1 Constant Real Rate C 1
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1 Consan Real Rae. Real Rae of Inees Suppose you ae equally happy wh uns of he consumpon good oday o 5 uns of he consumpon good n peod s me. C 5 Tha means you ll be pepaed o gve up uns oday n eun fo 5 uns n one peods me. Ths mples ha he eal ae of nees ( ) equals 5%.. Nomnal Rae of Inees Say a un of he consumpon good coss $ oday and you have $ oday. If he fully ancpaed ae of nflaon (π ) s expeced o be % fo nex peod how many dollas wll you need nex peod o buy 5 uns of he consumpon good? 5 * ( ) * (.) 5.5 Theefoe we need $55.5 nex peod o buy 5 uns of he consumpon good and he equed nomnal eun (R) s gven by: 5.5 R. 5.5% Wha s he elaon beween he nomnal eun and he eal eun? Les go back o ou calculaons above. dvde by cuen wealh 5( )(.) (. ).555 (.5)(.).55 ( )( π ) ( R) ()
2 If euns ae connuously compounded + π R () Ths s he Fshe equaon. In a wold of unceany hs s no a auology and n fac educes o: [ ] [ ] R E + E π (3) Fama (75) Fama ess equaon (3) unde he assumpon ha E E a consan Makes ae effcen. [ ] [ ] Ths mples fom equaon (3) ha [ ] [ ] R E + E π (3 ) Ths mples ha a one pecen ncease n expeced nflaon leads o a one pecen ncease n nomnal euns. Thus nomnal euns ae fully evealng as o expeced nflaon.. Auocoelaons of π Fama s nees n esng make effcency, ha s he make makng use of all avalable nfomaon n fomng expecaons of nex peods nflaon. Pehaps hee s no avalable nfomaon e nflaon. Tha s, pehaps nflaon s unpedcable. Is nflaon pedcable? Table shows us ha he answe s yes. The auocoelaons of ρ π ae hgh and sgnfcan (.36).. Auocoelaon of Snce s assumed o be a consan auocoelaons of should be zeo fo all lags. Table epos he auocoelaon of R π fo lags o. Each auocoelaon s small (.) and nsgnfcan. Queson Answe Is hee a way o es he jon sgnfcance of hese auocoelaons? The hypohess o be esed s ha
3 E k [ ] Ths mples ha [ ] E k Le us mulply by Z whee Z epesens elemens n he agens nfomaon se a me. Clealy Z ncludes j fo any j >. Thus we have k ( k) ( k) E ( k ) ( k) 3 Now we can pefom Hansens GMM o oban he value of he ovedenfyng sasc whch n hs case wll be χ. Of couse we could have chosen Z o be any elemen of he nfomaon se and no esced ouselves o pas values of. Of couse, you could un he auocoelaon ess ha we saw las week n opc 5, Seal Dependence. Would such a jon es ejec he consan eal ae hypohess? You wll fnd ou n you nex assgnmen ha he answe s yes. In addon Nelson and Schwe (977) pon ou ha Fama s es based on sample auocoelaons of has lle powe agans alenaves whch specfy economcally plausble vaaon and auocoelaon n expeced eal nees aes. To see hs defne he condonal expeced eal eun Thus E [ ] + ε Suppose he expeced eal eun s coelaed as follows φ + ω
4 Ths mples ha ( ) ( ) cov, cov, φ σσ, σ ( ) cov, φσ Now and + ε + ε ( ) ( ε ε ) ( ) ( ) ( ε ε ) ( ε ) ( ) ( ) φσ cov, cov +, + cov, cov, + cov, + cov, cov, cov, In addon va ( ) va ( + ε ) σ + σ ε Thus ρ ( ) cov, φσ, σ σ + σε Table n Nelson and Schwe (977) show ha he obseved value of ρ,. s conssen wh expeced eal aes changng ove me. Fom daa n we have ha ρ,. σ 4.3 Le us y a φ.4 ε ρ φσ, σ + σε σ. σ σ.44 σ. (.4) + 4.3
5 Now.3 Regessons Fom equaon (3 ) we see ha φ + w σ φσ + σw σw σ φσ σ w ( ) ( ) [ ] [ π ] [ π ] [ ] E[ ] R R E + E E E + R π + + ε and hus we can hnk of pefomng he egesson π α + α R (5) Takng expecaons of equaon (5) and equang coeffcens wh he model gven by equaon (4) mples E[ ] Thus he esable mplcaon s ha α fo he peod of hs sudy Fama fnds ˆ α.98 and can no ejec ha α Queson Pevous sudes had π as he explanaoy vaable. Why dd Fama swch he egesson so ha R was he explanaoy vaable?
6 Fama also uns he egesson π α + αr + απ (6) Once agan akng expecaons of equaon (6) and equang wh he model gven by equaon (4) we have E[ ] Thus he esable mplcaons hee ae ha α and ha α. Fo he peod unde sudy Fama fnds a ˆ.87 and ˆ α. and canno ejec he null hypohess of α and α. Nelson and Schwe ake ssue wh Fama s egesson equaon (6) above. They clam ha he powe of such a es wll be low f π conans lle nfomaon abou π. Nelson and Schwe use he mehodology of Box and Jenkns o consuc an opmal ˆ π nsead of. me sees pedco ( ) π Assume he obseved ae of nflaon π akes on he followng fom π µ + u + υ + ξ Also assume ha he opmal me sees esmao ˆ π s ˆ π µ + υ and ha he make foecas of nflaon R s R µ + u +. Thus he opmal me sees foecas leaves ou u and he make foecas leaves ou υ. Assume ha u, υ, and ξ ae uncoelaed and ha boh R and ˆ π ae unbased. When we un he egesson
7 π β βπ ε ˆ a+ R + + Now snce he explanaoy vaables ae unbased we have ha and snce β β we have β β ( R ˆ) ( R ˆ ˆ π π π) va ( R ˆ π ) ( u + υ u + ξ) va ( u + υ ) π ˆ π α + β π + ε cov, β cov, β β σ u συ + σu + σ β σ + σ υ συ + σu + σ Fama assumes ha σ he ex ane eal eun s consan and συ, π s an opmal foecas. and checks fo β and β. Bu even f σ υ we wll see β as σ affecs β Thus hs demonsaes clealy ha Fama s esng a jon hypohess. Foecass ae aonal and Real ae s consan. Nelson and Schwe sugges ha he acual daa shows ha he opmal foecas of π s gven by a fs ode MA on fs dffeences of π ha s ( L) π ( θl) e ( θl) ( L) π e π π + θ( π π) + θ ( π π3) + e 3 π π ( θ) π θ( θ) π θ ( θ) π θ ( θ) e 3 4 ( ) e π θ θ π +
8 Thus ( ) ˆ π θ θ π whee θ.89. Noe ha he wegh gven o π n hs foecas s vey small. Fama s es had lle powe as π was hus a saw man..4 Wha f π s non-saonay If we un he egesson π α + α R Then unde he consan eal ae hypohess R s fully evealng as o expeced nflaon and hence s self nonsaonay. When we calculae α ( XX) XY we assume ha p lm XX s a consan. Wh non saonay hee s no eason o expec T hs quany o convege and we ae hus pesened wh a poblem fo asympoc nfeence on an uncondonal bass. To ge d of non-saonay Nelson and Schewez un Now he model gven by equaon (4) s ( R ) π π α + α π + e [ π ] [ ] [ ] [ ] E E + R E π π E + R π Thus when we equae coeffcens s sll he case ha E[ ] and So hs s fne.
9 Wha abou unnng π π α α ) + R R + e Tha s akng fs dffeences and hen unnng he egesson. Is sll he case ha α Consde he case whee π follows a andom walk ( π π + a E π [ π ] and snce [ π ] [ ] R E[ ] E E + R + π In hs case ( R R π π ) va ( R R ) cov ( π π, a) va ( π π ) cov ( a, a ) cov, α α α σ a α snce ( a a ) cov,. So f you un π π α α ) + R R + e You have los sgh of he model. The model s no a model abou dffeences of nflaons vesus dffeence of R. The model says and hs does no mply ( [ π ] [ ] E E + R [ π π ] [ ] E E + R R
( ) ( )) ' j, k. These restrictions in turn imply a corresponding set of sample moment conditions:
esng he Random Walk Hypohess If changes n a sees P ae uncoelaed, hen he followng escons hold: va + va ( cov, 0 k 0 whee P P. k hese escons n un mply a coespondng se of sample momen condons: g µ + µ (,,
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