I) Til: Raional Expcaions and Adapiv Larning II) Conns: Inroducion o Adapiv Larning III) Documnaion: - Basdvan, Olivir. (2003). Larning procss and raional xpcaions: an analysis using a small macroconomic modl for Nw Zaland. Discussion Papr Sris, Rsrv Bank of Nw Zaland, DP2003/05. - Evans, Gorg W., Honkapohja, Sppo. (200). Larning and Expcaions in Macroconomics, Princon Univrsiy Prss. - Lucas, Robr E. (973). Som Inrnaional Evidnc on Oupu-Inflaion Tradoffs. Amrican Economic Rviw, Vol. 63, No. 3.
Raional Expcaions and Adapiv Larning Tabl of conns:. Inroducion 2. Th Lucas aggrga supply modl 3.) Economric Larning 4.) Expcaional Sabiliy 5.) Convrgnc 5.) Rcursiv las squars 5.2) Sochasic rcursiv algorihms 5.3) Applicaion o h Lucas modl 2
. Inroducion Th raional xpcaions hypohsis has ofn bn criicizd for bing oo srong an assumpion. I has bn suggsd o rlax his assumpion by modlling h bhaviour of conomic agns as h bhaviour of saisicians whn hy produc forcass abou h fuur sa of h conomy. This approach is calld adapiv larning as h agns upda hir forcasing rul as nw obsrvaions bcom availabl. Dspi his boundd raionaliy of h agns, a raional xpcaions quilibrium migh b larnd by hm. W will prsn h main idas and rsuls of h adapiv larning liraur by applying hm o h Lucas aggrga supply modl. 2. Th Lucas aggrga supply modl Considr h modl basd on Lucas (973) wih aggrga supply funcion givn by y = y+ π p p + ζ (0.) ( ) whr y dnos h log of oupu, y is h log of long run oupu, p h log of currn prics, p h xpcaions of prics for im basd on informaion up o im, ζ is a whi nois shock and π > 0 is a paramr. Th aggrga dmand in logs is m + v = p + y (0.2) whr m is h mony supply, following h policy rul m = m+ ρ w + u (0.3) and v is a vlociy shock drmind by v = μ + γ w + ξ (0.4) whr w is an n vcor of xognous variabls wih E [ w ] = 0, E [ ww ] = Ω and boh u and ξ ar whi nois shocks. Solving for h rducd form yilds p = + π m y+ v + π p ( ) ( ) = + m+ y + + p + + + w ( π) ( μ ) π ( π) ( π) ( ρ γ) ( π) ( u ξ ζ ) + + + = ϕ+ αp + δ w + η No ha w hav by dfiniion ( ) 0< α = π + π <. (0.5) Th raional xpcaions assumpion uss p = E p Ω whr Ω dnos h informaion s and by applying h condiional xpcaion opraor o (0.5) w g E p Ω = ( α) ( ϕ+ δ w ) (0.6) Anohr implicaion of h raional xpcaions assumpion is ha p E p Ω = ε whr ε is a whi nois rror. W can obain h raional xpcaions quilibrium (REE) from (0.6) 3
p ( ) ( ) = α ϕ+ α δ w + ε = a + bw + ε W conclud ha his modl has a uniqu REE givn by (0.7). (0.7) 3.) Economric Larning W now rlax h hypohsis of raional xpcaions and assum ha agns form hir xpcaions by conomric larning. Knowing ha h Lucas modl has a uniqu REE, i will b of mos inrs o know whhr i is larnabl or no in h sns ha i convrgs undr conomric larning o h REE. Suppos ha firms bliv ha prics follow h procss p = a+ bw + ε (0.8) corrsponding o h RRE (0.7), bu ha h ru paramrs a and b ar unknown o hm. Possibl rasons migh ihr b ha firms do no know h srucur of h conomy bu corrcly assum ha p dpnds linarly on w, or ha hy know h srucur bu no h srucural paramrs π, ρ and γ. Equaion (0.8) is calld h prcivd law of moion (PLM) and firms us las squars rgrssion o sima h paramrs of i. Dno by a and b h LS simas basd on informaion availabl a im and compud wih h sandard LS formula a = zi zi zi pi b (0.9) i= i= whr z i = ( w i). Th xpcd pric is hn givn by h forcas p = a + b w (0.0) Equaions (0.5), (0.9) and (0.0) fully spcify a dynamic sysm. A firms produc simas a and b, which hy us o forcas p. A h bginning of im, givn w and η, p is drmind by (0.5). Firms hn produc simas a and b... Th qusion of inrs now is if lim a = a and limb = b. 4.) Expcaional Sabiliy Firs w nd o mak sur ha h REE is sabl undr larning. In our xampl w assum ha h agns us h PLM (0.8) o form hir pric xpcaions. No ha w usually ak h PLM o b of h sam form as h REE of inrs. If w subsiu h pric forcas back in (0.5), w can solv for h acual law of moion (ALM) p = ( ϕ + αa) + ( αb+ δ) w + η (0.) Th ALM dscribs h sochasic procss followd by h conomy if forcass ar mad by using h PLM. This implicily dfins h mapping T from h PLM o h ALM for h cofficins a and b 4
a ϕ + αa T = (0.2) b δ + α b No ha h uniqu REE is h uniqu fixd poin of h T-map. Considr h diffrnial quaion d a a a = T (0.3) b b b Th REE is said o b xpcaionally sabl or E-sabl if h REE is locally asympoically sabl undr (0.3). E-sabiliy hus drmins h sabiliy of h REE undr h las squars larning rul which is usd o gradually adjus h PLM paramrs a and b in h dircion of h implid ALM paramrs. Th REE ( ab, ) is sabl if small displacmns from ( ab, ) ar rurnd o ( ab, ) undr h larning rul. Combining (0.2) and (0.3) w can drmin h E-sabiliy of h Lucas modl da = ϕ+ ( α ) a (0.4) dbi = δi + ( α ) bi for i=, Kn Th quaions in (0.4) imply ha h modl is E-sabl if and only if α <. By rcalling 0< α = π + π <, w can conclud ha h Lucas modl is E-sabl. ( ) 5.) Convrgnc 5.) Rcursiv las squars Th nx qusion of inrs is whhr h modl convrgs o h REE undr las squars larning. W assum ha agns us rcursiv las squars (RLS) o upda hir simas a and b as nw obsrvaions of p and w bcom availabl. W us anohr vrsion of h RLS formula w hav sn in his class so far. P = p K p b h Z = z K z L ( ) whr ( ) z = w b h k vcor of ndognous variabls, l ( ) 0 marix of xplanaory variabls and = ( ) φ a b h k vcor conaining simad cofficins. From h sandard LS formula for a rgrssion of p on z (cf (0.8)) w can driv h updaing quaion for φ as follows whr = ( ) givn by ( ) φ = ZZ ZP ( p ) ( p ) ( ) ( p ) ( ) ( ) = ZZ Z P + z = ZZ Z Z φ + z ( ) ( ) = ZZ ZZ z z φ + z ( ) ( p ) ( p ) = φ + ZZ z z φ = φ + R z z φ R ZZ dnos h momn marix for z. Th updaing formula for R is (0.5) 5
( ) ( ) + ( ) ( ) ( z z R ) R = ZZ = Z Z + z z Z Z z z Z Z = = R + (0.6) Rcalling (0.5) and (0.0) w hav p = ( ϕ + αa ) + ( δ + αb ) w + η (0.7) = T ( φ ) z + η Combining (0.5) and (0.6) wih (0.7) w g h sochasic rsursiv sysm φ = φ + R z z T φ φ + η (0.8) ( ( ( ) ) ) ( ) R = R + z zr (0.9) I rmains o drmin whhr his sochasic rcursiv sysm convrgs as. W would lik o find ha T φ = φ h uniqu REE is h uniqu φ φ bcaus from ( ) fixd poin of h map w could hn conclud ha h pric procss convrgs o h REE! A his poin i bcoms mos obvious whr h boundd raionaliy of h agns coms from: Th ru procss followd by p, which is givn in (0.7), has im varying paramrs, bu h agns ar simaing a modl (0.8) wih consan paramrs. This modl misspcificaion maks h agns no o ac fully raional. No howvr ha if h larnd cofficins convrg o h REE his diffrnc disappars in h limi. 5.2) Sochasic rcursiv algorihms W firs show som gnral rsuls and hn apply hos o h Lucas modl. Considr h sochasic rcursiv algorihm (SRA) θ = θ + γ Q(, θ, x ) (0.20) whr θ is h vcor of simas, γ is a drminisic squnc of gains, and x is h sa vcor which may ihr follow an xognous procss or a VAR procss whr h cofficins dpnd on θ. Th funcion Q ( ) drmins how θ is updad as h obsrvaion of h las priod bcoms availabl. Sochasic approximaion rsuls show ha h bhaviour of a SRA is wll approximad by an ordinary diffrnial quaion (ODE) for larg dθ h( ( τ )) lim E Q(,, ) = θ = θ x (0.2) if ha limi xiss. Possibl limi poins of h SRA corrspond o locally sabl quilibria of h ODE. Undr suiabl assumpions, if θ is a locally sabl quilibrium poin of h ODE, hn θ is a possibl poin of convrgnc of h SRA. If θ is no a locally sabl quliibrium poin of h ODE, hn θ is no a possibl poin of convrgnc of h SRA, i.. θ θ wih probabiliy 0. 6
Th ncssary chnical assumpions in ordr o obain his convrgnc condiions ar in paricular ha w nd rgulariy assumpions on Q ( ), condiions on h ra a which γ 0, and assumpions on h propris of h sochasic procss followd by x. W jus no ha all hos condiions ar m by h Lucas modl. Parnhsis: Shor rviw of ODE propris: d / = h - θ is an quilibrium poin of h ODE θ ( θ ) if h ( θ ) = 0. - θ is locally sabl if ε > 0, δ > 0 such ha θ( τ ) θ < ε for all ( ) < - θ is locally asympoically sabl if i is locally sabl and θ( τ ) θ for all ( 0) θ 0 θ δ. θ in h nighbourhood of θ. - θ is locally unsabl if i is no locally sabl. h θ provids informaion on h local sabiliy Furhr rcall ha h Jacobian Dh( θ ) of ( ) of θ : - If all ignvalus of Dh( θ ) hav ngaiv ral pars, hn θ is a locally sabl quilibrium poin of d / = h( ) θ θ. - If som ignvalu of Dh( θ ) has a posiiv ral par, hn θ is no a locally sabl quilibrium poin of d / = h( ) θ θ. 5.3) Applicaion o h Lucas modl Firs, w nd o pu h SRA givn by quaions (0.8)-(0.9) in sandard form φ = φ + S z z T φ φ + η (0.22) ( ( ( ) ) ) ( ) S = S + z zs (0.23) + whr w had o chang noaion S = R sinc h sandard form allows only laggd valus on h RHS of h quaions (0.22) and (0.23) L θ = vc( φ S ), x = ( w w η) and γ = hn w can wri h wo Q, θ, x as componns of h funcion ( ) Q (,, ) T( ) φ ( ( ) η ) θ x = S z z φ φ + QS (, θ, x) = vc ( z z S) + Nx, compu h associad ODE. To do so w hav o fix a valu for θ and ak h xpcaion ovr x o g (, ) lim ( ) 7 ( ( ) η ) (0.24) hφ φ S = E S z z T φ φ + (0.25) hs ( φ, S) = lim E ( z z ) + S 0 E zz = E z z = = M, no ha E[ z η ] = 0 and lim =, hn 0 Ω + L [ ] [ ]
h h φ S ( ) (, ) T( ) ( φ, S) = MS φ S = S M φ φ (0.26) Th associad ODE is hrfor dφ = S M ( T ( φ ) φ ) (0.27) ds = M S This sysm is rcursiv and h scond quaion is globally sabl, hus S M from any saring poin. This implis S M I if S is invribl along h pah and w hrfor only hav o look a T ( φ) φ in ordr o find ou if h sysm (0.27) is sabl. From h dfiniion (0.2) w find ϕ T ( φ) φ= + ( α ) Iφ (0.28) δ α I. No ha all which is a linar diffrnial quaion wih cofficin marix ( ) ignvalus of ( α ) I hav ngaiv ral pars if α <. Sinc his is h cas w find ha φ is a globally sabl quilibrium poin of (0.28) and h SRA rsuls hrfor imply ha for h SRA (0.22)-(0.23) ( φ, S ) ( φ, M ) wih probabiliy. Th Lucas modl is found o b E-sabl undr larning and i will always convrg o h REE undr larning. 8