Sequential Unit Root Test

Transcription

1 Sequenal Un Roo es Naga, K, K Hom and Y Nshyama 3 Deparmen of Eonoms, Yokohama Naonal Unversy, Japan Deparmen of Engneerng, Kyoo Insue of ehnology, Japan 3 Insue of Eonom Researh, Kyoo Unversy, Japan Emal: nshyama@kerkyoo-uajp Keywords: Un roo es, Sequenal Analyss, Bessel proesses EXENDED ABSRAC Consder a salar AR() proess = β + ε, where { ε } are d dsurbanes When β = ±, he seres s sad o possess a un roo ess for he esene of un roos n eonom me seres have been one of he man ssues of neres n eonomers sne he mddle of 98 s In prae, eonomerans fous on esng he null hypohess of β = agans he alernave of β < sne he eplosve ase β > and negave un roo β = are very unlkely n eonom seres he mos sandard un roo esng proedures mus be Dky-Fuller ess (DF ess hereafer) and s varans We esmae β by he OLS and onsru es sass by suably normalzng I s well known, however, ha he DF es sass do no have a good power propery for small and medum sample szes Also, he lmng dsrbuon s non-sandard beause he OLS esmaor of β s no normally dsrbued, whh s nonvenen n prae La and Segmund (983) (LS hereafer) and Shryaev and Spokony (997) respevely show ha he OLS esmaor of AR() oeffen, β, s asympoally normally dsrbued even f he rue value of β equals o uny and s greaer han uny n fa under a sequenal samplng sheme hs sequenal proedure nvolves a soppng rule suh ha one sops samplng a me f he generalzed nformaon I σ =, or s esmae n prae, eeeds a predeermned onsan he asympo normaly of sequenal OLS esmaor mples ha one an onsru a es sas for he un roo possessng a sandard normal lm I s obvously praally onvenen ompared wh DF ess havng non-normal lmng dsrbuon In hs samplng sheme, he number of observaons s also a random varable dependng on he realzaon of he me seres unlke he sandard samplng ase We may lke o know he sasal properes of he soppng me beause wll be of some help o deermne an approprae value of, and also beause may provde eran amoun of addonal nformaon n un roo esng o ha from β In hs paper, we derve he jon asympo dsrbuon of β and he soppng me suably normalzed he margnal dsrbuon of β s, of ourse, normal, whle / s shown o have a non-sandard dsrbuon haraerzed by a funonal of he Bessel proess wh he dmenson 3/ under he null We also oban he jon dsrbuon under loal alernaves where he margnal dsrbuon of he soppng me s represened n erms of a Bessel proess wh drf Usng hese asympo jon dsrbuons, we an onsru a lkelhood rao ype es sas for un roo We fnd ha he sas does no depend on he soppng me, and hus he soppng me arres no addonal nformaon o he OLS esmae of he AR() oeffen n erms of esng for un roo he followng seon revews DF es and sequenal AR() parameer esmaon, as well as he sequenal un roo es based on LS nludng some smulaon resuls Seon presens he jon dsrbuon of β and under he null, whle Seon 3 provdes ha under a loal alernave Seon 4 eplans lkelhood rao ype sas and he sequenal un roo es as well as he LAN propery under he normal dsurbanes Seon 5 onludes 33

2 INRODUCION Dky-Fuller es Suppose { } s generaed from = β + ε, () where { ε } are d dsurbanes When β =, he proess s alled a un roo proess and s behavour s very dfferen from ones wh β < Some maroeonom me seres are sad o have a un roo based on he resuls from DF es We frs brefly revew he DF -sas Gven a sample {, L, }, le he OLS esmaor of β and an esmaor of σ be β = ( ), σ = ( β ) Also denoe W (s) as a sandard Brownan moon hen, as, we have he followng asympo resuls: N(,) f β < ( β ) W ( s) dw ( s) β d f β = σ W ( s) ds Cauhy oherwse herefore o es he null of un roo agans he alernave of saonary, we use he value ( β ) /{ σ } whh onverges o a funonal of W(s) above under he null, whle able Rejeon rae of DF es (nomnal sze=5%) Bea= (sze) Bea=95 (power) = = = eplodes under he alernave able shows he sze and power of he es by smulaon he sze seems o be aepable, bu he power s unsasfaory for sample szes for =5~5 Sequenal Esmaon of AR() Parameer And he Soppng Rule Sequenal analyss was orgnally onsdered by Wald (947) he dea s as follows Suppose we an oban one observaon a day, say We sample every day and when we aumulae suffen ' nformaon, hen we sop samplng and make a sasal deson (esmaon or esng) How suffen ' s deermned by he researhers hrough some user-deermned parameer, whh onrols he auray of he resuls How we sop samplng s alled he soppng rule and he me when we sop samplng s alled he soppng me ypally, we are beer off f we an oban onlusons earler due o some os of samplng or akng me here ess a rade-off beween auray and os of samplng LS nvesgae he sasal properes of he sequenal esmaor of he AR() parameer n model () Formally, for a predeermned onsan, her soppng rule s defned as where = nf{ > / σ β = ( ), σ = ( β ) We sop samplng when he esmaed nformaon I / σ = eeeds a eran predeermned value, whh onrols he auray of esmaon hrough he sample sze We wre he soppng me as o emphasze ha depends on he hoe of onrols for he auray of he esmaon n he sense ha he varane of he esmaor, I, s } guaraneed o be smaller han here ess a rade-off beween he auray of esmaon and he os of observaons If we se large, wll end o be also large by onsruon, whh wll yeld a more aurae esmae If we se small, 33

3 samplng wll sop relavely earler, bu he auray wll be lower Noe ha self s a sas dependng on he observaons Usng hs soppng me, we alulae sequenal esmaors by ) β = (, σ = ( β ) LS prove he asympo normaly of β ase of β : I d ( β ) N(,) β n he Furher, Shryaev and Spokony (997) oban he same resul n he eplosve ase of β > under he assumpon of normal dsurbanes We an drely apply hs resul for un roo es whh we all a sequenal un roo es (SUR) able Properes of SUR (sze=5%) =6 Bea= (sze) Bea=95 (power) Rejeon rae E() E(Bea) n able wh hose n able, hey are mosly sasfaory In omparng he power, we need o be areful he SUR proedure requres more sample szes o sop samplng under he alernave, hus we anno drely ompare hem wh fgures under he null One pon we an make s ha DF es under he sandard (fed) samplng, we anno onlude a un roo ess even f he null s no rejeed from a sample of small or medum sze (hough seems o preval n eonom leraure) Under sequenal samplng, however, researhers wll be auomaally fored o wa unl a suffen amoun of nformaon s aumulaed boh under he null and alernave hypoheses he hoe of obvously beomes an mporan faor here o hoose suably, we need o sudy he sasal properes of whh also deermnes he power of he SUR proedure We lasly show he dsrbuon of he soppng me from a Mone Carlo smulaon Fgure shows he densy funons of / for β = d / wh d =, d = 8 = when and respevely orrespond o he ases of un roo and saonary he dsrbuons are learly que dfferen eah oher and he saonary ase requres more sample sze (or nformaon) n makng a sasal deson Fgure Denses of soppng me under he null (d=) and he alernave (d=) Sd() Sd(Bea) =5 Bea= Bea=95 (sze) (power) Rejeon rae E() 8 59 E(Bea) Sd() Sd(Bea) 9 able shows Mone Carlo resuls of SUR for =6 and 5 onrols for he auray of nferene hrough he soppng me We se =6 so ha he average soppng me under he null s abou 5, whle =5 yelds he average soppng me of under he null We wll show a heoreal relaonshp beween and he average soppng me (sample sze) laer Comparng he sze resuls JOIN DISRIBUION OF AR() PARAMEER ESIMAOR AND HE SOPPING IME UNDER HE NULL he followng heorem presens he jon dsrbuon of β and under he null HEOREM { ε }, =,, Suppose L s a saonary and ergod marngale dfferene sequene wh 333

4 E ( ε = ) σ < wh = as, where and β and =, and are generaed by () hen, we have, d ( β ), B, ds X s B s a sandard Brownan moon on [, ) X s a Bessel proess sasfyng δ dx = d + db X wh he dmenson δ = 3/ and he nal X value = We noe ha β s asympoally normally dsrbued, whh s onssen wh he resul of La and Segmund (983), and ha s O p ( ) Due o he resuls n Borodn and Salmnen (, p386), we know he analyal epresson of he jon densy: Usng hs jon densy, we an oban he asympo momens of, for eample, E E X ds = s Γ 9 Γ(5/ 4) (3/ 4) We se =6 n he smulaon presened n able suh ha he epeed sample sze s abou 5, namely E( ) 9 = 5, and smlarly for =5 Fgure shows he jon densy funon and s onour plo generaed from a smulaon he wo sass are obvously hghly orrelaed Fgure Jon densy under he null ( ) P B dz, X s ds du = (u+ z) ( u z) k Γ ( / + k + l) π k!l! Γ( / + k) k= l= {( ) } 4 3/ + k ( ) ( ) ep + k+ lu+ z / D + k+ lu+ z where D () s he parabol ylnder funon defned by ( ) = p/ ( ) Dp z ep z / p z p 3 z πf ; ; πzf ; ; p p Γ Γ wh he hypergeomer funon 3 JOIN DISRIBUION UNDER LOCAL ALERNAIVES We onsder he followng loal alernave; H : β = vs H : β =, where s a posve onsan We beleve hs s a naural loal alernave seng as we onsder he asympos of he followng heorem provdes he jon densy of AR() oeffen esmaor and he soppng me under he above loal alernaves ( α L α β Lβ ) F,, ;, ;z p q p q and = ( α) = α( α + )( α + ) L ( α + ) n ( α ) L( αp ) n ( β ) L( βq ) n= n n n n zn n! HEOREM Suppose he same ondons on ε s saed n heorem hold Under he loal alernave, we have, () as, 334

5 d ( β ), + B, ds X s X where s he Bessel proess wh a drf solvng he sohas dfferenal equaon δ dx ( ) d + db = + X whδ = 3/ and X = () Le f be he jon densy under he null shown n heorem, hen ha under he loal alernaves s gve by f = ep( z ) f hs ndaes ha he soppng me does no arry any addonal nformaon n esng he null of un roo o he nformaon arred by he AR() oeffen esmae hs s a naural resul n he ase of normal dsurbanes, bu s also rue for non-normal ases We noe ha hs lkelhood rao s no ealy he lkelhood rao n he ordnary sense beause does no presen he lkelhood of he observaons hemselves, bu only her funons, namely β and A es based on hs lkelhood rao may be a reasonable approah espeally when we do no know he dsrbuon of he dsurbanes as we anno wre down he ordnary lkelhood 4 Loal Asympo Normaly Suppose ha he dsurbanes are normally ndependenly dsrbued, ε ~ dn(, σ ) hen he log lkelhood rao of he observaons s Fgure 3 Conours of jon denses under he null ( = ) and he alernave ( = ) Λ(, L, = σ ; / ) ( ) σ Obvously, = redues o heorem Fgure 3 ompares he onours of f and f he frs erm on he rgh s a marngale and asympoally normally dsrbued, and he seond erm onverges o a onsan as due o he defnon of he soppng me herefore, possesses he LAN propery he pons are ha we sop samplng when he summand of he seond erm hs and ha hs quany ondes wh he quadra varaon of he frs marngale erm n he lm hough we are no sure f he LAN propery mples some opmaly n makng nferenes n sequenal samplng seup as n he sandard samplng, mgh be lkely We need furher researh on hs 5 CONCLUDING REMARKS 4 LIKELIHOOD RAIO YPE ES AND LAN PROPERY 4 Lkelhood Rao Usng heorem (), we an onsru a lkelhood rao ype es sas based on he wo sass: log f f = z + hs paper onsders esng for he esene of a un roo under he sequenal samplng proposed by La and Segmund (983) and Shryaev and Spokony (997) We oban jon dsrbuons of AR() oeffen esmaor and he soppng me boh under he null and loal alernaves he null dsrbuon of soppng me s haraerzed by a Bessel proess wh dmenson 3/, whle he dsrbuon under he loal alernaves s represened n erms of he same Bessel proess wh a drf 335

6 hough sequenal samplng suaon may no be very lkely n mos eonomer me seres eep some ases where we need o make a, say, poly deson as soon as possble, he proposed sequenal un roo es proedure may be a good alernave o he ommon DF es n erms of power I s known ha DF es does no have a suffen power under small or medum sample sze, bu he SUR proedure auomaally le eonomerans wa unl suffen nformaon s aumulaed o make a sasal deson I wll be possble o apply hs proedure o, for nsane, he deson makng of fund managers who may lke o know f a seres has a un roo or no, namely sable or no, as early as possble We provde a lkelhood rao ype es sas, whh s shown o be ndependen of he soppng me n he frs order asympos I may beome mporan n he seond order We also show ha has a LAN propery when he dsurbanes are normally dsrbued here are some possbles of eenons for fuure researh Frsly, we may need o ompare he SUR wh he sequenal probably rao es (SPR) whh s a sandard esng proedure under sequenal samplng o he bes of our knowledge, here has been onsdered no suh es or s asympo heory n me seres sengs Also, SPR requres a spefaon n dsrbuon, whh we hnk may be oo resrve Seondly, we use he soppng me proposed n La and Segmund (983), whh uses he generalzed nformaon I s a suable hoe of soppng rule n he ase of esmaon sne ondes wh he varane of he esmaor, so ha onrollng hs quany means onrollng he varane n fa However, may no be he bes approah for he sake of esng sne may be more approprae o onrol he auray of deson, or sze and power here s a possbly of pursung dfferen soppng rules for esng 6 REFERENCES Borodn, AN and P Salmnen (), Handbook of Brownan Moon Fas and Formulae, Seon edon, Brkhӓuser Chang,Y and JY Park(4), akng a New Conour:A Novel Vew on Un Roo es, mmeo Ferguson, S (967) Mahemaal Sass: A Deson heore Approah, Aadem Press, New York La, L and D Segmund (983), Fed auray esmaon of an auoregressve parameer,' Annals of Sass,, Lnesky, V (4), he speral represenaon of Bessel proesses wh onsan drf: Applaons n queueng and fnane, J Appl Probab, 4(), Revuz, D and M Yor (994), Connuous Marngale and Brownan Moon, nd ed, Sprnger-Verlag, New York Shryaev, A N and VG Spokony (997), On Sequenal Esmaon of an Auoregressve Parameer, Sohass Sohass Rep, 6, 9-4 anaka, K(996), me seres analyss, John Wley and Sons hrdly, we rea he smples ase of salar AR() whou onsan or drf erms We may be able o rela hese resrons We ould eend he proedure o AR(p) proesses, seres wh drf or rend, or long memory proesses Also, may be more praally useful o onsder ess for sruural break or hange pon problems Researh oward hese dreons s urrenly under way 336