WORKING PAPER SERIES (UN)NATURALLY LOW? SEQUENTIAL MONTE CARLO TRACKING OF THE US NATURAL INTEREST RATE NO 794 / AUGUST 2007

Transcription

1 WORKING PAPER SERIES NO 794 / AUGUST 2007 (UN)NATURALLY LOW? SEQUENTIAL MONTE CARLO TRACKING OF THE US NATURAL INTEREST RATE by Marco J. Lombardi and Silvia Sgherri

2 WORKING PAPER SERIES NO 794 / AUGUST 2007 (UN)NATURALLY LOW? SEQUENTIAL MONTE CARLO TRACKING OF THE US NATURAL INTEREST RATE 1 by Marco J. Lombardi 2 and Silvia Sgherri 3 In 2007 all publicaions feaure a moif aken from he 20 banknoe. This paper can be downloaded wihou charge from hp:// or from he Social Science Research Nework elecronic library a hp://ssrn.com/absrac_id= To a large exen, he paper was wrien while boh auhors were a he European Cenral Bank. We hank Gianni Amisano and Maeo Ciccarelli for useful suggesions and Güner Coenen, Ricardo Mesre, Frank Schorfheide, Frank Smes, and Michael Woodford for insighful feedback on a preliminary version of his paper. We are also graeful for commens o seminars paricipans a and DNB, he 2006 Conference on Compuaion in Economics and Finance, he 17h EC 2 Conference, he 2nd Ialian Congress of Economerics and Empirical Economics, and he Royal Economic Sociey Annual Conference. The views expressed in his paper are personal and do no reflec hose of any policy insiuion. The usual disclaimers apply. 2 Universiy of Pisa, Lungarno Pacinoi 43, Pisa, Ialy; mjl@ec.unipi.i 3 De Nederlandsche Bank and Inernaional Moneary Fund; Posal address: De Nederlandsche Bank, Posbus 98, 1000 AB Amserdam, Neherlands; s.sgherri@dnb.nl

3 European Cenral Bank, 2007 Address Kaisersrasse Frankfur am Main, Germany Posal address Posfach Frankfur am Main, Germany Telephone Inerne hp:// Fax Telex ecb d All righs reserved. Any reproducion, publicaion and reprin in he form of a differen publicaion, wheher prined or produced elecronically, in whole or in par, is permied only wih he explici wrien auhorisaion of he or he auhor(s). The views expressed in his paper do no necessarily reflec hose of he European Cenral Bank. The saemen of purpose for he Working Paper Series is available from he websie, hp:// eu/pub/scienific/wps/dae/hml/index. en.hml ISSN (prin) ISSN (online)

4 CONTENTS Absrac 4 Non-echnical summary 5 1 Inroducion 6 2 A generalized Neo-Wicksellian framework Demand side Supply side Moneary policy Model soluion Prior specificaion 16 3 Likelihood evaluaion and Bayesian esimaion A sequenial Mone Carlo approach Parameers esimaes Ou-of-sample forecass 22 4 Empirical resuls Srucural parameers: poserior disribuions Unobservable sae variables: A real-ime policy assessmen Ou-of-sample forecass and model evaluaion 27 5 Conclusions and prospecs 30 References 33 Tables and figures 38 European Cenral Bank Working Paper Series 48 3

5 Absrac Following he 2000 sockmarke crash, have US ineres raes been held "oo low" in relaion o heir naural level? Mos likely, yes. Using a srucural neo-keynesian model, his paper aemps a real-ime evaluaion of he US moneary policy sance while ensuring consisency beween he speci caion of price adjusmens and he evoluion of he economy under exible prices. To do his, he model s likelihood funcion is evaluaed using a Sequenial Mone Carlo algorihm providing inference abou he ime-varying disribuion of srucural parameers and unobservable, nonsaionary sae variables. Tracking down he evoluion of underlying sochasic processes in real ime is found crucial (i) o explain poswar Fed s policy and (ii) o replicae salien feaures of he daa. JEL Classi caion Numbers: E43, C11, C15 Keywords: Naural Ineres Rae; DSGE Models; Bayesian Analysis; Paricle Filers 4

6 Non-echnical summary Afer he 2000 sockmarke bubble burs, he Fed has successfully managed o reduce business cycle volailiy (resuling in he mildes US recession ever!) by reducing shor-erm raes o a 45-year low of 1 percen, while e ecively convincing he bond markes ha raes would have been kep so low for a considerable period. Ye, i remains an open quesion wheher or no he Fed has cu ineres raes by oo much and lef hem low for oo long in relaion o he curren economic condiions. Possibly, he excepionally low cos of capial a a ime when he expeced worldwide reurn on capial has likely increased migh have encouraged excessive borrowing, allowing nancial imbalances o build up. Inferring he sance of moneary policy from prevailing ineres raes requires some sor of benchmark. Economiss end o assume ha such a benchmark role is played by he equilibrium, or naural, real rae of ineres. Pu simply, he naural real rae of ineres is he rae ha keeps oupu a is poenial and in aion sable, once any shocks o he economy have played ou. According o his concep, one can hus gauge he sance of, say, he US moneary policy by comparing he acual level of he federal fund rae wih he naural rae. Ye, o make life ricky for boh policymakers and marke paricipans, he naural rae of ineres is no observable and may vary over ime, in line wih changes in he rae of reurn on capial or households rae of ime preference. If hese movemens are su cienly large, any consan long-erm average would be a poor predicor of he naural rae of ineres. By drawing on a srucural model feauring explici heoreical foundaions, our paper proposes an original economeric approach o esimae he evoluion of an economy under equilibrium condiions. The mehodology makes i possible o rack down and inerpre he di eren sources of variaion and uncerainy in ineres rae seing such as shifs in labour produciviy, preference shocks, or households degree of risk aversion wih su cien precision and in real ime. Our esimaes show how he Fed o deal wih he economic downurn following he sock marke crash has been driving he cos of capial signi canly below he naural rae for over four years. The resuls also provide some suppor o he belief ha poswar US in aion s rise and fall was due o Fed s inabiliy o disenangle in a imely manner ransiory from permanen shifs in produciviy. In fac, over 85 percen of he uncerainy surrounding fuure moneary policy sance is found o sem from imprecision surrounding inferences of he underlying equilibrium posiion. 5

7 1 Inroducion Afer he US sockmarke bubble burs, he Fed has successfully managed o reduce business cycle volailiy by reducing shor-erm raes o a 45-year low of 1 percen, while e ecively convincing he bond markes ha raes would have been kep so low for a considerable period. The quesion is o deermine wheher in he Fed has cu ineres raes by oo much and lef hem low for oo long in relaion o heir underlying equilibrium level. Alhough opimal moneary policy is no associaed wih keeping he acual real rae close o is normal level on a period-by-period basis, a prolonged period of excepionally low cos of capial a a ime when he expeced reurn on capial has likely increased migh have encouraged excessive borrowing, conribuing o he buildup of nancial imbalances. Inferring he sance of moneary policy from prevailing ineres raes requires some sor of benchmark. Following Wicksell s seminal idea, economiss end o assume ha such a benchmark role is played by he naural real rae of ineres. Pu simply, he naural real rae of ineres is he rae ha keeps oupu a is poenial and in aion sable, once any shocks o he economy have played ou. According o his concep, one can hus gauge he sance of, say, he Fed s moneary policy by comparing he acual level of he federal fund rae wih he nominal naural rae. If he rae se by he cenral bank is lower han he naural rae, he economy and in aion would be expeced o accelerae, as here will be excessive invesmen and borrowing, while households will no save enough. Conversely, if he Fed keeps ineres raes above he naural rae, policy would rein in he economy and in aion would evenually slow. Ye, o make life ricky for boh policymakers and marke paricipans, he naural rae of ineres is no observable and may vary over ime, in line wih changes in he rae of reurn on capial or households rae of ime preference. There are indeed reasons o hink ha he Wicksellian rae may have shifed over he las decade. The surge in produciviy growh owing o he informaion echnology revoluion is likely o have boosed expecaions abou fuure pro s and invesmen opporuniies, moving upwards he demand curve for invesmen funds, ceeris paribus. Meanwhile, in indusrial counries, households seem o have become more impaien and decided ha hey need o save less han hey used o, on he belief ha rising asse and house prices will provide hem wih adequae resources o nance heir reiremen. As a resul, he so-called IS (Invesmen=Saving) curve he equilibrium line showing he negaive relaionship beween spending and real ineres rae may have moved righwards, implying a higher naural rae of ineres for any level of poenial oupu. Srucural forces 6

8 such as enhanced compeiion from emerging markes, deregulaion, and faser produciviy growh may have also helped o hold down (he equilibrium level of) in aion, whils cenral banks success in aming in aion (around such an equilibrium level) has anchored in aionary expecaions. This has mean ha moneary policy can now be eased more freely o deal wih economic ucuaions, provided ha in aion remains subdued. From a heoreical perspecive, racking down he evoluion of he naural rae of ineres requires o pin down he behavior of he economy under equilibrium condiions. In our paper, he problem is ackled wihin Woodford s (2003) neo-wicksellian framework, which has he propery of ensuring consisency beween he speci caion of price adjusmens and he (unobservable) behavior of he economy under exible prices. 1 Speci cally, he model enails raional expecaions and neo-keynesian price rigidiies, while guaraneeing a zero oupu gap in seady sae, in compliance wih he naural rae hypohesis. Reconciling shor-erm and equilibrium dynamics becomes paricularly relevan in sysem esimaion. In his conex, each change o he speci caion of price adjusmens a ecs he likelihood-maximizing vecor of srucural parameers which ogeher wih he dynamics and he disribuions of shocks deermine he (unobservable) behavior of he economy under exible prices. Originally, our prooypical dynamic opimizing sicky-price model has been generalized along hree imporan dimensions. Firs, i accouns for possible nonsaionariy in he seadysae level of in aion and real oupu. Second, i allows for shifs in he underlying sochasic processes, as well as in he parameerizaion of he model. Third, i has been esimaed in real ime, e.g. by using observable, non-derended daa which are readily available a he ime hey are needed. 2 To do his, he model s likelihood funcion is evaluaed using a Sequenial Mone Carlo algorihm, providing join esimaes of he srucural parameers and he unobservable, ime-varying, nonsaionary sae variables. The idea beneah his approach is o represen any probabiliy law by a large number of random samples, or paricles, evolving over ime on he 1 Noable examples of dynamic, opimizing models examining he behaviour of he naural levels of oupu and ineres are McCallum and Nelson (1999), Roemberg and Woodford (1999), Galí, López-Salido, Vallés (2003), Woodford (2003), Giammarioli and Valla (2003), Neiss and Nelson (2003), Smes and Wouers (2003), Amao and Laubach (2004), Andrés, López-Salido, Nelson (2005). 2 The issue of real-ime esimaion of unobservable equilibrium variables has received increasing aenion in he conex of moneary policy analysis. See, among ohers, Laubach (2001), Larsen and McKeown (2002), Orphanides and Williams (2002), Laubach and Williams (2003), Basdevan, Björksen, and Karagedikli (2004), Cuaresma, Gnan, and Rizberger-Grunenwald (2004), Mésonnier and Renne (2006), Sevillano and Simon (2004), Garnier and Wilhelmsen (2005). Overall, real-ime measures of he ime-varying naural rae of ineres like hose of he equilibrium level of oupu, unemploymen, and in aion are found o be highly imprecise, limiing heir relevance for policy. 7

9 basis of a simulaion-based updaing scheme, so ha new observaions are incorporaed in he ler as hey become available. By racking down he whole disribuion of he naural rae of ineres in real ime, i is possible o inerpre he di eren sources of variaion and uncerainy in ineres rae seing such as shifs in labour produciviy, volailiy of preference shocks, or households degree of risk aversion in a imely and helpful manner. To anicipae our conclusions, our esimaes show how he Fed o deal wih he slowdown following he sock marke crash has been cuing ineres raes aggressively, driving he cos of capial signi canly below he naural rae for over four years. In he mos recen quarers, however, repeaed moneary policy ighening has managed o correc his disequilibrium while slowing down growh. Ye, he in aion gap is found o be sill posiive a he end of From a posiive perspecive, our ndings are broadly in line wih Cogley and Sbordone s (2005) general argumen ha ignoring ime variaion in he underlying evoluion of he economy under exible prices may aler he esimaes of srucural parameers. While he volailiy of preference shocks is likely o have fallen over ime, srucural pricing parameers are essenially ime-invarian once shifs in rend growh and in aion are also accouned for. From a normaive perspecive, our resuls provide some suppor o he hypohesis ha over he poswar period Fed s inabiliy o promply disenangle permanen from ransiory shifs in produciviy may have ranslaed ino persisen movemens in he in aion rae iself. By he same oken, uncerainy abou fuure ineres rae seings is found o be primarily due o vagueness in he esimae of he underlying sochasic equilibrium raher han o imprecision abou he srucural parameerizaion of he model economy or he policy rule. The res of he paper is organized as follows. Secion 2 lays down a dynamic sochasic model consisen wih he neo-wicksellian approach o price deerminaion. Secion 3 presens he mehodology adoped o evaluae he likelihood of such a model, esimae poserior densiies for is parameers, and rack down he ime-varying disribuion of is unobservable sae variables. Secion 4 discusses esimaion and forecasing resuls, while Secion 5 concludes he paper by providing an oulook on fuure work. 2 A Generalized Neo-Wicksellian Framework Our srucural model has a prooypical raional expecaions speci caion, similar o he one used by Boivin and Giannoni (2003), Giannoni and Woodford (2005), and described in Woodford (2003). On he demand side, we ake ino accoun habi persisence in he level of aggregae expendiure, assuming ha for each household i in period, uiliy depends no only on curren 8

10 expendiure, bu also on he level of expendiure in he previous period. On he supply side, we allow for price indexaion for rms ha are no allowed o se heir price opimally in a given period, in order o generae more realisic level of in aion ineria. 3 The seup is hen originally enriched by leing in exogenous non-saionary processes describing (labor) produciviy and arge in aion dynamics, respecively as well as idiosyncraic demand, supply, and policy shocks. 2.1 Demand Side We assume a model economy producing a coninuum of goods indexed by j and populaed by a coninuum of households indexed by i, uniformely disribued over he [0; 1] inerval. By considering he limiing case of a cashless economy and hereby absracing from real balances each household i seeks o maximize he following in nie discouned sum of fuure uiliies: E P1 = " u 1 1 Ci C i % Hi (j) 1+% where C i is an index of he household s consumpion of he di ereniaed goods supplied a and H i (j) denoes he amoun of hours supplied by each household i for he producion of each good j. In addiion, = 1=(1 + r) 2 (0; 1) is he households discoun facor, is he coe cien of relaive risk aversion or he inverse of he ineremporal elasiciy of subsiuion, and % represens he inverse of he elasiciy of work e or wih respec o he real wage. The parameer 0 1 measures he degree of habi formaion in consumpion. Consumers uiliy hence depends posiively on deviaions of consumpion C i from an exising sock C i 1, and negaively on he oal labor supplied. 4 (1) Equaion (1) also conains a shock o he discoun rae ha a ecs he ineremporal preferences of households, " u, disribued as a log-normal wih mean equals o 1. Households maximize heir objecive funcion (1) subjec o an ineremporal budge consrain ha is given by: b B i P = Bi 1 P + (W i H i + A i ) C i (2) 3 This exension has been recenly proposed by Chrisian, Eichenbaum, and Evans (2005). In aion indexaion has been also used, for insance, in Smes and Wouers (2005), Milani (2005), and Andrés e al. (2005). 4 The consumpion habi is assumed here o depend on he household s own pas level of expendiure, and no on ha of oher households. 9

11 implying ha households hold heir nancial wealh in he form of one-period securiies, B T, having price b T. Toal income consising of labor income and he ne cash in ow from paricipaing in sae-coningen securiies, i.e., Y i = W i H i + A i is solely used for consumpion purposes. 5 Under he assumpion of complee nancial markes and e cien risk sharing, sae-coningen securiies would insure households agains household-speci c ucuaions in labor income, so ha each household faces an idenical ineremporal budge consrain and an idenical marginal uiliy of oal income. As a resul, he superscrip i can be omied wihou loss of generaliy. Wih habi formaion, he maximizaion of he objecive funcion (1) subjec o he budge consrain (2) wih respec o consumpion can herefore be wrien as: h = " u (C C 1 ) " u +1 (C +1 C ) i (3) where he marginal uiliy of real income a ime (i.e., he Lagrange muliplier for he budge consrain, ) depends upon boh oday s and omorrow s expeced marginal uiliy of consumpion, as well as preference shocks occurring a ime and + 1. In addiion, he marginal uiliy of income sais es he rs order condiion for he opimizing consumer s problem wih respec o bonds holdings, yielding: where (1 + i ) = 1 b = E +1 (1 + i ) P P +1 denoes he riskless one-period (gross) nominal rae of reurn on bonds and E h P+1 P i = (1 + E +1 ) de nes he expeced (gross) rae of in aion. Using (3) o subsiue for he 0 s in (4), we can hus derive he generalized Euler equaion in he presence of habi formaion, namely: (4) " u (C C 1 ) E h " u +1 (C +1 C ) i E h " u +1 (C +1 C ) i E h " u +2 (C +2 C +1 ) i = (1 + i ) (1 + E +1 ) (5) In our basic neo-wicksellian closed economy, all ineres-sensiive privae expendiure is modeled as if i were nondurable consumpion, hereby absracing from he e ecs of variaions in privae expendiure on he evoluion of producive capaciy. This simpli caion has hree implicaions. Firsly, given our scal policy assumpion, he aggregae equilibrium relaion C = Y holds; secondly, he parameer 1 Uc C U cc > 0 indicaes he ineremporal 5 Assuming ha he govermen implemens a Ricardian scal policy while levying lump sum axes, scal policy can be ignored given ha i has no e ec on model aggregaes. 10

12 subsiuabiliy of privae expendiure as a whole, raher han ha of nondurable consumpion alone (Roemberg and Woodford, 1997); and, hirdly, he habi persisence considered here can be deemed as a proxy for adjusmen coss in boh invesmen and consumpion spending, raher han solely as a descripion of household preferences wih regard o personal consumpion. Along a balanced growh pah in which hours worked are consan, aggregae expendiure is growing a is equilibrium rae say g and he marginal uiliy of real expendiure is falling a rae (g ), log-linearizaion of Euler equaion (5) around seady sae yields a generalizaion of he IS relaion of he form: ey = E ey +1 [ (1 )] h i 1 bi E b +1 r + e u ; (6) where ey = (by by 1 ) E (by +1 by ) (7) e u = [ (1 )] 1 u E +1 u : (8) Here, u denoes normally disribued aggregae demand shocks and haed variables indicae (log-)deviaions from corresponding (ime-varying) equilibrium levels, e.g.: by = y y (9) bi = i i (10) b = (11) Noe ha as! 0, expression (5) collapses o he sandard Euler equaion for consumpion, while expression (6) reduces o he well-known log-linear forward-looking IS curve, where he oupu gap is expressed in erms of deviaions of acual oupu from is equilibrium pah. 2.2 Supply Side We assume a coninuum of monopolisic compeiive rms indexed by j and uniformely disribued over he [0; 1] inerval. Each rm is a monopolisic supplier of good j, which is produced according o he producion echnology Y (j) = Z H (j);where H (j) is labor inpu and Z describes an exogenous nonsaionary produciviy process common o all rms. Indicaing wih g = ln(g ) he saionary growh rae of he produciviy process, wih z idiosyncraic 11

13 innovaions o is level, and wih g idiosyncraic innovaions o is growh rae, he log of he produciviy process, z = ln(z ), evolves according o: ln(z ) = ln(z 1 ) + ln(g 1 ) + z (12) ln(g ) = # ln(g 1 ) + (1 #) ln(g ) + g (13) hereby inducing a sochasic rend in aggregae expendiure. As 0 # < 1, i follows ha our model economy evolves along a sochasic growh pah, feauring a g equilibrium rae. The quaniy of capial available for he producion of each individual good is regarded as xed, leaving labor as he only variable facor of producion. 6 compeiive facor marke, a he hourly real wage W. Labor is hired in a perfecly Monopolisic compeiive rms face a common demand curve Y (j) = Y p(j) P for heir producs, where he parameer > 1 represens he elasiciy of demand for each good j. In 1R 1 his conex, Y = Y (j) 1 dj is aggregae oupu and P is aggregae price boh aken 0 as given. Hence, all rms face idenical maximizaion problem and, if allowed o choose heir own price, se a common price p : Following Calvo (1983), however, we suppose ha only a fracion 0 < (1 ) < 1 of rms are allowed o change heir price in a given period, while he remaining fracion simply adjuss prices according o an indexaion rule. In conras o baseline Calvo price seing, we posulae ha hose price seers who canno rese heir prices oday, have heir prices auomaically raised by a percenage 0 < < 1; so ha represens he degree of indexaion o pas in aion. Under sandard properies of he pro funcion of each supplier j, and assuming ha pro s are discouned using a sochasic discoun facor equal on average o, Woodford (2003) demonsraes ha he resuling in aion dynamics can be log-linearized as: where b b 1 = E (b +1 b ) + f!by + [(1 ) ] ey g + (14) = (1 ) (1 ) ; (1 +!) is a whie noise exogenous aggregae supply shock, and ey, by, and b are de ned as in (7), (9), and (10), respecively. 7 6 Woodford (2003) shows ha he consan-capial model can be recovered as a limiing case of a model wih endogenous capial accumulaion, in he limi of very high adjusmen coss for invesmen. 7 For algebraic deails, see Woodford (2003). 12

14 Noe ha, in he presence of habi formaion ( > 0), he log marginal uiliy of oal real income enering he Phillips curve (14) is wrien as a linear funcion of ey ; raher han simply as a linear funcion of by (which would be he case for = 0). 8 Moreover, indexaion o pas in aion ( > 0) in he form speci ed in (14) ensures ha he NRH is also sai ed, albei locally: on average, oupu equals is poenial level for an exogenous seady-sae in aion rae meaning ha dynamic indexaion only describes emporary ucuaions of he economy around is ime-varying equilibrium. In he absence of nominal rigidiies, he equilibrium level of aggregae oupu is deermined by ln(y ) = 0 + ln(z ); (15) while is evoluion over ime is governed by he produciviy process described by (12). 9 I should be sressed ha in our modeling framework he conceps of poenial", equilibrium", seady-sae and exible-price level of oupu coincide: hey are hence inerchangably denoed by y and described by he same ime-varying, unobservable, nonsaionary process, (15), hereby shaping he oupu gap (9). The corresponding exible-price naural real rae of ineres (NRI) he real ineres rae in an equilibrium where price rigidiies are ruled ou and he oupu gap is zero a all imes is de ned as: 10 ln (1 + r ) = r + [(1 ) ] ln! " u E [Y+1] E " u +1 Y (16) wih ln (1 + r ) ln " # (1 + i ) : (17) 8 In he absence of habi formaion, he aggregae dynamics of in aion wih indexaion would reduce o b b 1 = E (b +1 b ) + (! + ) by + ", where (! + ) measures he elasiciy of he average real marginal cos wih respec o he aggregae oupu. 9 The consan 0 which is inversely relaed o he degree of marke disorion, does no a ec he law of moion of he aggregae level of oupu and is, herefore, unideni able. 10 Following Neiss and Nelson (2003), we de ne he NRI as he equilibrium rae of reurn ha would prevail if he prices had always been exible in he pas and expeced o be so in he fuure. Under his condiion, even if he model is exended o allow for endogenous capial accumulaion, he exising capial sock (and, hereby, pas moneary policy) would no maer for he deerminaion of he NRI. 13

15 2.3 Moneary Policy Moneary policy is inroduced in he model hrough a linearized Taylor rule wih parial adjusmen and ime-varying arge for in aion: i = i 1 + (1 ) [i + ' 1 b + ' 2 by ] + r (18) where is he degree of ineres rae smoohing, ' 1 and ' 2 denoe he long-run ineres rae responses o deviaions of in aion and oupu from heir respecive (ime-varying) seady-sae levels, while r accouns for unanicipaed deviaions from he sysemaic moneary policy rule. A similar rule allows for shifs in he equilibrium level of in aion, while he sysemaic policy responses have no changed over ime. An alernaive speci caion of he moneary rule is also considered, according o which deviaions of he policy rae from equilibrium are allowed o adjus for deviaions of in aion and oupu growh from heir equilibrium values. i = i 1 + (1 ) [i + ' 1 b + ' 2 (by by 1 )] + r (19) Several sudies on policy rule opimaliy nd ha growh rules such as (19) end o be more robus han gap rules such as (18) in he face of uncerainy (e.g. mispeci caion) abou he naural level of oupu (or unemploymen). 11 Our paper compares he performance of hese wo rules by esimaing hem in real ime, joinly wih he evoluion of poenial oupu, is growh rae, and heir relaed uncerainy. Exogenous shifs in he moneary policy arge are believed o follow a Markov-swiching process such as: = 1 + k (20) whose ransiion marix is given by: P ( = 1) = p u (21) P ( = 1) = p d (22) P ( = 0) = 1 p u p d (23) given k = 0:50 percen: The idea here is o capure rue shifs in he objecive of he cenral bank, under he presumpion ha he cenral bank arge is revised discreely over ime wih a sep of 50 basis 11 See, for insance, Levin, Wieland and Williams (2004). 14

16 poins. This is a relaively new approach in he lieraure, as in mos cases he arge in aion is assumed o follow a Gaussian random walk (see, for example, Amisano and Trisani (2006), Smes and Wouers (2004)). 12 Clearly, he fac ha he ransiion marix has only hree saes migh be viewed as a resricion, bu his limiaion can be easily generalized. 2.4 Model Soluion Equaions (6) o (23) form a linear raional expecaion sysem in he sae variables h i 0 x = by b br y r g ; h i 0. driven by innovaions = u r z g This raional expecaion sysem has o be solved and cas in sae space before he DSGE model can be esimaed. I is imporan o noe ha, even hough all variables in our model can be re-wrien in erms of deviaions from equilibrium, we do no induce saionariy by consrucing a log-linear approximaion of he model around he seady-sae for he derended variables (see, for insance, Alig e al. (2002)). Insead, we prefer o ake our model direcly o readily available, non-derended daa and accoun for possible shifs in heir underlying rends in real ime. In his way, we also avoid imposing any co-rending resricion, which are hardly suppored by empirical evidence (Canova e al. (1994), Del Negro e al. (2004)). The soluion of he raional expecaion sysem hence akes he form: x = F (x 1 ; ; ); (24) wih F being a general funcion of he vecor of srucural parameers of our DSGE model,, de ned as: h = 1 ' 1 ' 2 p u p d # g u r z g i 0 ; where u,, r, z, and g are he sandard deviaions of he shocks o he aggregae demand, supply, moneary policy, (sochasic) produciviy level and corresponding growh rae We hink of our discree Markov swiching process as a quie realisic represenaion of policymakers behavior over he spell of a quarer. Noneheless, he model has been also esimaed using a more convenional random walk in aion arge. For he sake of breviy, resuls are omied, alhough available upon reques. 13 In order o deal wih ideni caion issues, he values of he parameers and! have been xed o 7.52 and.898, respecively, in line wih Giannoni and Woodford (2003) and Milani (2005). We are aware ha his procedure may creae disorsions in he esimaed disribuion of he remaining parameers, unless he chosen values happen o be he correc ones (Canova and Sala, 2006). 15

17 Depending on he parameerizaion of he srucural model, here are hree possibiliies: no sable raional expecaion soluion exiss; he sable soluion is unique (deerminacy); or here are muliple sable soluions (indeerminacy). We will focus on he case of deerminacy and resric he parameer space accordingly. For linear models like he one a hand, several soluion algorihms are available. 14 For he subsequen analysis, we use Klein s (2000) generalized Schur form o solve our linearized DSGE model (6) o (23). Wihin a nonlinear framework, he ideni caion of srucural parameers becomes challenging given he nonlinear mapping from he srucural form of he DSGE model ino is sae-space represenaion. In paricular, under hese condiions, rs-order approximaions may no su ce any longer, and higher-order re nemen have o be brough ino play. 15 The model is compleed by de ning a se of measuremen equaions ha relae he elemens h i 0, of x o a se of observable variables, y = y i comprising quarerly daa over he period 1957:I o 2006:IV on real GDP, annualized quarerly rae of change of he CPI less food and energy, and he e ecive federal funds rae. 16 Namely: y = G(x ; ; ); (25) The general sae-space model (24)-(25) forms he basis for he likelihood esimaion of he srucural parameers, which ogeher wih he disribuion of he shocks deermine he underlying behavior of he economy under exible prices. 2.5 Prior Speci caion One of he hardes asks in implemening Bayesian esimaion is how o specify sensible priors abou he deep parameers of he model. While, in principle, priors should re ec song beliefs abou he validiy of underlying microeconomic heory, in pracice mos priors are ne-uned o mach imporan properies of he aggregae daa. Table 1 summarizes informaion abou our priors over he vecor of srucural parameers. For convenience, all parameers are assumed a priori o be independen. The habi and indexaion parameers, and, are assumed o follow bea disribuions wih mean 0.5 and 0.9, and sandard deviaions 0.3 and 0.05, respecively. The mean prior for he coe cien governing he degree of price sickiness is chosen equal o 0.2 based 14 For insance, Blanchard and Kahn (1980), Uhlig (1999), Klein (2000), Chrisiano (2002), and Sims (2002). 15 Algorihms o consruc second-order accurae soluions o DSGE models have been developed, for insance, by Schmi-Grohe and Uribe (2004) and Klein (2005). Amisano and Trisani (2006) explore he advanages of using higher order soluion mehods o sudy persisence in euro area in aion. 16 The series were obained from FRED, he daabase of he Federal Reserve Bank of Sain Louis. 16

18 on micro-evidence on price seing behavior, implying re-opimizaion of prices approximaely every quarer (for insance, Bils and Klenow (2004)). The prior for he degree of ineres smoohing in he moneary policy rule,, is cenered around 0.8, while he prior means of he feedback coe ciens, ' 1, and ' 2, are chosen o mach values ypically associaed wih he Taylor rule, namely 1.5 and 0.5, respecively. To guaranee values in R +, we posulae lognormal disribuions for he sandard deviaion of shocks, u,, r, z, and g, whereas for he ineremporal elasiciy of subsiuion, 1 ; we assume a gamma disribuion wih mean equal o 0.4, close o he corresponding esimae in Roemberg and Woodford (1997). In addiion, he implied annual discoun rae r is cenered around 2 percen, he roo of he auoregressive process characerizing he growh rae of produciviy is believed o be disribued around a mean value of 0.8, while he equilibrium rae of produciviy growh has a prior mean of 3 percen per annum. Finally, since we do no have srong prior beliefs for p d and p u lacking previous esimaes in he lieraure we prefer o represen hem hrough equally cenered bea disribuions, implying an overall 50 percen probabiliy of swiching. As for he saring values of he sae variables, by and large hey are chosen based on informaion a he beginning of he sample. In paricular, he prior mean of he year-on-year growh rae of produciviy, ln(g 0 ); is 4 percen, while he iniial arge of in aion is aken o be cenered a 2 percen. Finally, given he priors for he vecor of srucural parameers,, equaion (16) implies a prior mean for he annual NRI around 2.8 percen. 3 Likelihood Evaluaion and Bayesian Esimaion To he solved DSGE model o he vecor y of observable variables, we employ a Bayesian esimaion echnique based on he likelihood funcion generaed by he srucural sysem of equaions. More speci cally, he model s likelihood funcion is evaluaed using a Sequenial Mone Carlo algorihm or paricle ler providing join esimaes of he srucural parameers and he unobservable, ime-varying, nonsaionary sae variables. To our knowledge, his is a very innovaive way o bring DSGE models o daa. Up o now, in he seing of DSGE models, paricle lers have been used o consruc he likelihood, bu no o perform parameer esimaion. This is due o he compuaional burden in solving he model for each paricle and for each ime samp. We propose an indirec way of proceeding, which yields parameer esimaes direcly from he lering procedure, hrough a single soluion of he DSGE model for each paricle. Our idea is o impose priors on he srucural parameers of our DSGE model, solve i, esi- 17

19 mae he corresponding sae-space sysem using he paricle ler recursion, and map back he poserior disribuion of he srucural raional expecaion model s parameers. Speci cally: Given he se of overidenifying resricions implied by he model, we impose priors on he vecor and map he srucural parameer space ino he parameer space of he sae-space model, say ; The paricle ler is hen used for inference on he sae-space model, where underlying nonsaionary processes and saic parameers are reaed as sae variables; Each se of M draws from he lering disribuion of he saic parameers is hen mapped back ino he parameer space of he srucural model o ge he poserior disribuion of : The idea of mapping one parameer space ino anoher is somewha similar o ha of indirec inference (Smih (1993), Chrisiano e al. (2005)), alhough furher work is currenly in progress o invesigae his parallel. From a echnical sandpoin, he mapping employed in our analysis implies a direc knowledge of he binding funcion. Conrary o plain indirec approaches, however, no opimizaion algorihm is called for by he lering mehodology adoped here, as he esimaes of he sae-space model do already exhibi he necessary saisical properies. If a unique sable soluion exiss, he sae-space model is derived univocally from he srucural economic model: we can hus reasonably expec he misspeci caion bias o be relaively small. 3.1 A Sequenial Mone Carlo Approach The paricle lering approach is an exremely powerful framework for inference in sae-space model (Douce e al. (2000)). Alhough hese mehods have originaed in he engineering lieraure (Gordon e al. (1993), Kiagawa (1996)), hey have recenly been employed also for nancial (Pi and Shephard (1999)) and macroeconomic applicaions (Fernández-Villaverde and Rubio-Ramírez (2004a, 2004b), An and Schorfheide (2005), Amisano and Trisani (2006)). The idea is o represen he disribuion of he sae variables by a Mone Carlo approximaion consruced via a large number of random samples, or paricles, evolving over ime on he basis of a simulaion-based updaing scheme. In his way, new observaions are processed by he ler as hey become available. Each paricle is assigned a weigh, which is updaed recursively. Given he sae-space model de ned by (24)-(25), he lering problem lies in he deerminaion of p(x jy 1: ; ). This can be performed in wo seps: 18

20 1. Projecion: Z p(x +1 jy 1: ; ) = p(x +1 jx ; )p(x jy 1: ; )dx 2. Updaing: p(x +1 jy 1:+1 ; ) = p(x +1jy 1: ; )p(y 1:+1 jx +1 ; ) (26) p(y 1:+1 jy 1: ; ) Z p(y +1 jy 1: ; ) = p(x +1 jy 1: ; )p(y 1: jx +1 ; )dx +1 (27) Under he assumpion ha G and F in (24)-(25) are linear funcions and he shocks are Gaussian, a closed-form soluion o he lering problem is yielded by he Kalman ler. In he presence of nonlineariies and deviaions from Gaussianiy, hings become more complex and approximaed soluions are needed. One possibiliy is o linearize he model by means of Taylor series expansions. This approach is someimes referred o as he Exended Kalman Filer (Harvey (1990)). Condiions for convergence are ofen quie di cul o esablish, however, and in wha follows we will concenrae on a simulaion-based approach o lering which has he very useful feaure o be exremely general. The idea of a Sequenial Mone Carlo ler as anicipaed is o represen he disribuions of ineres (and herefore is momens and quaniles) by Mone Carlo approximaions, namely by a se of M paricles. Le suppose o be a ime 1 and generae M draws from p(x jx 1 ; ); hereby plainly simulaing from he sysem of sae equaions. Nex, le assign o each paricle m = 1; : : : ; M a weigh proporional o is likelihood: w ;m = p(y jx ; ): Weighs can be hen normalized as: ~w ;m = w ;m P M m=1 w ; ;m The weighs can hus be used o compue Mone Carlo inegrals via imporance sampling. I urns ou ha, afer a cerain number of seps, he ler migh evenually degenerae, i.e. assign all he weigh o a single paricle namely he one relaively mos likely in view of he observed daa. In order o avoid his, paricles can be resampled: afer having produced a se of paricles and assigned o each one an appropriae weigh, we associae o each paricle m a number of o spring O m such ha P M m=1 O m = M. Afer his selecion sep, o spring paricles 19

21 replace he original paricles and he imporance weighs are rese o 1=M, so ha he se of paricles can be hough of as a random sample. The resampling sep can be implemened a every ime inerval (Gordon, Salmond and Smih, 1993), or i can be employed whenever he se of paricles crosses a cerain degeneracy hreshold. A measure of degeneracy of he algorihm is he e ecive sample size (Liu and Chen, 1998), de ned as: This quaniy can be esimaed by: M e = ^M e = M 1 + Var(w ) : 1 P M ; m=1 ~w2 ;m when ^M e drops below a cerain hreshold, he resampling akes place. In our analysis, we compue he resampling in a compuaionally sraighforward manner. Speci cally, he number of o spring is aken o be proporional o he imporance weigh and generaed via simulaion of a se U of M uniformly disribued random variables on [0; 1], by using he cumulaed sum of he normalized weighs mx q m = ~w ;j ; and hen seing O m equal o he number of poins in U ha fall beween q m 1 and q m. j=1 3.2 Parameers Esimaes I is also possible o employ a paricle ler o perform parameer esimaion. To do his, one can simply preend ha saic parameers are indeed ime-varying saes, by adding a noise erm a each ime inerval. The problem however is ha, by doing so, relevan informaion is hrown away and addiional variabiliy generaed. Liu and Wes (2001) propose a mehod o quanify such a loss of informaion, by allowing for an ari cial parameer evoluion scheme immune o his problem. Inroducing ari cial parameer evoluion is equivalen o consider a model in which is replaced by is ime-varying analog ; which evolves according o: = 1 + ; N(0; ); where is a diagonal marix. In a siuaion in which is xed, he poserior disribuion p(jy 1: ) can be characerized by is Mone Carlo mean and variance and s 2, wih s 2 denoing he vecor of empirical variances of each elemen in. I is immediae o observe ha, in he case of ari cial parameer 20

22 evoluion, he Mone Carlo variance increases o s 2 +. In fac, he Mone Carlo approximaion can be expressed as kernel smoohed densiy of he paricles as: p(jy 1: ) MX j=1 (j) N whereas he arge variance s 2 can be expressed as: +1 j (j) ; ; (28) s 2 = s Cov( 1 ; ): By choosing Cov( 1 ; ) = 2 he loss of informaion can be easily avoided. A simple way o achieved his is o consider 1 = s 2 1 ; where is a discoun facor in (0; 1]. 17 If we de ne d = 3 1 2, he condiional densiy evoluion becomes: where p( +1 j ) N +1 jd + (1 d) ; h 2 s 2 ; (29) h 2 = 1 d 2 = ; 2 so ha sampling from (29) is equivalen o sampling from a kernel smoohed densiy in which he smoohing parameer h is conrolled via he discoun facor. Clearly, he choice of he number of paricles M a ecs boh he approximaion of he predicion error disribuion and he performance of he resampling algorihm. Our choice of 10,000 paricles has been driven, on he one hand, by he need o have enough paricles o capure he ails of p(jy 1: ); on he oher hand, by he evidence ha afer a cerain hreshold gains in noise reducion from he use of more paricles become very small (Fernández-Villaverde and Rubio-Ramírez (2004b), Lombardi and Godsill (2005)). To wrap up, all he necessary informaion abou he sae-space model parameers is summarized by he Mone Carlo approximaion (28) of p(jy 1: ); generaed by M paricles. Each paricle is hence mapped back ino he parameer space of he srucural model, enabling us o devise Mone Carlo approximaions o he poserior probabiliy disribuions of he srucural paramaers colleced in. 17 Liu and Wes (2001) sugges o choose a value of around

23 3.3 Ou-of-Sample Forecass The sequenial naure of he paricle ler lends iself paricularly well o perform ou-of-sample forecasing of boh sae and observable variables. 18 Indeed, he goal of he projecion sep is o consruc random samples from he Mone Carlo approximaion of he forecas disribuion x +1 ;compued using he poserior disribuion of he sae-space model parameers ; e.g.: p(x +1 jx ; ): Each random sample x +1 from he projeced (or forecas) disribuion is hen urned ino a random sample from he lered disribuion as weighs are assigned and resampling is performed. In order o obain muliple-sep-ahead forecass, i is herefore su cien o keep projecing ahead he paricles wihou any weighing or resampling. In his way, he Bayesian esimaion mehodology provides a useful ool for calculaing he full probabiliy disribuion around he median (uncondiional) forecas. 4 Empirical Resuls 4.1 Srucural Parameers: Poserior Disribuions Table 2 summarizes end-of-sample informaion abou he medians and he 95 percen probabiliy inervals of he srucural paramaers poserior disribuions under he wo alernaive moneary policy rules considered. All in all, our resuls indicae ha under he assumpion of fully raional expecaions and serially uncorrelaed disurbances a subsanial degree of srucural rigidiy has o be inroduced in prooypical small-scale DSGE models o explain he amoun of ineria in acual daa. In paricular, our esimaes indicae ha he degree of price sickiness,, and he degree of indexaion o pas in aion, ; are unlikely o be lower han 0:94, while he role of habi formaion,, is in he range 0:51 0:79. Alhough inconsisen wih microeconomic evidence on he imporance of habi formaion in consumpion (Dynan (2000)) and on price seing behavior (Bils and Klenow (2004)), such high degrees of srucural persisence appear comparable wih hose found in oher papers drawing on similar neo-keynesian models esimaed on derended daa Please noe ha whenever we refer o he ou-of-sample performance of our DSGE model, we always consider he sae-space represenaion of he DSGE model, and no is VAR approximaion. 19 For insance, Roemberg and Woodford (1997) and Giannoni and Woodford (2003) using a similar model esimae = 1and = 1, while xing = :66. Chrisiano e al. (2004) and Alig e al. (2005) esimae o 22

24 In line wih previous empirical sudies, shocks o produciviy growh are also found o be highly correlaed wih serial correlaion, #; unlikely o be lower han 0:91 while esimaes of he moneary policy reacion funcion poin o a sizable degree of ineres rae smoohing ( is ranging beween 0:80 and 0:89), a high responsivenness o in aion (' 1 is over 1:58) and a sizable degree of acivism (' 2 is over 0:63). The aggregae expendiure s elasiciy of ineremporal subsiuion, 1, is esimaed o be signi canly below 1, implying a coe cien of risk aversion beween 1:29 and 2:06 a resul which is also quie common in he lieraure. Perhaps more ineresingly, he equilibrium rae of he economy, g, is esimaed o be lower han 2:7 percen per annum a he end of he sample, whereas he poserior disribuions of p u and p d show ha changes in he seady-sae rae of in aion occur frequenly, wih on average a 15 percen higher probabiliy of downward over upward revisions. Our esimaes also allow o characerize he srucure of he economy which is mos likely o be associaed wih a cerain moneary policy rule, given he inheren uncerainy abou srucural parameers and shock processes governing he evoluion of he economy under equilibrium condiions. As o be hoped in he conex of a srucural raional-expecaions model, esimaes of he deep parameers are broadly invarian o he policy rule speci caion: overall, he magniude of he changes in poserior means and probabiliy inervals is small and saisically insigni can. Ineresingly, hough, an oupu gap rule is likely o feaure a higher degree of acivism, ' 2, a sronger response o in aion, ' 1, and if anyhing a lile more ineria,, han is oupu growh counerpar, while facing subsanially greaer uncerainy abou he seady-sae growh rae of he economy, g. Tellingly, under an oupu gap rule, he 95 percen probabiliy inerval of g poserior disribuion is esimaed o be posiively skewed and imes as large as under an oupu growh rule. The use of a sequenial esimaion echnique is also helpful o rack down ime variaion in he disribuion of policy and srucural parameers, as well as insabiliy in error variances. In his respec, Figure 1 plos he evoluion over he whole sample of he esimaed policy response o in aion and oupu, he smoohing parameer, he probabiliy of revising up and down he in aion arge, as well as he sandard deviaion of moneary shocks, r, under he wo moneary rules considered. Quaniaively, shifs in he corresponding poserior medians seem be :65 and = :82, while xing = 1. Under raional expecaions, Milani (2005) obains = :89, = :91, and = :92. Smes and Wouers (2004) esimae = :66, = :69, and = :87 on a sample. Their esimaes are somewha lower han oher papers, bu sill surprisingly large if we consider ha hey are obained in a rich model, incorporaing, besides habis, sicky prices, and indexaion, also wage sickiness, capial formaion, adjusmen coss, and several highly auocorrelaed shocks. 23

25 o be limied under boh policy reacion funcions, o ering scarce suppor o he convenional wisdom abou he loss of moneary conrol in he 1970s and is subsequen reemergence 20. However, sysemaic moneary policy responses appear less sable under an oupu gap rule han under an oupu growh rule. Indeed, he feedback coe ciens of he growh rule are broadly consan over he sample, if we excep a drasic one-o increase in he oupu response over he second half of he 1970s. In conras, under a gap rule speci caion, he in aion response is likely o have fallen a he beginning of he 1970s, o have recovered in he mid 1970s, and o have seadily increased from he early-1980s onwards. Similarly, he oupu response appears o have risen quickly over he second half of he 1970s and o have sabilized a such a higher level since he early 1980s. Ineresingly hough, a growh rule feaures a higher degree of ime variaion in he unsysemaic par of moneary policy han a gap rule. Indeed, while he volailiy of moneary shocks appears o be roughly consan under he oupu gap rule, i looks hump-shaped under he growh rule, wih a peak a he urn of he 1980s he ime of he Volker experimen. Time variaion in medians and probabiliy inervals of he srucural paramaers poserior disribuions can be observed in Figure 2. In his respec, resuls are less clear-cu. On he one hand, pricing parameers are essenially ime-invarian once shifs in rend growh and in aion are accouned for. Equally, he decline in he size of supply shocks is found o be small and saisically insigni can. On he oher hand, boh volailiy of ineremporal preference shocks and elasiciy of ineremporal subsiuion are likely o have fallen signi canly over ime, while he role of habi formaion and shocks o rend produciviy have somehow become larger. Our ndings seem hence o suppor Cogley and Sbordone s (2005) argumen ha allowing for heeroskedasiciy and ime variaion in he underlying sochasic processes governing he evoluion of an economy under exible prices permis o validae a srucural, consan-parameer, neo-keynesian model of price adjusmens. We are, however, unable o exend he same argumen o a srucural, neo-keynesian model of adjusmens in aggregae expendiure. Conceivably, disinguishing he evoluion of privae invesmen from privae and public consumpion migh be enlighening in his respec. 20 Several papers among which Taylor (1999) and Clarida, Galí, and Gerler (2000) nd ha during he grea in aion period he U.S. Federal Reserve pursued a policy ha accommodaed in aion and induced insabiliy in he economy, by lowering real ineres raes when expeced in aion increased and viceversa. These sudies sugges ha his perverse pracice ended wih Paul Volker s appoinmen, when he policy response o expeced in aion became su cienly aggressive o resore moneary sabiliy. 24

26 4.2 Unobservable Sae Variables: A Real-Time Policy Assessmen Le us now urn he aenion o he main policy quesions raised a he ouse. Has he Fed over cu ineres raes by oo much in relaion o he expeced reurn on capial? Have he laes Fed s rae hikes compleely removed policy accomodaion, hereby allowing he real Fed fund rae o rise again o he NRI? The answers are o be found in Figure 3. Speci cally, drawing on our neo-wicksellian framework embedding an oupu growh moneary policy rule, Figure 3 racks down in real ime he median and he 95 percen probabiliy inervals of he implied disribuions of NRi (op panel), oupu gap (second panel), rend growh (hird panel) and equilibrium in aion (boom panel) vis-á-vis heir acual values, whenever hey exis. According o he gap beween he acual real rae of ineres and our real-ime measure of he NRI aking ino accoun ler uncerainy, he Fed o deal wih he economic downurn following he sockmarke crash has been cuing ineres raes aggressively, hereby driving he cos of capial signi canly below he NRI for over four years from 2001Q3 o 2005Q3. Over he mos recen quarers, however, repeaed moneary policy ighening has managed o fully correc he disequilibrium, while slowing down growh and realigning real oupu wih is esimaed poenial level. A he end of he sample, he poserior median of he NRI is esimaed a 2:9 percen, wih a 95 probabiliy inerval ranging from 2:1 o 3:7 percen. A he same ime, he poserior median of he rend growh rae is esimaed a percen, wih a 95 percen probabiliy inerval ranging from 2:3 o 5:2 percen. Acual growh appears o be below such a rend, as he oupu gap is closing, in line wih recen indicaions ha a slowdown is currenly underway in he U.S. economy. Ye, core in aion appears o have creeped up since 2004, and is esimaed o have risen slighly above is arge during he las year. Using he same benchmark for a rerospecive assessmen, i appears ha U.S. moneary policy has been signi canly loose in oher four episodes: during he ve years running beween 1974Q4 and 1979Q3 when he real ineres rae has reached a record-low of 650 basis poins beneah is appropriae level and, more brie y, over he lae 1960s, in and in In all hese evens, he negaive real ineres rae gap has been preceeded by episodes of slowdown in rend produciviy growh and associaed wih episodes of rising in aion arge. Low-frequency ucuaions in poenial oupu growh are racable in he hird panel of Figure 3. Their suiabiliy is indirecly validaed by panel wo, where resuling roughs denoing he end of periods of economic conracion in 1970Q1, 1975Q1, 1991Q1, and 2001Q4 nicely mach available evidence abou NBER s daing of he US business cycle. The esimaes of 25