On the power of direct tests for rational expectations against the alternative of constant gain learning

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1 Bank i Krdy 41 (6), 2010, On h powr of dirc ss for raional xpcaions agains h alrnaiv of consan gain larning Vicor Bysrov*, Anna Saszwska-Bysrova # Submid: 1 Augus Accpd: 19 Novmbr Absrac In his papr w sudy h powr of dirc ss for raional xpcaions agains h consan gain larning alrnaiv. Th invsigaion is by mans of a Mon Carlo sudy. Th ss considrd us quaniaiv xpcaions daa and qualiaiv survy daa ha has bn quanifid. Th main finding is ha h powr of ss for raional xpcaions agains consan gain larning may b vry small, making i impossibl o disinguish h hypohss. Kywords: adapiv larning, ss for raional xpcaions, quanificaion mhods, consan gain las squars JEL: D84, D83, C12 * Univrsiy of Lodz, Insiu of Economics; -mail: mfvib@uni.lodz.pl. # Univrsiy of Lodz, Chair of Economric Modls and Forcass; -mail: mfans@uni.lodz.pl.

2 72 V. Bysrov, A. Saszwska- Bysrova 1. Inroducion Th raional xpcaions hypohsis has dominad h macroconomics liraur sinc h 1970s. Howvr, in many rcn modls i has bn rplacd by a mor plausibl adapiv larning hypohsis assuming ha agns form xpcaions by simaing and updaing a forcasing funcion (for an ovrviw s.g. Evans, Honkapohja 2001). Applicaions of adapiv larning hav providd nw insighs o ky issus of h monary policy, businss cycls, and ass pricing (Evans, Honkapohja 2009). On imporan conclusion from h adapiv larning liraur is ha policis which ar opimal undr raional xpcaions may no longr b opimal whn agns us a larning mchanism (in h conx of h monary policy s.g. Orphanids, Williams 2008). Givn h cnral rol ha xpcaions play in modrn macroconomic hory i is imporan o b abl o sudy mpirically h way in which xpcaions ar formd. A popular way o invsiga raionaliy is by mans of dirc ss (s.g. Kan, Runkl 1990; Lovll 1986; Psaran 1987; Zarnowiz 1985 and also Łyziak 2003; Tomczyk 2004; Osińska 2000). This approach is prfrrd o indirc sing which focuss on cross quaion paramr rsricions applid o a paricular paramric conomic modl. Th dirc approach uss daa obaind from consumr and businss ndncy survys which can b ihr quaniaiv or qualiaiv, wih h lar yp prvailing (s Psaran, Wal 2006). Th qualiaiv daa provid an xpcd dircion of chang for a givn conomic variabl. For h purpos of mpirical sudis h qualiaiv daa nd o b ransformd o figurs by mans of on of h many convrsion procdurs which hav bn proposd (s Bachlor, Orr 1988; Brk 1999; Carlson, Parkin 1975; Psaran 1987; Siz 1988; Smih, McAlr 1995). In a rcn ovrviw Nardo (2003) summarizs h conradicing rsuls of sandard raionaliy ss whn survy daa ar usd in s rgrssion. Th aim of his papr is o sudy h powr of ss for raional xpcaions whn hs ar applid o xpcaions daa consisn wih h adapiv larning hypohsis. Expcaions ar drivd from a forcasing funcion simad by consan gain las squars (CGLS). W focus on h fficincy and orhogonaliy ss. Th ss ar applid o quaniaiv and quanifid sris. Th propris of h raionaliy ss ar analysd by mans of Mon Carlo xprimns. Alhough h ss ar ofn applid, hir powr has no bn horoughly invsigad. In paricular, hr ar no sudis of h propris of h raionaliy ss whn hs ar applid o xpcaions consisn wih h consan gain larning hypohsis. Our conribuion is qui uniqu for wo mor rasons. Firs, w sudy h powr of raionaliy ss for h daa gnrad from h procss allowing for fdback from xpcaions o h ralizaions of h forcas variabl. Scond, w apply h ss no only o quaniaiv bu also o quanifid daa. Th propris of ss using quanifid sris wr prviously sudid by Common (1985) who xamind h siz and powr of h srial corrlaion s agains h alrnaiv of adapiv xpcaions; h quanificaion mhods h considrd wr h balanc saisics mhod and h Carlson and Parkin (1975) mhod. Th main finding of h papr is ha ss for raional xpcaions may hav vry low powr agains h consan gain larning boh for quaniaiv and quanifid daa. Th ss ar hnc no wll suid for making mpirical disincion bwn h wo yps of xpcaions. Low powr mans ha if h null hypohsis of raionaliy is no rjcd i is no saf o conclud ha xpcaions ar raional as hy migh hav bn gnrad by larning agns. Fals conclusion concrning raionaliy may consqunly lad o a non-opimal choic of policy dsign.

3 On h powr of dirc ss for raional xpcaions 73 Th oulin of h papr is as follows. Scion 2 dscribs h yps of xpcaions daa and scion 3 h quanificaion procdur usd. Scion 4 prsns h alrnaiv xpcaions formaion schms. Scion 5 conains h dscripion of ss for raional xpcaions. Th dsign of h Mon Carlo xprimns and h rsuls obaind ar givn in scions 6 and 7 rspcivly. Conclusions ar prsnd in scion Th xpcaions daa Daa on xpcaions ar usually obaind from consumr and businss ndncy survys. Survy daa can b ihr quaniaiv or qualiaiv. In h firs cas, agns provid a numrical valu for h variabl and in h scond h xpcd dircion of chang. Whil h quaniaiv daa can b dircly usd in conomric sudis, h qualiaiv rsponss nd o b firs convrd ino figurs. In h simulaions w gnra h xpcaions of N rspondns. Th daa hav boh h quaniaiv and quanifid form. Th qualiaiv xpcaions hav h shap ypical of daa on inflaionary xpcaions collcd from consumr survys carrid ou in h OECD counris. In hs survys rspondns sa whhr hy xpc prics o ris fasr han a prsn, ris a h sam ra, ris mor slowly, say a hir prsn lvl or go down. In h simulaions w assum ha survy xpcaions ar formulad wih rspc o h nx priod. Blow, h qualiaiv answrs collcd in priod 1 concrning xpcaions for im ar summarizd by h fracions of rspondns ha answrd ris fasr han a prsn, ris a h sam ra, ris mor slowly, say a hir prsn lvl and go down and ar dnod by RF, SR, RS, S and D, rspcivly. Th qualiaiv daa ar hn usd o driv quaniaiv masurmns of xpcaions by mans of h probabiliy convrsion procdur dscribd in scion Quanificaion procdurs Th qualiaiv rsponss ar quanifid using a vrsion of h probabiliy approach applicabl o daa obaind from survys wih fiv rspons cagoris. Th probabiliy approach rss on svral assumpions which hav bn horoughly rviwd in h liraur (s.g. Bachlor, Orr 1988; Brk 1999; 2002; Forslls, Knny 2004 and Łyziak 2003) and will no hus b dscribd hr in dail. In h cas of inflaionary xpcaions, h ida of h mhod is ha answrs of individual rspondns (from i = 1,..., N) ar formd dpnding on wo snsiiviy inrvals, on cnrd on 0 and h ohr cnrd on h currn prcivd inflaion ra. Boh h prcivd inflaion ra and h nd poins of h indiffrnc inrvals ar assumd o b fixd among h rspondns. Th prcivd ra is furhr assumd o b known and qual o h currn ra of inflaion. Thn h inrvals hav h form: r, r and s, + s. Th rsponss ar formulad as follows. In h cas whn h xpcd inflaion for h i-h rspondn falls wihin h inrval: r, r h rspondn rpors ha prics ar going o say h sam. If h xpcd inflaion is smallr han h lowr nd poin of his indiffrnc inrval, i.. -r, h xpcd dcras in prics is rpord. For h xpcaions falling bwn h valus r and s rspondns claim ha prics will,

4 74 V. Bysrov, A. Saszwska- Bysrova ris mor slowly. Th prics will ris a h sam ra answr is givn if h xpcaions ar covrd by h scond inrval s, + s and h ris fasr rspons rpord in h cas h xpcd inflaion is largr han + s. Givn hs assumpions i can b shown ha: P P P P { r Ω} = F ( r) = D { r r Ω} = F ( r) F ( r) = S { r s Ω} = F ( s) F { s + s Ω} = F ( + s { + s Ω} = 1 F ( + s) = RF P 1 ( r) = RS ) F ( s) = SR Ω whr Ω 1 is h union of individual informaion ss and F 1 is h cumulaiv disribuion funcion of 1. W assum ha F 1 is h cumulaiv sandard normal disribuion. Thn h avrag xpcd ra of pric changs, is givn by: = Φ Φ (1 RF SR RS ) + Φ ( D ) (1 RF SR RS ) + Φ ( D ) ( Φ (1 RF ) + Φ (1 RF SR )) (1) Φ whr 1 Φ is h invrs of h cumulaiv sandard normal disribuion. In wha follows h probabiliy mhod is applid o survy daa gnrad in h Mon Carlo xprimns. 4. Expcaions formaion schms Th main focus of h papr is on h powr of h raionaliy ss agains consan gain larning. For comparison, h powr agains adapiv xpcaions and h siz of h ss ar also considrd. Aloghr hr diffrn yps of xpcaions sris ar mployd: raional xpcaions, xpcaions gnrad as in CGLS adapiv larning and adapiv xpcaions. Th raional xpcaions hypohsis of Muh (1961) assums ha xpcaions ar ssnially h sam as h prdicions of h rlvan hory bu may b subjc o idiosyncraic rrors. In h xprimns, h raional xpcaions ar gnrad so ha all h prdicion rrors mad by agns ar Gaussian whi nois. W assum ha agns know h form of h quaion which gnras h acual daa and is paramr valus. In h cas of adapiv larning, h conomic agns form hir xpcaions on h basis of a forcasing funcion. Th paramrs of h funcion hav o b simad and hy ar updad whn nw daa bcom availabl. Th updaing procds by consan gain las squars. Suppos ha xpcaions for im ar formd a im 1 according o h quaion: ˆ x = '

5 On h powr of dirc ss for raional xpcaions 75 whr is an xpcd valu of h variabl, x is a vcor of prdicors and ˆ is a vcor of paramr simas obaind on h basis of informaion availabl a im 1. Thn h consan gain las squars paramr updaing rul can b wrin as (s Evans, Honkapohja 2001): ˆ = ˆ R = R + γr + γ( x x x' ( x' R ) Th consan gain procdur discouns λpas obsrvaions a a gomric ra 1 γ. CGLS is rasonabl whn mark paricipans bliv ha h conomic nvironmn is changing ovrim bu do no know whn h changs occur. Th las yp of xpcaions i.. adapiv xpcaions ar rvisd in lin wih pas forcas rrors according o: 1 ˆ ) (2) = + λ( 1) (3) whr λ 0,1. λ 5. Tss for raional xpcaions Th hypohsis of raionaliy is ypically invsigad by mans of h orhogonaliy s considrd o b h mos comprhnsiv s for raional xpcaions (s Psaran 1984). Th s has bn applid boh o quaniaiv and quanifid xpcaions. I consiss in rgrssing h xpcaions rror on informaion known a im 1 and sing whhr h informaional variabls ar significan. Th s quaion has hus h following form: ( ) = + 1I + ε 0 whr sands for ihr quaniaiv or quanifid xpcaions, I 1 rprsns informaional variabls known a im 1 and h null hypohsis of raional xpcaions is givn by 0 0 = and 1 = 0. Thr is a spcial cas of h orhogonaliy s concrnd wih h fficin us of h informaion conaind only in h hisory of changs in h variabl undr invsigaion. This fficincy s is basd on h quaion: (4) ( ) = + + ε 0 1 (5) α δ ε and h null hypohsis of raionaliy is 0 = 0 and 1 = 0. Mos commonly h abov hypohss ar sd on h basis of h ordinary las squars (OLS) simaion of (4) and (5) using h F-s. In wha follows w rpor h OLS s rsuls.

6 76 V. Bysrov, A. Saszwska- Bysrova 6. Dsign of h Mon Carlo xprimns To analys h propris of survy-basd ss for raional xpcaions w conduc h Mon Carlo sudy. Th daa gnraing procsss (DGPs) usd o gnra acual ralisaions of variabls ar basd on srucural bivaria modls. Th variabls dnod by and y can b inrprd as an inflaion ra and h dviaion of h acual oupu from is ponial lvl, in which cas h modl bcoms h invrs Lucas supply modl (Lucas 1973). Ohr inrpraions ar, howvr, also possibl. In h xprimns, quaniaiv xpcaions wih rspc o h valus of of N = 1000 rspondns ar gnrad using h alrnaiv xpcaions formaion schms dscribd in scion 4. In ordr o obain h qualiaiv sris, h daa ar convrd ino survy answrs. Thn, h procdur dscribd in scion 3 is usd o pu h qualiaiv rsponss ino numrical form. Th raionaliy ss dscribd in scion 5 ar applid o boh h quaniaiv and quanifid xpcaions. Sris of 100, 200 and 400 obsrvaions ar considrd. Sampls ar gnrad using random iniial valus of h variabls. In h xprimns 1000 rplicaions ar usd. Th daild dscripion of h xprimns is h following: 1. Th psudo-daass ar gnrad from h following gnral procss: y = 0.5y = α + + δy + η + ε (6) Svral valus of h paramr ar invsigad including 0, 0.3, 0.5, 0.7 and 0.9 rsuling in fiv alrnaiv DGPs. Each valu of implis a diffrn wigh wih which xpcaions influnc h acual valus of. For = 0 hr is no fdback from xpcaions o acual ralizaions of. Th ohr paramr valus ar chosn so ha in ach cas h modl corrsponds o h sam rducd form undr h raional xpcaions hypohsis. Hnc α is s o 4(1 ) and δ is pu qual o 1. Th rrors ar assumd o b normally disribud wih h covarianc marix Expcaions of 1000 agns ar hn γ gnrad. γ Th paramrs of h funcions usd o gnra xpcaions consisn wih h CGLS adapiv larning and adapiv xpcaions i.. γ and λ ar s qual o and 0.1 rspcivly. Similar valus ar commonly mployd in horical sudis of many conomic procsss. Sargn (1999) applis hs valus in his simulaion sudy of inflaionary xpcaions. Orphanids and Williams (2008) considr γ = 0.02 o b a rasonabl bnchmark. In an mpirical sudy, analysing US quarrly daa Branch and Evans (2006) obain γˆ = for h GDP growh ra and γˆ = for h CPI inflaion. To xamin h powr of h raionaliy ss mor fully w addiionally considr a much highr valu of γ qual o 0.1. This valu, maning ha pas obsrvaions ar discound gomrically wih h ra 0.9, would imply ha agns prciv h conomic nvironmn as ofn changing. Th raional xpcaions of h i-h agn, i = 1,..., 1000 for ar gnrad as: i = y 1 u, u ~ N(0,6) i i

7 On h powr of dirc ss for raional xpcaions 77 Ths raional xpcaions ar drmind by h forcass of in h raional xpcaions quilibrium condiional on informaion availabl a 1. To obain prdicions of individual rspondns w augmn hs forcass wih individual rrors u i. For CGLS adapiv larning xpcaions h following quaion is usd: αˆ 0 + αˆ 1y 1 ui, ui ~ N(0,6) i = + whr α α α ξ 1 and α 2 ar paramr simas obaind for h quaion = α 0 + α1y + ξ for obsrvaions up o priod 1 using (2) wih γ qual o or 0.1. Ths simas do no convrg o h paramr valus of h raional xpcaions quilibrium. Finally, adapiv xpcaions ar obaind on h basis of: i = i + ( ) + 0.1( i( 1) ) u, u ~ N(0,6) 3. Th quaniaiv avrag xpcaions daa a im ar calculad as N = 1/ N 4. Th qualiaiv survy xpcaions daa ar obaind for h indiffrnc inrvals: 0.3, 0.3 and 0.3, Th convrsion is don in h following mannr: all h individual xpcaions smallr han -0.3 ar xprssd as go down answrs, valus falling ino h inrval 0.3, 0. 3 as say a hir prsn lvl, valus bwn 0.3 and 0. 3 as ris mor slowly, valus covrd by 0.3, as ris a h sam ra and finally, valus xcding as ris fasr han a prsn rsponss. Th quaniaiv y daa ar hn summarizd by calculaing h proporions of ach rspons a im 1, i.. by obaining h RF, SR, RS, S and D saisic valus. 5. Th qualiaiv daa ar convrd o figurs by mans of h probabiliy mhod. 6. Th 5% orhogonaliy and fficincy ss ar hn applid o boh quaniaiv and quanifid xpcaions daa. In h orhogonaliy s 1 and y ar usd as h informaional variabls. 7. Th rjcion ra of h null hypohsis in h raionaliy ss in 1000 rplicaions is calculad. i= 1 i i i 7. Mon Carlo rsuls Th rsuls of h Mon Carlo xprimns ar givn in h Appndix. Ths ar prsnd as proporions of 1000 Mon Carlo rplicaions in which h raional xpcaions hypohsis was rjcd by h fficincy and orhogonaliy ss a h 5% significanc lvl. Th s oucoms ar givn for h quaniaiv xpcaions and for h sris quanifid by mans of h probabiliy mhod. Rsuls ar rpord for fiv DGPs diffring by h valu of qual o 0, 0.3, 0.5, 0.7 or 0.9 and hr sampl sizs of 100, 200 and 400 obsrvaions. Tabl 1 shows h rsuls concrning h simad siz of h alrnaiv ss for raional xpcaions. Undr raional xpcaions h fiv DGPs ar idnical and so only on s of rsuls

8 78 V. Bysrov, A. Saszwska- Bysrova is givn for ach sampl siz. I can b sn ha in ach cas h simad siz of h ss is rasonably clos o h nominal valu. Tabl 2 rpors h powr of h raionaliy ss agains h adapiv xpcaions alrnaiv. Th abiliy of h ss o rjc h null hypohsis is vry good in h majoriy of cass. Th powr of h orhogonaliy s is much br han ha of h fficincy s. For h smalls sampl siz h probabiliy of rjcing h null hypohsis by a las on of h ss, for 0 xcds 0.7, and in h cas of = 0 is of h ordr of 0.5. For sampls of 200 obsrvaions h powr is largr han 0.8 for all h cass and in sampls of 400 obsrvaions h fals null hypohsis is almos always rjcd. Th ss using quanifid daa prform similarly o h ss using quaniaiv daa. Th powr of h raionaliy ss agains h consan gain larning alrnaiv wih γ = is givn in Tabl 3. Th powr of all h ss for all sampl sizs is gnrally smallr γ han h nominal siz. Ths rsuls indica ha using h raionaliy ss i is no possibl o disinguish raional xpcaions from xpcaions formd by agns using his paricular form of CGLS simaion. Th prospcs of rjcing h null hypohsis for largr valus of γ can b valuad by looking a rsuls from Tabl 4 corrsponding o γ = For T = 200, h adapiv larning xpcaions γ ar sill undisinguishabl from h raional xpcaions. Th powr of all h ss is clos o h nominal siz. For T = 200 h probabiliis of rjcing h null hypohsis ar sill qui low and do no xcd For h largs sampl siz h prospcs of rjcing h null hypohsis dpnd on h valu of h paramr. For valus of qual 0.3, 0.5 or 0.7 h powr is highr han 0.5 whil for = 0 or = 0.9 i is smallr han 0.3. Th rsuls show ha h powr of h ss agains consan gain larning is low boh for h cas in which xpcaions influnc h ralisaions of h forcas variabl and for h no-fdback cas. Th rsuls of h ss basd on quanifid daa rproduc hos obaind for quaniaiv sris qui wll. 8. Conclusions Expcaions play a major rol in modrn macroconomic analysis. Conclusions from policy sudis ofn dpnd crucially on h paricular assumpions concrning h way in which xpcaions ar formd i.. dpnd on whhr h raional xpcaions hypohsis, h adapiv xpcaions hypohsis or h adapiv larning hypohsis is usd. Survy daa ar ofn usd for sing h raional xpcaions hypohsis. So-calld dirc ss ar basd on quaniaiv xpcaions daa or quanifid qualiaiv daa. Th ss invsiga h opimal propris of raional xpcaions and hy do no spcify a paricular alrnaiv. In his papr w hav sudid h powr of such ss whn h alrnaiv is consan gain larning. Th main finding of h papr is ha ss for raional xpcaions may hav vry low powr agains h CGLS adapiv larning for valus of h larning paramr ypically considrd in h liraur. Thr is a disurbing implicaion, viz. ha if h null hypohsis of raionaliy is no rjcd i is no saf o conclud ha xpcaions ar raional for hy migh hav bn gnrad by larning agns. As h way in which agns form xpcaions mars,.g. for h opimal policy dsign, fals conclusion concrning raionaliy may hav srious consquncs.

9 On h powr of dirc ss for raional xpcaions 79 Rfrncs Bachlor R.A., Orr A.B. (1988), Inflaion Expcaions Rvisid, Economica, 55 (219), Brk J.M. (1999), Masuring inflaion xpcaions: a survy daa approach, Applid Economics, 31, Brk J.M. (2002), Consumrs Inflaion Expcaions and Monary Policy in Europ, Conmporary Economic Policy, 20 (2), Branch W.A., Evans G.W. (2006), A simpl rcursiv forcasing modl, Economics Lrs, 91, Carlson J.A., Parkin J.M. (1975), Inflaion xpcaions, Economica, 42, Common M. (1985), Tsing for raional xpcaions wih qualiaiv survy daa, Manchsr School of Economic and Social Saisics, 53 (2), Evans G.W., Honkapohja S. (2001), Larning and xpcaions in macroconomics, Princon Univrsiy Prss. Evans G.W., Honkapohja S. (2009), Larning and Macroconomics, Annual Rviw of Economics, Spmbr, 1, Forslls M., Knny G. (2004), Survy xpcaions, raionaliy and h dynamics of Euro Ara inflaion, Journal of Businss Cycl Masurmn and Analysis, 1, Kan M.P., Runkl D.E. (1990), Tsing h Raionaliy of Pric Forcass. Nw Evidnc from Panl Forcass, Amrican Economic Rviw, 80 (4), Lovll M.C. (1986), Tss of h Raional Expcaions Hypohsis, Amrican Economic Rviw, 76 (1), Lucas Jr. R.E. (1973), Som Inrnaional Evidnc on Oupu-Inflaion Tradoffs, Amrican Economic Rviw, 63 (3), Łyziak T. (2003), Consumr Inflaion Expcaions in Poland, Europan Cnral Bank, Working Papr Sris, 287. Muh J.A. (1961), Raional Expcaions and h Thory of Pric Movmns, Economrica, 29 (6), Nardo M. (2003), Th Quanificaion of Qualiaiv Survy Daa: A Criical Assssmn, Journal of Economic Survys, 17 (5), Orphanids A., Williams J.C. (2008), Larning, xpcaions formaion and h pifalls of opimal conrol monary policy, Journal of Monary Economics, Ocobr, 55 (Supplmn), S80 S96. Osińska M. (2000), Ekonomryczn modlowani oczkiwań gospodarczych, Wydawnicwo Uniwrsyu Mikołaja Koprnika w Toruniu. Psaran M.H. (1984), Expcaions formaions and macroconomic modlling, in: P. Malgrang, P.A. Mu (ds), Conmporary Macroconomic Modlling, Basil Blackwll, Oxford. Psaran M.H. (1987), Limis o Raional Expcaions, Basil Blackwll, Oxford. Psaran M.H., Wal M.R. (2006), Survy Expcaions, in: Handbook of Economic Forcasing, vol. 1, Elsvir. Sargn T.J. (1999), Th Conqus of Amrican Inflaion, Princon Univrsiy Prss. Siz H. (1988), Th simaion of inflaion forcass from businss survy daa, Applid Economics, 20,

10 80 V. Bysrov, A. Saszwska- Bysrova Smih J., McAlr M. (1995), Alrnaiv procdurs for convring qualiaiv rspons daa o quaniaiv xpcaions: an applicaion o Ausralian manufacuring, Journal of Applid Economrics, 10, Tomczyk E. (2004), Racjonalność oczkiwań. Mody i analiza danych jakościowych, Monografi i Opracowania, 529, SGH, Warszawa. Zarnowiz V. (1985), Raional Expcaions and Macroconomic Forcass, Journal of Businss and Economic Saisics, 3, Acknowldgmns Suppor from h EU Commission hrough MRTN-CT COMISEF is grafully acknowldgd. Th auhors would lik o hank wo anonymous rfrs for commns and suggsions which hlpd o improv his papr.

11 On h powr of dirc ss for raional xpcaions 81 Appndix Tabl 1 Esimad siz of h 5% fficincy and orhogonaliy ss for raional xpcaions Quaniaiv daa Quanifid daa Efficincy s Orhogonaliy s Efficincy s Orhogonaliy s T = T = T = Nos: Esimad siz of h 5% fficincy and orhogonaliy ss is givn for boh quaniaiv xpcaions and xpcaions quanifid by mans of h probabiliy mhod. Th ru indiffrnc inrvals ar givn by: 0.3, 0. 3 and.3, 0.3. Rsuls ar rpord for sampl sizs T of 100, 200 and 400 obsrvaions. 0 + Tabl 2 Esimad powr agains adapiv xpcaions wih λ = 0. 1 of h 5% fficincy and orhogonaliy ss for raional xpcaions Quaniaiv daa Quanifid daa Efficincy s Orhogonaliy Orhogonaliy Efficincy s s s T = 100 = = = = = T = 200 = = = = = T = 400 = = = = = Nos: Esimad powr agains adapiv xpcaions wih λ = 0. 1 of h 5% fficincy and orhogonaliy ss is givn for boh quaniaiv xpcaions and xpcaions quanifid by mans of h probabiliy mhod. Th ru indiffrnc inrvals hav h form: 0.3, 0. 3 and 0.3, Rsuls ar rpord for fiv DGPs givn by (6) wih valus of of 0, 0.3, 0.5, 0.7 and 0.9 and sampl sizs T of 100, 200 and 400 obsrvaions.

12 82 V. Bysrov, A. Saszwska- Bysrova Tabl 3 Esimad powr agains CGLS adapiv larning wih γ = ss for raional xpcaions Quaniaiv daa of h 5% fficincy and orhogonaliy Quanifid daa Orhogonaliy Orhogonaliy Efficincy s Efficincy s s s T = 100 = = = = = T = 200 = = = = = T = 400 = = = = = Nos: Esimad powr agains CGLS adapiv larning wih γ = of h 5% fficincy and orhogonaliy ss is givn for boh quaniaiv xpcaions and xpcaions quanifid by mans of h probabiliy mhod. Th ru indiffrnc inrvals hav h form: 0.3, 0. 3 and 0.3, Rsuls ar rpord for fiv DGPs givn by (6) wih valus of of 0, 0.3, 0.5, 0.7 and 0.9 and sampl sizs T of 100, 200 and 400 obsrvaions.

13 On h powr of dirc ss for raional xpcaions 83 Tabl 4 Esimad powr agains CGLS adapiv larning wih γ = 0. 1 of h 5% fficincy and orhogonaliy ss for raional xpcaions Quaniaiv daa Quanifid daa Orhogonaliy Orhogonaliy Efficincy s Efficincy s s s T = 100 = = = = = T = 200 = = = = = T = 400 = = = = = Nos: Esimad powr agains CGLS adapiv larning wih γ = 0. 1 of h 5% fficincy and orhogonaliy ss is givn for boh quaniaiv xpcaions and xpcaions quanifid by mans of h probabiliy mhod. Th ru indiffrnc inrvals hav h form: 0.3, 0. 3 and 0.3, Rsuls ar rpord for fiv DGPs givn by (6) wih valus of of 0, 0.3, 0.5, 0.7 and 0.9 and sampl sizs T of 100, 200 and 400 obsrvaions.

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