Neutral interest rate in the Czech Republic: Alternative approach based on application of TVP-VAR model

© 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