Monetary Policy Rules and the U.S. Business Cycle: Evidence and Implications

Transcription

1 WP/4/164 Moneary Policy Rules and he U.S. Business Cycle: Evidence and Implicaions Pau Rabanal

2 24 Inernaional Moneary Fund WP/4/164 IMF Working Paper Wesern Hemisphere Deparmen Moneary Policy Rules and he U.S. Business Cycle: Evidence and Implicaions Prepared by Pau Rabanal 1 Auhorized for disribuion by Tamim Bayoumi Sepember 24 Absrac This Working Paper should no be repored as represening he views of he IMF. The views expressed in his Working Paper are hose of he auhor(s) and do no necessarily represen hose of he IMF or IMF policy. Working Papers describe research in progress by he auhor(s) and are published o elici commens and o furher debae. This paper esimaes Taylor-ype ineres raes for he Unied Saes allowing for boh ime and sae dependence. I provides evidence ha he coefficiens of he Taylor rule change significanly over ime, and ha he behavior of he Federal Reserve over he cycle can be explained using a wo-sae swiching regime model. During expansions, he Federal Reserve follows a rule ha can be characerized as inflaion argeing wih a high degree of ineres rae smoohing. During recessions, he Federal Reserve arges oupu growh and conducs policy in a more acive manner. The implicaions of conducing his ype of policy are analyzed in a small scale new Keynesian model. JEL Classificaion Numbers: C22, E58 Keywords: Swiching Regime Models, Time-Varying Coefficiens, Taylor Rule Auhor s Address: prabanal@imf.org 1 I am hankful o Tam Bayoumi and Calvin Schnure for very deailed commens, and o Jordi Galí and Chrisopher Towe for useful discussions. I am also hankful o Alfred Go for providing ediorial assisance. All remaining errors and omissions are mine.

3 - 2 - Conens Page I. Inroducion...3 A. Moivaions...3 B. How Well Does he Taylor Rule Fi he Daa?...3 II. III. IV. Time-Varying Parameers...6 A Sae-Dependen Taylor Rule...9 Relaing he Swiching Regime Resuls o Federal Reserve Chairmen...14 V. Implicaions of a Swiching Regime Rule in a Small Scale Macroeconomic Model...17 A. The Model...18 B. Impulse Responses...19 C. Second Momens...23 VI. Concluding Remarks...24 References...25 Tables 1. Esimaes of Swiching Regime Model Esimaes of he Taylor Rule, Swiching Regime Model, Using Quarerly Annualized Inflaion and Growh Raes Esimaes of he Taylor Rule, Swiching Regime Model, Using Annual Inflaion and Growh Raes Wald Tess on a Two-Sae Swiching Regime Model wih Dummy Variables Relaed o Fed Chairmen (196 23) Esimaes of he Swiching Regime Model wih Dummy Variables Volailiy of Inflaion, Oupu, and Ineres Raes...23 Figures 1. Original Taylor Rule Esimaes of he Taylor Rule: Time-Varying Coefficiens, Smoohed Series Probabiliies of Being in Recession and NBER Recession Daes Impulse Response o a Demand Shock Impulse Response o a Cos-Push Shock Impulse Response o a Demand Shock wih a Sae-Dependen Rule Impulse Response o a Cos-Push Shock wih a Sae-Dependen Rule...22

4 - 3 - I. INTRODUCTION A. Moivaion Since he beginning of he 21 s cenury, when a new era of price sabiliy was reached, several facors have revived ineres in moneary policymaking in he Unied Saes. The inflaion rae kep falling afer he 21 recession, and core CPI inflaion reached 1.1 percen by end-23, causing Federal Reserve officials o show public unease abou an unwelcomed risk of deflaion. A he same ime, concerns abou nominal ineres raes hiing he zero caused he Fed o use mos of is ammuniion preempively and reduce he federal funds rae arge sharply o 1 percenage poin, he lowes in 4 years, and o keep i a ha level for welve monhs (beween he monhs of June of 23 and 24). These evens have riggered a new debae on how moneary policy should be conduced under exreme bu low probabiliy oucomes, from he use of unconvenional moneary policy ools if he federal funds rae were o hi zero, o he call for a risk managemen approach by Federal Reserve Chairman Alan Greenspan. An imporan consequence of all hese recen developmens is ha he so-called Taylor rule, ha relaes he policy insrumen of he cenral bank o a measure of inflaion and deviaion of real aciviy from a long-erm value, seems o have los power as an approximaion o acual policymaking. More generally, cenral bankers consisenly emphasize ha seing moneary policy is a complex process, involving a range of judgmenal facors ha canno be condensed ino a parameric approach. This paper provides esimaes of Taylor-ype moneary policy rules for he Unied Saes where asymmeric and nonlinear behavior is allowed, and ries o answer wo main quesions: firs, have coefficiens of he Taylor rule changed over ime? And second, do he coefficiens of he Taylor rule change when economic condiions change? The answer o boh quesions is yes. The resuls sugges ha, during expansions, he Federal Reserve arges inflaion and conducs moneary policy in a highly inerial way, while during recessions, i arges oupu growh and moves he policy rae more sharply. The las par of he paper analyzes he consequences of parameer change in a small-scale macroeconomic model. The main resul is ha oupu volailiy can be furher reduced under a sae-dependen rule when a cos-push shock his he economy, bu a he cos of higher inflaion. B. How Well Does he Taylor Rule Fi he Daa? The debae on wheher cenral banks should follow a rule raher han discreionary moneary policy has no been seled beween academic and policymaking circles. From he academic lieraure, a radiion of papers saring wih Kydland and Presco (1977) and Barro and Gordon (1983) sugges ha a rules-based approach reduces he inflaionary bias in moneary policy. More recen research wihin he class of models ha use ineres rae rules (i.e., Woodford, 23 and Clarida, Galí, and Gerler, 1999) suggess ha a cenral bank can

5 - 4 - manage expecaions by commiing o a moneary policy rule, hereby facing a more favorable inflaion-oupu sabilizaion rade off. In pracice, no major cenral bank wans o ie is hands and commi is fuure moneary policy acions; discreionary moneary policy is he rule. In recen speeches, 2 Federal Reserve officials have suggesed ha i would make lile sense for any cenral banker o mechanically follow a simple rule. Such rules migh perform fairly well under normal condiions, and can be useful as a benchmark, bu hey offer no guidance when he economy poenially faces high risks or abnormally large shocks, because of nonlineariies associaed wih exreme evens. Chairman Greenspan is of he view ha, given our limied undersanding of relevan economic phenomena, a cenral bank, in conducing moneary policy, should apply judgmen and principles of risk managemen o avoid low-probabiliy exreme evens, raher han follow a mechanical rule. Even when cenral banks announce ha hey conduc policy in a discreionary or mosly judgmenal way, however, i sill migh be possible o characerize heir average behavior via a relaively simple economeric model. In a highly influenial paper, Taylor (1993) showed ha during he period from 1987 o 1992, a simple ineres rae rule fis he daa prey well for he Unied Saes. Taylor suggesed he following ineres rae rule, relaing he federal funds rae, wih a measure of a seady sae nominal ineres rae, consising of he real ineres rae and inflaion, and a reacion o deviaions from inflaion ( π ) and oupu ( y ) o * heir long-erm (or arge) values ( π ) and y ) respecively: ( * i * * = r + + γ ( π π ) + γ ( y π π y y * ) The federal funds rae (i ) is specified in annualized erms, inflaion consiss of he las four quarers of he GDP deflaor inflaion, and he oupu gap is measured as percen deviaions of acual oupu from long-erm (rend) oupu. 3 In order o keep hings as simple as possible, Taylor s paper did no esimae he rule bu raher suggesed some values for he parameers: * * he seady sae real ineres rae r was se a 2 percen; he inflaion arge π, was also se a 2 percen; implying ha he seady sae nominal ineres rae is 4 percen. The coefficien of he reacion of ineres raes o he inflaion deviaion γ π, was se a.5; and he same value was used for he coefficien of he reacion of he ineres rae o he percen deviaion of oupu from is poenial value γ y,.5. 2 See, for insance, Greenspan (24). 3 Two main problems arise when using his rule for acual policymaking. Firs, o obain he seady sae value for he nominal ineres rae, esimaes of seady-sae values for he real ineres rae are needed. Taylor assumed a seady-sae real ineres rae value of 2 percen. The second limiaion is ha poenial oupu is an unobservable variable. Taylor s guess was a rend growh rae of 2.2 percen beween 1984:Q1 and 1992:Q3.

6 - 5 - Figure 1 presens he simulaion of he Taylor rule using quarerly daa beween 1982:Q4 and 23:Q4, using Taylor s parameerizaion, excep for rend growh, which is assumed o be 2.8 percen (he sample mean for he period). The Taylor rule does a good job of explaining he behavior of he Fed beween 1987 and 1994, exending by wo years Taylor s original sudy. However, in oher periods i does no fi he daa so well. For insance, during and afer he 21 recession, he Fed eased moneary policy much faser han wha he rule would sugges, and in he pas hree quarers in paricular, he disance exceeds 15 basis poins. For he las quarer of 23, he Taylor rule would sugges a value of 3.1 percen for he federal funds rae, while he acual value is 1 percen. These imporan deviaions have led several analyss as well as Federal Reserve officials o sugges ha he Taylor rule migh no be a good approximaion o acual policymaking afer all. Figure 1. Original Taylor Rule 12 1 Taylor's Original Period 8 6 Federal Funds Rae 4 2 Taylor Rule Sources: Federal Reserve; and Fund saff esimaes. This paper seeks o help reconcile he more nuanced characerizaion of moneary policy espoused by many policymakers wih rules ypically esimaed in academic analysis. In paricular, he paper examines a policy rule whose parameers are no consan, wih a focus on wheher here is evidence ha moneary responses vary sysemaically over he business cycle. Such behavior would be consisen wih he noion ha policymakers consider more facors han are capured in a ypical moneary response regression,which also capures nonlineariies in he Taylor rule. The remainder of his paper is organized as follows. Secion II esimaes a model wih imevarying coefficiens for he Taylor rule, and we find ha he Fed s olerance o deviaions of inflaion from is arge increases during periods of recession or when oher risks come ino play. In Secion III, we esimae he Taylor rule in a swiching regime framework, and we find ha he Fed behaves quie differenly in booms and recessions. In Secion IV, we relae

7 - 6 - he esimaes of he swiching regime model o he various chairmen ha he Fed has had since 196. In Secion V, we explore he implicaions of such behavior in a small-scale macro model, and Secion VI concludes and provides some direcions for fuure work. II. TIME-VARYING PARAMETERS The empirical lieraure on moneary policy rules has been ypically concerned abou srucural breaks in he Taylor rule, wih an emphasis in he pre- and pos-greenspan years. A ypical resul, as in Clarida, Galí, and Gerler (2), is ha he policy rule places more weigh on inflaion sabilizaion afer However, he resuls ha come from his approach do no allow o discern wheher he Fed behaved differenly because he preferences of is chairmen were differen, or because he economic evens or shocks were also differen, which required differen moneary policy acions, possibly of a nonlinear and asymmeric naure. 5 Therefore, in his secion and he nex secion of he paper, we examine he following quesions: 1. Have he parameers of he Taylor rule changed significanly over ime? 2. Does he Fed behave differenly when he sae of he economy changes? I migh well be ha he parameers of he observed rule do no change significanly when averaged overime, bu he Fed uses very differen rules in booms and recessions. We sudy wheher he Fed uses one rule in all saes of he economy, or i uses various rules depending on is percepion of he sae of he economy. In his secion, we allow he coefficiens of he rule o change no only across sub samples, as is cusomary in he lieraure, bu a any poin in ime. We also inroduce some modificaions o he rule ha have been shown o improve is empirical fi. Firs, we inroduce lagged ineres raes o capure he endency of cenral banks o conduc moneary policy in a smooh way. Second, since he oupu gap can be measured in a variey of ways, we assume ha he Federal Reserve arges oupu growh, which is an observable variable, insead. 6 4 See also Boivin and Giannoni (23). 5 For insance, Dolado, María-Dolores, and Naveira (24) find evidence of nonlinear behavior in he Taylor rule, which hey relae o opimal responses o a convex-shaped Phillips curve. 6 For his specificaion of he Taylor rule, see Erceg and Levin (23).

8 - 7 - The model o be esimaed is as follows: i = ρ i + 1 ρ )( c + γ π π + γ y ) + ε, (1) 1 (, g, where ρ is he coefficien measuring ineres rae smoohing, y is oupu growh, and γ π, and γ g, are he ime-varying long-run elasiciies of he ineres rae rule wih respec o inflaion and oupu growh. ε is a Normally disribued, zero mean iid error erm. * * * * The ime-varying inercep is c = r ( γ π, 1) π γ g, g y,. I is no possible o esimae separaely he real rae of ineres, he inflaion arge of he cenral bank, and he arge * oupu growh rae ( g y, ) because he Federal Reserve does no announce explici inflaion or oupu growh arges. Hence, we have o calibrae wo of he hree parameers o obain he oher. However, he esimaes of he consan erm do no affec he slope elasiciies, as long as he free parameers are no over calibraed. To conduc he esimaion, i is assumed ha all he parameers follow a random walk: ρ = ρ c γ γ π, = c g, 1 1 = γ = γ + ε + ε π, 1 g, 1 c ρ + ε π g + ε. Equaion (1) and he processes for he parameers can be wrien in space-sae form. In a firs sep, he parameers of he model (he variances of he five shocks) are esimaed via maximum likelihood. Then, he Kalman filer is used o obain he one-sep ahead forecass and he smoohed series for he coefficiens, as in Hamilon (1994). Figure 2 presens he smoohed series of he parameer esimaes, where he shaded areas are he Naional Bureau of Economic Research (NBER) recession daes. 7 The ime-varying model favors a specificaion where he elasiciy of he Taylor rule o he inflaion rae and he ineres rae smoohing coefficien swing grealy o explain ineres rae movemens. Excep for he Volcker disinflaion period, he elasiciy of he Taylor rule wih respec o inflaion decreases during or afer he recession (his seems o be paricularly he case afer he and he 21 recessions). During he Volcker disinflaion, he coefficien reaches values above 2, and his a maximum of 2.3 in 1984:Q3. The model also suggess ha ineres rae smoohing was declining during he 198s, i became almos zero during he Volcker disinflaion, and i increased aferwards, jus o decline during he las hree years. 7 Bayoumi and Sgherri (24) presen similar evidence on ime-varying coefficiens in he Taylor rule.

9 - 8 - Flucuaions in he inercep and he reacion o oupu growh are less imporan quaniaively. 8 I is imporan o noice ha, in addiion o NBER recession daes, in periods where he Fed was concerned abou oher poenial high risks in he economy, such as he sock marke crash of 1987, he Iraq invasion of Kuwai (fall of 199), he Russian defaul (fall of 1998), he Sepember 11, 21 aacks, and more recenly, he risk of deflaion, he elasiciy of he Fed Funds rae wih respec o inflaion falls, showing more olerance wih inflaion when oher risks come o play. In fac, in he las cyclical episode, he coefficien on inflaion declined markedly and became negaive afer he firs quarer of 21, showing ha he Fed was no responding o curren inflaion, bu raher using is ammuniion preempively o avoid, firs, he recession and he impac of he erroris aacks and, second, an unwelcomed fall in he inflaion rae. This behavior of he Fed explains why a radiional Taylor rule will no fi he daa in he mos recen period. Figure 2. Esimaes of he Taylor Rule: Time Varying Coefficiens, Smoohed Series Consan Gamma_Pi Gamma_y Rho Source: Fund saff esimaes. 8 Indeed, he shor-erm responses ( 1 ρ ) c and ( 1 ρ ) γ g, are consan.

10 - 9 - III. A STATE-DEPENDENT TAYLOR RULE The previous secion has provided some anecdoal evidence regarding he behavior of he Federal Reserve in periods of recession and in periods where fighing inflaion was he mos imporan goal for moneary policy. This secion implemens a furher sep in he economeric analysis o discern wheher he Federal Reserve behaves consisenly is rule whenever economic condiions change. I migh well be ha according o he views ha he Fed has over he economy, i follows differen sae-coningen rules which, averaged over ime, deliver a rule wih no changing coefficiens, or, on he conrary, wih large swings in he coefficiens. Therefore, we esimae a regime swiching model for he Taylor rule. The esimaion is conduced in wo sages. Firs, we esimae a swiching regime model for oupu growh, as in Hamilon (1994). In a second sep, he probabiliies obained using Hamilon s mehod are used in a weighed leas squares regression o obain he coefficiens of he rule in each sae. The idea is o capure how policymakers assign probabiliies of being in an expansion or a recession, and how hey respond o he sae of he economy accordingly. The swiching regime model is more flexible because i allows o examine wha happens when he Fed (or an exernal observer) believes he economy is in an expansion or recession sae wih cerainy, or when, for insance, i assigns a ¼ probabiliy of being in a recession and a ¾ probabiliy of being in an expansion. 9 Formally, in he firs sep, we esimae he following model for oupu growh 1 : y = c( s ) + ε 2 where s = {1,2} and ε ~ iidn(, σ ) where he sae variable s follows a firs order Markov process. The sae s =1 denoes expansion, while sae s =2 is recession. The only parameer ha changes wih he sae is he mean of he growh rae. We idenify periods of expansion when he mean of oupu growh is high, and, conversely, periods of recession when he mean is low. We define he ransiion probabiliies as follows: p ij = = Pr{ s = i s 1 j}. 9 This wo-sep procedure is implemened because we are rying o capure he behavior of he Fed in periods of oupu expansion and recession. Esimaing a swiching regime model and he coefficiens of he rules simulaneously delivers saes of high inflaion and saes of low inflaion, which is no wha we are looking for. 1 Originally, his model was conceived wih auoregressive erms, bu his specificaion seems o be more robus o daa revisions. I am hankful o James Hamilon for his commens on his issue.

11 - 1 - Therefore, p p = Pr{ s = Pr{ s = 1 s 1 = 2 s 1 = 1}, = 1}, and p and p = 1 p = 1 p Table 2 presens he esimaes. The average growh rae in booms is 4.62 percen, while i is -.57 percen in recessions, boh cases annualized. Booms las longer han recessions. Booms las for an average of 11 quarers (1/(1-p 11 )), while recessions las, on average, roughly four quarers. Also, his esimaes sugges ha he uncondiional probabiliy of being in an expansion is.75, while he probabiliy of being in a recession would be.25. Table 1. Esimaes of Swiching Regime Model Variable Coefficien Sd. Error µ 1 =c(s =1) µ 2 =c(s =2) p p σ Source: Fund saff esimaes. Figure 3 presens he smoohed probabiliies of being in an expansion. Reassuringly, and as originally poined ou by Hamilon (1994), periods where he probabiliy is close o one coincide wih NBER recession daes. In oher cases, he model assigns moderae probabiliies of recession. Figure 3. Probabiliies of Being in Recession and NBER Recession Daes Probabiliy of Recession Sources: Naional Bureau of Economic Research and Fund saff esimaes.

12 In he second sage, he Taylor rule is assumed o be similar o equaion (1), bu wih coefficiens changing according o he sae of he economy: i ( s ) i = ρ 1 Var( ε ) = σ ( s ) + (1 ρ( s ))( c( s ) + γ ( s ) π + γ ( s ) y 2 π g ) + ε, Hence, he probabiliies of being in each sae are used o esimae a weighed leas squares regression, by minimizing he following loss funcion: where ( s = i) L = T 2 2 { Pr( s = 1) [ i x β ( s = 1) ] + Pr( s = 2) [ i x β ( s = 2) ] } = 1 β is he vecor of parameers (for each sae i=1,2) and x he vecor of explanaory variables (a consan erm, inflaion, oupu growh, and he lagged nominal ineres rae). As a resul, he esimaes for he parameers for each sae consis in a weighed OLS regression, using he square roos of he probabiliies as weighs. An alernaive way o derive he esimaes for he parameers of he Taylor rule in each sae is as follows. The densiy of he ineres rae condiional on he righ hand side variables and he sae is (2) f ( i x, s 2 [ i ( )] x β s = j 2[ σ ( s = j) ] 1 j, β ) = exp. 2πσ ( s = j) = 2 For each observaion, he uncondiional densiy of he ineres rae (wih respec o he sae) is a mixure of he condiional densiies as follows: f ( i 2 x, β ) = Pr( s = j) f ( i x, s = j, β ). j= 1 By maximizing he log-likelihood of he observed daa, [ f ( i x, )] T = 1 log β, we arrive a he same resul: he parameer esimaes of he Taylor rule on each sae are a weighed OLS regression, using he probabiliies of being in each sae as a weigh. 11 Table 2 repors he esimaes using quarerly annualized raes. The sample period remains 196 o 23, using quarerly daa. The specificaion in equaion (2), using conemporaneous values for inflaion and oupu is esimaed wih weighed OLS. The disincion beween shor-erm and long-erm response o oupu and inflaion is no so naural as in he case of 11 See Hamilon (1994, page 696).

13 fixed coefficiens. If we know ha recessions end o be shor-lived, i is difficul o inerpre he meaning of long-erm responses. However, for comparabiliy wih previous work, we presen boh shor-erm and long erm coefficiens. 12 Table 2. Esimaes of he Taylor Rule, Swiching Regime Model, Using Quarerly Annualized Inflaion and Growh Raes Shor Run Coefficiens Long Run Coefficiens OLS Recession Expansion OLS Recession Expansion Consan (.2) (.24) (.17) (3.85) (2.86) (5.85) Inflaion (.4) (.4) (.3) (.54) (.34) (1.48) Oupu (.2) (.3) (.2) (.59) (.61) (.58) Smoohing (.3) (.4) (.3) (.3) (.4) (.3) Wald Tess (p-value) 31.6 (.) 12.7 (.2) Source: Fund saff esimaes. Noes: Bold means significan a he 5 percen level, bold and ialics means significan a he 1 percen level. Focusing firs on he resuls of he shor-run coefficiens, hree main differences arise: firs, he coefficien on he reacion o inflaion is almos wice as large as in he case of expansions (.17) han in recessions (.1) and, in boh cases, he coefficiens are significan a he 5 percen level. Second, he coefficien of he reacion of he Taylor rule o oupu growh is four imes as large as in he case of recessions (.13) han in expansions (.3). Moreover, in he case of expansions his coefficien is only significan a he 1 percen level. Third, he coefficien on ineres rae smoohing is larger in expansions han in recessions, alhough in boh cases hey are fairly large (.95 versus.9). Excep for he consan erm, he OLS coefficiens lie in beween he wo rules. Individually, only he coefficien on oupu growh is significanly differen in he wo regimes, bu a Wald es suggess ha he null hypohesis of all coefficiens being equal in he wo rules is rejeced. The qualiaive inerpreaion of hese resuls is ha he Fed places much more weigh on inflaion sabilizaion in expansions, while i shifs is focus o oupu sabilizaion in recessions. This shows ha he Fed is ready o ease faser in recessions han ighen in booms, because, in recessions, i is precisely when oupu growh ends o be in negaive erriory. Finally, ineres rae smoohing is higher in expansions ha in recessions, as he Fed acs quicker in he laer case. ~ 12 Tha is, he shor-erm response coefficiens are γ π = (1 ρ) γ π and γ g = (1 ρ) γ g, while he longerm responses are given by γ π and γ g. ~

14 The long-run properies of he wo rules are quie differen: he coefficien of he reacion of he Taylor rule o inflaion is much higher in booms (3.45) raher han recessions (1.), suggesing ha i is in expansion imes ha he srong ani-inflaionary sance of he Federal Reserve is implemened. In recessions, he Taylor principle whereby he real ineres rae moves more han one-o-one wih inflaion is no implemened (in fac, he poin esimae is.9972). The long-run coefficien on oupu sabilizaion increases in recessions (i is.71 bu no significan in expansions, and 1.26 in recessions), again suggesing ha when oupu growh is negaive, he Fed eases by a larger amoun han when he economy is in expansion. 13 The fac ha he rule reacs srongly o (negaive) oupu growh in recessions makes he real ineres rae counercyclical. Table 3 presens he same esimaions bu using annual growh and inflaion raes insead. The main difference wih respec o he resuls of Table 2 is ha he coefficiens of he reacion of he Taylor rule o oupu growh are higher, and hey are significan a he 1 percen level for he long run coefficien. Also, he Taylor rule, in recessions, does respec he Taylor principle, wih a poin esimae of Finally, he Wald es rejecs he null hypohesis of equal coefficiens in he wo saes, boh for he shor-run and long-run coefficiens. Table 3. Esimaes of he Taylor Rule, Swiching Regime Model, Using Annual Inflaion and Growh Raes Shor Run Coefficiens Long Run Coefficiens OLS Recession Expansion OLS Recession Expansion Consan (.22) (.25) (.19) (4.71) (3.82) (6.77) Inflaion (.4) (.5) (.3) (.49) (.34) (1.21) Oupu (.3) (.5) (.3) (.91) (1.8) (1.) Smoohing (.3) (.4) (.3) (.3) (.4) (.3) Wald Tess (p-value) 3.67 (.1) (.2) Source: Fund saff esimaes. Noes: Bold means significan a he 5 percen level, bold and ialics means significan a he 1 percen level. 13 We also considered a forward-looking version of he rule, by replacing π wih E 1, and y wih E y + 1, where E denoes he mahemaical expecaions operaor wih informaion up o ime. We esimaed such rule wih one period ahead expecaions of oupu growh and inflaion using weighed wo-sage leas-squares (TSLS), wih he probabiliies of expansions and recessions as weighs, and four lags of he Federal Funds rae, inflaion and oupu growh as insrumens. The qualiaive resuls were similar o he rule ha reacs o curren values, wih he anomaly ha he coefficien of he reacion of he Taylor rule o expeced oupu growh urned ou o be negaive. π +

15 IV. RELATING THE SWITCHING REGIME RESULTS TO FEDERAL RESERVE CHAIRMEN In his secion, we relae he resuls of he swiching regime model o he chairmen ha he Federal Reserve has had since 196, and we ry o obain a paern beween cyclical behavior and Fed chairmanship. In order o fix ideas abou he ess we conduc in his secion, we sar wih a simple example. Suppose ha we assume ha he wo-sae Taylor rule only reacs o inflaion, and we use only a pos-83 dummy variable. Then, he model would be: i ( s = 1) = ( a( s = 1) + d83( s = 1)) π i ( s = 2) = ( a( s = 2) + d83( s = 2)) π. We run hree ess: (i) All coefficiens (including dummies) are joinly equal: a ( s 1) = a( s = 2) and = d 83 ( s = 1) = d83( s = 2). If we canno rejec his hypohesis, hen, he rule followed by he Fed is always he same, regardless of is chairmen and of he sae of he economy. (ii) Only he dummy coefficiens are equal: d 83 ( s = 1) = d83( s = 2), bu he level of he coefficiens for each sae are allowed o change wih each chairman. In his case, we assume ha he cyclical behavior of he rule is differen, bu ha he shif in he coefficiens across saes for each chairman is he same. (iii) If we canno rejec (ii), we fix he dummy coefficien across saes, such ha d83( s = 1) = d83( s = 2) d83 and es wheher he level is equal = = 1) = a( s 2) a ( s =. Basically we re-run (i) afer imposing he resricion in (ii). Tables 4 and 5 summarize he resuls of exending he regressions in he previous secions, using dummy variables for he following periods, when: Arhur Burns and William Miller (197:Q1-1979:Q2) were Fed chairmen, The Volcker disinflaionary period using non-borrowed reserves argeing ook place (1979:Q3-1982:Q4), The Fed moved o an ineres rae arge, wih chairmen Volcker and Greenspan (1983:Q1-23:Q4). Therefore, we leave he 196s as he period wihou dummy variables. In he firs row of Table 4, we repor he Wald Tes (and p-value) of assuming ha he coefficiens of he wo

16 rules in he wo regimes (including he se of dummy variables) are he same. 14 We rejec he null hypohesis a he 5 percen level, boh using quarerly annualized raes and using annual raes for inflaion and oupu growh. Nex, we assume ha he dummy effecs on each sae are he same across chairmen. The Wald es on his resricion is presened in he second row and we find ha we canno rejec such resricion. Tha is, he coefficiens of he rule do in fac change across chairmen, bu hey move by he same magniude in expansions and in recessions. Nex, afer imposing ha he dummy effecs be he same for each period across saes, we es wheher he levels of he coefficiens are joinly differen across saes in he hird row, and we find ha we canno rejec such resricion. The nex hree rows of Table 4 show ha he resuls also hold when only wo ses of dummy variables are inroduced o conrol for he Burns-Miller and he Volcker-Greenspan periods. Table 4: Wald Tess on a Two-Sae Swiching Regime Model wih Dummy Variables Relaed o Fed Chairmen (196 23) Using Dummies for Burns-Miller, Nonborrowed reserves argeing, pos-83 periods Null Hypohesis Using Quarerly raes Using Annual Raes Wald Wald Saisic p-value Saisic p-value All Coefficiens are equal Dummies for each period across saes are equal Once dummies for each period across saes are held equal, levels of coefficiens are equal Using Dummies for Burns-Miller, Volcker-Greenspan periods Null Hypohesis Using Quarerly raes Using Annual Raes Wald Wald Saisic p-value Saisic p-value All Coefficiens are equal Dummies for each period across saes are equal Once dummies for each period across saes are held Source: Fund saff esimaes. Table 5 shows he esimaes wih dummy variables for he Burns-Miller period and he Volcker-Greenspan period, and afer having dropped he coefficiens ha are no saisically significan. 15 The main differences arise for he 197s period wih respec o he 196s and he Volcker-Greenspan period. The consan erm and he reacion o inflaion are fairly 14 The Wald ess are always performed on he shor-run coefficiens, since during he Burns-Miller period, a uni roo on he ineres rae canno be rejeced and hence, he long-erm parameers canno be uncovered. 15 A Wald es rejecs ha he four level coefficiens (inercep, inflaion, oupu growh and lagged ineres rae) are he same across saes.

17 similar across saes during he 196s, and he pos-83 period. However, in he 197s, he shor-run reacion o inflaion becomes basically zero. The coefficien of he reacion o oupu is saisically differen beween recessions and expansions, wih no differences beween chairmen. Table 5: Esimaes of he Swiching Regime Mode wih Dummy Variables Shor Run Coefficiens Recession Expansion Consan (.25) (.16) Inflaion (.7) (.5) Inflaion*Burns-Miller -.52 (.8) Oupu (.5) (.3) Smoohing (.6) (.5) Smoohing*Burns-Miller.35 (.7) Smoohing*Volcker-Greenspan.7 (.3) Source: Fund saff esimaes. Noes: Bold means significan a he 5 percen level, bold and ialics means significan a he 1 percen level. Finally, he degree of ineres rae smoohing is lower under recessions han under expansions, and is saisically differen in he wo saes. Compared o he 196s, boh he Volcker-Greenspan and he Burns-Miller periods reflec higher and saisically significan ineres rae ineria. The Volcker-Greenspan rule is more sabilizing han in he 196s because for a given shor-run coefficien on inflaion, a higher ineres rae smoohing value delivers a larger long-run response. On he oher hand, he Burns-Miller period delivers coefficiens on he reacion of o he lagged ineres rae higher han one: as a resul i is no possible o derive he shor-erm responses (which, in any even, are likely o be very close o zero for inflaion). Therefore, he conclusions from his secion are: firs, he coefficiens of he rule shif beween expansions and recessions, even when we accoun for differen rules according wih Fed chairmen. And second, he Burns-Miller period is characerized by a much smaller reacion o inflaion and a larger reacion o lagged ineres raes when compared o he 196s, or afer 1983.

18 V. IMPLICATIONS OF A SWITCHING REGIME RULE IN A SMALL SCALE MACROECONOMIC MODEL So far, his paper has provided evidence ha i is possible o characerize he behavior of he Federal Reserve as a wo-sae Taylor rule, wih differen coefficiens depending on wheher he economy is in a boom or in a recession. In wha follows, we explore he implicaions ha such rules imply in a small-scale macro model ha joinly explains he behavior of oupu, inflaion, and nominal ineres raes. This secion presens he simples version of a New Keynesian model, as in Clarida, Galí, and Gerler (1999), which will be modified o allow for backward looking behavior in boh he inflaion and oupu equaions, and regime swiching in he nominal ineres rae rule. In addiion o he moneary policy rule, he model has wo equaions ha characerize he dynamics of inflaion and oupu. These equaions can be raionalized wih a micro founded general equilibrium model, where preferences by households are assumed o have habi formaion in consumpion, while on he supply side monopolisically compeiive firms face nominal rigidiies when seing heir price, and use backward looking indexaion. 16 In he las decade, a growing par of he macroeconomic lieraure has emphasized he ineracion beween nominal rigidiies and moneary policy rules, in a variey of conexs. 17 However, mos resuls ha come from his class of models have been obained assuming ha informaion is complee: he cenral bank observes he value of all variables, shocks, and parameers in he economy and is ready o reac opimally o heir flucuaions. Similarly, i is common o assume ha he moneary policy rule is perfecly observable and credible, and ha he coefficiens are sable over ime. Hence, i is no surprising ha, parallel o he developmen of he New Keynesian lieraure, many papers have emphasized he imporan implicaions of deparing from he assumpion ha informaion is complee a all levels. One branch of he lieraure assumes ha he privae secor does no have perfec knowledge or observabiliy abou he moneary policy rule. This assumpion is used o model lack of credibiliy of he cenral bank's policy: Erceg and Levin (23) and Schorfheide (23) look a he effecs of a change in he moneary auhoriy s inflaion arge ha agens learn over ime. Rabanal (22) sudies he properies of an economy where agens learn he moneary policy rule using discouned leas squares. Mispercepions abou he policy rule migh imply addiional volailiy and persisence in oupu and inflaion. The idea ha we capure in he nex subsecions is ha while long-erm expecaions are anchored by he hisorical behavior of he cenral bank, when a given shock causes a 16 In order o save space, he equaions show variables as linear approximaions o heir seady sae values. Microfounded versions of he New Keynesian model can be found in Woodford (23). 17 See he exensive survey in Clarida, Galí, and Gerler (1999), and he book by Woodford (23).

19 recession o he economy, he cenral bank swiches o recession mode o simulae he economy. The informaion is perfec, in he sense ha all agens know ha he Fed has swiched o recession mode, which becomes public knowledge. We will no be dealing wih credibiliy issues as hey do no seem o be relevan o he case of he Unied Saes. Noneheless, we would compleely agree ha hese issues are of cenral imporance for counries wih cenral banks ha do no have a repuaion of being hawkish on inflaion in he long run. A. The Model The inflaion equaion is he so called New Phillips Curve, or also known as he AS curve: π γ π + 1 γ ) E π + λy + u, (2) = b 1 ( b + 1 where π is he inflaion rae, y is he oupu gap, E denoes he expecaions operaor using informaion up o ime, and u is a cos push shock. The parameer γ b is relaed o he amoun of price seers ha follow a backward looking rule when seing prices. This equaion can be derived as a firs order approximaion o he opimal price chosen by a firm ha keeps is price fixed a random ime inervals, as in Calvo (1983), and wih backward looking indexaion. The second equaion reflecs he dynamics of he oupu gap: ( 1+ b ) y = by 1 + E y+ 1 (1 b)( i Eπ + 1) + g, (3) where i is he nominal ineres rae and g is a demand shock. b denoes degree of habi formaion in consumpion, and also relaes oupu o he real ineres rae. 18 This equaion is derived as a linear approximaion o a consumpion Euler equaion. The demand shock reflecs componens of GDP ha do no reac o he real ineres rae, such as governmen expendiures. To see he imporance of he moneary policy rule in he model, le s assume ha he model is enire forward looking (ha, is, ha γ b and b are se o zero) and ha all shocks are se o zero. Then, he dynamics of oupu and inflaion become: π = λ y = i= i= E E ( i y + i + i E π + i+ 1 ). 18 In order o preserve balanced growh, preferences are logarihmic in he quasi-difference of consumpion. Tha is, he uiliy funcion reads u(c,c -1 )=log(c -bc -1 ).

20 Clearly, no only does he curren value of he nominal ineres rae maer, bu also he way he policy rule is perceived by he privae secor. The way he privae secor perceives he policy rule affecs is expecaions, which feed back ino he curren values of he wo endogenous variables. Hence, i is imporan for he privae secor o undersand which par of he value of i belongs o sysemaic responses of he cenral bank o oupu and inflaion flucuaions, and which par belongs o unexpeced moneary policy shocks, when forming expecaions abou he fuure pah of moneary policy. Obviously, he higher is he coefficien of he reacion of oupu o he real rae of ineres, he more imporan he expecaions channel will be. The calibraion ha we use o conduc simulaions in his secion is as follows: we use a coefficien of γ b =. 5, consisen wih evidence ha sugges ha inflaion dynamics in he U.S. equally weighs forward and backward looking behavior (Fuhrer and Moore, 1995). The range of esimaes for λ in he lieraure is wide, so we choose a somewha inermediae value of.2, suggesed by Roemberg and Woodford (1997). We use a value for he habi formaion parameer of b=.5, consisen wih he esimaes of Galí and Rabanal (24). We assume ha he Federal Reserve follows a Taylor-ype rule like he one ha we modeled and esimaed in secion IV. We ake he coefficiens from Table 3, rounding hem bu always saying in he 95 percen confidence inerval. In recession, he coefficiens are ρ.9, γ π = 1, γ = 1.5 ; in expansion ρ.95, γ π = 3.5, γ = ; and in seady sae we use = y = y ρ =.9375, γ π = 2.5, γ y =.6. The laer comes from he fac ha, according o Hamilon s swiching regime model esimaes, he economy is hree-fourhs of he ime in expansion, and one-fourh in recession. B. Impulse Responses This subsecion presens impulse responses o cos push and demand shocks under he wo ype of rules ha were esimaed in secion IV, ha we label recession and expansion rules, assuming ha he Fed does no swich rules. In boh cases, he auoregressive coefficien of he shocks is se o.8. We also presen he behavior under a seady-sae rule, which would be he one ha agens expec o hold in he long run. Figure 4 shows he impulse response o a demand shock. The recession rule induces more volailiy in he sysem. By no reacing enough o inflaion flucuaions, he cenral bank ends up creaing more volailiy in oupu and inflaion, which is ranslaed ino higher volailiy of he nominal ineres rae. No surprisingly, he dynamics under he seady sae rule are closer o he expansion rule, because he former also includes a subsanial amoun of inflaion argeing.

21 - 2 - Figure 4. Impulse Response o a Demand Shock Percen Deviaion from Seady Sae Percen Deviaion from Seady Sae Response of Inflaion Quarers Afer Shock Response of Nom. In. Expansion Recession Seady Sae Quarers Afer Shock Percen Deviaion from Seady Sae Percen Deviaion from Seady Sae Response of Oupu Quarers Afer Shock Response of Demand Quarers Afer Shock Source: Fund saff simulaions. Figure 5. Impulse Response o a Cos-Push Shock Percen Deviaion from Seady Sae Percen Deviaion from Seady Sae Response of Inflaion Quarers Afer Shock Response of Nom. In. Expansion Recession Seady Sae Quarers Afer Shock Percen Deviaion from Seady Sae Percen Deviaion from Seady Sae Response of Oupu Quarers Afer Shock Response of Cos Push Quarers Afer Shock Source: Fund saff simulaions.

22 In he impulse response o a cos-push shock (Figure 5), oupu volailiy is quie similar under he wo rules, bu inflaion becomes more volaile under he recession rule. Moreover, since he recession rule arges oupu growh when i is below poenial, i is able o reduce he iniial jump of oupu. However, he maximum loss is quaniaively similar o he expansion case, only o be delayed by one quarer. Ineresingly, he seady sae rule does no generae oupu and inflaion dynamics ha lie beween he wo exreme cases, even hough he ineres rae does so. Wih he seady sae rule, inflaion is more volaile and oupu is much less volaile han in he oher wo cases, showing he sabilizaion properies of a rule ha implies srong inflaion argeing wih some scope for oupu sabilizaion. Figures 4 and 5, while being very illusraive on he properies of each policy rule, have some limiaions. The main one is ha, under all cases considered, i does no maer if he shock generaes a boom or a recession, because he response in all cases is symmeric. Tha is, a negaive demand shock or cos push shock would generae he same oucome bu wih a reversed sign. In Figures 6 and 7, we display he case when he cenral bank conducs policy in an asymmeric way. In order o focus on he relevan cases, we generae he shocks ha cause a drop in oupu (i.e., a negaive demand shock and a posiive cos-push shock). An imporan feaure of hese impulse-responses is ha agen s expecaions include going back o an average or seady sae rule in he long run. If he Fed abandons is ani-inflaionary sance in he shor run, i does no mean, if i has credibiliy, ha i will abandon i in he fuure. 19 Figure 6 displays he effecs of a negaive demand shock, and he reacion of he Fed under he sae dependen rule. We also have ploed, for comparison purposes, he seady sae or long-run rule, which simply behaves as he mirror of he one in Figure 4. The main conclusion from his figure could somehow be anicipaed from previous resuls: since he recession rule creaes more volailiy, he sae-dependen rule under recession also creaes more volailiy in inflaion and oupu. Boh oupu and inflaion fall for one addiional period afer he negaive shock his he economy. The recovery, however, akes he same ime, abou four periods, and he rebound is larger under he sae-dependen rule han oherwise. The dynamics of he sae-dependen rule become more ineresing when a posiive cos-push shock his he economy, displayed in Figure 7: ineres raes increase on impac (because of he iniial focus on inflaion) bu decrease aferwards (because of he focus on oupu growh). As a resul, oupu seems o recover faser from he shock, bu evenually inflaion picks up, he cenral bank ighens again, and oupu ends up displaying an oscillaory behavior. 19 Obviously, his would be an exremely ineresing area of research, bu ouside he scope of his paper.

23 Figure 6. Impulse Response o a Demand Shock wih a Sae-Dependen Rule Percen Deviaion from Seady Sae Percen Deviaion from Seady Sae Response of Inflaion Quarers Afer Shock Response of Nom. In. Sae-Dependen Seady Sae Quarers Afer Shock Percen Deviaion from Seady Sae Percen Deviaion from Seady Sae Response of Oupu Quarers Afer Shock Response of Demand Quarers Afer Shock Source: Fund saff simulaions. Figure 7: Impulse Response o a Cos-Push Shock wih a Sae-Dependen Rule Percen Deviaion from Seady Sae Percen Deviaion from Seady Sae Response of Inflaion Quarers Afer Shock Response of Nom. In. Sae-Dependen Seady Sae Quarers Afer Shock Percen Deviaion from Seady Sae Percen Deviaion from Seady Sae Response of Oupu Quarers Afer Shock Response of Cos-Push Quarers Afer Shock Source: Fund saff simulaions.

24 C. Second Momens This subsecion complemens he impulse-response analysis, by conducing a simulaion o examine how much volailiy (or sabilizaion) each rule provides, condiional on each shock. In every case, we run 1 simulaions of lengh 18 each (corresponding o 45 years). The resuls are displayed in Table 6. Table 6. Volailiy of Inflaion, Oupu and Ineres Raes (depending on each rule) Cos Push Shock Demand Shock Seady Sae Seady Sae Rec. Exp. Sae Dependen Rec. Exp. Sae Dependen Inflaion Oupu In. Raes Source: Fund Saff simulaions. Wih cos-push shocks presen, he expansion rule does a beer job in sabilizing inflaion, which is no surprising since i is an inflaion argeing rule, bu he recession rule sabilizes he volailiy of oupu beer. The seady sae rule delivers a somewha inermediae resul. Ineresingly, he sae-dependen rule does no deliver an inermediae resul: inflaion volailiy is he highes (alhough only marginally larger han he recession rule) bu oupu volailiy is he lowes. This resul is of relevance because i will be up o he cenral bank (or sociey as a whole) o decide if i is worhwhile o accep higher inflaion volailiy in order o induce lower oupu volailiy a he imes of recession, or no. Also, he sae-dependen rule also delivers he leas volailiy of he nominal ineres rae under cos-push shocks, which sugges ha policy announcemens and he managemen of expecaions have as much powerful impac as adjusing he value of he policy ineres rae. On he oher hand, under demand shocks, he ranking of rules is clearer. The sric inflaion argeing expansion rule is he one ha delivers he smalles variabiliy in all variables. The seady sae rule does a good job, while he swiching rule and he recession rule perform, in his order, much worse. This resul is no surprising given ha, in his class of models, he opimal rule under demand shock can indeed sabilize boh inflaion and oupu perfecly, and he rade-off beween inflaion and oupu comes from cos-push shocks. 2 2 See Clarida, Galí, and Gerler (1999).

25 To conclude his secion, we would like o menion ha we repeaed he previous exercises assuming a purely forward looking model ( b =, γ b = ), wih only habi formaion bu pure forward looking inflaion, ( b =. 5, γ b = ) and wih only backward looking behavior in he inflaion equaion ( b =., γ b =.5), finding very similar resuls from a qualiaive poin of view. VI. CONCLUDING REMARKS Much debae has been devoed recenly o he suiabiliy of he Taylor rule in characerizing he behavior of he Federal Reserve, especially in abnormal imes. This paper has presened evidence ha he coefficiens of he Taylor rule do, indeed, change boh wih ime and wih economic condiions. In paricular, during expansions he Fed operaes as a srong inflaion argeer wih a high degree of policy ineria, while during recessions i arges oupu growh and moves faser. We have relaed he swiching regime model wih recen moneary policy hisory. We found ha during he Burns-Miller period, he degree of response of he nominal ineres rae o inflaion was very low, and he degree of ineres rae ineria was much higher han wih respec o he 196s or he pos-1983 period. The second par of he paper presened some simulaion resuls, which, in he conex of he model, poin a sric inflaion argeing as he bes policy rule. When he economy is experiencing a recessionary supply shock, however, i migh be worhwhile adoping a swiching regime rule, as i is he opion ha minimizes he variabiliy of oupu in periods of recession; an oucome ha he cenral bank migh wan o consider, even a he cos of higher inflaion variabiliy. Inroducing nonlinear elemens ino he model seems o be worh pursuing in fuure research. We have assessed he behavior of nonlinear rules in he conex of linear models. I is well known ha models of opimal moneary policy wih quadraic loss funcions and linear consrains deliver linear rules as opimal, even under uncerainy. I would be worhwhile exploring he opimaliy of swiching regime rules when eiher he loss funcion is no quadraic or when he consrains are nonlinear. Dolado, María-Dolores, and Ruge-Murcia (23) and Dolado, María-Dolores, and Naveira (24) have sudied opimal nonlinear Taylor rules under asymmeric preferences or nonlinear Phillips curves. The lieraure on consumpion and precauionary savings (Deaon, 1992), suggess a role for nonlinear Euler equaions in oupu.

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