Nonlinear cointegration: Theory and Application to Purchasing

Transcription

1 Nonlnear conegraon: heor and Applcaon o Prchasng Power Par Ahor: Yan Zhao Deparmen of Economcs and Socal Scences Dalarna Uners Dae: Jne 3, 7 Spersor: Changl He D-leel hess n Sascs, 7

2 Absrac In hs paper, we sd a smooh-ranson pe of nonlnear conegraon among a dnamc ssem, n whch he proposed defnon ness Engle and Granger (987) s lnear conegraon. Based on he Smooh ranson Aoregresse (SAR) models, a ranglar represenaon for he nonlnearl conegraed ssem s nrodced. Frhermore, wo ess for nonlnear conegraon are dered n hs paper: one s a resdal-based es for esng he nll hpohess of no nonlnear conegraon and he oher s a es for esng he hpohess of lnear conegraon agans nonlnear conegraon. he sascal properes of he second es are nesgaed. An emprcal example s llsraed b applng he frs esng procedre o he dollar/lra real exchange rae. I s fond ha here s no lnear conegraon b a nonlnearl conegrang relaon n he nesgaed prchasng power par (PPP). he PPP pzzle n Hamlon (994) can be soled b applng or nonlnear conegraon. Ke words: Dcke-Fller es, Logsc smooh ranson, Nonlnear conegraon

3 . Inrodcon Snce nrodced n Engle and Granger (EG) (987)'s creae arcles, conegraon has become an nflenal ndsr n modelng macroeconomc me seres n economercs. he EG s lnear conegraon has been well deeloped n lnear models for modelng long-rn relaonshp among economc arables and has been wdel appled n emprcal sdes. Howeer, b nmercal emprcal fndngs, man economc me seres, sch as nemplomen raes, GDP growh, neres raes, alwas exhb nonlneares. For a sre of nonlnear dnamc models, see an Djk, eräsra and Franses (). On he oher hand, economc relaonshps ofen dspla nonlneares sch as srcral change and regme-swchng. Recenl, man researchers hae alread aemped o exend EG s conegraon o a nonlnear framework. For nsance, Johansen s (6) research allows for nonlnear shor-rn dnamcs n error correcon models. Balke and Fomb (997) nrodce a hreshold conegraon model ha accommodaes for nonlnear adjsmens owards a long-rn eqlbrm and Enders and Falk (998) emplo a smlar model o nesgae he prchasng power par. In hs paper, we propose a new nonlnear conegraon, whch s a jon analss of nonlnear and conegraon. For a nonlnear consderaon we adop one of he mos poplar nonlnear dnamc models, he smooh ranson aoregresse (SAR) model, whch ness a lnear aoregresse model and conans a regme-swchng srcre. For a dealed dscsson of SAR models, see eräsra (994). Frhermore, a consan conegrang ecor defned n Engle and Granger (987) cold be nerpreed as a specal case f a hreshold pe of conegraon s eden n he macroeconomc me seres ssem. herefore, he conegrang ecor can be specfed n a form of a smooh ranson fncon f here s a regme-swchng n he ssem. Emprcal sppor for he heor of prchasng power par (PPP) has been relael

4 mxed. he nll hpohess of no lnear conegraon of EG s (987) can no be rejeced a 5% sgnfcan leel when applng a resdal-based esng procedre o he dollar/lra real exchange rae. See Hamlon (994) for a dealed dscsson. he PPP pzzle can be soled b nrodcng a nonlnear conegraon concep and he emprcal resls show ha he economc hpohess ha he PPP ssem s conegraed s acceped. he paper s organzed as follows. In Secon we nerpre or nonlnear conegraon b an example of a barae ssem ha s nonlnearl conegraed. Secon 3 ges he general defnon of nonlnear conegraon. Secon 4 presens he examnaon of he long-rn eqlbrm relaon among nonsaonar arables, for specfc nonlneares whn conegrang ecor and propose a procedre of esng lnear conegraon agans nonlnear conegreaon. We recall he emprcal example of prchasng power par and appl or esng nonlnear conegraon procedre n Secon 5. he las secon s he conclson.. An example of nonlnear conegraon In hs secon, we llsrae a barae aoregresse process ha each nddal process s I() and her nonlnear combnaon s weakl saonar as follows: Le ( ), be a barae process defned as α + ε + ε, () where ε nd(,),,, E( ε ε ) ~ α + G ( + ( Δ ) G exp ) s a logsc smooh ranson fncon

5 Fgre plos realzaons of and nddall for and ndependen G N (, ) arables ε and ε Fgre : Realzaons of and generaed from eqaon () Obosl, s a lnear I(), snce s a pre random walk n eqaon (). If ends o be an I() process, we wold conclde ha here exss a conegraon n hs ssem where he conegrang ecor s me arng raher han smple consans. From Fgre, we obsere ha here mgh be me rend n. We appl Dcke-Fller es o proe ha wheher s nonsaonar. We es 3

6 H + :, agans H, + : ρ, where ~ d (, ) nll hpohess s ρ and he alernae hpohess s ρ <.. Hence he he sasc of Dcke-Fller ρ es s defned as ( ρ ) ; he sasc of Dcke-Fller ρ es s defned as, where ρ s he sample sze, ρ s he esmae of he coeffcen ρ and s he sandard error of ρ n he esmaed ρ model + ρ,. he esmaon resl of hs example s.974, (.358) In hs example, and he ale of he es sascs are ( ) (.974). 86 ρ ρ ρ he asmpoc dsrbons of boh sascs are consrced nder he n assmpon of ρ. Comparng he ale of each sasc a a 5% sgnfcan leel of able B.5 or able B.6 n Hamlon (994) (), we fnd ha neher he ales of he aboe es sascs falls no he rejec regon. here s no sffcen edence o rejec he nll hpohess, we conclde ha s an I() process. hen we plo he realzaons of and ogeher n Fgre and nspec he relaonshp beween hese wo me seres. 4

7 Fgre : Comparson of realzaon of and From Fgre, we noe ha eher seres { }, wll flcae randoml far from he sarng ale. he moe smlaneosl b no dencall. I can be seen ha shold reman some relaonshp wh, whch comes from he prodng nonlnear regresson coeffcen α. In oher words, a me-arng ecor as α (, α ) leads he nonlnear combnaon of wo me seres o be a saonar process. From he aboe example, we conclde ha here exss a conegraon wh a 5

8 me-arng ecor raher han smple consans. 3. Defnon of nonlnear conegraon Engle and Granger (987) s defnon of conegraon refers o lnear combnaon of nonsaonar arables and descrbes a long-rn eqlbrm relaon exsng n a lnear dnamc ssem. In hs secon, we shall generalze sch lnear conegreaon o a nonlnear conegraon based on a nonlnear dnamc ssem. Defnon Le he n-dmensonal ecor be an I() process, ha s for an, L, n, ~ I( ) defnon.. Here, he I() ma follow from Engle and Granger (987) s { } s sad o be nonlnearl conegraed f here exss a me-arng ecor α ( α, α L α ) n whch sasf he frs em α s a nonzero-consan ha a normalzed α, α L α ; ( ) n { },, L n α are well-defned fncon of a random arable sch ha for, each, α has a logsc smooh ranson fncon form: S α α + G ( { ( )}) where G + exp γ S c, α, γ and c are parameers, γ ( ) α ~ I,.e., a nonlnear combnaon of s I(). Here, α s called nonlnear conegrang ecor. Fgre 3 plos a realzaon of G for some and we can see ha hs seres ares 6

9 along he me rend beween and. Fgre 3: Plo of a realzaon of G where γ. 5 and c. he logsc smooh ranson fncon s lzed de o ewng G as a fncon of S, wh γ and c fxed, s bonded as G and ewng as a G fncon of γ, wh S and c fxed, he lms of G as γ and γ are consans. In hs paper, we assme S Δ n laer dscsson. 7

10 Noes: he aboe defnon of nonlnear conegraon ness Engle and Granger (987) s defnon f seng G, for all, L, n I s possble o es, nder he nll hpohess of γ, lnear conegraon agans nonlnear conegraon among a dnamc ssem { }. 4. esng nonlnear conegraon 4. es no nonlnear conegraon agans nonlnear conegraon In some economc relaonshps, no lnear conegraon has been esed so we are srongl cros abo wheher an nonlnearl conegrang relaon exss. In hs sbsecon, we assme he conegrang ecor α s known we assme frher { } has a ranglar represenaon formlaed as: + α + L+ α n n z (3) where ~ I() for each,, L, n In he ssem (3), we wsh o es no nonlnear conegraon agans nonlnear conegraon and we appl a resdal-based es. he nll hpohess s ha z ~ I( ) whereas he alernae s z ~ I( ) gen, he classcal Dcke-Fller ess appl.. hs, as he nonlnear conegrang ecor s o carr o hs es, we rn an aoregresson of z and dere he Dcke-Fller ρ es and Dcke-Fller es afer calclang he seres of resdals he known ecor drecl. z b sng 8

11 here are dfferen cases n esng n roo wh or who drf or me rend, see Hamlon (994). he common sed hpoheses are case: H : z H : z z ρ z + + H : ρ H : ρ < case : H : z H : z z + α + ρ z + H : α, ρ H : α, ρ < case 4 : H : z H : z α + z α + ρ z + + δ + H : ρ, δ H : ρ <, δ where ~ d (, ) he es sasc of Dcke-Fller ρ es s defned as ( ) ρ for he aboe hree cases, where sands for he sample sze and ρ sands for he esmae of he coeffcen ρ.he es sasc of Dcke-Fller es s defned as ρ, ρ where ρ represens he esmaed sandard error of ρ. If he nll hpohess of z ~ I() s rejeced, we conclde ha here exss a nonlnear conegraon n he ssem, oherwse here s no nonlnear conegraon n he ssem. he asmpoc dsrbon of Dcke-Fller ρ es and es s smmarzed n he able B.5 and able B.6 n Hamlon (994). Check he crcal ale nder he specal case and he sample sze and compare wh he obaned resls of he es sascs. 9

12 he smaller ales of es sascs ndcae ha he coeffcen ρ < s sgnfcan and we commen ha he nonlnear combnaon of I() processes s I() whch mples he exsence of nonlnear conegraon. 4. es lnear conegraon agans nonlnear conegraon Sppose we proe ha he conegraon s lnear, s eas o follow he dscsson n Hamlon (994) and Johansen (6) for eher known or nknown conegrang ecor cases. In hs sbsecon, we race he procedre of generang es and r o recognze whch pe of conegraon s, lnear or nonlnear. Le he ssem { } be nonlnearl conegraed sch ha has a nonlnear conegraon wh where,, L, n b { },,, L, n s a sbssem who an conegraon. he ranglar represenaon of { } s: + α + L+ α n n z (4) () where ~ I for each,, L, n, z ~ I( ) for each, L, n, α has a logsc smooh ranson fncon form defned n Secon 3. When γ, α are redced o consan erms. hs he ssem (4) cold be sed o esng lnear conegraon agans nonlnear conegraon. For a conenen dscsson we consder a barae ssem as follows: α + +, (5) where he error erms of and sasf:

13 ~ d (, ) wh 4 > and E < ; ~ d (, ) wh > and E < and E for an and τ ; 4 τ α β + G ( Δ ; c), G ( Δ ; γ, c) γ,, γ. + exp { γ ( Δ c) } he frs eqaon n model (5) s expressed as ( Δ;γ c) β + (6) + G, In eqaon (6), β sands for a lnear conegraon par and G ( Δ ;γ, c) sands for a nonlnear conegraon par. he feare ha he lm of G ( Δ ; c) γ, s zero, as γ gong o zero redces he nonlnear regresson o lnear regresson. Hence, he am of esng wheher he nonlnear par s essenal n he conegrang procedre s nerpreed n he followng hpoheses: H : H : β β + + G : γ ( Δ ; γ, c) + H : γ > H In order o aod he denfcaon problem whch arses from he seng of γ, we adop a frs-order alor expanson of γ arond n he ranson fncon G ( Δ ; γ c, nrodced b He and Sandberg (6)., ) he resl of he frs-order alor approxmaon s G ( Δ γ, c) ( Δ c) γ ; R( γ ) (7) 4 + where R ( γ ) s a remander erm whch depends on γ and conerges o zero as γ

14 gong o zero. We sbse eqaon (7) no he model n eqaon (6), afer mergng erms, we oban γ c γ β + Δ + R( γ ) + (8) 4 4 We rewre eqaon (8) as β + φ Δ + (9) where γ β c β, 4 γ 4 φ, R( γ ) + herefore, he nll hpohess of γ s dencal o H : β β, φ (), hen we rn o consder he axlar regresson β + φ Δ + () he asmpoc dsrbon can be dered b Fnconal Cenral Lm heorem, d 4 Δ d [ W () r ] dr [ W () r ] dr Δ Δ d p {[ W () ] } d {[ W () ] }

15 Under he nll hpohess n () ( ) () [ ] () [ ] () [ ] { } () [ ] { } Δ Δ Δ Δ 4 W W dr r W dr r W d φ β β hs, ( ) () [ ] () [ ] { } ( ) [ ] { } () [ ] dr r W W W dr r W d β β () () [ ] () [ ] { } ( ) [ ] { } () [ ] 4 dr r W W W dr r W d φ (3) Besdes he esmaed ales of and, assocaed esmaed error erms are deermned from axlar regresson n (), β φ Δ φ β Under he nll, we achee, hen he esmaed sandard deaon of s û ( ) ( ) ar ar E From model n eqaon (5), Δ, hen he esmaed sandard deaon of s ( ) ( ) Δ Δ Δ ar ar E herefore, we consrc he adjsed es sascs afer () and (3) as follows: 3

16 d ( ) {[ W () ] } β β d φ [ W () r ] {[ W () ] } [ W () r ] dr dr (4) (5) I s clear ha eher asmpoc dsrbon of adjsed es sascs s eqalen o he asmpoc dsrbon of Dcke-Fller ρ es of Case, see Hamlon (994). he crcal ales of aboe wo es sascs for dfferen sample sze are ablaed n able. able : Crcal Vales for Adjsed es Sasc based on OLS Esmaon of Axlar Regresson Sample Probabl ha Adjsed es Sasc s less han enr Sze Gen he neresed ale of β, we calclae he ales of boh adjsed es sascs. If eher repored ale s smaller han he crcal ale, he nll hpohess of lnear conegraon s rejeced. I s srongl sggese of he exsence of nonlnear conegraon. 5. Emprcal analss------prchasng Power Par 5. heor of prchasng power par A classcal economc example of conegraon nerpreaon s he prchasng power 4

17 par. he basc proposal of prchasng power par s frs deeloped b Gsa Cassel n 9. I s a mehod based on he law of one prce, n whch he dea s ha, n an effcen marke, he dencal goods ms hae onl one prce. he rao of he prces n dfferen crrences s he exchange rae. he prchasng power par exchange rae eqalzes he prchasng power of dfferen crrences n her home conres for a gen baske of goods. hese knd of specal exchange raes are ofen sed o compare he sandards of lng of he dfferen conres. Here, sng prchasng power par exchange raes s preferred o sng marke exchange raes becase of he sgnfcan dfference beween prchasng power par exchange rae and marke exchange rae. Marke exchange raes flcae freqenl de o he need of he marke whle man economss belee ha prchasng power par exchange raes are characerzed b a long-rn eqlbrm. he measremen of prchasng power s he domesc prce leel n each conr. he prce leel growh means ha he prchasng power of he crrenc of hs conr decreases and he crrenc s dealed a he same rae, and ce ersa. In fac, he measremen of prchasng power par s complcaed snce ha here s no smpl nform prce leel dffered among conres. I s necessar o compare he cos of baskes of goods and serces consmed b people from dfferen conres sng a prce ndex. Apparenl, he consmer prce ndex organzed monhl s he bes choce o assess he prchasng power. In economcs, a consmer prce ndex s a sascal me seres of a weghed aerage of prces of a specfed se of goods and serces prchased b he consmers. I s a prce ndex ha racks he prces of a specfed baske of consmer goods and serces. 5. Daa descrpon he fndamenal dea of prchasng power par holds ha P S P, where P denoes he domesc prce ndex, P denoes he foregn prce ndex and 5

18 S denoes he exchange rae beween he crrences of wo conres. In Hamlon s book, P denoes a prce ndex n Uned Saes (n dollar per good), P a prce ndex n Ial (n lra per good) and S he rae of exchange beween he crrences of Uned Saes and Ial (n dollar per lra). If we ake he logarhms of boh sdes of P S P, rns o s + p p. In fac, along wh he ransporaon coss and errors n assessng prces de o dfferen home conres, he prchasng power par cold hardl keep exacl a eer me pon hrogh he me rend. herefore, consder he economerc model z p s p, where z represens he a deaon from prchasng power par. A weakl expresson of prchasng power par s ha z s saonar een hogh he nddal elemens of { s, p p, } are all I(). For analss objeces, he raw daa are ransformed a frs as presened: p [ log( P ) log( )] P 973: p [ log( P ) log( )] P 973: s [ log( S ) log( )] S 973: Sch ranson s o make sre ha aboe hree me seres hae same sarng ales, zero a Janar 973. Mlplng b roghl represens he percenage dfference beween he ale a crren me pon and he sarng ale a Janar 973. Fgre 4 plos he monhl daa of he prce leel p n Uned Saes, he prce leel p n Ial and he exchange rae s beween dollar n Uned Saes and lra n Ial from Janar 973 o Ocober 989 afer he ranson. 6

19 Fgre 4: Monhl daa of p, s and p from 973 o 989 afer ranson 5.3 No Lnear conegraon n PPP We reew he example of prchasng power par dscssed n Hamlon s book. In Fgre 4, appears ha each me seres mgh be a I() process. he problem solng process sars from esng he nddal elemens of { p s, p }, are all I() process. For he monhl daa, we appl agmened Dcke-Fller es o proe ha we can no rejec he nll hpohess of nonsaonar. he resls of he esmaon and he calclaon of he es sascs are saed n deals n Hamlon (994) s. 7

20 Based on he prchasng power par, he consderaon of he relaon among he consmer prce ndexes and he exchange rae can be ndced o nesgae he ( ) ' exsence of he connegraon for a gen connegrang ecor α,,. hs, he ke sse of he dscsson s o es he saonar of z p s p. he plo of z s shown n Fgre 5, whch descrbes a rogh eqlbrm. Fgre 5: z, lnear combnaon of p, s and p he ops of he examnaon of he long-rn eqlbrm do no prode effece edence o rejec he hpohess of nonsaonar, hence he connegrang ecor 8

21 ' (,, α ) does no resl n a saonar. z From he research n Hamlon s, expresses ha he heor of prchasng power par s no sasfed n he example of Uned Saes and Ial. hs s wdespread dsagreemen of he prchasng power par. In nex secon, we wll brng n he nonlnear connegraon concep and nrodce a parclar ranson fncon, n order o erf he applcabl of he prchasng power par b he adjsmen of he connegrang ecor. 5.4 Exsence of nonlnear conegraon n PPP Afer sdng he Hamlon (994) s emprcal example of prchasng power par beween Uned Saes and Ial from 973 o 989, we appl he defnon of nonlnear conegraon and go frher analss o check wheher here s an conegrang connecon n he ssem. Accordng o he defnon n Secon 3, he neres n hs secon s moaed b he possbl of a specalzed gen nonlnear conegrang ecor, for whch, we appl he es procedre saed n Secon 4.. Sppose he parclar nonlnear conegrang ecor s α (, α α ) ' { ( Δ )} where, 3 α for, 3 s apponed as a logsc smooh ranson. + exp he demonsraon of I() process for each me seres of he consmer prce ndexes and he exchange rae has been arged n Secon 5.. he nonlnear combnaon of he consmer prce ndexes and he exchange rae s calclaed b z p s + exp { ( Δs )} + exp{ ( Δp )} p 9

22 Graph he nonlnear combnaon z n plo whch prodes a srong descrpon of he exclson of me rend n Fgre 6. Fgre 6: z, nonlnear combnaon of p, s and p Recall he knowledge n Secon 4., we es he nll hpohess H z z + agans he alerae hpohess H : z + ρ z + : α, where ( ~ d, ). he esmaon resl of hs example s z z (.888) (.65) In hs example and he ale of he es sascs are

23 ( ) (.599) ρ ρ ρ he dsrbons of sascs are generaed nder he compondng hpohess of he n assmpon of ρ and he zero assmpon of he nercepα. A a 5% sgnfcan leel, compare he ale of each sasc wh he crcal ale from he able B.5 or able B.6 n Hamlon s for a sample sze, we fnd ha he ρ es sasc s negae wh a larger absole ale whch belongs o he rejec regon and he ale of es sasc also prodes a sffcen edence o rejec he nll hpohess, ρ and α ( ) hrs, z ~ I means he resdals of he regresson p on s and p s saonar, whch eqals o he deaon from prchasng power par s saonar. { p s, p, } s conegraed b, + exp, { ( Δs )} + exp{ ( Δp )} ' whch s relaed o he dfference of s and p. he resls sgges ha alhogh all consmer ndexes and exchange rae exhb a n roo process, oer a long-rn, he nonlnear conegrang ecor e he separae I() processes ogeher, and ends o dspla a long rn eqlbrm arbe. Hence, prchasng power par s warraned n he US-dollar and Ial-lra nesgaon. 6. Conclson In hs paper, we nrodce a smooh-ranson pe of nonlnear conegraon wh he conegrang ecor hang a dnamc srcre, so ha Engle and Granger (987) s lnear conegraon s a specal case here. Based on he Smooh ranson

24 Aoregresse (SAR) models, a ranglar represenaon for he nonlnear conegrang ssem s sed, and sch a dnamc srcre s flexble snce a lnear conegrang regresson s nesed n he ssem. In order o aod spros regressons, wo ess of esng for nonlnear conegraon are creaed. he resdal-based es for esng he nll hpohess of no nonlnear conegraon s eas o carr o f he economc conegrang ecor s known. he second es s based on he SAR model ha conanng a sngle nonlnear conegraon and s sed o es he lnear conegraon nder he nll hpohess and alernael he model s a nonlnearl conegrang regresson. Asmpoc dsrbon of hs es for a barae ssem s dered and s showed ha he asmpoc dsrbon for he adjsed es sasc s dencal o he dsrbon of classcal Dcke-Fller ρ es nder Case. An emprcal example s consdered b applng or nonlnear conegraon heor o he dollar/lra real exchange rae. he llsraon of hs example shows ha here s no lnear conegrang relaon n he PPP ssem, howeer, here s or nonlnear conegraon edence sppored b or emprcal analss for he PPP example. I s sggesed ha he PPP ssem ma conan nonlnear feares beween he economc arables whch cold case he PPP pzzle n Hamlon (994).

25 References Balke, N. and. B. Fomb (997), hreshold Conegraon. Inernaonal economc Reew 38, Enders W. and B. Falk (998), hreshold-aoregresse, medan nbased, and conegraon ess of prchasng power par. Inernaonal Jornal of Forecasng 4, Engle, R. F. and C. W. J. Granger (987), Co-negraon and error-correcon: Represenaon, esmaon and esng. Economerca 55: 5-76 Hamlon, J. D. (994), me seres analss. Prnceon Uners Press He, C. and R. Sandberg (6), Dcke-Fller pe of ess agans nonlnear dnamc models. Oxford bllen of economcs and sascs 68: He, C.,. eräsra and A. González (7), esng parameer consanc n ecor aoregresse models agans connos change. Economerc Reews Johansen, S. (6), Conegraon: An oerew. Palgrae Handbooks of economercs: Vol. Economerc heor. Palgrae MacMllan eräsra. (994), Specfcaon, esmaon, and ealaon of smooh ranson aoregresse models. Jornal of he Amercan Sascal Assocaon 89, 8-8. an Djk, D.,. eräsra and P. H. Franses (), Smooh ranson aoregresse models - a sre of recen deelopmens., Economerc Reews, -47 3