COINTEGRATION: A REVIEW JIE ZHANG. B.A., Peking University, 2006 A REPORT. submitted in partial fulfillment of the requirements for the degree

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1 COINTEGRATION: A REVIEW by JIE ZHANG B.A., Peking Universiy, A REPORT submied in parial fulfillmen of he requiremens for he degree MASTER OF SCIENCE Deparmen of Saisics College of Ars And Sciences KANSAS STATE UNIVERSITY Manhaan, Kansas Approved by: Major Professor Shien-Shien Yang

2 Copyrigh JIE ZHANG

3 Absrac Many nonsaionary univariae ime series can be made saionary by appropriae differencing before ARMA models are fied o he differenced series. However, when i comes o nonsaionary vecor ime series, he siuaion is more complex. Since he dynamic of a mulivariae ime series is mulidimensional, even if we can make each componen saionary by appropriae differencing, he vecor process of he differenced componens may be sill nonsaionary. However, i is possible ha he projecions of a nonsaionary vecor ime series in some direcions may resul in a saionary process. Engle and Granger() formally demonsraed ha i is possible for some linear combinaions of he componens of nonsaionary vecor ime series o be saionary. They called his phenomenon Co-Inegraion. This concep of coinegraion urned ou o be exremely imporan in he modeling and analysis of non-saionary ime series in economics. Alhough economic variables individually may exhibi disequilibrium behaviors, ofen ime, due o economic forces, hese disequilibrium economic variables corporaely form a dynamic equilibrium relaionship. Specifically, cerain linear combinaions of nonsaionary ime series may appear o be saionary. Engle and Granger developed saisical mehod for deecing and esimaing his equilibrium relaionship. They also proposed he so called error correcion model o model Co-Inegraed vecor ime series. In his repor, I give a deail review on he concep of coinegraion, he -sep esimaion procedure for he error correcion models, and he ypes of ess for esing coinegraion. Since he es saisics for esing coinegraion do no follow any known disribuion, criical values were obained based on wo models by Engle and Granger. Augmened Dickey-Fuller and Dickey-Fuller ess were recommended as i is believed ha heir disribuions are independen of he under lying process model. The criical values able presened in heir paper is widely used in esing coinegraion. In his repor, we ll consruc ables of criical values based on differen models and compare hem wih hose obained by Engle and Granger. Also, o demonsrae he pracical usage of coinegraion, applicaions o currency exchange raes and US sock and Asian sock indexes are presened as illusraive examples.

4 Table of Conens Lis of Figures... v Lis of Tables... vi Acknowledgemens... vii CHAPTER - Inroducion... CHAPTER - Vecor Time Series... Secion. Weak Saionariy... Secion. Some Vecor Time Series Models... Secion. Nonsaionariy... CHAPTER - Coinegraion... Secion. Definiion of Coinegraion... Secion. MA, AR Represenaions and Error Correcion Model... Secion. Esimaing Coinegraion Sysem... CHAPTER - Tes of Coinegraion... Secion. Seven Types of Tess... Secion. Criical Values Simulaed for oher sample sizes... Secion. Criical Values Simulaed for oher models... CHAPTER - Applicaions o Finance Daa... References And Bibliography...

5 Lis of Figures Figure. Monhly high and low quoes of Dow Jones indusrial average... Figure. Log USDvsGBP exchange rae (-)... Figure. Log CNYvsGBP exchange rae (-)... Figure. Log CNYvsGBP and USDvsGBP (-)... Figure. The Plo of Residuals of Coinegraing Regression (LDIFF)... Figure. Plos of Log Adjused Close quoes of Dow Jones and His-hang Seng... v

6 Lis of Tables Table. The Tes Saisics: Rejec for large values... Table. Criical Values: Rejec for large values... Table. Criical Values for differen sample sizes... Table. Criical Values for differen models... Table. Regression of LAdjcloseDJ on LAdjcloseHS... vi

7 Acknowledgemens The accomplishmen of his paper is due o he effor of my major advisor: Dr. Yang, ogeher wih my commiee members: Dr. Song and Dr. Nelson. I would like o hank hem for heir advices and suppors. Special hanks o Mom and Dad. I won be here if i were no for hem. vii

8 CHAPTER - Inroducion Many nonsaionary univariae ime series can be made saionary by appropriae differencing before ARMA models are fied o he differenced series. However, when i comes o nonsaionary vecor ime series, he siuaion is more complex. Since he dynamic of a mulivariae ime series is mulidimensional, even if we can make each componen saionary by appropriae differencing, he vecor process of he differenced componens may be sill nonsaionary. However, i is possible ha he projecions of a nonsaionary vecor ime series in some direcions may resul in a saionary process. Engle and Granger() formally demonsraed ha i is possible for some linear combinaions of he componens of nonsaionary vecor ime series o be saionary. They called his phenomenon Co-Inegraion. This concep of coinegraion urned ou o be exremely imporan in he modeling and analysis of non-saionary ime series in economics. Alhough economic variables individually may exhibi disequilibrium behaviors, ofen ime, due o economic forces, hese disequilibrium economic variables corporaely form a dynamic equilibrium relaionship. Specifically, cerain linear combinaions of nonsaionary ime series may appear o be saionary. Engle and Granger developed saisical mehod for deecing and esimaing his equilibrium relaionship. They also proposed he so called error correcion model o model Co-Inegraed vecor ime series. This paper gives a deail review of he concep of coinegraion. The second chaper briefly inroduces basic noaion, represenaions of vecor ime series and definiion of saionariy. Chaper saes he definiion of coinegraion, several represenaions of coinegraed vecor processes and he wo-sep mehod for esimaing he coinegraing sysem proposed by Engle and Granger (). Chaper discusses ypes of coinegraion ess for bivariae CI (,) case and provides criical values based on several null hypohesis generaing models. Two applicaions of coinegraion on finance daa are presened in Chaper as examples. Codes for simulaions and examples are provided in he appendix.

9 CHAPTER - Vecor Time Series Time series daa in many empirical sudies, especially hose involved in economics and finances, consis of observaions from several variables. The inerrelaion and dynamic among all he variables are of grea ineres. For example, as economic globalizaion and inerne communicaion acceleraing he inegraion of world financial markes in recen years, financial markes are more and more dependen on each oher han ever before. Hence, o undersand he dynamic srucure of he global finance, one need o consider several financial and economical variables simulaneously represening he behaviors of differen markes. In such cases, mulivariae ime series models are used o describe inerrelaionships among several ime series variables. This chaper is organized as following. Secion defines weak saionariy and correlaion srucures of a vecor ime series. Secion discusses wo widely used models of mulivariae ime series: Moving average and Auoregressive represenaions. Secion inroduces he definiions of nonsaionariy and some of he sudies on mulivariae ime series. Secion. Weak Saionariy ime series Consider a m-dimensional ime series Z = [ Z,, Z,,..., Z m, ], =, ±, ±,.... A vecor Z is weakly saionary if is firs and second momens are ime-invarian. In paricular, he mean vecor of a weakly saionary series is consan over ime and he crosscovariance beween Zi, and Z, absolue value of he ime difference s- j s, for all i=,, m and j=,, m, are funcions only of he For a weakly saionary ime series Z, we define is mean vecor o be E ( Z ) = μ and he lag-k covariance marix as: Γ( k) = cov{ Z, Z } = E[( Z μ)( Z μ ) ] + k + k where γ( k) γ m( k) =, γm ( k) γmm( k) γ ( k) = E( Z μ )( Z μ ), ij i, i j, + k j

10 for k =, ±, ±..., i=,,..., m, and j =,,..., m. As a funcion of k, Γ ( k) is also called he covariance marix funcion for he vecor process Z. Γ() is called he conemporaneous variance-covariance marix of he vecor process Z. And γ ( k) is he auo-covariance funcion ii for he ih componen process Z i, ; γ ij ( k) is he cross-covariance funcion beween Z i, and Z j, for i j. The correlaion marix funcion is defined by ρ D Γ D / / ( k) = ( k) = [ ρij ( k)] where D is he diagonal marix wih γ ii (), (i=,,,m) as is diagonal elemens. The ih diagonal elemen of ρ ( k), ρii( k) is he auocorrelaion funcion for he ih componen series Z i, whereas he off-diagonal elemen ρ ( k), i j, represens he cross-correlaion funcion beween Z i, and j, Z. ij Like he univariae auocovariance and auocorrelaion funcions, he covariance and correlaion marix funcions are also semi-posiive definie in he following sense n n αγ i ( i j) α j i= j= and n n αρ i ( i j) α j i= j= for any se of ime poins,,, n and any se of real vecors α, α,, αn. Since γ ij ( k) = E( Zi, μi )( Z j, + k μj) = E( Z j, + k μj)( Zi, μi ) = γ ji ( k), Γ( k) and ρ( k) have he following symmery propery: Γ( k) = Γ ( k) ρ( k) = ρ ( k) Someimes, he covariance and correlaion marix funcions are also called auocovariance and auocorrelaion funcions. Secion. Some Vecor Time Series Models

11 An m-dimensional vecor process Z is said o be a purely nondeerminisic vecor process if i can be wrien as a weighed sum of sequence of m-dimensional whie noise random process. Namely, Z = μ + a + Ψ a + Ψ a + = μ+ s= Ψ a s s where Ψ = I is he mxm ideniy marix, he Ψ s j are mxm coefficien marices, and he a s are m-dimensional whie noise random vecors wih zero mean and covariance marix Σ, if k= E [ aa ' + k] =, if k wih Σ being a mxm symmeric posiive definie marix. Hence, even hough he componens of a a differen imes are uncorrelaed, hey migh be conemporaneously correlaed. Using he backshif operaor B, and wih Z = Z μ, he equivalen represenaion of he above model can be wrien as s Z = Ψ( B) a = Ψ s B a. This presenaion is known as he vecor moving average or Wold represenaion. s Le Ψ s = [ ψ ij, s] and Ψ ( B) = [ ψ ij ( B)] where ψ ij ( B) = ψ ij, sb. If he coefficien s= s= marices Ψs is square summable, in he sense ha each of he mxm sequences ij, s ψ is square summable, i.e., ψ s= ij, < for i=,,,m, j=,,,m, hen we say he vecor process is s saionary. Anoher useful represenaion of a mulivariae ime series is ha apar from a whie noise process a, Z is a linear funcion of is pas: In erms of backshif operaor, Z = Π Z + Π Z + +a s= - - = Π Z +a s -s Π( B ) Z = a where

12 and s Π( B) I Π B = s s= Π are m m auoregressive coefficien marices. The above represenaion is called vecor auoregressive (VAR) represenaion. Combined he wo represenaion, he widely used vecor auoregressive moving average VARMA(p,q) process expressed in backshif operaor is of he form where and Φ ( B) Z = Θ ( B) a p q Φ ( ) Φ Φ Φ Φ p p B = B B pb Θ ( ) Θ Θ Θ Θ q q B = B B qb are he auoregressive and moving average marix polynomials of orders p and q respecively. We assume ha he wo marix polynomials have no lef common facors; oherwise, we can simplify he model. When Σ (he covariance marix of a ) is posiive definie, wihou loss of generaliy we can also assume ha Φ = Θ = I, an mxm ideniy marix. By aking p= or q=, i is easily seen ha moving average and auoregressive processes are jus special cases of ARMA represenaion. The process is saionary if he zeros of he deerminanal polynomial Φ ( B ) are ouside he uni circle. In his case, wriing Ψ Φ Θ ( B) = [ p( B)] q( B) hen he equivalen moving average represenaion is and he sequence Ψs is square summable. = ( B) = s s= Z Ψ a Ψ s B s When a vecor ime series is saionary, and a model is idenified, he fiing of he model can be obained by maximizing he likelihood funcion if we assume he vecor ime series is a Gaussian process. However, when he ime series is no saionary, maximum likelihood procedure is no direcly applicable. p

13 Secion. Nonsaionariy In he analysis of ime series, i is no unusual o observe series ha exhibi nonsaionary behavior. One useful and mos frequenly used way o reduce nonsaionary univariae ime series o saionary series is by appropriae differencing. For example, in univariae ime series, a s d nonsaionary series Z can be reduced o a saionary series ( B ) Z for an appropriae choice of d> and s >, so ha we can wrie s d φ ( B)( B ) Z = θ ( B) a p q wih φ ( B ) a saionary AR operaor. A naural, an exension o he vecor process is i.e., p s d Φ ( B)( I IB ) Z = Θ ( B) a p q s d Φ ( B)( B ) Z = Θ ( B) a p q This exension implies ha all componen series are differenced he same number of imes, which is unnecessary and undesirable in mos cases. To be more flexible, we assume ha be reduced o saionary vecor series by applying a differencing operaor D ( B), where s d ( B ) s d ( B ) ( B) D = sm d m ( B ) and ( d, d,, dm )&( s, s,, sm ) are wo ses of nonnegaive inegers such ha we have a nonsaionary vecor ARMA model for Z Φ ( B) D( B) Z = Θ ( B) a p q for which he zeros of Φ ( B ) are ouside he uni circle. p Z can However, compared wih univariae case, differencing on vecor ime series is much more complicaed. Over differencing may lead o complicaions in model fiing. And Box and Tiao () shows ha when he orders of differencing for each componen series are he same, i may lead o a noninverible represenaion. Hence, one should be paricularly careful when handle he nonsaionary vecor processes by differencing. Box and Tiao () also poins ou ha when zeros of Φ ( B ) approach values on he uni circle, a canonical ransformaion can p

14 decompose Z ino wo pars, one of which follows a saionary auoregressive process, while he oher par approaches nonsaionariy. They suggesed ha, in analyzing muliple ime series, i is useful o enerain he possibiliy ha he dynamic paern in he daa may be due o a small subse of nearly nonsaionary componens and ha here may exis sable conemporaneous linear relaionships among he variables. Hence, differencing of he original series could lead o complicaions in he analysis. Especially, when a linear combinaion of he componen series is saionary, a model purely based on differences may no even exis. For example, suppose we have a bivariae model x = x + a, x = β x + a ( ) Each series individually is nonsaionary, bu he linear combinaion of he wo componens x β x is saionary. By differencing he wo series, we ge: i.e. w = ( B) x = a, w = ( B) x = β a + a a ( ) w a a w = β a + a The differenced series can no be expressed in he form of bivariae saionary auoregressive process any more, making he analysis more complicaed. Hence, o idenify and esimae he possible saionary linear combinaion of he componens and build an esimable model for his ype of ime series are of grea imporance in mulivariae ime series analysis. (Find an example, each componen is saionary bu joinly nonsaionary) CHAPTER - Coinegraion As saed in he las chaper, mos saisical heory applied in building, esimaing and esing ime series models are based on he assumpion ha he ime series in he models are saionary. Saisical inference associaed wih a ime series process is no valid if he assumpion of saionariy is violaed. However, nonsaionariy is a common propery o many ime series. Especially in macroeconomic and financial processes, ofen ime a process has no clear endency o reurn o a consan value or a linear rend. By appropriae differencing, one can

15 achieve saionary componens bu i migh complicae he srucure of he ime series. Forunaely, alhough individual ime series can wander exensively, some subse of hese series may move in a paern so ha hey do no drif oo far apar from each oher. Such phenomenon can be found in financial and economic ime series daa for examples, indexes of differen sock markes or exchange raes among differen currencies. We consider such phenomenon as exisence of an equilibrium relaionship beween he nonsaionary ime series. To describe his phenomenon, Clive Granger firs inroduced he concep of coinegraion, which was hough of as a grea breakhrough and has changed he way empirical models of macroeconomic relaionships are formulaed oday. As a review of Engle and Granger () s work, he firs secion of his chaper inroduces he definiion of coinegraion. Secion presens he wo equivalen models of coinegraed ime series: Granger s represenaion and error correcion model. Secion provides mehods for esimaing coinegraed sysems. Secion. Definiion of Coinegraion I is well known from Wold s heorem ha a single saionary ime series wih no deerminisic componens has an infinie moving average represenaion. If in addiion, i is inverible, hen i can be approximaed by a finie auoregressive moving average process. Many nonsaionary ime series can be made saionary by appropriae differencing. The following definiion formally defines such a class of nonsaionary ime series. Definiion: A series wih no deerminisic componen which has a saionary, inverible, ARMA represenaion afer differencing d imes, is said o be inegraed of order d, denoed by x ~ Id ( ). Under his noion, x ~ I() is saionary while x ~ I () is nonsaionary bu has a saionary change. There are subsanial differences in behavior beween a series ha is I () and anoher which is I (). Suppose x ~ I() wih zero mean, hen (i) he variance of x is a finie consan; (ii) an innovaion has only a emporary effec on he value of x ; (iii) he auocorrelaions, ρ k, decrease rapidly in magniude as k increases, so ha he infinie sum of

16 hem is finie. Whereas, if ~ () x I wih x =, hen (i) variance of x goes o infiniy as goes o infiniy; (ii) is innovaion has a permanen effec on he value of x, since x is he sum of all previous changes; (iii) he heoreical auocorrelaions, ρk for all k as. More discussion can be found in Feller () or Granger and Newbold (). Due o he relaive sizes of he variances, i is always rue ha he sum of an I () and I () will be I (). Generally, if a and b are consans, b, hen a+ bx is I( d) if x is I ( d ); However, if x and y are boh I ( d ), hen i is possible ha he linear combinaion z = x ay will be I( d b), b >. Consider he case when d = b=, so ha x and y are boh I () wih dominan long run componens, bu heir linear combinaion z is I (), a saionary series. This is a special consrain on he long-run componens of he wo series. However, i is worh noicing ha i is no generally rue ha here exiss such an a ha makes z ~ I (). To formalize he ideas above, he following definiion from Granger() and Granger and Weiss() is inroduced. Definiion: The componens of he vecor x are said o be co-inegraed of order db,, denoed x ~ CI( d, b), if (i) all componens of x are I( d ) ; (ii) here exiss a vecor α( ) so ha z = α ' x ~ I( d b), b>. The vecor α is called a co-inegraing vecor. As an illusraion, consider he vecor ime series in secion., x = x + a, x = β x + a. ( ) Clearly, each componen is an I() process, since hey become saionary afer firs differencing: w = ( B) x = a, w = ( B) x = β a + a a. ( ) However, wih α ' = [ β, ], he linear combinaion x α' x = [ β, ] βx x βx βx a βa a x = = = is I (). Hence, he vecor ime series x is coinegraed wih a coinegraion vecor α ' = [ β, ].

17 Someimes, co-inegraion vecor is also called he long-run parameer. I is clearly no unique. Because if α'x is saionary, hen so oo is cα'z for any nonzero consan c. Hence cα is also a coinegraing vecor. If x is a vecor of economic variables, hen hey are said o be in equilibrium when he following linear consrain is saisfied. α'x = Of course, in realiy, he equilibrium holds only approximaely in he sense ha z = α'x is a I() process, where z is called he equilibrium error. Concenraing on he bivariae ime series and d=, b= case, coinegraion would mean ha if he componens of vecor ime series x were all I (), hen he equilibrium error would be I (), so ha z will rarely drif far from zero if i has zero mean and will cross zero line ofen. I means ha he equilibrium or a leas a close approximaion will occur ofen; whereas if x is no coinegraed, hen for any vecor α, z = α ' x will always wander widely and equilibrium would be rarely reached, which suggess ha in his case equilibrium concep is no applicable. The phenomenon of coinegraion can be found in many economic sudies. For example, as shown in Figure., he monhly highes quoaion (he solid line) and lowes quoaion (he doed line) of Dow Jones indusrial average are boh individually nonsaionary. However, he difference of he wo series (he disconinuous line a he boom) is I (), which indicaes ha alhough each series can wander wildly, hey can no drif oo far apar from each oher. Figure. Monhly high and low quoes of Dow Jones indusrial average High

18 More generally, if he vecor series x conains p componens, each being I (), hen i is possible for several equilibrium relaions o govern he join behavior of he componens of x. So here may be k (<p) linearly independen coinegraion vecors α, α,..., αk such ha α'x is a saionary (kx) vecor process, where α ' ' ' α α = αk ' α' is called he coinegraing marix. If α' is a coinegraing marix, hen for any qxk marix C, Cα ' is also a coinegraing marix. Hence, α is no unique. If for any oher (xp) vecor linearly independen of he rows of α ', we have ha b ' ha is b'x is nonsaionary, hen x is said o be coinegraed of rank k. The vecors α', α',..., α k ' form a basis for he space of he coinegraing vecors which is called coinegraion space. Secion. MA, AR Represenaions and Error Correcion Model Suppose ha each componen of x is I (), hen wihou loss of generaliy we can assume ha he change in each componen is a zero mean purely nondeerminisic saionary sochasic process, since any known deerminisic componens can be subraced before he analysis is begun. I follows ha here will always exis a mulivariae Wold represenaion: Δ x = ( B) x = Ψ( B) a Where j Ψ( B) = = Ψ jb, Ψ() = I and he coefficien marices Ψ j are absoluely summable j since ( B) x is saionary. a is he vecor whie noise process wih mean and covariance marix E( aa' ) =, s s = σ, = so ha only conemporaneous correlaions can occur. The moving average polynomial can be expressed as s

19 j j= j j= j j= j j Ψ( B) = Ψ B = Ψ + Ψ ( B ) j i = Ψ() + ( B)( Ψ ( B )) j= j i= = Ψ() + ( B) ( Ψ ) B * = Ψ() + ( B) Ψ ( B) j= i= i+ j j Where * j ( B) = * Ψ Ψ j jb and = * = * Ψ j Ψ = i+ j. If Ψ( B) is of finie order, hen Ψ ( B) will i * be of finie order. If Ψ () is idenically zero, hen a similar expression involving defined. Based on he expression, x can be wrien in he form Denoe y = Ψ * ( B) a, hen x x =Δ x+δ x + +Δx = Ψ( B)( a + a + + a ) = [ Ψ() + ( ) Ψ ( )]( a + a + + a ) = Ψ()( a + a + + a ) + ( ) Ψ ( )( a + a + + a ) * B B * B B ( B) can be Hence and * ( B) Ψ ( B)( a + a + + a ) = ( B)( y + y + y ) =Δ y +Δ y + +Δy = y y x = x + Ψ()( a + a + + a ) + y y α ' x = α '( x y ) + αψ ' ()( a + a + + a ) + α ' y Obviously, b'( a+ a + + a ) is no saionary for any nonzero (xp) vecor b '. Therefore, will be saionary if and only if αψ ' ()= This indicaes ha a coinegraion marix is perpendicular o Ψ (). Thus he coinegraion space α ' x spanned by he rows of α ' is a complemen space of he column space of Ψ (). The deerminan Ψ( B ) = a B = ; hence he process is no inverible and we can never inver he MA represenaion Δ x = Ψ( B) a o represen a coinegraed process wih a vecor AR form in erms of Δx. The vecor AR represenaion of a coinegraed process mus be in erms of x direcly.

20 The represenaion Δ x = Ψ( B) a and he above resricion ogeher is he MA represenaion of a coinegraed vecor process. Suppose ha x is nonsaionary and can be represened as AR( p) model Φp ( B ) x = a such ha Φ ( B ) = conains some uni roos, where Φp ( B) = I ΦB ΦpB ( B) on boh sides, p ( B) Φ ( B) x = ( B) a. If each componen of x is I(), hen, he MA represenaion ( B) x = Ψ( B) a can be ransformed o p ( B) Φ ( B) x = Φ ( B) Ψ( B) a. p p p. Muliply Comparing he above wo, we have ( B) a = Φ p ( B) Ψ( B) a holds for any a. I implies ha for any B. Hence, if we ake B =, ( B) I = Φ p ( B) Ψ ( B) Φp () Ψ() =. Φ () p is perpendicular of Ψ (), so i mus belong o he coinegraion space spanned by he rows of α '. Tha indicaes Φp () = γα ' for some (pxk) marix γ. The model Φ ( B ) x = a and he above resricion ogeher is he AR presenaion of a coinegraed vecor process. p Noice in he AR represenaion, Φ I Φ Φ I λ Φ Φ p * * p ( B) B pb ( B) ( B p B p = = + + )( B), * where λ = Φ + + Φp and Φ j = ( Φ j+ + + Φp) for j =,,, p. Hence he AR represenaion Φp ( B ) x = a can be wrien as

21 * * ( I λb) x ( p Φ B+ + Φ p B ) Δ x = a or Subrac x on boh sides, hen x = λx + Φ Δ x + + Φ Δ x + a. * * p p+ Δ x = δx + Φ Δ x + + Φ Δ x + a * * p p+ where δ= λ I = Φ () = γα ' based on he resricion in AR represenaion. Therefore, p Δ x = γz + Φ Δ x + + Φ Δ x + a * * p p+ for some (pxk) marix γ, where z = α ' x is a (kx) saionary process. This represenaion implies ha he differenced series Δx of a coinegraed process x can no be described using only he values of is own lagged differences. The model mus include an error correcion erm, γz = γα ' x. Consider he relaion Δx in erms of is own pas lagged values as a longrun equilibrium, hen he erm z can be aken as an error from he equilibrium and he coefficien marix γ is an adjusmen for his error. Wriing he above represenaion in an AR(p) form, we have he definiion of error correcion represenaion as below. Definiion: A vecor ime series x has an error correcion represenaion if i can be expressed as: ( B)( B) = z + Φ x γ a where a is a saionary mulivariae disurbance, wih Φ() = I, Φ() z = α ' x and γ. has all elemens finie, The error correcion represenaion was firs proposed by Davidson e al.() and has been used widely in economic sudies. For a wo variable vecor process, a ypical error correcion model would relae he change in one variable o pas equilibrium errors, as well as o pas changes in boh variables. The relaionship beween error correcion models and co-inegraion was firs poined ou in Granger (). A heorem showing ha co-inegraed series can be represened by error correcion models was saed and proved in Granger () and herefore is called he Granger

22 Represenaion Theorem. Analysis of relaed bu more complex cases is covered by Johansen () and Yoo (). Secion. Esimaing Coinegraion Sysem Besides maximum likelihood esimaion procedure, wih differen represenaions for coinegraed sysems, oher esimaion procedures have been proposed. The mos convenien mehods use he error correcion form, especially when we can assume here is no moving average erm. Two of hese mehods are decribed below. The presenaion of error correcion model: Δ x = δx + Φ Δ x + + Φ Δ x + a (..) * * p p+ naurally leads o a mehod of regression o ge he esimae of δ. Johansen () inroduced a hree-sage regression procedure:. Regress Δx on Δx,, Δx p+ o obain residual marix e.. Regress x on Δx,, Δx p+ o obain residual marix e,.. Regress e on e, o obain he esimae of marix δ.. Then esimae model (..) wih δ fixed a he esimaed value obained in sep o ge esimaes of he * Φ j s. Noice δ= γα ', so by examining he rank of δ hrough is eigenvalues, we can also esimae and es he rank of coinegraing space. Engle and Granger () suggesed anoher esimaion mehod which is called wo-sep esimaor. In he firs sep coinegraion vecor is esimaed. And hen he esimaed coinegraion vecor is used in he error correcion form o esimae he dynamics of he process. These wo seps boh require only ordinary leas squares and he resul is consisen for all he parameers. This esimaing procedure is convenien in he sense ha he dynamics do no need o be specified unil he error correcion srucure has been esimaed, and i also provides some es saisics useful for esing for coinegraion. If he p-dimensional vecor process x' = [ x,, x,,, xp, ] is CI(,) wih single coinegraing vecor, here is a nonzero px vecor α ' = [ c, c,, c p ] such ha α ' x is saionary. Wihou loss of generaliy, say c. Then ( / c ) α ' is also a coinegraion vecor, wih he firs

23 elemen. Thus i is naural o consider he following regression model wih x, as he dependen variable and x,,, xp, he predicors: x = φ x + + φ x + ε.,, p p, This regression is called he coinegraing regression. I aemps o fi he long run equilibrium relaionship wihou worrying abou he dynamics. I provides an esimae of he elemens of he coinegraing vecor. Such a regression has been called a spurious regression by Granger and Newbold () since he sandard errors of he esimaed regression coefficiens are incorrec. So here we only seek coefficien esimaes o use in he second sage of esimaion and for ess of he equilibrium relaionship. Furher discussions abou more general cases of more han one coinegraion vecors can be found in Engle and Granger (). The esimaed coinegraing vecor obained by regression mehod provides a good approximaion o he rue coinegraing vecor because i seeks vecor wih minimal residual variance. Asympoically all linear combinaions of x will have infinie variance excep hose which are coinegraing vecors. A poin need o be made is ha we esimae he coinegraing vecor by normalizing he firs elemen o be uniy. However, we can normalize any nonzero elemen ci and regress xi, on oher variables in esimaing he regression coefficiens. The resuls are invarian of he choice of x i, as he dependen variable in he regression for mos of he cases, bu could be inconsisen someimes. This is a weakness of his approach. Bu due o is simpliciy, i is sill commonly used. In he second sep, he remainder of he parameers of he coinegraed sysem are esimaed by regressing he difference vecor series on is lagged series and he equilibrium error erm z wih α fixed a he esimaed value in he compuaion of z = α ' x. This simplifies he esimaion procedure by imposing cross-equaion resricions and he dynamics of he sysem does no have o be specified in order o esimaeα. Surprisingly, he wo-sep esimaor has excellen properies. As saed in he heorem below, i is as efficien as he maximum likelihood esimaor based on he known value of α. Under some regular condiions he esimaor is asympoically normal. This heorem is firs saed and proved by Engle and Granger ().

24 Theorem The wo sep esimaor of a single equaion of an error correcion sysem, obained by aking ˆα from he coinegraing regression as he rue value, will have he same limiing disribuion as he maximum likelihood esimaor using he rue value of α. Leas squares sandard errors will be consisen esimaes of he rue sandard errors. A simple example will illusrae his esimaion procedure. Suppose wo series are generaed according o he following model: x + β x = u, u = u + ε, (..) x + αx = u, u = ρu + ε, ρ < where ε and ε are whie noise processes. In he usual sense, α and β are unidenifiable since here are no exogenous variables and he errors are conemporaneously correlaed. Aso, noice ha u ~ () I and u ~ () I. By simply rearranging erms, x and x can be expressed as linear combinaions of u and u, so hey are boh I () x. The second equaion suggess ha + α x is a saionary series. Thus x and x are CI (,). We will esimae he parameers by he wo sep approach. Firs sep, a linear leas squares regression of x on x provides a good esimae of α. This is called coinegraing regression. All linear combinaion of x and x excep x + α x defined in he model will have infinie variance. Therefore, i makes sense ha regression of x on x by mehod of leas square will give good esimae of α. For series generaed by model (..), he reverse regression of x on x has he same propery and will give a consisen esimae of /α. Once he parameer α has been esimaed, he ohers can be esimaed in many ways condiional on he esimae of α. Le δ = ( ρ) / ( α β), hen he generaing model can be wrien in he auoregressive represenaion as Δx βδ αβδ x η = + (..) Δx δ αδ x η where he η s are linear combinaions of he ε s and hus are whie noise hemselves. Le z = x + α x. Then model (..) can be wrien in he error correcion represenaion form:

25 Δx βδ η = z +. Δx δ η There are unknown parameers in he original model (..). Now he error correcion form has only unknown parameers lef. Once α is esimaed in he firs sep, here is no consrains in he error correcion model, hus we can ge esimaors for he dynamics sysem by simple regression or MLE. Noice ha when ρ, he series are no longer coinegraed, bu correlaed random walks. CHAPTER - Tes of Coinegraion I is usually of ineres o es wheher a se of variables are coinegraed. This may be desirable because of pracical inquiries such as wheher a sysem is in some form of equilibrium in he long run, and wheher i is sensible o idenify coinegraion before esimaing a mulivariae dynamic model. Unforunaely, he seup of coinegraion sysem renders direc applicaion of likelihood base es impossible. The esing of coinegraion is closely relaed o ess for uni roos in observed series as formulaed by Fuller () and Dickey and Fuller (, ). I is also relaed o he problem of esing when some parameers are unidenified under he null hypohesis as discussed by Davies () and Wason and Engle (). vecor In esing for coinegraion in x, someimes we are paricularly ineresed in a marix or α ' based on some heoreical consideraion. Then we can simply formulae he null hypohesis o es wheher he process z = α ' x conains a uni roo so ha Dickey and Fuller es or Augmened Dickey and Fuller es is applicable. The disribuion in his case is already nonsandard and was obained hrough a simulaion by Dickey (). We will conclude ha x is coinegraed if he null hypohesis of uni roos is rejeced. However, when α ' is unknown and esimaed from he daa, he Dickey-Fuller es ends o rejec he null hypohesis oo ofen. The reasons are ha when he series is no coinegraed, α ' is no idenifiable and ha he variaion of he esimaed α ' is no accouned for.

26 Secion. Seven Types of Tess Suppose he rue sysem is a bivariae linear vecor auoregression wih Gaussian errors where each of he series is individually I() denoed by ( x, y ), Engle and Granger () inroduced seven ypes of ess. Each ype is useful under some assumpions.. CRDW. Afer running he coinegraing regression, he Durbin Wason es is carried ou o see if he residuals appear saionary. If hey are nonsaionary, he Durbin Wason saisic will approach zero and hus he es rejecs non-coinegraion null hypohesis if DW is oo big. This was firs proposed by Bhargava () for he case when null and alernaive are firs order models.. DF. This ess he residuals from he coinegraing regression by running an auxiliary regression as described by Dickey and Fuller. I also assumes ha he model is of only firs order.. ADF. The augmened Dickey Fuller es allows for more lagged erms in he regression and is appropriae o use when higher order lags are needed.. RVAR. The resriced vecor auoregression es is closely relaed o he wo sep esimaor. Based on he esimae of he coinegraing vecor from he coinegraion regression, he error correcion represenaion is esimaed. Then wheher he error correcion erm is significan is esed. Firs order sysem is assumed in his case.. ARVAR. The augmened RVAR es comes wih he same idea as RVAR bu allows higher order sysem.. UVAR. The unresriced VAR es is based on a vecor auoregression in he levels which is no resriced o saisfy he coinegraion consrains.the es is simply wheher he lagged levels would appear a all, or wheher he model can be expressed enirely in changes. This es assumes firs order model.. AUVAR. This is a higher order version of UVAR es. The es saisics of he above seven ypes of ess are saed in Table.. They are all compuable by leas squares. The criical values were esimaed for each saisics by simulaion using, replicaions by Engle and Granger () under he null hypohesis of wo independen I() series. Using hese criical values, he power of he es saisics were compued by simulaions under various alernaives.

27 In he more complicaed bu realisic case ha he sysem is of infinie order bu can be approximaed by a p h order auoregression, he saisics will only be asympoically similar. Therefore, ess, and are asympoically similar if he p h order model is rue, whereas ess,,, and are no asympoically similar since hese ess omi he lagged erms in regression. For his reason, Engle and Granger () suggesed one should no use he laer ess unless firs order assumpion is appropriae. Wheher i is preferable o use a daa base selecion of p for hese esing procedures needs furher invesigaion. Furhermore, by comparing he criical values and powers for he seven ess under firs order sysem and fourh order sysem assumpions, hey decided ha CRDW es is he bes in power for firs order case bu oo sensiive o changes of parameers in he null hypohesis. However, due o is simpliciy, Table. The Tes Saisics: Rejec for large values y = α x + c+ u.. The Coinegraing Regression Durbin Wason:. ξ = DW. Under null hypohesis DW = Δ u = φu + ε.. Dicky Fuller Regression: ξ = τ : he saisic for φ. φ. Augmened DF Regression: Δ u = φu +Δ u + +Δ u p + ε. ξ = τφ. Δ y = β u + ε, Δ x = β. u + γδ y + ε. Resriced VAR: ξ = τ + τ. β β. Augmened Resriced VAR: Same as () bu wih p lags of Δ y and Δ x in each equaion. ξ = τ + τ β β. Unresriced VAR: y, βy βx c ε ξ = [ F + F ] Δ = Δ x = β y + βx + γδ y + c + ε. where F is he F saisic for esing β and β boh equal o zero in he firs equaion; and F is he F saisic for esing β and β boh equal o zero in he second.. Augmened Unresriced VAR: Same as () bu wih p lags of Δx and Δ y in each equaion. ξ = [ F + F ]

28 CRDW is frequenly used as a quick approximae resul. Considering ha realisically, one could no know which criical value o use, he ADF es wih relaive high power and quie consisen criical values for boh firs order and fourh order cases was recommended by Engle and Granger () and has been widely used for esing coinegraion. However, is power is slighly lower han DF es when firs order can be assumed o be rue. The criical values obained by Engle and Granger for CRDW, DF and ADF es saisics are lised in Table.. The criical values lised here have only been esimaed by simulaion for he bivariae case for one sample size and from wo specific models under null hypohesis. More general cases are remained o be discussed. Neverheless, he criical values given in Table. have been used widely as a rough guide in applied sudies. Table. Criical Values: Rejec for large values Firs Order Model: Δ x, Δy independen sandard normal, observaions,, replicaions, p= Criical Values Saisics Type of Tes % % % CRDW... DF... ADF... Higher Order Model: Δ y =.Δ y + ε, Δ x =.Δ x + η ε, η independen sandard normal, observaions,, replicaions, p= Criical Values Saisics Type of Tes % % % CRDW... DF... ADF...

29 Secion. Criical Values Simulaed for oher sample sizes To discuss he effec of sample sizes on he simulaion resuls, criical values of DF and ADF ess are obained by simulaion under he same null hypoheses as in Table. bu wih various sample sizes. The independen series were generaed according o he models under null hypohesis, hen es saisics were calculaed as saed in Table.. The procedure was replicaed for, imes and he ( α) h perceniles were recorded as he criical values. Resuls is shown in Table.. Table. Criical Values for differen sample sizes Firs Order Model: Δ x, Δy independen sandard normal, observaions,, replicaions, p= Type of Tess DF ADF Criical Values Criical Values Sample Size % % % % % % Higher Order Model: Δ y =.Δ y + ε, Δ x =.Δ x + η ε, η independen sandard normal, observaions,, replicaions, p= Type of Tess DF ADF Criical Values Criical Values Sample Size % % % % % %

30 I can be seen from Table. ha he esimaed criical values sabilized owards a limiing value as n approaches and one should be cauious when use he criical values for n= if he sample size is less han. Surprisingly, he criical values for sample size are slighly larger han he criical values provided by Engle and Granger (). Thus, based on criical values in Table., i would be harder o rejec he null hypohesis and hus conclude coinegraion less ofen han based on criical values in Table.. Consequenly, he power of he es is lower. Since he algorihm of simulaion has been checked carefully, i is possible ha Engle and Granger () used slighly differen es saisic from he regression. Secion. Criical Values Simulaed for oher models In realiy, we are no likely o know beforehand which model is appropriae and hus which criical value o use. Hence, an ideal es saisic should be consisen under various kinds of null hypohesis. As he mos widely used coinegraion es DF and ADF, i is desirable o know heir behavior under models wih differen coefficiens, differen lags and differen forms. In his secion, he criical values of ADF and DF ess obained by simulaion under various null hypoheses are abulaed in Table.. Here five models were used as null hypoheses: he firs one has only lag erm wih coefficien. (his is he one used in Table.); he second model includes all lagged erms wih orders lower or equal o ; he hird one conains only lag erm; he fourh model is he same as model wih differen coefficien; and he las one is an inverible moving average model. I can be seen ha he criical values of DF es vary dramaically wih differen models. Since DF es does no include any lagged erm in he es regression, so no surprisingly i is sensiive o changes in lags. For ADF es, adding lower order lagged erms ino he model doesn affec he criical values very much. Bu wih higher order lag, he criical values decrease, so ha i would be easier o rejec null hypoheses and deec coinegraion. Consequenly, if we were o use he criical values from Table. o deec coinegraed sysem wih order higher han four, we migh fail o rejec he null hypohesis someimes. Forunaely, auoregressive ime series wih more han lags are no common.

31 Table. Criical Values for differen models observaions,, replicaions, p=, ε, η independen sandard normal Type of Tess DF ADF Criical Values Criical Values Models under Null Hypohesis % % % % % % Δ y =.Δ y + ε, Δ x =.Δ x + η (. B) y ε Δ =, (. B) Δ x = η Δ y =.Δ y + ε, Δ x =.Δ x + η Δ y =.Δ y + ε, Δ x =.Δ x + η Δ y = (. B)(. B) ε, Δ x = (. B)(. B) η I is worh noicing ha when he independen differenced series under null hypohesis are inverible moving average process insead of auoregression process, he criical values are much larger han hose for model one. Hence, when he rue sysem involves moving average erm, ess based on criical values provided by Engle and Granger () would rejec he null oo ofen wih oo many false posiives. This discussion is sill based on he bivariae case and leaves many quesions unanswered. Criical values for more variables and sample sizes were calculaed by Engle and Yoo () using he same general approach. Research on he limiing disribuion heory by Phillips () and Phillips and Durlauf () migh lead o alernaive approach wih beer performance. We should be cauious if he srucure of pracical ime series is no auoregressive or he es saisics is on he edge of criical values when applying es of coinegraion. CHAPTER - Applicaions o Finance Daa Nowadays, due o economic globalizaion, decisions and aciviies aking place in one par of he world have significan impac for people and communiies elsewhere in he world. This close relaionship among differen pars of he world can be seen in various kinds of economic and financial indexes and crieria.

32 To invesigae he currency relaionship beween Unied Saes and Asia, he monhly log exchange raes of US Dollar (USD) vs Briish Pound (GBP) and he log exchange rae of Chinese Yuan (CNY) vs Briish Pound (GBP) from Jan o Dec were obained. Iniial plos of he wo ime series Figure. and Figure. show ha hey are boh nonsaionary bu share similar rend over ime as seen in Figure. (doed line represens USD, and solid line represens CNY). Figure. Log USDvsGBP exchange rae (-) lusd Figure. Log CNYvsGBP exchange rae (-) lcny Then i was checked ha boh series are I (). ADF es was run for log USDvsGBP exchange rae (denoed by LUSD) wih lag. I gave a -saisic -. which suggess he exisence of uni roo. Running he same es for he firs difference of he series wih lag yielded a -saisic -. indicaing ha firs difference is saionary. For log CNYvsGBP

33 Figure. Log CNYvsGBP and USDvsGBP (-) lcny exchange rae (denoed by LCNY), same ess were used and wo -saisics were -. and -. respecively. Hence boh series are () I. I is of ineres o know if he raio of USD and CNY remains saionary over ime. Then a es for wheher log raio=lusd-lcny (denoed as LDIFF) is saionary or no could be conduced. In his case, he coinegraion vecor for esing is known as (, -). Thus an ADF or DF es on he series of difference beween log USDvsGBP exchange rae and log CNYvsGBP exchange rae would be sufficien. Surprisingly, ADF es wih firs lag gave a -saisic. Figure. The Plo of Residuals of Coinegraing Regression (LDIFF) ldiff ime

34 indicaing ha one canno rejec he null hypohesis, hus he wo series were no coinegraed by vecor (, -). The LDIFF series is ploed o show is behavior. I seems ha here was a gradually drop in he log raio beween USD and CNY from s June ill he end of as shown in Figure.. Therefore, i is possible ha equilibrium exiss bu has been violaed by an even happened around June. By removing he observaions since s June ill now, he remain wo series are sill I (). (ADF ess wih lags for LUSD and LCNY yielded saisics -. and -. indicaing ha boh series are nonsaionary. And DF ess for he firs differenced series gave -saisics -. and -. suggesing ha hey are saionary afer once difference.) LDIFF was esed again for saionariy by ADF es wih firs lag. The es saisic urned ou o be -. suggesing ha one should rejec he uni roo null hypohesis and conclude saionariy. Now LDIFF=LUSD- LCNY is saionary, we can conclude ha LUSD and LCNY are coinegraed wih coinegraion vecor (, -) unil June. I is found ha before, he value of China s Currency, he Yuan has been linked o US Dollar hrough governmen adjusmen. However, in he June of, China's poliical leadership acively menioned he hough of breaking such a link and insead, ying Yuan s value o a group of currencies as Euro, Yen ec. Since hen, he value of CNY has been slowly bu seadily going up causing he log raio of USD over CNY decreased and broke he equilibrium of he pas en years. Alhough losing he equilibrium wih USD, CNY is very likely o be coinegraed wih average values of he dollar, yen, euro and possibly oher currencies like he Briish pound. Oher han currency exchange raes, he performance of sock markes also presens cerain relaionships in economics and finance beween differen disrics of he world. Here, he monhly average of Adjused Close quoes (The adjused close adjuss for dividends and sock splis for he sock and will be a differen number han he close.) of Dow Jones Indusrial Average and he Hong Kong HSI-HANG SENG from Jan o Dec were pu ogeher o check he underlying relaionship beween he wo sock indexes. Alhough hey are boh going up, here is no clear common rend can be deeced from he graph. Hence, hey migh no be coinegraed. Tess were conduced o see if coinegraion can be deeced.

35 Firs, he wo series were aken log o sabilize he variances and ploed ogeher wih each oher. The solid line presens he behavior of log Adjused close quoes of Dow Jones Indusrial Average, while he doed line presens ha of HSI-HANG SENG. Figure. Plos of Log Adjused Close quoes of Dow Jones and His-hang Seng ladjdj They are denoed by LAdjDJ and LAdjHS. Then hey were checked by ADF ess for saionariy. The ADF es wih lag of he original series gave -saisics -. for LAdjDJ and -. for LAdjHS suggesing neiher of hem is saionary; afer he firs order difference, hey were esed again by ADF ess wih lag, which gave a -saisic -. for LAdjDJ and -. for LAdjHS indicaing ha afer firs difference, boh series urned ou o be saionary. Thus, he wo ime series of ineres are boh I (). Coinegraing regression was hen run and DW urned ou o be. which is no even close o he criical value lised in Table.. A regression of he differenced residual series on one lagged residual and lags of he differenced erms was hen run and he resuls of boh regressions are shown in Table.. The ADF es saisic is -.. Based on he criical values lised in Table., we canno rejec he non-coinegraion null hypohesis and hus failed o deec coinegraion beween Log AdjDJ and Log AdjHS. Since all he lagged erms appear o be significian a leas under. level, using DF es o seek for higher power is no appropriae in his case. Furher, when he coinegraing regression was reversed, by regressing Log AdjHS on Log AdjDJ, similar resuls were obained and no coinegraion was idenified. In conclusion, he log Adjused Close quoes for Dow Jones Indusrial Average and ha for Hong Kong HSI-HANG SENG are no coinegraed. I is no so

36 Table. Regression of LAdjcloseDJ on LAdjcloseHS Dependen Independen Variables, esimaes and -sas Variables c LAdjHS Res(-) Δ Res(-) Δ Res(-) Δ Res(-) Δ Res(-) LAdjDJ. (.). (.) Δ Res -. (-.). (.) -. (-.). (.) -. (-.) surprisingly in he sense ha insead of close relaionship beween jus wo sock markes, muually impacs and consrains are expeced as economics globalized. Hence, we migh expec ha if more variables such as he quoes for Shang Hai Sock Marke or Nasdaq were available, hen hey migh be coinegraed. Furhermore, i is clear from he plos in Figure. ha he Hong Kong sock marke is more volaile han he US marke and he invesmen paerns of he wo eareas are differen due o differen ypes of invesors. This may also explain why he wo series are no coinegraed.

37 References And Bibliography Bhargava, Alok (): On he Theory of Tesing For Uni Roos in Observed Time Series, manuscrip, ICERD, London School of Economics. Box, G. E. P. and Tiao, G. C. (): A Canonical Analysis of Muliple Time Series, Biomerika (),,, -. Davidson, J. E. H., David F. Hendry, Frank Srba, and Seven Yeo (): Economeric Modelling of he Aggregae Time-series Relaionship Beween Consumer s Expendiure and Income in he Unied Kingdom, Economic Journal,, -. Davies, R. R. (): Hypohesis Tesing When a Nuisance Parameer is Presen Only Under he Alernaive, Biomerika,, -. Dickey, David A. (): Esimaion and Hypohesis Tesing for Nonsaionary Time Series, PhD. Thesis, Iowa Sae Universiy, Ames. Dickey, David A, and Wayne A. Fuller (): Disribuion of he Esimaors for Auoregressive Time Series Wih a Uni Roo, Journal of he American Saisical Assoc.,, (): The Likelihood Raio Saisics for Auoregressive Time Series wih a Uni Roo, Economerica,, -. Engle, Rober F., and Byung SamYoo (): Forcasing and Tesing in Co-inegraed Sysems, U.C.S.D. Discussion Paper. Engle, Rober F., and Granger C. W. J. (): Co-inegraion and Error Correcion: Represenaion, Esimaion, and Tesing, Economerica, Vol., No. (Mar., ), -. Feller, William (): An Inroducion o Probabiliy Theory and Is Applicaions, Volume I. New York: John Wiley. Fuller, Wayne A. (): Inroducion o Saisical Time Series. New York: John Wiley. Granger, C. W. J. (): Some Properies of Time Series Daa and Their Use in Economeric Model Specificaion, Journal of Economerics, (): Co-Inegraed Variables and Error-Correcing Models, unpublished UCSD Discussion Paper -.