The Relaionship beween Real Ineres Raes and Inflaion Michał Brzoza-Brzezina * Absrac In he recen decade, a huge amoun of papers, describing moneary policy rules based on nominal ineres raes, has been wrien. As i is, however, well known, i is in fac he real and no he nominal ineres rae, ha can influence spending decisions of enerprises and households and hus inflaion. One way, o describe he relaionship beween real ineres raes and inflaion, is based on our experience wih he moneary heory of he price level. The quaniy heory of money can be used under cerain assumpions as a good descripion of he long-run relaionship beween money and prices. In his respec he bes known empirical applicaion is probably he P-sar model of Hallman, Porer and Small (1991). In his paper we use wo simple descripions of he long run link beween real ineres raes and inflaion, and subsequenly es heir empirical performance, using similar echniques as employed in P-sar modeling. In an empirical sudy, based on coinegraion analysis, we show ha he gap beween he real and naural rae of ineres does no deermine inflaion, as i is ofen posulaed, bu is growh rae. We find ha his relaionship describes reasonably well he long run influence of he ineres rae gap on inflaion. Simulaneously we calculae he average naural rae of ineres. JEL: E31, E40. Keywords: Inflaion, Naural Rae of Ineres *) Research Deparmen, Naional Bank of Poland and Chair of Moneary Policy (Warsaw School of Economics). The views expressed are hose of he auhor and do no necessarily reflec hose of he Bank. I graefully acknowledge commens on his paper received from J.Amao, J.Borowski, L.Chrisiano, M.Dudek, R.Harrison, R.Kokoszczyński, L.Mahadeva, Z.Polański, P.Robinson, he paricipans of he Summer School on Moneary Theory and Policy (Bonn, Augus 2001) and he CEFTA Workshop (Prague, Sepember 2001). Commens are welcome: Michal.Brzoza-Brzezina@nbp.x400.ne.pl
Conens 1 Inroducion...3 2 The models...5 2.1 Model 1: ineres rae gap as a deerminan of inflaion...5 2.2 Model 2: ineres rae gap as a deerminan of inflaion growh...8 3 Empirical resuls...10 3.1 The daa...10 3.2 Inegraion ess...12 3.3 Coinegraion ess - model 1...13 3.4 Coinegraion ess - model 2...17 4 Conclusions...22 Appendix 1...24 References...25 2
1 Inroducion For many years money has been a cenral issue in moneary policy making. Cenral banks used o se moneary arges and academics used o each moneary policy, as a sory abou how cenral bankers adjuss he money supply. Even he name of he main aciviy of cenral banks ook is origins from he word money. Thus, i is no wonder ha many economic papers describing inflaionary phenomena sill assume ha cenral banks conrol he money supply. Thanks o is imporan role in moneary policy, a lo of research has been done on esing he long-run relaionship beween money and inflaion. The probably bes known sudy, is based on he quaniy heory of money, and called P-sar 1. The model shows ha he quaniy equaion, being a very simplified descripion of he relaionship beween money and prices, can be used for moneary argeing and inflaion forecasing, provided ha some addiional assumpions are fulfilled. The mos imporan one is relaed o he long-run sabiliy (or a leas predicabiliy) of velociy. A posiive verificaion of he quaniy equaion saes ha here is a long-run pah for he general price level, deermined by he quaniy of money ha he acual price level is coinegraed wih. However, he world is changing, and argeing moneary aggregaes becomes less and less fashionable. The main reason is probably he growing insabiliy of money demand funcions. In reacion, moneary auhoriies move from argeing he money supply owards conrolling nominal ineres raes a he money marke. As a resul, in he recen decade, a huge amoun of papers, describing moneary policy rules based on nominal ineres raes, has been wrien. As i is, however, well known, assuming here is no money illusion, i is in fac he real and no he nominal ineres rae, ha can influence spending decisions of enerprises and households. Moneary auhoriies can aler real raes (a leas in he shor run) as long as prices and inflaionary expecaions are sicky 2. Thus, i is crucial for a cenral banker no only o look a he level of nominal ineres raes, bu also o monior he behaviour of real raes. 1 See J. Hallman, R.Porer, D.Small (1991), Deusche Bundesbank (1991) or M.Brzoza-Brzezina, J.Kołowski (2001). 2 A simple, bu comprehensive descripion of hese mechanisms is given by A.Blinder (1998). 3
Despie he growing imporance of ineres rae oriened policies, our knowledge on his opic is sill unsaisfacory. The firs approach o describe he relaionship beween real ineres raes and inflaion is ofen ascribed o K.Wicksell (1898, 1907). However already 100 years earlier, wo Briish economiss, H.Thornon and T.Joplin, described economic processes resuling from he cenral bank s influence on he real rae of ineres (T.M.Humphrey 1993). Neverheless, no much has been done in his field since. Recen papers, among ohers by M.Woodford (1999, 2000), revived he (now called) Wicksellian idea of inflaionary processes being deermined by he gap beween he real and naural 3 raes of ineres. In a very recen sudy K.Neiss and E.Nelson (2001) use a sochasic general equilibrium model o examine he properies of he ineres rae gap as an inflaion indicaor. The above menioned sudies are srongly in favour of using he gap as a measure of he sance of moneary policy ha could be used by cenral bankers in heir day-o-day (or raher monh-o-monh) policy seing. This paper aims o es, wheher a simple equaion, of he form inroduced o he economic lieraure by he quaniy heory, can be found and empirically verified for he long-run relaionship beween he real ineres rae gap and inflaion. In oher words, we will check wheher here is a long-run pah for inflaion, deermined by he ineres rae gaps ha acual inflaion is coinegraed wih. When describing he relaionship, we will naurally ignore shor-run dynamics, and he influence of exernal shocks, which should be he reasons for emporary divergences beween acual and equilibrium inflaion raes. A brief look ino he relaed lieraure, in search for an appropriae equaion, reveals ha a leas 2 differen specificaions should be considered. In some descripions a closed gap (real ineres rae equal o he naural one) resuls in a sable price level, in ohers in sable inflaion. A deailed descripion of he wo models will be presened in secion 2 of he paper. The res of he paper is srucured as follows. In secion 2, wo models are described, one relaing he ineres rae gap o inflaion, he oher one - o is growh rae. 3 The naural rae of ineres is someimes called he neural rae, see A.Blinder (1998), Economis (1999), E.Reing (1999). 4
Empirical verificaion of he models finds place in secion 3 of he paper. As we are looking for long-run relaionships, coinegraion analysis will be used for assessmen, if any of he models fis he daa. The esimaion resuls will allow us calculae among ohers he average naural rae of ineres for he US. Secion 4 concludes, and an appendix presens some of he empirical resuls in deail. 2 The models As i has already been noed, various descripions (definiions) of he naural rae of ineres can be me across economic papers. This secion describes wo mos frequenly posulaed models ha relae he ineres rae gap o inflaion and o is growh rae respecively. I should be noed ha hese descripions can be equally reaed as definiions of he naural rae. Thus, we are no going o check wheher so defined naural raes exis (because hey do by definiion), bu wheher any of hem is sable enough o become a useful benchmark for moneary policy. In wha follows we will verify he exisence of he wo naural raes under he idenifying assumpion of consancy 4. 2.1 Model 1: ineres rae gap as a deerminan of inflaion The basic propery of he model described in his secion is ha he gap beween he real and naural raes of ineres deermines, afer all lags have worked hemselves ou, he rae of inflaion 5. This kind of long-run relaionship can be described by means of a simplified equaion: (1) π p p = ( r * r ) ψ > 0, + 1 + 1 ψ 4 Useful as i is for our purpose, his idenifying assumpion canno be saisfying as a horough research on he naural rae. However, I hope ha a paper devoed exclusively o esimaing he hisorical ime series of he NRI using he Blanchard-Quah mehod will proceed soon. 5 This kind of relaionship is wha K.Wicksell (1907) probably hough abou he influence of he ineres rae gap on inflaion (see J.Amao (2001)). A similar equaion (alhough in forward-looking form) is presened by M.Woodford (1999, pp. 40-41) as soluion o a general equilibrium model. See also W.Kerr and R.King (1996) for a broad discussion of various sysems of macroeconomic equaions. Noe ha his relaionship (as well as equaion 6) can be considered as a definiion of he naural rae. 5
where π is he inflaion rae, p he log price level, r* he naural rae of ineres 6 and r he real rae of ineres. I is worh noing ha a popular descripion 7 of he relaionship beween he ineres rae gap and inflaion, of he form π = απ + ψ ( r * r ), 0 < α < 1 exhibis he same 1 seady sae properies as equaion (1): - when he ineres rae gap is closed, inflaion is zero and prices are sable. Loose moneary policy (r*>r) will evenually cause inflaion, resricive moneary policy (r*<r) will induce deflaion 8. - permanenly higher raes of inflaion are relaed o a permanenly lower real rae of ineres 9 (assuming ha he naural rae is quie sable, as has been posulaed by K.Neiss and E.Nelson 2001). Table 1. Properies of model 1 Model 1 π p = ψ ( r * r) r=r* r>r* r<r* p = cons. π = 0 π = 0 p π < 0 π = 0 p π > 0 π = 0 6 Throughou he res of he paper i will be assumed ha he naural rae of ineres is consan. Alhough his is cerainly no rue (i depends among ohers on he marginal produc of capial and on he subjecive discoun rae of agens), our assumpion will be based on he empirical resul of K.Neiss and E.Nelson (2001), who found ha he variance of r* is much smaller han he variance of r. This means ha he ineres rae gap is o a grea exen deermined by changes in r and hus r* can be assumed for simpliciy consan. 7 See for insance he model esimaed by K.Neiss and E.Nelson (2001, p. 30-32). 8 I am however, fully aware ha measuring he sance of moneary policy relying only on ineres raes is a simplified view of he cenral bank business. 9 This seems o be in line wih empirical sudies done among ohers by R.King and M.W.Wason (1992, 1997). 6
Thus he model 1 economy works like a car driven by he cenral banker. When he presses he acceleraor (i.e. opens he ineres rae gap: r*>r), he car goes faser (i.e. inflaion picks up), when he pus he gear sick ino neural (i.e. closes he gap: r*=r) he car will sar slowing down unil i sops (i.e. inflaion falls o zero). Sable speed (sable inflaion) necessiaes a permanenly pressed acceleraor (permanenly open ineres rae gap). Basic properies of model 1 are presened in able 1. To prepare he model for empirical analysis, some ransformaions will have o be performed. This is because, as i has already been noed, his paper is o describe he long run equilibrium and uses coinegraion analysis. As he ineres rae gap is expeced o be saionary 10, equaion (1) has o be ransformed one level of inegraion upwards, o allow for order 1 inegraion of he variables. This conclusion is a resul of he model s heoreical specificaion. Empirical inegraion ess will be presened in secion 3. The price level can be calculaed from he definiion of π: (2) p p 1 + π = p0 + π i, i= 1 and yields afer subsiuing from equaion (1): (3) p = p0 + ψ ( r * r i ). i= 0 1 Before empirical analysis is conduced, one more fac has o be noed. Equaion (3) has been posulaed for a saionary economy, wih a consan level of poenial oupu. However, as i is widely acceped, he permanen growh of poenial oupu will ceeris paribus lower he general price level. This fac is for example incorporaed ino he QTM equaion hrough he presence of Y. Accordingly our model has o be enlarged by he poenial oupu growh impac on prices 11 : 10 The main reason is arbirage beween invesmen in financial insrumens (reurn r) and physical capial (reurn f (k)=r*). 11 For simpliciy we ignore he relaionship beween poenial oupu (especially beween he produciviy growh rae) and he naural rae of ineres. 7
(4) p = p0 + ψ ( r * ri ) y *. = 0 i 1 Anoher imporan assumpion has o be made in order o empirically esimae he equaion. As he naural rae of ineres is no observable, we will assume ha i is a saionary variable and ha is variance is small as relaed o he variance of he real ineres rae 12. This allows us keeping r* consan and aking i ou from below he sum: (5) p 1 * * 0 + r ψ ri y i 0 = p ψ, = where denoes he ime rend. An implici message of equaion (5) is ha he general price level will depend on he whole hisory of ineres rae gaps. This specificaion can be subjec o coinegraion analysis ha will be presened in secion 3. 2.2 Model 2: ineres rae gap as a deerminan of inflaion growh The basic propery of he model described in his secion, will be ha he gap beween he real and naural raes of ineres deermines, afer all lags have worked hemselves ou, he change of he inflaion rae 13. This kind of long-run relaionship can be described by means of a simplified equaion: (6) π = ψ r * r ). ( 1 This model exhibis he following properies: - when he ineres rae gap is closed, inflaion growh is zero and inflaion is sable. Loose moneary policy (r*>r) will sar he process of inflaion acceleraion, resricive moneary policy (r*<r) will reduce inflaion. The bigger he gap he faser will he inflaion rae change. 12 This resul has been described by E.Nelson and K.Neiss (2001). 13 This kind of relaionship has been advocaed among ohers by J.C.Fuhrer and G.R.Moore (1995), T.Henckel, A.Ize and A.Kovanen (1999) as well as J.Andres, R.Mesre and J.Valles (1997). 8
- if he cenral bank wans o lower inflaion, i has o raise ineres raes o open he gap on he resricive side, and wai for inflaion o fall o he desired level. Once his has happened, he gap should be closed again. An undeniable advanage of model 2, is is accordance wih he principle of only shor run influence of he cenral bank on he real rae. The model 2 economy works like a spacecraf driven by he cenral banker. When he presses he acceleraor (i.e. opens he ineres rae gap: r*>r), he spacecraf goes faser (i.e. inflaion rises), bu once he engines are urned off (i.e. he gap is closed: r*=r) he shule will fly a a consan speed (i.e. inflaion will say sable). In his specificaion expecaions are he (implici) driving force behind inflaion persisence. If inflaion (for whaever reason) sabilized a a cerain level, raional agens observe he behavior of he cenral bank. If hey do no see any sign of policy ighening (i.e. opening he gap) hey expec inflaion no o change in he nex period and hus increase wages and prices by he inflaion rae. Basic properies of model 2 are presened in able 2. Table 2. Properies of model 2. Model 2 π = ψ ( r * r) r=r* r>r* r<r* p =? π = cons. π = 0 p =? π π < 0 p =? π π > 0 As before, he model has o be ransformed o he coinegraing represenaion. As wih respec o model 1, i is assumed ha he ineres rae gap is a saionary variable. This implies ha he model has o be ransformed one level of inegraion upwards : 9
1 (7) π = π 0 + ψ ( r * r i ). i= 0 Proceeding furher as in case of equaions (3) and (4), he model is enlarged by he influence poenial oupu exers on inflaion. Addiionally, we keep r* consan as in equaion (5): 1 * (8) * π = π 0 + ψ r ψ ri y. = 0 i This equaion can be subjec o coinegraion analysis. 3 Empirical resuls In wha follows we will use coinegraion analysis 14, o verify wheher any of he described models is a reasonable descripion of he long run relaionship beween he real ineres rae gap and inflaion (or is growh rae). 3.1 The daa As long run relaionships will be esed, i is desirable o work wih respecively long daa series 15. The exchange rae regime being anoher obsacle, only a few counries in he world have such long and reliable ime series. In his siuaion he US seem o be a good candidae. In his big, open economy, he influence of exchange rae flucuaions has only a small impac on he domesic price level. As an addiional advanage, conrary o he European counries, he US has had a floaing exchange rae regime since i abandoned he Breon Woods sysem in he early 70 s. For empirical sudies, half-yearly daa for he period 1954-1999 was used. Following raw daa series were uilised: 14 All he economeric ools used in his paper have been described in deail in M.Brzoza-Brzezina, J.Kołowski (2001). 15 In a recen sudy D.Hendry (1999) used ime series saring in 1874 o asses he impac money has on prices. Neverheless he complains abou no being able o reach deeper. 10
Table 3. Raw daa Variable CPI DEF FED TBOND5 CPIEX GDP Descripion Consumer price index (fixed basis) GDP deflaor (fixed basis) Federal funds rae Yield on 5 year T-bonds Expeced inflaion (Livingsone) Real GDP (fixed prices) In he nex sep auxiliary ime series were consruced, conaining he daa for esing equaions (5) and (8). Nominal ineres raes were deflaed 16 wih inflaionary expecaions from Livingsone polls 17. Poenial oupu was esimaed by means of he Hoddrick-Presco filer 18. Table 4. Daa used in he models Variable Transformaion LCPI ln (CPI) LDEF ln (DEF) RFEDFUND (1+FED)/(1+CPIEX) RTBOND5 (1+TBOND5)/(1+CPIEX) RFEDFUNDSUM Σ ln (RFEDFUND) RTBOND5SUM Σ ln (RTBOND5) LGDP ln (GDP) LGDPTREND HP filer (100) (LGDP) 16 Calculaion of real ineres raes is no an easy or sraighforward ask. The mehod employed is only one of he possibiliies. For more informaion on calculaing real raes see ECB 1999, pp. 16-18, and N.Anderson, J.Sleah 2001. 17 Daa from Federal Reserve Bank of Minneapolis. 18 This is cerainly a shorcu and i could be ineresing o see how he models behave under alernaive specificaions of poenial oupu. However, in our sudy he mulipliciy of models o esimae would have been overwhelming. 11
3.2 Inegraion ess For inegraion analysis wo ess have been chosen: Augmened Dickey-Fuller (ADF) and Phillips-Perron (PP). The lag order for ADF has been chosen o accoun for auocorrelaion of residuals, and for PP according o he Newey-Wes crierion, which poined a 3 lags in our case. The resuls 19 are presened in able 5. Table 5. Uni roo ess Variable Lag order in ADF ADF PP saisics PP saisics ADF es saisics saisics wihou wih wihou wih consan consan consan consan LCPI 4 0,09-0,17 d(lcpi) 3-1,56-2,96** -1,42-2,97** dd(lcpi) 3-6,18*** -6,15*** -9,20*** -9,15*** d(ldef) 0-1,02-1,94-1,06-2,06 dd(ldef) 0-9,43*** -9,38*** -9,43*** -9,38*** RFEDFUNDSUM 1 2,41-0,03 7,21 0,58 d(rfedfundsum) 1-1,08* -3,41** -1,85* -3,65*** dd(rfedfundsum) 1-6,04*** -6,01*** -9,50*** -9,45*** RTBOND5SUM 2 0,53 1,15 d(rtbond5sum) 2-1,06-3,27** -1,02-2,88* dd(rtbond5sum) 1-6,54*** -6,51*** -7,75*** -7,70*** LGDPTREND 4-0,91 d(lgdptrend) 4-0,15-2,59* -0,23-1,87 dd(lgdptrend) 3-4,33*** -4,31*** -2,88*** -2,88* * denoes rejecion of H 0 a 10%. ** denoes rejecion of H 0 a 5%. *** denoes rejecion of H 0 a 1%. The analysis of he resuls is no an easy ask. According o he uni roo ess, i canno be unambiguously decided, wha he level of inegraion of mos variables is. The inflaion rae may be saionary, if measured wih CPI, or I(1), if measured wih he GDP deflaor. However 19 ADF and PP criical values come from R.Davidson, J.MacKinnon (1993). 12
i has o be born in mind ha he sample incorporaes he 1973-79 period, when oil price shocks induced a huge rise in inflaion raes. Their influence can resul in imprecise es resuls ha may oversae he number of uni roos. Taking his addiional handicap ino accoun, i canno be said precisely, wheher he general price level in he US is I(1) or I(2). Similar conclusions can be drawn for real ineres raes and for poenial oupu. According o our esimaes real ineres raes are probably saionary, alhough here is a possibiliy ha hey are I(1). Poenial oupu is probably an I(2) variable, bu even his canno be saed definiely. The uni roo ess disappoin no only because hey impede he choice of appropriae economeric ools for daa analysis. I is worh noing ha knowing wih cerainy he inegraion level of prices could help eliminaing he wrong model wihou furher esimaion. As i has been noed before, economic heory predics ha he ineres rae gap should be a saionary variable. I follows from equaion (1) ha for model 1 o comply wih he daa, he LHS should also be I(0), which means ha inflaion should be saionary. In conrary, for model 2 o be consisen wih he daa se, inflaion should be I(1), and is growh rae π saionary. However, as he real level of inegraion of he price level canno be saed unambiguously from able 5, none of he models can be rejeced on he basis of uni roo ess. The inegraion level of he GDP deflaor poins a model 2, bu he resuls obained for he CPI can be complian wih boh models. I does no however seem o be reasonable o disinguish he models assigning each of hem is respecive price index. The price indices are only imperfec approximaions of wha economiss call he general price level and which should be explained by he heory, and so he ambiguiy should be raher explained as a resul of imperfecion of he indices or low power of inegraion ess. 3.3 Coinegraion ess - model 1 As i has been previously noed, his paper aims a finding he long-run relaionship beween he ineres rae gap and inflaion (or is growh rae). Thus appropriae economeric ools have o be chosen, ha can es for coinegraion in economic sysems. One of he possible echniques, and probably he mos popular one a presen, has been proposed by Johansen (1991). In our case, i will be based on he basis of a Vecor Error Correcion Model (VECM) 13
build for hree variables, he price level (p), he sum of all previous real ineres raes ( r i ), i= 0 and poenial oupu (y*). In wha follows, a shor descripion, of wha will be done furher in his secion is presened: 1 1. The lag order for he VECM will be specified on he basis of informaion crieria and sequenial es. 2. The coinegraing relaionship for equaion (5) will be found. This means finding a vecor [1, ψ, 1] wih ime rend and consan such, ha he residuals ε from equaion (9) are saionary. 1 (9) p = p + ψ r ψ * 0 * ri y + ε, = i 0 3. The signs of vecor componens will be verified o comply wih he heoreical model. 4. The parameer on y* will be resriced o 1 (as in he heoreical model), he validiy of resricion will be esed. 5. The average naural rae of ineres will be calculaed and compared o esimaes from oher papers. 6. The error correcion adjusmen coefficiens will be esed o show wheher he causaliy is complian wih heory (he only significan error correcion mechanism should obain in he inflaion equaion) Only models passing all sages of verificaion will be considered a proof for he exisence of a long run relaionship, connecing he price level o he sum of ineres rae gaps. Of course if he analysis breaks down a some poin (for insance no coinegraing relaionships will be found), subsequen seps will be cancelled. According o he uni roo ess, from he formal poin of view, CPI is he only price index ha can be used for empirical ess of his model. This is because he GDP deflaor is an I(2) variable and, as such, canno be consisenly included ino he sysem. Moreover, i has been decided, ha wo differen daa ses will be used, a long sample (1954-1999) and a shor one (1972-1999). This is because of he exchange rae regime change in he early 70 s 20, which could have caused imporan changes in he way, in which ineres raes ransmi o prices. 20 More on he breakdown of he Breon Woods regime see in H.R.Wüffli (1979). 14
Boh ineres raes, he federal funds rae and he 5 year T-bond rae were included. In addiion, as i was difficul o decide how long one lag is (ineres raes ener equaion (9) wih a lag), wo differen approaches were used, where he lag was inerpreed as half a year and one year, respecively. Therefore, 8 differen models were esed, based on: - 1 price index, - 2 daa samples, - 2 ineres raes, - 2 differen lags. In he firs sep, he lag order of he VAR model was deermined, based on he sequenial es (LR), Akaike informaion crierion (AIC), Schwarz crierion (SC) and Hannan-Quinn crierion (HQ) 21. I was arbirarily assumed ha he maximal lag order should no exceed 6 and han he lag indicaed by mos crieria was chosen. In case of an ambiguous resul, he smaller lag was chosen, provided ha he VECM residuals did no show auocorrelaion. The resuls are presened in Appendix 1. As he nex sep, coinegraing relaionships were searched. Table 6 conains he maximum eigenvalue and race es saisics for he 8 models 22. According o equaion (9), a consan and ime rend have been included. As i can be seen, for he long sample, boh ess rejeced he hypohesis of 2 in favour of 3 coinegraing vecors, which would normally indicae ha he variables are saionary. As hey cerainly are no, his case will no be examined any furher. Table 6. Coinegraion ess - model 1 Variables Hypohesis H 0 H 1 LCPI; RFEDFUNDSUM(-1); LGDPTREND r = 0 r = 1 (r 1) r = 1 r = 2 (r 2) r = 2 r = 3 Trace es saisics 1954-1999 1972-1999 52,76** 57,07** 29,04* 26,21* 12,89* 12,08 Maximum eigenvalue saisics 1954-1999 1972-1999 23,71 30,86** 16,15 14,12 12,89* 12,08 21 The crieria are described in deail in H.Lükepohl (1995). 22 Criical values come from M.Oserwald-Lenum (1992). 15
LCPI; RFEDFUNDSUM(-2); r = 0 r = 1 r = 1 (r 1) r = 2 (r 2) 63,24** 30,28* 64,69** 23,99 32,95** 16,54 40,80** 14,44 LGDPTREND r = 2 r = 3 13,74* 9,76 13,74* 9,76 LCPI; RTBOND5SUM(-1); r = 0 r = 1 r = 1 (r 1) r = 2 (r 2) 63,74** 32,66** 55,74** 26,75* 31,08** 19,05* 28,98* 17,44 LGDPTREND r = 2 r = 3 13,60* 9,30 13,60* 9,30 LCPI; RTBOND5SUM(-2); r = 0 r = 1 r = 1 (r 1) r = 2 (r 2) 72,29** 36,10** 57,94** 26,09* 36,18** 22,80* 31,85** 15,27 LGDPTREND r = 2 r = 3 13,26* 10,82 13,26* 10,82 r denoes he number of coinegraing vecors * denoes rejecion of H 0 a 5%. ** denoes rejecion of H 0 a 1%. As regards he shor sub-sample, i seems possible o find a long run relaionship in he daa. In all four cases, he maximum eigenvalue es indicaed one coinegraing vecor, whereby he race saisic indicaed one or wo vecors. In wha follows, he esimaed coinegraing relaionships will be presened. In all cases he exisence of one vecor has been assumed. Table 7 presens he resuls: Table 7. Coinegraing vecors 23 RFEDFUNDSUM(-1) RFEDFUNDSUM(-2) RTBOND5SUM(-1) RTBONDSUM(-2) LCPI 1,00 1,00 1,00 1,00 LGDPTREND 10,77 12,44 8,63 7,60 Ineres rae -0,71-0,46-0,67-0,05 TREND -0,17-0,26-0,13-0,14 CONSTANT -83,57-96,47-67,44-60,06 As i can be easily noiced, all four relaionships fail o fulfil he crierion on coefficien signs. In all cases he elemens of he coinegraing vecor sanding wih he sum of ineres raes are negaive, which implies a posiive relaionship beween real raes and inflaion and hus conradics he heoreical model. In his respec, furher analysis of model 1 seems purposeless, and will be given up, making room for he empirical verificaion of model 2. 23 In he following ables numbers in bold denoe resuls no complian wih he model. 16
3.4 Coinegraion ess - model 2 As before, Johansen ess will be used for he analysis of coinegraing relaionships and he 6- sep procedure will be adoped: 1. The lag order for he VECM will be specified on he basis of he sequenial es and informaion crieria. 2. The coinegraing relaionship for equaion (8) will be found. This means finding a vecor [1, ψ, 1] wih ime rend and consan such, ha he residuals ε from equaion (10) are saionary. 1 * (10) * π = π 0 + ψ r ψ ri y + ε. = 0 i 3. The signs of vecor componens will be verified o comply wih he heoreical model. 4. The average naural rae of ineres will be calculaed, and compared o esimaes from oher papers. 5. The error correcion adjusmen coefficiens will be esed o show wheher he causaliy is complian wih heory (he only significan error correcion mechanism should obain in he inflaion equaion). 6. The parameer on y* will be resriced o 1 (as in he heoreical model), he validiy of resricion will be esed. As before, 2 differen measures of ineres raes were aken, and he ess were conduced separaely for 2 daa samples. As earlier, wo differen lag srucures have been adoped, bu in conrary o model 1, wo differen measures of he price level could be inroduced. This is because for coinegraion analysis of equaion (10) inflaion has o be an I(1) variable, and as i can be seen from able 5, according o uni roo ess, boh he CPI level and he GDP deflaor can be inegraed of order 2, which means ha he inflaion measures can be I(1). Thus model 2 will be esed in 16 cases, consising of: - 2 price indices, - 2 daa samples, - 2 ineres raes, - 2 differen lags. 17
Table 8. Coinegraion ess - model 2 Variables Hypohesis H 0 H 1 D(LCPI); RFEDFUNDSUM(-1); LGDPTREND D(LCPI); RFEDFUNDSUM(-2); LGDPTREND D(LCPI); RTBOND5SUM(-1); LGDPTREND D(LCPI); RTBOND5SUM(-2); LGDPTREND D(LDEF); RFEDFUNDSUM(-1); LGDPTREND D(LDEF); RFEDFUNDSUM(-2); LGDPTREND D(LDEF); RTBOND5SUM(-1); LGDPTREND D(LDEF); RTBOND5SUM(-2); LGDPTREND r = 0 r = 1 r = 2 r = 1 (r 1) r = 2 (r 2) r = 3 r = 0 r = 1 (r 1) r = 1 r = 2 (r 2) r = 2 r = 3 r = 0 r = 1 (r 1) r = 1 r = 2 (r 2) r = 2 r = 3 r = 0 r = 1 (r 1) r = 1 r = 2 (r 2) r = 2 r = 3 r = 0 r = 1 (r 1) r = 1 r = 2 (r 2) r = 2 r = 3 r = 0 r = 1 (r 1) r = 1 r = 2 (r 2) r = 2 r = 3 r = 0 r = 1 (r 1) r = 1 r = 2 (r 2) r = 2 r = 3 r = 0 r = 1 (r 1) r = 1 r = 2 (r 2) r = 2 r = 3 Trace es saisics Maximum eigenvalue saisics 1954-1999 1972-1999 1954-1999 1972-1999 56,16** 43,94* 39,79** 24,30 16,36 19,63 13,78 10,13 2,58 9,50 2,58 9,50 41,74 38,60 25,22 19,58 16,52 19,02 13,65 11,69 2,877 7,32 2,87 7,32 61,36** 39,87 43,03** 20,72 18,32 19,15 14,80 12,42 3,52 6,72 3,52 6,72 50,17** 39,94 36,31** 28,39* 13,85 11,54 9,62 7,42 4,23 4,12 4,23 4,12 44,97* 41,00 26,32* 20,02 18,64 20,97 15,96 12,60 2,68 8,37 2,68 8,37 41,07 35,59 25,29 20,90 15,782 14,68 12,22 9,22 3,55 5,46 3,55 5,46 52,08** 46,33* 31,40** 21,86 20,67 24,46 17,91 16,82 2,75 7,64 2,75 7,64 45,10* 38,66 29,94* 23,71 15,15 14,95 11,90 9,91 3,25 5,04 3,25 5,04 r denoes he number of coinegraing vecors. * denoes rejecion of H 0 a 5%. ** denoes rejecion of H 0 a 1%. 18
As before, he lag order of he VECM augmenaions, has been derived from four informaion crieria, which are presened in Appendix 1. Table 8 conains he oucome of coinegraion ess. Le us sar describing he resuls wih he cases based on he federal funds rae. As i can be seen only in 3 ou of 8 cases, a coinegraing vecor has been repored. This resul seems poor, bu i sill migh be worh aking a look a he esimaed vecors (ab. 9). Table 9. Coinegraing vecors for RFEDFUNDSUM D(LCPI) RFEDFUNDSUM(-1) 1954-1999 D(LDEF) RFEDFUNDSUM(-1) 1954-1999 D(LCPI) RFEDFUNDSUM(-1) 1972-1999 Inflaion 1,0000 1,000 1,000 D(LGDPTREND) 3,467910 3,855123 2,774886 Ineres rae 0,082327 0,059435 0,084818 TREND -0,001953-0,001304-0,001931 CONSTANT -0,134388-0,129226-0,131249 Error correcion coefficien in he inflaion equaion (sandard deviaion) Implied average naural rae of ineres -0,559009-0,232074 (0,09282) (0,08812) -0,830450 (0,17431) 2,4% 2,2% 2,3% Following he above-oulined seps of analysis, we can sae ha: 1. All parameer signs are as expeced. 2. The naural rae of ineres amouns o respecively 2,2%, 2,3% and 2,4% and hus is similar o oher repored esimaes (A.Blinder (1998)). 3. The error correcion mechanism is quie srong, and significanly differs from zero. 4. The parameers sanding wih poenial oupu amoun o 3,46, 3,85 and 2,77, respecively. Wheher hese numbers are significanly differen from one, will be shown in able 10. The es has been proposed by Johansen and is based on he likelihood raio saisics. I can be clearly seen from able 10 ha only in one case he hypohesis of significan difference of he parameer from one could no be rejeced a he 5% level. 19
Table 10. Tesing he validiy of resricions imposed on he y* parameer of he coinegraing vecor. Model D(LCPI) RFEDFUNDSUM(-1) 1954-1999 D(LDEF) RFEDFUNDSUM (-1) 1954-1999 D(LCPI) RFEDFUNDSUM (-1) 1972-1999 Saisics 18,15566 10,33139 2,336606 p-value 0,000020 0,001308 0,126365 Le us now move o he coinegraing equaions esimaed wih he long ineres rae variable RTBOND5SUM. The 6 esimaed vecors are presened in ables 11 and 12. Table 11. Coinegraing vecors (1954-1999) D(LCPI) RTBOND5SUM(-1) D(LCPI) RTBONDSUM(-2) D(LDEF) RTBOND5SUM(-1) D(LDEF) RTBONDSUM(-2) Inflaion 1,0000 1,0000 1,0000 1,0000 D(LGDPTREND) 1,373842 1,434585 2,835506 2,544106 Ineres rae 0,083813 0,082602 0,053273 0,058477 TREND -0,002541-0,002505-0,001500-0,001664 CONSTANT -0,091262-0,089200-0,102605-0,098730 Error correcion coefficien in he inflaion equaion (sandard deviaion) Implied average naural rae of ineres -0,751238-0,609649 (0,12032) (0,14298) -0,252958-0,277906 (0,09064) (0,10797) 3,0% 3,0% 2,8% 2,9% Table 12. Coinegraing vecors (1972-1999) D(LCPI) RTBOND5SUM(-2) D(LDEF) RTBOND5SUM(-1) Inflaion 1,0000 1,000 D(LGDPTREND) 1,506398 1,891945 Ineres rae 0,049769 0,019799 TREND -0,001236-0,000112 CONSTANT -0,092560-0,096108 Error correcion coefficien in he inflaion equaion (sandard deviaion) -0,969133 (0,27029) -0,688883 (0,15521) Implied average naural rae of ineres 2,5% 0,6% 20
Proceeding as before, i can be saed ha: 1. In all six cases he signs of parameers are as expeced, 2. Wih one excepion, he naural raes of ineres are of reasonable size (2.8-3.0%), 3. In all six cases here is significan error correcion in he inflaion equaion, 4. As i can be seen from ables 13 and 14, in four cases he y* parameer does no significanly differ from one (a he 5% level). Table 13. Tesing he validiy of resricions imposed on he y* parameer of he coinegraing vecor (1954-1999) D(LCPI) RTBOND5SUM(-1) 1954-1999 D(LCPI) RTBOND5SUM(-2) 1954-1999 D(LDEF) RTBOND5SUM(-1) 1954-1999 D(LDEF) RTBOND5SUM(-2) 1954-1999 Saisics 0,033257 0,697099 5,857772 6,678978 p-value 0,855297 0,403760 0,015508 0,009756 Table 14. Tesing he validiy of resricions imposed on he y* parameer of he coinegraing vecor (1972-1999) D(LCPI) RTBOND5SUM(-2) 1972-1999 D(LDEF) RTBOND5SUM(-1) 1972-1999 Saisics 0,234689 0,539415 p-value 0,628068 0,462675 Summing up he resuls 24, i can be noed ha from among sixeen cases under consideraion four of he saed coinegraing relaionships fulfilled all he crieria imposed on he model. Three successful models are based on he long erm ineres rae, whereby one includes he federal funds rae. I also seems imporan ha in all four models wih imposed resricions he adjusmen coefficiens show a significan error correcion mechanism in he inflaion equaion and no error correcion in he wo remaining equaions. This implies a causal relaionship of he ype we would have expeced, going from real ineres raes o inflaion. I hus can be said ha he relaionship described in model 2, linking inflaion o he hisory of 24 Though our uni roo es resuls srongly suppor he I(2) resul for rend GDP, i is ofen argued ha his variable is I(1). If so, D(LGDPTREND) has o be excluded from he CV in Model 2. However, he resuls of such an exercise give comparable suppor o he concep described by Model 2 and place he naural rae of ineres in he range 2.2-3.0% for he long rae and 1.5-1.9% for he federal funds rae. 21
real ineres rae gaps and implicily making he growh rae of inflaion deermined by he gap, describes he macroeconomic relaionship beween cenral bank insrumens and inflaion. 4 Conclusions This paper aimed a esing, wheher a simple equaion, of he form inroduced o he economic lieraure by he quaniy heory, can be found and empirically verified for he longrun relaionship beween he real ineres rae gap and inflaion. Our resuls, based on coinegraion analysis, show ha such a sable long-run equaion links he growh rae of inflaion o he ineres rae gap: π = ψ r * r ). ( 1 Thus, a leas wih respec o inflaionary processes, he economy seems o work like a space shule, ha once acceleraed, will cruise a a sable speed wihou he use of engines. The cenral bank can open he ineres rae gap o accelerae inflaion, and once his has happened he gap can be closed and inflaion will remain a he higher level. The naural rae of ineres, alhough cerainly no consan, is sable enough, o allow us deermine he ineres rae gap by means of changes in he real ineres rae. Using he calculaed average value of he naural rae we can easier guess wha he curren sance of moneary policy is. This does no mean ha esimaing more precisely he naural rae of ineres would no be helpful for moneary policy. We sill do no know much abou he behavior of his variable. Is deerminans, among ohers he marginal produc of capial, he produciviy growh rae and he subjecive discoun rae of privae agens are only hardly observable. Wih beer esimaes of he naural rae of ineres, cenral banks could influence economic behavior wih more precision. Disinflaing counries would know, a wha level o se real ineres raes, afer disinflaion has been finished, wihou aking he risk of reflaing he economy again. As already menioned, he adoped research echnique only allowed us o calculae he average level of he naural rae; he acual ime series is sill unknown. There are various ways o proceed furher. One possibiliy is o build a general equilibrium model of he economy, calibrae i and calculae he flexible price equilibrium level of real ineres raes. 22
Anoher possibiliy is he use of advanced ime-series echniques like he Kalman filer, o disinguish beween permanen and emporary changes of real raes. The hird soluion could be based on inroducion of he echnique described by O.J.Blanchard and D.Quah (1989). This ask however, will be lef for anoher paper. 23
Appendix 1 The lag order chosen from he sequenial es and informaion crieria: Variables in he model Sample LR AIC SC HQ Choice MODEL 1 LCPI; RFEDFUNDSUM(-1); LGDPTREND 1954-1999 5 5 4 5 5 LCPI; RFEDFUNDSUM(-2); LGDPTREND 1954-1999 5 6 4 5 5 LCPI; RTBOND5SUM(-1); LGDPTREND 1954-1999 5 5 5 5 5 LCPI; RTBOND5SUM(-2); LGDPTREND 1954-1999 5 6 5 5 5 LCPI; RFEDFUNDSUM(-1); LGDPTREND 1972-1999 5 5 5 5 5 LCPI; RFEDFUNDSUM(-2); LGDPTREND 1972-1999 5 5 5 5 5 LCPI; RTBOND5SUM(-1); LGDPTREND 1972-1999 5 5 5 5 5 LCPI; RTBOND5SUM(-2); LGDPTREND 1972-1999 5 5 5 5 5 MODEL 2 D(LCPI); RFEDFUNDSUM(-1); LGDPTREND 1954-1999 4 4 4 4 4 D(LCPI); RFEDFUNDSUM(-2); LGDPTREND 1954-1999 5 5 4 5 5 D(LCPI); RTBOND5SUM(-1); LGDPTREND 1954-1999 4 4 4 4 4 D(LCPI); RTBOND5SUM(-2); LGDPTREND 1954-1999 5 5 4 4 4 D(LDEF); RFEDFUNDSUM(-1); LGDPTREND 1954-1999 4 4 4 4 4 D(LDEF); RFEDFUNDSUM(-2); LGDPTREND 1954-1999 5 5 4 4 4 D(LDEF); RTBOND5SUM(-1); LGDPTREND 1954-1999 4 4 4 4 4 D(LDEF); RTBOND5SUM(-2); LGDPTREND 1954-1999 5 5 4 4 4 D(LCPI); RFEDFUNDSUM(-1); LGDPTREND 1972-1999 4 4 4 4 4 D(LCPI); RFEDFUNDSUM(-2); LGDPTREND 1972-1999 5 6 4 5 5 D(LCPI); RTBOND5SUM(-1); LGDPTREND 1972-1999 4 6 4 4 4 D(LCPI); RTBOND5SUM(-2); LGDPTREND 1972-1999 5 6 3 5 5 D(LDEF); RFEDFUNDSUM(-1); LGDPTREND 1972-1999 4 4 4 4 4 D(LDEF); RFEDFUNDSUM(-2); LGDPTREND 1972-1999 4 4 3 4 4 D(LDEF); RTBOND5SUM(-1); LGDPTREND 1972-1999 4 4 4 4 4 D(LDEF); RTBOND5SUM(-2); LGDPTREND 1972-1999 4 4 3 4 4 24
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