© 2022 Published by VŠB-TU Os a a. All igh s ese ed. ER-CEREI, Volume 25: 87–99 (2022).
ISSN 1212-3951 (P in ), 1805-9481 (Online)
Neu al in e es a e in he Czech Republic: Al-
e na i e app oach based on applica ion o
TVP-VAR model
Lukáš JURSA a
*
, Sá a HAMOVÁ b
a Depa men o Economics, VŠB – Technical Uni e si y o Os a a, Sokolská ř. 33, Os a a 70200, Czech Republic.
b Depa men o Economics, VŠB – Technical Uni e si y o Os a a, Sokolská ř. 33, Os a a 70200, Czech Republic.
Abs ac
Ou objec i e is he es ima ion o he neu al in e es a e in he Czech Republic spanning Janua y 2002 o De-
cembe 2020. Employing he Bayesian Time-Va ying Pa ame e s (TVP) model wi h local p io sh inkage, ou
no el con ibu ion o he exis ing knowledge o neu al in e es a es in ol es he de elopmen o an es ima ion
me hod u ilizing he TVP-VAR model. This app oach ep esen s a mode n and lexible pa adigm applied wi hin
he Czech Republic's economic en i onmen , whe e i has no been employed p e iously. No ably, ou chosen
econome ic amewo k elimina es he o en-challenging ask o de ending heo e ical links. The TVP-VAR
model elies on modes economic assump ions conce ning he s uc u e o a gi en economy, wi h es ima es
de i ed om empi ical economic ela ionships among a iables ha con ibu e o he neu al in e es a e. While
he heo y o neu al in e es a es boas s a ich his o ical de elopmen ancho ed in he wo k o economic lumina -
ies, ecen yea s ha e un eiled i as a compelling subjec o ongoing esea ch. Despi e being an unobse ed
a iable, we illumina e he de elopmen al ends o he neu al in e es a e wi hin he domes ic economy. In he
Czech Republic, ou indings indica e a sus ained and g adual decline in he neu al in e es a e, se ling a a
nega i e le el o app oxima ely -0.5%. Rema kably, he neu al in e es a e exhibi s a s able ajec o y. Compa a-
i e analysis wi h analogous su eys conduc ed in he eu o a ea e eals a obus co-mo emen be ween he neu al
in e es a es in he Czech Republic and he eu o a ea.
Keywo ds
Bayesian Econome ics, Czech Republic, Mone a y Policy, Neu al Ra e o In e es , NRI, TVP model
JEL Classi ica ion: C22, E43, E52, E58
*
lukas.ju[email p o ec ed]
This pape was inancially suppo ed wi hin he VŠB–Technical Uni e si y SGS g an p ojec No. SP2021/50 (Cu en Chall-
enges o Economic De elopmen ).
88 Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 25, 2022
© 2022 Published by VŠB-TU Os a a. All igh s ese ed. ER-CEREI, Volume 25: 87–99 (2022).
ISSN 1212-3951 (P in ), 1805-9481 (Online)
Neu al in e es a e in he Czech Republic: Al-
e na i e app oach based on applica ion o
TVP-VAR model
Lukáš JURSA, Sá a HAMOVÁ
1. In oduc ion
In his manusc ip , we seek o con ibu e ou esea ch
indings o he discou se su ounding he neu al
in e es a e and i s pi o al ole in shaping mone a y
policy. The p ope assessmen o in e es a es, aligned
wi h mac oeconomic heo y, necessi a es a me iculous
unde s anding o he neu al a e o in e es . Essen ial-
ly, he neu al in e es a e (NRI) mi o s economic
g ow h in andem wi h po en ial ou pu , ee om
undue p essu e on he p ice le el. Consequen ly, any
de ia ion o he eal in e es a e om his neu al
benchma k se es as a c ucial indica o o mac oeco-
nomic imbalance, signaling ei he an o e hea ing o a
ecession in he economy. In an economy cha ac e -
ized by lexible p ices and a ional expec a ions,
de ia ions be ween eal in e es a es and he neu al
a e a e a ibu ed o ei he inadequa ely p ocessed
in o ma ion o an e oneous calib a ion o mone a y
policy by cen al banks.
The p onounced su ge in he p ominence o his
mac oeconomic a iable occu ed in he a e ma h o
he global inancial c isis om 2008 o 2009, p omp -
ing cen al banks o adop uncon en ional mone a y
policy measu es. Ins umen s such as quan i a i e
easing, ze o o nega i e in e es a es, and o eign
exchange in e en ions we e delibe a ed upon and
subsequen ly employed. As mone a y policies g adu-
ally e e o he con en ional amewo k ollowing an
ex ended pe iod o uncon en ional ins umen s,
cen al banks a e inc easingly inclined owa ds quan i-
ying he le el o he na u al in e es a e.
The ocal poin o his esea ch is he es ima ion o
he ajec o y o he neu al in e es a e in he Czech
Republic spanning om Janua y 2002 o Decembe
2020. Fo his es ima ion, we employ he Bayesian
ime- a ying pa ame e s (TVP) model wi h local
sh inkage o p io s—a con empo a y app oach o
es ima ing he neu al in e es a e ailo ed o he
Czech Republic's economic con ex . Ou indings
e eal a sus ained and g adual decline in he neu al
in e es a e, se ling a a nega i e le el o app oxi-
ma ely -0.5%. Howe e , despi e his decline, he
neu al in e es a e exhibi s a s able de elopmen al
pa e n. In compa ison wi h analogous su eys con-
duc ed in he eu o a ea, ou analysis indica es a co-
mo emen be ween he neu al in e es a es in he
Czech Republic and he eu o a ea.
The s uc u e o his pape un olds as ollows: Sec-
ion 2 delinea es he o e a ching heo e ical ame-
wo k, while Sec ion 3 in oduces he sh ink TVP-
VAR model, ou empi ical model, and he pe inen
da a. Sec ion 4 p esen s and discusses he esul s o
ou es ima ion. Finally, Sec ion 5 p o ides concluding
ema ks.
2. Theo e ical Backg ound o he Neu al Ra e o
In e es
The e m "na u al a e o in e es " is a ibu ed o he
Swedish economis Knu Wicksell (1898), widely
ega ded as i s p ogeni o o in oducing his concep-
ual amewo k. Wicksell (1898), in his seminal s udy,
delinea ed he na u al in e es a e as a speci ic in e es
a e on loans ha main ains neu ali y conce ning
commodi y p ices, he eby exe ing no disce nible
in luence on hei ascen o descen . He s aigh o -
wa dly de ined he na u al a e as he eal a e le el
ha aligns wi h a s able ajec o y o commodi y
p ices. Acco ding o Wicksell, any de ia ion o he
eal in e es a e om i s na u al coun e pa would
esul in a co esponding inc ease o dec ease in he
gene al p ice le el (Wicksell, 1898).
2.1 Neu al Ra e o In e es
The neu al a e o in e es , o en deno ed as * o -
s a , is synonymous wi h he na u al in e es a e o
eal equilib ium in e es a e. B oadly de ined, his
in e es a e ep esen s he sho - e m eal in e es a e
consis en wi h ou pu a i s po en ial, ull employ-
men , and a s able in la ion a e, commonly e e ed o
as p ice s abili y (Kaplan, 2018). Cen al banks e-
quen ly in oke he concep o he neu al a e in hei
decision-making p ocesses conce ning he bank a e,
as i se es as he c ucial dema ca ion be ween expan-
siona y and con ac iona y ( es ic i e) mone a y
policies. In layman's e ms, i is he a e ha ac s as an
equilib ium o he economy. Consequen ly, i he
cen al bank se s an in e es a e p ecisely a he
neu al le el, i nei he igge s an "o e hea ing" no
induces a "cooling down" e ec . The e o e, he
L. Ju sa, S. Hamo á – Neu al in e es a e in he Czech Republic: Al e na i e app oach based on applica ion o TVP-VAR
model
89
neu al a e se es as a aluable me ic o cen al
banks, o e ing a s anda d agains which o e alua e
he o ien a ion o mone a y policy. I is impo an o
no e ha he neu al a e canno be di ec ly obse ed
and necessi a es es ima ion (Wood o d, 2003).
Mo eo e , he neu al a e cons i u es a pi o al a -
iable o ming he co ne s one o mac oeconomic
models employed by cen al banks o o ecas ing
pu poses. Wi hou knowledge o a leas an es ima e o
he neu al in e es a e, i becomes challenging o
gauge whe he mone a y policy is accommoda i e o
es ic i e. Like he in la ion a e, he neu al a e is
con ingen on he dynamics o he eal economy and
he shocks impac ing i . Unde speci ic condi ions, a
5% a e may be conside ed " elaxed," while a o he
imes, ze o o e en a sligh ly nega i e a e could be
deemed excessi ely " igh ." The e o e, he alue o
his a iable mus exhibi cons ancy and p edic abili y
o ensu e he eliabili y o assessing he " igh ness" o
mone a y policy (Powell, 2020).
No ably, he neu al a e is a om s a ic; i is a
dynamic a iable in luenced by a ious economic and
inancial ma ke ac o s (Kaplan, 2018). A consensus
p e ails ega ding he subs an ial ole played by he
ageing popula ion and a decele a ion in p oduc i i y
g ow h in shaping he long- un Neu al Ra e o In e -
es (NRI). Simul aneously, eal GDP g ow h expe i-
ences a slowdown in de eloped coun ies, con ibu ing
o a decline in he equilib ium in e es a e (Rachel,
Summe s, 2019). Se e al s udies highligh iscal
policy as a signi ican de e minan a ec ing he sho -
e m neu al eal in e es a e, pa icula ly in ela ion
o go e nmen and i s budge . The de eloped wo ld
aces wo p ima y obs acles p e en ing he descen
in o nega i e in e es a e e i o y—go e nmen deb
and social sys ems. The expansion o go e nmen deb
heigh ens he demand o a ailable sa ings and
capi al, subsequen ly d i ing up hei p ice, i.e., he
in e es a e. Con e sely, he wel a e s a e, ope a ing
on he supply side, diminishes he necessi y o
accumula ing subs an ial p uden ial and pension
sa ings (Laubach, Williams, 2003).
2.2 Empi ical Li e a u e Re iew
Nume ous cen al banks a e diligen ly engaged in he
pu sui o es ima ing he neu al in e es a e; howe e ,
he endea ou o assess his a e p esen s a ple ho a o
ques ions. De e mining he sui able econome ic
me hod o es ima ing he neu al in e es a e (NRI)
and deciding upon he ype o NRI o be es ima ed, be
i sho - e m o long- e m, pose in ica e challenges,
gi en he p e alence o bo h empo al dimensions in
esea ch. Va ious me hodologies ha e been p oposed
in he academic li e a u e. In his in es iga ion, pa ic-
ula emphasis is placed on he sho - e m neu al
in e es a e. No ably, mone a y policymake s adi-
ionally wield sho - e m in e es a es as a policy ool,
as highligh ed by Wood o d (2003). A sho - e m
neu al a e p o es ins umen al in disce ning whe e
policymake s migh choose o main ain in e es a es
while s i ing o maximum sus ainable employmen
in he sho e m.
As elucida ed ea lie , he neu al a e can be
gauged ac oss di e en ime ho izons. Es ima es o
sho - e m neu al a es ypically exe in luence o e
non-mone a y d i e s o sho - e m GDP g ow h, such
as shi s in p e ailing iscal policy. This neu al a e is
ins umen al in s abilising he economy on a pe iod-
by-pe iod basis. Con e sely, es ima es o he 'longe -
e m' neu al a e end o in luence non-mone a y
ac o s e lec ing mo e endu ing conside a ions, such
as he an icipa ed medium- e m g ow h o he labou
o ce and he expec ed g ow h a e o labou p oduc-
i i y. This a e se es o s abilise he economy o e
he long un (Wynne, Zhang, 2017).
No ewo hy esea che s, including Laubach and
Williams, and Hols on (Robe s, 2018), ha e dedica -
ed hei e o s o es ima ing he equilib ium alue o
sho - e m in e es a es, such as he ede al unds a e
and he o e nigh a e.
The li e a u e e iew p ima ily cen es on he my -
iad econome ic me hods and models employed in
es ima ing he NRI. Se e al dis inc me hodologies
exis o es ima ing he eal neu al in e es a e, each
wi h i s inhe en me i s and d awbacks. In ou es ima-
ion o he neu al in e es a e, we op o he Bayesi-
an ime- a ying pa ame e s (TVP) model wi h local
sh inkage o p io s. Signi ican ly, his TVP-VAR
model, wi h i s mode n and adap able app oach
ailo ed o he Czech Republic's economic landscape,
ep esen s a no el applica ion in he ield.
Laubach and Williams (2003) in oduced hei ap-
p oach, u ilising a s a e-space model o es ima e he
unobse able neu al a e ac oss wen y ad anced
economies om 1961 o 2015. This semi-s uc u al
model, oo ed in he neoclassical g ow h model,
es ablishes a connec ion be ween he in e es a e,
po en ial g ow h, and consume p e e ences. Hols on
e al. (2016) co obo a ed Laubach and Williams'
indings o he Eu o A ea, UK, and Canada, e ealing
co-mo emen and unde sco ing he ole o global
ac o s in shaping NRI. Lubik and Ma hes (2015a),
Pesca o i and Tu unen (2015), and Kiley (2015)
es ima ed NRI based on ime- a ying VAR models.
Lei B ubakk, Jon Ellingsen, and Ø jan Robs ad
(2018) app oached he es ima ion o he neu al eal
a e o he No wegian economy h ough wo empi i-
cal models: he ec o au o eg essi e model wi h ime-
a ying pa ame e s (TVP-VAR) and a S a e-Space
90 Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 25, 2022
(SS) model akin o he Laubach-Williams model.
Aligning wi h in e na ional e idence, hei es ima es
indica ed a declining end.
Ano he a enue o es ima ing he na u al in e es
a e in ol es s uc u al models, as exempli ied by
Hlédik and Vlček (2018) in he case o he Czech
Republic. Thei semi-s uc u al model inco po a es
a ional expec a ions and a o wa d-looking in e es
a e ule, iden i ying he neu al a e as he eal a e
consis en wi h ou pu a i s equilib ium le el and
in la ion a he a ge .
Ka h yn Hols on, Thomas Laubach, and John C.
Williams (2017) applied he Laubach-Williams
me hodology o es ima e he na u al a e o in e es o
he Uni ed S a es, Canada, he Eu o A ea, and he
Uni ed Kingdom, e ealing subs an ial declines in
end GDP g ow h and neu al a es o in e es o e
he pas 25 yea s.
S e ański (2018) u ilised a simila amewo k o
he eu o a ea and h ee Cen al and Eas e n Eu opean
economies: Poland, he Czech Republic, and Hunga y.
Howe e , his model, o iginally designed o la ge
economies, equi ed augmen a ion o small open
economies.
B zoza-B zezina (2006) employed a s uc u al
VAR model and he Kalman il e o es ima e he
na u al a e o in e es in Poland, highligh ing he
in o ma i e ole o NRI o cen al banke s seeking o
s abilise in la ion and o coun ies aspi ing o join a
mone a y union.
The na u al a e o in e es : es ima es, d i e s, and
challenges o mone a y policy, a wo king pape by
Claus B and, Ma cin Bielecki, and Ad ian Penal e
(2018) unde he auspices o The Eu opean Cen al
Bank, documen ed a sus ained decline in he neu al
a e o in e es in ad anced economies. The au ho s
a ibu ed his decline o waning p oduc i i y g ow h,
ising ma k-ups, an ageing popula ion, and heigh ened
isk a e sion ollowing he global inancial c isis.
Pikha and F oňko á (2019) adop ed Taylo 's
ule-based app oach o condi ions in a small and open
economy, indica ing a signi ican decline in he
na u al a e o in e es since 2010. Thei me hodology
in ol ed de eloping a Mone a y Condi ions Index
(MCI) as a single gap index cap u ing changes in eal
in e es a es and he eal exchange a e.
A compa a i e analysis o he s udies e eals a
subs an ial decline in in e es a es, a end no solely
a ibu able o cen al banks' e o s o main ain low
and s able in la ion. As cen al banks phase ou un-
con en ional mone a y policy ools, he c ucial ole o
in e es a es as a mone a y policy ins umen is
eins a ed. The pe sis en ly low, o e en nega i e, eal
in e es a es, coupled wi h p olonged supp essed
in la ion, necessi a e a ho ough explo a ion o he
neu al in e es a e's le el (Hols on e al., 2017). I he
na u al a e has diminished compa ed o i s p e-c isis
le el, cen al banks mus keep nominal a es lowe o
acili a e eal g ow h. Con e sely, i he na u al a e
emains consis en wi h i s p e-c isis le el, mone a y
policy may be o e ly accommoda i e. The e o e, he
inqui y in o he le el o he neu al in e es a e as-
sumes pa amoun impo ance in he ealm o policy-
making.
This a icle is p ima ily ocused on iden i ying a
speci ic le el o NRI, conside ing well-es ablished
global ac o s, wi hou del ing in o o he coun y-
speci ic de e minan s. The TVP-VAR me hod se es
as he chosen app oach o accu a ely es ima ing NRI
in he Czech Republic, di e en ia ing ou s udy wi hin
he li e a u e e iew, which speci ically delinea es he
indi idual me hods employed o NRI calcula ions.
3. Me hods and Da a
3.1 VAR-Model wi h Time-Va ying Pa ame e s
The econome ic app oach we use in he pape is
called ime- a ying pa ame e s VAR (Cogely and
Sa gen , 2001; P imice i, 2005). Lubik and Ma hes
(2015a) hen a gue ha his is a lexible amewo k
ha is sui able o acking di icul links be ween
mac oeconomic ime se ies and da a. TVP-VAR is a
ime se ies model ha explains he mo emen s o
economic a iables. Each o hese a iables is ex-
plained by ano he lagged endogenous a iable o he
sys em and by lagged alues hemsel es (Lubik and
Ma hes, 2015a; Nekajima, 2011; B ubakk, e al.,
2018). B ubakk, e al. (2018) a gue ha he undamen-
al di e ence om s anda d VAR models is ha a
pa ame e ec o can change o e ime. By hese
pa ame e s, we mean he lagged coe icien s and he
a iance o he economic shock. Lubik and Ma hes
(2015b) s a e ha he a ac i eness o TVP-VAR
models is based on he inding ha mos mac oeco-
nomic ime se ies show a ce ain deg ee o nonlinea i-
y.
Acco ding o Lubik and Ma hes (2015a, 2015b),
he main ad an age o he TVP-VAR model is ha he
lag coe icien s and he a iance o he economic
shock can change o e ime. The econome ic passage
is, he e o e, able o cap u e nonlinea de elopmen s
o e ime. These a e, o example, asymme ic mo e-
men s o mac oeconomic a iables in di e en phases
o he economic cycle. The change in eal in e es a e
ola ili y and he achie emen o a ze o-lowe bound
(ZLB) o he nominal in e es a e deepens he non-
linea i y o mac oeconomic ies. This inc eases he
a ac i eness o using he TVP-VAR model. The used
econome ic amewo k cap u es co-mo emen
L. Ju sa, S. Hamo á – Neu al in e es a e in he Czech Republic: Al e na i e app oach based on applica ion o TVP-VAR
model
91
be ween a iables in a lexible concep wi hou he
need o use a s uc u al amewo k and deepen he
heo e ical ounda ions. This di e s om he LW
model (Laubach and Williams, 2003; Hols on e al.
2017), which is based on he s ong ela ionship o key
a iables.
Howe e , Bi o and F ühwi h-Schna e (2019)
and Cadonna e al. (2020) s a e ha ime- a ying
pa ame e (TVP) models ha e se e al limi a ions. One
is he p oblem wi h o e i ing. I is possible o use a
sui able global-local sh inkage o p io s, which will
allow app oaching ime- a ying pa ame e s owa ds
s a ic ones. In a speci ic o m, we use he Bayesian
app oach and sh inkage p io s o he TVP-VAR
model (Bi o and F ühwi h-Schna e , 2019; Cadona
e al., 2020).
We i s de ine he basic p inciples o he TVP
model (1) and hen discuss i s ec o o m (2). Bi o
and F ühwi h-Schna e (2019) s a e ha he s a e
space o m o he TVP model o 𝑡 = 1,…,𝑇, has he
ollowing o m:
𝑦𝑡= 𝑥𝑡𝛽𝑡+ 𝜀𝑡,
(1)
𝛽𝑡= 𝛽𝑡−1 + 𝜔𝑡,
whe e 𝑦𝑡 is he uni a ia e esponse a iable and
𝑥𝑡=(𝑥𝑡1,𝑥𝑡2, … ,𝑥𝑡𝑑) is a d-dimensional ow ec o
ha includes eg esso s a ime 𝑡, wi h he co espond-
ing 𝑥𝑡1 in e cep . I is also necessa y o speci y he
e o e ms 𝜀𝑡~𝒩(0,𝜎𝑡
2) and 𝜔𝑡~𝒩
𝑑(0,𝑄). Bi o and
F ühwi h-Schna e (2019) assume ha he ini ial
alue o 𝛽0 ollows he no mal dis ibu ion, i.e.,
𝛽0~𝒩
𝑑(𝛽,𝑄) wi h he de aul mean alue o 𝛽 =
(𝛽1,…,𝛽𝑑).
Bi o and F ühwi h-Schna e (2019) u he s a e
ha he sh inkTVP model can model he homoscedas-
ic (𝜎𝑡
2≡ 𝜎2) esiduals o 𝑡 = 1,…,𝑇 and he e osce-
das ic esiduals using he s ochas ic ola ili y (SV)
speci ica ion model. The second case is ha he log-
ola ili y ℎ𝑡=𝑙𝑜𝑔𝜎𝑡
2 ollows he au o eg essi e
model AR. The s ochas ic ola ili y (SV) model
applied o he obse ed e o can p e en he de ec ion
o alse de ia ions in TVP coe icien s by cap u ing
pa o he a iabili y in he e o e m (Nekajima,
2011).
Acco ding o Lubik and Ma hes (2015b), he main
challenge in applying he TVP-VAR models is in e -
ence. In his pape , we will use he Bayesian app oach
o es ima e TVP-VAR. Nex , we will ely on he
Ma ko chain Mon e Ca lo 𝑀𝐶𝑀𝐶 𝐺𝑖𝑏𝑏𝑠 𝑠𝑎𝑚𝑝𝑙𝑒𝑟,
which is designed o make i easy o calcula e mul i-
dimensional densi ies. The key inding is he di ision
o a compu a ionally unsol able p oblem in o se-
quences o easible s eps, i.e., 𝑀𝐶𝑀𝐶 𝐺𝑖𝑏𝑏𝑠 𝑠𝑎𝑚𝑝𝑙𝑒𝑟
(Bi o and F ühwi h-Schna e , 2019). P imice i
(2005) adds ha 𝐺𝑖𝑏𝑏𝑠 𝑠𝑎𝑚𝑝𝑙𝑒𝑟 is a special a ian o
he Ma ko chain Mon e Ca lo (MCMC) algo i hm.
The TVP model (1) can be e ec i ely used o es-
ima e he neu al in e es a e acco ding o B ubakk e
al. (2018) and Lubik and Ma hes (2015a) as ollows
using he TVP-VAR model:
𝑋𝑡= 𝜃𝑡𝑋𝑡−1 + 𝑒𝑡,
(2)
𝜃𝑡= 𝜃𝑡−1 + 𝑢𝑡,
(3)
whe e 𝑋𝑡 is a ma ix o endogenous a iables, 𝜃𝑡
deno es he ec o o ime- a ying pa ame e s and 𝑒𝑡
and 𝑢𝑡 a e e o e ms. Subsequen ly, we use he
app oach o B ubakk e al. (2018). Fi s , he es ima es
o ime- a ying pa ame e s 𝜃𝑡 a e ob ained based on
he whole obse a ion pe iod o 𝑡 = 1, … , 𝑇. Fo each
𝑡, we hen c ea e a o ecas o he eal in e es a e o
he pe iod when he empo a y shocks subsided – o
gene a e an es ima e o he neu al eal in e es a e
(Ruch, 2021) om he TVP-VAR, we used i e a ed
o ecas s o he eal in e es a e i e yea s ahead
(𝑟𝑡
∗= 𝑟𝑡+60), simila o Lubik e al (2015). Tha is, 60
pe iods in he case o mon hly obse a ions. Once he
ansi o y shocks ha e dissipa ed, he sho - e m eal
in e es a e should s abilize a i s na u al le el.
3.2 Model Desc ip ion
We c ea e a TVP-VAR model o he pe iod Janua y
2002 o Decembe 2020, mon hly pe iodici y, when
𝑇 = 228. The model includes a o al o h ee a ia-
bles: he a iable o he ou pu o he economy (𝑦),
he p ice le el (𝑝), and he eal in e es a e (𝑟). Each
o hese quan i ies is explained in he VAR model by
he lagged alues o hemsel es and he lagged alue
o ano he endogenous a iable. Using models (4) and
(5), we es ima e he de elopmen o he neu al in e -
es a e o he Czech Republic.
Ou model is based on he modi ied TVP-VAR
model (B ubakk e al., 2018; Lubik and Ma hes,
2015a, 2015b) and in he o m o one lag wi hou le el
cons an , he model can be de ined as ollows:
𝑦𝑡= 𝜃1,1𝑦𝑡−1 + 𝜃1,2𝑝𝑡−1 + 𝜃1,3𝑟𝑡−1 + 𝑒𝑡,
𝑝𝑡= 𝜃2,1𝑦𝑡−1 + 𝜃2,2𝑝𝑡−1 + 𝜃3,3𝑟𝑡−1 + 𝑒𝑡
,
𝑟𝑡= 𝜃3,1𝑦𝑡−1 + 𝜃3,2𝑝𝑡−1 + 𝜃3,3𝑟𝑡−1 + 𝑒𝑡
,
(4)
and o he ime- a ying pa ame e :
𝜃𝑡= 𝜃𝑡−1 + 𝑢𝑡.
(5)
The empi ical model (4) will be cons uc ed wi h a
lag 𝑝 = 3, i.e., by h ee mon hs. The accu acy o he
speci ica ion is e i ied using in o ma ion c i e ia:

92 Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 25, 2022
SBIC (Schwa z-Bayesian In o ma ion C i e ion),
HQIC (Hannan-Quinn In o ma ion C i e ion), AIC
(Akaike's In o ma ion C i e ion), and FPE (Akaike's
Final P edic ion E o ). Howe e , we assess he esul s
o he LR c i e ion (sequen ial modi ied LR es
s a is ic) as mo e signi ican because we use ou
model o long- e m o ecas ing and au oco ela ion
would ha e a signi ican impac on des abilizing he
econome ic amewo k (Appendix 1). Pa ame e 𝜃𝑡
hen deno es he ec o o ime- a ying pa ame e s
and 𝑒𝑡 and 𝑢𝑡 a e he e o e ms. As desc ibed by
B ubakk e al. (2018), we i s ob ain he es ima es o
ime- a ying pa ame e s 𝜃𝑡 based on he whole
obse a ion pe iod o 𝑡 = 1,…,𝑇 acco ding o model
(4) and model (5). Fo each 𝑡, we hen c ea e a o e-
cas o he eal in e es a e o he pe iod when he
empo a y shocks subside. The pe iod equi ed o he
shock o subside is commonly de ined as a long pe iod
o i e o en yea s (B ubakk e al., 2018; Lubik and
Ma hes, 2015a). In ou pape , we will use a pe iod o
i e yea s, i.e., 60 pe iods. These a e hen 𝑡 es ima es
o he neu al in e es a e. I is a long- e m ahead in
sample o ecas o eal in e es a e (in each 𝑡) based
on ime- a ying pa ame e s (5). Lubik and Ma hes
(2015a) p opose as a measu e o he na u al a e o
in e es he condi ional long-ho izon o ecas o he
obse ed eal a e. Ou chosen ime ho izon is i e
yea s, and he o ecas is compu ed o each da a poin
since 2007M1. We will discuss he inpu da a below,
bu i should be men ioned ha he chosen p ocedu e
leads o a loss o obse a ions. The s abili y o he
models is e i ied using he alues o he oo s o he
cha ac e is ic polynomial (Appendix 2)
In his pape , we will use he Bayesian app oach o
es ima e TVP-VAR (4). Nex , we will ely on he
𝑀𝐶𝑀𝐶 𝐺𝑖𝑏𝑏𝑠 𝑠𝑎𝑚𝑝𝑙𝑒𝑟 o allow us o ewo k he
nonlinea TVP-VAR model in o a condi ional s a e
space model o each s ep o he
𝑀𝐶𝑀𝐶 𝐺𝑖𝑏𝑏𝑠 𝑠𝑎𝑚𝑝𝑙𝑒𝑟 p ocedu e (P imice i, 2005;
Lubik and Ma hes, 2015b). We apply Gibbs sampling
echniques and he Ca e -Kohn algo i hm in he
BVAR es ima ion. This makes i possible o obse e a
complex nonlinea in e ence p oblem. We will use a
modi ica ion o he TVP-VAR model in he o m o
he sh inkTVP-VAR model (Bi o and F ühwi h-
Schna e , 2019; Cadonna e al., 2020). Tha is, we
will use sh inkage p io s in ou es ima es.
In he case o a speci ic se ing o he econome ic
amewo k, we ollow he p ocedu e o Bi o and
F ühwi h-Schna e (2019) o he sh inkTVP model.
We es ima e a sh inkTVP-VAR model wi h a ully
hie a chical no mal-gamma (NG) p io o 60000
i e a ions, wi h a hinning o 10 and a bu n-in o
10000, so we will keep 5000 pos e io i d aws. Fully
hie a chical no mal-gamma (NG) p io has he ollow-
ing hype pa ame e s: 𝑑1= 𝑑2= 𝑒1= 𝑒2= 0.001,
𝛼𝛼𝜉= 𝛼𝛼𝜏= 5 and 𝛽𝛼𝜉= 𝛽𝛼𝜏=10. The mean alue
o he p io is 𝐸(𝛼𝜉) = 𝐸(𝛼𝜏)= 0.1. We assume a
homoscedas ic e o e m and he s ochas ic ola ili y
(SV) speci ica ion model is no used. Fo homoscedas-
ic e o e m, we use he ollowing hype pa ame e s
o p io : 𝑐0= 2.5, 𝑔0= 5 and 𝐺0= 𝑔0/(𝑐0− 1).
3.3 Da a
We a e c ea ing he ime se ies o he pe iod Jan-
ua y 2001 o Decembe 2020 o he Czech Republic.
We ob ain da a wi h mon hly pe iodici y. Ou da a se
con ains a o al o 240 obse a ions. Fo he coun ies
o Cen al, Eas e n, and Sou h-Eas e n Eu ope, i is
ad isable o a oid da a om he 1990s due o he
u bulen pe iod, which was cha ac e ized by many
s uc u al b eaks. The e o e, ou ime se ies s a s in
2001 and he da a sample hen con ains he cu en
da a. We do no a oid de elopmen s du ing he
co ona i us c isis. The sou ce o da a is he da abase
Eu os a (2021).
In he analysis, we use undamen al mac oeconom-
ic a iables o he Czech Republic. These a e he eal
ou pu o he economy (𝑦), which is app oxima ed by
he indus ial p oduc ion index, he ha monized index
o consume p ices (𝑝), and he sho - e m eal in e es
a e (𝑟). The eason o using he indus ial p oduc ion
index is p ima ily enough obse a ions, as his index
is published mon hly. The mon hly pe iodici y hen
signi ican ly inc eases he numbe o obse a ions.
Ra azzolo and Vespignani (2015) claim ha indus ial
p oduc ion has been widely used as a measu e o eal
economic ac i i y, among o he s, Mullineaux (1980),
G illi and Roubini (1996), and Kim (2001) ha e used
indus ial p oduc ion as a p oxy o eal economic
ac i i y. Mo eo e , IPI is qui e common in VAR
models (Billio e al., 2016). The desc ip ions and
sou ces o each a iable a e in Table 1.
L. Ju sa, S. Hamo á – Neu al in e es a e in he Czech Republic: Al e na i e app oach based on applica ion o TVP-VAR
model
93
We es he ime se ies o analysis a iables o he
exis ence o uni oo s using he ADF and KPSS uni
oo es (Appendix 3). The eason is ha mac oeco-
nomic ime se ies a e cha ac e ized by hei end
componen and show a g ow h ajec o y. The ime
se ies o he p ice le el (𝑝) and he eal in e es a e
(𝑟) is no s a iona y a hei le els. Howe e , we will
no u he adjus he eal in e es a e and keep i a i s
le el. The eason is a p oblema ic in e p e a ion
because he in e es a e is calcula ed pe annum.
Howe e , his is common p ac ice (B ubakk e al.,
2018; Lubik and Ma hes, 2015a). We adjus he ime
se ies o ou pu (𝑦) and p ice le el (𝑝) o yea -on-
yea changes. We do no use mon h-on-mon h changes
because long- e m in o ma ion is los . This adjus men
will also achie e weak s a iona i y. We belie e ha
lea ing only a non-s a iona y ime se ies o he eal
in e es a e will no dis up he econome ic ame-
wo k. Time se ies wi h a uni oo ha en e s VAR
models a e commen ed on, o example, by Sims
(1980) and Sims e al. (1990).
We c ea ed he TVP-VAR model o he pe iod
Janua y 2002 o Decembe 2020. The eason is he e-
o e he adjus men o he da a sample by yea -on-yea
changes, speci ically o he ou pu o he economy
(𝑦) and he p ice le el (𝑝), and he loss o 12 obse a-
ions. The da a sample con ained 240 obse a ions and
ou esul ing model con ained 228 obse a ions (𝑇 =
228). Because o he me hod used, ou es ima es o
he neu al in e es a e s a om 2007M1.
The de elopmen o he nominal and eal in e es
a es in he Czech Republic is hen shown in Figu e 1.
In pa icula , he peak o he nominal in e es a e
om 2008 o 2009 is e iden . Du ing his pe iod, he e
was a cyclical o e hea ing o he economy, bu also a
ela i ely dynamic g ow h o he p ice le el. The
cen al bank esponded by igh ening mone a y policy
and ising main in e es a es. The di e ence be ween
he nominal and eal in e es a es is hen gi en by he
in la ion a e. In 2018, he mone a y policy igh ened
again, ollowed by easing in esponse o he Co id-19
c isis. F om 2013 o 2017, he nominal in e es a e
app oached he ze o loo . Howe e , only Figu e 1 is
unable o p o ide us wi h knowledge o he neu al
in e es a e ( *).
Figu e 1. De elopmen o nominal and eal in e es
a es in he Czech Republic, Janua y 2001 o Decem-
be 2020 (in pe cen )
-3
-2
-1
0
1
2
3
4
5
2002 2004 2006 2008 2010 2012 2014 2016 2018 2020
Nominal in e es a e
Real in e es a e
Pe cen
Mon hs in indi idual yea s
Sou ce: Eu os a (2021), own calcula ions
4. Resul s and Discussion
We es ima e he neu al in e es a e using a me h-
od called he sh inkTVP-VAR model (Bi o and
F ühwi h-Schna e , 2019; Cadonna e al., 2020). The
me hod elimina es he sho comings o he o iginal
TVP-VAR models (P imice i, 2005; Cogley and
Sa gen , 2001), especially he p oblem wi h model
o e i ing. We build on he wo k o B ubakk e al.
(2018) and Lubik and Ma hes (2015a), who apply he
TVP-VAR model o es ima e he neu al in e es a e.
Howe e , we de elop he me hod wi h new
knowledge in he ield o de elopmen o TVP-VAR
models and use an app oach ha is based on he
global-local sh inkage o p io s.
Table 1. Desc ip i e s a is ics and sou ce o a iables (2001M01-2020M12)
Va iable
Sou ce
Dec ip ion
Min
Median
Max
Sde
𝑦
Eu os a (2021)
Indus ial p oduc ion index (IPI),
2015=100, s.a.
67.40
92.90
114.90
13.07
𝑝
Eu os a (2021)
Ha monized index o consume
p ices (HICP), 2015=100, s.a.
75.70
93.58
112.16
10.46
𝑟
Eu os a (2021)
PRIBOR 3M, mon hly a e age (in
%), de la ed by he a e o in la ion.
-3.65
-0.23
3.45
1.44
No es: Fo seasonal adusjmen , we use Census X-13 a ailable in he EViews10 so wa e packed om U.S.
Census Bu eau. Fo a mo e de ailed desc ip ion o he me hod, see Findley e al. (1998). The indus ial p oduc ion
index is al eady ob ained as seasonally adjus ed (Eu os a , 2021).
Sou ce: own calcula ion
94 Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 25, 2022
The esul s a e in Figu e 2 and Figu e 3. Excep o
2008, he neu al in e es a e is nega i e and declin-
ing. The neu al in e es a e eached a maximum
alue o 0.195%, a minimum alue o -1.204%, and a
mean alue o -0.586%. Ou esul s, he e o e, show
ha he neu al in e es a e was a ound he alue o -
0.5%. Hlédik and Vlček (2018) apply a semi-s uc u al
model, a modi ied Laubach and William (LW) me hod
(Laubach and Williams, 2003), and es ima e ha he
neu al in e es a e de eloped a ound 1% be ween
2000 and 2017. Especially wi h he decline in he
pe iod om 2008 o 2010 and hen om 2016 o 2017.
Ou esul s con i m he conclusion ha in hese
pe iods he e is a decline in he neu al in e es a e. In
ou wo k, we no e h ee majo peaks in he neu al
in e es a e. In pa icula , i is he pe iod be o e 2008,
when he economy peaked, and he economy o e -
hea ed (Appendix 5) and he neu al in e es a e also
esponded wi h g ow h. Subsequen ly, i is he pe iod
be o e 2012, when he e is also a ela i ely signi ican
g ow h in he ou pu o he economy in he Czech
Republic. This is ollowed by a peak a he end o
2014 and he beginning o 2015, which e lec s he
inc easing pe o mance o he economy. This peak is
also epo ed by Hlédik and Vlček (2018) and du ing
2012 hey eco ded a decline in he neu al in e es
a e. The inc ease in he neu al in e es a e a he
beginning o 2015 is due o a signi ican inc ease in
he ou pu o he economy, bu also, acco ding o
Hlédik and Vlček (2018), he in low o in es men s is
due o he use o EU s uc u al unds.
Using a combina ion o me hods, Pikha and
F oňko á (2019) show ha he neu al in e es a e in
he Czech Republic is declining and has been in
nega i e e i o y since 2012. Since he i s qua e o
2012, he neu al in e es a e has oscilla ed a ound -
0.5%. Hlédik and Vlček (2018) ge in o nega i e
alues when es ima ing he neu al in e es a e only
when applying he HP il e . Ou esul s o he HP
il e and i s compa ison wi h he TVP-VAR me hod
a e in Figu e 3. Howe e , es ima es using he TVP-
VAR model a e mo e s able compa ed o he HP il e
me hod. In addi ion, we an icipa e he disappea ance
o s uc u al changes. The e may also be si ua ions
whe e he esul s a e sca e ed o a sho ime because
es ima es o neu al in e es a es using he HP il e
espond p ima ily o sho - e m economic shocks.
Es ima es using he TVP-VAR model a e mo e
ine ial, which we also expec a a neu al in e es a e.
The na u al a e o in e es should exhibi some
known co-mo emen wi hin he global en i onmen
(Hols on e al., 2017). I is use ul o compa e he
esul s wi h o he Eu opean coun ies whe e simila
esea ch has been ca ied ou . In he Czech Republic,
he neu al in e es a e is nega i e, and ou esul s a e
close o he alues o he neu al in e es a e in he
eu o a ea. Hols on e al. (2017) es ima e ha he
neu al in e es a e in he eu o a ea has anged om
0% o -1% since 2011. K upko ic (2020) shows in he
example o Slo akia ha he neu al in e es a e has
been nega i e since he second qua e o 2009 and is
declining. In 2019, i eached a alue o app oxima ely
-1.5%. Bencik (2009) claims ha he neu al in e es
a e in Slo akia shows a declining end and eached
low alues in 2006 and 2007. B zoza-B zezina (2006)
es ima es ha he neu al in e es a e in Poland anged
e y low om 1998 o 2003, wi h 0% and be ween
0.12% and 0%. Howe e , he esul s o S e ański
(2018) sugges ha he neu al in e es a e when using
he Kalman il e is o de s o magni ude highe . In
1997 and 2008, i eached a le el o up o 6%. And in
2000 and 2012, S e ański (2018) eco ded i s decline.
Since 2012, he neu al in e es a e in Poland has been
a ound 2%. S e ański (2018) con i ms he conclusions
o he pape in he case o he e alua ion o he Czech
Republic. Acco ding o him, he neu al in e es a e in
he Czech Republic has been nega i e since 2008 and
has subsequen ly isen sha ply du ing 2014 and 2015.
The neu al in e es a e in he Czech Republic has
been ollowing he de elopmen o he neu al in e es
a e in he eu o a ea since 2008. Fu he wo k by
Fio en ini e al. (2018) and B and and Mazelis (2018)
sugges ha he neu al in e es a e in he eu o a ea
has been nega i e since 2008, wi h a declining end.
This is con adic ed by (B and e al., 2018), who claim
ha he neu al in e es a e in he eu o a ea is in
posi i e alues excep o 2013 and has been a ound
0.5% since 2014.
Figu e 2. De elopmen o neu al a e o in e es
(NRI) and eal in e es a e in he Czech Republic (in
pe cen )
-3
-2
-1
0
1
2
2008 2010 2012 2014 2016 2018 2020
Neu al a e o in e es
Real in e es a e
Pe cen
Mon hs in indi idual yea s
No es: The neu al in e es a e is es ima ed using he
TVP-VAR model
Sou ce: own calcula ions
Ou pape hen con i ms S e ański's (2018) con-
clusion ha he neu al in e es a e in he Czech
Republic shows a co-mo emen wi h he neu al
in e es a e in he eu o a ea. And so, i is nega i e.
L. Ju sa, S. Hamo á – Neu al in e es a e in he Czech Republic: Al e na i e app oach based on applica ion o TVP-VAR
model
95
Hols on e al. (2017) a gue ha he wo ld's neu al
in e es a e is in luenced by domes ic ac o s such as
demog aphic changes and end changes in p oduc i i-
y, bu also by global ac o s. Co-mo emen s be ween
indi idual coun ies a e ypical o neu al in e es
a es.
In Figu e 3, we p esen he de elopmen o he
neu al in e es a e, which was es ima ed using he
ime- a ying pa ame e s VAR (TVP-VAR) and he
Hod ick – P esco il e (H-P il e ). We will con inue
o discuss he dispa i ies be ween hese speci ic
es ima ion me hods. I he economy is going h ough
signi ican economic p oblems o s uc u al changes,
he e a e di e en de elopmen s be ween hese es i-
ma es. These a e, o example, he yea s 2008, 2009,
and 2012. O he cu en de elopmen in 2020. The
es ima e using he TVP-VAR model shows mo e
s able and ine ial esul s, which we expec om he
de elopmen o neu al in e es a es, which a e
mainly a ec ed by demog aphic changes, p oduc i i y
ends, and global ac o s (Hols on e al., 2017). The
g ow h o he neu al in e es a e acco ding o he
TVP-VAR model is mainly due o he ela i ely
dynamic de elopmen in p e ious yea s. P ecisely
because o s abili y and ine ia, he neu al in e es a e
is s ill ising sligh ly in 2020. Howe e , he eal
in e es a e in he Czech Republic ell sha ply.
Figu e 3. De elopmen o eal in e es a e and neu al
in e es a e (NRI) acco ding o gi en me hods in he
Czech Republic (in pe cen )
-3
-2
-1
0
1
2
2008 2010 2012 2014 2016 2018 2020
Real In e es Ra e
Neu al Ra e (HP)
Neu al Ra e (TVP-VAR)
Pe cen
Mon hs in indi idual yea s
No es: The neu al in e es a e is es ima ed using TVP-
VAR model and H-P il e (𝜆 = 14 400)
Sou ce: own calcula ions
We also calcula e he in e es a e gap (Appendix
4). Using he in e es a e gap, we assess he quan i a-
i e di e ences be ween eal and heo e ically op imal
mone a y policy. In pa icula , he in e es a e gap is
g owing in si ua ions o c isis economic de elopmen .
Howe e , i canno be said ha he cen al bank
should main ain mone a y policy a a neu al le el a
all cos s. Fo example, in 2008 he e is a signi ican
gap o up o -4 pp. Howe e , he cen al bank has
al eady esponded o he down u n in economic
de elopmen (Appendix 5) and i is no en i ely
possible o main ain a igh mone a y policy. Howe -
e , we s a e ha a e lea ing he exchange a e com-
mi men in 2017, he economy in he Czech Republic
could ha e highe a es. In 2017, he cen al bank had
he oppo uni y o c ea e ese es wi hin in e es a es
o he coming pe iod.
5. Al e na i e model speci ica ion and obus ness
checking
We also used a di e en speci ica ion om ou o igi-
nal TVP-VAR model (4). This me hodology is used
by Lubik and Ma hes (2015a) o es ima e he NRI o
he US, B ubakk, e al. (2018) o he case o No way,
and Teodo u and Tok onalie a (2020) o he Ky gyz
Republic. Fu he mo e, Ca illo e al. (2018) apply
Bayesian Vec o Au o eg ession (BVAR) wi h ime-
a ying pa ame e s (TVP) o es ima e he neu al
in e es a e in Mexico. We do no u he modi y
hese app oaches bu apply hem o calcula e he
neu al in e es a e o he Czech Republic. Thus, we
abandon he o iginal p ocedu e o using local sh ink-
age o p io s and use p imal ini ializa ion o p io s and
hen he o he p io s co espond o he me hod (P im-
ice i, 2005). The only excep ion is he end o 2010,
when he na u al a e o in e es was sligh ly posi i e
due o highe le els in he p e ious pe iod. To gene -
a e neu al eal in e es a e es ima es (Ruch, 2021)
om he TVP-VAR, i e-yea ahead i e a ed (in
sample) o ecas s o he eal in e es a e a e used.
The sys em is based on h ee endogenous a ia-
bles. We use he pe cen age g ow h o eal GDP
(𝐺𝐷𝑃), he pe cen age change in he p ice le el as
measu ed by he CPI (𝑖𝑛𝑓) and he sho - e m eal
in e es a e in le els (𝑟). Thus, we use yea -on-yea
pe cen age changes o (𝐺𝐷𝑃) and (𝑖𝑛𝑓). Da a a e
ob ained on a qua e ly basis om a da abase (Eu o-
s a , 2021). The obse a ion pe iod o he analysis is
2000Q1 o 2020Q1 in case o al e na i e model
speci ica ion. We a e a oiding he Co id-19 global
pandemic and eplacing he indus ial p oduc ion
index wi h he g ow h a e o eal GDP. We e i ied
he s a iona i y o he a iables using uni oo es s
(Appendix 6).
In a simpli ied o m wi h one lag, he TVP-VAR
model is es ima ed based on he ollowing equa ions:
𝐺𝐷𝑃𝑡= 𝛼1𝑡 + 𝛽1,1𝐺𝐷𝑃𝑡−1 + 𝛽1,2𝑖𝑛𝑓𝑡−1+𝛽1,3𝑟𝑡−1
+ 𝑣1𝑡