Modeling Inflation Expectations: The Case of Iran

Transcription

1 Modling Inflaion Expcaions: Th Cas of Iran Dissraion zur Erlangung ds Grads Dokor dr Wirschafswissnschaf (Dr. rr. pol.) dr Jurisischn und Wirschafswissnschaflichn Fakulä dr Marin-Luhr-Univrsiä Hall-Winbrg vorglg von Shahram Faahi Gakih M.A. aus Iran Hall (Saal) 008 urn:nbn:d:gbv: [hp://nbn-rsolving.d/urn/rsolvr.pl?urn=nbn%3ad%3agbv%3a

2 Guachr dr Dissraion: 1. Guachr: Prof. Dr. Hinz P. Gallr. Guachr: Prof. Dr. Gunr Sinmann Tag ds öffnlichn PromoionsKolloquiums: 17. Juli 008

3 To my family

4 Tabl of Conns 1. Inroducion 1. Modling xpcaion formaion 5.1 Thorical concps Adapiv xpcaions 5.1. Raional xpcaions Larning procsss Educiv larning Adapiv larning Raional larning Expcaion formaion using saisical prdicors Paramric prdicion modls ARIMA modling Sa-spac modling Nonparamric prdicion modls Nonparamric Smoohrs Local Avraging Krnl Smoohr Lowss Smoohr Splin Smoohr Nonparamric Modls Addiiv modl Mulipl Adapiv Rgrssion Splins Projcion-Pursui Rgrssion Nural Nworks Basics of nural nworks Larning in nural nwork modl Linar paramr simaion Nonlinar paramr simaion Larning of raional xpcaions using a nural nwork 68

5 5. Empirical Rsuls Simpl forcas A muli-quaion modl Raional xpcaions Backward-looking xpcaions Adapiv xpcaions Forming xpcaions using a mix of xrapolaiv and rgrssiv xpcaions Forming xpcaions using larning modl Forward-looking xpcaions Summary and Conclusions 108 Lis of Tabls, Figurs and Acronyms 11 Lis of Tabls 11 Lis of Figurs 113 Lis of Acronyms 114 Rfrncs 115 Appndics 13

6 1. Inroducion Expcaions ar cnral o our undrsanding of h bhavior of h conomy and any xplanaion of inflaion dynamics nds o xamin h procss of xpcaion formaion. Economiss hav rcognizd ha xpcaions play a drmining rol in conomic horis. For xampl, Kynsian bliv ha h IS curv is volail bcaus firm s xpcaions abou h fuur probabiliy of hir invsmn projcs ar hmslvs highly volail; hy ar subjc o animal spiris. In his prmann incom hypohsis, Fridman (1957) srssd h rol of xpcd fuur incoms in drminaion of consumpion xpndiur. In fac, many imporan macroconomic rlaionships includ lmn of xpcaions. Whn such rlaionships ar combind o build a full macroconomic modl, policy implicaions of ha modl will dpnd on how xpcaions ar bing spcifid. Th conomic oucoms ha agns can xpc from conomic policy ar affcd by h way xpcaions ar formd and how hy vary ovr im. I mars whhr agns form hir xpcaions by looking a h pas or by looking forward by ihr rusing conomic policymakrs promiss or forcasing conomic condiions. On h ohr hand, policy makrs nd o ak xpcaion of conomic agns ino accoun whn dciding on policy acions. For ha purpos, an undrsanding of xpcaion formaion is ndd. Thrfor, failur o invsiga hs issus fully could lad o flawd conomic policy. Public xpcaions abou h cnral bank s objcivs ar imporan for pric sabiliy. If priva agns ar no sur ha h cnral bank prfrs lowr o highr inflaion, xpcaions abou fuur policy acions and fuur inflaion will highly bcom snsiiv o h cnral bank s inflaion arg and hus rsul in conomic insabiliy (Brnank, 003). Furhrmor, h rpuaion of a cnral bank has an impac on inflaionary xpcaions. Th chang of h cnral bank s rgim givs ris o a chang in h lvl of inflaion xpcaions. Changing parns of inflaion xpcaions formaion may b rsuling from larning procss abou nw monary rgims. Any unfavorabl conomic shock raiss acual inflaion and causs priva agns o rais forcass of fuur inflaion. Highr inflaion xpcaions will in urn incras inflaion. In his siuaion, policymakrs nd o hav policy ools o anchor 1

7 xpcaions. Som conomiss bliv ha if h cnral bank announcs an xplici arg for inflaion and crdibly dmonsras ha i will ak acions o rurn inflaion o h arg whn conomic shocks occur, firms and housholds ar lss likly o incras hir long run inflaion xpcaions vn if a shock incrass inflaion for a coupl of monhs. Th rsul is ha wih inflaion xpcaions wll-anchord, any givn shock, whhr i is from aggrga dmand or supply, will no lad o incras in inflaion bu only o a chang in rlaiv prics. Inflaion xpcaions ar vry unsabl in Iran s conomy bcaus h Cnral Bank is unabl o adhr o an inflaion arg in pracic. Thus, inflaion xpcaions ar no wll-anchord and any oil pric incras, which sms apparnly o b a favorabl shock, rsuls in mony craion, fuld by govrnmn spnding ou of oil rvnus, and inflaion and causs priva agns o rais inflaion xpcaions. This in urn will incras inflaion. As a rsul, poor anchord inflaion xpcaions mak pric sabiliy much mor difficul o achiv in h long run and dcras h Cnral Bank s abiliy o sabiliz oupu and mploymn in h shor run. This rsarch ris o xamin how mark paricipans form hir inflaion xpcaions in h Iranian conomy ovr h priod Th Iranian conomy dpnds largly on oil rvnus so ha any chang in oil prics can dircly affc all conomic scors. An incras in oil prics will rsul in mony craion and inflaion. Furhrmor, h larg numbr of govrnmn-conrolld nrpriss, bnfid from subsidis, which incras budg dfici hrough borrowing from h Cnral Bank and hus hav incrasd h monary bas. During his priod, mony supply has bcom 1017 fold whil ral GNP rcordd only a 10 fold incras, rsuling in a rlaivly high inflaion wih an avrag inflaion ra of abou 15 prcn. Wih such vry high liquidiy, any dcision or nws announcd by h govrnmn or h Cnral Bank could svrly chang disribuion of rsourcs in h conomy. In such circumsancs, i mars for h Cnral Bank o know how priva agns form hir xpcaions. Morovr, opimal monary policy dpnds considrably on h assumd naur of xpcaions formaion procss. Empirical analyss on h formaion of xpcaions can b dividd ino wo cagoris: firs, hos sudis ha hav bn don by asking popl abou h fuur valus of inflaion (survy sudis). Scond, hos sudis ha hav rid o xrac xpcaions from pas daa, on h assumpion ha popl look o pas xprinc as a guid o h fuur. This sudy will go h lar way.

8 This sudy compars wo approachs o modling inflaion xpcaions: simpl forcas and a muli-quaion modl. In h firs cas, paramric and nonparamric mhods ar applid and hn i is valuad whhr nonparamric modls yild br simas of inflaionary xpcaions han do paramric alrnaivs. Th agns ar assumd o us an opimal paramric auorgrssiv moving avrag (ARMA) modl or nonparamric modls including Addiiv, Mulipl Adapiv Rgrssion Splins, Projcion-Pursui Rgrssion, and Nural Nworks for forcasing. In fac, ou-of-sampl simas of inflaion gnrad by h paramric and nonparamric modls will b compard. In h cas of a muli-quaion modl, his sudy will focus on h srucural modl of Phillips curv. Th xpcd inflaion gnrad by h raional, nar raional and larning schms will b xamind in h augmnd Philips curv quaion. Th main focus of his sudy is on h following gnral qusions: (I) Do inflaion xpcaions play a main rol in drmining h wags? (II) How do priva agns form hir xpcaions? Ar hy raional, nar raional, or do hy us a larning mchanism? (III) Ar nural nworks br suid for modling xpcaions han nonparamric alrnaivs? (IV) Wha implicaions aris from (II)? (V) Wha conclusions can b drawn basd on h findings abov? This hsis is organizd as follows. Following his inroducion in chapr on, an ovrviw of h horical concps of xpcaion formaion including adapiv xpcaions, raional xpcaions and larning approach will b givn in chapr wo. Th mris and dmris of h ach approach ar discussd in dails. In chapr hr, xpcaion formaion using saisical prdicors is xamind. Paramric modls including auorgrssiv moving avrag (ARMA) modls, saspac modl, h Kalman filr, and nonparamric modls including h addiiv modl (AD), mulipl adapiv rgrssion splins (MARS), and projcion-pursui rgrssion (PPR) will b discussd. 3

9 An innovaion basd on compuaional inllignc has bn h us of nural nworks as a smi paramric approach o dscrib larning procdurs. This is prsnd in chapr four. Th basics of nural nworks ar firs xplaind. Thn h procss of larning in hs modls using h backpropagaion algorihm is dmonsrad. Th inrs is in xamining whhr raional xpcaions ar larnabl by us of nural nworks. Chapr fiv prsns h rsuls of an mpirical analysis. Th daa as wll as a background of Iranian conomy ar dscribd. In his chapr, firs simpl saisical prdicors will b usd for forcasing and a hn muli-quaion modl including h augmnd Phillips curv quaion will b usd o xamin inflaion xpcaions gnrad by h raional, nar raional and larning approachs. Finally, chapr six prsns a brif summary, conclusions and policy implicaions. 4

10 Modling xpcaion formaion In his chapr, diffrn approachs o modling inflaion xpcaions ar prsnd. Firs, horical concps of adapiv xpcaions ar dmonsrad. Thn, h raional xpcaions hypohsis is discussd in dails. Th mris and dmris of raional xpcaions as wll as diffrn vrsions and diffrn ss of his hypohsis ar also considrd. Finally, h larning approach and is rol in macroconomics ar xplaind. Approachs o larning including duciv larning, adapiv larning, and raional larning ar also illusrad..1 Thorical concps.1.1 Adapiv xpcaions On of h mos familiar radiional modls of xpcaion formaion is adapiv xpcaions. This modl can b sad using h following quaion, whr P is his priod s xpcd inflaion; P 1 is las priod s xpcd inflaion; and P 1 las priod s acual inflaion: P = P + λ( P P ) (1) wih λ bing a valu bwn 0 and 1. According o his hypohsis, currn xpcaions of inflaion rflc pas xpcaions and an rror-adjusmn rm. Th paramr valu of λ dpnds on wha w hink abou h likly sourc of las priod s rror. If i was a prmann shif in h procss forming P, hn w s λ = 1so ha P = P. This is saic xpcaions: his yar s inflaion is xpcd o 1 b h sam as las yar s. If las priod s rror was jus du o a random vn, w sλ = 0, so hr is no adjusmn, and w should no chang xpcaions a all ( P P 1 = ). Popl will chang xpcd inflaion if hr is a diffrnc bwn wha hy wr xpcing i o b las priod and wha i acually was las priod. In fac, xpcd inflaion is rvisd by som fracion of mos rcn forcas rrors. If h xpcd inflaion was, say 5 prcn, bu h acual inflaion 10 prcn, popl rais hir xpcaions by som fracion λ of h diffrnc bwn 5 and 10. Using h Koyck ransformaion, h quaion (1) can b ransformd ino 5

11 P = (1 λ) P + λ(1 λ) P + λ (1 λ) P + λ (1 λ) P... () Now w can xamin h rlaionship bwn bn consan for a long im a P 0 P and P. Suppos ha P has. Thn, suppos ha a im priod T, h inflaion jumps up o P and says hr indfinily. A T, all h rms on h righ-hand sid 1 of quaion () ar qual o P 0, so h xpcd inflaion for T is givn by P 0 is PT = P : 0 P = (1 λ) P + λ(1 λ) P + λ (1 λ) P + λ (1 λ) P... = P T , ha Onc T is ovr, xpcaions ar formd by quaion () wih s qual o T+1. Thrfor, h firs rm on h righ-hand sid for priod T+1 is P 1 : 3 PT + 1 = (1 λ) P1 + λ(1 λ) P0 + λ (1 λ) P0 + λ (1 λ) P0... Sinc P 1 > P 0, i is asy o vrify ha P1 > PT + 1 > PT = P. Thr is som 0 corrcion in T+1 for h rror mad a T, bu is no compl. A h sar of following priod, wo of h righ-hand rms of quaion () includ P 1.Th rmaining rror is again parly corrcd bu h absolu valu of corrcion is lss. This procss coninus unil h scond rm on h righ-hand sid of quaion (1) diminishs o mak h diffrnc ( P P ) arbirarily small Thr ar mris and dmris of h adapiv xpcaions hypohsis (AEH). On h on hand, h hypohsis has h advanags of bing simpl o opra as a rul of humb. I is a h bs appropria in a sabl nvironmn whr h pric lvl movs up and down in a fairly random fashion, wih h possibiliy of somwha mor prmann shifs in h background. On h ohr hand, howvr, i has wo disadvanags: firs, i is a backward-looking approach (no accoun of fullyannouncd fuur policis). Scond, i has sysmaic rrors basd on h prvious forcas wih som corrcion for prvious forcas rrors. Individuals do no sysmaically larn from prvious forcas rrors, hy do ignor informaion ha would hlp hm improv h accuracy of hir forcass. Thus, h AEH assums subopimal bhavior on h par of conomic agns. For xampl, considr h Phillips curv quaion: P = P ( U U *) + ε 1 1 P = Acual inflaion a im 6

12 U * = Naural of unmploymn Assum ha (for simpliciy): hn U * = U = U = U... P = P + ε wih adapiv xpcaions: P = λ P + (1 λ ) P 1 1 If λ = 0.5 = λ P + (1 λ )( λ P + (1 λ ) P ) P = 0.5P + 0.5P P = 0.5P + 0.5[ P ε ] [ P ε ε ] +... (3) Equaion (3) shows ha h AEH ignor pas forcas rrors in forming xpcaions. Undr adapiv xpcaions, if h conomy suffrs from consanly rising inflaion ras, popl would b assumd o squnially undrsima inflaion. This may b rgardd unralisic- surly raional popl would soonr or lar raliz h rnd and ak i ino accoun in forming hir xpcaions. Morovr, modls of adapiv xpcaions nvr rach an quilibrium; insad hy only mov oward i asympoically..1. Raional xpcaions Th ris of Raional Expcaions Th raional xpcaions hypohsis rsponds o his criicism by assuming ha individuals us all informaion availabl in forming xpcaions. During h la 1960s, raional xpcaions conomics sard changing h fac of macroconomics. Robr Lucas, Tomas Sargn, and Nil Wallac sard o domina h macroconomic discussion. Noions such as h Lucas criiqu, h Lucas supply curv, and h Sargn-Wallac policy irrlvanc proposiion bcam ingral pars of h macroconomics discours. 7

13 Thr ar diffrn rasons bhind h ris of raional xpcaions (RE). Sn (1998) argus ha h main facors ar as follows: 1. Expiraion of h Phillips curv: in h la 1960s o arly 1970s, policy makrs usd a rad-off bwn inflaion and unmploymn o lowr unmploymn. Howvr, hy facd high inflaion ras accompanid by high unmploymn ras in h 1970s. In ohr words, h rsul of policy making was highr inflaion wih no bnfis in rms of lowr unmploymn. Raional xpcaions conomiss wr abl o xplain h xpiraion of h Phillips curv. Thy, using h raional xpcaion hypohsis, dmonsrad ha govrnmn acions causd an advrs shif of h Phillips curv.. Policy irrlvanc: orhodox prscripions of conomic policy crumbld, sinc much of h ffcivnss of hs policis wr basd on h govrnmn s abiliy o fool popl. Raional xpcaions conomiss assrd ha popl can foil govrnmn policis by larning hir misaks. Thy jusifid h inffcivnss of govrnmn inrvnion in h conx of h failur of radiional Kynsian policis in h 1970s. Also, hy rcognizd h limiaions of hir profssion mainaining ha h conomy would basically b sabl if i wr no subjcd o h shocks adminisrd by h govrnmn. 3. Using availabl chniqus: raional xpcaion conomiss usd sophisicad mahmaical chniqus in ordr o prdic. Thy larnd and usd h chniqus of inrmporal opimizaion dvlopd by mahmaicians and conrol sciniss. Thy also improvd h ools of opimal prdicion and filring of sochasic procsss. Som of hs chniqus such as classical linar prdicion hory¹ was dvlopd in 1940s o1950s bu did no immdialy bcom par of conomiss oolkis. Howvr, Pr Whil mad mor accssibl o conomiss his hory ha was havily usd by raional xpcaion conomiss. This dlay xplains h laggd ffc of Muh s conribuions. Thus raional xpcaion conomiss wr abl o calcula raional xpcaion quilibria using nw chniqus. 4. Rsoring symmry: h hypohsis of adapiv xpcaions had bn usd havily up unil h la 1960s. According o his hypohsis, individuals usd forcasing rrors in rvising hir xpcaions. Economricians wr prsumd o b fully knowldgabl whras h agns wr assumd o mak sysmaic Th mahmaical hory for inrpring disribud lags in rms of conomic paramrs and incorporaing h raional xpcaions hypohsis in conomic modls. 8

14 forcasing rrors priod afr priod. Thus hr was an asymmry among conomiss or conomricians and h agns in ha conomricians fi modls ha forcas br han agns. Raional xpcaions hypohsis (REH) rmovd his asymmry making h conomricians par of h agns bhavior. Thrfor, raional xpcaion conomiss placd conomricians and agns on an qual fooing by posulaing ha forcass mad by h agns wihin h modl wr no wors han hos h conomricians who had h modl. 5. Opimizing ovr informaion: according o REH, opimizaion ovr prcpions implid agns did h bs hy could and formd hir viws of fuur using availabl informaion, including hir undrsanding of how h conomy works. Raional xpcaion horiss xndd xpcaion hory ino h opimizing bhaviors hory. If prcpions wr no opimally chosn, unxploid uiliy or profi-gnraing possibiliis would xis wihin h sysm. Hnc, hs conomiss insisd on h disapparanc of all such unxploid possibiliis. 6. Endognizing xpcaions: Kyns (1936) doubd ha xpcaions could b modld accuraly. So h considrd xpcaions as givn. Also, Kyns followrs assumd ha popl mad gusss abou h fuur by looking xclusivly backward. In fac, h hypohsis of adapiv xpcaions is backward-looking in ha i allows h possibiliy of sysmaic forcasing rrors for many priods in succssion. This is a subopimal us of availabl informaion and is no consisn wih h ida of opimizaion. Evn hough popl usd adapiv xpcaions, no widly accpd conomic hory was offrd o xplain h amoun of h adjusmn paramr. Th mchanism of raional xpcaions formaion is ndognously moivad and xpcaions or forcass ar corrc on avrag if rrors individuals rmain saisfid wih hir mchanism. This hypohsis assrd ha h rsuling prdicions migh sill b wrong, bu h rrors would b random. If rrors follow a parn, hy conain informaion ha could b usd o mak mor accura forcass. Thrfor rrors wr prsumd o cancl ou whn all individual xpcaions ar addd oghr. 7. Making public prdicions: som auhors blivd ha h ris of raional xpcaions could figh h hra of indrminacy of conomic oucoms. This indrminacy rsuld from his fac ha making boh slf-falsifying and slf-fulfilling prdicions abou popl was possibl. Sinc oucoms dpndd parly on wha popl xpcd hos oucoms o b if popl s bhavior dpndd on hir 9

15 prcpions, conomic sysms wr hough o b slf-rfrnial. This ld som conomiss o dspair ha conomic modls could produc so many oucoms ha hy wr uslss as insrumns for gnraing prdicions. Raional xpcaions,howvr, was a powrful hypohsis for rsricing h rang of possibl oucoms sinc i focusd only on oucoms and sysms of blifs ha wr consisn wih on anohr. Undr raional xpcaions, corrc public prdicions could b mad bcaus raional xpcaions prdicions wr prsumd o b ssnially h sam as h prdicions of h rlvan conomic hory. Also, h hypohsis consisd of xpcaional rspons of h agns and h influnc of prdicions on bhavior of h agns. 8. Counring boundd raionaliy: raional xpcaions hory was born a h sam im in h sam siuaion as h concp of boundd raionaliy, namly, in h 1960s a h Gradua School of Indusrial Adminisraion (GSIA) a Carngi Mllon Univrsiy. Hol, Modigliani, Muh, and Simon wr collagus and workd on h Planning of Conrol of Indusrial Opraion Projc, which consisd of dvloping and applying mahmaical chniqus o businss dcision making. Though Simon and Muh had boh paricipad in h projc, Simon saw h srong assumpion undrlying his projc as an insanc of saisfying, whras Muh saw his spcial cas as a paradigm for raional bhavior undr uncrainy. Som argu ha Muh, in his announcmn of raional xpcaions, xpliciy labld his hory as a rply o h docrin of Simon s boundd raionaliy. 9. Rsricing disribud lags: in h la 1960s, raional xpcaion conomiss wr confrond wih horiical modls ha analyzd individual bhavior in a conx wihou uncrainy and randomnss. A h sam im, sinc hy rad hir daa probabilisically, hy had o incorpora uncrainy and randomnss in opimizing conomic hory and using h oucom o undrsand, inrpr, and rsric h disribud lags ha aboundd in h dcision ruls of dynamic macroconomic modls. Thy promisd o ighn h link bwn hory and simaion. 10. Incorporaing vcor auorgrssion: h final causal background of raional xpcaions is rlad o h blif ha i crad a conncion bwn vcor auorgrssions and conomic hory. Som argu h REH was abl o rviv hory by showing ha vcor auorgrssions was no ncssarily ahoriical and could provid a saisical sing wihin which h rsricions implid by horical modls could b imposd. In paricular, raional xpcaion horiss xploid cross- 10

16 quaion rsricions o connc h vcor auorgrssiv paramrs of dcision ruls wih horical paramrs of as, chnology, and ohr sochasic nvironmns. Raional xpcaions and procsss Th raional xpcaion hypohsis (REH) assums conomic variabls ar gnrad by rcurring procsss (Afild al, 1991). Ovr im, conomic agns larn h procss drmining a variabl and hy will us his knowldg and all informaion availabl (ha is rlad o h variabl) o form xpcaions of ha variabl. As a rsul, h agns subjciv probabiliy disribuion coincids wih h objciv probabiliy disribuion of vns¹. In ohr words, h xpcaions of agns will b h sam as h condiional mahmaical xpcaions basd on h ru probabiliy modl of h conomy. For xampl, suppos h valu of variabl Y in priod is drmind by is own laggd valu and by laggd valus of ohr variabls X and Z in h following way: Y = α + αy + α X + α Z (4) whr α 0, α 1, α and α 3 ar consan cofficins. Considr a prson who, a h nd of priod -1, is rying o form an xpcaion abou h valu ha Y is going o ak in priod. Sh knows ha h procss drmining Y is givn by quaion (4): knowldg of his procss is said o b par of hr informaion s a h nd of priod -1. Sh also knows h valus of all laggd variabls of X, Y, Z, ha also ar par of hr informaion s a h nd of priod -1. If sh is raional, hr xpcaion of wha Y is going o b in priod, on h basis of hr informaion s a h nd of priod -1, will b formd as follows: E Y = α + αy + α X + α Z (5) whr E 1 is h xpcaion of Y formd on h basis of h informaion availabl a h nd of priod -1. Th raional xpcaion of Y formd a priod -1 (dnod as This is h srong vrsion of h raional xpcaions hypohsis, du o Muh, (Psaran, 1987). 11

17 E[ Y I 1] is h mahmaical xpcaion of Y condiional on h availabl informaion a h nd of priod -1 ( I 1 ). If Y dos indd coninu o follow h procss shown in quaion (5) hn his prson s xpcaion will b prfcly accura, h prson s forcasing or xpcaional rror is zro. This rsul is no gnral bcaus in his cas w assumd h procss drmining Y is drminisic. Howvr, mos procsss in ral world ar sochasic; ha is, hy includ an unprdicabl lmn of randomnss in human rsponss. On way o incorpora his lmn in quaion (4) is o add o i a random rm ( v ): Y = α + αy + α X + α Z + v (6) v may b posiiv or ngaiv. Sinc his variabl is sn as h rsul of a larg numbr of random facors affcing human bhavior, i is naural o hink of small valus of v rahr han larg valus. In fac, w assum ha variabl v has a probabiliy disribuion cnrd a zro and a consan, fini varianc { σ }. Th valu of v in priod is unknown a h nd of priod -1; i is no par of h informaion s a priod -1. Bu i is clar ha a raional forcasr has o form som xpcaion of h valu ha v is going o ak in priod. Th raional xpcaion of Y in accordanc wih quaion (6) is as follows: v E Y = α + αy + α X + α Z + E v (7) whr E 1vis h xpcaion of v formd on h basis of all h informaion availabl a h nd of priod -1. Th bs guss a raional agn can mak of v is ha i will qual is man valu E 1v = 0. Thus, h raional xpcaion of Y in priod, basd on informaion availabl a h nd of priod -1, can b wrin as: E Y = α + αy + α X + α Z (8) Thus h raional xpcaion of h variabl Y in priod is is mahmaical xpcaion givn h availabl informaion. Raional xpcaions, as Muh (1961) xplaind, should b gnrad by h sam sochasic procss ha gnras h variabl o b forcas. 1

18 In quaion (8), if h procss drmining Y rmains unchangd, i follows ha h xpcaional rror will b h random componn v of Y: Y E Y = v (9) 1 Th gnral characrisics of Raional Expcaions A numbr imporan implicaions follow from h fac ha, if h procss drmining Y is undrsood, h rror of raional xpcaion of Y is h sam as h random componn of h procss drmining Y. Thy ar as follows: (a) Th rrors of raional xpcaions ar on avrag zro I is clar from quaion (9) ha onc h procss drmining Y is allowd o b sochasic h raional xpcaion of Y will no always b prfcly accura, for h random componn v is inhrnly unprdicabl. Th bs a raional forcasr could do is xpc h man valu of v and ha is dfind o b zro. In fac, h rror may b posiiv, ngaiv or zro. Bu on avrag or ovr a larg numbr of priods h ngaiv rrors will cancl ou wih h posiiv ons, laving an avrag rror of zro. (b) Th rrors of raional xpcaions xhibi no parn If xpcaions ar raionally formd, h forcasing rror will qual h random lmn in h procss bing forcas. This random variabl, and hnc forcasing rror, may b surpriss or nws in h sysm. If i xhibis no parn, hn h forcasing rror dos no xhibi any parn ihr. Bu wha happn if v xhibis a parn in h following way: v = β v + ε (10) 1 1 Th currn valu of v is linkd o h prvious priod s valu of v. ε is a random rror wih zro man which can no b prdicd on h basis of any informaion availabl a h nd of priod -1; β 1 is a cofficin, h valu of which lis bwn - 1 and +1. If v is bing drmind according o quaion (10) hn raional popl will form hir xpcaion of currn priod s valu of v in accordanc wih ha procss. 13

19 And sinc h valu of v in h prvious priod, -1, will b par of h availabl informaion a h nd of priod -1, i follows ha h forcas of v will divrg from h acual valu of v by an unknown, unprdicabl lmnε. Th rror rm ( ε ) xhibis no parn and has a man valu of zro. Thus vn if v dos xhibi a parn, h raional forcas of Y would, on avrag, sill b corrc and h forcasing rror would xhibi no parn. As for h iming of a chang in h mhod of forming xpcaions, h raional xpcaions hypohsis suggss ha as long as hr is no chang in h procss drmining a variabl, h mhod of forming xpcaions will no chang. Bu if h acual procss drmining a variabl is known o hav changd, hn h mhod by which xpcaions ar formd will chang in lin wih i. (c) Raional xpcaions ar h mos accura xpcaions Raional xpcaions is h mos fficin mhod of forcasing in ha h varianc of h forcasing rrors will b lowr undr raional xpcaions han undr any ohr mhod of forcasing or forming xpcaions. Bcaus forcass of a variabl on h basis of raional xpcaions hypohsis will us all availabl informaion on h procss drmining ha variabl. In ohr words, as xpcaions ar formd h unprdicabl par of Y can no rgularly b prdicd. So any mhod of xpcaion formaion will b inaccura o a dgr drmind by h likly rang of valus ha v can ak. Bu i is possibl o b vn mor inaccura by forcasing wihou rfrnc or wih only parial rfrnc o h procss drmining h variabl. Gnral criiqu of h raional xpcaions hypohsis Criicisms of h REH ar as follows (Afild al, 1991): (a) Th plausibiliy of raionaliy REH assums popl o us all h informaion abou h procss drmining a variabl whn forming xpcaions. Is i rally plausibl? Can w rally assum ha all dcision-makrs ar inllign nough o us and fully undrsand all h availabl informaion? In raliy popl ofn ignor conomic mars. This criicism is ha a major assumpion bhind raional xpcaions is implausibl. 14

20 Th advocas of REH rspond o his criicism in his way: firs of all h ida ha h ypical individual is capabl of making h bs of opporuniis opn o him is a common on in conomics. For xampl, in dmand hory i is assumd ha h ypical prson chooss o consum goods a a poin givn by h angncy of an indiffrnc curv and a budg consrain. Th mahmaics bhind his choic sragy is highly sophisicad for mos popl. Y i is assumd ha popl ac as if hy undrsand i. If such assumpion lads o a hory which maks accura prdicions, hn h assumpion of mahmaical awarnss is hrby shown o b a usful on. Popl forming xpcaions us firms- who spcializ in or provid h srvic of making conomic forcass- or govrnmn bodis-who mak forcas public. Som conomiss also criiciz h rol of raionaliy in REH. Advocas of h hypohsis sa ha h rol of raionaliy has bn usd in REH in ha h procss of acquiring informaion has bn carrid ou up o h poin whr h marginal cos of acquiring mor informaion quals h marginal bnfi of making mor accura forcass. Bu his poin dos no ncssarily corrspond o h poin a which h forcasing rror is qual o h purly random componn of h drmining procss. I may b ha knowldg abou som drmining variabl could b obaind and xra accuracy hrby achivd, bu only a a pric which i is no worh paying. In ha cas h forcasing rror will nd o b absoluly grar han h random lmn in h drmining procss. Advocas of REH accp his criicism bu hy assr ha for mos purposs i is no of gra significanc. Th rason for his is ha forcasing rrors hmslvs ar obsrvd a no cos. For xampl, any rror in your forcas abou h lvl of prics is obsrvd as a coslss sid-ffc of shopping. In ohr words, i mus b worhwhil o xploi his informaion fully unil is marginal bnfi is zro. (b) Th availabiliy of informaion REH assums ha h procss Y is known and ha h valus of variabls in ha procss ar known a h nd of priod -1. Bu wha happns if w do no know h procss drmining h variabl (Y) and if w ar no abl o acquir h ncssary informaion? Advocas of h REH sa ha i is ru ha popl canno auomaically know which variabls ar imporan in h procss drmining Y bu i is also ru ha h REH dosn claim ha hy do. Wha h hypohsis argus is 15

21 ha on avrag and afr a priod of im, conomic agns will larn from pas xprinc wha h procss is. Thy will combin his dvlopd knowldg wih currn availabl informaion o form hir xpcaions¹. For xampl, if, a h nd of priod -1 h raional agn dos no know h ru valu of X in priod -1, and if h valu of X in priod -1 drmins h valu Y in his priod, h agn will hav o form xpcaions of h valu X in priod -1. Suppos h procss drmining Y is as follows: Y = α + αy + α X + α Z + v (11) Suppos ha h valu of X 1 is unknown a h nd of priod -1. And l h procss drmining X in any priod as follows: X = β + βv + β W + ε (1) whr V and W ar ohr variabls, h β s ar cofficins, and ε is a random rror rm wih man zro. Th raional forcas of h unknown valu of X in priod -1 will b as follows: E E X = β + βv + β W (13) X 1 1 will b usd in plac of 1 raional xpcaions of Y in priod will b: X in quaion (11). Thus if X 1 is unknown h E Y = α + αy + α ( β + βv + β W ) + α Z (14) Th forcasing rror will hrfor b givn by: Y E Y = v + α ε (15) Sinc 1 1 v and 1 ε ar random rrors wih mans of zro, nihr of which can b vn parly prdicd on h basis of any informaion availabl a h nd of priod - 1. Th raional forcas or xpcaion of Y in quaion (14) is, in gnral, h mos accura forcas Fridman(1979), criicizing h REH, assrd ha wha is ypically missing in raional xpcaions modls is a clar oulin of h way in which conomic agns driv h knowldg which hn hy us o formula xpcaions ming rquirmn. 16

22 (c) Limis o h applicabiliy of raional xpcaion Many imporan conomic vns can b sn as uniqu, or a las xcpional or unusual du o h paricular poliical circumsancs of day. In wha sns can h REH b said o apply o hs xcpional cass? Th advocas of raional xpcaions assr ha h REH can bs b applid o variabls or vns which can b sn as a par of rcurring procss. Howvr, his class of vns may b a largr on han is commonly hough. For xampl, govrnmns dsir o hav a high lvl of conomic aciviy a h im of gnral lcions and may swich som policis. Such swichs of policy could b sn as par of a fairly rgular and rasonably prdicabl procss. So an vn which could b porrayd as uniqu from anohr viwpoin may wll b par of an undrlying rcurring procss. (d) Tsabiliy of REH Som conomiss hav criicizd ha REH is no sabl. Raional xpcaions horiss sa ha hr ar svral layrs o his criicism. Firs, if REH is akn rahr loosly o imply ha popl mak h bs of hir availabl informaion, hn i may always b possibl o dfin h availabl informaion so ha h hypohsis bcoms immun o falsificaion. This criicism is valid if ss of REH ndd o mploy h loos form of h hypohsis. Bu if hy nd o mploy srong vrsions of h hypohsis in which popl s knowldg of h procss drmining a variabl is assumd o b h sam as h bs sima ha can b mad of ha procss by conomric chniqus hn his criicism is hardly a srong on. Bcaus his assumpion lads o prdicions which ar boh clar and diffrn from h prdicions drivd from ohr horis abou xpcaions. An imporan criicism is ha xpcaions abou a variabl ar almos always only par of a modl. Thus hr ar join ss of h REH islf and h rs of h modl. If h modl fails h ss o which i is subjcd on can always rscu h REH by arguing ha i is h rs of h modl which is wrong. I is a ims possibl o disinguish bwn h rsricions imposd on h daa by REH islf and h rsricions imposd by h rs of h modl. Howvr, h usfulnss of h REH, in his way, can b sd informally and lss saisfacory. If, im afr im, his kind of modls wr rjcd hn w can rjc h REH. 17

23 Th final yp of criicism of sabiliy of REH is wha is known as obsrvaional quivalnc. For many raional xpcaions modls which fis h daa hr will always b a non-raional xpcaions modl which fis h daa qually wll. Th daa hmslvs canno discrimina bwn wo horis, which ar hrfor said o b obsrvaionally quivaln. Th implicaion of his is ha, vn if a raional xpcaions modl passs convnional mpirical ss, his dos no ncssarily imply ha on should accp h hypohsis. Whhr you do or do no, dpnds on whhr you find i mor plausibl han h non-raional xpcaions modl on som ohr unspcifid grounds. () Muli raional xpcaions quilibria Th modls of Muh and Lucas assum ha a any spcific im, a mark or h conomy has only on quilibrium (which was drmind ahad of im), so ha popl form hir xpcaions around his uniqu quilibrium. If hr is mor han on possibl quilibrium a any im hn h mor inrsing implicaions of h hory of raional xpcaions do no apply. In fac, xpcaions would drmin h naur of h quilibrium aaind, rvrsing h lin of causaion posid by raional xpcaions horiss. (f) Abiliy of agns in acion In many cass, working popl and businss xcuivs ar unabl o ac on hir xpcaions of h fuur. For xampl, hy may lack h bargaining powr o rais nominal wags or prics. Alrnaivly, wags or prics may hav bn s in h pas by conracs ha canno asily b modifid. (In sum, h sing of wags and prics of goods and srvics is no as simpl or as flxibl as in financial marks.). This mans ha vn if hy hav raional xpcaions, wags and prics ar s as if popl had adapiv xpcaions, slowly adjusing o conomic condiions. Diffrn vrsions of RE Many dfiniions of raional xpcaions (RE) hav bn proposd sinc Muh (1961) publishd his sminal aricl on his concp. In is srongr forms, RE opras as a coordinaion dvis ha prmis h consrucion of a rprsnaiv agn having rprsnaiv xpcaions. Gnrally, wo dfiniions for RE is usd ar applid rsarch: h wak form and h srong form. 18

24 Wak-form RE Th wak vrsion of RE is indpndn of h conn of h agn s informaion s. Suppos hr ar N agns (i=1,...,n) in an conomy and E,i Y +k dno agn i s subjciv (prsonal) xpcaion formd a h nd of priod of Y +k (k 1). Also l E [Y +k I,i ] dno h objcivly ru xpcaion for Y +k condiional on h informaion availabl o agn i a h nd of priod (I,I ).Th agns ar said o hav wak-form raional xpcaions for variabl Y +k if h following condiion holds: For ach i = 1,, N, E,i Y +k = E[Y +k I,i ] + є,i whr є,i ar srially and muually indpndn fini-varianc rror rms ha saisfy E[є,i I,i ] = 0. Wak-form RE has som faurs. Firs, i is applicabl only if hr ar objcivly ru condiional xpcaions. Wak-form RE assums ha agns mak opimal us of all availabl informaion. Scond, i is consisn wih h ida of conomically raional xpcaions, proposd by Fig and Parc (1976), in which agn s informaion ss ar h rsul of cos-bnfi calculaions by h agns rgarding how much informaion o obain. Finally, many conomiss ar willing o us his vrsion, as a usful bnchmark assumpion consisn wih h ida ha agns ar arbiragurs who mak opimal us of informaion. Srong-form RE Muh (1961) usd a srongr vrsion of RE in ha h placd a rsricion on h informaion ss of agns in horical conomic modls. This vrsion guarans h xisnc of objcivly ru condiional xpcaions bu a h cos of ransforming RE ino an incrdibl concp in rlaion o h form of xpcaions ha ral conomic agns could rasonably b supposd o hav. Agns in a horical modl of a muli-agn conomy will b said o hav srong-form RE if hy hav wak-form RE and, in addiion, hir informaion ss a h nd of priod conain h following informaion: 19

25 (a) (b) (c) (d) Th ru srucural quaions and classificaion of variabls for h modl, including h acual dcision ruls usd by ach priva and public (govrnmn) agn o gnra acions and/or xpcaions; Th ru valus for all drminisic xognous variabls of h modl; Th ru probabiliy disribuions govrning all xognous sochasic rms; Ralizd valus for all ndognous variabls obsrvd by h modlr hrough h nd of priod. Srong-form RE has som inrsing faurs. Firs, i is assumd ha agns ar smar and as wll informd abou h conomy. Th issu ha agns know a priori h acual dcision ruls usd by ach ohr agn is incrdibl. This vrsion can hrfor b inrprd as an idalizd Nash quilibrium¹ bnchmark for agns xpcaions ha agns may (or may no) vnually arriv a hrough som procss of rasoning and/or larning. Scond, in pracic horiss modling conomic sysms assum ha hy hav an xraordinary amoun of informaion abou h ru working of h conomy. As a rsul, undr srong-form RE, conomic agns ar prsumd o hav a gra dal mor informaion han would acually b availabl o any conomrician who ampd o s hs modls agains daa (Sargn, 1993). Third, many conomiss ar uncomforabl wih h mor common assumpion in h srong-form RE. Nvrhlss, his vrsion bcoms mor accpabl if i is viwd as a possibl idal limiing poin for h xpcaions of bounddly raional agns wih limid informaion who ngag in larning in succssiv im priods. Finally, considring prfc forsigh² RE is inrsing. Agns in a horical modl of a muli-agn conomy will b said o hav prfc forsigh RE if h following wo condiions hold: If hr is a s of sragis wih h propry ha no playr can bnfi by changing hr sragy whil h ohr playrs kp hir sragis unchangd, hn ha s of sragis and h corrsponding payoffs consiu h Nash Equilibrium.. I mus b nod ha prfc-forsigh RE diffrs from h prfc forsigh assumpion usd in Walrasian quilibrium modls. In h lar kind of modls, prfc forsigh is h assumpion ha housholds and firms corrcly fors h mark-claring lvls and solv hir opimizaion problms condiional on hs lvls. 0

26 (a) (b) Agns hav srong-form RE; Thr ar no xognous shock rms affcing h conomy, so ha all xpcaions ar corrc wihou rror,.g. E,i Y +k = Y +k Thr ar som implicaions of srong-form RE. Firs, if hr is a chang in h way a variabl movs, hn h way in which xpcaions of his variabl ar formd also changs. For xampl, a chang in h govrnmn s monary policy rul lads o a chang in h movmns of h Fd Funds ra. Scond, forcass ar no always xacly corrc, bu forcas rrors ar no prdicabl in advanc and hy avrag ou o zro. Third, wo rasons why xpcaions can fail o b raional in h srongform sns: (a) agns fail o us all availabl rlvan informaion and (b) agns fail o mak opimal us of all availabl rlvan informaion. An xampl of srong-form RE Suppos an conomy is dscribd by h Lucas Modl (Caplan, 000): (IS) y = -ar + u (1) (LM) m -p = by -ci +v () (Fishr quaion) i = r +E p +1 -p (3) (AS) y = y* + α (p - E -1 p ) (4) (Monary Policy Rul) m +1 = m +φ +1 (5) (Srong-Form RE) E p +1 = E [p +1 I ) (6) Whr y = oupu, p = pric lvl, m = mony supply, r = ral inrs ra, i = nominal ra, u, v, and φ = random variabls wih man 0, y* = ponial oupu, E p +1 = h subjciv forward-looking xpcaion of rprsnaiv agn a im rgarding h pric lvl in priod +1, E [p +1 I )= h objcivly ru condiional xpcaion, I = informaion s ha is availabl o h rprsnaiv agn a h nd of priod whos conns ar assumd o b consisn wih srong-form RE. 1

27 All variabls ar logs of hir lvl valus. In h priod prdrmind variabls ar m and E -1 p for > 1. Th xognous variabls ar: y*, u, v and φ ; h posiiv xognous consans a, b, c, and α ; an iniial valu m 1 =m 0 + φ 1 for h priod 1 mony supply m 1, whr m 0 is xognously givn, and iniial valu for E 0 p 1. Modl quaion (6) is incompl as i sands, in ha h ru condiional xpcaion on h righ hand sid nds o b drmind in a mannr consisn wih srong-form RE. Tha is, givn his xpcaion, h subsqun way in which h pric lvl for priod +1 is acually drmind by h modl quaions mus conform o his xpcaion in h sns ha h objcivly ru I -condiiond xpcaion of h modl-gnrad soluion for h pric lvl in priod +1 mus coincid wih h xpcaion assumd for his pric lvl in modl quaion (6). To compl his modl wih srong-form RE, w mus solv a fixd poin problm of h form f(x) = x, whr x = E [p +1 I )¹. To drmin h ndd xpcaional form, E [p +1 I ), h mhod of undrmind cofficins is usd. Conjcur a possibl soluion form for p as a paramrizd funcion of ohr variabls, whr h paramr valus ar unknown. Thn, drmin valus for hs unknown paramrs ha nsur srong-form RE. For simpliciy assum ha y* = 0. Combining modl quaions (1) hrough (4) plus (6) lads o I mus b nod ha hr is a problm for h RE soluion, i is no uniqu. In fac, muli raional xpcaions ar likly o xis for modls ha includ quaions ha ar nonlinar in h ndognous variabls. This sprads som doubs abou h raionaliy of hs RE soluions. For xampl, considr h following modl of an conomy: y = a + b E -1 y + є, (1) 1, a >0, 0<b<1, E [є I -1] = 0 If a rprsnaiv agn forms his xpcaions for y in priod -1 in accordanc wih srong-form RE, ha is, E -1 y = E [y I -1 ] () In his cas h y gnraing procss in (1) aks h form y = a + b E [y I -1 ] + є, 1 (3) Th righ sid of quaion (3) can b xprssd as a funcion M(x) of x, whr x = E [y I -1 ]. Taking condiional xpcaions of boh sids of (3), on can obains a rlaion of h form x = E [M(x) I -1 ] f (x), 1 (4) Suppos ha h RE soluion for oupu of a modl conomy in priod saisfis a fixd poin problm having form (4) and ha wo disinc soluions x' and x" xis- ha is, f (x') = x' and f (x") = x". Thus, if all agns in h conomy a h nd of priod -1 anicipa oupu lvl x' for priod, h objcivly ru xpcd oupu lvl for h conomy in priod will b x'; and if insad, all agns in h conomy a h nd of priod -1 anicipa oupu lvl x" for priod, h objcivly ru xpcd oupu lvl for h conomy in priod will b x".

28 p = (1/1+c) m + (c/1+c) E [p +1 I ] β [p - E [p I -1 ]]+ w (7) whr β=α[(b+c/a)/(1+c)]; w = (1/1+c)[(c /a)u v ] (8) Suppos i is conjcurd ha h soluion for p aks h form p = q 1 m + q w + q 3 φ, 1, (9) Lad quaion (9) on priod and aking condiional xpcaion of boh sids: E[p +1 I ] = q 1 E[m +1 I ], 0, (10) Taking condiional xpcaion of boh sids of quaion (5) lads o E[m +1 I ] = m, hnc E[p +1 I ] = q 1 m, 0 (11) Now lag quaion (11) on priod and lag quaion (5) on priod o subsiu m - φ in for m -1, hus obaining E[p I -1 ] = q 1 [m φ ], 1 (1) Combining quaions (9) and (1), on hn has p - E[p I -1 ] = [q 1 + q 3 ] φ + q w, 1 (13) Using quaions (11) and (13) o subsiu ou for h xpcaions in h pric quaion (7) and combining rms lads o p = [(1/1+c) +(c/1+c)q 1 ] m + [1-βq ]w β[q 1 +q 3 ] φ, 1 (14) Now w hav wo disinc quaions-quaions (9) and (14) ha sa p as linar funcion of m, w, and φ. To mak hs quaions consisn, s h hr cofficins in (9) qual o h hr cofficins in (14). I yilds: q 1 = 1; (15) q = (1/1+β); (16) q 3 = - (β/1+ β); (17) 3

29 Thus i follows ha on possibl soluion for p consisn wih srong-form RE is: p = m + (1/1+β)w - (β /1+ β) φ (18) Equaion (18) shows ha h pric lvl is dircly proporional o h mony supply, a posiiv funcion of invsmn shocks, a ngaiv funcion of mony dmand shocks, and a ngaiv funcion of unxpcd mony supply incrass. Th corrsponding srong-form RE for p, o b subsiud in on h righ hand sid of modl quaion(6), is hn found by aking h I -condiional xpcaion of ach sid of quaion (18) bumpd up on priod, which yilds E[p +1 I ] = E[m +1 I ] = m, 1 (19) Combining modl quaion (4) (wih y* = 0) wih (18) and (19), i follows ha h soluion for priod oupu consisn wih srong-form RE is givn by y = α[ (1/1+β) φ + (1 /1+β) w ] (0) Oupu is an incrasing funcion no of mony, bu of unxpcd mony shocks as wll as of shocks u and v o h IS and LM curvs. Equaion (0) has som conclusions for conomic policymaking. From Lucas poin of viw, if h Cnral Bank dcids o lowr h unmploymn ra by an xpansionary monary policy, hn according o h REH h policy will b inffciv. Popl will s wha h Cnral Bank is doing and rais hir xpcaions of fuur inflaion. This is in urn will counrac h xpansionary ffc of h incrasd mony supply. All ha h Cnral Bank can do is o rais h inflaion ra, wih a mos mporary dcrass in unmploymn. Diffrn ss of REH 4

30 Following Sargn (1993), four diffrn ss of Muhian raionaliy may b disinguishd. Ling k variabl X mad a im -k. x signify h xpcaion rpord in h survy for a 1. Unbiasdnss: h survy xpcaion should b an unbiasd prdicor of h variabl. Tha is, a rgrssion of h form x = a + b x + ε k Should yild cofficin simas a=0 and b=1. This is ncssary condiion. A sufficin condiion is as follows x x = E = μ + ε k Th hypohsis o s is μ = 0. Efficincy: h survy xpcaion should us informaion abou h pas hisory of h variabl in h sam way ha h variabl acually volvs hrough im. Tha is, in h wo rgrssions, x k = a 1 X -1 + a X a n X -n + є X = b 1 X -1 + b X b n X -n + u I mus b ru ha a i = b i for all i. This s is calld orhogonaliy s. Anohr possibiliy for xamining h fficincy propry is ha h forcas rror is sd for srial corrlaion. 3. Forcas rror unprdicabiliy: Th forcas rror, ha is, h diffrnc bwn h survy xpcaion and h acual ralizaion of h variabl, should b uncorrlad wih any informaion availabl a h im h forcas is mad. 4. Consisncy: whn forcass ar givn for h sam variabl a diffrn ims in h fuur, h forcass should b consisn wih on anohr. For xampl, in h rgrssions, x x = c c X c n X -n + є 5

31 1 x = a 1 X -1 + a X a n X -n + u I mus b ru ha c i = a i for all i. Ths ss ar diffrn ways of sing propris of condiional xpcaions in ha whhr h rpord survy xpcaions ar consisn wih bing condiional xpcaions. For xampl, considr h fficincy s and suppos ha a 1 b 1. Subsracing h firs quaion from h scond yilds h xprssion x X - 1 = forcas rror = (a 1 - b 1 ) X -1 Sinc, by hypohsis a 1 b 1, his implis ha h forcas rror is corrlad wih X -1, which violas h orhogonaliy of condiional xpcaions as long as X -1 is conaind in h informaion s. Alhough i would b dsirabl for any xpcaion mchanism o saisfy a las som of hs four propris, condiional xpcaions mus saisfy all of hm..1.3 Larning procsss Rol of larning in macroconomics Larning in macroconomics rfrs o modls of xpcaion formaion in which agns rvis hir forcasing ruls ovr im as nw daa bcoms availabl. Larning plays a ky rul in macroconomics. Raional xpcaions can b assssd for sabiliy undr diffrn kinds of larning such as las squars larning. Larning can b usful whn hr is a srucural chang in conomy. Suppos a nw govrnmn appars. Agns nd o larn abou h nw rgim. Bsids, larning can b usd as a slcion cririon whn a modl has mor han on quilibrium soluion. (Bullard, 1991) L us illusra his poin using a modl of hyprinflaion. Assum a govrnmn prins mony o financ a consan budg dfici, hn PG M M 1 = (1) whr P is h pric lvl, G = Gis h consan ral dfici, and M is h mony sock. Suppos h dmand funcion for mony is as follows 6

32 M P M P Ep + whr = f ( Ep ) () 1 Ep < E P = is h xpcd ra of inflaion and ral oupu has (log( )) P bn assumd consan. Considring quilibrium in h mony mark and subsiuing () ino (1) will rsul in p G = f( E p ) f( E p ) (3) (sinc P P p log = p, = ) P 1 P 1 I can b shown ha quaion (3) has wo RE quilibria: h high inflaion quilibrium and h low inflaion quilibrium. If w assum raional xpcaions, h high inflaion quilibrium is locally sabl and h lowr on is unsabl. Ths rankings will b rvrsd if w assum adapiv xpcaions. If i is considrd ha sabiliy is no h appropria slcion criria in a raional xpcaions modl hn hr is no mchanism o choos bwn h wo quilibrium soluions. In such cass, larning provids a slcion cririon. Rsarchrs hav frqunly facd h issu of mulipliciy of RE quilibria in nonlinar modls. Assum a nonlinar modl y = F( y + 1) has h S-shap shown blow 7

33 y F ( y + ) 1 45 o y + 1 Figur 1.1: Mulipliciy of soluions in nonlinar modls Th mulipl sady sas y = F( y) occur a h inrscion of h graph of F (.) and 45-dgr lin. This possibiliy can appar in modls wih monopolisic compiion, incrasing rurns o scal producion or xrnaliis. Ohr spcificaions of his modl can prsn mulipl prfc forsigh quilibia aking h form of rgular cycls in addiion o a sady sa or sunspo qilibria, aking h form of a fini sa Markov procss (Evans and Honkapohja, 001). An inrsing qusion may now b posd: which of h sady sas ar sabl undr larning. Approachs o larning Following Evans and Honkapohja (1999, 001), h approachs o larning can b cagorizd ino hr groups: duciv larning, adapiv larning, and raional larning Educiv larning In h duciv approach, w xamin whhr xpcaions convrg o h raional xpcaions quilibrium hrough a procss of rasoning. Considr h following xampl basd on Dcanio(1979) Considr h dmand and supply in a mark ar givn by q = a bp + w (4) 8

34 Hr q = c+ dp + v (5) q and p ar h acual quaniy and pric lvl, w and v ar random disurbancs which ar assumd o b whi nois and a, b, c, and d ar consan. Dmand is downward-sloping linar funcion of h mark pric and supply dpnds posiivly and linarly on xpcd pric du o a producion lag. p dnos h xpcaions of h rprsnaiv supplir (avrag xpcaions). Th good is assumd o prishabl and marks clar. Th rducd form for h prics is givn by whr p = A Bp + u (6) A a b c =, B d b =, and u w b v =. Firs w xamin h modl undr RE. Th RE hypohsis can b formally sad as p = E( p I ) = E p (7) -1 1 So ha xpcaions ar h ru mahmaical condiional xpcaions, condiional on availabl informaion a h nd of priod -1. Th informaion s includs pas daa { u 1, u,,p -1, P -, }=I -1 and knowldg of h modl. W can compu RE by subsiuing (7) in (6) and obain p = A BE p + u (8) 1 E p = A BE p so ha Taking condiional xpcaions E -1 of boh sids yilds 1 1 xpcaions ar givn by A E 1p = 1 + B And h uniqu RE soluion is of his form: p A = + v. 1+ B Th RE quilibrium for h modl is a random variabl ha is of h form consan plus nois. Undr RE h appropria way o form xpcaions dpnds on h sochasic procss followd by h xognous variabls, v in his cas. Now w considr h modl undr duciv larning. Suppos agns form hir xpcaions iniially in an arbirary mannr, for xampl, saic xpcaions E p = p (9)

35 Th qusion is whhr hy can modify hir bhavior so ha raional xpcaion A quilibrium, givn by, would b aainabl. Plugging (9) ino (8) rsuls in h 1 + B acual voluion of prics p = A Bp + u (10) 1 I is assumd ha afr som passag of im agns raliz (rason or dduc) ha prics ar volving according o (10) and form nw xpcaion E p = A Bp (11) Th voluion of h sysm is changd by his nw xpcaion p = A B( A Bp ) + u = A BA+ B p + u 1 1 Obsrving h nw voluion of prics, agns rvis hir xpcaions o E p = A BA+ B p (1) 1 1 So ha prics volv as follows by plugging (1) ino (8) p = A B( A BA+ B p ) + u = A BA+ B A B p + u (13) If w rpa his procss, h xpcaions afr n iraions will b E p = A BA+ B A B A B A+ B p (14) n 3 n n 1 1 = (1... n n A B B B B ) B p 1 Sinc ( 3 1 n n 1 B + B B B = ) and B p 1 0 for B <1 and larg n, 1 + B xpcaions will convrg o raional xpcaions n A E 1p = 1 + B Th raional xpcaions, in his cas, is said o b iraivly E-sabl. I is clar ha convrgnc o raional xpcaions is no guarand if B >1. Gusnri, 199; Evans, 1985, 1986; Pl and Chappll, 1986; and Bullard and Mira, 000), mploying h iraiv xpcaions mhod in diffrn modls, xamind convrgnc o raional xpcaions Adapiv larning Agns would larn from daa via rgrssion abou h modl and h policy rgim. Alhough his would produc xpcaions formaion vry similar o adapiv 30