for Model Selection in the AR(1)

Transcription

1 Loal o Uniy, Long-Hoizon Foeasing hesholds fo Model Seleion in he AR John L. une # Absa: he pape develops a famewok fo analyzing long-hoizon foeasing in he AR model using he loal o uniy speifiaion of he auoegessive paamee. I epo new asympoi esuls fo he disibuions of he foeas eos when he AR paamees ae esimaed wih odinay leas squaes OLS, and also fo he disibuions of Random Walk RW foeass. hee exis funions, elaing loal o uniy dif o foeas hoizon, suh ha OLS and RW foeass shae he same expeed squae eo. RW foeass ae pefeed on one side of hese foeasing hesholds, while OLS foeass ae pefeed on he ohe. I idenify hese foeasing hesholds, use hem o develop novel model seleion ieia and show how hey help a foease edue eo. JEL Classifiaion: C Keywods: foeasing, auoegessive, nea-non-saionay, long-hoizon Sepembe 3 # Depamen of Eonomis, ey College of Business, Univesiy of Geogia, Books Hall 5 h Floo, Ahens, GA 36. el: Fax: jlune@ey.uga.edu. I hank Chisophe Ook, im Vogelsang, Jonahan Wigh, Bill Lasapes, wo anonymous efeees and semina paiipans a he Univesiy of Viginia fo hei helpful ommens.

2 ime seies foeasing of nea non-saionay, possibly I, auoegessive poesses emains pooly undesood despie he fequen use of suh models in he eonomei and maoeonomi lieaue. Reseah has foused on wo foeasing models o hoose fom, he VAR famewok and he I speified o-inegaion famewok. Howeve, neihe uni oo no foeasing eseah has yielded peise heoeial ules fo seleing beween hese wo models. Conside he univaiae AR model, fo whih he I famewok speifies he Random Walk model RW and he VAR famewok uses a simple odinay leas squaes OLS egession o esimae he paamees of he model. I is lea ha, in finie samples, RW foeass have lowe mean-squae eo MSE han OLS foeass when he auoegessive paamee is suffiienly lose, bu pehaps no equal, o uniy see, e.g. Sok 996 and Diebold and Kilian. As a esul, RW foeass an be pefeed even whee he Random Walk model iself epesens a misspeifiaion of he ue ime seies poess. In suh ases, RW podues biased foeass. Leing he auoegessive paamee shink away fom uniy, his bias ineases, diving up he eo of RW foeass elaive o OLS. Evenually, fo suffiienly small values of he auoegessive paamee, he MSE of RW foeass beomes equal o, hen lage han, he MSE of OLS foeass. I em he value of he auoegessive paamee whee MSERW = MSEOLS a foeasing heshold, sine on one side of he heshold RW is a pefeed foeasing model, while on he ohe side OLS is pefeed. Sine he heshold value of he auoegessive paamee hanges wih sample size, I appeal o loal o uniy heoy o idenify asympoi Piman dif hesholds ha ae obus o hanges in sample size. he idenifiaion of hese hesholds is impoan fo hee pimay easons. Fis, i exends Sok 996 in a way ha allows us o auaely pedi he elaive pefomane of RW and OLS foeass. Seond, i helps explain how downwad bias in he OLS esimae of he auoegessive paamee affes foeasing. Mos impoanly, hough, i efames he quesion of how bes o sele beween RW and OLS. his pape shows ha endeing a saisial judgmen as o he pesene o absene of a uni oo should no be a majo onen of a foease.

3 . Inoduion Conside he anonial AR model: y d u ε ~ iid, σ u = = d = µ + θ = ρu + u + ε, ρ [,] While he model is only univaiae wih a simple innovaion suue, i is noneheless vey impoan o eonomiss. heoy ofen pedis ha eain maoeonomi aggegaes have an auoegessive suue, and empiial eseah onfims he fis-ode auoegessive endenies see, e.g., Hall 978. Given his, i is impoan fo eonomiss o undesand fully how bes o foeas his model. Addiionally, he lessons leaned fom he AR model an hen be exended in moe geneal VAR and ARMA speifiaions. I onside hee diffeen vesions of : Case I: =, θ = Case II:, θ = Case III:, θ µ y = ρ y + ε µ y = α + ρy + ε µ y = α + δ + ρy + ε 3 whee α = µ ρ + θρ, δ = θ ρ. Cases II and III ae widely appliable o foeasing eonomi vaiables. Case I is of mosly heoeial inees bu is elaively saighfowad o analyze. I onside i pimaily fo exposioy puposes. Seveal pevious papes have expessed analyially he disibuion of foeass of ases I and II when he model s paamees ae esimaed by leas-squaes. Box and Jenkins 97 deive ude, analyially aable, appoximaions fo he disibuion of AR foeass fo Case I, while Phillips 979, Case I, Fulle and Hasza 98, Case II and Magnus and Pesaan 989, Case II deive moe peise, analyially umbesome, appoximaions fo he disibuions of hese foeass. his pas eseah, whih employs adiional pobabiliy heoy, offes onsideable insigh ino AR foeass. Howeve, he developmen and disseminaion of loal o uniy My speifiaion of u = follows Sok 99, fo easons ha will beome lea lae in he pape. 3

4 asympoi heoy has laid he foundaion fo signifianly moe poweful analysis of foeasing. I speifies he auoegessive paamee ρ as being in an O/ neighbohood of uniy: ρ = + 4 Fo asympoi analysis, he dif is held onsan as appoahes infiniy. Using he speifiaion in 4 and he funional enal limi heoem, i is possible o explain why nealy non-saionay AR poesses eain some impoan popeies of I poesses. Many auhos use his speifiaion o develop uni oo ess and median unbiased esimaion poedues Cf. Bobkoski 983, Chan and Wei 987, Phillips 987, Sok 99, Ellio e al 996 and Ellio 999 while Sok 996, Phillips 998, Kemp 999 and Ng and Vogelsang use i o analyze foeasing. Sok 996, Phillips 998 and Kemp 999, who onside Case I, onside boh onesep-ahead and long hoizons. Fo he lae ase, hey ea he foeas hoizon asympoially as a faion η of he sample size of he ime seies i.e. η = h/, whee h is he foeas hoizon and is he sample size. Sok 996 shows ha he fis ode asympoi pefomane of long-hoizon foeass depend on whehe one hooses a VAR o o-inegaion famewok. An impliaion of his esul is ha, as ineases o infiniy, OLS and RW foeass end o dispese he same elaive o eah ohe if he dif and he hoizon η ae held onsan. his pape adops boh he loal o uniy and he long-hoizon speifiaions. I exend Sok 996 o show ha, fo a given foeas hoizon η, sine OLS foeas eos impove monoonially elaive o RW foeass as falls, hee is a value of suh ha he expeed squaed foeas eos fom hese models ae asympoially idenial. Speifially, fo a given foeas hoizon η, hee is a value of dif, *η, suh ha RW and OLS ae equally auae. I hus say ha fo a given value of η, *η is he foeasing heshold of he AR model a foeas hoizon η. hen, if > *, RW is a pefeed model and if < *, OLS is pefeed. In his pape, I idenify foeasing hesholds fo ases I, II and III. Sine Sok 996, Phillips 998 and Kemp 999 eah epo loal o uniy, long-hoizon asympoi I also allows a eseahe o absa fom some disibuional assumpions 4

5 disibuions of Case I foeas eos fo RW and OLS, I mus only apply hei heoems o build he heshold funion *η fo Case I. I do so in seion 3. Fo ases II and III, howeve, muh moe wok is needed. Ng and Vogelsang epo loal o uniy asympoi disibuions fo one-sep-ahead OLS foeass and pove he invaiane of foeass o he values of he end paamees µ and θ. hey do no epo longhoizon disibuions, howeve. I give heoems desibing loal o uniy, long-hoizon asympoi disibuions fo Case II and III foeas eos fo boh RW and OLS in seion. I hen use hem o build foeasing hesholds fo ases II and III in seion 3. he idenifiaion and analysis of he suue of hese foeasing hesholds foms a body of eseah apable of speaking o numeous impoan quesions abou foeasing and abou he AR in paiula. I fous on hee poins in his pape. Fis, he failue of OLS o oupefom RW when RW is misspeified bu is vey lose o is due, in pa, o bias in OLS esimaes of he auoegessive paamee bu is due pimaily o he addiional esimaion eo pesen in OLS foeass. his illusaes lealy he fundamenal advanage of using pasimonious models fo foeasing. Seond, i is lea ha opimal ieia fo hoosing beween RW and OLS ae linked o he foeasing hesholds bu no o he pesene o absene of a uni oo. In fa, he uni oo hypohesis and es ae only indiely elaed o seleing a foeasing model. hus, in geneal, one should no expe a 5% uni oo es o hoose opimally beween RW and OLS foeass. In seion 5, I use a foeasing example o show he supeioiy of a novel model seleion ieion ha uses hese foeasing hesholds expliily. Speifially, I popose using median unbiased esimaion of he dif, inepeed wih espe o he foeasing hesholds, o sele he model mos likely o minimize foeas eo. his offes subsanial impovemen in foeasing ove a seleion ieion using a 5% uni oo es. hid, alhough I only pesen heoeial esuls fo he AR, he mehods in his pape ae likely o be useful in sudying foeasing in nea-non-saionay AR poesses of lage ode. In seion 6, I give some peliminay Mone Calo esuls ompaing foeass fom an OLS-esimaed ARq o foeass obained by speifying a uni oo and esimaing he paamees of he ARq unde ha esiion. I is lea boh ha he lessons leaned fom he AR help one undesand he foeasing of hese poesses and ha muh moe wok is needed o undesand i fully. 5

6 he balane of his pape is as follows. In seion, I deive asympoi disibuions of he foeas eos fo RW and OLS foeass fo ases I-III. In seion 3, I use ompue simulaion o idenify foeasing heshold funions *η fo all ases. In seion 4, I disuss he esuls of seions and 3 in deail. In seion 5, I demonsae he usefulness of he foeasing heshold funions in a foeasing example. In seion 6, I offe some peliminay esuls on foeasing hesholds fo ARq poesses wih q > and give bief onluding emaks.. Analysis In his seion I deive he asympoi disibuions of foeas eos fom he Random Walk RW and OLS-esimaed foeasing models. Ulimaely, I wish o ompae he meansquae eo MSE of foeass fo hese models. o deive limiing disibuions fo he foeas eos Sok s heoems and heoems -3, I use loal o uniy asympoi heoy. he foeas eos ae funions of Onsein-Uhlenbek poesses and, fo he OLS-esimaed model, he limiing disibuions ae quie ompliaed. Indeed, losed-fom expessions fo squaed eos ae mahemaially inaable. Reall ha he loal o uniy assumpion equaion 4 speifies he auoegessive paamee ρ as a funion of sample size. When he dif paamee is a negaive numbe, ρ is less han uniy, whih would seem o imply ha he auoegessive poess is sily saionay. Howeve, fo a finie sample size, auoegessive poesses wih ρ lose o bu slighly below uniy eain some evoluionay popeies of he non-saionay andom walk poess =. Pimaily, he lose is o zeo, he less is he endeny of he ime seies o mean-eve. I sa wih he following definiion of eminology. Definiion: Le J be defined as follows: dj = J d db + 5 whee B = σw and W is a sandad Weine poess. his is a sandad Onsein- Uhlenbek poess, and J ~ e N, σ 6 6

7 I is well known ha, fo he daa geneaing poess : 3 and [ ] ε = σ u [ ] = [ ] = W = B [ ] + ε J whee [w] denoes he inege pa of w fo any eal w and he symbol denoes onvegene in disibuion. All of he asympoi esuls used in his pape o build foeas eo limiing disibuions deive fom hese wo. he oal eo in a single foeas an be boken down ino wo pas: foeas eo and innovaion eo. Fo he h-peiod-ahead foeas y + h, his oal eo is given as: whee y + h y + h + h + h + h + h + h y = y y + y y 7 is he infeasible oe foeas, using he model s ue paamees. he seond em on he igh hand side of 7 is he innovaion eo : h h y + h y + h = ρ ε + + ρ ε ρε + h + ε + h... 8 Clealy, his eo is ommon o foeass fom boh he RW and OLS models. he expeed squae of his em is saighfowad o evaluae: E h ρ y = + h y + h σ ρ he fis em in paenheses on he igh-hand side of 7 is he diffeene beween he feasible foeas and he infeasible foeas ha would be made if he model s paamees wee known wih eainy. Fo OLS and RW, espeively, we define he foeas eos fo Case K: K OLS K S + h y + h y + h = 9 K RW K R + h = y + h y + h My use of he em foeas eo is he same as ha in Ng and Vogelsang. Clealy, if a foeas has a lowe expeed squaed foeas eo, i has lowe expeed squaed oal eo. 3 See, e.g., Phillips

8 Case I { y = } Fo Case I OLS foeass, ρ is esimaed using a leas squaes egession of { on. he foeas is hen: OLS I h h + = ρ y y y} = whee he ba ove ρ denoes he OLS esimae. Fo foeass fom his model, he foeas eo is hus given: I OLS I h h S + h = y + h y + h = ρ ρ y he following esul was fis epoed by Sok 996. I is also a speial ase of a heoem due o Kemp 999 and is simila o a esul in Phillips 998. heoem Sok 996: 4 Le he daa be geneaed by. Le I S +η be he foeas h eo fo he OLS-esimaed foeas equaion fo hoizon η =. whee: I S + η e η e η J = lim ρ ρ = J db J d he asympoi disibuion of only. is non-nomal and non-enal. Is momens depend upon Fo ases I and II, he RW model imposes ρ = and foeass all fuue values of he ime seies by using he final available value of he ime seies: 4 his is equivalen o equaion 4 fom Sok 996, p

9 RW I, II y = + h y he foeas eo is hus defined as follows: I, II RW I, II h R + h = y + h y + h = ρ y heoem Sok 996: 5 Le he daa be geneaed by o. Le I, II R +η be he h foeas eo fo he andom walk foeas equaion fo hoizon η =. I, II R + η η e J Noe ha no esimaion eo is pesen in he foeas eo. All poenial eo is due o bias and if =, RW is oely speified and he foeas eo is zeo. Case II Fo Case II OLS foeass, ρ and α ae esimaed using a leas-squaes egession of { y} = on { and {,,,}. he foeas is hen: y } = y ρ h OLS II ρ h + h = α + ρ y 3 he onsuion of he foeas eo S + h is analogous o S + h. I now give he following heoem. II I heoem : Le he daa be geneaed by. Le II S +η be he foeas eo fo he h foeas fom he OLS-esimaed model equaion 3 fo hoizon η =. 5 his is equivalen o equaion 6 fom Sok 996, p

10 II S + η whee: B + η e η η J e J db B J d = J d J d Poof: See Appendix d + e J his limiing disibuion appeas o inlude moe eo han he limiing disibuion fo he Case I OLS foeas eo. Sine he RW model uses he same foeas fo Case II as in Case I, i is o be expeed ha i should pefom even bee agains he OLS-esimaed model fo Case II. Case III Fo Case III, he fom of RW is known as he Random Walk wih Dif RWD and ollapses o: y = θ + y + ε 4 o foeas h seps ahead wih his model, a eseahe esimaes θ by aking he mean of he fis diffeenes of he ime seies and hen uses he foeas: RW III y = θ h + y 5 + h whee he ba ove θ denoes he esimaed value. heoem : Le he daa be geneaed by 3. Le III R +η be he foeas eo fo he h andom walk wih dif foeas equaion 5 fo hoizon η =. III R + η Poof: See Appendix η + η e J

11 Noe ha, wheneve η >, he expeed squae of he above em will be lage han fo he andom walk wihou dif ases I and II. his addiional eo sems exlusively fom he eo in esimaing he dif θ. Fo Case III OLS foeass, ρ, α and δ ae esimaed using a leas squaes egession of on {, {,,,} and {,,,}. he foeas is hen: y } { = y } = h h j j h III OLS h y j h y ρ ρ δ ρ ρ δ α = = + 6 he following heoem gives he asympoi disibuion fo he foeas eos. heoem 3: Le he daa be geneaed by 3. Le be he foeas eo fo he foeas fom he OLS-esimaed model equaion 6 fo hoizon III S +η h = η. S III +η { d J d J db e η η d J d J db e η + } J e e η η whee: = d J d J d J d J d J d J d J db db J d J d J B Poof: See Appendix 3. Simulaion of Foeasing hesholds In his seion, I idenify foeasing hesholds fo he RW/OLS deision in wo ways: pedied hesholds using he fomulas in he heoems of seion ; and aual hesholds using h-peiod-ahead foeass fom RW and OLS models. I ely upon simulaion; fo

12 eah Mone Calo ial I use Gaussian eos fo { ε, wih =, in he DGP given in. I onside he foeasing hoizons η = {.,.,.5,.,.,.3,.4} and dif values [, ], and use a gid seah o deemine he heshold values of dif * as a funion of hoizon η. Fo eah,η pai I use 5, ials fo peise esimaes of he expeed squaed foeas eo fom RW and OLS. I hen use linea inepolaion o esimae he funion *η whee he expeed squaed foeas eos ae idenial fo hese wo models. he esimaed funions fo ases I, II and III ae shown in Figues, and 3 espeively. I also ompue aual foeass fom he RW and OLS models fo η = {.,.5,.,.,.4} fo a ypial empiial sample size fo pos-wa quaely US maoeonomi aggegaes =. I ompae he mean-squaed eo of foeasing, whih inludes boh he foeas eo and he innovaion eo, fo hese wo models, o desibe he elaive diffeene in foeasing effiieny. Fo simpliiy, I epo simulaed measues of: MSE RW,, η R, η = 7 MSE OLS,, η fo eah and η. hus, if his value is below, RW is a pefeed foeasing model. Figue and ables a and b give he esuls fo Case I. he egion above he uves in Figue is whee RW foeass have lowe foeas eo and hus lowe MSE han OLS foeass. In he gaph, he dif is ploed agains he foeas hoizon faion η. he diffeene beween he pedied and aual hesholds is small, indiaing ha he asympoi appoximaions given in Sok s heoems ae vey auae. Noe ha, fo eah foeas hoizon, RW is always pefeed fo vey lose o, and OLS is pefeed fo fa fom. Fo insane, a η =., so ha h =, he heshold * -.8. hus, RW is pefeed wheneve ρ } = 6 6 Simulaion of he pedied foeas eos, using he heoems of seion, onsiss of simply building paial sums ha onvege o he vaious piees of he asympoi disibuions and hen ombining hem aoding o he heoems. Fo example, o simulae, use J d u. 3 = Fo simulaion of aual foeas eos, fo eah ial I build he h-peiod-ahead foeas he aual h-peiod-ahead value y + h y + h, ompae i o, and ompue he squae. Aveaging ove all ials gives he expeed squae.

13 .997,] and OLS is pefeed wheneve ρ <.997. Howeve, he heshold value of, whee RW and OLS have he same MSE, hanges wih he hoizon η. he simulaed funion *η ahieves a minimum a η =., and ises wih η unil η =.3, hen i delines again. In able a, I epo he heshold funions *η, whih gives he numbes used o build he gaphs in Figue. Noe he pedied and aual values of *η neve diffe by moe han.4. In able b, I epo values of R,η fo seleed and η. A = -, η =.5, fo insane, R =.95, so ha RW has 5% lowe MSE han OLS. his aio deeases wih he foeas hoizon wheneve RW is lealy pefeed = o = -, and ineases wih he foeas hoizon wheneve OLS is lealy pefeed -3. Hene, he expeed loss fom hoosing he wong foeasing model is geae a longe foeasing hoizons. When = -, RW is pefeed a some hoizons, while OLS is pefeed a ohes. Fo Case II, *η is shown in Figue. his funion shaes seveal feaues wih ha of Case I. Again *η ahieves a minimum a η =. and gows wih η. In his ase, howeve, hee is no peak; he funion ahieves is lages value a η =.4. Noe ha he funion is beneah ha of Case I a all foeas hoizons. As expeed, RW has a geae advanage agains his model. Nex onside ables a and b. Again he pedied and aual values of *η ae lose, neve diffeing by moe han.9. Noe ha he loss fom using OLS, when is vey lose o zeo, is geae hee han in Case I. Fo insane, we epo in able b ha R-,.5 =.9, fo a diffeene of 8%, as ompaed o a 5% diffeene in Case I. Again, he expeed loss fom hoosing he wong foeasing model is geae a longe foeasing hoizons. Fo Case III, *η, as shown in Figue 3, is well below he Case I and Case II funions. I, oo, ahieves a minimum a η =. and is onave, bu peaks a η =.. In ables 3a, he pedied and aual values of *η ae again lose. In able 3b, he loss fom using OLS when is vey lose o zeo is shown o be geae han in ases II and III. Fo insane, R-,.5 =.88, fo a diffeene in MSE of %. he expeed loss fom hoosing he wong foeasing model is usually magnified a longe foeasing hoizons, alhough no beween η =. and η =.4 when = -7, -8, -9 and -, whee *η is deeasing in η. 3

14 4. he Impoane of Downwad Bias Inuiively, RW foeass should be pefeed when is vey lose o. hee ae fewe paamees o esimae and bias due o misspeifiaion is also small. I should no be supising, hen, ha hee ae even heoeial ases whee RW is pefeed o OLS when he auoegessive paamee is known wih eainy. 7 Howeve, downwad bias in he OLS esimae of he auoegessive paamee, a well-known phenomenon, does mae. 8 In fa, his lagely explains why RW foeass ae pefeed moe ofen a sho hoizons in my analysis. Conside a vesion of Case I whee <, so ha ρ <. he RW eo is ρ h u, whih lealy ineases monoonially in h. Fo mos values of ρ <, usually he ase fo he h h OLS esimae, is eo, ρ ρ u also gows in absolue value wih h. 9 Bu even if i gows, i does so moe slowly han does elaive o RW, as h ineases. ρ h u. hus, OLS eos ypially impove, h h Beause of downwad bias, ρ ρ u will fequenly be lage in absolue value han ρ h u when is lose o. If ρ is below ρ, he ill effes of his bias will be mos damaging, elaive o RW, when h =. hus, holding onsan, he elaive impovemen of 7 In Case II, fo insane, if OLS mus esimae α bu no ρ, RW is pefeed a all foeas hoizons wheneve - < <. his is simple o show analyially. 8 Phillips 979 gives an exellen disussion of downwad bias. Fo Case II, an of-ied ude appoximae fomula fo downwad bias in he OLS esimae of ρ is E ρ ρ + 3ρ. e.g., Mak I will only shink iniially if ρ is nea. Fo insane, suppose ρ =.98. If ρ =.96 is he OLS esimae, hen ρ.98u is he same in absolue value. u fo he RW as fo he OLS model. A h = 4, howeve, his em is.9 u =. 78u in he RW model bu only u =. 73 in he OLS model. A h =, his em is. 667u =. 333u in he RW u model, bu only u =.5u in he OLS model. hus, i is inuiive ha he OLS model should impove, elaive o RW, as he foeas hoizon ineases. Sine he em eos fo ases I, II and III, his effe holds in all hee ases. h ρ ρ h u appeas in he foeas 4

15 OLS foeass as η ineases, as illusaed by he posiive slope of he foeasing hesholds in Figues -3, is magnified by downwad bias in he OLS esimae of ρ. 5. Seleing Beween RW and OLS he pedied hesholds in his pape help infom he deision beween RW and OLS foeass. As was menioned in he inoduion o his pape, if one had song evidene ha ρ is vey lose o o equal o, RW would be a good hoie fo a foeasing model. Uni oo ess an povide suh evidene. Reen eseah has invesigaed foeasing saegies of he model in ha use uni oo ess o deide whehe o no o speify a RW model. Sok 99, Campbell and Peon 99, Sok 996 and Diebold and Kilian have invesigaed he Mone Calo auay of hese pees saegies aoss a onsideable ange of paameeizaions of his model, and found ha hese saegies have some obvious value elaive o unifom saegies ha eihe always o neve use RW. In all ases, he size of peess hosen has been eihe 5% o %. Sizing peess aoding o onvenional hypohesis es sizes is due o wo fas: ha 5% and % ae geneally aeped hesholds fo saisial signifiane in hypohesis esing; and he iial values ae eadily available fo hese sizes. As Diebold and Kilian poin ou, howeve, hese sizes saisfy no ieion of opimaliy fo foeasing. Indeed, as his pape has shown, he uni oo hypohesis is fa sie han a hypohesis ha says ha RW will be a pefeed foeasing model. In fa, wha maes is whehe is above o below *η. Unfounaely, is no onsisenly esimable in a univaiae famewok. hus, i is impossible o es, using adiional asympoi heoy, whehe is above o below *η. Howeve, Sok 99 shows ha i is possible o find esimaes of ha ae median unbiased. hese values an be used o pedi, fo a given ime seies, whih side of *η he ue mos While he heshold funions fo ases I and II beome nealy fla afe η., he funion fo Case III has a ponouned peak a η =. and delines signifianly as he hoizon ineases fuhe. his appeas o be due o he need o esimae he end in he OLS model. In a hypoheial vesion of his ase whee ρ is known wih eainy, i an be shown ha he heshold is nea * -5.9 a η, and delines monoonially wih η, eahing * -8.5 a η =.4. In Figue 3, as η ineases fom o., his downwad pessue on he heshold funion is dominaed by upwad pessue on ρ ρ u heshold does indeed deline. h h desibed ealie in his seion. Afe η eahes., howeve, he foeasing 5

16 likely falls on. Hene, i is possible o hoose he foeas ha is mos likely o minimize he foeas eo. We now es he usefulness of his poposiion. Figue 4 shows he elaive pefomane of wo inuiive saegies in foeasing Case II of he AR. One saegy uses he adiional saisial ieion, employing a 5% Dikey-Fulle es and hoosing OLS only if he DF es ejes he uni oo null. he seond saegy obains a median unbiased esimae of dif, hen ompaes his value o he appopiae foeasing heshold. OLS is used if his esimae is below he heshold. his onsiss of fis building he Dikey-Fulle -saisi: ρ τ = 8 σ ρ whee σ ρ is he usual OLS sandad eo. A median-unbiased esimae of an be found fo τ using he ables in Sok 99. Fo foeas hoizon η, if ha esimae is below *η, OLS is used. his deision ule essenially podues a iial value fo he -saisi τ ha is diffeen fom he 5% Dikey-Fulle iial value of I epo he iial Cases II and III beneah Figue 4. I employ a Mone Calo simulaion o esimae he MSE of eah saegy. in Figue 4 measues he aio of MSEs of hese wo saegies: MSE Saegy,, η R *, η = 9 MSE Saegy,, η τ* * values fo he sufae Wheneve he sufae in Figue 4 is above, Saegy has lowe MSE han ou poposed saegy, whih uses foeasing hesholds and median-unbiased esimaion. his only ous when is lage han 4, and in hese ases R* is neve moe han.5. When is less han 4, on he ohe hand, Saegy is dominan and by a lage amoun. R* sinks o.8 when = -3, η =., and eahes.7 a η =.4. his is an exemely signifian gain in foeas auay by Saegy. Fo example, onside an AR wih a non-zeo mean and ue ρ =.935. If one had a ime seies of lengh geneaed by his poess, Saegy would edue he MSE of - sep-ahead foeass by nealy %. =, numbe of ials =,. 6

17 Clealy, saegy has an advanage ove saegy if hee is any eason o believe ha is less han he foeasing heshold. Moeove, i is simple o employ, and amouns essenially o using a moe poweful DF es. In sum, he pope use of uni oo ess o sele foeasing models mus aoun fo he fa ha he RW model an be a bee foeasing model even if hee is no uni oo. hus, hee is no agumen fo geneally using uni oo ess, wih adiional 5% and % saisial ieion, as peess o sele foeasing models. Indeed, he seleion ieia inodued hee epesens an obvious impovemen. 6. hesholds fo he ARq wih q > In foeasing an empiial ime seies wih auoegessive endenies, a eseahe will seldom know wih absolue eainy whehe he poess is in fa an AR. A ypial mehod fo handling his poblem is o ea he seies as an ARq and esimae q diely. Hene, i is an impoan paial mae o ask how he hesholds in his pape would hange if q >. Fo ase II, hen, I speify he ime seies as follows: whee: y p = α + ρi yi + ε i= α = µ p ρ i i= I efe o foeass buil fom esimaing as OLS foeass. Unde he ARq assumpion, he seies is saionay wheneve he pesisene of he ime seies, as defined by he sum of he auoegessive paamees, is less han one in absolue value. 3 Hee, hen, I model he pesisene as loal-o-uniy: p i= ρ i = + 3 I also assume ha, wih he exepion of he = ase, he sysem is sable, in he sense ha all eigenvalues ae less han one in modulus. 7

18 Noe ha if =, RW is no longe neessaily he oely speified model when q >. 4 his ase, howeve, sine i will always be ue ha he sum of he auoegessive oeffiiens is, hee exiss a esiion of he seies in ha is a naual qh-ode analog o RW: y ρ... + ε p = y + ρ y + + ρi y p i= As is he ase wih RW, hee is no need o esimae he paamee α ha is pesen in. In addiion, hee is one less ρ paamee o esimae. hus, his uni oo UR speified ARq i esimaes wo fewe paamees han he full speifiaion of. his is he same diffeene as beween RW and OLS in he AR ase. his suggess ha he foeasing hesholds fo he ARq may shae some feaues of he AR hesholds. As we now demonsae in a Mone Calo analysis, if he ue poess is aually an AR bu a highe ode ARq is used, he foeasing hesholds ae vey simila o he AR hesholds. Figue 5 shows esimaed heshold funions fo he AR, AR and AR5 ases ogehe. he levels of he esimaed funions ae vey simila, alhough he shape of he AR and AR5 hesholds is diffeen a vey sho hoizons. While he AR heshold is a is minimum a η =., he AR and AR5 hesholds fall fom η =. o η =.. When q >, he fa ha he pesisene of he seies does no uniquely deemine he auoegessive paamees affes he hesholds. In he AR, fo insane, as ρ moves, holding onsan, he hesholds move as well. his is demonsaed in Figue 6, whih shows he esimaed AR heshold fom Figue 5 along wih wo addiional speifiaions of ρ : -. and.. All heshold funions have sho-hoizon feaues ha ae diffeen han hose in he AR hesholds. A mos hoizons, he hesholds fo ρ = -. favo OLS a moe values of han in he AR while he hesholds fo ρ =. favo UR a moe values of. he shapes of hese funions ae simila, bu he levels ae lealy diffeen a long hoizons, moe so han he heshold funions in Figue 5. We fom wo impoan bu peliminay onlusions fom hese esuls. Fis, i appeas ha if he auoegessive poess is loal o uniy, he fis auoegessive paamee is vey lose In 4 RW is oely speified only when = and ρ =. Even wih his addiional eason ha RW may be misspeified, i may noneheless oninue o pefom well as a foeasing model. We disuss his in moe deail lae in his seion. 8

19 o uniy and he ohe auoegessive paamees ae vey lose o zeo, he AR hesholds fom his pape povide exellen appoximaions o ue ARq hesholds. ha is, if he ue seies is vey nealy an AR bu an AR5 is used fo foeasing, he AR hesholds auaely fame he hoie beween OLS and UR. Seond, he faos affeing foeasing fo highe-ode AR poesses ae poenially many and omplex. Clealy, i emains o explain he sho-hoizon dip in he AR and AR5 heshold funions. Moeove, alhough he AR hesholds appea o be quie easonable appoximaions fo AR hesholds when ρ is lose o bu no equal o zeo, boh he magniude of ρ and is sign lealy affe he hesholds. his is no supising, given ha he eigenvalues of he sysem, whih deemine is dynamis, ae no uniquely deemined by he loal-o-uniy pesisene. 5 hus, in ode o moe fully undesand hese esuls, hallenging heoeial wok may be neessay. Howeve, suh eseah is likely o lead o a fa geae undesanding of he foeasing of nea non-saionay AR poesses. Given he pevalene of suh poesses in sudies of empiial maoeonomi and finanial daa, his eseah is likely o be poduive. I will be ineesing also o invesigae he elaive foeasing pefomane of RW o he OLS esimaed ARq. I is well esablished ha, in foeasing pesisen eonomi and finanial ime seies, RW is vey diffiul o bea. If he advanages of RW foeass exend o nea-non-saionay ARq poesses heoeially, suh eseah may help explain why. 5 Fo insane, Hamilon 994, pp. 3-8 shows ha, in he AR, he dynami muliplie follows diffeen paens depending upon he eigenvalues of he sysem. Namely, if he eigenvalues ae eal and of less han uni modulus, whih is ue wheneve ρ and ρ ae boh geae han zeo, he dynami muliplie follows a paen of geomei deay. If hey ae omplex and of less han uni modulus, whih may hold if ρ is below zeo, he dynami muliplie follows a paen of damped osillaion. 9

20 8. Refeenes Bobkoski, M.J. Hypohesis esing in Non-saionay ime Seies, unpublished PhD disseaion, Depamen of Saisis, Univesiy of Wisonsin, 983. Campbell, John Y. and Piee Peon. 99. Pifalls and oppouniies: wha maoeonomiss should know abou uni oos. in O. Blanhad and S. Fishe eds., NBER Maoeonomis Annual, Boson., MA. Chan, N.H. and Wei, C.Z Asympoi infeene fo nealy non-saionay AR poesses. Annals of Saisis, 5,5-63. Diebold, Fanis X. and Luz Kilian.. Uni oo ess ae useful fo seleing foeasing models. Jounal of Business and Eonomi Saisis 8 3, Ellio, Gaham, Rohenbeg,.J. and J.H. Sok Effiien ess fo an auoegessive uni oo. Eonomeia 64, Ellio, Gaham Effiien ess fo a uni oo when he iniial obsevaion is dawn fom is unondiional disibuion. Inenaional Eonomi Review 4, Fulle, Wayne A. and David P. Hasza. 98. Popeies of Pedios fo Auoegessive ime Seies. Jounal of he Ameian Saisial Assoiaion 76, Hall, Robe E Sohasi impliaions of he life yle pemanen inome hypohesis: heoy and evidene. Jounal of Poliial Eonomy 86, Hamilon, James D ime Seies Analysis. Pineon Univesiy Pess, Pineon, NJ. Kemp, Godon C.R he behavio of foeas eos fom a nealy inegaed AR model as boh sample size and foeas hoizon beome lage. Eonomei heoy 5, Magnus, Jan R. and M. Hashem Pesaan he exa muli-peiod mean-squae foeas eo fo he fis-ode auoegessive model wih an ineep. Jounal of Eonomeis 4, Mak, Nelson Exhange aes and fundamenals: Evidene on long-hoizon pediabiliy. Ameian Eonomi Review 85, -8. Ng, Seena and imohy J. Vogelsang.. Foeasing auoegessive ime seies in he pesene of deeminisi omponens. Eonomeis Jounal 5, Phillips, P.C.B he sampling disibuion of foeass fom a fis-ode auoegession. Jounal of Eonomeis 9, 4-6. Phillips, P.C.B owad a unified asympoi heoy fo auoegession. Biomeika 74, Phillips, P.C.B Impulse esponse and foeas eo vaiane asympois in nonsaionay VARs. Jounal of Eonomeis 83, -56.

21 Sims, Chisophe A., James H. Sok and Mak W. Wason. 99. Infeene in linea ime seies wih some uni oos. Eonomeia 58, Sok, James H. 99. Uni oos in eonomi ime seies: do we know and do we ae? A ommen. Canegie-Rohese Confeene Seies on Publi Poliy 3, Sok, James H. 99. Confidene inevals fo he lages auoegessive oo in US maoeonomi ime seies. Jounal of Moneay Eonomis 8, Sok, James H VAR, eo oeion, and pees foeass a long hoizons. Oxfod Bullein of Eonomis and Saisis 58,

22 Case I Figue - Foeasing hesholds Dif - -3 Pedied * Aual * -4 Hoizon ea able a - Foeasing hesholds Hoizon η Pedied * Aual * able b - MSERW / MSEOLS Hoizon η Dif

23 Case II Figue - Foeasing hesholds Dif -4-6 Pedied * Aual * -8 Hoizon ea able a - Foeasing hesholds Hoizon η Pedied * Aual * able b - MSERW / MSEOLS Hoizon η Dif

24 Case III Figue 3 - Foeasing hesholds Dif Pedied * Aual * - Hoizon ea able 3a - Foeasing hesholds Hoizon η Pedied * Aual * able 3b - MSERW / MSEOLS Hoizon η Dif

25 Figue 4 {MSESaegy / MSESaegy} Case II Saegy : Use a 5% Dikey-Fulle Pees o deide beween RW and OLS. hus use OLS if he Dikey-Fulle -saisi τ is less han he 5% iial value -.86 fo Case II and -3.4 fo Case III and RW ohewise. Saegy : Ge a Median Unbiased MU esimae of, using ables fom Sok 99. Fo foeasing hoizon η, use OLS if he MU esimae of is less han he foeasing heshold *η given in ables a and 3a Cases II and III espeively and use RW ohewise. his oesponds o using OLS if he ~ Dikey-Fulle -saisi τ is less han he * given below. ~ τ ~ Hoizon η Case II ~ τ* Case III ~ τ*

26 Figue 5 - ARp Foeasing hesholds, ue Poess is an AR * Hoizon ea AR5 AR AR Figue 6 - AR Foeasing hesholds Dif ho = -. ho = ho =. Hoizon ea 6