BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION. Anna Bykhovskaya and Peter C. B. Phillips. September 2017

Transcription

1 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION By Aa Bykhovskaya a Pete C. B. Phillips Septembe7 COWLES FOUNDATION DISCUSSION PAPER NO. 38 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 88 New Have, Coecticut

2 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION ANNA BYKHOVSKAYA AND PETER C. B. PHILLIPS Abstact. This pape stuies fuctioal local uit oot moels FLURs i which the autoegessive coefficiet may vay with time i the viciity of uity. We exte covetioal local to uity LUR moels by allowig the localizig coefficiet to be a fuctio which chaacteizes epatues fom uity that may occu withi the sample i both statioay a explosive iectios. Such moels ehace the flexibility of the LUR famewok by icluig beak poit, teig, a multi-iectioal epatues fom uit autoegessive coefficiets. We stuy the behavio of this moel as the localizig fuctio iveges, theeby etemiig the impact o the time seies a o ifeece fom the time seies as the limits of the omai of efiitio of the autoegessive coefficiet ae appoache. This bouay limit theoy eables us to chaacteize the asymptotic fom of powe fuctios fo associate uit oot tests agaist fuctioal alteatives. Both sequetial a simultaeous limits as the sample size a localizig coefficiet ivege ae evelope. We fi that asymptotics fo the pocess, the autoegessive estimate, a its t statistic have bouay limit behavio that iffes fom staa limit theoy i both explosive a statioay cases. Some ovel featues of the bouay limit theoy ae the pesece of a segmete limit pocess fo the time seies i the statioay iectio a a egeeate pocess i the explosive iectio. These featues have mateial implicatios fo autoegessive estimatio a ifeece which ae examie i the pape. Keywos a phases: Bouay asymptotics, Fuctioal local uit oot; Local to uity; Sequetial limits; Simultaeous limits; Uit oot moel JEL Classificatio: C, C65. Itouctio Time vayig coefficiet moels have bee extesively use i applie ecoometic wok a povie a atual mechaism fo a moel to evolve ove time. Vaious appoaches have bee stuie i the liteatue, icluig ealy wok oigially publishe i 97 by Swamy o aom coefficiets, explicit paametic time seies fomulatios Havey 99, time vayig pobability measues that ae implie i Bayesia autoegessios Aa Bykhovskaya: Yale Uivesity Pete C. B. Phillips: Yale Uivesity, Uivesity of Auckla, Southampto Uivesity, Sigapoe Maagemet Uivesity aesses: aa.bykhovskaya@yale.eu, pete.phillips@yale.eu. Date: Septembe 6, 7. Phillips acowleges suppot fom the NSF ue Gat No. SES

3 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION Phillips a Plobege 994 a ecet o-paametic wok usig keel egessio methos Gao et al. 8, Kistese, Giaitis et al. 4. The latte evelopmets have emphasize the flexibility of opaametic fomulatios a smooth tasitio appoaches of captuig tempoal coefficiet evolutio. Both these ieas have bee use i pactical ecoometic wok. They also povie a mechaism fo moelig ostatioaity though the vehicle of ealy itegate time seies without isistig o a fixe local uit oot stuctue, theeby accommoatig epatues fom uity i both statioay a explosive iectios that ca evolve ove time Phillips a Yu ; Geeaway-McGevy a Phillips 6. Such moels ae calle fuctioal local uit oot moels FLURs. They wee ecetly stuie i Bykhovskaya a Phillips 7 i the cotext of poit optimal uit oot tests, showig how iffeet the powe evelope ca be whe the epatues fom a uit oot ae time vayig. The avatage of FLUR moels compae to the staa local uit oot LUR moel Phillips 987, Cha a Wei 987 is that they explicitly allow the autoegessio coefficiet θ to vay with time, while etaiig poximity to a uit oot. This poximity is achieve though the specificatio θ t = + ct/ which ivolves a localizig time vayig coefficiet fuctio c t epeet o the positio of obsevatio t withi the sample of size. With this FLUR mechaism we ca moel ecoomic a fiacial ata that ae well escibe i pats of the sample as uit oot pocesses a yet subject to episoes of booms, busts, a ecoveies at othe times uig the same sample peio. Bykhovskaya a Phillips 7 evelope a limit theoy fo the FLUR pocess a aalyze some of the popeties of fuctioal poit optimal uit oot tests i compaiso with staa scala poit optimal tests, showig that the latte elives powe that is ofte well below the optimal fuctioal powe evelope. This powe eficiecy of the staa poit optimal test eflects the limitatio of specificatios that ivolve costat uiiectioal epatues fom uity thoughout the sample peio whe the ata ivolves moe complex foms of behavio, such as peios of itemittet epatues fom a uit oot o peios of fiacial exubeace a collapse. The peset pape stuies the same FLUR moel as Bykhovskaya a Phillips 7 a examies behavio as the localizig fuctio c t iveges, theeby etemiig the impact o the popeties of the time seies as the limits of the omai of efiitio of the autoegessive coefficiet ae appoache. This bouay limit theoy eables us to chaacteize the asymptotic fom of powe fuctios fo associate uit oot tests whe the alteatives ivolve time vayig fuctioal foms of iffeig types. The esults theefoe exte the oigial wok o bouay limit behavio of LUR moels as a scala localizig coefficiet c appoaches the limits of its omai of efiitio. I cotast to this ealie wok, the asymptotics fo the pocess itself, the autoegessive estimate, a its associate t statistic ae all fou to have bouay limit behavio that iffes fom staa limit

4 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 3 theoy i the explosive a statioay iectios. Novel featues of this bouay limit theoy i the fuctioal case ae the pesece of a segmete limit pocess fo the time seies i the statioay iectio, a egeeate pocess i the explosive iectio, a moe complex fuctio-epeet ceteig a staaizatio i the limit theoy fo the autoegessive coefficiet estimato. These esults iffe makely fom the bouay limit theoy that applies as the scala localizig coefficiet c ± i the staa LUR moel evelope i Phillips 987. The pape is ogaize as follows. Some pelimiay limit theoy is give i the followig sectio. Sectio 3 evelops the bouay limit theoy as the localize coefficiet fuctio c ±, which eables us to exploe popeties of the pocess at the limits of its omai of efiitio. Sectio 4 cosies some milly itegate Phillips a Magalios 7 FLUR cases a the coespoig limit behavio at the bouay. This famewok is of paticula iteest because it eables the aalysis of asymptotics as c a joitly, which has pove to be paticulaly useful i the stuy of uifom ifeece Giaitis a Phillips 6, Mikusheva 7. Implicatios of the fiigs ae iscusse i Sectio 5. Poofs ae give i the Appeix.. Pelimiaies To fix ieas, we cosie a time seies geeate by the moel X t = θ t X t + u t, t =,,...,, whee the autoegessive coefficiet θ t = exp ct/ + ct/, the pocess X t is iitialize at X = o p a the istubaces u t ae zeo mea statioay with vaiace σ a patial sums that satisfy the fuctioal law t= u t B, a Bowia motio with vaiace ω = Eu + Eu u h, pimitive coitios fo which ae wiely available h= e.g., Phillips a Solo 99. Time seies geeate by ae ea itegate aays with a localizig coefficiet fuctio c t that allows fo vaiatio i the autoegessive coefficiet accoig to the positio i the sample while etaiig poximity to uity. The moel is theefoe a time vayig coefficiet moel i the viciity of uity. It is a paticulaly useful famewok fo stuyig the effects of epatues fom simple uit oot a LUR moels to moe complex time seies behavio. Bykhovskaya a Phillips 7 show that upo staaizatio the pocess X t satisfies the metioe below fuctioal law with a Gaussia limit pocess which extes the limit theoy fo LUR time seies. I what follows, we cofie attetio to fiite vaiace

5 4 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION pocesses, Gaussia limit pocesses a fuctioals of them. Cetai extesios to ifiite vaiace pocesses a stable limit pocesses ae also possible but will ot be pusue hee. Lemma. X F c := ωk c = ω e s caa W s, whee W is staa Bowia motio a K c satisfies the followig oliea iffusio equatio K c = ck c + W. Whe c = c is costat thoughout the sample, the limit pocess K c euces to the liea iffusio ec sk W s stuie i Phillips 987 a Cha a Wei 987. I this LUR case, locally statioay a locally explosive time seies occu accoig to the sig of c. Moeove, as c ±, LUR asymptotics of the cete least squaes estimate of θ a its t atio tasitio to the asymptotics fo statioay a explosive time seies. This tasitio povies a likage betwee the limit theoy fo uit oot, local uit oot, statioay, a explosive moels. The followig sectios exploe the behavio of cetai fuctioals of K c as c appoaches the limits of its omai of efiitio. This limit behavio is of iteest because it escibes the liks betwee ea-itegate time seies of the FLUR class a time seies that tasitio betwee uit oot, statioay, a explosive pocesses. Coespoigly, this limit theoy captues the limitig foms of the powe fuctios of uit oot tests at the limits of the omai of efiitio of c. I paticula, whe c ±, the limit theoy etemies whethe uit oot tests ae cosistet agaist cetai fuctioal alteatives to a uit oot i both statioay a explosive iectios a the ole of fuctioal shape i etemiig powe. The fixe coefficiet autoegessio ca be viewe as a special case of the FLUR moel with c = cost a cost. Thus, takig limits as c ± may be viewe as eliveig a appoximate oute to staa autoegessio at least whe c fo all. Ou pimay iteest i the peset pape, howeve, coces cases i which c = a c occu ove complemetay subpeios, theeby allowig fo fiite sample episoes of uit oot a FLUR behavio withi the same sample of obsevatios. Bouay limit theoy as c ± the eveals the asymptotic impact of these subpeio extemes of statioaity a explosiveess. As usual i multiimesioal asymptotics Phillips a Moo, 999, thee ae two possibilities: sequetial a simultaeous limit theoy. The followig sectio cosies sequetial limits, whe fist goes to ifiity a the c goes to eithe plus o mius ifiity. This limit theoy extes to the FLUR eviomet the sequetial asymptotics fo LUR moels Phillips 987. Late we evelop pathwise joit limit theoy that povies simultaeous asymptotics ue the coitio that k =

6 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 5 /c ±. These pathwise joit limits implemet i the FLUR settig the cocept of mil itegatio/mil explosiveess that was evelope i Phillips a Magalios 7. Ou iscussio cocetates o the use of oiay least squaes OLS egessio o. This focus is useful because the limit theoy both as a the bouay limit theoy whee c ± shows the impact o the staa OLS estimato a associate tests of episoes of ea itegatio that take a geeal fuctioal fom of epatue fom uity. The theoy also povies asymptotic powe fuctio behavio of uit oot tests agaist such geeal alteatives i which thee may still exist peios of uit oot behavio. The pesece of ea itegatio i the geeatig pocess of X t is uow a, i pactice, uowable give that the localizig coefficiet is ot cosistetly estimable. It is theefoe of wie iteest to uesta the popeties of staa OLS egessio ue geeal fuctioal epatues fom uity. Issues of cofiece iteval costuctio a the potetial fo uifom ifeece i the pesece of such fuctio epatues ae cosiee i othe ogoig wok Phillips Limit istibutios as c ± Oe avatage of the FLUR specificatio is that use of a localizig fuctio c athe tha a costat c i chaacteizig epatues fom uity accommoates subsample uit oot behavio wheeve the localizig fuctio is zeo. As might theefoe be expecte, asymptotic behavio ca vay cosieably epeig o the specific fom a popeties of c. Regios of zeo a o-zeo values of c tu out to be paticulaly impotat i the limit theoy as they switch uit oot behavio o a off uig the sample. The impact of such switches ae atually magifie as c appoaches the limits of its omai of efiitio. This sectio ivestigates the impact of switchig behavio o the limit theoy by cosieig localizig fuctios c that switch fom zeo ove some iteval [, ] to o-zeo values ove, ] a switch back to zeo o, ] fo < < <. This specificatio eables us to stuy athe geeal foms of subpeio ea itegatio a ea explosiveess i the ata o the asymptotic behavio of FLUR autoegessios. To captue bouay behavio we moel passage to the limit c ± via the specificatio ct = c ft, whee c is a scala that passes to ±, a f is a give itegable fuctio of costat sig, esigatig the iectio of the epatue fom uity. The moel theefoe has the fom with time vayig coefficiet 3 θ t =, t τ =, + cft/, t τ, τ ], τ =, t > τ,

7 6 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION whee the fuctio f is zeo outsie [, ], itegable o [, ], a stictly positive o,. Simple examples iclue level patial epatues fom uity such as fx = {x [, ]}, tiagula epatues such as fx = x {x [, ]} + x {x, ]} o vaious smooth vaiats of such fuctios. Usig 3 a iteative back-substitutio fo X t i we have the solutio X t = + { X + { t s=τ + } t u s {t τ } + s= u s + X τ } {t > τ }, { t τ j= u t j j l= t τ θ t l, + X τ l= θ t l, } {t τ, τ ]} showig the pesece of two peios of uit oot behavio that occu at the stat a at the e of the sample. Itemeiate betwee these peios is a episoe i which the pocess evolves with a time vayig paamete i the viciity of uity. The followig subsectios examie limit behavio i two bouay cases coespoig to statioay c < a explosive c > epatues fom uity as the extet of the epatue c while etaiig the same fuctioal fom fo the localizig coefficiet fuctio f. Sequetial limits ae use i the followig thee theoems a these limits employ the otatio c, seq fo limits i which followe by c, a similaly c, seq eotes limits i which followe by c. We look fist at the statioay bouay. 3.. Sequetial limits whe c. Theoem. The staaize pocess X, the least squaes estimate ˆθ OLS of θ t, a the associate t-atio cetee o uity have the followig limit behavio ue sequetial limits i which followe by c : X c, seq Ba { } + B b { < }, ˆθ OLS c, seq ˆθOLS c, seq cff c F c B a B a + B b B b + λ B a +, B b B b B b + λ ω B a + B b,

8 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 7 7 tˆθols c, seq B b B b + λ ω σ B a + /, B b whee B a a B b ae iepeet Bowia motios with vaiace ω, a λ = ω σ. The limit 4 is a segmete Bowia motio pocess which vaishes o the iteval whee f > a has iepeet Bowia motio segmets o the itevals [, ] a, ] whee the localizig fuctio f =. Whe the oe-sie log u covaiace λ =, the limit istibutio of the aom-cete least squaes estimate ˆθ OLS give by 5 has the simple fom ˆθ OLS cff c F c c, seq B a B a + B b B b B a +, B b which we call a segmete uit oot limit istibutio because of the excisio of the subpeio [, ] i the umeato a eomiato itegals. Whe = o = thee is o episoe of ea itegatio a this istibutio coespos to the staa uit oot istibutio. As is appaet fom the fom of 5, the OLS estimate ˆθ OLS has aom ceteig that ivolves the compoet cff c / F c. As show i the poof, whe c, this ceteig ca be eplace by uit ceteig, but with a impact o the limit istibutio as eviet i the fom of the limit esity fo the o-aom uit-cete esity give i 6. Figue shows the asymptotic esity of the uit-cete 6 OLS estimate ˆθ OLS fo λ =, ω =, = /3, = /3 alog with the esities of the coespoig segmete 5 a staa uit oot esities fo compaiso. All these esities ae skewe a have a typical uit oot istibutioal shape with a log left tail. The staa uit oot esity has the lagest skewess a most ispesio, the segmete uit oot has the least skewess a ispesio, a the uit-cete OLS esity is the most left shifte, showig how misceteig accetuates the owwa bias i the limit istibutio. These shapes become moe istict as a. These esults eveal the substatial impact that FLUR specificatios have o ea uit oot limit theoy. I the staa LUR moel as the localizig scala paamete c, the coectly cete a scale OLS estimate has a bouay limit omal istibutio that coectly epouces the staa statioay case limit theoy. This uifomity i the limit theoy plays a impotat ole i the costuctio of uifom ifeece poceues Mikusheva 7;Mikusheva ;Phillips 4. I the FLUR moel, the bouay limit theoy has geate complexity that eflects featues of the

9 8 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION.4.35 segmete UR esity ˆθ limit esity staa UR esity Figue. Desities of the OLS estimate, uit oot, a segmete uit oot fo λ =, ω =, = /3, = /3. localizig coefficiet fuctio eve i the limit as c a this o loge geeally epouces the statioay limit theoy. I paticula, the fom of the localizig coefficiet fuctio plays a ole i coect ceteig of the istibutio, whe this ceteig is aom. Whe thee ae episoes of uit oot behavio i the pocess, these episoes cotiue to impact the limit theoy at the bouay. The bouay limits 6 a 7 show that both the coefficiet-base a t-atio uit oot tests ae icosistet agaist the alteative of beaks that ivolve subpeios of statioaity. Whe thee ae subpeios of uit oot behavio i the ata, the tests o ot ivege a theefoe fail to etect the existece of statioay episoes i the sample with pobability oe as c. 3.. Sequetial limits whe c. Theoem. Upo appopiate staaizatio, the pocess X, the least squaes estimate ˆθ OLS of θ t, a the associate t-atio cetee o uity have the followig limit behavio ue sequetial limits i which followe by c : 8 e c faa X c, Ba { }, seq 9 e c faa ˆθ OLS ˆθOLS cff c F c c, seq c, seq, C,

10 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 9 e c faa tˆθols c, seq ω σ N,, whee B a is a Bowia motio with vaiace ω, a C is staa Cauchy. As show i the poof of the theoem i 44, the limit behavio of X as has the fom X F c = B a,, J c ;, <, B b + J c ;, < ; with iepeet Bowia motio compoets B a B b a oliea iffusio compoet J c ; = e c s faa Bs = e c faa B a + e c s faa Bs. As c the FLUR peio of explosive behavio omiates though the pesece of J c ;. I paticula, we have X e c faa B a { } a, as i 8, X e c faa B a { }, showig that the pocess iveges expoetially fom the level B a whe > Thus, ove the peio [, ] the staaize pocess / X evolves as a uit oot pocess a eaches the limit value B a at =, at which poit a beak occus i the geeatig mechaism a the pocess evolves i a explosive FLUR way that iflates the iitial coitio eache at =. This behavio cotiues util = whe uit oot behavio e-commeces but fom a explosive iitial coitio give by J c ;. Coespoigly fom 9, a appopiately cete OLS estimate ˆθ OLS has a explosive ate of covegece with ate e c faa. I this case, the ceteig is the aom quatity + cff c / F c, which epes o the scale coefficiet c, the fuctio f a the stochastic pocess F c. Whe c, as show i the poof of the theoem, the aom compoet of this ceteig elemet appoaches a costat, satisfyig cff c F c c, which leas to. Thus, the coefficiet-base uit oot test ˆθOLS = O p as c, seq a theefoe the test fails to ivege i the pesece of a iteal subpeio, ] of explosive behavio i the FLUR moel eve at the bouay as c. Howeve, because the limit >, the test oes have o-tivial powe at the bouay limit c a test powe cotiues to icease as a the peio of explosive behavio expas i the sample. This esult povies aalytic cofimatio of the simulatio esults i Evas 99 that showe how full sample peio uit oot tests pefome pooly i

11 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION the etectio of peioically collapsig bubbles withi sample. O the othe ha, as show i Phillips, Wu a Yu, ecusive uit oot tests ae cosistet agaist such subpeio explosive alteatives. Recusive mechaisms of etectio theefoe have cosieable avatage i this cotext, paticulaly i the pesece of multiple bubbles Phillips et al. 5a, Phillips et al. 5b. Note that the limit theoy 9 fo the cete OLS estimate is a scale Cauchy istibutio. Impotatly, the scale coefficiet { } / iveges whe eithe o. This is explaie by the fact that the covegece ate chages as the limits of the egio of explosive behavio ae eache. Such cases ivolve iffeet asymptotics a iffeet ates of covegece that accout fo the shape behavio of the fuctioal coefficiet f at the limits of the omai of efiitio. They ae epote i etail i ogoig wok Phillips 7 a oe such esult is give i Theoem 5 i the followig sectio. Iteestigly, the asymptotic istibutio of ˆθ OLS is egeeate whe cetee o uity, as eviet i the limit of the coefficiet-base uit oot test. Moeove, it is isufficiet to simply ecete agai usig the costat. I fact, i the spiit of the poof of Theoem, we ca show that whe f >, c ˆθ OLS c, seq 4 f, leaig to a futhe egeeate istibutio. The limit theoy equies moe pecise appoximatio tha of the aom ceteig that is peset i 9. Ou ext theoem gives the coect o-egeeate asymptotics with etemiistic ceteig. Theoem 3. If f C > fo all [, ], the OLS estimate ˆθ OLS has the followig limitig Cauchy istibutio afte appopiate etemiistic ceteig a scalig 3 e c faa ˆθOLS e c faa + e c faa c, seq C 3. Itiguigly, Theoems a 3 both lea to vey simila Cauchy limit istibutios. The oly iffeece besies the eceteig is the umeical coefficiet 3 i Theoem 3. The explaatio fo this simple scala iffeece i the limits lies i the eplacemet of a aom ceteig i 9 with accuate o-aom ceteig i 3. The ituitio is as follows: i Theoem the cetee statistic 9 has aitioal vaiability because of the aom ceteig, which leas to lage ispesio i the Cauchy limit theoy tha i Theoem 3 whee the ceteig is costat a ispesio eceases.

12 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION As is appaet i the poof of Theoem 3, the easo fo the specific umeical costat 3 is that ajustmet of the ceteig pouces a emeae Bowia motio i the limit theoy athe tha a Bowia motio, with a coespoig euctio i the vaiace of the umeato. I paticula, i Theoem the limit istibutio is govee by the atio, wheeas i Theoem 3 the goveig atio is B b B b B a 4 B b B b B b. B a The umeato i 4 is the emeae fom of the Bowia motio iffeetial B b B b a, coespoigly, the umeato vaiace euces by the scale facto 3, theeby leaig to the umeical coefficiet / 3 which appeas i Mil FLUR Moels a Simultaeous Asymptotics The moels cosiee so fa i the pape all follow with a time vayig coefficiet i the local viciity of uity that has the geealize LUR fom θ t = + ct/. I oe to wie the viciity of uity ue aalysis, this sectio cosies coefficiets θ t that pass to uity at a slowe ate O tha O whee k a. The autoegessive coefficiets have the fom 5 θ t = + ct/ k, fo some fixe fuctio c. The fomulatio 5 falls i the class of milly itegate/milly explosive pocesses cosiee by Phillips a Magalios 7. With this specificatio, wie epatues fom uity may be cosiee a it is possible to evelop simultaeous asymptotics whee the paametes c, may joitly pass to ifiity. I this passage to ifiity what mattes is the atio k = a, as above, we assume that c k +. Fo example, we may have k = α with α, o k = / log. Sice the paamete settig 5 leas to autoegessive coefficiets that ae close asymptotically to the statioay zoe tha those of the FLUR moel whe c <, it is coveiet to evelop the limit theoy ue statioay matigale iffeece eos {u t }, a settig that is bette suite to that cotext Phillips a Magalios 7; Giaitis a Phillips 6. Futhe, the fuctios c pemitte i this sectio coespo to some of those use i Phillips 7. Specifically, we assume the fuctio c o [, ]. Moe pecisely, i the explosive case whee c > we assume c is o-zeo i some fixe egios of the oigi a uity, so that the FLUR pocess is active i those egios. As will become clea i the followig aalysis, behavio i those egios is paticulaly impotat i the explosive case because they play a sigificat ole i the behavio of the time seies a, i cosequece, the limit theoy also. Fially, it is coveiet to set the iitial coitio i at t = a assume that X = o p k, which ules out iitial

13 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION coitio effects, although this coitio may be elaxe as i Phillips a Magalios 9 with some atteat iffeeces i the asymptotics. Moels such as with time vayig autoegessive coefficiets of the fom 5 ae milly itegate/explosive fuctioal local uit oot MIFLUR, MEFLUR moels. As will become clea, MEFLUR specificatios lea to asymptotics whee thee ae epeecies o specific fuctio values, such as the oigiatio a e poit values c, c as well as the fuctio c ove its full omai [, ]. Fo the MEFLUR case we also cosie a seco specificatio fo the coefficiet fuctio i place of 5, viz., 6 θ t = + ct/k k, whee θ t coveges to uity at the slowe ate O a the time vayig coefficiet fuctio c is ow scale cosoatly i /k uits athe tha / uits. I this case, the limit theoy epes o the coefficiet fuctio c ove its etie omai, which is ow [,, a c is accoigly assume to be itegable ove this omai. 4.. Milly Explosive FLUR. We stat fom the fist specificatio 5 of θ t with c >. Solvig the system yiels 7 X t = θ t X t + u t = t e j= t j l= c t l+ uj + e t j= c j X. It is coveiet to costuct the two staaize pocesses without employig aay otatio 8 Xt = e X t t j= c j = t j= e u j t l=t j+ c t l+ + X, a Ỹt = t j= u je t c l=j l. The time seies X t upweights ealy iovatios {u j : j =,,...} because of the smalle umbe of compoets that ete the summatio t l=t j+ c t l+ i the expoet whe the iex j is small. I a simila way, the time seies Ỹt owweights ealy iovatios {u j : j =,,...} because of the lage umbe of compoets that ete the summatio t l=j c l i the expoet. The pocess Xt is theefoe weighte i favo of the oigiatio poit of the obsevatios, a Ỹt, as a mio image, is weighte i favo of the temial poit t of the obsevatios i the sum. The limit theoy whe t k as fo these two staaize pocesses is give i the followig theoem which eveals the impotace of the epoit coitios.

14 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 3 Theoem 4. If thee exists ε > such that fo all [, ε] [ ε, ], c C >, the fo all, ] as k + 9 X Ỹ X c = N, Y c = N, a the limit vaiates X c a Y c ae iepeet. σ, c σ, c Thus, the staaize pocess X / k tes fo all, ] to the same aom vaiable X c whose istibutio epes o c a o othe value of the fuctio c. The explaatio is that sice c > the time seies X t is explosive, which meas that iitial shocks a iitial coitios ae magifie, as is appaet i the solutio 7. { Moe paticulaly, the coefficiet exp t j k l= c t l+ } is positive a iceasig as j eceases, so that i 7, the ealy shocks {u, u,...} have the lagest coefficiets a the geatest impact o X t comes fom the ealy pat of the seies. Coespoigly, the staaize pocess X t i 8 is omiate by the ealy k elemets of the seies, which leas to the commo cetal limit theoem give i Theoem 4 fo X, whose vaiace ivolves oly c athe tha the full fuctio c. Aalogous mio-image ituitio applies to the staaize pocess Ỹ / k whee the temial k elemets of the seies lea to a commo limit theoy that epes o the e poit c athe tha the full fuctio c. Next we tu to the limit behavio of the autoegessive coefficiet estimate ˆθ OLS. Afte suitable ceteig a scalig we obtai the followig esult. Theoem 5. Suppose c is itegable ove [, ] a c C > fo all [, ε] [ ε, ] fo some ε >. The, as k +, the least squaes estimate ˆθ OLS has the followig commo limit theoy with aom a etemiistic ceteig k e k e caa caa ˆθ OLS ˆθ OLS whee C is a staa Cauchy vaiate. t= ct/ k X t Xt t= ct/e ccc, cj/ j=t t= k e cj/ j=t t= ccc,

15 4 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION The poof of Theoem 5 is legthy a ivolves complex calculatios which ae give i full i the poof i the Appeix. Impotatly, although a iffe i tems of thei espective ceteig, the limit istibutio behavio emais the same. As show t= i the poof, the aom ceteig quatity ct/x t that appeas i equals j=t t= ct/e cj/ k j=t t= e cj/ k t= X t plus a aom compoet of smalle oe that is too small to affect ceteig but which, as show i the poof, still comes ito play i calculatig the limit behavio of the escale a ecete estimato. As the esult shows, the aom ceteig ca be eplace by a etemiistic ceteig that is still epeet o the sample size but without affectig the Cauchy limit istibutio o the scale of this istibutio. Futhe, as k + we fi that 3 t= ct/e t= e j=t cj/ = j=t cj/ t= ct/e j=t cj/ t= e j=t cj/ c, givig a simple limitig fom of the e-ceteig elemet 3. This simple fom caot, howeve, be use iectly i the limit theoy because the expoetial ate of covegece k e caa plays a key ole i efiig those compoets that ive the asymptotics, as explaie i the poof of Theoem 5. Obsevig that θ = c k { + o }, θ = c k { + o } a Π t=θ t e k Σ c t= t e caa, we may wite i the equivalet fom Π 4 t=θ t ct/e cj/ j=t t= [θ θ ] ˆθ OLS cj/ C, k t= e which is suggestive of ealie wok i simple cases of explosive pocess autoegessio. I paticula, 4 shows that, upo suitable staaizatio which i this cases elies o the time vayig autoegessive coefficiet, cetal limit theoy hols fo autoegessive estimatio i the fuctioal LUR with the same Cauchy limit theoy as hols i i the fixe coefficiet explosive case ue Gaussia iovatios with o ivaiace piciple, a ii i the milly explosive case ue cetal limit theoy Phillips a Magalios 7. Impotatly, both the covegece ate a the ceteig epe o the fuctioal coefficiet c thoughout the [, ] iteval. I the special case whee c = c > is costat a θ t t =,...,, 4 euces to the fom 5 θ θ ˆθ OLS c k j=t C, = + c k =: θ fo all

16 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 5 the esult obtaie oigially i Phillips a Magalios 7 fo a costat milly explosive pocess X t with autoegessive coefficiet θ = + c k fo which θ e c. I θ that case the covegece ate of 5 is e θ c. The limit theoy give i 4 c the specializes to this ealie theoy as k +. I the peset case, the pimay implicatio of a fuctioal localizig coefficiet is to ajust the ate of covegece i to emboy the aggegate impact of the fuctio c ove its full omai via the itegal c a a, so that the staaizatio facto is k e caa athe tha k e c. It is the the accumulative mil FLUR epatue fom uity that etemies the covegece ate of the estimato ˆθ. Iteestigly, a secoay implicatio of the ew limit theoy i is that the limit aom vaiable ccc epes explicitly o the behavio of the localizig fuctio at the oigiatio a temiatio ates via the pai c, c. This epeece is a cosequece of the magificatio of ealy a late shocks that takes place i the limitig pocess escibe above fo a milly explosive time seies. The e-staaizatio by [θ θ ] i 4 ajusts fo these iitial a temial effects a the epeece is elimiate. Next cosie the seco specificatio 6 with c >. I this case, time is measue i /k uits athe tha / uits i the localizig coefficiet fuctio c a sice k the omai of the fuctio is [,, leaig to the followig limit theoy fo the staaize pocess whe c > is itegable ove [, ] 6 X X c, = N, σ e a cpp a. The limit i 6 emais the same aom vaiable fo all values of, just as i Theoem 4 above. But i the peset case, as is clea fom 6, the limit vaiace epes o fuctio values c > ove the full omai [, athe tha the sigle fuctio value c at the oigi. We may also cosie the case whee the staaize time seies is measue i segmets of legth Ok athe tha legth O. The famewok the matches the usual FLUR moel of Bykhovskaya a Phillips 7 but ove a much wie ifiite omai. Moe specifically, whe we focus o the pocess X t with t = k istea of t =, we have the followig limit theoy fo the staaize pocess X. 7 X X c, = σ e a cpp W a. Impotatly, i 7 the omai of is the half lie [,, theeby accommoatig limit behavio of the pocess X t fo t k. Thus, the limit X c, i 6 may be itepete as the limit of the stochastic pocess X c, as. The covaiace keel of X c, is γ, s = σ mi,s e a cpp a, which euces to the usual expessio fo the covaiace

17 6 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION keel of a liea iffusio whe c is costat o to that of a Bowia motio whe c =. 4.. Milly statioay FLUR. We focus o the fist specificatio 5. As show below, this fomulatio leas to a limit pocess fo a staaize vesio of X t= whee thee is explicit epeece o the localizig fuctio value c at the sample factio coespoig to the sample poit t. This outcome iffes fom 9 a i the milly explosive case whee thee is epeece o the e poit values c a c. Suppose that c C <. By 7 we have the epesetatio X t = t e j= t j l= c t l+ uj + e t cj/ j= X. a the followig limit theoy the hols fo X t afte suitable staaizatio. Theoem 6. Give ay fixe, ] fo which c C <, the as 8,,, ]. X f X c = N σ c Diffeet values of lea to iepeet aom vaiables. Impotatly, 8 gives a fiite imesioal limit istibutio fo each fixe, ot a fuctioal law. This is sigifie i 8 by the affix f i place of weak covegece ove [, ]. As the theoem iicates, the limit vaiates X a X s ae iepeet fo all s. While the limit aom vaiable X exists fo each fixe, the limitig stochastic pocess X o, ] has pathological path popeties because the iepeece of abitaily ajacet compoets X a X s implies a egee of local vaiability that is uealizable. 5. Some Implicatios of Bouay Limit Theoy Local uit oot limit theoy eable aalysis of the powe popeties of uit oot tests a helpe exploe the passage to statioay a explosive behavio by examiig bouay behavio i the asymptotics. The LUR methoology has sice bee use extesively i the ecoometic aalysis of tests i uit oot moels, coitegate systems, a peictive egessio. I FLUR moels, epatues fom uity allow fo fuctioal, time epeet foms that vay ove the sample peio. Coespoigly, i FLUR specificatios both the limit theoy a the asymptotic powe popeties ivolve iche possibilities that accommoate ealistic empiical situatios whee uit oot behavio may be iteupte by episoes of ea-statioay o ea-explosive behavio i the ata. The passages to statioay a explosive behavio at the bouay of fuctioal specificatios become

18 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 7 similaly moe complex a have implicatios fo pactical wok with ifeece fo time seies ata. This pape has exploe these implicatios i autoegessios whe o allowace is mae fo time vaiatio i the paametes i estimatio a ifeece, as happes i pactice whe a ivestigato pocees with paametic autoegessio a staa testig poceues i igoace of the geate complexity of the geeatig mechaism. I movig to the statioay bouay as might be expecte i a time vayig eviomet, the omiatig compoet of the limit theoy is ay emaiig ostatioay episoe i the ata. Fo the pocess itself, fo the autoegessive estimate, a fo uit oot test statistics, the bouay asymptotics epe o the iteval that efies this episoe, leaig to a fom of segmete uit oot limit theoy. These esults iffe sigificatly fom those of LUR bouay asymptotics which ae well ow to lea to staa omal asymptotics at the statioay bouay Phillips 987; Giaitis a Phillips 6; Mikusheva 7. The implicatio is that fuctioal epatues fom uity ca have a majo effect o limit theoy a test pefomace. Likewise, movig to the explosive bouay pouces mateial chages i the asymptotics. I this case, the omiatig compoet of the limit theoy comes fom the explosive episoe i the ata. Agai, the bouay limit theoy epes o the egio that efies the episoe. I the explosive iectio, the bouay limit theoy is cete i the explosive egio. But while uit oot tests have o-tivial powe at the bouay they ae ot cosistet, which patly explais the poo pefomace of ight-sie uit oot tests i the etectio of peioic episoes of bubbles a the ee fo ecusive egessio methos of etectio which have geate sesitivity to local epatues fom uity. Fuctioal local alteatives such as those cosiee hee i the uit oot cotext obviously have wie applicatios i statistical limit theoy a powe fuctio aalysis beyo those of uit oot moels, although thee seems to have bee little use o metio of them i the liteatue to ate. They ae also useful i the costuctio of fuctioal poit-optimal test poceues, whee thee ae potetial gais fom the cosieatio of explicit fuctioal alteatives athe tha fixe alteatives. Fo istace, Bykhovskaya a Phillips 7 examie some of the implicatios of fuctioal epatues fo uit oot testig with a focus o the popeties of poit optimal poceues. A futhe applicatio of these bouay asymptotics that is elevat to empiical wok is the impact of local time vaiatio of the type cosiee hee o uifom ifeece i autoegessio. That subject is ivestigate i othe ogoig wok Phillips 7.

19 8 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION Poof of Theoem. 6. Appeix Poof. Fom a with log u vaiace ω = h= pocess fo X t as has the followig segmete fom X ωk c = F c = with B a, B b = BMω, B a B b a J c ; = e s cfaa Bs = = e c faa B a + Eu o u h, we euce that the limit B a,, J c ;, <, B b + J c ;, < ; e cfaa Bs + e c s faa Bs. e s cfaa Bs Because f is stictly positive o,, we ow that faa > fo all s > s, > s. Thus, e c s faa mootoically i s as c, so that J c ; as c fo, ]. Theefoe, the limit pocess F c i coveges to F = B a { } + B b { < }. We may ow calculate the limit istibutio of the OLS estimate ˆθ OLS = t= XtX t t= X t i sequetial asymptotics as passes to ifiity followe by c passig to mius ifiity, which we wite as c, seq.. Wite 9 ˆθOLS = X t X t = t= Xt t= X t u t + t= Xt t= t= cft/ X t Xt t= a ote fom Lemma that we may euce the joit weak covegece cft/xt t= t= X t F c, cffc = ω cfkc. To calculate the limit of t= X t u t i the umeato of the fist membe of 9, squae, sum ove t, a scale by, givig, X = t u t + t X t u t + t cft/ Xt + O p.5,

20 so that BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 9 X t u t ω Kc / σ / ω t= cfk c. Stochastic iffeetiatio of Kc gives Kc = + cfk c + K cw, fom which we euce that X t u t t= ω K cw + λ, whee λ = Eu o u h = ω σ, B = B a { }+B { < }+B b h= { < }. Thus, as we fi that ˆθ OLS cffc F c F cb + λ F, c 3 ˆθOLS F cb + λ + cffc F. c The limits of F cb + λ a F c as c ae staightfowa a we fi the followig cetee limit theoy i the bouay asymptotics cffc 3 ˆθ OLS c, seq. F c B a B a + B b B b + λ B a +. B b Note that the ceteig of the limit theoy fo ˆθ i 3 is stochastic a ivolves the weighte quatity cffc whose limit behavio is complicate. We pocee to calculate this limit to evelop a o aom ceteig i place of 3. 3 c ffc = ω c f e c faa W a + = ω cw a fe c faa + ω cw a + ω c f fe c faa e c e c e c s faa W s. s faa W s s faa W s We evaluate the limit of each of the thee tems i Eq. 3. Fist, eotig :=,

21 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 33 ω cw a = ω W a fe c faa = ω cw a e c faa c Ba. c ec faa The seco tem has mea [ E cw a fe c ] faa e c s faa W s =, as W a a W ae iepeet. We ow show that the vaiace of this tem coveges to zeo as c, so that the tem tes i pobability to zeo. We have 34 [ E cw a [ = 4 c E [ = 4 c E = 4 c = 4 c = s fe c faa e c s faa W s] fe c faa s c e c e c s faa W s] fe c faa e c s faa W s fe c faa e c s faa s e c faa+c s faa+c s ] faa e c s faa s faa + e c s faa e c faa c, a hece 35 ω cw a fe c faa e c s faa p W s. c We ae left with the thi tem. As with the seco tem, we show that the vaiace coveges to zeo, a thus the whole tem coveges to the limit of its expectatio. We

22 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION fist calculate this limit as follows 36 E ω c f e c [ = ω c fe = ω c fe c s = ω e c s faa W s s ] s faa W s = ω c f e c s faa s faa ω s = e c s faa s e c faa ω s s c ω. We pocee to show that the vaiace of the thi tem goes to zeo. We use the fact that fo a stochastic pocess ξ t with fiite seco momets we have V Theefoe, ξ t t = E = E = ξ t t ξ t t ξ s s E ξ t t E ξ t ξ s st E ξ t te ξ s s Eξ t Eξ s st = cov ξ t, ξ s st. 37 [ V c = c f e c s faa W s ] ff cov e c s faa W s, e c s faa W s. Fo this calculatio, we ee the covaiace betwee two squae omal vaiables with zeo mea. By Isselis s theoem, if ξ, ξ ae omally istibute possible epeet with zeo mea, the covξ, ξ Eξ ξ Eξ Eξ = Eξ ξ.

23 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION Thus, we ca ewite Eq. 37 as 38 [ V c f e c s faa W s ] = c ff E e c s faa W s = c ff mi, e c s faa W s e c s faa e c s s faa = 4c ff e c faa e c s s faa 39 = 4c = 4c = c s = c s maxs,s ff e c faa e c s faa e c s faa s s e c s faa s fe c faa f e c s e c s faa s e c faa f e c s e c s faa e c faa f e c s faa s faa s faa s s, with 4 c e c s faa f e c maxs,s = = = e c faa s e c e c s s faa faa e c s s faa e c maxs,s e c s s faa s faa s s faa s s e c s s s faa e c s faa s s e c s s faa s + s se c s s s faa s,

24 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 3 a 4 c = e c maxs,s e c s s faa f e c s faa s s faa+c s faa e c maxs,s s faa+c maxs,s s faa s s = e c s s s faa e c s s faa s s. Usig 4 a 4 i 39 gives [ V c f s faa W s ] 4 = + s e c e c e c s s faa s s s faa s s c, e c e c s faa s s se c as each tem goes to zeo. It follows that 43 ω c f s faa p W s c ω. a usig 33, 35, a 43 i 3, we obtai cffc ω + B a. c s faa s Fially, takig the limit as c i 3 leas to a fom of segmete uit oot limit istibutio as follows F cb + λ + cffc F c c = B a B a + B b B b + λ ω + B a B a + B b B b B b + λ ω B a + B b. Usig these esults a wokig i a simila way, we ca eive the bouay limit behavio of the uit oot t statistic associate with ˆθ OLS, i.e., = ˆθ Xt ˆθOLS OLS t= :=. tˆθols ŝˆθols X t ˆθ OLS X t t=

25 4 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION Sice ˆθ OLS = + O p = + cft// + O p, it follows that a we have aleay show that ˆθOLS F c B + λ X t ˆθ OLS X t p t= σ, F cb + λ + cffc F, c B a B a + B b B b + λ, c cff c ω + B a, c F c B a + c Combiig these limits yiels the esie esult B b, tˆθols c, seq = B a B a + B b B b + λ ω + B a σ B a + / B b B b B b + λ ω σ B a + /. B b Poof of Theoem. Poof. By Eq., the limit pocess fo X t has the segmete fom 44 X F c = B a,, J c ;, <, B b + J c ;, < ; with B a B b a 45 J c ; = e c s faa Bs = e c faa B a + e c s faa Bs.

26 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 5 So 46 e c faa J c ; = B a + = B a + e c e c faa e c s faa Bs s faa Bs = B a + o p. Thus, fo all, e c faa B = e c faa O p, c a, fo all < while Thus e c faa X e c faa J c ; = e c faa B a + o p, c e c faa J c ; c Ba. c, seq Ba { }. By the same agumet as i the poof of Theoem, ˆθOLS = X t u t + t= Xt t= t cft/ X t X t t F cb + λ + cff c F, c so that ˆθ OLS cff c F c F cb + λ F. c Usig Eq. 46, we ca calculate the limit of the eomiato F c as c goes to ifiity. Fist, 47 F c = B a = O p.

27 6 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION The, 48 Fially, F c = = = J c ; = [ e c faa B a + [ e c faa B a + e c e c faa B a + o p = B a F c = = Hece, usig Eq. 45, we have 49 e c faa e c faa + o p B b + J c ; B b + e c ] s faa Bs B b J c ; + F c = o p + B a e c faa Combiig Eq. 47, Eq. 48, a Eq. 49 the yiels 5 e c faa e c faa. ] s faa Bs J c ;. B b + B a = B a + o p. F c = o p + B a e c faa + o p e c faa + B a = B a + o p. I a simila way, we ca aalyze the fist pat of the umeato, F cb + λ. Note that F c B = + = + B a B a + B b + J c ; B b B a B a + J c ; B B b B b J c ; B + J c ; B b B b.

28 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 7 Thus, 5 [ e c faa + e c ] F c B + λ = o p + B a B b B b faa B a + o p B = B a B b B b + o p. Combiig Eq. 5 a Eq. 5, we obtai e c [ faa ] F cb + λ e c faa F c c B a B b B b B a = Bb B b B a = N, N, = C. Thus e c faa cff c ˆθ F c c, seq C, leaig to a scale Cauchy istibutio i the limit. We ae left to aalyze the seco pat of the umeato, cff c. The aalysis is almost ietical to the case c stuie i the poof of Theoem. By Eq. 3, c ffc = ω cw a + ω cw a + ω c f By Eq. 33, fo the fist tem we have fe c faa fe c faa e c e c s faa W s. s faa W s 5 e c faa ω cw a = ω W a fe c faa e c faa c Ba.

29 8 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION Next, fo the seco tem, Ee c faa [ cw a = cew a E fe c a by Eq 34, [ E e c faa cw a = Fially, by Eq. 36 a 4, [ a V [ = E e c s faa + e c = ω e c faa ω c s e c e c faa c e c faa fe c faa faa e c e c s + ] s faa W s faa W s =, fe c faa e c s faa W s] faa c s faa s ω faa e c faa f e c s faa W s ] e c f e c s faa W s ] e c s faa s e c faa + e c faa se c s faa s s Combiig these thee tems gives 53 e c faa e c s + s cff c faa c, s e c faa s s c. c s Ba. Futhe, combiig Eq. 5, Eq. 5, a Eq. 53, we get e c [ faa F cb + λ + c ] ff c c e c faa F c Ba B a =, c. faa s s

30 so that BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 9 ˆθ p c, seq, givig a costat i the limit. The limit behavio of the t statistic fo ˆθ OLS follows as i pevious esults. I paticula, we have ˆθ c, seq, leaig to e c faa tˆθols e c faa c, seq / F c B a. c B a σ = ω σ N,. Poof of Theoem 3. Poof. Fom the poof of Theoem, we ow that e c J c ; = B a + o p. Thus, J c ; has the lagest stochastic oe amog the J c ; fo. It is coveiet to ewite J c ; i tems of J c ; as 54 J c ; = e c s faa Bs = e c faa J c ; e c s faa Bs, whose fist tem is expoetially lage a whose seco tem is O p c.5, sice it is Gaussia with zeo mea a vaiace e c s faa s +ε e c s fa s = cf e cfε. Defie ξ cf := e c s faa Bs cf N, a ewite both umeato a eomiato of as follows. 55 F c B = B a B a + F cb+λ+ cffc Fc J c ; B + = J c ; B b B b + J c ; i tems of J c ; B b + J c ; B b e c faa Bs + Op, cff c = cfj c ; = J c ; c cj c ; c fe c faa ξ fe c faa + ξ,

31 3 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION whee E ξ fe c faa = E ffte c faa e c t faa t = ξ ξ t ffte c faa e c t fe c faa faa t c f. Hece, ξ fe c faa = O p /c. Notig that c fe c faa = e c faa, we the have 56 cff c = e c faa J c ; + J c ; O p / c, a 57 Fc = B a + Jc ; + B b + J c ; = Jc ; e c faa J c ; ξ e c c f ξ faa + c f + Jc ; + J c ; B b + O p = Jc ; + e c faa + J c ; B b + J c ; O p /c c + O p, whee c ξ fcc ξ = O p /c. Combiig Eq. 55, 56, a 57, we get Eξ ξ t Eξ Eξ t = =.

32 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 3 58 F cb + λ + cffc F c = e c faa Jc ; + J c ; B b B b Jc ; + faa + J c ; B b e c faa + e c e c faa e c faa + e c + faa B b e c faa B b B b c faa e J c ; B b + e c faa J c ; J c ; B b B b e c. faa J c ; Rewitig Eq. 58 a usig the fact that J c ; = e c faa B a + O p, we get 59 e c faa B a F cb + λ + cffc e c faa F c + e c Bb B b e c B b faa + faa + e c 6 B b B b B a B b. faa e c faa e c faa Because B b B b B b is Gaussia a iepeet of B a the limit vaiate i 6 has a Cauchy istibutio. We pocee to calculate the scale coefficiet of

33 3 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION this istibutio. The compoet tems i the vaiace of the umeato ae as follows: E B b B b B b = 4 EB = 4ω B = ω, E B b B b = 4 ω, E B b = so that B b B b B b e c faa c + It follows that e c faa 4 = 4ω = 4ω 3 3 EBsBs mis, s = 4 3 ω, N, 4 3 ω a thus F cb + λ + cffc F c 4/3 C = ˆθ OLS + 3 C. e c faa e c faa e c faa +,c seq + e c faa 3 C. Poof of Theoem 4. Poof. The poof of 9 follows by the matigale CLT fo tiagula aays by establishig the stability a Liebeg coitios. Fist cosie the vaiace of X, fo

34 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 33, ]. Usig the matigale iffeece popety of {u t } t, we have 6 X = σ k j= σ { e l= j+ σ l+ c } + o e { z cx x}z + o e zc z = σ c, which is costat. Next suppose that s > a ote the covaiace X, X s = σ k j= σ e σ c l+ s + l= j+ l= s j+ c s l+ + o p e z cx x z + o e zc z = σ c = X = X s, is also costat. The Liebeg coitio is establishe as follows. Take δ >, a otig that c a ove a [, ] a c a C > fo all a [, ε] [ ε, ] fo some ε >, we obtai E j= j= e l= j+ l+ c u j k [ u j > δ k e l= j+ l+ c ] = e l+ c l= j+ E {u k j [ u j > δ k e l+ c l= j+ ]} = e l+ c l= j+ E {u k [ u > δ ]} k k j= e l+ c l= j+ E {u [ u > δ ]} k j= k e s { caa s E u [ u > δ ]} k { c E u [ u > δ ]} k

35 34 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION sice E u < a e s ε caa s e cs s = k e cε c k c. Thus, the Liebeg coitio hols a 9 hols fo all, ]. Result follows i a elate way a the poof is omitte. Poof of Theoem 5. Poof. The poof is simila to the lie of easoig use i the poof of Theoem 3 but the calculatios ae cosieably moe complex. Fist, it is useful to e-omalize the time seies X t as X t, a the ewite eveythig i tems of the last obsevatio, X. Note that 6 X t = e t t l+ l= c Xt = e t c j j= Xt = X + = X j=t+ e We stat with the ecompositio 63 k ˆθOLS = j l= cl/ uj. t= X t X t k t= X t = t j= c t k Xt t= k t= X t e + j cl/ l= uj t= X t u t k t= X t a ewite the compoet sums i tems of X usig 6. We cosie each tem sepaately, statig with the commo eomiato. i Deomiato of 63 Upo scalig by e c j j=, we wite the eomiato as 64 e c j j= = = + t= t= X j=t X t= t= e j=t c j Xt = e j cl/ l= uj t= e c j j=t j=t e t Xt e c j= j e c j j= e j=t c j X j cl/ l= t= u j e j=t c j j=t e =: D, + D, + D,3., j cl/ u j l=

36 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 35 Fom Theoem 4, we ow that X = O p. It is easy to see that i the last pat of 64 the fist tem D, has the lagest oe of magitue a the seco tem D, has the seco lagest oe. Moe specifically, the thee tems of 64 take the followig foms as k a 67 D, = X X D, = ε t= e c j j=t X X e c s s = X t= X k e j=t c j e z caa = X e caa k = X e caa k X = X k = O p e e caa caa k e caa D,3 = k = k e = e t= = O p e, z j=t e e s caa s k e cε = O p c j cl/ u j l= e s caa Bsz e z caa e s s caa e s caa cs Bs e c j j=t e z caa caa caa z caa. z e s z caa Bsz e s z caa zbs e s caa cs Bs, z k j=t e j cl/ u j l= e s caa Bs z e s z caa Bs z z e s z caa Bs z,

37 36 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION Note that the stochastic itegal epesetatio e s caa Bs that appeas cs i the Eq. 66 elies as o the fuctio cs oly i the immeiate eighbohoo of the e poit s =, so asymptotically the coitio c > is sufficiet fo the asymptotics to hol. ii Numeatos of 63 Stat by cosieig the umeato i the fist tem o the ight sie of 63. Upo scalig by e c j j=, as fo the eomiato, a by expaig the expessio i tems of X, we obtai 68 e c j j= = + X t= t= t= t Xt c = t= t c e c j=t j t c e c j=t j =: N A, + N A, + N A,3. j=t e t Xt c e c j=t j X t= j cl/ u j l= t c e j=t c j j=t e j l= c l u j By the same logic employe with the eomiato, i 68 the fist tem has the lagest oe of magitue, a the seco has the seco lagest oe of magitue. Note, i paticula, that 69 N A, = X t= X k t c e j=t c j cze z caa = X e caa k = X e caa k X = X k = O p e e caa caa k e caa, z j=t e j c l l= u j e s caa Bsz cze z caa e s s caa e s caa Bs, e s caa Bs, z e s z caa Bsz cze s z caa zbs,

38 BOUNDARY LIMIT THEORY FOR FUNCTIONAL LOCAL TO UNITY REGRESSION 37 which has pecisely the same fom a cosequetly the same oe as 66. Simila agumets to those of the eomiato apply to the othe two tems i this umeato expasio. Next cosie the umeato of the seco compoet of 63. Scalig this tem by k e j= c j, we have 7 e c j j= = t= = X X t= =: N B, + N B,. X t t= j e l= j=t u t = c l uj t= X t u t e u t e j=t c j u t e c j j=t u t e t= t j= c j e j= c j j=t c j j=t e j l= c l u j The fist tem of 7 is appoximately sice N B, = X k e s caa Bs = O p N, k k σ = O p, c V k σ k ε e s caa Bs = σ e s caa s e c s s = The seco tem of 7 is appoximately σ e cε c σ c. 7 N B, k = e e z caa caa = O p k e k z z caa, e s caa BsBz e s z caa BsBz which is of smalle oe tha the tem N B,. Futhe, upo multiplicatio by k e c j j= to match the scalig of the eomiato a the fist pat of the umeato, both tems of 7 ae evietly of smalle oe tha 64 a 68. It is