UNIVERSITY OF NOTTINGHAM

Transcription

1 UNVERSY OF NONGHAM SCHOOL OF ECONOMCS DSCUSSON PAPER NO. 99/7 Surious rjcions by Prron ss in h rsnc of a mislac or scon brak unr h null a-hwan Kim, Shn J. Lybourn, Paul Nwbol School of Economics, Univrsiy of Noingham, Noingham NG7 RD, UK Absrac is known ha Dicky-Fullr ss can la o surious rjcions of h uni roo null hyohsis whn h ru gnraing rocss is iffrnc-saionary wih a brak. Suos now ha an unsuccssful am is ma o allow for a brak, ihr hrough mislac ummy variabls or hrough nglcing a scon brak. is monsra ha surious rjcions can now occur for a broar s of ru brak as han woul b h cas if h ossibiliy of a brak was ignor.

2 . nroucion n a sminal ar, Prron 989 monsra ha Dicky-Fullr ss may hav lil owr whn h ru gnraing rocss is saionary aroun a brokn linar rn s also Monans an Rys 998. Convrsly, Lybourn al 998 show ha whn h ru gnraing rocss is iffrnc saionary, bu wih a brak, rouin alicaion of Dicky-Fullr ss can yil surious rjcions of h uni roo null hyohsis whn h nglc brak is rlaivly arly. his is so for boh a brak in lvl an a brak in rif, alhough hr ar imoran iffrncs bwn h wo cass. Prron 989, 99, 994 an Prron an Voglsang 99, 99 iscuss moificaions of h ss ha circumvn hs ifficulis by incororaing ummy variabls a h brak a. For h rocurs consir hr, h Prron ss hav null isribuions ha ar invarian o h magniu of h brak whn hr is a singl brak a, xognously an corrcly rmin. n his ar, w concnra on h cas of an xognously chosn brak a, bu nrain h ossibiliy ha an incorrc choic is ma. n fac, h Prron s saisics consir ar invarian o any brak in h gnraing rocss a h assum brak a. Our rsuls hrfor aly qually o h ossibly mor racically imoran cas of a gnraing rocss wih wo braks, only on of which is scifically accoun for in h analysis. As in Lybourn al 998, w fin ha a nglc rlaivly arly brak can la o surious rjcions of h uni roo null hyohsis. Morovr, for all bu on of h ss analys, surious rjcions now also aris if a ru brak occurs rlaivly soon afr h assum brak a. hus, for xaml, if h ru gnraing rocss is iffrnc-saionary wih wo rlaivly clos braks, a rjcion of h null hyohsis is likly o occur if ummy variabls accouning for only h arlir of hs braks ar incorora ino h analysis. his hnomnon ariss for boh braks in lvl an braks in rif, hough, as in Lybourn al 998, hr ar imoran iffrncs bwn h wo cass, which w analys rscivly in scions an. W consir boh aiiv oulir an innovaional oulir varians of h Prron saisics, rsricing our horical analysis o h formr. Simulaion vinc suggss only minor iffrncs in rsuls for corrsoning varians of h wo ys of ss.

3 . Braks in lvl Consir h siml iffrnc-saionary gnraing rocss for h sris y y = + ν, ν = ν + ε, =,..., whr ε ar i.i.. isurbancs wih man an sanar viaion σ. A lvl brak of magniu σb, occurring a fracion, hrough h sris, is scifi as / = σb = σk [ > ]. n fac, if b is hl consan as saml siz incrass, h asymoic isribuions of h s saisics ar invarian o brak magniu. Howvr, allowing brak magniu o grow a ra wha is acually foun in mora-siz samls., / gnras limiing isribuional rsuls ha ric h aiiv oulirs varian of h Prron s, wih assum brak fracion, is bas on wo rgrssion ss an whr y = αˆ + βˆ + γˆ + = ρ + + φ ω 4 ω in 4 is an rror rm, an is a ummy variabl allowing a lvl brak a fracion hrough h sris, = [ > ]. 5 Of cours, if a brak is assum a h corrc lac an hr is no aiional brak, so ha =, his is h su analys by Prron 989, 99, 994 an Prron an Voglsang 99. n aricular, h limiing null isribuion of h s saisic, akn hr o b h -raio associa wih h sima of ρ in 4, an no AO lv, is invarian o h brak magniu. Our concrn now is wih h cas whr h assum brak fracion iffrs from h ru fracion. can hn b shown ha, for fix k in, h limiing isribuion of h Prron s saisic is of h form AO E + E + E lv σ + + / / k [ F + F F ] 6 whr is h inicaor funcion = [ ], so ha in h cas =, 6 rucs o / σ F E, which noaion w us for h funcionals in h limiing isribuion givn by Prron 99 an Prron an Voglsang 99. Mor gnrally

4 in xrssion 6, whn hs funcionals n on h assum brak fracion. Also, in 6 E an F ar ranom variabls wih mans, whil h quaniis E an F ar rminisic. n fac, as w s blow, i is hs rminisic quaniis ha omina h bhaviour of h limiing isribuion for moraly larg k, laing o ricions of surious rjcions. Figur shows E an F whn h assum brak fracion is =. 5, σ =, an k =. Algbraic xrssions for hs quaniis ar givn in h Anix. Also, for comarison w show h mans of E + E an F + F. From Figur, i can b sn ha h imac of E is o inuc a vry larg rucion in h man of h numraor in 6 whn h ru brak fracion is ihr vry small or a vry lil mor han h assum brak fracion of.5. Morovr, as can b sn from h grah of F in Figur, i is rcisly for ru braks in hs lacs ha h osiiv quaniy F conribus virually no aiional comnsaing incrmn o h nominaor of 6. W migh hrfor conjcur ha surious rjcions of h uni roo null ar likly o occur boh for vry arly ru braks an for brak fracions a lil highr han. his conjcur is confirm by Figur, which grahs h rsuls of som simulaion xrimns. Sris of = obsrvaions wr gnra from, wih normally isribu innovaions ε an σ =. Brak magnius b of wr s a 5 an. A squnc of ru brak fracions was analys, bu in all xrimns h assum brak fracion was s a.5 in 5, an h s saisic AO lv was calcula from h rgrssions an 4. Nominal 5% significanc lvls wr foun by sing = =. 5. hn, for ohr valus of, Figur shows h rcnags of rjcions of h uni roo null hyohsis a nominal 5%- lvls. Hr an hroughou h ar, simulaions wr bas on, rlicaions. hs rsuls confirm our ricions. Vry frqun rjcions of h null hyohsis occur for boh arly ru braks an for ru braks a lil afr h assum brak a. h firs of hs hnomna is simly h analogu of ha ror by Lybourn al 998 for Dicky-Fullr ss in h rsnc of a nglc brak, whil h scon is an aiional rgion of surious rjcions gnra by h mislac insrion of ummy variabls ino h sima mols. Noic ha h surious rjcion

5 4 hnomnon is almos qually svr in h wo rgions - abou % rjcions for a brak of 5 sanar viaions, an ovr 7% for a brak of sanar viaions. h innovaional oulirs varian of h s is bas on h rgrssion y = + β + γ + φ + ρy α + ω 7 whr ω is an rror rm an is fin in 5. h s saisic O lv is h -raio associa wih h sima of ρ in 7. Using xacly h sam gnraing rocss as for h aiiv oulirs varian of h s, w again sima rjcion rcnags for h O lv s wih s a.5. hs ar also shown in Figur, an ar virually inical o hos of h rgions, hough hr ar sligh iffrncs ousi hs rgions. AO lv s in h mos inrsing is worh r-mhasising ha, whil our rsuls ar bas on h gnraing rocss an, hy woul coninu o hol in h rsnc of an aiional ru brak a im. hy can hus b inrr as h consquncs of allowing for only on brak whn wo ar rsn as wll as of mislacing a singl ru brak.. Braks in rif W now consir h cas whr h ru gnraing rocss is iffrnc-saionary, bu wih a brak in rif. h simls gnraing rocss of his form is y = + ν, ν = ν + ε, =,..., 8 whr ε ar i.i.. isurbancs wih man an sanar viaion σ, an = σb [ > ]. 9 Svral varians of h Prron s migh b mloy whn such a brak is susc. h mos gnral ossibiliy allows for a brak in boh lvl an slo a im h aiiv oulirs varian of h s is hn bas on h wo rgrssion ss, y = αˆ + βˆ + γˆ ˆ + + δ. follow by h rgrssion 4. W consir h s saisic AO r givn by h - raio associa wih h sima of ρ. n, h ummy variabl is fin in 5, an = [ > ].

6 5 nsigh ino h bhaviour of h s saisic whn h assum brak fracion iffrs from h ru fracion can b obain via h robabiliy limi lim. n fac, h -raio ivrgs, bu, as shown in Voglsang an Prron 998, h lim of / AO r ρˆ = [ Var ˆ ρˆ] / os xis. Our inrs hr is o xn hir rsuls o inify h rcis rgion in, sac whr his lim is ngaiv. hrough an aroach skch in h Anix, w fin for h numraor of ha for any nonzro fix brak amoun b of 9, whn, ρˆ whr = [ < ]. hn, sinc h nominaor of is always osiiv, h lim of has h sam sign as h numraor of. hn follows ha h Prron s saisic AO r ivrgs o in h rgion givn by: R = < [ + > < ] hus, h siz of h Prron s ns o % in his rgion for any fix brak amoun b. n aricular hn noic, as for h braks in lvl cas of h rvious scion, hr ar wo ss of valus of h ru brak fracion for which surious rjcions can b xc whn h assum brak fracion. h xrssion for h lim of h nominaor of is consirably mor involv an os n on b, σ 4 an rahr han rovi h algbra w rsn in Figur a a hr-imnsional lo of h lim of / AO r for σ = an b =. For as of inrraion h funcion is runca abov, so ha only valus for h rgion R of 4 ar shown. his is of mos inrs as i is h rgion whr surious rjcions migh b xc for sufficinly larg samls. Byon rvaling h rgion 4, h sha of Figur a is inrsing, as h h of h lim shoul giv an inicaion of h likly rlaiv svriy of h surious rjcion hnomnon. Firs, noic ha h funcion falls an hn vnually riss boh as incrass from an as incrass from. his suggss ha h mos svr cass of surious rjcions ar likly o occur whn is a lil grar han, an again whn is a lil grar han. his is in conras o h rsuls of h rvious scion, whr w foun, for xaml, a "wors cas" occurs whn h ru brak is immialy afr

7 6 h assum brak. Scon, which of h wo isjoin ss of valus conains h svrs cas of surious rjcions woul sm o n on h locaion of in aricular on whhr h assum brak fracion is mor or lss han on-half., an hs ricions ar closly vrifi by h simulaion rsuls rsn in Figur 4, bas on sris of = obsrvaions gnra from 8, 9 wih ε normally isribu wih σ =, an valus of.5 an. for b. Prcnags of rjcions of h uni roo null hyohsis for nominal 5%-lvl vrsions for h AO s ar shown. n ar a of h figur, h assum brak fracion is =. 5. r wo rgions of srious surious rjcions ar rval - on whr h ru brak occurs qui arly in h sris, an h scon whr h ru brak is soon afr h assum brak. h osiion is virually inical in hs wo rgions, h mos svr cass occurring a roorion. from h bginning of h sris an h sam amoun from h assum brak. n ar b of Figur 4, h assum brak fracion is =.85. n ha cas, h iniial rgion of surious rjcions is broar an mor svr, whil h scon rgion is narrowr an lss svr han in h =. 5 cas. h innovaional oulirs varian of h s is bas on h rgrssion y = + β + γ + φ + δ + ρy α + ω 5 whr ω is an rror rm. Figur 4 also shows simulaion vinc on h s saisic O, h -raio associa wih h sima of ρ from 5. h r icur is broaly similar o ha for h aiiv oulirs varian of h s, hough i aars ha h surious rjcion hnomnon is a lil mor svr an xns o a somwha broar rang of valus of h ru brak fracion for h innovaion oulirs s. h ss iscuss so far in his scion rmi braks in boh lvl an slo. An aiiv oulirs s incororaing jus h lar is bas on h wo rgrssion ss an y = αˆ + βˆ + δˆ + 6 = ρ + ω 7 W hn consir h s saisic AO r, h -raio associa wih h sima of ρ from 7. As no for xaml in foono 4 of Prron 994, h

8 7 corrsoning innovaional oulir varian of his s is usually no rcommn as h asymoic null isribuion of h s saisic is no invarian o h magniu of any corrcly scifi brak unr h null hyohsis. W invsiga h lim of / AO r, again for h gnraing rocss 8, 9. Wriing h quaniy of inrs in h sam form as h righ-han si of, i can b shown ha, for, + [ + + ] ρ ˆ. 8 [ ] h funcion 8, an hrfor h lim of / AO r, is ngaiv in h rgion R = { < [ < + ]} { > [ < ]}. 9 follows ha h s saisic AO r ivrgs o in h rgion 9, so ha for fix brak magnius b, surious rjcions can b xc in his rgion for sufficinly larg saml sizs. h algbraic xrssion for h lim of / AO r is xrmly lnghy. h funcion, runca abov, is grah for σ = an b = in Figur b. h conras wih Figur a is qui sriking, suggsing ha h surious rjcion hnomnon will sm qui iffrn in h wo cass. n aricular, noic ha h rgion associa wih > - ha is, h scon lmn in 9 - is boh qui small an shallow, xning only ovr rlaivly low valus of. ha suggss ha, unlss h ru an mislac braks ar qui arly, a scon rgion of surious rjcions will no b foun. n, vn in hs cass, surious rjcions will occur on ihr si of h assum brak a an b rsric o qui low valus of. Figur b also suggss ha, h grar is, h broar will b h inrval in which surious rjcions ar foun, an h mor svr will b hs rjcions. Noic ha, as aroachs, R of 9 ns o <, whil as aroachs, R ns o <. h corrsoning rgion for surious rjcions by Dicky- Fullr ss in h rsnc of a brak unr h null is shown by Lybourn al, 998, o b <.. hus, whil h insrion an locaion of a mislac brak in h AO r s can b xc o hav som imac on h surious rjcion hnomnon, ha imac shoul b far lss ramaic han for h AO r s.

9 8 Figur 5 shows simulaion rsuls for h alicaion of h h sam gnraing mols as for h AO r s, using AO r s in Figur 4. Again, as in Figur 4, assum brak fracions of =. 5 an.85 wr us. As ric, only on rgion of surious rjcions is foun in hs cass - ha rgion incrasing in wih, an ning in svriy as incrass. Noic also ha h surious rjcion hnomnon is a lil mor svr for h AO r s han for h AO r s of Figur 4, in h sns ha h rjcion frquncy is somwha highr in h wors cass for h formr han for h lar. 4. Summary W hav shown horically an by simulaion ha Prron ss bas on a mislac srucural brak can gnra wo rgions of surious rjcions of h uni roo null hyohsis. h firs of hs is whr h ru brak in h sris is rlaivly arly, mirroring h finings of Lybourn al 998 of surious rjcions from Dicky- Fullr ss in h rsnc of a brak unr h null. Howvr, surious rjcions now also occur if h ru brak in h sris is rlaivly soon afr h assum brak. his hnomnon can b qually svr in h scon rgion as in h firs. f a im sris has a srucural brak, hn i is nirly ossibl, if h assum brak is a lil arlir han h ru brak, for surious rjcions of h uni roo null hyohsis o occur hrough alicaion of h Prron s in cass whr his hnomnon woul no b foun hrough rouin alicaion of h Dicky-Fullr s, ignoring h ossibiliy of a brak. h rsuls can also b inrr as a monsraion of h onial for surious rjcions of h uni roo null whn scific allowanc in h analysis is ma for only h firs of wo braks clos in im. Of cours, hs rsuls shoul no b inrr as suggsing ha uni roo ss, such as h Prron ss, ha allow for h ossibiliy of srucural braks ar of limi valu. On h conrary, our rsuls oin in h ohr ircion, inicaing h snsiiviy of h s oucoms o scificaion of boh h numbr an locaion of any braks. is qui clar ha any rminaion of h issu of uni auorgrssiv roos is inimaly link wih succssful rsoluion of hs issus concrning braks. Finally, w also no ha our finings also xn o ss such as ha of Zivo an Anrws 99 which nognis h brak oin, an ar bas on h mos ngaiv valu of h Prron-y uni roo s across all ossibl valus for. h

10 9 criical valus of hs ss ar calcula assuming no brak unr h null. Howvr, whn a brak xiss, our rsuls suggs ha h acual criical valus will b much mor ngaiv han whn no brak xiss. hus, surious rjcions of h uni roo null will aris if h convnional criical valus for such ss ar mloy. Rfrncs Monans, A., Rys, M., 998, Effc of a shif in h rn funcion on Dicky-Fullr uni roo ss. Economric hory 4, Lybourn, S.J., Mills,.C., Nwbol, P., 998, Surious rjcions by Dicky Fullr ss in h rsnc of a brak unr h null. Journal of Economrics 87, 9-. Prron, P., 989, h Gra Crash, h oil ric shock an h uni roo hyohsis. Economrica 57, 6-4. Prron, P., 99, h Gra Crash, h oil ric shock an h uni roo hyohsis: Erraum. Economrica 6, Prron, P., 994, rn, uni roo an srucural chang in macroconomic im sris. n B.B. Rao,., Coingraion for h Ali Economis, Macmillan, -46. Prron, P. an.j. Voglsang, 99, Nonsaionariy an lvl shifs wih an alicaion o urchasing owr ariy. Journal of Businss an Economic Saisics, -. Prron, P. an Voglsang,.J., 99, A no on h asymoic isribuions of uni roo ss in h aiiv oulir mol wih braks. Rvisa Economria, 8-. Voglsang,.J. an Prron, P., 998, Aiional ss for a uni roo allowing for a brak in h rn funcion a an unknown im. nrnaional Economic Rviw, 9, 7-. Zivo, E. an Anrws, D.W.K., 99, Furhr vinc on h Gra Crash, h oil ric shock an h uni roo hyohsis. Journal of Businss an Economic Saisics, 5-7.

11 Anix Braks in lvl A horical ricion of surious rjcions foun for h aiiv oulirs varian of h Prron s is bas on h following rsuls, which w sa wihou roof: Consir h gnraing rocss, wih k fix. hn h s saisic AO lv, h -raio associa wih h las squars sima of ρ in 4, has h limiing null isribuion givn by 6. n ha xrssion, E an F, hr akn o b funcions of, ar h ranom variabls givn in h corrsoning rsul of Prron 99 an Prron an Voglsang 99 for h cas =. Also, E an F ar ranom variabls funcions of h Winr rocss in C [,] wih mans. h fix numbrs E an F ar givn by an whr E σ k + σ k / H / D + σ k H / D F + D σ k + H / + σ + k H / D σ k H / D σ k + H / D + σ k HH / D D / σ k [ H / σ k ] A. A. H / σ k + [ + h funcions E an F givn by A. an A. ar grah in Figur for k =, σ = an assum brak fracion =.5. is h bhaviour of hs funcions for mora k ha suggss h ossibiliy of surious rjcions whn h Prron s is ali wih a mislac brak. ]. Braks in rif W skch how o obain h lim givn by. Consir h firs rgrssion s. h Frisch-Waugh-Lovll horm givs us θˆ = A b + b

12 whr δ γ β θ ˆ ˆ ˆ ˆ, A, b, v v v b, i i, i i, an all variabls ar xrss as viaions from hir saml mans. can b shown ha A AD D, b b D, b D whr D, D, A is a non-singular marix, b is a vcor. hrfor, ˆ b A D θ. n orr o obain h lim for ach comonn in θˆ, firs riv all lims in A an b, an hn us h Symbolic Mah oolbox in MALAB o comu b A symbolically. hn w hav h following rsuls: / ˆ + β A. / ˆ γ / ˆ δ. From h scon s, on can comu ha F E / ˆ = ρ whr + ] [ ] [ E an ] [ F. Wih sraighforwar bu ious algbra, on can show using h rsuls in A. ha / E + / F + which livrs h sir rsul in.