Irrelevant but highly persistent instruments in stationary regressions with endogenous varables containing near-to-unit roots.

I ele an bu highly pe sis en ins umen s in s a iona y eg essions wi h endogenous a iables con aining nea - o-uni oo s. Nicolaos Kou ogenis∗Eka e ini Panopoulou†Niki as Pi is‡ No embe 2005 Abs ac This pape sugges s ha IV es ima o s, u ilizing i ele an bu pe sis en ins umen s may p oduce eliable in e ences, in small samples, in cases whe e he endogenous a iables con ain au o eg essi e oo s nea uni y. In such cases, hese es ima o s appea o ou pe o m IV es ima o s wi h s ong ins umen s as well as some asymp o ically efficien coin eg a ion es ima o s. JEL classi ica ion: C12, C13, C22 Keywo ds: Ins umen al a iables es ima o , pe sis en ins umen s, nea - o-uni oo s. Acknowledgemen s:Weacknowledge inancial suppo om he G eek Minis y o Educa ion and he Eu opean Union unde “H aklei os” g an . We a e g a e ul o Guido Ku es eine and A is Spanos o help ul commen s. The usual disclaime applies. ∗Depa men o Banking and Financial Managemen , Uni e si y o Pi aeus. †Depa men o Banking and Financial Managemen , Uni e si y o Pi aeus, G eece and Depa men o Economics, Na ional Uni e si y o I eland Maynoo h. ‡Depa men o Banking and Financial Managemen , Uni e si y o Pi aeus. Co espondence o: Niki as Pi is, Depa men o Banking and Financial Managemen , Uni e si y o Pi aeus, 80 M.Ka aoli and A. Dimi iou s . 18534 Pi aeus, G eece. E-mail: [email p o ec ed] 1 In oduc ion Selec ing app op ia e ins umen s in he con ex o an Ins umen al Va iables (IV) p o- cedu e is o pa amoun impo ance o p oducing eliable in e ences on he s uc u al pa ame e s o in e es . I is now well unde s ood ha i he ins umen s a e only weakly co ela ed o he endogeneous a iables, hen IV es ima o s a e likely o a e no be e han he O dina y Leas Squa es (OLS) es ima o (see Nelson and S a z 1990a, 1990b, Buse 1992, Bekke 1994, Bound, e . al. 1995, Du ou 1997, S aige and S ock 1997 and Wang and Zi o 1998). The li e a u e on he ‘weak ins umen s’ issue implici ly e e s o cases whe e he eg esso is ei he se ially unco ela ed o exhibi s a e y low deg ee o pe sis ence. This is due o he ac ha a pe sis en eg esso is always accompanied by s ong ins umen s, namely i s own lagged alues. I , o example, he eg esso , x , ollows an AR(1) p ocess, wi h coefficien ρx, hen he lagged alue o he eg esso , x −1,is eadily a ailable as an ins umen o x .In such a case, he ‘weak ins umen ’ p oblem is no an issue, unless ρxis close o ze o. Mo eo e , he highe is he alue o ρx, he s onge is x −1as an ins umen o x .Howe e , his is ue only o alues o ρxless han one. I ρx=1, he eg esso is an I(1) p ocess, pa icipa ing in a coin eg a ing eg ession. In such a case, he OLS es ima o is supe -consis en , which in u n implies ha ‘ i s -o de ’ asymp o ic bias effec s dissapea . In such a case, an IV p ocedu e, such as he wo-s ages leas squa es (TSLS) es ima o , is inapp op ia e since i is designed o deal wi h a p oblem ha no longe exis s. The asymp o ic p oblems in he coin eg a ion case a e o diffe en na u e, usually e e ed o as ‘second-o de ’ effec s (see, o example, Phillips 1988, Pa k and Phillips 1988, Phillips and Lo e an 1991). To deal wi h hese p oblems, one has o employ an asymp o ically efficien coin eg a ion es ima o , a he han a s anda d IV one. I one insis s on using IV p ocedu es in he case o coin eg a ion, hen she ends up wi h an es ima o whose asymp o ic dis ibu ion suffe s om nuisance pa ame e dependencies (second-o de effec s) a ising no only om he co ela ion be ween he eg ession e o and he eg esso , bu also om he co ela ion be ween he ins umen and he eg esso ! In o he wo ds, he p oblem o ‘weak ins umen s’ is e e sed. In he 1 case o coin eg a ion, a weak, o e en mo e so, an i ele an ins umen is bene icial, since i simpli ies he nuisance pa ame e dependencies in he asymp o ic dis ibu ion o he IV es ima o wi hou affec ing he consis ency o his es ima o . This may be hough o as a bene icial a i ac o he spu ious eg ession heo y (see Phillips and Hansen 1990). The p eceding discussion implies ha he issue o ‘weak ins umen s’ should be ex- amined in conjuc ion wi h he ime se ies p ope ies o he da a in hand. I is ue ha a weak ins umen is likely o be a p oblem in a low-pe sis ence en i onmen , bu i is also ue ha a s ong ins umen may c ea e mo e p oblems han i sol es in a ‘high- pe sis ence’ o ‘nea - o-uni - oo ’ amewo k. As ρxmo es om he s a iona y o he uni - oo egion, i s -o de effec s a e declining bu second-o de effec s a e eme ging. Al hough he asymp o ic heo y has p o ided clea answe s on he p ope ies o IV es- ima o s o he wo pola cases |ρx|<1and ρx=1,i is o li le help o sugges he op imal es ima ion p ocedu e, in ini e samples, o he cases ha ρxis less han bu close o uni y. To pu i diffe en ly, i is no clea whe he i s o second o de effec s a e p edominan in he case ha ρxis in he iscini y o uni y. This pape examines hese issues in some de ail. Speci ically, we add ess he ollowing ques ions: Wha is he op imal way o es ima e he s uc u al pa ame e o in e es , o samples o ypical sizes, when he eg esso is a s a iona y bu highly pe sis en p ocess, co ela ed wi h he e- g ession e o ? Is i s ill op imal o employ an IV p ocedu e ha u ilizes he s onges a ailable ins umen (s), as he ele an asymp o ic heo y sugges s? O is i be e o ea he eg ession as a nea ly-coin eg a ed one and employ an asymp o ically efficien coin eg a ion es ima o ?1This pape offe s simula ion e idence agains hese op ions. Bo h me hods a e ou pe o med by a TSLS es ima o ha u ilizes i ele an bu highly pe sis en ins umen s. The pape is o ganized as ollows. Sec ion 2 in oduces he DGP and b ie ly e iews he ele an heo y. Sec ion 3 epo s he simula ion indings and Sec ion 4 concludes he 1Ellio (1998) examines he p oblems wi h employing s anda d coin eg a ion es ima o s in cases whe e he se ies in ol ed in he eg ession con ain nea - o-uni oo s. He demons a es ha commonly applied hypo hesis es s on he pa ame e s o in e es suffe om se e e size dis o ions, when slowly mean e e - ing p ocesses a e app oxima ed by ones wi h uni oo s. He also shows ha using lags o he eg esso s as ins umen s is inapp op ia e in his case. 2 pape . 2 The Model, and Some Backg ound Theo y Conside he eg ession equa ion: y =θx +u1 (1) whe e he eg esso is gene a ed ia an AR(1) p ocess: x =ρxx −1+u2 (2) We also assume he p esence o a hi d a iable, z , ha migh se e as an ins umen o iden i ying θ, which also ollows an AR(1) p ocess, z =ρzz −1+u3 (3) The e o ec o u =[u1 ,u 2 ,u 3 ]|is assumed o be no mal, independen and iden ically dis ibu ed wi h ze o mean and co a iance ma ix Σ.Speci ically,       u1 u2 u3       ˜NIID       0 0 0             σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33       (4) Le us i s e iew some use ul esul s om he exis ing li e a u e o he s a iona y and coin eg a ing eg ession cases, de ined by |ρx|<1and ρx=1, espec i ely, s a ing om he o me . S a iona y Reg ession We i s assume ha |ρx|and |ρz|a e less han one, which means ha he ins umen and he eg esso a e I(0) p ocesses. I σ12 6=0, he OLS es ima o , b θLS , esul s in asymp o ic bias gi en by he ollowing exp ession: 3 plim ³b θLS −θ´=σ12 σ22 (1 −ρ2 x)(5) I can be seen ha he asymp o ic bias o b θLS is p opo ional o he deg ee o co ela- ion be ween he eg ession e o and he e o ha d i es he eg esso , and in e sely p opo ional o he deg ee o pe sis ence o he eg esso . Nex , assume ha σ12 6=0and σ13 =0.In such a case, θcan be consis en ly es ima ed by TSLS.The se o a ailable ins umen s can be iden i ied by conside ing he i s -s age eg ession, implied by he DGP unde s udy. This can be ob ained by i s no ing ha u2 =σ23 σ33 u3 +ν and hen subs i u ing his exp ession back in o equa ion (2), o ob ain, x =ρxx −1+σ23 σ33 z −ρz σ23 σ33 z −1+ν (6) The i s -s age eg ession implies ha he e a e h ee a ailable ins umen s, namely x −1, z and z −1.In hecase ha ρx=σ23 =0, he mean o he TSLS es ima o employing all he h ee a ailable, bu i ele an , ins umen s is he p obabili y limi o he OLS es ima o . Coin eg a ing Reg ession Le us now ocus a en ion on he case ρx=ρz=1.Equa ions (1) - (2) o m a iangula coin eg a ion sys em, pu o wa d by Phillips (1988). In such a case, b θLS is T-consis en , e en i σ12 6=0. Howe e , i σ12 6=0,‘long- un endogenei y’ p oblems (second-o de effec s) a e s ill encoun e ed wi hin he OLS es ima ion me hod. S anda d IV p ocedu es a e no designed o deal wi h such effec s. Ins ead, an asymp o ically efficien coin eg a ion es ima o , such as he Fully Modi ied Leas Squa es (FMLS), o he Dynamic OLS (DOLS) es ima o should be used. (see Phillips and Hansen 1990, S ock and Wa son 1993). Phillips and Hansen (1990) examine he beha iou o IV es ima o s in a coin eg a ion amewo k, and show ha , due o he non-diagonali y o Σ, he p esence o ele an ins umen s makes he asymp o ic dependence o he IV es ima o on nuisance pa ame e s mo e complica ed han ha o he OLS es ima o . I , howe e , he ins umen and he eg esso e o a e s ochas ically independen , ha is when σ23 =0, he nuisance 4 pa ame e dependencies a e educed. In o he wo ds, asymp o ic heo y sugges s ha i ele an ins umen s a e p e e able o s ong ones, in he case ha IV p ocedu es a e applied on a coin eg a ing eg ession. S a iona y Reg ession wi h nea - o-uni Roo s Finally, le us assume ha ρxis close o bu less han uni y, o example ρx=0.95. Wha is he op imal p ocedu e o es ima ing θin his case? Asymp o ically, he p oblem alls in o he ca ego y o eg essions wi h s a iona y a iables, whe e only i s -o de effec s, a ising om σ12 6=0,a e p esen . In ini e samples, howe e , second-o de effec s, a ising om he ac ha he eg esso esembles a uni - oo p ocess a e also likely o appea . The p esence o bo h i s and second o de effec s sugges s he adop ion o an IV es ima o wi h i ele an bu e y pe sis en ins umen s. Such ins umen s may be spu iously co ela ed wi h he eg esso , hus (pseudo) dealing wi h he i s -o de effec s and, a he same ime, minimizing he second-o de effec s. 3 Mon e Ca lo Resul s The se s o ins umen s, used in he i s -s age eg ession, a e {z },{x −1},{z ,z −1,z −2} and {z ,x −1,z −1}, esul ing in he IVZ, IVX, IVZZ and IVZX es ima o s, espec i ely. We also include he OLS es ima o o compa ison pu poses, and wo asymp o ically efficien coin eg a ion es ima o s, namely FMLS and DOLS ha a e expec ed o pe o m bes in he exac coin eg a ion case (ρx=1).The au o eg essi e pa ame e s, ρxand ρz, ake alues in he in e als [0, 0.8] and (0.8, 1], by s eps o 0.1 and 0.02, espec i ely. In he i s se o expe imen s we assume ha ρx=ρz.Fo each alue o ρx(= ρz),we gene a e 2000 se ies o leng h 150 (350) s a ing wi h u10 =u20 =0, and hen disca d he ini ial 50 obse a ions, hus gene a ing a sample size o 100 (300). The accu acy o he se en es ima o s, in oduced abo e, is assessed by means o he median bias, since o IVZ and IVX he uncondi ional mean does no exis . To examine he effec s o pe sis en ins umen s on hypo hesis es ing on θ, we also epo he mean, s anda d de ia ion, skewness and ku osis coefficien s o he es ima o s’ -s a is ics. The pe o mance o hese es s is assessed by compa ing he 2.5% ( 0.025)and he 97.5% ( 0.975)poin s in he 5 empi ical dis ibu ions o he ele an -s a is ics wi h hose om he s anda d N(0,1). Finally, we epo he (a e age) F-s a is ics om he i s -s age eg essions. As o he es o he pa ame e s, we se θ=1,σ 11 =σ22 =1,σ 12 =0.7and σ13 =0, ha is, we in oduce a a he s ong ‘endogenei y’ effec and main ain he o hogonali y condi ion o z .Finally, he key pa ame e , σ23,is se , h oughou , equal o ze o. This means ha IVZ and IVZZ u ilize solely i ele an ins umen s o all he alues o ρxand ρz. Fo b e i y, we do no epo he ull se o esul s. Ins ead, we p esen he esul s o he cases ρx=ρz=0,ρ x=ρz=0.5,ρ x=ρz=0.96 and ρx=ρz=1 o a sample size equal o 100, in Tables 1A o 1D, espec i ely. The esul s may be summa ized as ollows: (i) When he eg esso and he ins umen exhibi ze o deg ee o pe sis ence, ha is when ρx=ρz=0,all he IV es ima o s employ i ele an (and se ially unco ela ed) ins umen s and he esul s a e simila o hose ob ained in he s anda d ‘weak ins u- men s’ li e a u e: The F-s a is ics om he i s -s age eg essions a e e y close o uni y, and he median bias o each o hese es ima o s is almos iden ical o he OLS one. The empi ical dis ibu ions o he associa ed -s a is ics a e skewed and shi ed o he igh , meaning ha he - a io is expec ed o be la ge e en i he null hypo hesis is ue. Fo example, he 5% empi ical sizes o IVZZ and IVZX a e 27.4% and 28.1%, espec i ely. (ii) When he eg esso and he ins umen exhibi a mode a e deg ee o pe sis ence, ha is when ρx=ρz=0.5, he esul s a e, o a la ge ex en , consis en wi h he ele an heo y. The bes pe o ming es ima o is IVZX, whose median bias is smalle han ha o OLS by a ac o o wen y, ollowed by IVX. Fo his le el o pe sis ence, IVZ and IVZZ s ill ollow, o a la ge ex en , he beha iou o OLS. Howe e , some small bu impo an diffe ences be ween his and he p e ious case a e isible: Fi s , he F-s a is ics o IVZ and IVZZ ha e inc eased om 0.98 o 1.70 and om 0.99 o 1.28, espec i ely, despi e he ac ha hei popula ion analogues, emain ixed o ze o. Second, he median bias o IVZ as a a io o ha o OLS has dec eased om 1.004, in he ze o pe sis ence case, o 0.87 in he p esen case. Thi d, he dis ibu ional di e gencies o he IVZ and IVZZ -s a is ics om he s anda d no mal, ha e sligh ly dec eased. 6 (iii) As he deg ee o pe sis ence ises, he pe o mance o IVZ and IVZZ imp o es mono onically. Fo ρx=ρz=0.96, he ins umen s, employed by hese es ima o s, do no appea o be i ele an a all! The co esponding F-s a is ics a e now as la ge as 20.94 and 8.28, espec i ely, hus hea ily o e -es ima ing hei popula ion analogues, which emain equal o ze o. This means ha ‘spu ious’ eg ession effec s in he i s -s age eg essions a e clea ly in place, despi e he ac ha he se ies in ol ed a e s ill I(0). Howe e , hese effec s u n ou o be qui e bene icial as a as s a is ical in e ences on θa e conce ned. The median bias o IVZ (IVZZ), as a a io o he median bias o OLS, is as low as 0.42 (0.57). Mo eo e , he dis ibu ion o he IVZ -s a is ic is loca ed close o ze o (a ound 0.307) as opposed o ha o OLS, loca ed a ound 2.16. In ac , IVZ p oduces he bes - cen e ed -s a is ic o all he es ima o s unde conside a ion. Fo example, he mean alue o he IVZ -s a is ic is close o ze o han ha o he IVX -s a is ic, which eaches he alue o -0.592. In o he wo ds, he empi ical dis ibu ion o he -s a is ic p oduced by an IV es ima o u ilizing an i ele an ins umen is be e cen e ed han ha o an IV es ima o , employing an ex emely s ong ins umen . Mo eo e , he IVZ -s a is ic is, in gene al, be e app oxima ed by a s anda d N(0,1), han any o he es ima o ’s -s a is ic. Fo example, he 0.025 and 0.975 poin s o IVZ a e -1.21 and 2.09 espec i ely, hus esul ing in an empi ical size o 3.6%. On he o he hand, he co esponding pai s o OLS, IVX,DOLSandFMLSa e(0.588,3.776),(−2.37,1.332),(−1.219,3.773),and (−0.679, 3.773), esul ing in empi ical sizes o 59.1%, 7.05%, 25.5% and 35.95%, espec i ely. This in u n implies ha IVZ ou pe o ms no only IVX, bu also FMLS and DOLS, as a as hypo hesis es ing on θis conce ned. (i ) In he ex eme case ρx=ρz=1,IVZZ and, especially, IVZ con inue o pe o m su p isingly well. In his case he dominance o IVZ o e IVX is clea in all aspec s o s a is ical in e ence. Fo example, he mean alues o he IVZ and IVX -s a is ics a e 0.040 and -1.019, espec i ely and he ( 0.025, 0.975)pai s a e (−1.603,1.739),and (−2.645,0.840), espec i ely. I is in e es ing o no e ha he pe o mance o IVZ is compa able e en o ha o he coin eg a ion es ima o s, FMLS and DOLS, which now ope a e in hei na u al en i onmen . 7 The effec s desc ibed abo e a e summa ized in Figu es 1 and 2, ha desc ibe he median bias o IVZ and IVX, espec i ely ela i e o ha o OLS, o sample sizes o 100 and 300. I can be seen ha he ela i e bias o IVZ, as opposed o ha o IVX, ends o ze o as ρx(= ρz) ends o one. I can also be seen ha as he sample size inc eases, and he ele an asymp o ic heo y o s a iona y eg essions becomes mo e ele an , he ‘i ele an ins umen s’ effec weakens. Howe e , he a e a which his effec declines appea s o be ex emely slow. In all he expe imen s, so a , we ha e e ained he assump ion ρx=ρz, ha is, he ins umen s and he eg esso exhibi he same deg ee o pe sis ence. How many o he abo e esul s emain alid when ρx6=ρz?To answe his ques ion, we un ano he se o expe imen s, whe e he alue o ρxis kep ixed o a pa icula alue om he se I={0,0.1,...1}.Fo his alue o ρx,ρ z akes sequen ially all he alues o I. We epea he same p ocedu e un il all he alues o ρx∈Ia e exhaus ed. O e all, we un 121 simula ions, plus some addi ional, mo e speci icones, o ρxin he neighbo hood o uni y. The esul s (no epo ed) sugges ha he gene al pic u e, desc ibed abo e, emains he same o he cases ha he ins umen s and he eg esso exhibi diffe en deg ees o pe sis ence, p o ided ha he diffe ence |ρx−ρz|is no e y la ge. Fo example, when ρx=0.96, hen IVZ pe o ms sa is ac o ily well o a alue o ρzas low as 0.8(and, o cou se, as la ge as uni y). 4 Conclusions Ou conclusions om he in es iga ion o he beha iou o he TSLS p ocedu e, unde al e na i e deg ees o pe sis ence o he eg esso and he ins umen s used, a e he ol- lowing: Fi s , he pe o mance o he es ima o , u ilizing solely i ele an ins umen s, imp o es mono onically, as he deg ee o pe sis ence o he eg esso and ha o he in- s umen s, inc eases. Second, in he case whe e he eg esso and he ins umen s a e nea - o-uni oo p ocesses, he es ima o ha u ilizes a single i ele an ins umen , ou pe o ms IV es ima o s wi h s ong ins umen s, as well as asymp o ically efficien coin eg a ion es ima o s. 8