SMALL SAMPLE POWER OF BARTLETT CORRECTED LIKELIHOOD RATIO TEST OF COINTEGRATION RANK

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1 SALL SAPLE POWER OF BARTLETT CORRECTED LIKELIHOOD RATIO TEST OF COINTEGRATION RANK PIOTR KĘBŁOWSKI 3 ay 5 Asrac I s well-ocumee pheomeo ha he asympoc sruo of he lelhoo rao es of coegrao ra s ffere from he small sample sruo. I he afermah sze a power of he es are susaally sore a he ull hypohess s rejece oo ofe whe s rue. Oe of he saar soluos of such prolem s a correco of es sasc so ha he small sample sruos a s cumulas were closer o he asympoc couerpars. The Barle correco for he race es erve y Johase for he mulvarae case cosues a covee ool for plausle ferece o coegrao ra. However he ecle of emprcal sze of he es eals he ovous ecrease of power. The paper vesgaes he power of he Barle correce race es wh referece o he power of he usually apple ucorrece oe a res o aswer for whch hypoheses sample szes a parameer s spaces s sasfacory.. INTRODUCTION The vecor auoregressve framewor a he supercosse maxmum lelhoo esmaor has ecame a popular meho of moellg of vecor sochasc processes egrae of orer oe (see Egle a Grager (987) Johase (988)). However a sore po of he coegrao aalyss s he ferece o he coegrao ra especally f he samples are shor a he aa o very formave. Am all ess use o scrmae ewee ffere hypoheses o he coegrao ra he race es s mosly use a some cases exame y Lüepohl () has eer small sample properes whe a umer of coegrag relaos exs as compare wh compeve maxmum lelhoo es.

2 The lm sruo of he race es sasc s gve as he race of sochasc marx whch epes o mulmesoal Browa moo a eermsc compoes prese he coegrao space. I s approxmae y meas of smulaos for log sample szes (see Johase (988 99) Ah a Resel (99)). I shoul e oe ha he race es s cosse a follows ha asympocally he sze of he es for he rue ull hypohess s maae f he op ow sraegy of ferece s apple (see Juselus (3)). However ure ou ha he sze a he power of he es are severely sore small samples whe he sasc s juxapose wh he asympoc crcal values (see Toa (995) cf. Haug (996) for maxmum egevalue es a ohers). The soros resul from he fac ha small sample sruo of sasc a s expece value epes o oly o he umer of commo sochasc res a he eermsc compoe u also o he umer of oservaos a he rue parameers of he vecor auoregressve moel. Therefore a umer of soluos were propose a employe o reuce he evao of he expece value of he race es sascs small sample from he expece value he lm. They ca e geeral ve o wo groups oe ryg o esmae he acual crcal value for gve sample sze a aoher oe ryg o rg closer he expece value of he sasc o he asympoc crcal value y some correco of he sasc. To he former group fall he respose surfaces (see Cheug a La (993) acko e. al. (998)) as well as he oosrap a he Egeworh expasos (see Roheerg (984) Gerserge (996)) whereas o he laer oe fall Barle a Barle-Type correcos (see Barle (937) Crar-Neo a Corero (996)) a oher correco facors (see Remers (99)). I he paper we shall focus o he Barle correco for he race es a he emphass wll e la o small sample properes of correce race es whe he alerave hypohess s rue. The movao resuls from he fac ha he ucorrece race es suffers from he lac of power shor samples (see Toa (995)) whereas he Barle correco for he race es has o resul a ecrease of power as he emprcal sze of he es s reuce mos cases (alhough he correco facor ca ae also fracoal value as well as usasfacorly he egave oe). Thus for plausle ferece seems o e crucal o ow for whch sample szes whch hypoheses a whch values of parameers he Barle correce race es s a powerful ool. The ea of he Barle correco s que uve a amous o rgg closer he expece value of sasc fe sample o expece value of sasc he lm wh hope ha he oher momes of he sruo are also eer approxmae. Therefore suppose ha he lelhoo rao sasc coverges o s asympoc crcal value wh a error of orer a mos T - (a s fulflle ee).e.: E [ ] O( T ) [ l( LR) ] = E lm( l( LR) ) ()

3 he he Barle correco reles o he exsece of such expaso ha E [ ] O( ) [ l( LR) ] = E lm( l( LR) ) T T () where cosa ca e cossely esmae uer he ull hypohess a o he [ ] covergece of l( LR) E[ l( LR) ] o lm( l( LR) ) E lm( l( LR) ) as T (see Barle (937)). Hece he Barle correco ca e erve from he approxmae equaly: lm ( l( LR) ) l( LR) E [ lm( l( LR) )] E[ l( LR) ]. (3) Neverheless he assumpo ha whole shape of sruo s eer approxmae y Barle correco has o e verfe. I fac ule he classcal ferece whe regulary coos hols he LR sascs s χ srue a all quales are approxmae wh a error of orer T -3/ (see Lawley (956)) vecor auoregressve framewor wh u roos he mproveme s o so specacular however for some cases a eer f exs (see Bravo (999)). I he ex seco we wll recall he Barle correco for he race es a o some remars o s small sample properes. Seco 3 clue he esg of several oe Carlo expermes he resuls cocerg he power of he Barle correce race es a some hs for ecoomercas wllg o apply he correce es. Seco 4 gves he summary.. THE BARTLETT CORRECTION FOR THE TRACE TEST The Barle correco for he race es sasc varae auoregressve framewor has ee wore ou y Jacoso a Larsso (999) whle he formula for he geeral mulvarae auoregressve moel wh u roos was erve y Johase () who suggese he approxmao of he expece value of he sasc uer he ull hypohess whch aes he form: ( θˆ ) E[ l( LR) ] = f ( T ) (4) T where eoes umer of commo sochasc res efes he maxmum power of lear re resrce o he coegrao space a ( θˆ ) expresses he aforemeoe cosa epeg o esmae parameers of he moel uer he ull. Sce [ lm ( l( LR) )] = f ( ) E a ( T ) ( θˆ ) he correco facor s gve as: (5) T 3

4 4 where ( ) ( ) ( ) f T f T a =. Johase proposes o calculae (5) y meas of smulao a preses he esmae coeffces of he approxmao whch are closely relae o respose surfaces regresso. The erm ( ) θˆ s erve aalycally a afer eglecg oe sgfca compoe s gve as: ( ) ( ) ( ) ( ) ( ) ( ) 3 ˆ g c c c h c = θ (6) where ( ) h a ( ) g are aga calculae y meas of smulaos whle c c 3 c are he coeffces whch are calculae o he ass of he compao form parameers of he vecor auoregressve moel uer he ull whch ur s as follows: ( ) ε Φ Γ ρ β α = = = (7) a he compao form of (7) s: Qε PY Y =. (8) Afer omg he eermsc erms as sgfca for he formula gve y equao (6) (see Johase ()) he compao form ca e wre as: r ε I β β I I Γ Γ Γ α β Γ β Γ β Γ β α I β = L L L L where eoes umer of sochasc varales a efes umer of lags vecor auoregressve moel. I shoul e hghlghe ha he Barle correco chages he small sample properes of he race es oher maer ha he correcos erve y Ah a Resel as well as y Remers o. As presee y Johase sze of he ucorrece race es grows owars oe as he sochasc process ecomes almos egrae seco orer whle sze of he Barle correce race es es o zero such crcumsaces. Ths s ue o he fac ha he Barle correco hales he soro whch resul from oh small sample as a epeece o he parameers uer he ull whereas he aove-meoe correcos hale oly he former cause. Such propery of he Barle correce race es has o pose he queso wha s he power of he es small samples a how he shor erm yamcs as well as oher parameers of he moel uer he ull hypohess fluece he power.

5 To he es owlege of he auhor o evece o power of he Barle correce race es s avalale apar from wo fgures calculae y Johase () whch relae o a emprcal moel. 3. POWER OF THE BARTLETT CORRECTED TRACE TEST The aalyss of power of he Barle correce race es was couce hree sc secos. Frs of all he power was calculae for wo ffere sample szes: T = 5 a T = ue o he epeece of he sruo of he sascs o he sample sze (see Johase (996) a ()) a accorace wh he assume sample szes he laer of he aforemeoe papers whch small sample properes of he Barle correce race es are cosere wh referece o he proaly of ype I error. Secoly he power was aalyse for ffere saces ewee he assume ull hypohess a he rue alerave ecause of he fac ha he more o-zero egevalues s prese he sasc he easer he ull shoul e rejece. Fally he power for ffere umer of coegrao relaos uer he ull hypohess was cosere orer o vesgae wheher he es has he same power he small sample whe he alerave ha oe more coegrao relao exss s rue rrespecve of umer of coegrao relaos he ull hypohess assumg ha he umer of commo res s cosa. The oe Carlo echque was employe for he calculaos. For each experme housa replcaos were compue. The vecor auoregressve moel uer he alerave a he assume aa geerag process s as follows (cf. Johase ()): ( β ρ ) α ( β ρ ) Γ Φ ε = α. (9) I all smulaos he oly oe eermsc compoe was cosa resrce o he coegrao space.e. = umer of lags for shor erm srucure was assume as = a varace-covarace marx was Ω = I. = = 3.. The power for sc sample szes As meoe aove he sruo of he race es sascs epes o he sample sze a he Barle correco reuces he sze of race es geeral. Therefore s of crucal mporace o ow for whch sample szes he Barle correce race es s powerful ool comparso o he ucorrece es. However shoul e sresse ha he laer ofe leas o false coclusos as he sze of he es s serously sore. 5

6 The power was calculae for he se of = 5 sochasc varales he DGP was gve y DGP he equao (9) rue coegrao ra was r = whereas ese ull hypohess was r = versus a he mos r = 5 uer he alerave (oce ha for he sae of he way he asympoc crcal values are calculae he hypoheses are revere comparso wh he lelhoo rao sasc for whch he ull hypohess have o esure he maxmum value of he sasc). The parameers of DGP were gve as follows: α = [ α ] β = [ ] = [ ] ρ α = [ α ] β = [ ] [ ] ρ = Γ = ξ I5. A he ouse sample sze T = 5 was assume ex T = for ffere values of loags a auoregressve coeffces.. Tales a prese he resuls Tale. α \ ξ (.4) (.) (.8) (.5) (.4) (.4). 8 (.3) 7. 9 (.) (.9) (.7) (.5) (.5) (.3) (.6) 48.. (.) (.) (.9) (.9) (.38) (.33) 6 7. (.9) (.7) (.6) (.6) 7. 3 (.54) (.48) (.44) (.43) (.63) (.4) (.) (.34) (.) (.4) (.) (.9) The fgures fraco prese he smulae power of he ucorrece (aove) a Barle correce (elow) race es respecvely for omal ype I error equal.5. The Barle correco facor s parehess. Tale. α \ ξ (.3) (.) (.8) (.7) (.7) (.6). 8 (.4) 3. 5 (.) (.9) (.8) (.7) (.7) (.6) (.3) (.) (.) (.9) (.9) (.9) (.6) (.4) (.3) 9 (.3) 9 (.3) (.6) (.3) (.) 9 (.) 9 (.) 9 (.) (.64) (.6) 9 (.6) 9 (.59) 9 (.58) 9 (.58) The fgures fraco prese he smulae power of he ucorrece (aove) a Barle correce (elow) race es respecvely for omal ype I error equal.5. The Barle correco facor s parehess. 6

7 As ure ou whe esg he coegrao ra for T = 5 (see Tale ) he power of he Barle correce race es s raher usasfacory for such shor sample a f here ca e cae such a parameer space ha he ucorrece race es has suffce low proaly of ype II error wh possly large proaly of ype I error (see Johase ()) here s almos o space for whch he resuls for he correce es coul e cosere as sasfacory. The creme of sample sze o T = esseally chages he coclusos (see Tale ). The Barle correce race es s farly powerful for moerae loags of aou. 5 a he larger he auoregressve coeffces are he hgher he power of he es s. For α = a ξ = we are ale o cae a rue alerave hypohess 99% whereas he proaly of ype I error s uer corol as he sasc was Barle correce. The usasfacory resuls cocerg he power of he Barle correce race es for T = 5 pu he queso wheher he ferece s more relale whe a sace ewee verfe ull hypohess a rue alerave s larger ha oe o-zero egevalue.e. wo or more coegrao relaos shoul e sll eece. 3.. The power for ffere saces ewee ull a alerave hypoheses The DGP s efe as equao (9) he umer of sochasc varales = 5 a he rue DGP coegrao ra r = rema uchage. The ull hypohess s verfe ha r = versus a he mos r = 5 uer he alerave hypohess. The parameers of DGP are as follows: α = α α = α β = a he resuls for T = 5 are repore Tale 3. ρ = Γ = ξ I5 Tale 3 α \ ξ (.6) -. 8 (.6) (.6) (.6) (.6) (.6) (.8) (.8) 4. 8 (.8) (.8) (.8) (.8) (.5) 9. 7 (.5) (.5) (.5) (.5) (.5) 78 (.37) (.37) (.37) (.37) (.37) (.37) 96 (.66) (.66) (.66) (.66) (.66) (.66) 99 (3.7). 99 (3.7) (3.7) (3.7) (3.7) (3.7) The fgures fraco prese he smulae power of he ucorrece (aove) a Barle correce (elow) race es respecvely for omal ype I error equal.5. The Barle correco facor s parehess. 7

8 As expece a aoal o-zero egevalue has cause a crease power of he Barle correce race es as well as power of he ucorrece oe. Therefore for hgh values of loags.e. a leas. 6 or eve. 8 accompae y o-zero auoregressve coeffces he power coul e cosere as suffce. Therefore ag o coserao he resuls presee Tales a 3 s almos mpossle o couc plausle a accurae ferece o he rue coegrao ra y meas of he Barle correce race es for shor samples of aou ffy oservaos ha s for such case ha he proaly of ype I error s corolle y he Barle correco he proaly of ype II error s suffcely low a o formao o he meso of coegrao space s gve a pror. However whe he ajusmes o he log-ru aracors are fas a he sochasc varales are auocorrelae he correce es s capale o gve some approxmao. I shoul e oe ha whe he auoregressve coeffces are close o a he process ecomes almos egrae of orer seco he he power of he correce es ecle ecause he sze of he correce es ecrease owars (see Johase ()) as s reveale y he las colum of Tale 3 a a par of he las colum of Tale. The ex queso whch shoul e sae s wheher for gve umer of commo res a cosa sace ewee he ull a alerave hypohess he proaly of ype II error rse f he coegrao ra uer he ull oes he same. 3.3 The power for ffere mesos of vecor sochasc processes I s well ow ha he race es s cosse as well as he Barle correce oe as he correco facor s oue. Thus he lm here s o fferece erms of power (a sze oo) whe we es ffere ull hypoheses f he umer of commo res s cosa a he alerave hypohess s rue (he ull for sze respecvely). oreover f he alerave hypohess s rue he regarless of he umer of commo sochasc res he power s he same a equal o. Neverheless he small sample he expecao o power of he es cao e he same. Therefore wo aoal marces of resuls of oe Carlo expermes were calculae o e compare wh he resuls presee Tale. Frsly he meso of process was se o = 4 he DGP was efe as DGP equao (9) he rue coegrao ra was r = he ull hypohess was r = a sample sze T = 5 so ha he umer of commo sochasc res was he same as he experme repore Tale. The parameers of DGP were as follows: α = = [ α ] α β = [ ] [ ] The resuls are presee Tale 4. 8 ρ = Γ = ξ I 4.

9 Tale 4 α \ ξ (.3) (.3) (.3) (.3) (.3) (.3) 5 (.4) (.4) (.4) (.4) 78 (.4) (.4) 34 (.) (.) (.) (.) 6. 9 (.) (.) 49 (.9) (.9) (.9) (.9) (.9) (.9) 75 (.5) (.5) (.5) (.5) (.5) (.5) 99 (.64).. 99 (.64) (.64) (.64) (.64) (.64) The fgures fraco prese he smulae power of he ucorrece (aove) a Barle correce (elow) race es respecvely for omal ype I error equal.5. The Barle correco facor s parehess. Tale 5 α \ ξ (.8) (.8) (.) (.9) -.8. (.6) (.6).. 8 (.3) (.9).4 3 (.3) (.9) (.8) (.7) 5 (.4).3 (.9) (.6) (.) (.) (.) 36 (.5) (.39).3 6 (.3) (.8) (.7) (.7) 79 (.65).. 8 (.53) (.47) (.44) (.44) (.44) (.53) (.3) (.9) (.9) (.6) (3.) The fgures fraco prese he smulae power of he ucorrece (aove) a Barle correce (elow) race es respecvely for omal ype I error equal.5. The Barle correco facor s parehess. Secoly he umer of sochasc varales was se as = 6 he DGP was efe as DGP equao (9) he rue coegrao ra was r = 3 he ull hypohess was r = a sample sze T = 5 so aga he umer of commo sochasc res was he same. The parameers of DGP were gve as: α = α α = α α β = ρ = Γ = ξ I 6 he resuls are Tale 5. Comparso of Tales: 4 a 5 respecvely reveals ha for every values of parameers α a ξ he power of he Barle correce race es mshes as he umer of he coegrao relaos uer he ull hypohess creases for researche hypoheses. The resul 9

10 oes o allow o coclue ha he power of he Barle correce race es ecles uformly a mooocally as he umer of coegrao relaos uer he ull hypohess rses however eales o sae ha shor samples s eer o ul coegrae sysems wh a lme umer of sochasc varales a coegrao relaos. The smlar remars ca e also mae for o repore here sample sze T =. 4. CONCLUSIONS The aalyss of he power of he Barle correce race es was couce hree recos. Frs he epeece of he proaly of ype II error o he sample sze was vesgae a was fou ha properes of he correce es erms of power are clearly eer a sasfacory for he moerae sample szes of aou oservaos coras o he case of he shor sample of aou 5 oservaos for whch he ferece s o plausle. Nex power of he correce es was exame for he case whe here s more ha oe o-zero egevalue ha shoul e eece as he alerave hypohess s rue comparso o verfe ull hypohess. The resul s ha here s some parameer space for whch he proaly of ype II error s suffcely small eve f here s oly 5 oservaos avalale. Fally power of he correce es s aalyse for ffere hypoheses o he coegrao ra uer he ull whereas he umer of commo sochasc res s cosa a he rue coegrao ra s hgher y oe ha he ull assumes. As ure ou s more lely o rejec he false ull hypohess f he meso of vecor sochasc process s small. A overall cocluso s ha for shor samples of aou 5 oservaos he resul of ferece wh he Barle correce race es shoul e cosere as a proxy for he rue coegrao ra a possly aoal a pror formao s eee. For moerae samples of aou oservaos f resuals are mulvarae whe ose process he ferece s val.

11 REFERENCES Ah S. K. G. C. Resel (99) Esmao for Parally Nosaoary ulvarae Auoregressve oels Joural of he Amerca Sascal Assocao Bravo F. (999) A Correco Facor for U Roo Tes Sascs Ecoomerc Theory Cheug Y.-W. K. S. La (993) Fe-Sample Szes of Johase s Lelhoo Rao Tess for Coegrao Oxfor Bulle of Ecoomcs a Sascs Crar-Neo F. G.. Corero (996) O Barle a Barle-Type Correcos Ecoomerc Revews Egle R. F. C. W. J. Grager (987) Coegrao a Error Correco: Represeao Esmao a Tesg Ecoomerca Gerserge N. P. A. va (996) Boosrapg he race sascs VAR moels: oe Carlo resuls a applcaos Oxfor Bulle of Ecoomcs a Sascs Haug A. A. (996) Tess for Coegrao: A oe Carlo Comparso Joural of Ecoomercs Jacoso T. R. Larsso (999) Barle correcos coegrao esg Compuaoal Sascs & Daa Aalyss Johase S. (988) Sascal Aalyss of Coegrao Vecors Joural of Ecoomc Dyamcs a Corol 3 54 Johase S. (99) Esmao a Hypohess Tesg of Coegrao Vecors Gaussa Vecor Auoregressve oels Ecoomerca Johase S. (996) Lelhoo-Base Iferece Coegrae Vecor Auoregressve oels Oxfor Uversy Press Oxfor Johase S. () A Small Sample Correco for he Tes of Coegrag Ra he Vecor Auoregressve oel Ecoomerca Juselus K. (3) The Coegrae VAR oel: Ecoomerc ehoology a acroecoomc Applcaos upulshe mauscrp Lawley D. N. (956) A Geeral eho for Approxmag he Dsruo of Lelhoo Rao Crera Bomera Lüepohl H. P. Saoe C. Treler () axmum Egevalue versus Trace Tess for he Coegrag Ra of a VAR Process Ecoomercs Joural acko J. G. A. A. Haug L. chels (998) Numercal Dsruo Fucos of Lelhoo Rao Tess for Coegrao upulshe mauscrp Remers H.-E. (99) Comparso of Tess for ulvarae Coegrao Sascal Papers

12 Roheerg T. J. (984) Approxmag he Dsruos of Ecoomerc Esmaors a Tes Sascs Haoo of Ecoomercs vol. Z. Grlches a. D. Irlgaor (Es.) Elsever Toa H. Y. (995) Fe Sample Performace of Lelhoo Rao Tess for Coegrag Ra Vecor Auoregresso Ecoomerc Theory 5 3