Testing for linear cointegration against nonlinear cointegration: Theory and application to Purchasing power parity

Transcription

1 Deparmen of Economics and Sociey, Dalarna Universiy Saisics Maser s Thesis D 2008 Tesing for linear coinegraion agains nonlinear coinegraion: Theory and applicaion o Purchasing power pariy Auhor: Xijia Liu Supervisor: Changli He 1

2 Absrac In his hesis, we sudy a smooh-ransiion ype of nonlinear coinegraion among a dynamic sysem. Base on he Logisic Smooh Transiion Auoregressive (LSTAR) models, he definiion of coinegraion which is exended form Engle and Granger (1987) s definiion of linear coinegraion is inroduced. Then saisical es for linear coinegraion agains nonlinear coinegraion is derived. The saisical properies of his es are invesigaed and he asympoic disribuion of saisics is obained by Mone Carlo simulaion. We apply his esing heory ino an empirical example which concern dollar/lira real exchange rae. We find ha here is nonlinearly coinegraing relaion in his purchasing power pariy (PPP) dynamic sysem and he PPP puzzle in Hamilon (1994) can be explained by nonlinear coinegraion defined in his essay. Key words: Nonlinear Coinegraion, OLS F es, LSTAR model, Mone Carlo Experimen. 2

3 Conen lis 1. Inroducion Definiion of nonlinear coinegraion The Model: Tesing procedure: Asympoic Disribuion by Mone Carlo Simulaion Empirical Example Purchasing Power Pariy Concluding remarks Reference...16 Appendix: R code

4 1. Inroducion Coinegraion which is inroduced in Engle and Granger (EG) (1987) s aricles is an economeric propery of ime series variables and i has been well applied in modeling macroeconomic ime series in economerics. The EG s definiion of coinegraion have been well developed in linear models for modeling relaionship among economic variable and also have been well applied in empirical problems. However, economic relaionships ofen presen in such nonlinear form as srucural change and regime-swiching. Thus, sudies for exension of his definiion of coinegraion ino nonlinear sysem are necessary. Recenly, many researches which aemped o exend EG s coinegraion ino nonlinear form have been sudied. The well known example of his approach is o manifess iself in he concep of hreshold coinegraion and is smooh versions sudied by Balke and Fomby (1997) and o employ a similar model o invesigae he purchasing power pariy by Enders and Falk (1998). In his essay, my goal is o es he null hypohesis of linear coinegraion agains nonlinear coinegraion. Firsly, we raised a definiion of nonlinear coinegraion which is developed from he EG s definiion of linear coinegraion and propose a esing procedure o es ha here exis linear coinegraion agains here exis nonlinear coinegraion in he dynamic sysem. For a nonlinear form, we adop one of he mos popular nonlinear dynamic models, he smooh ransiion auoregressive (STAR) model, which ness a linear auoregressive model and conains a regime-swiching srucure. For a deailed discussion of STAR models, see Terasvira (1994). Therefore, he coinegraing vecor can be specified in a form of a smooh ransiion funcion if here is a regime-swiching in he sysem. For esing procedure, we apply he mehod which is raised by He and Sandberg (2005a) o reduce he model ino linear form and employ he auxiliary regression o se up esing procedure. And hen, he able of criical F value will be produced by Mone Carlo simulaion experimen. A las, we will apply he definiion and esing procedures ino he empirical problem which concern he heory of purchasing power pariy (PPP). From Hamilon (1994), we can see ha he null hypohesis of no linear coinegraion of EG s (1987) can no be rejeced a 5% significan level when applying a residual-based esing procedure o he dollar/lira real exchange rae. In his essay, we sugges ha he PPP puzzle could be solved by he nonlinear coinegraion which is inroduced in previous secion. This essay is organized as follow: Secion 2 inroduces he definiion of his smooh ransiion ype nonlinear coinegraion; Secion 3 presens he model; Secion 4 inroduces he procedure of esing for linear coinegraion agains nonlinear coinegraion; Secion 5 presens he resul of asympoic disribuion properies by Mone Carlo simulaion; Secion 6 shows he sudying resuls of an empirical example abou PPP; Concluding remark will be illusraed a las secion. 4

5 2. Definiion of nonlinear coinegraion Engle and Granger (1987) have raised a definiion of coinegraion which conains a linear combinaion of nonsaionary variables and describes a long-run equilibrium relaion exising in a linear dynamic sysem. In his secion, I will generalize his concep o nonlinear coinegraion which based on a nonlinear saionary sysem. Definiion: Le y = ( y 1,..., y )' be he n-dimensional random vecor and for each n = 1,2,...,, ~ () 1, where he definiion of I ( 1) i n y linear I Granger (1987) s definiion. Then i y is said o be nonlinear coinegraion if here exiss a ime-varying follows from he Engle and vecor ( ) α α α α =,,..., ' such haα ' y ~ I(0). 1 2 n Where, ime-varying vecor should saisfy following condiion: (1) The firs iemα 1 should be a nonzero-consan such ha is can be normalized ( ) as α 1, α,..., α ' ; = 2 n (2) α, i = 2,..., nare well defined funcion of a random variable S such ha for each i, i has a logisic smooh ransiion form: αi = αi + Gi α i ( { } (3) Where, G 1 exp γ ( S c ) i i i i 1 ) = + α i, γ i and c are parameers, γ 0 i i The ransiion variables S i are weakly saionary or deerminisic variables. Here, α is called nonlinear coinegraing vecor. 3. The Model: In his essay, we should consider he rinomial sysem since we will apply he resul o he ppp daa. The model form as follow: 2 y1 = α2y2 + α3y3 + ε1 ε1 0 σ1 σ12 σ 13 2 y2 = y2 1+ ε2 ε = ε2 ~ N 0, σ2 σ23 2 y3 = y3 1+ ε 3 ε 3 0 σ 3 αi = αi + G γi, Δ yi d, ci, i= 2,3 Where ( ) The ransiion funciong( γ, Δy, c i i d i ) is defined as follows: 5

6 ( ) 1 1 ( γi i d i) { γi( i d i) } G, Δ y, c = 1+ exp Δy c, i= 2,3 2 The ransiion funcion is called he logisic smooh ransiion funcion and he model is called nonlinear coinegraion wih logisic smooh ransiion funcion. Where, γ i is a parameer which deermines he speed of ransiion from one regime o anoher a ime c and he speed of ransiion will be faser wih he increasing ofγ i. And we choose Abou he delay parameer d of ransiion variable Δ yi d Δ yi d as ransiion variable., i should be larger han 1 a leas since if d=0, hen here will be specificaion problem (see Tsay (1989)). In applicaion, he delay parameer d needs o be deermined from he daa. In his essay, we should apply he mehods which have been discussed by Tsay. Tsay (1989) suggess ha firs selec he order of auoregression p and hen deermine d by varying i and choosing he value minimizing he p value of his lineariy es. In he esing procedure, we fixed he delay parameer a Tesing procedure: In his essay, my goal is o es he null hypohesis of a linear coinegraion agains nonlinear coinegraion. The null hypoheses can be expressed as he following parameer resricion: H : γ = 0, γ = From he model which was menioned above, we can see haγ 2 = 0, γ 3 = 0 implies ha he ransiions funcion G( γ i, yi, 1, ci) Δ = 0. Tha is he model will degenerae ino linear model. ( i i, 1 i) Then we will apply Taylor expansion of gamma around 0 in ransiion funcion G γ, Δy, c. There are wo meris of his mehod o apply Taylor expansion o ge he auxiliary regression. Firs we don need o esimae he model before he esing procedure. Secondly he compuing echnical will be simplified. However, we should keep in mind ha he firs-order expansion will lead o low power if he ransiion akes place only in he inercep (see Luukkonen e al.(1988) and He and Sandberg (2005a)). The firs and hird-order Taylor expansion are as follows: ( Δy, 1 c ) γ i i i G1, i, ( γ i, Δ yi, 1, ci) = + r1 ( γ i) (4.1) 4 3 ( Δy, 1 c ) γ ( Δy, 1 c ) γi i i i i G3, i, ( γ i, Δ yi, 1, ci) = + + r3 ( γ i) (4.2) 4 48 Pu he above equaions ino he model which is under he aleraive hypohesis, and hen we can ge he auxiliary regression as follow: * ( )' ϕ ( )' ϕ μ y = y h + y h + (4.3) 1, 2, 2, 2 3, 3, 3 3 6

7 Where he parameers are defined as follows: For order I, m=1: h = ( 1, Δ y )' ; h3, = ( 1, Δ y3, 1) ' ; ϕ2 = ( ϕ20, ϕ21) '; ϕ3 ( ϕ30 ϕ31) 2, 2, 1 For order III: = ( 1, Δ, Δ, Δ )'; h3, = ( Δy3, 1 Δy3, 1 Δy3, 1 ) h y y y 2, 2, 1 2, 1 2, 1 = ( ) ( ) ϕ ϕ, ϕ, ϕ, ϕ '; ϕ = ϕ, ϕ, ϕ, ϕ ' * H0 : ϕ = 0, i > 0; m= 1, Then he corresponding auxiliary null hypoheses are: m mi 1,,, ' ; 5. Asympoic Disribuion by Mone Carlo Simulaion =, '. (1) We should define he daa generaing funcion. Under he null hypoheses, he model can be reduced as linear form, and hen we can ge he daa generaing funcion as follow: y = α y + α y + μ y2, = y2, 1 + μ2, y3, = y3, 1 + μ3, 1, 2 2, 3 3, 1, u ~ iid N 0,1 i= 1, 2,3 where ( ) i We can generae he daa wih desired sample size according o above funcion. (2) Given he auxiliary regression, we employ OLS o esimae he parameer and compue he value of F saisic according o following formula: T = ( ' ) 1 ' 1, Where, X ( h2,, h3, ) b X X X y =, ( )'( )/( ) 2 s y1 X bt y1 X bt T k =, 1 T / 2 1 ( )' ( ' ) ' ( ) F = Rb r s R X X R Rb r m, Where, R is a diagonal marix and r is a vecor which value is fixed according o he null hypoheses. Then repea sep (1) and (2) imes o simulae he asympoic disribuion of OLS F saisic. Le T=1,000,000 and replicaions are 10,000 o ge asympoic disribuion for F saisic. The simulaion resuls are repored in Table 1 and Table 2 (5.1) 7

8 Table 5.1 Probabiliy ha F saisic is greaer han enry T 99.0% 97.5% 95.0% 90.0% 10.0% 5.0% 2.5% 1.0% Criical values for F es, Order I Table 5.2 Probabiliy ha F saisic is greaer han enry T 99.0% 97.5% 95.0% 90.0% 10.0% 5.0% 2.5% 1.0% Criical values for F es, Order III Figure 5.1 Simulaion, ime=10000, when T=50 8

9 Figure 5.2 Sander F disribuion wih df (6,42) We compare he hisograms of he OLS F es wih sandard F (6, 42) es in he above graphs. We can see ha he sandard F (6, 42) disribuion urn ou o be heavier skewed o lef. 9

10 6. Empirical Example Purchasing Power Pariy 6.1 Theory of PPP The purchasing power pariy (PPP) heory which is a classical economic example of coinegraion uses he long-erm equilibrium exchange rae of wo currencies o equalize heir purchasing power. The basic idea of purchasing power pariy is firs inroduced by Gusav Cassel in I is based on he law of one price: he heory ha, in an ideally efficien marke, idenical goods should have only one price. A purchasing power pariy exchange rae equalizes he purchasing power of differen currencies in heir home counries for a given baske of goods. I is ofen used o compare he sandards of living beween counries, raher han a per-capia gross domesic produc (GDP) comparison a marke exchange raes. There is obvious difference beween purchasing power pariy exchange rae and marke exchange rae, because he marke exchange rae flucuaes frequenly according o he need of he marke. While many economiss believe ha purchasing power pariy exchange raes are characerized by a long-run equilibrium. In his essay, we will discuss he PPP heory base on he daa se of dollar/lira real exchange rae. Given fundamenal idea of he PPP heory, we know ha: P=P S where P denoes he domesic price index P is he foreign price index and S is he exchange rae beween hese wo counries. Thus, we ake logarihm s of boh sides of P=P S and urn i ino plus form: p=p +s. I is o say ha his dynamic daa se should be modeled as coinegraion model by coinegraing vecor (1,-1,-1). However, from he research in Hamilon s, i expresses ha he heory of purchasing power pariy is no saisfied in he example of Unied Saes and Ialy. This is widespread disagreemen of he purchasing power pariy and his problem is called PPP puzzle. We will explain he PPP puzzle in more deails in secion 6.3. Then, in las wo secions, we apply he esing procedure which is inroduced in previous secion o es here is linear coinegraion agains nonlinear coinegraion in his dynamic sysem and esimae he coinegraion model by NLS so ha we can check he residuals of model wheher saisfy he condiion which we expeced. Anoher problem abou he price level of cerain counry should be menioned. I is naure o use price level o measure he purchasing power in each counry. We can say ha he purchasing power of cerain counry decrease if he price level growh. Bu in fac here is no uniform price level can be provided o measure he purchasing power among differen counries. Thus i is necessary o compare he cos of baskes of goods and services consumed by people from differen counries using a price index and he consumer price index organized monhly is he bes choice o assess he purchasing power. 6.2 Daa descripion In his daa se, P denoes a price index in Unied Saes (in dollar per good) P is a price index in Ialy (in lira per good) and S is he rae of exchange beween hese wo counries. For analysis objecives, he raw daa are ransformed a firs as presened: 10

11 ( ) ( ) p = 100 log P log P 1973:1 ( ) ( ) p' = 100 log P' log P' 1973:1 ( ) ( ) s = 100 log S log S 1973:1 Such ransiion is o make sure ha above hree ime series have same saring values, zero a January Muliplying by 100 roughly represens he percenage difference beween he value a curren ime poin and he saring value a January Figure 6.1 plos he monhly daa of he price level and he exchange rae Ocober 1989 afer he ransiion. s p in Unied Saes, he price level p ' in Ialy beween dollar in Unied Saes and lira in Ialy from January 1973 o Figure 6.1 Monhly daa of p, s and p ' from 1973 o 1989 afer ransiion (Green is p ', Red is p ', Blue is ) s 6.3 PPP puzzle in Hamilon s book Given he fundamenal idea of PPP heory we have ha he real exchange and purchasing power is equilibrium in a longer erm. I is o say P =P S where P denoes he domesic price index P is he foreign price index and S is he exchange rae beween hese wo counries. Thus, we ake 11

12 logarihm s of boh sides of P =P S and urn i ino plus form: p =p +s. I is o say ha his dynamic daa se should be modeled as coinegraion model by coinegraing vecor (1,-1,-1). However when we consider his model ino dollar/lira real exchange rae daa se, we found ha i is no he case. Under ha daa se and given he coinegraing vecor, we compue he z =p -p -s and plo he rend figure as follow: Figure 6.2 Time series plo of residuals for coinegraion model From figure 6.2, we could suspec ha z is no nonsaionary. Then afer he esing by Dickey Fuller es and Philips Peron es, we can ge he common resul ha is we can rejec he null hypohesis, z is saionary. Thus, he resul by esing is opposie o he PPP heory. 6.4 Tesing for nonlinear coinegraion agains nonlinear coinegraion in PPP Afer sudying he Hamilon (1994) s empirical example of purchasing power pariy beween Unied Saes and Ialy from 1973 o 1989, we apply he definiion of nonlinear coinegraion and go furher analysis o check wheher here is any coinegraing connecion in he sysem. I) Tes p, p', sare all I(1) process: H p, p ', s are all nonsaionary. 0 : We apply augmened Dickey-Fuller es o prove ha we can no rejec he null hypohesis of 12

13 nonsaionary. The resuls of he esimaion and he calculaion of he es saisics are saed in deails in Hamilon (1994) s. Nex sep, we apply he esing procedure in secion 4 o check if here exis nonlinear coinegraion vecor such ha a nonlinear coinegraion which was defined in secion 2 can be expressed in his dynamic sysem. II) Tes linear coinegraion agains nonlinear coinegraion: H : γ = γ = Given he hird order auxiliary regression, we apply OLS o fi resriced regression and non-resriced regression respecively: Resriced regression: p = 1p' + 2s + β β μ Non-resriced regression: p = ϕ p' + ϕ p' Δ p' + ϕ p' Δ p' + ϕ p' Δ p' + ϕ s + ϕ sδ s + ϕ sδs ϕ sδ s + μ 3 * 30 1 Compue he sum square of he residuals for hese wo regressions respecively, and hen ge he value of F saisics: ( 0 1) / RSS / ( T k ) RSS RSS m F = = Compare wih criical value of F is 3.45 in able 5.2, we can rejec he null hypohesis which is here is linear coinegraion in his dynamic sysem a 1% level. By furher invesigae, we found ha for differen delay parameer we all can rejec he null hypohesis, bu we can ge he max F value if deermine he delay parameer d a 1. Thus, we will deermine d as 1 o esimae he nonlinear coinegraion model in nex secion. 6.5 Model esimae By NLS regression, we ge he esimaion of model. The esimaion of nonlinear coinegraion model as follow: ( 1 ( 1 exp { γ1( ' 1 1) }) ) ' 2 1 exp{ γ2( 1 2) } ( ( ) ) 1 1 Z = p a + + Δp c p + a + + Δs c s γ = 0.750, γ = 0.145, c = 3.648, c = , a = 0.398, a = where,

14 Figure 6.3 Residuals of model ime series plo Figure 6.3 shows he ime series plo for residuals of nonlinear coinegraion model which we esimaed above. Afer comparing wih Figure 6.2 and 6.3, we can see ha he residual of coinegraion model could be saionary. In oher words, he dynamic sysem of purchasing power pariy could be modeled as a coinegraion model by he corresponding coinegraing vecor. The nonlinear coinegraion which we defined in his essay may be a soluion for PPP puzzle. 14

15 7. Concluding remarks In his essay, we raised he definiion of nonlinear coinegraion which generalized Engle and Granger (1987) s definiion of linear coinegraion. In his definiion of nonlinear coinegraion, we are base on he Logisic Smooh Transiion Auoregression model o represen he nonlinear coinegraion sysem by riangular represenaion. Given his definiion, we sugges a esing procedure o es linear coinegraion agains nonlinear coinegraion. Under he definiion and esing procedure, we ge he criical value of F saisic able and asympoic disribuion by Mone Carlo simulaion. Compared wih sandard F disribuion, we can see ha he OLS F saisics disribuion is heavier skewed o lef. In empirical problem applicaion, we consider he Purchasing Power Pariy and employ he esing procedure and criical value able which we work ou before o analyze dollar/lira real exchange rae. From he esing and model esimaion, we can see ha here is no linear coinegraing relaion in he PPP sysem, however, here is our nonlinear coinegraion evidence suppored by our empirical analysis for he PPP example. I is suggesed ha he PPP sysem may conain nonlinear feaures beween he economic variables which could cause he PPP puzzle in Hamilon (1994). 15

16 8. Reference Hamilon, J. D. (1994), Time series analysis. Princeon Universiy Press Balke, N. and T. B. Fomby (1997), Threshold Coinegraion. Inernaional economic. Enders W. and B. Falk (1998), Threshold-auoregressive, median unbiased, and coinegraion ess of purchasing power pariy. Inernaional Journal of Forecasing. Engle, R. F. and C. W. J. Granger (1987), Co-inegraion and error-correcion: Represenaion, esimaion and esing. Economerica He,C.and Rickard Sandberg. (2006) Dickey-Fuller Type of Tess agains nonlinear Dynamic Models Oxford Bullein of Economics and Saisics, 68, supplemen (2006) He, C., T. Terasvira and A. Gonzlez (2007, Tesing parameer consancy in vecor auoregressive models agains coninuous change. Economeric Reviews Johansen, S. (2006), Coinegraion: An overview. Palgrave Handbooks of economerics: Vol. 1 Economeric Theory. Palgrave MacMillan Tsay, R. (1989), esing and modeling Threshold Auoregressive Processes, Journal of he American Saisical Associan. Timo Terasvira (1994), Specificaion, Esimaion, and Evaluaion of Smooh Transiion AuoRegressive Models, Journal of he American Saisical Associan. 16

17 Appendix: R code (1) se up funcion o generae daa and compue f saisics: sim=funcion(=25,a=1,b=1,d=0){ d1=5-d # daa generaion under H0 d2=-1-d =+5 # check he delay parameer d, and he diff_y3=c(diff_y30[d1:d2]) firs five value is... yy3=diff_y3*y3[6:] y2=rep(0,) yy31=diff_y3^2*y3[6:] y2[1]=rnorm(1) yy32=diff_y3^3*y3[6:] for(i in 2:){ ee=rnorm(1) y11=marix(y1[6:],-5,1) y2[i]=y2[i-1]+ee } y22=y2[6:] y3=rep(0,) y33=y3[6:] y3[1]=rnorm(1) for(i in 2:){ x=cbind(y22,yy2,yy21,yy22,y33,yy3,yy31,yy ee=rnorm(1) 32) y3[i]=y3[i-1]+ee b=solve((x)%*%x)%*%(x)%*%y11 } rr=marix(c(a,0,0,0,b,0,0,0),8,1) y1=rep(0,) s=((y11-x%*%b)%*%(y11-x%*%b))/(-8) for(i in 1:){ F_v=(b-rr)%*%solve(s[1]*solve((x)%*%x) ee=rnorm(1) )%*%(b-rr)/6 y1[i]=a*y2[i]+b*y3[i]+ee } reurn(f_v) # Compue he F Value } diff_y20=diff(y2) # simulaion: d1=5-d F_v=rep(0,10000) d2=-1-d for(i in 1:10000){ diff_y2=c(diff_y20[d1:d2]) F_v[i]=sim(25,1,1,0) yy2=diff_y2*y2[6:] } yy21=diff_y2^2*y2[6:] his(f_v,breaks=40) yy22=diff_y2^3*y2[6:] q_val=quanile(f_v,probs=c(99,97.5,95,90,10, 5,2.5,1)/100) diff_y30=diff(y3) q_val (2) Tesing for nonlinear coinegraion for PPP daa se p1=100*(log(pzunew)-log(pzunew[1])) p2=100*(log(pc6it)-log(pc6it[1])) dd=1 s=100*(log(1/exritl)-log(1/exritl[1])) diff_p2=diff(p2) c=lengh(diff_p2)-dd PPP.daa=s(cbind(p1,p2,s)) diff_p2=diff_p2[1:c] plo(ppp.daa,plo.ype="single",col=2:4) g=dd+2 abline(h=0, col = "gray60") pp2=p2[g:lengh(p2)] 17

18 p22=diff_p2*pp2 p23=diff_p2^2*pp2 p24=diff_p2^3*pp2 diff_s=diff(s) diff_s=diff_s[1:c] ss=s[g:lengh(s)] s2=diff_s*ss s3=diff_s^2*ss s4=diff_s^3*ss pp1=p1[g:lengh(p1)] lm0=lm(pp1~pp2+ss-1) lm1=lm(pp1~pp2+p22+p23+p24+ss+s2+s3+s 4-1) rss0=sum(residuals(lm0)^2) rss1=sum(residuals(lm1)^2) F_val=((rss0-rss1)/6)/(rss1/(lengh(pp1)-8)) F_val (3) esimaion for model by NLS regression diff_p2=diff(p2) c=lengh(diff_p2)-dd pp2=diff_p2[1:c] g=dd+2 p22=p2[g:lengh(p2)] diff_s=diff(s) ss=diff_s[1:c] s2=s[g:lengh(s)] pp1=p1[g:lengh(p1)] mm1=nls(pp1~(a1+1/(1+exp(-gamma1*(pp2- c1))))*p22+(a2+1/(1+exp(-gamma2*(ss-c2)))) *s2,daa=d,sar=lis(a1=1,a2=1,gamma1=0.5, gamma2=0.5,c1=1,c2=1)) summary(mm1) rr=residuals(mm1) plo.s(rr) abline(h=0) 18