Cointegration Analysis of Exchange Rate in Foreign Exchange Market

Transcription

1 Coinegraion Analysis of Exchange Rae in Foreign Exchange Marke Wang Jian, Wang Shu-li School of Economics, Wuhan Universiy of Technology, P.R.China, Absrac: This paper educed ha he series of exchange raes prices show heir firs-order difference saionary by he uni roo es. On he basis of he resul of uni roo es, we focuses on he issue of empirical analysis of coinegraion relaionship wih hese exchange raes, and find ou ha here is o exis long-erm sabiliy relaionships wih hem. The paper adops hisorical daa of daily closing prices of CAD/USD,GBP/USD, CF/USD currencies and carries ou uni roo es and coinegraion es. The empirical resul shows here is o exis coinegraion relaionship among he currencies, and here is o exis long-erm sabiliy relaionship. This means ha economy in he hree counries has a very close relaionship wih American economy. The sudy will benefi invesors o make decision on heir porfolio and o effecively avoid risk. Key words: Coinegraion analysis, Foreign exchange marke, Exchange Raes, Uni roo es Inroducion The correlaion of wo arbirary reurn series is impossible o exis if he wo series can be wihou relaionship. owever, he correlaion coefficien of he wo series can be go by measure formula. In fac, he correlaion coefficien is of no significance, moreover, he relaiviy is very insabiliy in he siuaion. The esimae of relaiviy has significance under he condiion ha he differen kinds of exchange raes have he long-erm sabiliy relaionship. Now he maor banks in China pu ino operaion he FX opions whose currencies based on dollars are he developed indusrialized counries. So he paper inroduces Coinegraion Theory (Engle & Granger, 987) o research coinegraion relaionship abou he several indusrialized counries currencies in order o find wheher hey have he long-erm sabiliy relaionship. Because he coinegraion relaionship can be depiced ino same rend, he sysem exiss long-erm sabiliy relaionship if he coinegraion relaionship of sysem Consising of several variables holds. And o esimae correlaion coefficien of reurn series is has significance only exising he long-erm sabiliy relaionship. Then his can benefi invesors o make decision on heir porfolio and o effecively avoid risk. Some scholar sudied he coinegraion problem of exchange raes of he maor developed indusrialized counries (akkio & Rush, Baillie & Bollerslev, Diebold, 99, 994, 994), bu heir conclusions are muual conradicion. The paper seleced CAD/USD,GBP/USD, CF/USD currencies in foreign exchange rading marke of home bank and carried hrough coinegraion research o find wheher hey have he long-erm sabiliy relaionship. 2 Research model 2. Uni roo es The precondiion of coinegraion es is ha ime series is insabiliy and has uni roo. So we firs carried ou he uni roo es on ime series of logarihmic price of exchange raes. The mos commonly used mehods of uni roo es are ADF and PP. 2..Augmened Dickey-Fuller uni roo es Dickey & Fuller(979)firsly pu forward DF es on he basis of firs order auoregression, and he expression is in equaion (). y () y As he developmen of DF es,dickey & Fuller (98) pu forward problem of higher order 595

2 ε. And he several lagged erms of y are added ino he equaion series relaion o emend error erm (). The es assumes daa generaion procedure of y is he process of AR(p), is expression of model is in equaion (2). y = + y + y + + y α ρ ξ ξ + ε L p p+ (2) 0 : ρ = 0 A : ρ 0 y has uni roo process of random walk and is insabiliy if null hypohesis comes ino exisence; y is sabiliy if null hypohesis is refused PP non-parameric es PP Non-parameric Tes is pu forward and developed by Philips & Perren in 988, which is a kind of uni roo es wih non-parameric mehod and doesn demand he error erm ε obeys independen idenical disribuion. I solves problem of higher order series relaion o emend error erm by saisic of firs order auoregression coefficien. Considering equaion () of firs-order auoregression process, he es has emended saisic of significance of coefficien ρ in equaion (). And he null ρ = hypohesis is he same as ADF es, i.e., 0, and he series exiss uni roo. Adoping Newey-Wes heeroskedasiciy esimae, he es saisic is / 2 2 q T ρ0 ρ ( ϖ ρ0) TSρ 2 PP = ϖ = ρ0+ 2 ( ) ρ ρ = ˆ ε ˆ ε ϖ 2ϖσ ˆ =, where, q+ T = +,and. ρ S and ρ ρ are respecively es saisic of he coefficien and sandard error. And σˆ is sandard error of esimaion in es equaion. T is ime span of es. q is lagged ails of inercepion. The null hypohesis 0 is acceped when PP is less han he criical value. And he series has uni roo and obeys random walk. Vice versa, he null hypohesis 0 is refused, and he series is no uni roo. 2.2 Coinegraion es The process of uni roo has no long-erm mean and no rend o reurn mean since he process has insabiliy. Is fuure value or presen value depends on all is pas. Thus any is innovaion will produce lasing impac. So one process of uni roo is imposible o describe he long-erm sabiliy relaionship. owever, a sysem consis of several variables having uni roo process, and if one linear combinaion of he variables has saionary, he saionary linear combinaion is named as coinegraion relaionship which describes he long-erm sabiliy relaionship of hese variables and delineaes he saionary characerisic of sysem. Currenly, one linear combinaion consising of he variables having I(d) process can make d decrease, hen he linear combinaion is called coinegraion relaionship. Now coinegraion research based on d= has been evolving since mos ime series of economic I(). And coinegraion relaionship is equivalen o long-erm sabiliy variables have process of relaionship among relaed lieraures abou coinegraion research. The paper adops Johansen coinegraion es, and coinegraion likelihood raio assumes 0 : a mos have r coinegraion relaionships; : have n coinegraion relaionships, i.e., full rank. Traced saisic of es Q r = T n i= r+ log( λ ) i λ, where, i is he ih eigenvalue. In fac, raced es ε 596

3 is exension of DF uni roo es. And alernaive hypohesis indicaes here is no any coinegraion relaionship since coinegraion vecor marix is impossible o be full rank, i.e., here are impossible o exis n coinegraion relaionships in he process of coinegraion es on sysem consising of n series. 3 Empirical analysis Using he above heory, and adoping hisorical daa of closing prices of CAD/USD,GBP/USD, CF/USD currencies and carrying ou uni roo es of logarihmic price of hose exchange raes, hen carrying hrough empirical analysis of coinegraion relaionship o find wheher hese logarihmic price series have long-erm sabiliy relaionship. 3. Daa Sample space adops hisorical daa of daily closing prices of CAD/USD,GBP/USD, CF/USD currencies from January, 999 o Sepember 8, 2003, need 228 observaions of daily closing prices. One currency daily logarihmic reurn is compued by logarihm difference, i.e., ri, = ln Pi, ln Pi, where P i, is closing price a he h rade day. 3.2 The resul of coinegraion es on exchange raes 3.2. The resul of uni roo es on logarihmic prices of exchange raes Three logarihmic prices series of CAD/USD,GBP/USD, CF/USD currencies have been carried hrough uni roo es, is resul shown in Tab.. Since daa figure of logarihmic prices of he hree currencies are similar o random walk, here are no ime rend in assumed es equaion of ADF and PP. From Tab., hree logarihmic prices series of CAD/USD,GBP/USD, CF/USD currencies have he same resul of exising uni roo and obeying random walk wih ADF es or PP es The resul of coinegraion es The vecor consising of hree series of CAD/USD,GBP/USD, CF/USD currencies can been I() carried hrough coinegraion es since he logarihmic prices series have process, i.e., exis uni roo. And among he given coinegraion vecor has no ime rend since each of series has no eviden ime rend in ADF es. To adop Johansen s maximum likelihood race es, is resul shown Tab. 2. The values of likelihood raio in Tab.2 show only he firs null hypohesis is refused under significance level 0.05, i.e., us one coinegraion relaionship for only he firs saisic of likelihood raio more han criical value under significance level And Johansen s coinegraion es sars on having no any coinegraion relaionship, hen a mos one coinegraion relaionship, ill a mos n coinegraion relaionship. Toal n imes ess have been carried hrough and alernaive hypohesis is invarian where n is variable number. Bu here is no coinegraion relaionship under significance level 0.0. Is esimaed values γˆ of weigh of corresponding coinegraion vecor are shown in Tab.3 From he above es resul, here is o exis coinegraion relaionship among CAD/USD,GBP/USD, CF/USD currencies under significance level This means ha economy in he hree counries has a very close relaionship wih American economy, and here is o exis long-erm sabiliy relaionship. Thus he correlaion coefficien among hem has significance. 597

4 Tab. ADF and PP es resul of he hree currencies(significance level is 0.05) Series ADF Saisic ( = 0 α ) Criical value Conclusion PP Saisic ( = 0 α ) Criical value Conclusion CAD/USD ADF(2) =-0.69 (-0.84) (-.94) I() (-0.94) (-.94) I() GBP/USD ADF(2) =-.65 (-0.29) (-.94) I() -.78 (-0.37) (-.94) I() CF/USD ADF(4)=-.28 (-0.26) (-.94) I() -.4 (-0.2) (-.94) I() Noe: p in ADF(p) is lagged orders and is selecion based on value of AIC and D.W. α = 0 shown no inercep in ADF equaion,,brackeed daa of corresponding row is value of ADF saisic when having no inercep. brackeed daa of row of criical value is criical value whenα = 0. Tab. 2 Johansen coinegraion es resul Eigenvalue Likelihood raio Significance level 0.05 Significance 0.0 level Assumed coinegraion number( * a mos one a mos wo Noe:* denoes o refuse he null hypohesis under significance level Tab.3 Esimaed value of coinegraion coefficien afer regularizaion CAD/USD GBP/USD CF/USD (0.32) (0.28) null ) 4 Conclusion Noe: brackeed value is asympoic sandard error of Esimaed value of coinegraion coefficien The paper adops ADF es and PP non-parameric es o udge wheher logarihmic prices series of exchange raes have saionary process, i.e., uni roo process. Then he paper inroduces Coinegraion Theory o research coinegraion relaionship o find wheher here is long-erm sabiliy relaionship among exchange raes. The conclusion from empirical resul as follows: Firsly, he hree daily logarihmic prices series of exchange raes have non-saionary process and exis uni roo using ADF es and PP non-parameric es. And he series are no saionary process having reurned o mean value. 598

5 Secondly, Johansen coinegraion es resul shows here is o exis coinegraion relaionship among he currencies, and here is o exis long-erm sabiliy relaionship. Thus o measure is covariance is significan. The research will benefi invesors o make decision on heir porfolio and o effecively avoid risk. A he same ime, his shows ha economy in he hree counries has a very close relaionship wih American economy. Reference [] Baillie R T, Bollerslev T. Coinegraion, Fracional Coinegraion, and Exchange Rae Dynamics[J]. Journal of Finance, 994, 49: [2] akkio C S, Rush M.. Coinegraion: ow shor is he Long Reurn[J]. Journal of Inernaional Money and Finance, 999, 0: [3] Dickey, D.A. and Fuller, W.A. Disribuion of he Esimaors for Auoregressive Time Series wih a Uni Roo [J]. Journal of he American Saisical Associaion, 979, 74: [4] Diebold F X, Gardeazabal J, Yilmaz K. On Coinegraion and Exchange Rae Dynamics[J]. Journal of Finance, 994, 49: [5] Engle, R.F. and Granger, C.W.J. Coinegraion and Error Correcion: Represenaion, Esimaion and Tesing [J]. Economerica, 987, 55: [6] Philips, P.C.B. and Perren, P. Tesing for Uni Roos in Time Series Regression [J]. Biomerika, 988, 75: