Adaptive Learning and Monetary Policy Design

Transcription

1 Adapive Learning and Moneary Policy Design George W. Evans Universiy of Oregon Seppo Honkapohja Universiy of Helsinki and Bank of Finland Deparmen of Economics, Universiy of Helsinki Discussion Papers No 566:2003 ISBN May6,2003;revised Absrac We review he recen work on ineres rae seing, which emphasizes he desirabiliy of designing policy o ensure sabiliy under privae agen learning. Appropriaely designed expecaions based rules can yield opimal raional expecaions equilibria ha are boh deerminae and sable under learning. Some simple insrumen rules and approximae argeing rules also have hese desirable properies. We ake up various complicaions in implemening opimal policy, including he observabiliy of key variables and he required knowledge of srucural parameers. An addiional issue ha we ake up concerns he implicaions of expecaion shocks no arising from ransiional learning effecs. Presened a he 2002 Journal of Money, Credi and Banking Conference on Recen Developmens in Moneary Macroeconomics. We hank John Duffy and Tom Sargen for heir commens. Financial suppor from US Naional Science Foundaion, he Academy of Finland, Yrjö Jahnsson Foundaion, Bank of Finland and Nokia Group is graefully acknowledged. 1

2 Key words: Commimen, ineres rae seing, adapive learning, sabiliy, deerminacy, expecaions shocks. JEL classificaion: E52, E31, D84. 1 Inroducion The conduc of moneary policy in erms of ineres rae or oher rules has been exensively sudied in recen research, for surveys see e.g. (Clarida, Gali, and Gerler 1999), (Woodford 1999b) and (McCallum 1999). This lieraure gives a cenral role for forecass of fuure inflaion and oupu. Empirical evidence on Germany, Japan and he US since 1979 provided by (Clarida, Gali, and Gerler 1998) suggess ha cenral banks are forward looking in pracice. Bank of England Inflaion Repors, see (Bank of England 2002), discuss privae secor forecass while he June and December Issues of he Monhly Bullein of he European Cenral Bank, see (European Cenral Bank 2002), presen boh inernal macroeconomic projecions and forecass by oher insiuions. However, he precise role of hese forecass in he decision making of hese cenral banks is no revealed. The quesion of wheher moneary policy should be forward looking has been subjec o discussion and debae. 1 Some formulaions of opimal moneary policy specify he ineres rae reacion funcion solely in erms of fundamenals. Insrumen rules such as varians of he Taylor rule are also widely considered, and discussions focus in par on wheher or no he shorerm nominal ineres rae should reac o forecass of inflaion and/or he oupu gap. Theoreical sudies have shown ha, because he privae economy is in any case forward looking, here are wo poenial difficulies ha moneary policy design mus confron. Firs, he proposed ineres rae rules may no perform well when he expecaions of he agens are ou of equilibrium, e.g. as a resul of srucural shifs. The consequences of emporary errors in forecasing, and he resuling correcion mechanisms, have been sudied in recen research using he adap- 1 General discussions someimes pose he quesion of wheher cenral banks should focus aenion on economic fundamenals or follow he markes, which someimes sray far from fundamenals, see e.g. pp of (Blinder 1998). (Hall 1984), p.146, suggesed some ime ago ha he Fed s inernal procedure should place some weigh on reliable ouside forecass. 2

3 ive learning approach. 2 For moneary policy (Evans and Honkapohja 2003a) and (Evans and Honkapohja 2002a) show ha cerain sandard forms of opimal ineres rae seing by he Cenral Bank can lead o insabiliy as economic agens unsuccessfully ry o correc heir forecas funcions over ime, wih he resul ha he economy may no converge o he desired raional expecaions equilibrium (REE). They also propose a new mehod of implemening opimal policy ha always leads o sabiliy under learning. (Bullard and Mira 2002) consider he sabiliy of equilibria when moneary policy is conduced using varians of he Taylor ineres rae rule. Bullard and Mira argue ha moneary policy making should ake ino accoun he learnabiliy consrains, which imply consrains on he parameers of policy behavior. 3 Second, moneary policy rules, including some formulaions for opimal seing of he insrumen and some Taylor rules based on forecass of inflaion and/or oupu gap, can lead o indeerminacy of equilibria, as discussed furher below. Under indeerminacy here are muliple, even coninua of REE and he economy need no sele in he desired REE. The possible res poins can be sudied using sabiliy under learning as a selecion crierion, see e.g. (Honkapohja and Mira 2001a) and (Carlsrom and Fuers 2001). We noe ha indeerminacy is no a criical problem if he fundamenal REE is he only sable equilibrium under learning. Moreover, indeerminacy need no arise if he forward-looking ineres rae rule is carefully designed, see (Bullard and Mira 2002), (Evans and Honkapohja 2003a) and (Evans and Honkapohja 2002a). In his paper we review recen resuls on he performance of ineres rae rules using sabiliy under learning as he key crierion (hough we also provide some discussion of deerminacy of equilibria). We consider boh arge rules ha are opimal eiher under commimen or discreion and also insrumen rules ha do no explicily aim for opimaliy. The laer 2 (Evans and Honkapohja 2001) is an exensive reaise on he analysis of adapive learning and is implicaions in macroeconomics. (Evans and Honkapohja 1999), (Evans and Honkapohja 1995), (Marimon 1997), (Sargen 1993) and (Sargen 1999) provide surveys of he field. 3 Oher papers on moneary policy using he learning approach include (Bullard and Mira 2001), (Mira 2001), (Honkapohja and Mira 2001a), (Honkapohja and Mira 2002), (Carlsrom and Fuers 2001), (Orphanides and Williams 2002), (Ferrero 2002), (Preson 2002b) and (Evans and Ramey 2001). An imporan predecessor o his work is (Howi 1992), hough he did no use he New Keynesian framework. 3

4 include exensions or varians of he rule proposed by (Taylor 1993) as well as approximae argeing rules suggesed recenly by (McCallum and Nelson 2000). Afer reviewing he heoreical resuls we ake up a number of pracical concerns ha can arise when forecas based rules for ineres rae seing are employed. The firs issue is observabiliy of relevan variables. Issues of non-observabiliy can arise in connecion wih privae forecass ha are needed for he desired implemenaion of opimal policy suggesed by (Evans and Honkapohja 2003a) and (Evans and Honkapohja 2002a), as well as wih curren daa as noed in (Bullard and Mira 2002) and (McCallum and Nelson 2000). Second, we inroduce expecaion shocks and sudy wheher hey affec he conclusions derived when hese shocks are absen. The hird concern is knowledge of he srucure of he economy ha is required for implemenaion of opimal ineres rae policies. We exend he analysis of opimal commimen policy under privae agen learning o a siuaion in which he cenral bank esimaes he srucural parameers of he economy and uses hese esimaes in heir rule for seing ineres raes. 2 The Model We use a linearized model ha is sandard in he lieraure, see (Clarida, Gali, and Gerler 1999) for he paricular formulaion used here. The original nonlinear framework is based on a represenaive consumer, and a coninuum of firms producing differeniaed goods under monopolisic compeiion. Firms are subjec o consrains on he frequency of price changes, as originally suggesed by (Calvo 1983). 4 The behavior of he privae secor is summarized by he wo equaions x = ϕ(i E π +1 )+E x +1 + g, (1) which is he IS curve derived from he Euler equaion for consumer opimizaion, and π = λx + βe π +1 + u, (2) which is he price seing rule for he monopolisically compeiive firms. Here x and π denoe he oupu gap and inflaion rae for period, respecively. i is he nominal ineres rae, expressed as he deviaion from 4 See e.g. (Woodford 1996) for he nonlinear model and is linearizaion. 4

5 he seady sae real ineres rae. The deerminaion of i will be discussed below. E x +1 and E π +1 denoe privae secor expecaions of he oupu gap and inflaion nex period. Since our focus is on learning behavior, hese expecaions need no be raional (E wihou denoes raional expecaions). The parameers ϕ and λ are posiive and β is he discoun facor wih 0 <β<1. For breviy we do no discuss deails of he derivaion of equaions (1) and (2). I should be poined ou ha he derivaion is based on individual Euler equaions under (idenical) subjecive expecaions, ogeher wih aggregaion and definiions of he variables, see (Evans and Honkapohja 2002a) for a furher discussion. The Euler equaions for he curren period give he decisions as funcions of he expeced sae nex period. Rules for forecasing he nex period s values of he sae variables are he oher ingredien in he descripion of individual behavior. Given forecass, agens make decisions according o he Euler equaions. 5 The shocks g and u are assumed o be observable and follow ( ) ( ) ( ) g g 1 g = F +, (3) u ũ where F = u 1 ( µ 0 0 ρ 0 < µ < 1, 0 < ρ < 1 and g iid(0,σ 2 g), ũ iid(0,σ 2 u) are independen whie noise. g represens shocks o governmen purchases and/or poenial oupu. u represens any cos push shocks o marginal coss oher han hose enering hrough x. For simpliciy, we assume hroughou he paper ha µ and ρ are known (if no, hey could be esimaed). The recen lieraure on moneary policy has focused on ineres rae seing by he Cenral Bank. 6 One approach examines insrumen rules 5 This kind of behavior is boundedly raional bu in our view reasonable since agens aemp o mee he margin of opimaliy beween he curren and he nex period. Oher models of bounded raionaliy are possible. Recenly, (Preson 2002b) has proposed a formulaion in which long horizons maer in individual behavior. For furher discussion see (Honkapohja, Mira, and Evans 2002). 6 We follow he common pracice of leaving hidden he governmen budge consrain and he equaion for he evoluion of governmen deb. This is accepable provided fiscal policy appropriaely accommodaes he consequences of moneary policy for he governmen budge consrain. The ineracion of moneary and fiscal policy can be imporan ), 5

6 ha specify i in erms of key macroeconomic variables wihou explici consideraion of policy opimizaion. A prominen example of his ype is he sandard (Taylor 1993) rule, i.e., i = π +0.5(π π)+0.5x, where π is he arge levels of inflaion and he arge level of he oupu gap is zero. (Recall ha i is specified ne of he real ineres rae, which in he sandard Taylor rule is usually se a 2%). More generally Taylor-ype rules are of he form i = χ 0 + χ π π + χ x x. For convenience (and wihou loss of generaliy) we will ake he inflaion arge o be π =0so ha his class of rules akes he form i = χ π π + χ x x where χ π,χ x > 0. (4) Variaions of he Taylor rule replace π and x by lagged values or by forecass of curren or fuure values, e.g. in he former case by i = χ π π 1 + χ x x 1 where χ π,χ x > 0. (5) Alernaively, ineres rae policy can be derived explicily o maximize a policy objecive funcion. This is frequenly aken o be of he quadraic loss form, i.e. E s=0 β s [ (π +s π) 2 + αx 2 +s], (6) where π is he inflaion gap. This ype of opimal policy is ofen called flexible inflaion argeing in he curren lieraure, see e.g. (Svensson 1999) and (Svensson 2001). α is he relaive weigh on he oupu arge and sric inflaion argeing would be he case α =0. The policy maker is assumed o have he same discoun facor β as he privae secor. The lieraure on opimal policy disinguishes beween opimal discreionary policy, in which he policy maker is unable o commi o policies for fuure periods, and opimal policy in which such commimen is possible. Wihou commimen policy is reopimized each period and reduces o a sequence of saic problems in which he Cenral Bank aims o minimize (π π) 2 + αx 2 subjec o (2). This leads o he firs-order condiion for he sabiliy of equilibria under learning, see (Evans and Honkapohja 2002b) and (McCallum 2002). 6

7 λ(π π) +αx =0. Again, for convenience and wihou loss of generaliy we se π =0so ha he opimaliy condiion is λπ + αx =0. (7) Under commimen he policy maker can do beer because of he effec on privae expecaions. Solving he problem of minimizing (6) subjec o (2) holding in every period, and assuming RE, leads o a series of firs order condiions for he opimal dynamic policy. This policy exhibis ime inconsisency, in he sense ha policy makers would have an incenive o deviae from he policy in he fuure, bu performs beer han discreionary policy. Assuming ha he policy has been iniiaed a some poin in he pas, and again seing π =0, he firs-order condiion specifies λπ + α(x x 1) =0 (8) in every period. 7 Neiher condiion (7) for opimal discreionary policy nor condiion (8) for opimal policy wih commimen is a complee specificaion of moneary policy, since one mus sill look for an i rule (also called a reacion funcion ) ha implemens he policy. I urns ou ha a number of ineres rae rules are consisen wih he model (1)-(2), he opimaliy condiion (7) or (8), and raional expecaions. However, and his poin is fundamenal, some of he ways of implemening opimal moneary policy lead he economy vulnerable o eiher indeerminacy or insabiliy under learning or boh, while oher implemenaions are robus o hese difficulies. The implemenaions of opimal policy ha we will consider can be divided ino fundamenals based and expecaions based rules. The fundamenals based rule depends only on he observable exogenous shocks g and u in he case of discreionary policy, i.e. i = ψ g g + ψ u u. (9) Under policy wih commimen he fundamenals based rule mus also depend on x 1 so ha i = ψ x x 1 + ψ g g + ψ u u, (10) 7 Treaing he policy as having been iniiaed in he pas correspond o he imeless perspecive described by (Woodford 1999a) and (Woodford 1999b). 7

8 where he opimal coefficiens are deermined by he srucural parameers and he policy objecive funcion. The coefficiens ψ i are chosen so ha he effecs of aggregae demand shocks g are neuralized and so ha for inflaion shocks u he opimal balance is sruck beween oupu and inflaion effecs. In (10) he dependence of i on x 1 is opimally chosen o ake advanage of he effecs on expecaions of commimen o a rule. Calculaion of he coefficiens ψ i requires firs calculaing he opimal REE and hen insering he soluion ino he IS curve (1) o obain he i rule of he desired form. These seps are summarized in Appendix 1. Expecaions based opimal rules are advocaed in (Evans and Honkapohja 2003a) and (Evans and Honkapohja 2002a). They argue ha fundamenals based opimal rules will ofen be unsable under learning, as discussed below. However, if privae expecaions are observable hen hey can be incorporaed ino he ineres rae policy rule. If his is done appropriaely he REE will be sable under learning and hus opimal policy can be successfully implemened. Opimal expecaions based rules ake he form under discreion or i = δ π E π +1 + δ x E x +1 + δ g g + δ u u, (11) i = δ L x 1 + δ π E π +1 + δ x E x +1 + δ g g + δ u u (12) under commimen. The specific coefficiens will be derived below. The essence of hese rules is ha hey do no assume raional expecaions on he par of privae agens, bu are designed o feed back on privae expecaions in such a way ha hey generae convergence o he opimal REE under learning. (If expecaions are raional, hese rules deliver he opimal REE.) Finally, we will also examine he ypes of rule inroduced by (McCallum and Nelson 2000), which aim o approximae opimal policy using an ineres rae rule based on x and π. (McCallum and Nelson 2000) consider insrumen rules of he form i = π + θ[π +(α/λ)(x x 1)], (13) where θ>0. From now on we will call his rule simply he approximae argeing rule. For reasons discussed below, hey also examine a forward looking version of his rule. Given an ineres rae rule we can obain he reduced from of he model and sudy is properies under raional expecaions. The paricular properies in which we are ineresed are deerminacy (uniqueness) of he RE 8

9 soluion and he sabiliy under learning of he REE of ineres. We now urn o hese issues. 3 Deerminacy and Sabiliy under Learning Consider he sysem given by (1), (2), (3) and one of he i policy rules (4), (5), (9), (10), (11), (12) or (13). Defining ( ) ( ) x g y = and v = π he reduced form can be wrien as u y = ME y +1 + Ny 1 + Pv (14) for appropriae marices M, N and P. In he case of rules (4), (9), and (11) we have N =0and hus he simpler sysem y = ME y +1 + Pv. (15) The firs issue of concern is wheher under raional expecaions he sysem possesses a unique saionary REE, in which case he model is said o be deerminae. If insead he model is indeerminae, so ha muliple saionary soluions exis, hese will include undesirable sunspo soluions, i.e. REE depending on exraneous random variables ha influence he economy solely hrough he expecaions of he agens. The possibiliy of ineres rae rules leading o indeerminacy was demonsraed in (Bernanke and Woodford 1997), (Woodford 1999b) and (Svensson and Woodford 1999), and his issue was furher invesigaed in (Bullard and Mira 2002), (Evans and Honkapohja 2003a) and (Evans and Honkapohja 2002a). The second issue concerns sabiliy under adapive learning. If privae agens follow an adapive learning rule, such as leas squares, will he RE soluion of ineres be sable, i.e. reached asympoically by he learning process? If no, he REE is unlikely o be reached because he specified policy is poenially desabilizing. This is he focus of he papers by (Bullard and Mira 2002), (Bullard and Mira 2001), (Evans and Honkapohja 2003a), (Evans and Honkapohja 2002a) and ohers. 9

10 3.1 General Mehodology Consider firs he reduced form (15) under RE. I is well known ha he condiion for deerminacy is ha boh eigenvalues of he 2 2 marix M lie ouside he uni circle. In he deerminae case he unique saionary soluion will be of he minimal sae variable (or MSV) form y = cv, where c is a 2 2 marix ha is easily compued. If insead one or boh roos lie inside he uni circle hen he model is indeerminae. There will sill be a soluion of he MSV form, bu here will also be oher saionary soluions. Nex consider he sysem under learning. Suppose ha agens believe ha he soluion is of he form y = a + cv, (16) bu ha he 2 1 vecor a and he 2 2 marix c are no known bu insead are esimaed by he privae agens. (16) is called he Perceived Law of Moion or PLM of he agens. Noe ha we now include an inercep vecor because, alhough for heoreical simpliciy we have ranslaed all variables o have zero means, in pracice agens will need o esimae inercep as well as slope parameers. Wih his PLM and parameer esimaes (a, c) agens would form expecaions as E y +1 = a + cf v, where eiher F is known or is also esimaed. Insering hese expecaions ino (15) and solving for y we ge he implied Acual Law of Moion or ALM, i.e. he law ha y would follow for a fixed PLM (a, c). Thisisgiven by y = Ma+(P + McF)v. We have hus obained an associaed mapping from PLM o ALM given by T (a, c) =(Ma,P + McF), and (0, c) is a fixed poin of his map. Under real ime learning privae agens have esimaes (a,c ) a ime, which hey use o form expecaions E y +1 = a + c Fv (assuming for 10

11 convenience ha F is known), and y is generaed according o (15). Then a he beginning of +1 agens use he las daa poin o updae heir parameer esimaes o (a +1,c +1 ), e.g. using leas squares, and he process coninues. (Secion 6 gives deails on he form of he leas squares algorihms.) The quesion is wheher over ime (a,c ) (0, c). I urns ou ha he answer o his quesion is given by he E-sabiliy principle, which advises us o look a he differenial equaion d (a, c) =T (a, c) (a, c), dτ where τ denoes noional ime. If he REE (0, c) is locally asympoically sable under his differenial equaion hen he REE is sable under real ime learning. Condiions for local sabiliy of his differenial equaion are known as expecaional sabiliy or E-sabiliy condiions. We will also refer o hese sabiliy condiions as he condiions for sabiliy under adapive learning or jus he condiions for sabiliy under learning. 8 For he reduced form (15) i can be shown ha he E-sabiliy condiions are ha (i) he eigenvalues of M have real pars less han one and (ii) all producs of eigenvalues of M imes eigenvalues of F have real pars less han one. I follows ha for his reduced form he condiions for sabiliy under adapive learning are implied by deerminacy bu no vice versa. This is no, however, a general resul. For some reduced forms E-sabiliy is a sricer requiremen han deerminacy and in oher reduced forms neiher condiion implies he oher. 9 Consider nex he reduced form (14). Sandard echniques are available o deermine wheher he model is deerminae. The procedure is o rewrie he model in firs-order form and compare he number of non-predeermined variables wih he number of roos of he forward looking marix ha lie inside he uni circle. In he deerminae case he unique saionary soluion akes he MSV form y = a + by 1 + cv, (17) 8 (Evans and Honkapohja 2001) describes he conceps and mehods for he sudy of adapive learning. The educive approach o learning, in which agens use menal reasoning, is also someimes used, see (Guesnerie 2002) for a review. The connecions beween sabiliy under adapive and educive learning are discussed in (Evans 2001). 9 If he model is indeerminae, so ha sunspo soluions exis, hen one can also ask wheher he sunspo soluions are sable under learning. For a general discussion see (Evans and Honkapohja 2001) and for an analysis of he model a hand see (Honkapohja and Mira 2001a) and (Carlsrom and Fuers 2001). 11

12 for appropriae values (a, b, c) =(0, b, c). (In he indeerminae case here are muliple soluions of his form, as well as non-msv REE). To examine sabiliy under learning we rea (17) as he PLM of he agens. Under real ime learning agens esimae he coefficiens a, b, c of (17). This is a vecor auoregression (VAR) wih exogenous variables v. The esimaes (a,b,c ) are updaed a each poin in ime by recursive leas squares. Once again i can be shown ha he E-sabiliy principle gives he condiions for local convergence of real ime learning. For E-sabiliy we compue he mapping from he PLM o he ALM as follows. The expecaions corresponding o (17) are given by E y +1 = a + b(a + by 1 + cv )+cf v, (18) where we are reaing he informaion se available o he agens, when forming expecaions, as including v and y 1 bu no y. (Alernaive informaion assumpions are sraighforward o consider). This leads o he mapping from PLM o ALM given by T (a, b, c) = ( M(I + b)a, Mb 2 + N,M(bc + cf )+P ), (19) E-sabiliy is again deermined by he differenial equaion d (a, b, c) =T (a, b, c) (a, b, c), (20) dτ and he E-sabiliy condiions govern sabiliy under leas squares learning. For furher deails see Chaper 10 of (Evans and Honkapohja 2001) and (Bullard and Mira 2002), (Bullard and Mira 2001), (Evans and Honkapohja 2003a) and (Evans and Honkapohja 2002a). 3.2 Resuls for Moneary Policy We now describe he deerminacy and sabiliy resuls for he ineres rae rules described in Secion 2. (Bullard and Mira 2002) consider Taylor ype rules and find ha he resuls are sensiive o wheher he i rule condiions on curren, lagged or expeced fuure oupu and inflaion. In addiion o assuming ha χ π,χ x 0, hey assume ha he serial correlaion parameers in F are nonnegaive. For he rule (4) he resuls are paricularly 12

13 sraighforward and naural. (Bullard and Mira 2002) show ha he model is deerminae and sable under learning if and only if 10 (using our noaion) λ(χ π 1) + (1 β)χ x > 0. In paricular, if policy obeys he Taylor principle ha χ π > 1, soha nominal ineres raes respond a leas one for one wih inflaion, hen deerminacy and sabiliy are guaraneed. If lagged or forward looking Taylor rules are used he siuaion is more complicaed. Full analyical resuls are no available, bu (Bullard and Mira 2002) invesigae he issues numerically using a calibraed version of he model. Under (5) hey find ha for χ π > 1 and χ x > 0 sufficienly small he policy leads o an REE ha is deerminae and sable under learning. For χ π > 1 bu χ x oo large he sysem is explosive. For χ π < 1 he possibiliies include regions of χ π,χ x space ha are deerminae bu unsable. (Bullard and Mira 2002) also look a forward looking versions of he Taylor rule, aking he form i = χ π E π +1 + χ x E x +1 where χ π,χ x > 0, (21) where we can inerpre E π +1 and E x +1 as idenical one sep ahead forecass, based on leas squares updaing, used by boh privae agens and policy makers. Again we find ha for χ π > 1 and χ x > 0 sufficienly small he policy leads o an REE ha is deerminae and sable under learning. Now for χ π > 1 and χ x large he sysem is indeerminae, ye he MSV soluion is sable under learning (while for χ π < 1 here are regions in which he sysem is indeerminae bu he MSV soluion is no sable). 11 The (Bullard and Mira 2002) resuls emphasize he imporance of he Taylor principle in obaining sable and deerminae ineres rae rules. A he same ime heir resuls show ha sabiliy under learning mus no be aken for graned, even when he sysem is deerminae so ha a unique saionary soluion exiss. The parameers of he policy rule χ π,χ x mus be appropriaely seleced by he policy maker when an insrumen rule describes policy. Sabiliy under learning provides a consrain for his choice. 10 Throughou we will assume ha we are no exacly on he border of he regions of deerminacy or sabiliy. 11 For ineres rae rule (21) here exis E-sable sunspo equilibria if λ(χ π 1) + (1 β)χ x > 0 and λ(χ π 1) + (1 + β)χ x > 2ϕ 1 (1 + β), see (Honkapohja and Mira 2001a). Thus policy under (21) should no be oo aggressive. 13

14 As oulined above, in (Evans and Honkapohja 2003a) and (Evans and Honkapohja 2002a) we focus on opimal moneary policy and obain sriking negaive resuls for fundamenals based policy rules (9), (10). Under opimal discreionary policy, wih i rule (9), he sysem is invariably unsable under privae agen learning (he sysem is also invariably indeerminae in his case). The basic inuiion for his resul is ha, say, upward misakes in E π +1 lead o higher π, boh direcly and indirecly hrough lower ex ane real ineres raes, which under learning ses off a cumulaive movemen away from REE. One migh hope ha he feedback from x 1 under (10), he fundamenals based i rule wih commimen, would sabilize he economy. However, we show ha wih his policy rule, as well, he economy is invariably unsable under learning. This is he case even hough wih his rule here are regions in which he opimal REE is deerminae. 12 The insabiliy of he fundamenals based rules, designed afer all o obain opimal policy, is deeply worrying and serves as a srong warning o policy makers no o auomaically assume ha raional expecaions will be aained. I is necessary o examine explicily he robusness of conemplaed policy rules o privae agen learning. In (Evans and Honkapohja 2003a) and (Evans and Honkapohja 2002a) we show how he problems of insabiliy and indeerminacy can be overcome if privae agens expecaions are observable, so ha ineres rae rules can be in par condiioned on hese expecaions. We now look for appropriae rules of he form (11), (12). We focus here on he case in which policy makers can operae under commimen. The desired rule is obained by combining he IS curve (1), he price seing equaion (2) and he firs-order opimaliy condiion (8), reaing he privae expecaions as given. Eliminaing x and π from hese equaions, bu no imposing he raional expecaions assumpion, leads o an ineres 12 In he case of policy wih commimen he learning sabiliy resuls are sensiive o he deailed informaion assumpions. Wih PLM (17) if agens can make forecass condiional also on y hen under he fundamenals based rule here are boh regions of sabiliy and insabiliy, depending on he srucural parameers. 14

15 rae equaion of he form (12) wih coefficiens δ L = δ π = 1+ α ϕ(α + λ 2 ), λβ ϕ(α + λ 2 ),δ x = ϕ 1, λ δ g = ϕ 1,δ u = ϕ(α + λ 2 ). Under opimal discreionary policy (7) is used insead and he coefficiens are idenical excep ha δ L =0. Under his expecaions based opimal rule we obain equally sriking posiive resuls. For all possible srucural parameer values he sysem is deerminae and he opimal REE is sable under privae agen learning. The key o he sabiliy resuls is he feedback from expecaions o ineres raes, so ha deviaions from raional expecaions are offse by policy and in such a way ha under learning privae agens are guided over ime o form expecaions consisen wih he opimal REE. Our expecaions based rule obeys a form of he Taylor principle since δ π > 1. Noe ha our opimal policy rule condiions on boh privae expecaions and observable exogenous shocks, as well as lagged oupu. 13 We also remark ha i is imporan for he cenral bank o use a correc srucural model of boundedly-raional individual behavior when compuing he opimal expecaions based rule. 14 Nex we consider he approximae argeing rules of (McCallum and Nelson 2000). In (Evans and Honkapohja 2002a) we numerically invesigae he rule (13) and find ha i is also invariably deerminae and sable under learning. Since i can be shown ha for large θ>0 he resuling REE will be close o he opimal REE, his also provides an aracive policy rule. As (McCallum and Nelson 2000) poin ou, a poenial difficuly wih his rule is ha i requires conemporaneous observaions of aggregae oupu and inflaion when seing he ineres rae. They herefore consider alernaive versions of heir approximae argeing rule and in paricular recommend a forward looking version, which is discussed below. 13 In he conex of opimal discreionary policy (Ferrero 2002) akes up he addiional issue of he speed of convergence under leas squares learning. There is a coninuum of expecaions based rules of he form (11) ha are consisen wih he opimal discreionary REE bu hese can have differen speeds of convergence. 14 See (Preson 2002a) for he formulaion of he expecaions-based rule when longhorizon forecass of privae agens influence heir behavior. 15

16 Finally, we remark ha our discussion has been limied o formulaions of moneary policy in erms of ineres rae rules. Alernaively, policy could be formulaed as a money supply rule, a prominen example of which is he Friedman proposal for k percen money growh. (Evans and Honkapohja 2003b) show ha Friedman s rule always delivers deerminacy and E-sabiliy in he sandard New Keynesian model. However, i does no perform well in erms of he policy objecive funcion. A more sysemaic sudy of he performance of money supply rules remains o be done. 4 Operaionaliy Many of he i rules discussed above have he poenial difficuly ha hey may no be operaional, as discussed in (McCallum 1999). For example, (McCallum and Nelson 2000) noe ha i may be unrealisic o assume ha policy makers can condiion policy on curren x and π. Similarly, one could quesion wheher accurae observaions on privae expecaions are available. We consider hese poins in he reverse order. 4.1 Observabiliy of Privae Expecaions Our recommended expecaions based rule requires observaions of curren privae expecaions of fuure variables. Survey daa on privae forecass of fuure inflaion and various measures of fuure oupu do exis bu here are concerns abou he accuracy of his daa. If observaions of expecaions are subjec o a whie noise measuremen error hen our sabiliy and deerminacy resuls are unaffeced. Furhermore, if measuremen errors are small hen he policy will be close o opimal. However, if measuremen errors are large hen his will lead o a subsanial deerioraion in performance. In his case one migh consider subsiuing a proxy for such observaions. Since we are assuming ha agens forecas by running VARs, he mos naural proxy is for he Cenral Bank o esimae corresponding VARs and use hese in (12). Clearly, if his precisely describes agens forecass hen i is equivalen o observing hese expecaions. However, we can consider less exreme cases in which agens and he Cenral Bank begin wih differen iniial esimaes and/or use daa ses wih differen iniial daes. For he case of opimal discreionary policy and forward based insrumen rules his issue was analyzed in (Honkapohja and Mira 2002). We here show 16

17 ha using VAR proxies can also achieve convergence o he opimal REE wih commimen. Before proceeding wih his analysis we noe ha in he case in which privae expecaions are observed and incorporaed ino our expecaions based rule (12), he reduced form is given by (14) wih ( 0 λβ M = 0 α+λ 2 αβ α+λ 2 ),N= ( α α+λ 2 0 αλ α+λ 2 0 ) and P = ( 0 λ α+λ 2 0 α α+λ 2 ), (22) and ha (Evans and Honkapohja 2002a) show ha his reduced form leads o sabiliy and deerminacy. When he privae agens and he Cenral Bank are separaely esimaing and forecasing using VARs, we mus disinguish beween heir expecaions. An exended E-sabiliy analysis can give he condiions for convergence of learning, as shown in (Honkapohja and Mira 2001b). In he curren conex his is done as follows. The equaions (1)-(2) depend on privae forecass E P π +1 and E P x +1 while he ineres rae rule (12) depends on Cenral Bank forecass E CB he reduced form ( ) x = π or in marix form π +1 and E CB ( 1 ϕ λ ( β + λϕ 1 ϕ λβ α+λ 2 λ λ(ϕ + λβ ( α α+λ 2 0 αλ α+λ 2 0 x +1. Combining hese equaions leads o )( ) E P x +1 E P π +1 α+λ 2 ) )( x 1 π 1 + ) ( E CB x +1 ) + E CB π +1 ( 0 λ α+λ λ2 α+λ 2 y = M P E P y +1 + M CB E CB y +1 + Ny 1 + Pv, ) + (23) ) ( ) g, u where y =(x,π ) and v =(g,u ). The PLMs and forecass of privae agens and he Cenral Bank ake he form y = a j + b j y 1 + c j v, and E j y +1 = (I + b j )a j + b 2 jy 1 +(b j c j + c j F )v where j = P, CB. I is easily verified ha he implied ALM is of he form y = a + b y 1 + c v, 17

18 wih he associaed map from he PLMs o he ALM } a P a a M P (I + b P )a P + M CB (I + b CB )a CB CB } b P b b M P b 2 P + M CB b 2 CB + N CB } c P c c M P (b P c P + c P F )+M CB (b CB c CB + c CB F )+P. CB Because he P and CB parameers are mapped ino he same ALM parameers, i can be shown ha he E-sabiliy condiions are idenical o hose ha obain if forecass are idenical in he model, i.e. in which he coefficien marix on he expecaions E y +1 is M P + M CB. This is idenical o he earlier reduced form under he ineres rae rule (12). Hence we have sabiliy here as well Non-Observabiliy of Curren Daa As poined ou by (McCallum and Nelson 2000), a difficuly wih he approximae argeing rule (13) is ha i presupposes ha he policy maker can observe curren oupu gap and inflaion when seing he ineres rae. (McCallum and Nelson 2000) recommend use of forward looking versions of approximae argeing rules. In his case he policy maker adjuss he curren ineres rae in response o he discrepancy from he opimaliy condiion (8) anicipaed for he nex period, i.e. i = E CB π +1 + θ[e CB π +1 +(α/λ)(e CB x +1 E CB x )]. (24) This requires specificaion of E CB (.) and we consider he case where boh he Cenral Bank and privae agens use forecass based on idenical esimaed VARs, i.e. E CB ( ) x = π (.) =E P (.) =E (.). The reduced form is hen ( )( 1 αϕθλ 1 ϕθ E x ) +1 λ αϕθ β ϕθλ E π + (25) +1 ( )( αϕθλ 1 0 E ) ( )( ) x 1 0 g αϕθ 0 E +. π λ 1 u 15 The corresponding real-ime learning sabiliy resul is somewha sensiive o he deailed assumpions, e.g. addiional resricions are required for sabiliy if privae agens esimae parameers using sochasic gradien echniques while he Cenral Bank uses leas squares. 18

19 I urns ou ha deerminacy and sabiliy under learning are no longer guaraneed if he rule (24) is employed. Numerical resuls indicae ha sufficienly large values of he policy parameer θ induce boh insabiliy under learning and indeerminacy. We focus here on he upper bound on θ required o avoid insabiliy and compare he performance of (24) in erms of welfare loss agains he welfare loss under opimal policy. We consider he performance of he rule (24) for differen numerical calibraions of he model, Calibraion W: β =0.99, ϕ =(0.157) 1 and λ =0.024; Calibraion CGG: β =0.99, ϕ =1and λ =0.3; Calibraion MN: β =0.99, ϕ =0.164 and λ =0.3; due o (Woodford 1999b), (Clarida, Gali, and Gerler 2000) and (McCallum and Nelson 1999), respecively. We also se α =0.3, ρ =0.4, µ =0.4, σ g =1and σ u =0.5. Herex and π are expressed in percenage unis. The values for σ g and µ are broadly consisen wih hose obained in (McCallum and Nelson 1999). The value for σ u is close o ha used by (McCallum and Nelson 2000). The laer paper uses a wide range of values for ρ and our choice is in he middle of his range. Seriously calibraing hese parameers o he daa would require heir values o be coningen on boh ϕ and λ as well he policy rule. Because we wan o compare he performance of differen policy rules for given srucural parameers, we mus in any case fix he underlying srucure of he model and herefore we simply adop he above values as a benchmark for numerical compuaions. For he differen calibraions he value of he welfare loss from he rule (24), wih θ se opimally under he E-sabiliy consrain 16, and for he opimal REE are as follows: Calibraion W AT W EB W CGG MN For he AT rule he opimal choice of θ is on he edge of he E-sabiliy consrain. However, we remark ha his choice is somewha risky since a small error in he model parameer values could lead o insabiliy. Moreover, his choice of θ could lead o slow convergence of real ime learning. 19

20 where AT and EB refer o approximae argeing and opimal expecaions based rules, respecively. (Technical deails for compuing he welfare loss are given in Appendix 2.) These resuls show ha here are someimes subsanial welfare losses from using he E-sable forward looking AT rule ha can be avoided if he EB rule is feasible. 5 Expecaion Shocks In his secion we inroduce anoher ype of observabiliy problem in privae expecaions. I is assumed ha, hough privae agens are learning as before, heir acual expecaions are affeced by addiional shocks of opimism or pessimism. We iniially assume ha he policy maker does no see hese shocks and bases is ineres rae policy only on he componen of expecaions ha comes from learning behavior. For example, he Cenral Bank could be running is own version of leas squares learning o generae he required forecass. We consider boh he expecaions based rule (12) and he forward looking approximae argeing rule (24). We hen conras hese resuls wih hose from he EB rule when privae expecaions are fully observable. We mus derive he modificaions o he reduced form. PLMs of privae agens and he Cenral Bank ake he familiar form y = a + by 1 + cv leading o Ẽy +1 =(I + b)a + b 2 y 1 +(bc + cf )v and Ẽy = a + by 1 + cv. However, he acual forecass of privae agens are now assumed o be E y +1 = Ẽy +1 + ε, where ε =(ε x,,ε π, ) is a shock o privae expecaions ha is for simpliciy assumed o be iid. The forecass of he Cenral Bank are E CB y +1 = Ẽy +1 + κε, where κ =0or 1, depending on wheher he Cenral Bank is able o obain precise informaion on he expecaion shocks. 20

21 For he case of he expecaions based rule (12) he modified reduced form is ( ) ( ) ( ) x 0 λβ α+λ = 2 Ẽ x +1 αβ + π 0 Ẽ α+λ 2 π +1 ( α )( ) ( )( ) 0 α+λ 2 x 1 0 λ α+λ αλ + 2 g α (26) 0 π α+λ u α+λ 2 ( ) 1 κ ϕ(1 κ) κλβ ( ) α+λ + 2 εx, λ(1 κ) β + λϕ(1 κ) κλ2 β. ε α+λ 2 π, For he approximae argeing rule (24) he reduced form (25) changes o ( x π ) = ( )( ) 1 αϕθλ 1 ϕθ Ẽ x +1 + λ αϕθ β ϕθλ Ẽ π +1 ( )( ) ( )( ) αϕθλ 1 0 Ẽ x 1 0 g + αϕθ 0 Ẽ π λ 1 u ( )( ) 1 ϕ εx, +. λ β + λϕ ε π, (27) We noe ha E-sabiliy and deerminacy properies of he models are no affeced by he expecaion shocks. From he reduced forms (26) and (27) we nex calculae welfare losses due o unobserved expecaions shocks. We assume ha he sandard deviaion of he expecaion shocks in oupu gap and inflaion are 0.1 and 0.05, respecively. The welfare losses for (12) and (24) wih he differen calibraions are Calibraion W AT W EB W CGG MN Comparing hese resuls o he preceding secion i is seen ha expecaion shocks resul in an increase in he losses under eiher rule. The compued 21

22 losses sugges ha he expecaions based opimal rule seems o perform beer han he forward looking approximae argeing rule. However, he comparison could go he oher way if here are big shocks o inflaion expecaions. We now increase he sandard deviaion for inflaion expecaion shocks firs o 0.1 and hen o 0.2. In hese cases he welfare losses are Calibraion W AT W EB W CGG MN when he sandard deviaion for he inflaion shock is 0.1, and Calibraion W AT W EB W CGG MN when he sandard deviaion for he inflaion shock is 0.2. I is seen ha (24) can deliver a smaller loss han (12) if he shocks o inflaion expecaions are sufficienly high; see he resuls for he W calibraion. 17 Finally, we noe ha if he Cenral Bank has accurae informaion on he expecaion shocks hen he welfare loss from he expecaions based rule (12) is dramaically improved. We illusrae his for he las case in which he sandard deviaion of he oupu expecaion shock is 0.1 and he sandard deviaion of he inflaion expecaion shock is 0.2. Formally, seing κ =1 we obain 17 For he oher calibraions he same reserval happens a higher values for he sandard deviaion of he inflaion expecaions shock. 22

23 Calibraion W EB W CGG MN These welfare losses remain somewha higher han he minimum losses wihou he expecaion shocks, which were, respecively, 0.755, and for he hree calibraions. Though he rule wih κ =1can neuralize fully he shocks o oupu gap expecaions (see (26)), his is no he case for he shocks o inflaion expecaions. These resuls noneheless illusrae he value of policy being able o condiion on accurae observaions of privae expecaions. 6 Esimaion of Srucural Parameers Implemenaion of our expecaions based opimal rule, as well as he approximae argeing rule, requires knowledge of srucural parameers. A quesion of considerable ineres is wheher policy makers can obain consisen esimaes of λ and ϕ if privae agens are learning. We ake up his issue in he conex of he expecaions based rule. In (Evans and Honkapohja 2003a) we showed ha consisen esimaion of he srucural parameers was possible in he case of opimal discreionary policy. Here we show how o carry ou his procedure in he conex of policy wih commimen and we do so in a seing ha requires insrumenal variable esimaion. Firs, we exend he model o allow for unobserved shocks. The IS and Phillips curves now ake he form x = ϕ(i E π +1 )+E x +1 + g + e x, (28) π = λx + βe π +1 + u + e π,, (29) where now x,π,e x, and e π, are no observable a ime. g,u are observable a and x,π are observed wih a lag. I is assumed ha he unobserved shocks (e x,,e π, ) are bivariae whie noise bu we will allow for he possibiliy ha he componens are conemporaneously correlaed. (g,u ) and (e x,,e π, ) are exogenous and are assumed o be muually independen. Privae expecaions are assumed o be observable and o be governed by 23

24 leas squares learning as above. We assume ha β and α are known, bu he key srucural parameers ϕ and λ mus be esimaed by he policy maker. The opimal REE now akes he form y = by 1 + cv + de, where e =(e x,,e π, ). Privae agen learning is as before, wih agens using leas squares o esimae he parameers a, b and c of he PLM (17). Thus, a ime agens use he PLM y = a + b y 1 + c v + η o forecas y +1 using he forecas funcion (18) wih (a, b, c) replaced by (a,b,c ). To sudy he sysem under real ime learning we express he leas squares esimaion in recursive form as follows. Define he marix of parameers ξ =(a, b, c) and he vecor of sae variables z =(1,y 1,v ). The recursive leas squares algorihm is 18 ξ = ξ R 1z 1(y 1 ξ z 1 1) R = R (z 1z 1 R 1). The R equaion updaes esimaes of he marix of second momens of he regressors z. The parameers ξ are updaed using his marix and he regression errors y 1 ξ z 1 1. Noehaξ is esimaed using daa hrough ime 1, which is he sandard assumpion in he lieraure. To esimae he srucural parameers he Cenral Bank consrucs he variables w x, = x E x +1 g w π, = π βe π +1 u r = i E π +1. Under he expecaions based rule (12), in he REE i depends on g,u, x 1, E x +1 and E π +1. Since E x +1 and E π +1 in urn depend on g, u, and x 1, i follows ha r depends only on g,u,andx 1. From(1)i 18 The recursive formulaions for he parameer esimaes vary slighly from leas squares since we have inroduced an addiional lag in he equaions for he second momens. This is analyically convenien. The same resuls apply if R 1 is replaced by R in he firs equaion, see Chaper 10 of (Evans and Honkapohja 2001). 24

25 also follows ha x depends only on g,u,x 1 and e x,.ife x, and e π, were known o be uncorrelaed, consisen esimaes of ϕ and λ could be obained by leas squares regressions of w x, on r and w π, on x, respecively. Thus he policy maker would esimae w x, = ϕr + e x, and (30) w π, = λx + e π, (31) using recursive leas squares. However, we can allow for he possibiliy ha he componens e x, and e π, are conemporaneously correlaed. We hus proceed as follows. The firs equaion (30) can be esimaed hrough leas squares, since in he REE r and e x, are uncorrelaed. 19 Formally, we can wrie ˆϕ = ˆϕ Rr, 1r 1(w x, 1 ( ˆϕ 1)r 1) (32) R r, = R r, (r 2 1 R r, 1). The second equaion (31) mus be esimaed by recursive insrumenal variables, since x and e π, are correlaed in he REE. The naural insrumen here is x 1. The recursive algorihm akes he form ˆλ IV = IV ˆλ (Rx, 1) IV 1 IV x 2(w π, 1 ˆλ x 1 1) (33) R IV = R IV x, x, (x 2x 1 R IV x, 1). The Cenral Bank is assumed o conduc moneary policy using he opimal expecaions based rule (12) wih esimaed values for he srucural parameers, i.e. i = δ L, x 1 + δ π, E π +1 + δ x, E x +1 + δ g, g + δ u, u (34) where α δ L, =, ˆϕ (α +(ˆλ IV ) 2 ) δ π, = 1+ δ g, = ˆϕ 1,δ u, = ˆλ IV β ˆϕ (α +(ˆλ IV ) 2 ),δ x, =ˆϕ 1, ˆλ IV ˆϕ (α +(ˆλ IV ) 2 ) 19 Noe ha his holds only if agens do no use conemporaneous values of x and π in forming expecaions E y +1. If insead hese conemporaneous values are used by privae agens, hen ϕ as well as λ would also have o be esimaed using insrumenal variables.. 25

26 The key resul is ha he economy converges o he opimal REE when he Cenral Bank uses he opimal expecaions based rule (34) and boh privae agens and policy maker learn using he specified algorihms. We remark ha convergence o equilibrium is local in he sense ha he iniial parameer esimaes can be chosen freely only wihin a neighborhood of he RE parameer values. Appendix 3 oulines he proof of his resul. As noed above, his resul is robus o he assumpion ha agens use conemporaneous values of endogenous variables in he forecass. In his case he equaion (30) also mus be esimaed using recursive insrumenal variables. Again x 1 can be used as he insrumen. 7 Conclusions The design of moneary policy needs o ake ino accoun he possibiliy ha he economy may no always be in a full ineremporal, i.e. raional expecaions, equilibrium. If economic agens updae heir forecasing procedures, he resuling process of learning may or may no lead he economy owards an REE. Convergence or non-convergence of he learning process depends criically on he policy rule followed by he Cenral Bank. We have reviewed recen research demonsraing ha sabiliy under learning is a serious concern. In paricular, some recenly proposed policy rules ha depend solely on fundamenals are no conducive o convergence o equilibrium. We recommend insead expecaions based opimal rules, i.e. ineres rae rules depending appropriaely on privae expecaions as well as fundamenals, which have been shown o lead o boh sabiliy under learning and deerminacy of equilibria. We have also noed ha, if non-opimal insrumen rules are used insead, sabiliy under learning imposes addiional consrains on moneary policy rules, and policy design should ake accoun of hese consrains. Implemenaion of ineres rae rules raises several pracical concerns. Firs, quesions of observabiliy can arise. These problems can relae o availabiliy of curren daa and o forecass of privae agens. We have discussed ways for dealing wih hese concerns. Second, he economy may in pracice be subjec o expecaion shocks, in addiion o ransiory shocks associaed wih learning. These will no usually aler he condiions for sabiliy under learning, bu if he shocks are sufficienly large he ranking of differen policy rules can be affeced. However, if privae expecaions are 26

27 fully observable, his provides an addiional moivaion for following our recommended expecaions based opimal rule. Finally, implemenaion of expecaions based or approximae argeing policy rules requires knowledge of key srucural parameers. If he Cenral Bank does no have his knowledge, he problem can be overcome by appropriaely esimaing hese parameers and using he esimaed parameers in he ineres rae rule. In paricular, we have shown ha his procedure is viable for he expecaions based rule, which is locally sable under simulaneous learning by privae agens and policy makers. We hope ha hese rules provide useful guidelines for he design of moneary policy. However, we recognize ha much furher research on hese issues remains o be done. 27

28 Appendix Appendix 1: Derivaion of Opimal REE and he Fundamenals Based i rule Consider he model under raional expecaions. I can be shown ha he aggregae supply curve (2), he equaion for u given by (3), and he opimaliy condiion (7) or (8) specify a unique nonexplosive soluion. Consider he sysem under commimen, i.e. (8). The unique saionary soluion akes he form x = b x x 1 + c x u, (35) π = b π x 1 + c π u, (36) where b x is he soluion b x < 1 o βb 2 (1 + β + λ2 )b x α x +1=0and where b π,c x,c π depend on he srucural parameers. To obain he fundamenals based i rule, he corresponding raional expecaions E x +1 and E π +1 are compued as linear funcions of x 1 and u. Using he RE assumpion, hese expecaion funcions and he above expression for x are subsiued ino he aggregae demand equaion (1) and solved for i. This gives he fundamenals based rule (10), where ψ x = b x [ϕ 1 (b x 1) + b π ], ψ g = ϕ 1, and ψ u = [b π + ϕ 1 (b x + ρ 1)]c x + c π ρ. Deails are given in (Evans and Honkapohja 2002a). In he case of discreion similar seps lead o hese coefficiens wih ψ x = b x = b π =0. Deails are given in (Evans and Honkapohja 2003a). Appendix 2: Welfare Calculaion We calculae he expeced welfare loss of he saionary REE, which is 1/(1 β) imes E(αx 2 + π 2 ). The REE soluion y = by 1 + cv can be wrien as ( ) ( )( ) ( y b cf y 1 c = + v 0 F I 28 v 1 ) ṽ,