Macroeconomic Stability of Interest Rate Taylor Rules with the Heuristics Switching Model

Transcription

1 Macroeconomic Sabili of Ineres Rae Talor Rules wih he Heurisics Swiching Model Alexander Schmi Facul of Economics and Business, Universi of Amserdam Supervised b dr. Tomasz A. Makarewicz Economerics Bsc. Thesis June 28, 26 Absrac This paper sudies he effec of differen monear policies on macroeconomic sabili and he convergence of he macroeconomic variables inflaion and oupu, using he New Kenesian model in combinaion wih he Heurisics Swiching Model. I use wo differen ineres rae Talor rules, argeing inflaion or price-level. The wo ineres rae Talor rules are opimized, upon which he performance of he rules is compared using domain of aracion, speed of convergence and oher crieria. The resuls show ha inflaion argeing has a more smooh convergence as well as a bigger domain of aracion, while price-level argeing has a faser convergence in he long run. Furhermore, he roles of he differen heurisics in macroeconomic sabili are sudied, where I find ha he anchoring-and-adjusing heurisic performs bes in an oscillaing econom, whereas he rend-following heurisics performs bes in an econom wih a smooh convergence. The adapive expecaions heurisic has a significan role as i weakens oscillaions. Suden number: , address: alexander@schmi.nl

2 Conens Inroducion 2 Design of he model 3 2. The New Kenesian model The Heurisics Swiching Model Monear polic and macroeconomic sabili 8 3. Inflaion Targeing rule Price-Level Targeing rule Comparison beween Inflaion Targeing and Price-Level Targeing 6 5 Roles of he Heurisics 8 5. Inflaion Targeing rule Price-Level Targeing rule Conclusions 22 Bibliograph 24 Appendix A Overview of he parameers and variables 26 Appendix B Time series figures 27 B. Inflaion Targeing rule B.2 Price-Level Targeing rule B.3 Excluding Heurisics wih he Inflaion Targeing rule B.4 Excluding Heurisics wih he Price-Level Targeing rule

3 Inroducion Over he pas wo decades, here has been a shif in he focus of research on monear polic. Especiall he 27 financial crisis made polic makers more concerned abou prevening he econom from falling ino a secular sagnaion, a prolonged period wih low inflaion and weak economic growh. This resuled in man cenral banks having lowered heir ineres raes close o zero, also called he zero lower bound (ZLB). Once he ineres rae of he Cenral Bank reaches he ZLB, he canno simulae economic growh an furher, which makes radiional monear polic inefficien. Cenral banks have even developed alernaive policies o simulae sagnaing economies, used when he ineres rae has reaches he ZLB. Expecaions pla a crucial role in macroeconomics, monear polic and fiscal polic. Individual decisions ha are made oda, as well as he monear polic of cenral banks, rel upon expecaions of he fuure sae of economic variables. Subsequenl, he expecaions feed back ino he fuure realizaion of he economic variables. The econom can hus be seen as an expecaion feedback ssem, where macroeconomic agens form heir expecaions of fuure economic variables, which deermine he acual realizaion. Therefore, undersanding he formaion of individual expecaions is of grea imporance for conducing monear polic, and his has widel been sressed, e.g. in Woodford (23, p. 5). Macroeconomics models ofen rel on he simplificaion ha all economic agens have idenical expecaions. Due o his assumpion models are unable o model heerogenei in expecaions, which resuls in an unrealisic view of he agens expecaions, and herefore of he econom (see e.g. Assenza e al., 24b, for a discussion abou heerogeneous expecaions). Anoher widel used assumpion is ha he agens form raional expecaions (RE). Since he seminal works of Muh (96), Lucas (972) and Sargen (973), he RE paradigm has become a mainsream approach in modeling expecaion formaion. The idea of RE relies on wo componens: firs, ha individual expecaion formaion can be modeled b maximizing an objecive funcion under given consrains; and second, ha expecaions do no ssemaicall differ from marke equilibria. The imporan advanage of he RE hpohesis, besides is simplici, is ha he number of parameers in a model are kep o a minimum b imposing srong consrains on individual forecasing behavior. On he oher hand, RE has been criicized for assuming ha agens have a perfec undersanding of he srucure of he economic environmen, which is unrealisic. Research conduced o show evidence agains RE is in abundance, and shows for example ha individual responses o simple economic decisions have ssemaic errors and pschological biases (see e.g. Greher and Plo, 979; Kahneman e al., 99; Tversk and Kahneman, 974; Tversk and Thaler, 99, for discussion). A widel used model for he analsis of monear polic is he New Kenesian (NK) model, which assumes a represenaive raional agen srucure (see e.g. Gali, 28; Woodford, 23). This is called a dnamic sochasic general equilibrium (DSGE) model, used o explain aggregae macroeconomic behavior. This approach however has been criicized for is unrealisic assumpion of represenaive raional agens, which makes i unfi for modeling monear pol-

4 ic. Hence economiss have made several adjusmens o he sandard NK model. The have incorporaed for example bounded raionali for he formaion of expecaions o make i more realisic (see e.g. Evans and Honkapohja, 998, 2; Sargen, 999, for discussion). The main idea of bounded raionali is ha he raionali of economic agens is limied b facors such as non-ransparenc, ime consrains and cogniive inabiliies. A popular approach of bounded raionali is he use of heurisics, which are simple rules of humb, o le individuals make decisions. Anoher exension of he NK model is he inclusion of heerogeneous expecaions (see e.g. Anufriev e al., 23; Cornea e al., 22; Massaro, 23, for discussion), relaxing he assumpion of idenical expecaions. Evidence of heerogenei in inflaion expecaions is found for example b Mankiw e al. (24). Anufriev and Hommes (22) propose a heurisics swiching model (HSM) o describe individual decision making under uncerain. Laboraor experimens have shown ha hese heurisics can beer describe decision making under uncerain han perfecl raional behavior (see e.g. Camerer and Fehr, 26; Kahneman, 23; Tversk and Kahneman, 974, for discussion). 2 The idea of he HSM is as follows. There are various heurisics (which include e.g. adapive and rend-following rules) from which agens can choose o forecas an economic variable. A ever ime period, forecass are made, upon which he realized value of he variable is formed, depending on is average marke forecas. Subsequenl he performances of he heurisics are measured, based on he accurac of heir forecass. Agens will pu higher weigh on heurisics ha perform relaivel beer han he oher heurisics. The HSM hus relaxes he represenaive agen assumpion and he raional expecaions assumpion, which makes i a more realisic approach o sud macroeconomics. In macroeconomic modeling, monear policies are ofen represened b a Talor rule (Talor, 993), which is used o seer inflaion owards a long erm arge. The are a guideline for how cenral banks should change he ineres rae, in response o changes in economic variables such as inflaion or nominal price level. The financial crisis of he US and Europe in 27 and especiall he long period of low inflaion and occasional deflaion in Japan in he mid 99s raised ineres in price-level argeing, as an alernaive o (he a ha ime mos commonl used) inflaion argeing. In his paper I sud he effec of monear polic hrough ineres rae Talor rules on macroeconomic sabili and he convergence of he macroeconomic variables. I specificall invesigae he difference beween he ineres rae Talor rules, which arge inflaion and nominal price level. The model I use is a New Kenesian (NK) model in combinaion wih he heurisics swiching model (HSM). I address he following quesions: How does monear polic hrough ineres rae Talor rules affec he sabili and convergence of macroeconomic variables? This model is buil upon he Adapive Belief Scheme proposed b Brock and Hommes (997), bu incorporaes also a performance measure and asnchronous sraeg updaing. 2 These experimens are ofen conduced o sud he formaion of expecaions, in paricular how he inerac and how he subjecs behavior influences economic variables (see e.g. Assenza e al., 24b; Massaro, 23, for discussion). 2

5 Wha are he opimal parameerizaions for he ineres rae Talor rules wih respecivel inflaion argeing and price-level argeing? Which roles do he differen heurisics pla in macroeconomic sabili and how imporan are he? Following exising lieraure on parameerizaions of he ineres rae rules, i is expeced ha wih inflaion argeing a weak ineres rae Talor rule, which responds less han one-o-one o changes in inflaion, will lead o a desabilizing econom wih oscillaor behavior. On he oher hand, a srong ineres rae Talor rule will sabilize he econom and dampen he oscillaions, following he research of Assenza e al. (24b). This paper makes he following conribuions. Firs, he effeciveness of monear polic hrough ineres rae Talor rules in he convergence of macroeconomic variables is sudied. More precisel, I deermine an opimal parameerizaion for he ineres rae Talor rule wih inflaion argeing ha resuls in he convergence of inflaion and oupu. Subsequenl, he opimal parameerizaion for he ineres rae Talor rule wih price-level argeing is deermined. Secondl, I compare he wo ineres rae rules. The comparison includes suding phase plos, domain of aracion, speed of convergence, oscillaor behavior and overall performance. Thirdl, he roles of he differen heurisics in macroeconomic sabili is sudied. This is done b excluding heurisics and suding he effec on he convergence of he model. The heurisics are hen compared on heir performance of providing macroeconomic sabili. This paper is organized as follows. Secion 2 gives a deailed descripion of he New Kenesian model and he Heurisics Swiching model, as well as he differen heurisics and he ineres rae Talor rules. Secion 3 discusses how monear polic affecs macroeconomic sabili, and proposes opimal parameerizaions for he wo ineres rae Talor rules. Secion 4 compares he wo ineres rae rules on heir abili in reaching macroeconomic sabili, using several crieria. Secion 5 discusses he roles of he differen heurisics in macroeconomic sabili, and discusses wha effec excluding a heurisic has on he model. Finall, secion 6 concludes. 2 Design of he model The model is based on he New Kenesian (NK) framework, in combinaion wih he heurisics swiching model (HSM). The HSM generaes expecaions of he inflaion (+) e and consumpion (c e +), which he NK model uses o generae he realized variables. The ineres rae is generaed b one of wo ineres rae Talor rules, he inflaion argeing rule (IFT), or he price-level argeing rule (PLT). Subsequenl, he HSM generaes new expecaions for he nex ime period, based on he new realized variables. This process resuls in a wodimensional DSGE model, described below (for an overview of he variables and parameers of he model, see Appendix A). 3

6 2. The New Kenesian model The agen based language, where agens forecas he economic variables inflaion + e and consumpion c e +, is modeled hrough he HSM (which is covered in he nex subsecion). Hereafer, he New Kenesian (NK) model generaes he realized values of he economic variables hrough a number of equaions. The NK model consiss of he consumpion (IS) equaion and he Phillips curve, which resuls from he opimizaion of e.g. consumer uili hrough a number of supplemenar equaions, such as budge consrains and producion funcions. These supplemenar equaions are also referred o as micro foundaions. 3 This resuls in he following equaions: Aggregae consumpion: ( ) e /σ +, () c = c e + βr where c denoes consumpion a period, c e + expeced consumpion a period +, R he ineres rae a period, β he discoun facor and σ he consumpion elasici. Realized oupu: (2) = c + ḡ, where denoes oupu a period and ḡ governmen spending (we assume governmen spending o be fixed (g = ḡ), since we invesigae onl monear polic). Phillips curve: (3) = Q ( K(c, e +) ), where and Q( ) = ( ), ˆk(c, +) e = β+( e + e ) + ν αγ (c + ḡ) (+ε)/α + ν γ (c + ḡ)c σ, ˆk(c K(c, +) e, +) e if ˆk(c, +) e > 5, = 5 else, where denoes inflaion a period, + e expeced inflaion a period +, ν demand s elasici of subsiuion, α oupu elasici, γ Roemberg price sickiness and ε labor suppl elasici. 3 See e.g. Massaro (23) for he developmen of a micro-founded framework for monear polic analsis consisen wih heerogeneous expecaions. 4

7 The ineres rae is deermined b one of he wo ineres rae Talor rules, which represen he monear polic of he Cenral Bank. The are given b he following equaions: Inflaion argeing rule: } (4) IFT: R = + max {, R + ψ ( e+ + e ) + ψ, where R denoes he ineres rae a ime, R he sead sae gross ineres rae, ψ he inflaion argeing parameer, he gross inflaion arge, ψ he oupu argeing parameer, e + he expeced oupu a period + and he sead sae oupu. Price level argeing rule for P / P = : (5) PLT: R = + max {, R P+ e + ψ P } + + e P + ψ, P + where P e + = e +P, P + = P, where P denoes he argeed price-level a period, P e + he expeced price-level a period +, P he realized price-level a period, ψ P he nominal price-level argeing parameer and ψ he oupu argeing parameer. The ineres rae rule specified in equaion (4) is he inflaion argeing rule (IFT), a linear varian of he Talor rule used in he paper of Hommes e al. (25). 4 The IFT has parameers ψ and ψ, which respecivel represen he responsiveness of he Cenral Bank o changes in he inflaion and oupu expecaions. The second ineres rae rule in equaion (5), ogeher wih he supplemenar price equaions, form he price-level argeing rule (PLT). The PLT has parameers ψ P and ψ, which respecivel represen he responsiveness o changes in he nominal price-level and oupu expecaions. Besides he HSM, his model is deerminisic, for he sake of simplici (his means here are no producivi shocks). Since governmen spending is assumed o be fixed (g = ḡ), oupu is full dependen on consumpion c. Paramerizaion of he variables is aken from Benhabib e al. (24) and is given in Table. 5 I is possible o show ha wih he given parameerizaion, he model has wo differen sead saes, which are specified in Table 2: A normal (full emplomen) sead sae wih posiive inflaion, and a zero-lower bound (ZLB) sead sae. In he normal sead sae, he inflaion converges o he argeed inflaion, where R = β (he Fisher equaion). In he ZLB sead sae, he inflaion converges o ZLB = β <, where R ZLB =. Oupu has o be found numericall. 4 Hommes e al. (25) use a more aggressive rule, which ses he ineres rae o he ZLB each ime he inflaion drops below a given hreshold, following Evans e al. (28). 5 This paper uses =.5 as inflaion arge insead of =.2 used b Benhabib e al. (24) o creae more space beween he ZLB sead sae and he normal sead sae in order o simplif convergence. 5

8 Table : Paramerizaion of NK Model Parameer Noaion Value Discoun facor β 9 Governmen spending ḡ Oupu elasici α.7 Roemberg price sickiness γ 35 Labor suppl elasici ε Demand s elasici of subsiuion ν 2 Consumpion elasici σ Gross inflaion arge.5 Sead sae gross ineres rae R.(6) Sead sae consumpion c Sead sae oupu 538 Table 2: Sead saes Sead sae Parameer Noaion Value Normal (full emplomen) Gross inflaion arge.5 Sead sae gross ineres rae R.(6) Sead sae consumpion c Sead sae oupu 538 Zero lower bound ZLB gross inflaion ZLB 9 ZLB gross ineres rae R ZLB ZLB consumpion c ZLB ZLB oupu ZLB The Heurisics Swiching Model The forecass are generaed b he heurisics swiching model (HSM), where we suppose he expecaions of he agens are heerogeneous, resuling in he aggregae final expecaion being a weighed sum of he expecaions formed b he heurisics. Each period, he performance of he heurisics is measured, upon which he fracion of he heurisics is updaed using he relaive performances. Subsequenl, he fracions of he heurisics are used in he deerminaion of he aggregae final expecaion in he nex period. I use he model where agens forecas wo variables, inflaion and oupu. Therefore here are wo inpu variables for he HSM, which resuls in he following equaions for he variable x, being inflaion or oupu : Aggregae final expecaion: (6) x e + = H h= n x h,x e h,+, where x e + denoes he aggregae final expecaion of x a ime +, n x h, he fracion of heurisic h in he forecas of x a ime, and x e h,+ he expeced value of x formed b heurisic h. 6

9 Performance measure: (7) U x h, = + x x e h, + ηu x h, 2, where Uh, x denoes he performance of heurisic h in he forecas of x a ime, he expeced value of x formed b heurisic h, and η [, ] he memor of he x e h, agens. Logi discree choice model for updaing he fracions: exp λu (8) n x h, = δn x h, x h, + ( δ) H h= exp λu, h, x where n x h, denoes he fracion of heurisic h in he forecas of x a period, and δ [, ] denoes he updaing parameer, which represens he ineria of he agens in updaing he fracions of he rules. The parameer λ denoes he inensi of choice, which indicaes he sensiivi of agens o differences in heurisics performance. The HSM can use an arbirar se of heurisics. In his paper I use he heurisics described b Anufriev and Hommes (22) o model individual forecasing behavior. Anufriev and Hommes (22) sud learning-o-forecas experimens (LFE s) conduced in he CREED laboraor a he Universi of Amserdam, based on an asse pricing model (see Hommes e al., 25, 28, for a deails). In hese experimens subjecs were asked o forecas asse prices, where he realized asse prices were deermined b he aggregae individual forecass. 6 Anufriev and Hommes (22) show ha raional expecaions is no a good explanaion of individual forecasing and observe srong coordinaion on a common predicion rule. The four rules given in Table 3 were observed in he experimens and esimaed, describing pical individual forecasing behavior. Table 3: Se of heurisics ADA adapive rule x e,+ = 5x +.35x e, WTR weak rend-following rule x e 2,+ = x + (x x 2 ) STR srong rend-following rule x e 3,+ = x +.3(x x 2 ) LAA learning anchoring-and-adjusmen rule x e 4,+ =.5(x av + x ) + (x x 2 ) The adapive rule (ADA) is he firs rule observed b Anufriev and Hommes (22), which he paricipans ofen used in he experimen showing convergence of asse prices. This rule is he opimized weighed average of he observed variable a period, denoed b x and he expecaion of x b ADA a ime, denoed b x e,. The rend-following rules (WTR and STR) were mosl observed b paricipans in he experimen showing permanen or dampened oscillaions. The are purel exrapolaing rules, as he forecas uses he las observaion and adjuss in he direcion of he las observed change 6 See Assenza e al. (24a) for a recen overview of LFE s and oher macroeconomic experimens. 7

10 in he variable. The degree of exrapolaion in he experimens varies highl, wih STR having high exrapolaing coefficien.3, and WTR having small exrapolaing coefficien. The learning anchoring-and-adjusing rule (LAA) is acuall also an exrapolaing rule, namel wih coefficien., bu also incorporaes he sample average, denoed b x av, as anchor. The LAA was mainl observed in he experimen showing permanen oscillaions. The learning parameers from he HSM, λ =, η =.7, δ =, are also aken from Anufriev and Hommes (22), which he obained afer rial and error simulaions. This paper follows heir specificaion o have an empiricall esed model. 3 Monear polic and macroeconomic sabili This secion discusses condiions under which ineres rae Talor rules sabilize or fail o sabilize macroeconomic variables. The parameers of he ineres rae Talor rule deermine he degree of posiive feedback in he New Kenesian model, which is crucial for he dnamics. Posiive feedback means ha he realized aggregae variable increases when he forecass of ha variable increase (see Assenza e al., 24a, for discussion). In he model, his can hus be of major imporance, since oo much posiive feedback can resul in exploding behavior, or in a decrease of he ineres rae o he zero-lower bound (ZLB), such ha he model falls ino a deflaionar spiral, evenuall collapsing he model. Because he model has man variables ha have influence on he dnamics of he model, i is difficul o show heir influence. The parameers o be varied include iniial values, and he polic parameers ψ, ψ for inflaion argeing (IFT) and ψ P, ψ for price-level argeing (PLT). 7 The learning parameers λ, η, δ specified b Anufriev and Hommes (22) can also be varied, bu are lef fixed o preserve he empirical validaion of he model. To examine he effec of hese parameers on he dnamics of he model, we use he following mehods: To sud he sensiivi on he iniial condiions of each ineres rae Talor rule, we look a he domain of aracion of he model. This shows how far from he argeed sead sae he iniial values of inflaion and oupu can lie while sill resuling in a convergence o he argeed sead sae. 8 A bigger domain of aracion is obviousl beer since ha means here are less iniializaions where he model avoids convergence. The effec of he polic parameers on he convergence of he model can also be sudied b using he domain of aracion. This illusraes which parameerizaions resul in a convergence of he model. 8 This mehod will be used o find he opimal parameerizaions for he ineres rae Talor rules. 7 Anufriev and Hommes (22) show ha he iniial fracions of he heurisics n h, are essenial for he convergence of he model, bu in his model he iniial fracions are equall disribued, i.e. n h, = 5 h. Fuure research can include suding he convergence of he model when varing he iniial fracions of he heurisics. 8 Convergence is said o be reached when he Euclidean norm of he aggregaes is less han % of he argeed sead sae (see Table 2 for he sead saes). 8

11 The polic parameers also affec he ampliude of he oscillaions in he model. This effec can be examined b suding he change in he ampliude of he oscillaions in model rajecories, while varing he polic parameers. Throughou his secion and he following secions, I will refer o Appendix B, which includes deailed figures of he model, showing ime series of inflaion and oupu, ime series of he fracion of he heurisics n h, for inflaion and oupu, and phase plos of he model. 3. Inflaion Targeing rule The sandard parameerizaion of he ineres rae Talor rule wih inflaion argeing (IFT) using he heurisics swiching model (HSM) is ψ =.5, ψ =., following Assenza e al. (24b), who show ha an increase of ψ from. o.5 sabilizes he macroeconomic variables in he long run. This parameerizaion is aken as saring poin. To sud he effecs of he iniial condiions on he convergence of he model, we consruc he domain of aracion for IFT, given in Figure. 9 This Figure shows he effec of he iniial values of inflaion, and oupu on he convergence of he model. The whie surface denoes he iniializaions resuling in convergence o he normal sead sae (denoed b he red do), while he black surface denoes no convergence, resuling in eiher a collapsing model or an exploding model. Wih convergence we mean ha he iniializaion aracs inflaion and oupu o he normal sead sae, such ha he Euclidean norm of he aggregaes is less han % of he normal sead sae. Suding Figure, i becomes clear ha he influence of he iniial inflaion is greaer han he influence of he iniial oupu. The Figure shows ha if is oo low he model does no converge, whereas onl has a sligh influence on his. Figure 4 in Appendix B. shows he model iniiaed ouside he domain of aracion. Ineresingl, he model does no converge o he zero-lower bound (ZLB) sead sae, bu insead collapses. This shows ha he ZLB sead sae is no sable, because i riggers he deflaionar rap if he model reaches i, evenuall causing o drop o.5, and o. Due o he srucure of he Phillips curve, hese values are he numerical minimum of he model. When we aler he polic parameers, causing he model o avoid he normal sead sae (ψ =, ψ =.), we can see ha he iniializaion deermines if he model collapses or explodes. If he model is iniiaed below he normal sead sae ( =.2, c =.7), he model will collapse (see Figure 5 in Appendix B.). On he oher hand, if he model is iniiaed above he normal sead sae ( =.6, c =.76), he model will explode (see Figure 6 in Appendix B.). Noe ha he share of he srong rend-following rule (STR) reaches almos % a he peak for boh inflaion and oupu, which explains he explosive behavior. When all agens use he srong rend-following rule, i resuls in an growing rend, 9 To consruc he domain of aracion, he iniial expeced values of he heurisics are se o he iniial values of he model ( e =, e = ). This means he adapive par of he ADA rule is se o he iniial value, he rends of he rend-following rules WTR and STR are equal o zero and he anchor of he LAA rule is se o he iniial value. 9

12 ha evenuall causes he model o explode. To sud he IFT rule wih convergence, we ake =.2, c =.7 as iniializaion for all following compuaions wih he IFT rule Figure : Domain of aracion (iniial values) wih IFT. The whie surface denoes he iniializaions resuling in a convergence o he normal sead sae (denoed b he red do) afer n = 4, while he black surface denoes no convergence. Fixed variables: ψ =.5, ψ =. To sud he effec of he polic parameers of he model, we firs discuss which parameerizaion of he Talor rule resuls in he convergence of he model o he normal sead sae. Figure 2 plos he domain of aracion of he polic parameers ψ and ψ, where he whie surface denoes convergence and he black surface denoes no convergence. The Figure clearl illusraes ha ψ has o be a leas greaer han approximael. in order for he model o converge, whereas ψ has no decisive effec besides narrowing he domain in erms of ψ. Besides he convergence of he model, he parameerizaion also has effec on he speed of convergence. Trial and error showed ha he closer ψ is o., he slower i converges, wih an opimum speed of convergence a ψ =.5, ψ =. (see Figure 7 and Figure 8 in Appendix B. for deails). This difference in speed of convergence can be explained b he fac ha a higher ψ reduces he degree of posiive feedback in he model, resuling in a quicker adjusmen of biased expecaions and hus a quicker convergence. The effec of he polic parameers on he ampliude of he oscillaions can be examined b suding he model rajecories. Figure 3 and Figure 4 respecivel show he effec of ψ and ψ on he rajecor of he model, using he same iniializaions. The Figures show muliple sacked phase plos of he model, each compued wih a differen polic parameer, visible on he z-axis. The dashed green line denoes he iniializaions, where he differen colors are o disinguish he sacked phase plos. Figure 3 shows ha if ψ. he model falls ino he liquidi rap, where drops o.5 and drops o (see Figure 5 in Appendix B.), which

13 ψ ψ Figure 2: Domain of aracion (polic parameers) wih IFT. The whie surface denoes he parameerizaion resuling in a convergence o he normal sead sae (denoed b he red do) afer n = 4, while he black surface denoes no convergence. Fixed variables: =.2, c =.7 is he numerical minimum of he model. Figure 4 shows ha when ψ, he models sars showing cclical oscillaor behavior, wih heav oscillaions around ψ = (see Figure 9 in Appendix B. for deails). Ineresingl, in Figure 9b and 9d i is visible ha he learning anchoring-and-adjusing rule (LAA) dominaes in he forecas of boh inflaion and oupu, and reaches shares of approximael 35% and 4% a he peak. LAA seems o perform bes when he model shows heav oscillaions, in conras o he rend-following rules, which seem o overshoo he rend reversals (see Figure 9a and 9c for deails). This is similar o he resuls of Anufriev and Hommes (22). Despie hese oscillaions, he variables sill converge o he normal sead sae in he long run, bu onl afer approximael 2 periods. Figure 4 also shows ha when ψ is oo high (ψ.5), he model sars o show non-cclical oscillaions (see Figure 2 in Appendix B. for deails). Figure 2e shows ha he oscillaions slow down he convergence of he model, creaing non-cclical oscillaions. Increasing ψ or ψ even more resuls in heavier oscillaions (see Figure 2 in Appendix B. for deails). When we compare Figures 3 and 4, i becomes clear ha ψ has no effec on he oscillaions, bu does have effec on he convergence of he variables, unlike ψ. In order o converge o he normal sead sae, we can hus sa ha ψ has o be greaer han, and in order for he model o have a smooh convergence, avoiding oscillaor behavior, ψ has o be beween.5 and. In he ideal case for he IFT rule, he parameerizaion would hus be ψ =.5, ψ =. (see Figure 8 in Appendix B. for deails), regardless of he iniial values. This resuls in a close o immediae convergence o he normal sead sae (see Figure 8e).

14 .5 ψ Figure 3: Trajecories of and agains ψ wih IFT. The Figure shows muliple sacked phase plos of and wih n = 2, paramerized wih differen ψ. The dashed green line denoes he iniializaion, he oher colors are o disinguish he differen phase plos. Fixed variables: ψ =., =.2, c = ψ Figure 4: Trajecories of and agains ψ wih IFT. The Figure shows muliple sacked phase plos of and wih n = 2, paramerized wih differen ψ. The dashed green line denoes he iniializaion, he oher colors are o disinguish he differen phase plos. Fixed variables: ψ =.5, =.2, c =.7 2

15 3.2 Price-Level Targeing rule To sud he model wih price-level argeing (PLT), we firs look a he dependenc on he iniial condiions. Because here is no behavioral lieraure e on he parameerizaion of he PLT rule, I deermined a parameerizaion ha converges o he normal sead sae b rial and error. This parameerizaion (ψ P = 3., ψ =.7) was hen used o compue he domain of aracion of iniial values. Figure 5 gives he domain of aracion for he PLT rule agains Figure 5: Domain of aracion (iniial values) wih PLT. The whie surface denoes he iniializaions resuling in a convergence o he normal sead sae (denoed b he red do) afer n = 4, while he black surface denoes no convergence. Fixed variables: ψ P = 3., ψ =.7 he iniial inflaion and iniial oupu. The whie surface denoes he iniializaions resuling in convergence o he normal sead sae (denoed b he red do), while he black surface denoes no convergence, resuling in eiher a collapsing model or an exploding model. The Figure clearl shows ha has o be smaller han approximael.9, bu greaer han a value approximael beween 9 and.3, depending on. The iniial oupu has no decisive effec and onl narrows he domain of aracion in erms of iniial inflaion. In order for he model wih PLT o have a convergence, we ake =.8, c =.74 as iniializaion for all following compuaions wih he PLT rule. Figure 6 and 7 respecivel show he influence of he polic parameers ψ P and ψ on he rajecories of and. The Figures show muliple sacked phase plos of he model, each compued wih a differen polic parameer, visible on he z-axis. The differen colors are o disinguish he sacked phase plos. Figure 6 shows ha a ever parameer value of ψ P, he model shows cclical oscillaions, bu if ψ P is lowered o approximael 2.2, he ampliude of hese oscillaions increases, and in he end he model collapses. Figures 22 and 23 in Appendix 3

16 3.5 3 ψp Figure 6: Trajecories of and agains ψ P wih PLT. The Figure shows muliple sacked phase plos of and wih n = 2, paramerized wih differen ψ. The dashed green line denoes he iniializaion, he oher colors are o disinguish he differen phase plos. Fixed variables: ψ = 2., =.8, c = ψ Figure 7: Trajecories of and agains ψ wih PLT. The Figure shows muliple sacked phase plos of and wih n = 2, paramerized wih differen ψ. The dashed green line denoes he iniializaion, he oher colors are o disinguish he differen phase plos. Fixed variables: ψ P = 3., =.8, c =.74 4

17 B.2 respecivel show plos of he model where ψ P = 2. and ψ P = 3., keeping ψ fixed a 2.. If we sud Figure 22, i is ineresing ha he model does no converge o he zero-lower bound (ZLB) sead sae. Insead i his a deflaionar spiral, which causes o drop o.5 and o, which is idenical o wha we saw wih inflaion argeing. Ineresingl, he learning anchoring-and-adjusing rule (LAA) is he mos dominan heurisic and reaches a he peak a share of approximael 8% (see Figure 22b for deails). This suggess ha coordinaion on LAA can cause he model o collapse. If we sud Figure 23, we can see ha increasing ψ P from 2. o 3. prevens he model from collapsing. However, i sill shows oscillaor behavior in combinaion wih periodic amplificaions of he oscillaions, causing he model o never full converge o he normal sead sae (see Figure 23e for deails). Figure 23d shows ha he adapive-expecaions rule (ADA) seems o dominae, however he periodic oscillaions sill cause he share of ADA o flucuae beween 3% and 6%. Figure 7 shows ha heav oscillaions appear when ψ is smaller han approximael.4 and greaer han 2.4, causing he model o collapse. Figures 24, 25 and 26 in Appendix B.2 respecivel show he model paramerized b ψ =.2, ψ = 2. and ψ = 2.5, keeping ψ P = 3. fixed. Figure 7 hus shows ha in order o avoid he model from collapsing, ψ has o be beween approximael.5 and ψ ψ P Figure 8: Domain of aracion (polic parameers) wih PLT. The whie surface denoes he parameerizaion resuling in a convergence o he normal sead sae (denoed b he red do) afer n = 4, while he black surface denoes no convergence. Fixed variables: =.8, c =.74 The effec of he polic parameers can also be sudied b looking a he domain of aracion for he polic parameers. Figure 8 plos he domain of aracion agains he polic parameers ψ P and ψ, where he whie surface denoes convergence o he normal sead sae and he black surface denoes no convergence. I is clearl visible ha when ψ drops 5

18 below approximael.4, he model does no converge. On he oher hand, if ψ is greaer han approximael.9, he model does no converge, bu has poins where here is sill convergence. Targeing oupu oo heavil can hus cause he model o avoid convergence. Seeing ha onl he model wih ψ P = 3., ψ =.7 (see Figure 27 in Appendix B.2 for deails) resuls in a convergence wihou periodic amplificaion of he oscillaions, makes i an opimal parameerizaion. I resuls in he convergence of and afer approximael periods and also resuls in he convergence of he shares of he heurisics. 4 Comparison beween Inflaion Targeing and Price- Level Targeing The choice of he ineres rae rule is imporan for he sabilizaion of economies. This secion herefore examines he differences beween inflaion argeing (IFT) and price-level argeing (PLT), using several crieria (a) Fixed variables: ψ =.5, ψ = (b) Fixed variables: ψ P = 3., ψ =.7 Figure 9: Phase plos of and wih IFT (a) and PLT (b). The figures represen he rajecories of he convergence o he normal sead sae (denoed b he red do) wih he opimized ineres rae Talor rules (n = 4). Looking a Figure 9, which shows he phase plos of boh he inflaion argeing (IFT) and price-level argeing (PLT) rule in is opimal form, is i clear ha he IFT converges more smooh o he normal sead sae han he PLT, in he sense ha i does no show oscillaor behavior. The model wih PLT shows a convergence o he normal sead sae hrough cclical flucuaions, while he model wih IFT has a smooh convergence. This resul is imporan because if an econom is subjec o frequen random shocks, i is desirable o have a smooh convergence o he normal sead sae of he econom. However, here is a rade-off beween smoohness and speed. The model wih PLT does have cclical flucuaions in he iniial periods, bu converges faser o he normal sead sae in he long run han he model wih IFT. Figure illusraes his difference in he speed of convergence. In his 6

19 .6.5 PLT IFT Disance o sead sae Figure : Speed of convergence of IFT (denoed b he blue line) and PLT (denoed b he red line). On he verical axes he Euclidean disance from he model o he normal sead sae, and on he horizonal axes period. Boh models are iniiaed wih =.8, c =.74 Figure we see he developmen of he disance from he locaion of he model o he normal sead sae (measured b he Euclidean norm), where he blue line denoes he model wih IFT, and he red line denoes he model wih PLT. I clearl shows ha he model wih IFT converges faser in he iniial periods, while he model wih PLT proves o be faser in is convergence in he long run. Hence boh ineres rae rules have heir advanages. The choice of he ineres rae rule will evenuall depend on he quani and gravi of random shocks in he econom. I will be desirable o have a smooh convergence when an econom is subjec o man shocks, because IFT has a faser speed of convergence in he iniial periods. On he oher hand, if an econom is subjec o few or weak shocks, PLT can be beer if i is desirable o have a faser convergence in he long run. Anoher imporan difference beween IFT and PLT is he sensiivi in iniial values. As we observed in he previous secion, PLT proved o be more sensiive han IFT in erms of iniial inflaion and iniial oupu. From he perspecive of a Cenral Bank, i is imporan o look a how well he model can deal wih shocks. Shocks can shif he model from is sead sae, risking falling ino a deflaionar spiral if he new iniial sae is below he zero-lower bound, causing he model o collapse. A bigger domain of aracion is hus preferable, since ha means here will be less iniializaions ha cause he model o avoid convergence. Figure gives he domain of aracion of respecivel he IFT and PLT rule in heir opimal forms. The whie surface denoes he iniializaions resuling in convergence o he normal sead sae (denoed b he red do), while he black surface denoes no To preven biased resuls, boh models are iniiaed b =.8, c =.74 o provide for he same iniial disance o he normal sead sae. 7

20 (a) Fixed variables: ψ =.5, ψ =..5. (b) Fixed variables: ψ P = 3., ψ =.7 Figure : Domain of aracion (iniial values) wih IFT (a) and PLT (b). The whie surface denoes he parameerizaion resuling in a convergence o he normal sead sae (denoed b he red do) afer n = 4, while he black surface denoes no convergence. convergence, resuling in eiher a collapsing model or an exploding model. This Figure clearl illusraes ha he model wih IFT has a bigger domain of aracion, especiall in erms of he iniial inflaion. Noice ha for boh ineres rae rules, has a decisive effec in he convergence of he model. If is oo low, he econom his a deflaionar spiral, causing boh models o collapse. This means ha if a shock shifs he model under he zero-lower bound (ZLB), IFT as well as PLT does no preven he model from falling ino a deflaionar spiral. I is also imporan o noice ha wih IFT, he iniial inflaion has no upper boundar. This means he model can be iniiaed wih an higher han he inflaion arge. Looking solel a he domain of aracion of boh models, IFT is beer suied for monear polic han PLT. 5 Roles of he Heurisics The heurisics ha are used in he heurisics swiching model (HSM) deermine he dnamics of he model. These heurisics represen individual forecasing behavior, observed in learningo-forecas experimens (LFE s). Because he aggregae of he heurisics expecaions form inpu for he realized variables, coordinaion on cerain heurisics can deermine if he model reaches a convergence. For hese reasons, i is ineresing o sud wha effec he heurisics have on he dnamics of he model, and o compare hem wih resuls from he LF experimens discussed b Anufriev and Hommes (22), o see if coordinaion on cerain heurisics in he experimens wih he asse pricing model resuls in he same convergence of he variables as in our model. To sud he effec of he differen heurisics on he dnamics of he model, we compue he model when excluding one of he four heurisics and see wha happens wih he convergence of he model. We ake he opimal parameerizaions of he ineres rae Talor rules we found in secion 3, wih iniializaions =.2, c =.7 for he inflaion 8

21 argeing (IFT) rule and =.8, c =.74 for he price-level argeing (PLT) rule. The big difference beween he wo ineres rae rules is ha he model wih IFT has a smooh convergence, while he model wih PLT shows oscillaor behavior. This makes suding he effec of he heurisics difficul, which is wh I will sud his separael for each ineres rae rule (more deailed plos of he following Figures in his secion can be found in Appendix B.3 for IFT, and in Appendix B.4 for PLT). 5. Inflaion Targeing rule Beginning wih he IFT rule, we sud he model when excluding he learning anchoring-andadjusing rule (LAA) (Figure 2a) and compare i o he model including all heurisics (Figure 2e). I shows ha when excluding LAA, he model converges faser and more direcl o he normal sead sae han i does when including all heurisics (convergence is reached in 25 periods when excluding LAA, whereas in he normal model convergence is onl reached afer approximael periods). Because LAA focuses on he mean realized values, i slows down he convergence of he model due o is anchoring in he iniial periods of he model. When respecivel excluding he srong and weak rend-following heurisics (STR and WTR) (Figure 2b and 2c), we noice ha he resuling models are ver much alike. Noiceable is ha hese models do no full converge, bu sele on a considerable disance from heir sead saes. Wihou one of he rend-following heurisics, he model seems o fail a converging full o is normal sead sae. Ineresingl, when excluding one of he rendfollowing rules, he oher rend-following rule akes over and reaches he highes share (see Figures 29b, 29d, 3b and 3d for deails). This shows ha he rend-following heurisics conribue o a smooh convergence. When removing he adapive expecaions heurisic (ADA) from he model (Figure 2d), we see i creaes weak oscillaions in he iniial periods of he model. Therefore, ADA seems o have a sabilizing effec on he oscillaions. This resul is also found b Anufriev and Hommes (22), who show ha coordinaion on ADA maches he behavior shown in experimens wih a convergence. This also suggess ha ADA has an effec in sabilizing he oscillaions. 5.2 Price-Level Targeing rule Now we sud he effec of he heurisics on he model wih PLT. When we look a he model excluding he learning anchoring-and-adjusing LAA rule (Figure 3a), and compare i o he model including all heurisics (Figure 3e), we noice ha excluding he dominaing LAA rule does no affec he convergence of he model. I does seem o have some effec on he ampliude of he oscillaions. The same resul is observed b Anufriev and Hommes (22), who show ha when LAA akes over and dominaes, large oscillaions are observed in heir model. Ineresingl, he share of he LAA rule is equall divided among he oher hree heurisics, resuling in heir shares saing relaivel he same (see Figures 27b, 27d, 32b and 32d for deails). 9

22 c (a) (c) c (e) c c (b) c (d) 5 Figure 2: Phase plos (wih he sead sae denoed b he red do) and ime series of and wih IFT, wih he model excluding LAA (a), excluding STR (b), excluding WTR (c), excluding ADA (d) and including all heurisics (e). Nex we sud he model when respecivel excluding one of he rend-following STR and WTR rules, which is respecivel displaed in Figure 3b and 3c. Ineresingl, excluding STR does no have he same effec as excluding WTR, which conrass o wha we observed 2

23 c (a) (c) c c (b) c c (d) (e) Figure 3: Phase plos (wih he sead sae denoed b he red do) and ime series of and wih PLT, wih he model excluding LAA (a), excluding STR (b), excluding WTR (c), excluding ADA (d) and including all heurisics (e). in he case of IFT. Excluding STR decreases he ampliude of he oscillaions, while excluding WTR increases he ampliude of he oscillaions. Because hese heurisics exrapolae rend, he end o overshoo rend reversals when he model is oscillaing. The STR rule, having a higher exrapolaing coefficien, hus overshoos he rend reversals more han he WTR, 2

24 which is wh he STR causes he model o show more oscillaions han he WTR. This means ha he rend-following rules work well in models wih a smooh convergence (wih IFT), bu perform poorl when he model oscillaes, resuling in he LAA rule aking over. Anufriev and Hommes (22) also show ha in heir model wih oscillaions, a greaer exrapolaing coefficien resuls in an increase in he ampliude of he oscillaions of he model. Finall we look a he effec of excluding he adapive expecaions (ADA) rule from he model. Figure 3d shows ha his resuls in an increase in he ampliude of he oscillaions, causing he model o fall ino a deflaionar spiral afer approximael 2 periods. This suggess ha he ADA rule has an imporan role in he model, bu when we compare his resul wih he model including all heurisics, i is ineresing o see ha he ADA rule is acuall one of he wors performing heurisics (see Figure 27 for deails). Since he ADA rule does no follow he oscillaions when forming is expecaion, i performs poorl in he model wih oscillaions. Despie being a poorl performing heurisic, i has a sabilizing effec on he model. This means ha leaving he LAA, WTR and STR rules in he model enforces he oscillaions. Again, his maches he resuls of Anufriev and Hommes (22), who observe ha coordinaion on ADA provides a good explanaion of he convergence in heir model. To summarize our findings from excluding heurisics, we can sa ha he ADA rule has an imporan role because i sabilizes oscillaions and prevens he model from falling ino a deflaionar spiral, despie being a poorl performing heurisic. Because IFT shows no oscillaor behavior, he role of he ADA rule hus depends on wheher we use IFT or PLT. Coordinaion on he rend-following rules resuls in a fas convergence in he model wihou oscillaions (using IFT), while he perform poorl in he model wih oscillaions (using PLT), because he overshoo rend-reversals. On he oher hand, he LAA rule performs bes in he oscillaing model wih PLT, bu slows down he convergence of he model when using IFT, due o is anchor. 6 Conclusions This sud provides an assessmen of he effec of he inflaion argeing rule and price level argeing rule on macroeconomic sabili under heerogeneous expecaions, embodied b he heurisics swiching model. The parameers of he opimized ineres rae Talor rules are in he proximi of ψ =.5, ψ =. for inflaion argeing, and ψ P = 3., ψ =.7 for pricelevel argeing. The resuls show ha inflaion argeing has a bigger domain of aracion in erms of iniial condiions, which makes i more fi o process random shocks in an econom. However, in boh models here is a risk of falling ino a deflaionar rap when he inflaion falls below a given hreshold. I should also be noed ha inflaion argeing can be iniiaed wih an value higher han he arge inflaion, and sill converge o he normal sead sae. The model wih price-level argeing fails o do his when he iniial inflaion is more han 4% above he arge inflaion. Besides a bigger domain of aracion, inflaion argeing also has a more smooh convergence, which is imporan for dealing wih random shocks in an econom. 22

25 On he oher hand, price-level argeing has a faser convergence han inflaion argeing in he long run. Though i shows significan cclical flucuaions in he iniial periods, afer approximael 5 periods i performs beer han inflaion argeing. This sud also analses he roles of he differen heurisics b compuing he model wih inflaion argeing and price-level argeing when excluding one of he four heurisics. The resul of excluding he learning anchoring-and-adjusing heurisic (LAA) from he model is ha LAA performs bes in he model wih cclical oscillaions. This is in conras o he rend-following heurisics, which end o overshoo he rend-reversals of he oscillaions. I also shows ha he LAA heurisic slows down he convergence of he model wihou oscillaions, due o is anchor. When suding he rend-following heurisics, he oucomes for he wo models are differen. Wih inflaion argeing, excluding he srong rend-following rule (STR) resuls in almos he exac same model as when excluding he weak rend-following rule (WTR), while wih price-level argeing he have a significanl differen effec on he ampliude of he oscillaions. The STR rule, having a higher exrapolaing coefficien, amplifies he oscillaions. The rend-following rules hus perform poorl in models showing oscillaor behavior (price-level argeing), because he exrapolae rend and herefore amplif he oscillaions, bu he perform well in models wih a smooh convergence (inflaion argeing). The adapive-expecaions heurisic (ADA) seems o have a sabilizing effec on he oscillaions of he model. This can be explained b he fac ha he ADA rule does no follow rends. This is observed when excluding he ADA rule from he model wih inflaion argeing, creaing weak oscillaions. Moreover, when excluding he ADA rule from he model wih price-level argeing, i causes he model o fall ino a deflaionar spiral, evenuall collapsing i. Ineresingl, in he model wih price-level argeing he ADA rule is one of he wors performing heurisics, bu is neverheless necessar for he model o ensure a convergence. 23

26 Bibliograph Anufriev, M., Assenza, T., Hommes, C., and Massaro, D. (23). Ineres Rae Rules and Macroeconomic Sabili under Heerogeneous Expecaions. Macroeconomic Dnamics, 7 : Anufriev, M. and Hommes, C. (22). Evoluionar Selecion of Individual Expecaions and Aggregae Oucomes. American Economic Journal: Microeconomics, 4 (4): Assenza, T., Bao, T., Hommes, C., and Massaro, D. (24a). Experimens on expecaions in macroeconomics and finance. Experimens in macroeconomics, 7 : 7. Assenza, T., Heemeijer, P., Hommes, C., and Massaro, D. (24b). Managing self-organizaion of expecaions hrough monear polic: A macro experimen. Technical repor, CeNDEF Working Paper, Universi of Amserdam. Benhabib, J., Evans, G. W., and Honkapohja, S. (24). Liquidi raps and expecaion dnamics: Fiscal simulus or fiscal auseri? Journal of Economic Dnamics and Conrol, 45 : Brock, W. and Hommes, C. (997). A Raional Roue o Randomness. Economerica, 65 (5): Camerer, C. and Fehr, E. (26). When Does Economic Man Dominae Social Behavior? Science, 3 (5757): Cornea, A., Hommes, C., and Massaro, D. (22). Behavioural Heerogenei in U.S. Inflaion Dnamics. Technical repor, Tinbergen Insiue Discussion Paper. Evans, G. W., Guse, E., and Honkapohja, S. (28). Liquidi raps, learning and sagnaion. European Economic Review, 52(8): Evans, G. W. and Honkapohja, S. (998). Economic Dnamics Wih Learning: New Sabili Resuls. Review of Economic Sudies, 65 (): Evans, G. W. and Honkapohja, S. (2). Prineon Universi Press. Learning and Expecaions in Macroeconomics. Gali, J. (28). Monear Polic, Inaion, and he Business Ccle: An Inroducion o he New Kenesian Framework. Princeon Universi Press. Greher, D. and Plo, C. (979). Economic Theor of Choice and he Preference Reversal Phenomenon. American Economic Review, 69 (4): Hommes, C., Massaro, D., and Salle, I. (25). Monear and Fiscal Polic Design a he Zero Lower Bound - Evidence from he Lab. Workshop on Socio-Economic Complexi, 23 :27. 24