The real interest rate gap as an inflation indicator

Transcription

1 The real ineres rae gap as an inflaion indicaor Kaharine S. Neiss* and Edward Nelson** * Srucural Economic Analysis Division, Moneary Analysis, Bank of England. kaharine.neiss@bankofengland.co.uk ** MPC Uni, Bank of England, and Cenre for Economic Policy Research, London. ed.nelson@bankofengland.co.uk The views expressed are hose of he auhors and do no necessarily reflec hose of he Bank of England, he Moneary Policy Commiee, or he Cenre for Economic Policy Research. The auhors hank Peer Andrews, Mark Gerler, DeAnne Julius, Rober King, Rober Kollmann, Thomas Laubach, Ahanasios Orphanides, Glenn Rudebusch, Chris Salmon, Frank Smes, Paul Tucker, Simon van Norden, Raf Wouers, Michael Woodford, and paricipans a Bank of England seminars and he Canadian Macro Sudy Group meeing, November 2000, for helpful discussions. Copies of working papers may be obained from Publicaions Group, Bank of England, Threadneedle Sree, London, EC2R 8AH, elephone , fax , mapublicaions@bankofengland.co.uk Working papers are also available a The Bank of England s working paper series is exernally refereed. Bank of England 2001 ISSN

2 Conens Absrac 5 Summary 7 1 Inroducion 9 2 The model 13 3 Model calibraion and properies 21 4 Model evaluaion 29 5 Indicaor properies of he real ineres rae gap 30 6 Empirical properies of he real ineres rae gap 34 7 Conclusions 37 Appendix: Calculaion of oupu and ineres rae gaps 38 References 42 3

3 Absrac A long-sanding area of research and policy ineres is he consrucion of a measure of moneary policy sance. One measure ha has been proposed as an alernaive o indices ha employ moneary aggregaes or exchange raes is he spread beween he acual real ineres rae and is flexible-price, or naural-rae, counerpar. We examine he properies of he naural real ineres rae and real ineres rae gap using a dynamic sochasic general equilibrium model. Issues we invesigae include: (1) he response of he gap and is componens o fundamenal economic shocks; and (2) he indicaor and forecasing properies of he real ineres rae gap for inflaion, boh in he model and in he daa. Our resuls sugges ha he real ineres rae gap has value as an inflaion indicaor, supporing he neo-wicksellian framework advocaed by Woodford (2000). 5

4 Summary In his paper, we invesigae he business cycle properies of he real ineres rae relaive o is naural value. Our invesigaion ino he naural real ineres rae is moivaed by he possibiliy of consrucing a measure of moneary policy sance based exclusively on ineres raes. Recen work by Michael Woodford has revived he ideas of Knu Wicksell by focusing on he gap beween he curren level of he naural rae of ineres and he ineres rae conrolled by he cenral bank as he key variable for he analysis of inflaionary or deflaionary pressures. In line wih his erminology, we describe he spread beween acual and naural real ineres raes as he real ineres rae gap. This paper examines a number of quesions involving he real ineres rae gap, including: Does he real ineres rae gap provide a useful ool for moneary policy analysis? Is he real ineres rae gap more difficul o measure han he oupu gap? How do empirical measures of he real ineres rae gap perform in forecasing UK inflaion? We develop a dynamic sochasic general equilibrium model wih sicky prices in order o examine he behaviour of he naural real ineres rae and he real ineres rae gap. In our model, household spending and asse accumulaion, and he prices ha firms se, are based on opimising behaviour. We build on he exising lieraure by including capial formaion (subjec o adjusmen coss), habi persisence in consumpion, echnology and demand shocks, and wo alernaive models of price sickiness. The baseline model of price sickiness ha we use is Calvo price-seing, which can be inerpreed as a sysem of saggered conracs for nominal prices. We calibrae he model o he UK economy, and examine he response of he naural real ineres rae o shocks o boh echnology and demand. Our focus is mainly on he indicaor properies of he real ineres rae gap, and so we examine how well he real ineres rae gap does in signalling fuure inflaion boh in response o specific shocks (which we examine using impulse response funcions) and when all shocks are hiing he economy simulaneously (which we examine using sochasic simulaions). Using our model as a guide, we also consruc empirical esimaes of he naural real rae and he real ineres rae gap from UK daa. Our key resuls include: The response of he naural real ineres rae o a echnology shock depends on wheher or no capial is included in he model and, if so, wheher or no here are capial adjusmen coss. We find 7

5 ha wih capial adjusmen coss, he naural real ineres rae can fall in response o a echnology shock. For a given acual real ineres rae, his leads o a rise in he real ineres rae gap. Conversely, he naural real ineres rae rises in response o a demand shock. For a given acual real ineres rae, his leads o a decline in he real ineres rae gap. The less firms and households are willing o adjus heir quaniies, he more he naural real rae needs o adjus o mainain equilibrium. Sochasic simulaions indicae ha he real ineres rae gap and oupu gap do equally well in forecasing inflaion. In addiion, he behaviour of he real ineres rae is a reasonable approximaion for he behaviour of he real ineres rae gap. By conras, oupu (or derended oupu) is no a good indicaion of he behaviour of he oupu gap. This suggess he value of consrucing measures of boh gaps insead of concenraing only on oupu gap measures. Finally, we es he predicive power of he real ineres rae gap for UK inflaion. On quarerly UK daa, he real ineres rae gap is closely relaed o fuure inflaion, wheher he relaionship is judged by correlaions or by he marginal predicive conen of he gap for inflaion in regressions. Our resuls sugges ha consrucing a real ineres rae gap series, using heory as a guide, can have value for evaluaing he sance of moneary policy and he prospecs for fuure inflaion, in keeping wih he neo-wicksellian framework of Woodford (2000). 8

6 1 Inroducion In evaluaing he consequences of moneary policy acions for he fuure behaviour of inflaion, i is ofen useful o consruc a measure of moneary policy sance. Typically, such a measure will use some indicaor of moneary or demand condiions, and express ha measure relaive o a baseline value (such as he value consisen wih price sabiliy). There are several candidaes for variables o include in a measure of moneary sance. Moneary aggregaes (or heir raes of growh) have hisorically been prominen candidaes. A common criicism of heir use for his purpose is ha hey can be disored by innovaions o he financial sysem. (1) In addiion, moneary aggregaes have been viewed as being oo far removed from he ypical insrumen of moneary policy, namely he shor-erm nominal ineres rae. This is paricularly so for broad moneary aggregaes. One measure of he sance of moneary policy ha does no rely on moneary aggregaes is he moneary condiions index (MCI). An MCI consiss of a weighed index of he ineres rae and he exchange rae. (2) A weakness of his approach is ha i relaes changes in he exchange rae o inflaion wihou idenifying he shock responsible for he change in he exchange rae. For example, MCIs end o inerpre exchange rae depreciaions as producing moneary ease, bu i is possible for he equilibrium real exchange rae o depreciae under condiions of price sabiliy (see King (1997)). More generally, he weigh given o exchange raes in he index may implicily assume a much closer and more mechanical link beween exchange rae depreciaions and inflaion han can be jusified empirically (see McCallum and Nelson (2000) and Sock and Wason (2000)). (3) In ligh of he argumens agains including moneary aggregaes or exchange raes in measures of moneary policy sance, one poenially fruiful opion is o consruc a measure of he moneary policy sance based exclusively on ineres raes. In his vein, Woodford (1999a, 2000) has advanced he use of he naural or equilibrium real rae of ineres (he real ineres rae ha prevails under price flexibiliy) in he analysis of price level deerminaion. Woodford s analysis revives he ideas of Wicksell (1898, 1906) wihin a dynamic sochasic general equilibrium model. He argues ha an analysis of price level deerminaion ha is based on he difference beween acual and naural real ineres raes is preferable o radiional analysis, which is based on he ineracion of money demand and supply. In models in which he insrumen of moneary policy is a nominal ineres rae, his neo-wicksellian analysis of price level deerminaion avoids he cumbersome procedure of solving for he implied money supply funcion. In Woodford s framework, he key variable for he analysis of inflaionary or deflaionary pressures is he gap beween he curren level of he naural rae of ineres and he ineres rae conrolled (1) See Svensson (1999) for a recen ariculaion of his and oher criicisms of moneary aggregaes. (2) Ericsson and Kerbeshian (1997) provide a deailed bibliography of he MCI lieraure. (3) One par of he problem is ha he relaive weighs on ineres raes and exchange raes in MCIs are ypically based on he esimaed relaive effecs of he wo variables on aggregae demand, raher han inflaion per se. See Eika, Ericsson and Nymoen (1996) for a discussion of MCI weighs. 9

7 by he cenral bank (Woodford (1999a, page 35)). In line wih his erminology, we define he real ineres rae gap as he spread beween he acual and naural real ineres raes, and his paper sudies his gap concep. The posiion of acual real ineres raes relaive o heir naural, neural, or equilibrium value, has been considered by policy-makers as well. For example, a he 9 10 December 1998 meeing of he Bank of England s Moneary Policy Commiee, []he Commiee discussed wheher i was helpful o hink abou he appropriae level of nominal ineres raes by reference o he concep of a neural level, which provides neiher simulus nor resrain o he economy. The oucome of his discussion was ha [w]hile some members of he Commiee found he concep of he neural rae useful in deciding on ineres rae policy, oher members found he uncerainy surrounding is level so large ha he concep was of lile use as a pracical guide o policy (Moneary Policy Commiee (1999, page 67)). Thus i is probably accurae o characerise he aiude by many policy-makers o he pracical usefulness of he real ineres rae gap concep as scepical. I is also very likely ha he real ineres rae gap is regarded as less easy o measure empirically han an oupu gap concep. Neverheless, we see hree key pracical reasons for reconsidering he imporance of looking a real ineres gaps: 1. Undersanding he behaviour of he naural real rae appears o be imporan for undersanding he empirical relaionship beween he real ineres rae and oupu. Consider Table A below, which gives correlaions beween he level of derended log GDP (y ) and he real ineres rae (curren and lagged) for quarerly US and UK daa: Table A: Correlaions beween derended GDP and he real ineres rae Corr(y, r k ) Unied Saes 1980 Q Q4 Unied Kingdom 1980 Q Q4 k = k = k = k = k = Noe: For boh he Unied Saes and he Unied Kingdom, y is obained from he residuals of a regression of log real GDP (seasonally adjused) on a quadraic rend, 1976 Q Q4, and he nominal ineres rae used in he calculaion of r is he quarerly average of he Treasury bill rae. For he Unied Saes, r is hen measured by he ex pos real ineres rae. For he Unied Kingdom, r is (as in Char 7 and Secion 6 below) he Treasury bill rae minus he forecass of nex-period annualised inflaion from a VAR(8). The inflaion series used in hese calculaions is he seasonally adjused quarerly log change in he CPI (US) or he RPIX (UK). 10

8 For he Unied Saes, he correlaion beween he wo variables is negaive and, herefore, perhaps inerpreable as reflecing he negaive effecs of he real ineres rae on aggregae demand. Bu for he Unied Kingdom, he correlaion is generally posiive, indicaing ha he real ineres rae is procyclical. (4) To idenify he negaive effecs of he real rae on aggregae demand suggesed by economic heory, i may be necessary o conrol for he flucuaions of he naural levels of oupu and he real rae in response o real shocks. 2. The real ineres rae gap poenially has sronger leading-indicaor properies han he oupu gap. In models where he oupu gap responds o he real ineres rae gap wih a lag and inflaion reacs o he oupu gap gradually, he real ineres rae gap should give advance informaion abou boh he oupu gap and inflaion. 3. I is possible ha he real ineres rae gap may acually be measured wih less uncerainy han he oupu gap. While measures of he oupu gap are very commonly used in policy analysis and forecasing, i is worh remembering ha many of hese measures are based on he assumpion ha poenial oupu evolves according o a deerminisic rend (such as a linear, quadraic, or broken-linear rend). Economic heory suggess, insead, ha poenial oupu, while cerainly conaining a rend componen, also flucuaes over he business cycle in response o all real shocks. The validiy of many sandard measures of he oupu gap herefore ress on he hypohesis ha he response of poenial oupu o hese shocks is relaively fla, so ha derended oupu provides a good approximaion of he oupu gap. (5) This hypohesis is difficul o jusify in ligh of he resuls from many dynamic general equilibrium models (including ours below) ha sugges ha poenial oupu flucuaes considerably. On he oher hand, i is possible ha he response of he naural real rae o shocks may be relaively fla, which would make i relaively sraighforward o consruc reliable measures of he real ineres rae gap even hough daa are no available on he naural rae. The above consideraions sugges ha here are benefis from furher sudy of he real ineres rae gap concep. To his end, his paper develops a sicky-price, dynamic sochasic general equilibrium (DSGE) model in order o examine he behaviour of he naural real ineres rae and he real ineres rae gap. Any general equilibrium model, wheher sicky or flexible-price, implicily provides a model of he naural real ineres rae and he real ineres rae gap. In flexible-price models, he behaviour of he real ineres rae gap is rivial i is zero each period by definiion, since he real ineres rae is he naural real rae. (6) In sicky-price models, (7) he real ineres rae gap is zero on average (provided ha he Phillips curve is verical in he long run), bu will no be zero every period (4) In erms of Boldrin, Chrisiano and Fisher (1999), he real ineres rae does no exhibi an invered leading-indicaor propery for real GDP in he Unied Kingdom. (5) Similarly, producion funcion based approaches o measuring poenial oupu ypically do no keep rack adequaely of he disincion beween acual and flexible-price values of capial and labour inpus. For example, measuring flexible-price labour supply by he oal labour force involves he assumpion ha labour supply is inelasic and ends o generae an overly smooh poenial oupu series. A general equilibrium model such as ours provides a way of keeping rack of flexible-price values of variables and of heir response o real shocks. (6) This is so even for models, such as hose in Beaudry and Guay (1996), ha enrich he dynamics of real business cycle models, bu mainain he assumpion of price flexibiliy. (7) Here we presume ha he sickiness of prices lass more han one period. 11

9 (excep in he very specific case in which he moneary auhoriy runs a policy ha eliminaes compleely he real effecs of price sickiness). In addiion o he Woodford papers menioned above, King and Wason (1996), Roemberg and Woodford (1997), Clarida, Gali and Gerler (1999), and Giannoni (2000) have focused on he behaviour of real raes relaive o heir naural values in general equilibrium sicky price models. Of hese, only King and Wason have explici capial formaion. (8) Their model, however, does no include preference shocks, which in heory (and in our own numerical resuls below) have a major influence on he naural rae. Our model builds on he exising lieraure by including boh capial formaion and preference shocks, as well as elemens absen from he aforemenioned papers, such as non ime separable preferences. Unlike he previous work, we also presen resuls for more han one model of price sickiness. An advanage of our use of a fully specified general equilibrium model is ha we can race he deerminaion of he naural real rae which we denoe r * back o he model s underlying economic shocks. Woodford (1999a, 1999b) assumes ha r * is a univariae AR(1) exogenous process. By making his high-level assumpion, such an analysis does no permi decomposiion of he underlying real shocks ha deermine r *. And analysis of he underlying shocks affecing r * could aid in reducing he uncerainy surrounding is level, which, as noed above, has inhibied he usefulness of he concep o policy-makers. The framework we adop also enables us o examine he effec of paricular shocks or model elemens on he variance of he naural real rae. This is imporan in ligh of he diverging esimaes of he variance of he naural real rae in recen papers. King and Wason (1996) find ha he naural real rae in a calibraed DSGE model has a variance well below ha observed in US daa for he acual (ex pos) real rae. The esimaes in Roemberg and Woodford (1997), however, sugges a more volaile naural real ineres rae series, wih a sandard deviaion of 3.9 percenage poins (annualised) on pos-1979 US quarerly daa (9) around 1 percenage poin greaer han ha observed empirically for he acual real rae in US and UK daa over ha period. Our paper is organised as follows. Secion 2 describes he model. Secion 3 analyses he response of he naural real ineres rae and he real ineres rae gap in he model o shocks. Secion 4 compares he model s dynamics o hose in he daa, and Secion 5 examines he properies of he real ineres gap in he model. Our focus is mainly on he indicaor properies of he real ineres rae gap, and so we examine how well he real ineres rae gap does in signalling fuure inflaion when all shocks are hiing he economy simulaneously (which we examine using sochasic simulaions). Using he heoreical model as a guide, Secion 6 hen consrucs esimaes of he real ineres rae gap series from UK daa and evaluaes he series forecasing abiliy for inflaion. Secion 7 concludes, and an appendix describes how we measure oupu and real ineres rae gaps in he model. (8) Woodford (2000) adds endogenous capial formaion o his model. One difference beween his model and ours below is ha our calibraion of preference parameers, which srongly affec he ineres elasiciy of aggregae demand, is more in keeping wih hose suggesed by economeric sudies such as Fuhrer (2000) and Ireland (2000). (9) See Woodford (1999b, foonoe 50). 12

10 2 The model This secion describes he DSGE model we employ in he paper. 2.1 Households The economy is inhabied by a large number of households, each of which has preferences defined over a composie consumpion good (denoed C in period ), leisure (l ), and real money balances (M / P ). Money eners he uiliy funcion direcly o capure he idea ha real balances provide a ransacions-faciliaing service. The represenaive household chooses a sequence of consumpion, leisure, nominal money and one-period bond (B +1 ) holdings, and capial (K +1 ), o maximise lifeime uiliy: E σ 1 C σ j σ j γ M j β + + λ + bl h j j 0 σ + + = 1 C+ j 1 1 ε P+ j 1 ε (1) subjec o a series of real period budge consrains: M B B M τ C X w N z K R X + j + j+ 1 G + j + j 1 + j η + j j = + j + j+ + j + j + + j 1 + φ + j, P+ j P+ j P+ j P+ j P+ j j = 0,1,..., (2) where b > 0, γ > 0, ε > 0, and β (0,1). X in equaion (2) is relaed o K by: ( δ ) X K K + j = + j j (3) where δ [0,1). In equaion (2), R G denoes he gross nominal ineres rae, N denoes labour supplied, τ denoes governmen ransfers, and z he reurn on capial. In he numerical resuls of Secion 5, we consider wo versions of he model: a simplified sripped-down version of he model wih no capial variaion (K = K ss and X = δ K ss for all ), and one wih capial accumulaion bu wih adjusmen coss operaive. If he model had no capial adjusmen coss, he variable X could be inerpreed as invesmen expendiure; wih capial adjusmen coss, we refer o X as quasi-invesmen. The size of capial adjusmen coss is deermined by he parameers ϕ (ϕ > 0) and η (1 < η < ). (10) When capial accumulaion is subjec o adjusmen coss here is a wedge beween he reurn on capial and he real ineres rae, beyond he consan wedge implied by he rae of depreciaion of capial (δ). (10) Our specificaion of he capial adjusmen cos funcion follows Abel (1983) and Casares and McCallum (2000). 13

11 The parameer σ in he consumer s uiliy funcion indexes he curvaure of he uiliy funcion: he larger σ, he more he household is willing o shif consumpion across periods in response o ineres rae changes. 1/(σε) is he seady-sae consumpion elasiciy of money demand, and γ deermines he seady-sae consumpion/money raio. Preferences over consumpion ake on a non ime separable form o capure he idea ha households may exhibi habi formaion in heir consumpion paerns. The parameer h [0,1) indexes he degree of habi formaion. If h = 0, he household exhibis no habi formaion in consumpion, and preferences are ime-separable. For 0 < h < 1, uiliy from curren consumpion depends parially on consumpion in adjacen periods. We work wih sricly posiive h in his paper in ligh of evidence ha doing so reduces some of he empirical shorcomings of quaniaive business cycle models (Boldrin, Chrisiano and Fisher (1999); Fuhrer (2000)). Finally, he uiliy funcion is augmened by a disurbance (λ ) o consumpion preferences, which we inerpre as an IS or real demand shock. The ime allocaion consrain of he household is normalised so ha labour and leisure sum o one: N + l = 1 (4) The composie consumpion good C is a Dixi-Sigliz aggregae of a mulipliciy of differeniaed goods, indexed by i [0,1]. Under his scheme, he composie consumpion good and price index are defined as: 1 1 ρ = ( ) 0 C C i di ρ = 1 ρ P ( ) 1 p i ρ di 0 1 ρ ρ (5) (6) which can be derived from wo-sage budgeing. The parameer ρ (0,1) deermines he degree of subsiuabiliy of differeniaed goods in consumpion. For example, if ρ is close o 1, goods are perfecly subsiuable in consumpion and firms are perfecly compeiive. The inverse of ρ is herefore he (gross) seady-sae mark-up of price over marginal cos. Leing ψ denoe he Lagrange muliplier on (2), opimal household behaviour yields he following condiions: C : σ 1 σ 1 C σ 1 C σ he h + 1 h = C 1 C C C λ β λ ψ (7) 14

12 l : b = wψ (8) ε M ψ + 1 M : γ = ψ βe P P + 1 P (9) ßψ 1 B +1 : 0 G E + = ψ R P P + 1 (10) η K +1 : ( 1 η 0 ψ 1 fη ) βeψ ( 1 δ)( 1 φη 1 ) = + X + X + z (11) M B+ 1 G B M 1 τ η ψ : C X = wn + zk + R 1 + φ X (12) P P P P P 2.2 Firms There is a coninuum of monopolisically compeiive firms, indexed by j [ 0,1]. Each firm j chooses price (p j ), labour (N j ), and capial (K j+1 ) o maximise is profis: Yj pj wn j zk j P (13) subjec o a demand funcion for is good, wrien as: p j P Y j = Y ( 1 ) ρ (14) and a producion funcion: Y = AN K α 1 j j j (15) where P and Y are he aggregae price and oupu levels, A is a echnology shock, and α (0,1). In a symmeric equilibrium (under which we can drop he subscrip j), he firs-order condiions for he represenaive firm are given by: l : Y G α = µ N w (16) 15

13 Y K : (1 α ) = µ K z G (17) Here, µ G is he gross mark-up of price over marginal cos; in seady sae his variable equals 1/ρ, as noed above. 2.3 Marke-clearing Finally, he economy is subjec o he following resource consrain: Y = C + X +φ X η (18) which differs from he usual closed-economy consrain due o he presence of capial adjusmen coss. 2.4 Equilibrium In order o invesigae he dynamics of he model, we log- linearise he above opimaliy condiions and echnological consrains around he seady sae. The resuling equaions are as follows (wih he superscrip ss denoing seady-sae values): Consumpion βh σ βh σ βh βhσ h σ σ(1 βh) σ(1 βh) σ(1 βh) 1 βh 2 2 ( 1) + 1 ( 1) 1 βhρ Ec 1 c c λ + = 1 ψ + λ (19) Labour marke equilibrium 0 = y n + ψ µ (20) Money demand = ψ rm ε ε R 0 R ss (21) Euler equaion Eψ = + ψ r 1 (22) 16

14 Fisher equaion Eπ + = R r 1 (23) Quasi-invesmen ( ) ( ) ss 1 α ρy ss 1 1 δ E x + K E ( y k µ ) = x + r η 1 φη η 1 φη + 1 η η 1 ss ( ) ( X ) ss ( ) ( X ) (24) Law of moion for capial k = 1 δx + (1 + δ) k (25) Resource consrain ss ss ss η C X +φη X y = c + x Y Y ss ss (26) Producion funcion ( 1 α) y = a + αn + k (27) G where y, c, k, n, rm, µ, and a are he log deviaions of Y, C, K, N, (M / P ), µ and A, respecively, from heir seady-sae values. Similarly, he Lagrange mulipliers should now be inerpreed as log-deviaions from he seady sae of he corresponding variables in he original non-linear model. π is he de-meaned quarerly ne inflaion rae, and R and r denoe he de-meaned ne nominal and real ineres raes respecively. Thus we have a log-linear model describing he pah of eleven endogenous variables: oupu (y ), capial (k ), consumpion (c ), quasi-invesmen (x ), he nominal ineres rae (R ), he real ineres rae (r ), he marginal uiliy of consumpion (ψ ), real balances (rm ), labour inpu (n ), he mark-up (µ ), and inflaion (π ). The sysem so far consiss of nine equaions. To complee he model, we need wo more equaions: a price-seing equaion and a moneary policy rule. We urn now o he firs of hese. 2.5 Price-seing This paper is concerned wih he real ineres rae gap, defined as he spread beween he acual and naural real ineres raes. We herefore need o specify he equilibria associaed wih each of he wo ineres rae conceps. The naural real ineres rae is he real rae ha prevails in he case of 17

15 fully flexible prices, whereas he acual real ineres rae is he real rae ha prevails under sicky prices. (11) Our procedure for obaining a sicky-price equilibrium consiss of wo seps: (a) Firs, we solve for naural oupu (in logs, y *) and he naural real ineres rae (r *) by obaining he flexible-price equilibrium of he log-linear model above. (12) This flexible-price equilibrium is characerised by a consan mark-up, so ha µ = 0 (see for example Ireland (1997)). The flexible-price equilibrium can herefore be obained by imposing his condiion. (b) We specify a model of gradual price adjusmen, describing how he price level responds o he oupu gap (y y *). Our baseline specificaion of gradual price adjusmen will be he Roemberg (1982) Calvo (1983) model, which, following Robers (1995), we wrie as he New Keynesian Phillips curve: β π π α µ E = µ (28) where α µ > 0. To examine he sensiiviy of our resuls o he price-seing specificaion, we repor supplemenary resuls using a differen model of price-seing, ha of Fuhrer and Moore (1995): 0.5Eπ = π 0.5 π + α µ µ (29) In boh models, he size of α µ governs he degree o which prices are sicky. The larger is α µ, he more flexible are goods prices. (13) 2.6 Shocks There are wo ypes of real shocks in his model: a echnology and a demand (IS) shock. Each of he wo shocks is assumed o follow an AR(1) process: a = ρ a + e a 1 a (30) λ = ρλ + e? 1? (31) where ρ a and ρ λ lie in [0,1], and e a and e λ are whie noise innovaions. (11) As in oher sicky-price DSGE models, he assumpion of monopolisic compeiion among firms provides groundwork for he assumpion of gradual price-seing. (12) A moneary policy rule needs o be appended o his sysem o complee he model, bu due o moneary neuraliy under price flexibiliy, he soluion for poenial oupu will be invarian o he rule seleced. (13) For he Calvo model, he flexible-price equilibrium is reached as α µ approaches infiniy. 18

16 2.7 Policy rule In an environmen of flexible prices, he acual real ineres rae will always equal he naural rae regardless of he moneary auhoriy s policy rule. On he oher hand, moneary policy has real effecs in an environmen of sicky prices. Therefore, he specificaion of he moneary policy rule will have implicaions for he real ineres rae gap. Taking his ino accoun, we have examined he properies of he real ineres rae gap under a variey of differen rules. Our resuls below will focus on a baseline policy rule esimaed on UK daa and described in Secion Open-economy consideraions Our model has no explici open-economy elemens. I is herefore worhwhile o discuss in wha sense he model s explanaion of real ineres rae deerminaion may be useful for small open economies in a world of inegraed capial markes. One inerpreaion of our model is ha i refers o he deerminaion of he world ineres rae, or he ineres rae in a large counry. We believe, however, ha our analysis has some value as a descripion of ineres rae deerminaion in a small open economy. The condiions under which a small open economy s real ineres rae is deermined compleely by global facors are acually quie sringen. One consequence of his may be ha in small open economies, domesic raher han solely global facors play a significan role in he deerminaion of real ineres raes. This poin is paricularly relevan for a model inended for moneary policy analysis. Inflaion-argeing cenral banks ypically operae wih a shor-erm nominal ineres rae insrumen, which, combined wih some ineria in inflaion, implies considerable shor-run influence by he domesic moneary auhoriy over he domesic shor-erm real ineres rae, even hough hese economies are highly open and are par of a global capial marke. Indeed, many observers would consider he following model feaures essenial for a realisic analysis of moneary policy: (i) cenral bank conrol of nominal ineres raes and shor-run influence over real raes; (ii) a considerable amoun of persisence in inflaion; and (iii) invesmen in physical capial being very imporan for cyclical flucuaions and being a major channel hrough which moneary policy affecs aggregae demand. Our closed-economy model can (under cerain seings) saisfy all hree crieria; ye very few exising open-economy DSGE models can. In he Obsfeld-Rogoff (1995) model, for example, real ineres raes are no affeced by domesic moneary policy and are equal o he foreign real rae every period. The failure of sandard open-economy models o mee crierion (ii) is documened in McCallum and Nelson (2000); and inroducing endogenous physical capial is ofen problemaic in open-economy models. We herefore proceed wih our use of a closed-economy model, bu ake he openness of he UK economy ino accoun in our calibraion of he model. 19

17 In addiion o heir effec on capial markes, open-economy elemens aler goods marke behaviour by adding ne expor demand, which is a funcion of foreign and domesic demand, o oal domesic aggregae demand. We can ake hese facors ino accoun in our closed-economy model by modifying he calibraion of he aggregae demand side of he economy. Consider he Euler equaion for consumpion in our model, for simpliciy absracing from habi formaion (h = 0). In his case, c = E c +1 σr + σ(1 ρ λ )λ. Ieraions on his condiion produce: c = σe S j=0 r +j + λ (32) so consumpion depends on curren and expeced fuure real shor raes. Now suppose ha ne foreign demand for domesic oupu is given by he log-linear equaion: nx = b 1 q + κ (33) where q is he real exchange rae (an increase signifying a depreciaion), κ is a shock o foreign demand, and b 1 > 0. Supplemening he model wih he real ineres pariy condiion: q = E q +1 r + u (34) where he shock u includes boh he foreign real ineres rae and he foreign exchange risk premium, one can wrie (33) as: nx = b 1 E S j=0 r +j + b 1 E S j=0 u +j + κ (35) Thus, aggregae non-invesmen demand is given by: s c c + s NX nx = (s c σ + s NX b 1 ) E S j=0 r +j + exogenous shocks (36) where s c and s NX are he seady-sae shares of consumpion and ne expors in GDP respecively. The effec of adding open-economy influences is herefore o raise he ineres elasiciy of (non-invesmen) aggregae demand from s c σ o (s c σ + s NX b 1 ). In our calibraion, we ake hese influences ino accoun by calibraing σ o a higher value han would be suggesed by he ineres responsiveness of consumpion alone. Finally, we discuss he relevance for our analysis of wo oher aspecs of open-economy analysis. Firs, he consumpion Euler equaion (19) in our model would sill hold if we made he model open. However, some open-economy models assume eiher finie horizons for consumers or an endogenous discoun facor, wih he effec of making he exernal asse posiion relevan for consumpion behaviour. (14) The evidence suggess, however, ha he business cycle frequency (14) See Kollmann (1991), Kim and Kose (2000), and Smes and Wouers (2000) for applicaions. 20

18 dynamics of endogenous variables he frequency wih which he presen paper is concerned are lile changed by he inroducion of hese feaures (see Kollmann (1991), and Kim and Kose (2000)). Second, openness makes impors a componen of he consumer price index; his may creae an exra channel hrough which shocks are ransmied o inflaion he direc exchange rae channel. This would appear o jusify he inclusion of separae exchange rae erms in he Phillips curve. However, as we discussed in he inroducion, he empirical relaionship beween exchange raes and inflaion is weak. There is lile empirical case for including exchange rae erms in he Phillips curve (see Sock and Wason (2000)), possibly because of slow or incomplee pass-hrough of exchange rae movemens o impor prices. This suggess ha our use of a Phillips curve wih no explici open-economy erms is a reasonable approximaion. 3 Model calibraion and properies In his secion we describe he responses of he naural real rae and he real ineres rae gap in a calibraed version of our model. We firs urn o our calibraion. 3.1 Calibraion The parameer values assigned o our model correspond o hose commonly found in he lieraure and are similar o hose found in earlier work on sicky-price general equilibrium models, including King and Wason (1996) and McCallum and Nelson (1999). These are presened in Table B, and we now discuss some of he key choices. The parameer σ indexes he curvaure of he uiliy funcion, and also deermines he ineres-elasiciy of he non-invesmen componen of aggregae demand. Due o he habi formaion in our uiliy funcion, our calibraed value for his parameer mus be beween 0 and 1. While many business cycle sudies selec values of σ near uniy, he esimaes of Euler and opimising IS equaions on US daa in Hall (1988), McCallum and Nelson (1999), Fuhrer (2000) and Ireland (2000) sugges a lower value, of around 0.2. (15) On he oher hand, as we discussed in Secion 2.8, he openness of he UK economy jusifies a higher value. We herefore choose σ = 0.6. (16) The habi formaion parameer is se o h = 0.8, in line wih Fuhrer s (2000) esimae. The capial adjusmen cos parameers are calibraed so ha (quasi-)invesmen is considerably more ineres-elasic han consumpion, bu no implausibly large. The capial adjusmen cos seings in (15) Higher values of σ have been repored by Roemberg and Woodford (1997) and Amao and Laubach (1999), bu, unlike he sudies menioned in he ex, hese are no based on convenional economeric esimaion procedures such as insrumenal variables or maximum likelihood. (16) By comparison, Boldrin, Chrisiano and Fisher ake logarihmic preferences (he limi as σ 1) and se h = 0.9. Beaudry and Guay (1996) impose logarihmic preferences and esimae h o be in he range. Woodford (2000) ses σ 1 and h = 0. All of hese parameer seings seem o us o impose an unrealisically high ineres elasiciy of consumpion, compared wih he esimaes cied above. 21

19 Table B imply a semi-elasiciy of invesmen wih respec o he shor-erm real ineres rae of abou 3.2%. (17) Table B: Model calibraion Parameer Descripion Quarerly value α Labour share 0.64 β Discoun facor 0.99 σ Parameer indexing he curvaure of he uiliy 0.6 funcion h Habi formaion parameer 0.8 δ Rae of depreciaion /σε Scale elasiciy of money demand 1 ϕ Capial adjusmen cos parameer 0.75 η Capial adjusmen cos parameer 2 1/ρ, µ Seady sae gross mark-up 1.25 ρ λ AR(1) parameer, IS shock 0.33 ρ a AR(1) parameer, echnology shock 0.95 Var(e a ) Variance of echnology innovaions (0.007) 2 Var(e λ ) Variance of IS innovaions (0.01) 2 N ss Seady-sae fracion of ime in employmen 0.33 α Degree of nominal rigidiy under sicky prices. µ Calibraed value corresponds o a 75% probabiliy ha firms will no be able o change heir price. 3.2 Model properies Response of he naural real rae o shocks Chars 1a and 1b plo he responses of he naural real ineres rae o echnology and real demand shocks for he calibraed model. (18) Three cases are considered: a seing of he model wih no capial; he model wih capial formaion bu wih adjusmen coss; and he model wih capial ha can be adjused coslessly. The no-capial and cosless capial adjusmen cases are presened o indicae he effec our specificaion of capial formaion has on he behaviour of he naural real rae. Char 1a indicaes ha a emporary 1% shock o echnology raises he naural real rae by abou 5 basis poins when capial adjusmen is cosless, bu ha i reduces he rae in he cases of no capial and capial wih adjusmen coss. In he no-capial case, a echnology shock raises oupu and (17) Capial adjusmen coss are also imporan for generaing realisic oupu behaviour under sicky prices. Wihou capial adjusmen coss, oupu exhibis an exremely elasic response o moneary policy shocks (see Casares and McCallum (2000)). (18) Deails on how we compued hese impulse responses are presened in he appendix. 22

20 consumpion oday by more han in fuure periods. Oupu and consumpion herefore jump oday and reurn laer o heir original levels. Households would like o smooh heir consumpion Char 1: Equilibrium real ineres rae responses o shocks (especially given he habi formaion in heir preferences), and aemp o shif resources away from he curren period o fuure periods. Bu, in equilibrium, all oupu mus be consumed oday. The naural real ineres rae declines o ensure ha his occurs. (19) When capial formaion is presen, invesmen demand goes up because he echnology shock has raised he profiabiliy of producion. By iself, his would push up he real ineres rae. Offseing pressure comes from households desire o save some of heir increased income, and he disincenive o rapid invesmen provided by adjusmen coss. The ne effec of hese pressures is again o reduce he real ineres rae, hough by less han in he no-capial case. Char 1b depics he effec of a emporary 1% shock o real demand (a shock o households preferences). An IS shock of his ype, unlike a moneary policy shock, affec he values of real variables such as he real ineres rae, even when prices are flexible. This shock raises households (19) Char 1a maches Woodford s (2001) resul ha emporary produciviy shocks (ie 0 ρ a < 1) lead o a couner-cyclical response of he equilibrium real ineres rae. If he echnology shock were compleely persisen (ie had a uni roo), and here were no habi formaion in preferences, he echnology shock would leave he equilibrium real rae unchanged, as in Clarida, Gali and Gerler (1999). 23

21 consumpion demand on impac. Wih fully flexible capial, firms are willing o adjus heir invesmen in response o higher aggregae demand wihou a large change in he real ineres rae; hence he limied response of he naural rae depiced for ha case in Char 1b. If here is no capial or if firms face large adjusmen coss, hen producion does no mee he higher demand, and so real ineres raes mus rise by a relaively large amoun o dampen he rise in consumpion. A common feaure of Chars 1a and 1b is ha he naural real ineres rae responds by more when he capial sock canno adjus coslessly. (20) A flexible-price, flexible-capial model would imply almos no variaion in he naural real ineres rae since quaniies bear he bulk of he adjusmen. Char 2: Poenial oupu responses o shocks The responses for poenial oupu (Chars 2a and 2b) are he mirror image of he real ineres rae responses. Alhough he capial adjusmen cos case coninues o hold an inermediae posiion, poenial oupu responds less o a echnology shock when here is no capial in he model and more in response o a real demand shock. Impedimens o adjusing capial inhibi he rise in invesmen ha a echnology shock would normally induce. And when he capial sock canno be varied a all, i is no possible for higher consumpion demand o be saisfied parly by a rise in consumpion s share of income (which a rise in real raes would normally induce, as invesmen is more (20) This is rue even if preferences do no exhibi habi formaion, alhough habi formaion magnifies he response of he real rae o boh echnology and real demand shocks. 24

22 ineres-elasic han consumpion). So he real demand shock leads o sharp increases in boh real ineres raes and oupu. In Char 3, we presen he naural real rae response o a domesic demand shock for a differen model, namely McCallum and Nelson s (2000) open-economy model. We argued in Secion 2.8 above ha our choice of σ = 0.6 in he closed-economy model sudied in he presen paper serves o approximae he effecs of openness on real ineres rae dynamics. The char gives he naural rae response in he McCallum-Nelson (MN) model o a shock o households preferences in he domesic Char 3: Equilibrium real rae response o domesic demand shock: open economy economy, where preferences are characerised by equaion (1) bu wih σ = 0.2. (21) As he MN model has no capial, Char 3 is an open-economy analogue of he no-capial case in Char 1a. The size and shape of he response in Char 3 closely resembles ha in Char 1a. Thus our approximaion of open-economy effecs appears o be a good one. (21) The model is idenical o ha in McCallum and Nelson (2000, Figures 2 4) excep ha: (i) flexible prices are assumed, o isolae he naural rae response; (ii) he seady-sae share of expors in GDP has been raised from 0.11 o 0.25, a more appropriae value for he Unied Kingdom; and (iii) habi formaion (wih parameer h = 0.8) has been included o make preferences idenical (oher han regarding he σ value) o hose in our model. 25

23 3.2.2 Response of he real ineres rae gap o shocks We now urn o he sicky-price case, and examine he response of he real ineres rae gap he spread beween acual and naural real ineres raes o shocks. Examinaion of he sicky-price case requires ha we choose a price-adjusmen specificaion and policy rule o close he model. For price adjusmen, we use he Calvo model, alhough in Secion 4 we will also presen resuls wih he Fuhrer-Moore model. For he moneary policy rule, an esimaed policy rule for he Unied Kingdom for would be desirable, as we will laer be comparing our model wih daa for ha sample period. Bu regime changes in UK moneary policy over mean ha he full-sample esimaes of a policy rule are no very reliable, and exhibi some problems of inerpreaion (for example, hey fail o saisfy he condiion ha he nominal ineres rae responds by more han one-for-one o expeced inflaion). Insead, we use he following policy rule esimaed by Nelson (2000) for he Unied Kingdom for he sub-sample 1992 Q Q1: 4*R = ρ R 4*R 1 + (1 ρ R )1.267E 4 p +1 + (1 ρ R )0.47y + 4*e R (37) where ρ R = 0.29, 4 p = S j=0 3 π j is he annual inflaion rae, and he sandard deviaion of he policy shock e R is The policy rule is similar in specificaion o he Taylor (1993) rules esimaed by Clarida, Gali and Gerler (1999). According o equaion (37), moneary policy adjuss he nominal ineres rae in response o expeced fuure annual inflaion (wih a greaer han one-for-one long-run response), and o derended oupu (y ). Derended oupu in he policy rule is mean o proxy for he oupu gap. Chars 4 o 6 display he response of he real ineres rae gap, derended oupu, and he difference beween oupu and poenial oupu o echnology, demand, and policy shocks respecively. The policy rule operaes according o equaion (37), and capial formaion is subjec o adjusmen coss. Char 4 depics he effec of a echnology shock. The shock generaes an increase in he real ineres rae gap an effecive policy ighening for wo reasons. Firs, he naural real rae falls, so for a given acual real ineres rae moneary policy becomes igher. Second, if policy responds o he level of oupu, a produciviy shock induces policy-makers o raise he nominal ineres rae, raising he acual real ineres rae in he process. In he case of a real demand shock, however, boh he acual and naural real rae increase (Char 5). The rise in he acual real rae, due o he moneary policy ighening in response o higher oupu, is less han he rise in he naural rae. The overall policy sance is herefore looser, once he naural rae movemen is aken ino accoun. 26

24 Char 6 simply illusraes ha a moneary policy ighening affecs only he acual real ineres rae. The real rae gap herefore rises one-for-one wih he increase in he acual real ineres rae. The oupu gap and oupu responses are idenical because moneary policy canno affec poenial oupu. The response of he real rae gap o differen shocks illusraes how a policy rule like (37), which responds o he level of oupu, can yield perverse resuls. In effec, he rule responds symmerically o posiive supply and demand shocks. This is because i does no ake ino accoun ha he naural values of boh he real ineres rae and oupu have been affeced by he real shocks. The naural real ineres rae falls in response o he echnology shock, and herefore he moneary policy response should be o lower, no raise, ineres raes. In he case of a real demand shock, on he oher hand, he naural rae increases. A policy aimed a minimising oupu gap and inflaion variaions would raise he ineres rae in line wih he increase in he naural rae. Bu in fac he policy rule (37) generaes a smaller increase in he real rae han is necessary o keep he ineres rae gap zero. One reason why responding o oupu appears o lead o counerproducive resuls, for boh echnology and real demand shocks, is ha derended oupu is no a good indicaor of oupu gap behaviour in our model. Chars 4 6 also demonsrae he mirror-image relaionship beween he oupu gap and he real ineres rae gap. We discuss hese issues furher in Secion 5. Char 4: Responses o echnology shock 27

25 Char 5: Responses o real demand shock Char 6: Responses o moneary policy shock 28

26 4 Model evaluaion We now presen some evidence on he mach beween our model and he daa by examining daa and model vecor auocorrelaions. The version of he model we consider includes Calvo price-seing, capial formaion subjec o adjusmen coss, and he policy rule (37) described in he previous secion. The model correlaions are averages of he oupu of 100 sochasic simulaions of he model (for 200 periods each ime). The corresponding daa correlaions are based on quarerly UK daa for he 1980 Q Q4 sample period. The daa consis of y (quadraically derended log real GDP), R (he quarerly average of he nominal Treasury bill rae), r (R minus he expeced value of nex quarer s seasonally adjused RPIX inflaion, π +1 ), (22) and annual inflaion, 4 p. Char 7 plos he correlaions for he model (solid line) and daa (broken line). The model succeeds in reproducing he auocorrelaions of he four series and, o a lesser exen, maching several oher dynamic relaionships, including he real / nominal ineres rae relaionship, and he close correlaion beween four-quarer inflaion and he nominal ineres rae. The correlaion beween he level of Char 7: Empirical and model vecor auocorrelaions (22) Expeced inflaion is calculaed from he VAR(8) described in Secion 6 below. 29

27 oupu and he lags of he real ineres rae is generally posiive in he model, which is qualiaively he relaionship observed in UK daa, as noed earlier in Table A. The main excepion is he conemporaneous correlaion beween he wo series, which is slighly negaive in he model. The model s mos serious weaknesses are is abiliy o generae negaive correlaions beween oupu and fuure ineres raes (boh nominal and real). These are a variance wih he posiive correlaions observed in he Unied Kingdom for Par of his problem may reflec he use in he model of a policy reacion funcion ha is esimaed over only a sub-sample of : he reacion of nominal ineres raes o oupu movemens may have differed subsanially in ha sub-sample ( ) from ha observed over he full sample. Bu he discrepancy beween he daa and he model also reflecs a propery of he srucure of he model: he model s forward-looking aggregae demand specificaion implies ha aggregae demand (y ) depends negaively on expeced fuure real ineres raes. The model also does poorly in accouning for he correlaion beween oupu and annual inflaion. The divergences beween daa and model in Char 7 sugges ha fuure work could focus on modifying he aggregae demand specificaion. (23) 5 Indicaor properies of he real ineres rae gap We now examine some dynamic properies of our model, focusing on saisics ha describe he relaionship beween he real ineres rae gap, aggregae demand, and inflaion. Table C gives seleced correlaions and sandard deviaions for four seings of he model (consan or flucuaing capial, wih Calvo or Fuhrer-Moore price-seing). The variables focused on are log oupu (y ), he real ineres rae (r ), log of naural oupu (y *), he naural real ineres rae (r *), he oupu gap (y y *), and he real ineres rae gap (r r *). A noable feaure of Table C is ha i reveals ha he real ineres rae gap and he oupu gap have quie differen properies. The wo series have he expeced srong negaive relaionship (he correlaion ranges from 0.63 o 0.92 depending on model seing), bu differences beween he wo series emerge when we analyse he conribuion of individual componens of each gap. In paricular, he behaviour of he real ineres rae (r ) is a reasonable approximaion for he behaviour of he real ineres rae gap. The correlaion beween hese wo series is relaively high, and he wo series are of roughly he same degree of volailiy. Correspondingly, he naural real rae varies much less han he acual real rae. This conrass wih Roemberg and Woodford s (1997) dynamic sochasic general equilibrium model, which generaes a sandard deviaion for he naural real rae, of 3.7%, ha is higher han ha observed empirically for he acual real rae. Bu i is consisen wih King and Wason (1996), whose DSGE model produced low variabiliy in he real ineres rae under price flexibiliy. Roemberg and Woodford s finding may have arisen parly from larger shock variances han are ypically found in oher sudies. (23) Oher model specificaions we analyse, including he no-capial and Fuhrer and Moore pricing specificaion, reain many of he weaknesses discussed here. 30