−5
0
5
10
M.5.s M.5. M.4.s M.4. M.1.s M.1. M.1.usc M.1.a M.1.uc
Figu e 2: In la ion o ecas s e o s o Aus ia.
The iolin plo ( wo-sided ke nel densi y plo s) summa izes he dis ibu ion o he o ecas e o s
(i.e., o ecas ed in la ion minus obse ed in la ion) o all included models o e all ho izons (h=
1,...,6). The scale o he e ical axis was limi ed o ±10 o exclude ex eme o ecas e o s (in
he case o M.4.s and M.4. models).
Densi ies a e mos ly highes o o ecas e o s jus below ze o. Nega i e o ecas e o s, ha
is, an unde es ima ion o in la ion, a e obse ed mo e o en han posi i e ones in ou e alua ion
sample. The wo unobse ed componen s models show a endency owa ds a bimodal o ecas
e o dis ibu ion.
5.2 Fo ecas s
This sec ion p esen s he in la ion o ecas s o he six coun ies conside ed. Figu e 3 p esen s
in la ion o ecas s o he (coun y-by-coun y) BVAR models M.5.s o Aus ia, Belgium, Ge -
many, Finland, I aly, and Slo akia o he pe iod June 2023 o Janua y 2026, i.e., o 32 mon hs
ahead. The solid black lines a e pos e io median es ima es based on 8,000 MCMC samples,
and he ou ypes o blue a eas ep esen he 90%, 60%, 50%, and 30% o ecas ing in e als,
espec i ely. As he ealized alues o in la ion span un il Feb ua y 2024, we can hus obse e
13

how well (o no ) in la ion was o ecas ed. No e ha o Aus ia and Slo akia he nine-mon hs
ahead o ecas s a e inside he 90% o ecas ing in e als, while o Finland and I aly none o he
in la ion o ecas s a e wi hin he 90% o ecas ing in e als.
Aus ia, Ge many, and I aly beha e e y simila ly ega ding pas and o ecas ed in la ion
a es. Wi h espec o he ealized alues we obse e in he case o I aly a sha pe decline o
in la ion, namely om app oxima ely 8% in June 2023 o app oxima ely 0.5% in Feb ua y 2024.
The highes in la ion a es a e obse ed in he ou h qua e o 2022 eaching a es be ween
11.6% (in Ge many and Aus ia) and 12.6% (in I aly). Also he o ecas s wi h ega d o he le el
o in la ion a he end o he o ecas ing ho izon as well as he o ecas ing in e als a e e y much
alike.
Belgium, Finland and Slo akia a e di e en wi h espec o pas as well as o ecas ed in la ion
a es. The s ong inc ease as well as he sha p dec ease o he Belgium in la ion a e migh be
a ec ed by Belgium HICP measu emen .9The o ecas o Belgium i s declines below he 2%
in la ion a ge o he Eu opean Cen al Bank (ECB) and hen app oaches his a ge om below.
Finland s ands ou wi h compa ably low in la ion a es. This does no come as a su p ise aking
in o accoun ha gas in Finland is used almos en i ely by he indus ial sec o 10 (e.g., pulp
p oduc ion) and only e y ma ginally by households.11 The in la ion de elopmen in Slo akia is
di e en , because i s peak is he la ges and occu s la e han in o he coun ies. Also he o ecas
s ands ou , as i has much b oade o ecas ing in e als wi h gene ally highe in la ion a es.
Figu e 4 p esen s six snapsho s o wo-yea s ahead in la ion o ecas s o Aus ia wi h o ecas s
s a ing a six di e en ime poin s, namely a July 2021, Janua y and July 2022, Janua y and
July 2023, as well as Janua y 2024. In all six cases in la ion is o ecas ed o decline and he
in la ion o ecas s dec ease as e when he s a ing poin s o in la ion o ecas s a e pa o he
mo e u bulen ime pe iod when in la ion in Aus ia was highes (July 2022 and Janua y 2023).
We also obse e ha he o ecas ing in e als a e la ges du ing mo e u bulen imes sugges ing
la ge o ecas unce ain y.
5.3 Impulse esponses
Figu es 5 and 6 p esen impulse esponse unc ions o in la ion wi h espec o a (posi i e) one-
s anda d-de ia ion shock in he oil p ice change (i.e., abou 10%) and wi h espec o a (posi i e)
one-s anda d-de ia ion shock in he gas p ice change (i.e., abou 16%) o e 24 mon hs o model
M.5.s . Es ima es a e ob ained using he gene alized impulse esponse analysis o ec o au-
o eg essi e models as p esen ed in Pesa an and Shin (1998).12 No e ha he calcula ion o he
9No e, ha in Belgium only new ene gy con ac s a e included in HICP measu emen and no all con ac s
(exis ing and new) (see, e.g., Jonckhee e, 2022).
10See, o ins ance, Vaden e al. (2022).
11Acco ding o Eu os a he HICP weigh o gas consump ion (i.e. by Finish households) is ze o.
12This app oach does no equi e o hogonaliza ion o shocks and is in a ian o he o de ing o he a iables in
he VAR.
14
gene alized impulse esponse unc ion equi es es ima es o he co a iance ma ix Σ , which is
ime dependen o a model wi h s ochas ic ola ili y. When applying he s ochas ic ola ili y
model we use he samples α(m)and he samples Σ(m)
o ob ain he gene alized impulse esponse
unc ion. The ime poin used is June 2023.13 The solid black lines a e again median es ima es
and he ou ypes o blue a eas ep esen 90%, 60%, 50% and 30% c edible in e als. We obse e
a posi i e hough small impac o an inc ease in he oil p ice on in la ion o all six coun ies, wi h
in la ion i s inc easing and hen g adually dec easing. The in la ion impulse esponses peak in
all coun ies in he i s yea (a e he shock), and he ea lies in la ion peaks occu o Ge many
and Finland, while he la es one occu s o Slo akia. Finally, he la ges unce ain y (in e ms o
he wid h o he c edible in e als) can be obse ed o Slo akia. In p inciple, he e ec s implied
by a shock in he gas p ice a e simila o he ones implied by a shock in he oil p ice, only smalle .
We obse e a posi i e impac o an inc ease in he gas p ice on in la ion. The e ec is la ges o
Slo akia al hough su ounded also by he highes unce ain y. No e ha he ecen ly obse ed
inc eases in in la ion a e much la ge han he shocks o one s anda d de ia ion applied in he
impulse esponse unc ions shown in Figu es 5 and 6. The p ices o oil and gas ose by o e 40%
du ing he mos u bulen imes, while he shocks assumed in he impulse esponse unc ions a e
a ound 10% and 16%, espec i ely.
13Fo he s ochas ic ola ili y model he impulse esponses depend on ime. We show hem o June 2023. The
ones o Feb ua y 2024 (las mon h) a e a he simila .
15
Aus ia Belgium
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01 2025-01 2026-01
-2
0
2
4
6
8
10
12
14
16
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01 2025-01 2026-01
-2
0
2
4
6
8
10
12
14
16
Ge many Finland
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01 2025-01 2026-01
-2
0
2
4
6
8
10
12
14
16
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01 2025-01 2026-01
-2
0
2
4
6
8
10
12
14
16
I aly Slo akia
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01 2025-01 2026-01
-2
0
2
4
6
8
10
12
14
16
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01 2025-01 2026-01
-2
0
2
4
6
8
10
12
14
16
Figu e 3: In la ion o ecas s and o ecas ing in e als.
The igu e shows in la ion o ecas s and o ecas ing in e als o 32 mon hs ahead o Aus ia,
Belgium, Ge many, Finland, I aly, and Slo akia om June 2023 o Janua y 2026. The es ima ion
sample is Feb ua y 2004 o May 2023. The o ecas s a e based on 8,000 samples, 2,000 bu n-in
s eps.
16
July 2021 Janua y 2022
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01
-2
0
2
4
6
8
10
12
14
16
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01
-2
0
2
4
6
8
10
12
14
16
July 2022 Janua y 2023
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01 2025-01
-2
0
2
4
6
8
10
12
14
16
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01 2025-01
-2
0
2
4
6
8
10
12
14
16
July 2023 Janua y 2024
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01 2025-01 2026-01
-2
0
2
4
6
8
10
12
14
16
5%-95% 20%-80% 25%-75% 35-65% Median Obse ed
2021-01 2022-01 2023-01 2024-01 2025-01 2026-01
-2
0
2
4
6
8
10
12
14
16
Figu e 4: In la ion o ecas s and o ecas ing in e als o Aus ia.
The igu e shows in la ion o ecas s and o ecas ing in e als o Aus ia o six di e en s a ing
poin s (July 2021, Janua y and July 2022, Janua y and July 2023, as well as Janua y 2024). The
es ima ion sample anges om Feb ua y 2004 o he mon h p e ious o he indica ed s a ing
poin s. The o ecas s a e based on 8,000 samples, 2,000 bu n-in s eps.
17
Aus ia Belgium
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
Ge many Finland
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
I aly Slo akia
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
Figu e 5: Gene alized impulse esponses o in la ion wi h espec o he oil p ice.
The igu e shows gene alized impulse esponse unc ions o in la ion (in %) wi h espec o a one-
s anda d-de ia ion shock in he oil p ice (i.e., ≈10%), o Aus ia, Belgium, Ge many, Finland,
I aly, and Slo akia o 24 mon hs. We apply samples o he co a iance ma ix Σ(m)
o he ime
poin June 2023.
18

Aus ia Belgium
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
Ge many Finland
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
I aly Slo akia
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
5%-95% 20%-80% 25%-75% 35-65% Median
0 6 12 18 24
-0.02
0
0.02
0.04
0.06
Figu e 6: Gene alized impulse esponses o in la ion wi h espec o he gas p ice.
The igu e shows gene alized impulse esponse unc ions o in la ion (in %) wi h espec o a one-
s anda d-de ia ion shock in he gas p ice (i.e., ≈16%), o Aus ia, Belgium, Ge many, Finland,
I aly, and Slo akia o 24 mon hs. We apply samples o he co a iance ma ix Σ(m)
o he ime
poin June 2023.
19
6 Conclusions
A se ies o mac oeconomic shocks hi many coun ies be ween he yea s 2020 and 2022, p ima ily
igge ed by he COVID-19 pandemic and Russia’s in asion in Uk aine, which aised in la ion
a es ac oss Eu ope o mul i-decade highs and pu well-documen ed ela ionships among mac oe-
conomic a iables unde sc u iny. In pa icula , in la ion o ecas ing became much mo e di icul .
We p opose a mixed- equency Bayesian ec o au o eg essi e model and, accoun ing o he ecen
ail e en s, we assume S uden - dis ibu ed inno a ions o , al e na i ely, s ochas ic ola ili y. We
include he a iables in la ion, indus ial p oduc ion, g oss domes ic p oduc , oil and gas p ices.
We o ecas in la ion in selec ed eu o a ea coun ies, which ha e been hea ily exposed o ene gy
supply om Russia and, hus, o he ecen oil and gas p ice shocks.
We compa e he o ecas pe o mance o ou model wi h he o ecas pe o mance o se e al
compe ing models o in la ion in he ou -o -sample pe iod om July 2021 o Feb ua y 2024. In he
ou -o -sample o ecas e alua ion i u ns ou ha wi h espec o log p edic i e densi y sco es he
mixed- equency BVAR models domina e he compe ing models. When he o ecas pe o mance
is measu ed by MAE and RMSE, hen he bes o ecas accu acy, excep o Ge many and Finland,
occu s again o mixed- equency BVAR models, hough uni a ia e models a e s ong compe i o s.
Agains p e-COVID-19 e idence (see, e.g., Koop and Ko obilis, 2019), BVAR o ecas s in a panel
se -up a e s ongly domina ed by ou coun y-speci ic BVAR models, which migh be due o he
desc ibed coun y he e ogenei y. Ou esul s a he suppo he ecen emphasis on a - ailed
noise e ms o in la ion modeling in he pos -pandemic wo ld as well as he as e idence ha
s ochas ic ola ili y is pi o al o in la ion o ecas ing.
In ou o ecas ing exe cise, we p esen in la ion o ecas s s a ing in June 2023, a ime o s ill
high in la ion in mos coun ies, un il Janua y 2026. Fo Aus ia, Ge many, Finland, and I aly
in la ion o ecas s beha e simila ly, hey slowly dec ease o a es be ween app oxima ely 2.5%
and 3% in Janua y 2026. The in la ion ajec o y o Belgium is di e en , since i alls below
2% and, a e wa ds, con e gences smoo hly owa ds le els close o he Eu opean Cen al Bank’s
2% in la ion a ge . Finally, he in la ion o ecas o Slo akia exhibi s he highes unce ain y.
When o ecas ing in la ion o Aus ia in di e en poin s in ime, we demons a e ha he highes
o ecas ing unce ain y occu s in he mos u bulen ime pe iods.
The me hodology de eloped in his a icle can be ex ended in se e al ways: Fi s , one can spli
up HICP in la ion in o i s componen s. Sepa a e modeling o hese componen s allows o in e
shocks o oil and gas p ices on hese componen s o in la ion. Fo example, he ansmission o
oil and gas p ices on he ene gy componen is o pa icula in e es . Second, ins ead o a educed
o m VAR one may conside a s uc u al VAR, wi h he goal o iden i y s uc u al shocks and
o model he ins an aneous e ec s, e.g., o ene gy p ices on in la ion in mo e de ail. Thi d, he
s uc u al s abili y o he ela ionship be ween he a iables conside ed can be u he in es iga ed,
o example, whe he he e a e signi ican changes in he ela ionship be ween ene gy p ices and
20
in la ion du ing u bulen imes.
21
A The Panel Model
This Appendix conside s a panel VAR. Due o he o ecas ing pe o mance o he panel model,
coun y speci ic models a e conside ed in he main ex . To ge he coun y speci ic analogs o
he panel model (10) simply se n= 1.
We conside a panel ec o au o eg essi e (VAR) model o o de pas a s a ing poin (in his
sec ion we mainly ollow L¨u kepohl, 2006; Kilian and L¨u kepohl, 2017):
yi =ai+
p
X
j=1
Ay,jyi −j+εi .(10)
The ime se ies dimension is = 1, . . . , T, while i= 1, . . . , n deno es he c oss-sec ional dimension.
The a iables yi ∈Rk, he in e cep e ms a e allowed o be coun y dependen , ai,i= 1, . . . , n,
while he au o eg essi e ma ices Ay,j a e he same o all coun ies i= 1, . . . , n. The noise e ms
a e εi ∈Rk,i= 1, . . . , n. In addi ion, we include common a iables yc ∈Rkc. In he empi ical
applica ion discussed in Sec ion 5, k= 3 and kc= 2. The ec o yi con ains g ow h a es in
indus ial p oduc ion, in la ion and he mon hly GDP g ow h a e. The common a iables a e
he g ow h a es in he oil and gas p ices.
Le y := y⊤
1 ,...,y⊤
n ,y⊤
c ⊤∈Rnk+kc,a:= a⊤
1,...,a⊤
n,a⊤
c⊤∈Rnk+kc, and ε :=ε⊤
1 ,...,
ε⊤
n ,ε⊤
c ⊤∈Rnk+kc.14 Then we desc ibe he coun y models including he common a iables by
one join VAR sys em, ha is
y =a+
p
X
j=1 (In⊗Ay,j)Ayc,j
0 Ac,j 
| {z }
Aj∈R(nk+kc)×(nk+kc)
y −j+ε ,(11)
whe e Ayc,j is a kn ×kcma ix. In (11) we imposed he simpli ying assump ion ha yi ,i=
1, . . . , n, do no G ange cause yc . The noise e m ε ollows an iid mul i a ia e S uden -
dis ibu ion wi h mean ze o and co a iance ma ix Σand νdeg ees o eedom. By ollowing,
e.g., Geweke (1993), by means o sampling om a no mal wi h mean ze o and co a iance ma ix
Σ =λ Σ, whe e 0 <Σ<∞and λ ∼ IG ν
2,ν
2, we ob ain samples om a mul i a ia e
-dis ibu ion. In addi ion, we also conside s ochas ic ola ili y (Kas ne , 2019).
The VAR sys em de ined in (11) esul s in he polynomial a(z) = Ik−A1z−· · ·−Apzp,z∈C.
We assume ha he s abili y condi ion ( he de e minan o a(z)= 0, o all |z| ≤ 1) is me . Le
Ldeno e he lag ope a o . Then, y =a(L)−1ε , ∈Z, p o ides us wi h he unique s a iona y
(and causal) solu ion o (1) (see, e.g., Deis le and Sche e , 2018, Theo em 4.4).
Le αbe ob ained by s acking a1,...,an,acand Ay,1,Ayc,1,...,Ay,p,Ayc,p,Ac,1,...,Ac,p col-
umn wise. By means o Zi := (y⊤
i −1,y⊤
c −1,...,y⊤
i −p,y⊤
c −p)⊤∈R(k+kc)p,Zc := (y⊤
c −1,...,y⊤
c −p)⊤∈
14In he main ex , n= 1, k= 3, kc= 2, and ˜
k=nk +kc= 5.
22
esul s in he ollowing s a e space o m, whe e we conside one slow low a iable (GDP). Fo
each coun y i he mon hly GDP g ow h a e is con ained in he hi d coo dina e o yi .16 Le us
in oduce he ollowing no a ion:
w+
=H x +Q η ,η ∼ N (0nk+kc,Ink+kc)
x +1 =Fx +1 +Gε ,ε ∼ N (0nk+kc,Σ ),whe e (19)
Hi =Hobs := Ik−10k−1×k20k−1×k(p−k−1)+1
01×k−111×k⊗e⊤
1k01×k(p−k−1)+1 
| {z }
[k×kp]
o ∈NZand e1k= (1,0,...,0)⊤∈Rk×1
Hi =Hno obs := Ik−10k−1×k(p−1)+1
01×k−101×k(p−1)+1 
| {z }
[k×kp]
o /∈NZ
Hc o dimension kc×kcpis ob ained con o mingly
H =In⊗Hi 0
0 Hc 
Qi =0[k×k] o ∈NZ,Qi = (0k×k−1ekk)
| {z }
[k×k]
o /∈NZ
whe e ekk = (0,...,0,1)⊤∈Rk×1.
No e ha Qc o dimension kc×kcis ob ained con o mingly; in ou applica ion Qc =02×2.
Q =In⊗Qi 0
0 Qc .(20)
Recall ha hose coo dina es o y obse ed e e y pe iod a e called he as a iables, while he
coo dina es only obse ed a NZ,N > 1, a e called he slow a iables. De ine x |T:= Exi |Y+
T,
Π |T:= Co x x⊤
|Y+
Tand Π , −1|T:= Co x x⊤
−1|Y+
T,θdeno es he model pa ame e s ( o
he Kalman il e as well as he Kalman smoo he , see, e.g., Shumway and S o e , 1982; Deis le
and Sche e , 2018). Le
K := FΠ | −1H⊤
Σ−1
| −1,
Π +1| =Vx +1 −x +1| =FΠ | −1F⊤+GΣG⊤−K Σ | −1K⊤
,
x +1| =Fx | −1+K w+
−w+
| −1,
w+
+1| =H +1x +1| ,
Σ +1| =Vw+
+1 −w+
+1| =H +1Π +1| H⊤
+1 +Q +1IQ⊤
+1 ,(21)
whe e V(·) deno es a a iance. The sys em is s a ed a some x1|0= (1,x0)⊤and Π1|0=
diag 1,1⊤
pnk+pkc, such ha w+
1|0=H1z1|0and Σ1|0=H1Π1|0H⊤
1+˜
Q1˜
Q⊤
1. No e ha o
adap ed in a s aigh o wa d way. In he ollowing we mainly ocus on ou applica ion, whe e one slow low a iable
is conside ed.
16Ma ices Hi and Qi a e cons uc ed o he case when only one slow a iable is conside ed, namely GDP
g ow h. This ( low) a iable is he hi d coo dina e o wi .
29

a iables which canno be obse ed, deno ed w+,no obs
j ,E(w+,no obs
j ) = 0, V(w+,no obs
j −w+,no obs
j | −1) =
V(w+,no obs
j ) = 1, and Co (w+,no obs
j , yi ) = 0, o all i=jand all , whe e Co (·,·) deno es he
co a iance ma ix. Fo hose ime poin s whe e w+
=w we ge V(w −w | −1) = H Π | −1H⊤
.
In he ollowing o ecas s we conside w +h(and no w+
+h), whe e H +h=In⊗Hobs 0
0 Hc 
o all h > 0. The h-s ep ahead o ecas s, h≥1, ollow om
x +h| =Fx +h−1| ,
Π +h| =Vx +h−x +h| =FΠ +h−1| F⊤+GΣG⊤,
w +h| =H +hx +h| ,
Σ +h| =Vw +h−w +h| =H +hΠ +h| H⊤
+h
such ha o h= 0 we ge
x | =x | −1+Π | −1H⊤
Σ−1
| −1w+
−w+
| −1,
Π | =Vx −x | =Π | −1−Π | −1H⊤
Σ−1
| −1H Π | −1.(22)
No e ha x | =Ex |Y+
and Π | =Vx −x | =Vx −Ex |Y+
. Fo he as a i-
ables we ge xj | =Exj |Y+
=xj =wj , ha is he condi ional expec a ion is he ac ual
obse a ion o he a iable j, while o he slow a iables we ge xj | =Exj |Y+
which is
ob ained by he abo e ecu sions. The lagged coo dina es con ained in x ollow om hese e ms
in a de e minis ic way. Only hose elemen s o Π | e e ing o co a iances o slow a iables a e
non-ze o. This di ec ly ollows om he p ope ies o condi ional expec a ion. In addi ion,
x | =x | −1+Π | −1H⊤
Σ−1
| −1w+
−w+
| −1
=x | −1+Π | −1H⊤
H Π | −1H⊤
−1w+
−w+
| −1
=1, w+
1 , w+
2 , x3 | −1, w+
4 , w+
5 , x6 | −1, . . . , w+
(n−1)k+1, , w+
(n−1)∗k+2, , xnk, | −1, w+
c1 , w+
c2 ⊤
=1, y1 , y2 , x3 | −1, y4 , y5 , x6 | −1, . . . , y(n−1)k+1, , y(n−1)∗k+2, , xnk, | −1, yc1 , yc2 ⊤.(23)
Fo hose coo dina es whe e wj is an obse ed as a iable, he condi ional expec a ion gi en
he pas and he cu en obse a ions is simply yj ( his ollows again om he p ope ies o
condi ional expec a ion). The a iables xj | −1(= Exj |Y+
as men ioned be o e) ollow om
he abo e ecu sion (23). The Kalman-smoo hing equa ions o =T−1,...,2,1 a e
B +1 =Π | −1F⊤−H⊤
K⊤
Π−1
+1|
x |T=x | +B +1 x +1|T−x +1| 
Π |T=Π | +B +1 Π +1|T−Π +1| B⊤
+1 .(24)
Since xj | =xj o he as a iables, also xjT| =xj and he co esponding a iance e ms in
30
Π |Ta e ze o. Since xj | =xjT | he ows o B +1 e e ing o as a iables ha e o be ze o.
Assuming ha he noise e ms condi ional on x0a e no mally dis ibu ed, i ollows om
F ¨uhwi h-Schna e (1994) o F ¨uhwi h-Schna e (2006)[p. 419] ha he missing alues can be
ecu si ely d awn om a (degene a ed) no mal dis ibu ion wi h mean ec o ¯
x |Tand co a iance
ma ix ¯
Π |T, =T, T −1,...,1,0. In he ollowing we sligh ly adap he p oo o F ¨uhwi h-
Schna e (1994) o sample y =Syx ;Syis a nk +kc×1 + nkp +kcpselec o ma ix (see also
(17)) whe e SyS⊤
y=Ink+kc.17 By he Bayes heo em we ge
πx |y +1,...,yT,Y+
,θ∝π(Syx +1|x ,θ)πx |Y+
,θ(25)
Fo /∈NZ+ 1, he las densi y πx |Y+
,θis a no mal densi y wi h mean ec o x | and
co a iance ma ix Π | . The densi y π(Syx +1|x ,θ) = π(y +1|x ,θ) is a no mal densi y wi h
mean ec o SyFx and co a iance ma ix SyGΣ G⊤S⊤
y. Since x +1 con ains y +1,y ,...,y −p+2,
[x +1](1+(nk+kc):1+p(nk+kc)) de e minis ically ollows om [x ](2:1+(p−1)(nk+kc)). Hence, x +1 ollows
a singula no mal dis ibu ion wi h mean ec o Fx and co a iance ma ix GΣ G⊤. F om
he appendix in F ¨uhwi h-Schna e (1994) we know ha by “comple ing he squa e” in he
co esponding s a e space model we a i e a a no mal dis ibu ion wi h a mean ec o o he o m
¯
x |T= (I−B +1F)x | +B +1x +1 and a co a iance ma ix o he o m ¯
P |T= (I−B +1F)Π | .
Fo ou applica ion his esul and he ela ionship be ween x and y ,...,y −N+1 shows ha y
ollows a no mal dis ibu ion wi h mean ec o ¯
y |Tand co a iance ma ix Sy¯
P |TS⊤
y. Hence, o
/∈NZ+ 1, samples o y ollow om:
y |y +1,...,yT,Y+
T∼ N ¯
y |T,Sy¯
P |TS⊤
y,whe e
¯
y |T=Sy(I−B +1F)x | +B +1ST
yy +1
=Syx | +SyB +1S⊤
yy +1 −SyFx | =Syx | +SyB +1S⊤
yy +1 −Syx +1| ,
¯
P |T= (I−B +1F)Π | ,
B +1 =Π | F⊤F Π | F⊤+GΣ G⊤−1,(26)
whe e in ou applica ion x di ec ly ollows om y ,...,y −N+1. By plugging in e ms ob ained
abo e and addi ional calcula ions we ge
¯
P |T= (I−B +1F)Π | =Π | −B +1FΠ |
=Π | −Π | F⊤F Π | F⊤+GΣ G⊤−1
| {z }
=Π−1
+1| by (22)
FΠ |
=Π | −Π | F⊤Π−1
+1| Π +1| Π−1⊤
+1| FΠ⊤
|
=Π | −B +1Π +1| B⊤
+1 =Π | −B +1Vx +1 −x +1| B⊤
+1 .(27)
Thus, o hose pe iods whe e /∈NZ+ 1, samples o y ollow om (26). Since Π | is a spa se
ma ix, we sample om a singula no mal dis ibu ion.
Finally, o ∈NZ+ 1, y +1,...,y +N−1and w+
+N−1=w +N−1allow o calcula e yj by
17To simpli y no a ion we o en do no dis inguish be ween samples ob ained by means o (26) and he andom
a iables y .
31
means o yj =wj, +N−1−yj +1 −· · ·−yj +N−1 o all slow coo dina es j. Hence, in o mal e ms in
his case he condi ional dis ibu ion o y |y +1, . . . yT,Y+
is a Di ac dis ibu ion wi h poin mass
on he obse ed as a iables and yj =wj, +N−1−yj +1 − · · ·−yj +N−1 o he slow coo dina es
j. Hence, o he pe iods ∈NZ+ 1 and slow a iables wi h index j( ha is, o he i s mon h
o he co esponding qua e in ou applica ion), we ge xj |T=Exj |Y+
T,x +1,x +2, . . . =
y+
j, +N−1−yj, +1 − · · · − yj, +N−1, o hose pe iods =s+ 1, ∈NZ, a e swhe e ws=w+
s.
The a iance o his e m is ze o. The lagged coo dina es con ained in x ollow om hese e ms
in a de e minis ic way.
Fo ou applica ion his implies: We obse e a qua e ly g ow h a e o GDP, e.g., o he i s
qua e . The mon hly GDP g ow h a es o Ma ch and Feb ua y ollow om samples ob ained
by means o (26). The mon hly GDP o Janua y di ec ly ollows om he qua e ly GDP g ow h
a e ( om Janua y o Ma ch) minus he mon hly g ow h a es sampled i s o Ma ch and hen
o Feb ua y.
B.2 Con e gence and Mixing
This sec ion analyzes he con e gence and mixing p ope ies o ou Bayesian sample o he
models wi h i e a iables. We conside he M1pos e io d aws θ(m)
j,m=M0+ 1, . . . , M,
M=M0+M1, o each coun y i= 1, . . . , n. In he ollowing M0= 2,000 and M1= 8,000.
Mixing o he Chain: To in es iga e he mixing beha io o he chain we de i e he e ec i e
sample size c
Me
jas, e.g., de ined in Gelman e al. (2013)[Chap e 11.5]; he e he coda package
in Rwas applied.
Table 3 p esen s he a e age e ec i e sample sizes (i.e., we ake he sample mean o he
e ec i e sample sizes ob ained o he co esponding pa ame e s con ained in α, ech(Σ), e c.).
In he Case 1, whe e he noise e ms ollow a -dis ibu ion, we conside he pa ame e sub ec o s
α, ech(Σ), λ, and ν. Fo ε gene a ed by a s ochas ic ola ili y model (Case 2) we conside
he pa ame e sub ec o αand samples o he ola ili y ma ix Σ a =T( oo keep he
amoun o MCMC ou pu o be s o ed low we only s o e he las alue o he ola ili y p ocess
( ech (Σ )) =0,1,...,T ). Fo some pa ame e s he e ec i e sample size is la ge han M1which can
be explained by he es ima ion o he long un co a iance ma ix o ob ain c
Me
j. In bo h cases
we obse e o he pa ame e s α, he ola ili y pa ame e s, and ν, ha he a e age e ec i e
sample size is la ge han 700 based on 8,000 MCMC d aws. Only o λ he e ec i e sample size
emained ela i ely low, which can be explained by he ela i ely high pe sis ence o he samples
o λ . When applying he s ochas ic ola ili y model we obse e ha he a e age e ec i e sample
size is a leas 5,000.
Con e gence: Since he main ocus o his pape is on o ecas ing, we un he Bayesian sample
wi h di e en seeds and compa e all he an cha s (= dis ibu ions o Bayesian poin o ecas s)
o he in la ion o ecas s by means o isual inspec ion. He e we obse ed ha he an cha s
s ongly o e lap also o di e en seeds. The e o e, we can conclude ha he Bayesian sample
has su icien con e gence and mixing beha iou .
C Unobse ed componen s model wi h s ochas ic ola ili y
We use he unobse ed componen s model wi h s ochas ic ola ili y desc ibed in K oese e al.
(2014) as one o he benchma k models in he o ecas e alua ion. The obse able a iable, in ou
32
Case 1: Case 2:
-dis ibu ed noise s ochas ic ol.
αΣλναΣT
Aus ia 1163 2012 101 1962 6261 7297
Belgium 719 1959 92 1962 5087 7023
Ge many 864 2060 101 1820 5611 6824
Finland 1034 2030 120 1962 6134 6829
I aly 843 1998 67 2051 8048 7906
Slo akia 1096 2001 159 1962 5258 6918
Table 3: A e age e ec i e sample size c
Me
j.
case In l , depends on he unobse ed componen , τ , and a s ochas ic ola ili y e m,
In l =τ +e˜
h /2ϵ ,(28)
whe e ϵ ∼iid N(0,1). The unobse ed componen and log- ola ili y, ˜
h , bo h ollow a andom
walk, ha is
τ =τ −1+u ,
˜
h =˜
h −1+ν ,(29)
whe e u ∼iid N(0, ω2
τ) and ν ∼iid N(0, ω2
˜
h). The s a e equa ions a e ini ialized wi h τ1∼
N(τ0, Vτ) and ˜
h1∼N(˜
h0, V˜
h), whe e τ1=˜
h1= 0 and Vτ=V˜
h= 9. We assume independen
in e se-gamma p io s o ω2
τand ω2
˜
h, namely
ω2
τ∼ IG(˜ατ,˜
λτ),
ω2
h∼ IG(˜α˜
h,˜
λh),(30)
wi h ˜ατ= ˜α˜
h= 10 and ˜
λτ= 0.252(˜ατ−1) and ˜
λ˜
h= 0.22(˜α˜
h−1). The s ochas ic ola ili y model
is es ima ed by auxilia y mix u e sampling, whe e he app op ia e Gaussian mix u e is chosen as
p oposed by Kim e al. (1998). We use he code UCSV.R o K oese e al. (2014) o ob ain 10,000
pos e io d aws a e a bu n-in o 2,000 d aws o each es ima ion. Fo u he de ails see K oese
e al. (2014).
33
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