多利率假说的协整检验

Jour nal of Sy stems Science and Systems Eng ineering2000,V o l.9,No.2,pp.227-235Cointegration Test of Expectation Hypothesisin Multi-Interest-Rates SystemC HAO J ian-xiongInternational Business School,Hunan University,Changsha410082,ChinaAbstract:　Th e expectation h ypoth esis of interest rate term s tructure th eory posits an implication:in a sys-tem of p in teres t rates of differen t matu rity there sh ould be one com mon trend driving interes t rates,corre-sponding to p-1cointegration vectors,and in each of these v ecto rs the coefficien ts sh ould s um to zero,giv-ing s tationary interest rate sp reads.This paper analyses the cointeg ration implications of th e ex pectation hypothesis on UK interes t rates.Tw o types of tests hav e been carried out in the analysis.The firs t is th e tes t of the implication that pairs oftwo interes t rates cointegrate into stationary s preads.Th e second is the tes t of th e coin teg ration implicationson a full s ystem of in teres t rates of different matu rity.This analysis h as been carried ou t using the V ARtech nique.Th e resu lts in gen eral favou r th e expectation hypothesis:th e UK term structure of in teres t rate isdriven b y on e comm on stochas tic trend,and th e in teres t rate s preads are g enerally found to be s tationary.Th e v alidity of the expectation hypo thesis s uggests th at th e central bank can have a direct effect on long in-teres t rates b y operating at the sh ort end of th e market.Keywords:　cointeg ration;expectaction h ypoth esis;interes t rates1　IntroductionThe m ost popular theo ry used to describe the term structure o f interest ra tes,the ex pecta-tion hypothesis with co nsta nt term premia,implies cointeg ratio n betw een long-term inter-est rates and short-term interest rates(Campbell and Shiller,1987).Suppose that lo ng rate R t and sho rt ra te r t a re both integ rated to order o ne,R t～I(1), r t～I(1).Then R t and r t are said to be cointeg ra ted if there exists a U such tha t R t-U r t is sta tio nary,R t-U r t～I(1).This is denoted by saying R t a nd r t a re CI(1,1).W ha t this m eans is that the regression equa tio nR t=U r t+X tm ake sense because R t and r t do no t drift too far apart fro m each other ov er tim e.Thus there is a long-run equilibrium rela tionship betw een them.If R t a nd r t a re not cointeg ra t-ed,tha t is,R t-U r t=X t is also I(1),they can drift apa rt from each o ther mo re a nd m ore as time g oes o n.Thus there is no lo ng-run equilibrium relationship betw een them.There-Received date:April26,1999fore ,cointegration betw een lo ng and shor t rates means suppo rt for ex pectatio n hypothe-sis .As a ll interest ra tes sho uld be co nsidered as endogenous v ariables,when testing the ex pectation hypo thesis,w e need to take into account the interdependence betw een interest ra tes of different ma turity .One w ay of doing this is to estim ate a simultaneous equatio n m odel with lag s in a ll the v ariables.Ho w ever,this form ula tion inv olves tw o steps :first,w e have to classify th e v ariables into tw o ca tego ries,endog enous and ex og enous and sec-o nd,w e hav e to im pose som e constraints on the pa ram eters to achieve identification.Bo th these steps invo lv e many arbitrary decisio ns .An alternativ e is the cointeg ration a nalysis w ith V ector Autoreg ressiv e model.This is the multiple tim e-series generaliza tion of the AR m odel.The virtue of the V AR m odel is that one do es no t hav e to w o rry abo ut deter-mining which v ariables a re endogenous and which ones ex og enous .All v aria bles in V AR m odel are endogeno us.2　Theory and Methodology2.1　Bivariate C ointegrationB iv ariate cointeg ratio n reg ressio n is specified as follo ws :R t =T 0+T 1t +T 2r t +X t (1)w here Xt is the residual va riable .The initial step invo lv es the determinatio n of tim e series property of each v ariable individually with the aid of unit roo t tests,popula rly know n as Dickey -Fuller (DF)a nd aug mented Dickey-Fuller (ADF)tests(Dickey and Fuller ,1981).The existence o f a unit root in a time series im plies non -statio na rity in levels and hence sta tio narity can be induced by the first differencing o f the lev el da ta.This requires running a t least one of the fo llowing reg ressio ns :R t =U R t -1+∑ki =1W i△R t -i (2)R t =_+U R t -1+∑k i =1W i△R t -i (3)R t =_+T t +U R t -1+∑ki =1W i △R t -i (4)w here △deno tes the first difference o perato r .The releva nt null hypo thesis is that |U|=1against its alternativ e that |U |<1.A failure to reject the null hypothesis show s that the time series is no n-sta tio nary in lev els a nd the first differencing of the data will restore sta-tionarity in the tim e series .The same will also be true for a time series on r t .There are some questio ns w ith respect to the co rrect specifica tion o f the equa tion.If the series is genera ted by a random walk w itho ut a drift and a mean,then the unit roo t tests a re based on Equa tion (2).If the series is g enerated without drift a nd with a no n -ze-ro m ean ,the tests are based o n Equatio n (3).If the tim e series has a non -zero drift and a non-zero mean,then the tests are based o n Equatio n (4).Fo r the tim e series da ta used in this paper,Equatio n (2)is found inapplicable.So,the unit root tests are based o n Equa-228J o urnal of Sy stem s Scie nce and Sy stems Engineering V ol .9N o .2　tions (3)and (4).It is wo rth mentioning tha t the abov e unit roo t tests are biased against the null hypothesis if the series has undergo ne a m ean or trend shift and that a wide va riety o f specificatio ns for th e deterministic part o f the process are possible (Cam pbell and Per-ro n,1991).Nex t ,ADF reg ressio n to be estimated is specified as fo llow s :X t =T 0X t -1+∑ni =1T i△X t -i +_t (5)w here _t is the w hite noise and n is the number o f lag s required to obtain w hite noise resid-ual terms .The ADF test statistic is the t v alue associated with T0.The ADF test is then applied o n |T0|.If it is no t sig nifica ntly different from unit then X t a re I (1),R t a nd r t a re no t cointegra ted .If it is sig nificantly different from unit then Xt are I (0),R t and r t a re coin-teg rated .2.2　Multivariate CointegrationIn a multiva ria te system of p interest ra tes of different ma turity,the ex pectation hy-poth esis then posits a number o f cointegration im plica tions :there sho uld be o ne co mmo n trend driving interest rates ,co rrespo nding to p -1cointeg ration v ectors ,and in each o f these vecto rs the coefficients should sum to zero,g iv ing statio nary spreads.The ex pecta tion hy pothesis can be ex pressed linearly as :r t (m )=1m ∑m -1j =0E t [r t +j (1)]+H (m )(6)w here the yield,r t (m ),a t time t o n a pure discount bo nd w ith time to maturity m is ex-pressed as an av erage of ex pected o ne-period yields.The ex pectatio n o perator,E t ,is co n-ditio nal on info rm atio n a t tim e t ,and H (m )is a time-inva riant but maturity dependent term premium .In the case of three interest rates r t (1),r t (2)and r t (3)of different maturity ,r t (2)=12[r t (1)+r t +1(1)],r t (3)=13[r t (1)+r t +1(1)+r t +2(1)].If w e ig no re the constant term ,U 1r t (1)+U 2r t (2)+U 3r t (3)=U 1r t (1)+12U 2r t (1)+r t +1(1)+13U 3[r t (1)+r t +1(1)+r t +2(1)]=U 1r t (1)+12U 2r t (1)+12U 2r t (1)-12U 2r t (1)+12U 2r t +1(1)+13U 3r t (1)+23U 3r t (1)-23U 3r t (1)+13U 3r t +1(1)+13U 3r t +2(1)=(U 1+U 2+U 3)r t (1)+12U 2[r t +1(1)-r t (1)]+13U 3[(r t +1(1)-r t (1))+(r t +2(1)-r t (1))].If w e consider p pure disco unt bo nds with time to maturity 1,m 2,…,m p ,then all pairs o f 229CHAO J ia n -xiong Cointeg ra tio n T est of Ex pec ta tio n Hy po thesis …yields,[(r t(1),r t(m2)],[r t(1),r t(m3)],…,[r t(1),r t(m p)],fulfill equations of type (6).Given the fact tha t,in g eneral,interest rates behav e like integ rated stochastic pro-cesses,this equatio n can be show n to hav e a num ber of cointeg ra tion im plica tions.These can be deriv ed by co nsidering the linear com bina tio n U1r t(1)+…+U p r t(m p).If w e inser t Equation(6)into this ex pression,w e o btain(apa rt fro m a co nsta nt term):U1r t(1)+…+U p r t(m p)=(U1+U2+…+U p)r t(1)+U2m2∑m2-1j=1E t[r t+j(1)]-r t(1)]+…+U pm p∑m p-1j=1E t[r t+j(1)]-r t(1)](7)If r t+j(1)is a non-statio nary I(1)process,i.e.a process w hich needs first-differencing to becom e statio nary,then E t[r t+j(1)-r t(1)]is sta tionary;therefo re the left hand side o f E-quatio n(7)is statio na ry if U1+U2+…+U p=0.This implies cointeg ration in the full sy s-tem o f p yields and that the sum of the cointeg ration coefficients equals zero.As this im-plication is v alid for any p≥2,there should be p-1independent cointeg ra tion v ecto rs, w hich is equiva lent to there being one comm on stochastic trend driv ing th e term structure. The p-1cointeg ra ting v ectors must each satisfy the zero-sum restrictio n.Therefo re,the p-1dim ensio nal processz t=U′x t(8) w herex t=r t(1)r t(m2)…r t(m p),　U=U11U12…U1p-1U21U22…U2p-1U p1U p2…U p p-1(9)should be stationa ry w ith the sum of the columns of U equaling zero.In Joha nsen′s(1988) terms,the columns of U span the cointeg ratio n space.It can be show n(Hall,1992)that,under the restrictions laid dow n by the ex pectatio n hypothesis,the cointeg ratio nspace isalso spa nned by the columns ofU=111 (1)-100 00-10 0000…-1(10)As these spreads v ecto rs are linearly independent,the cointeg ration space has rank(p-1).Therefo re under the ex pectatio n h ypo thesis,the p-1spreads,S t(i)=r t(m i)-r t(1), I=2,…,p,sho uld be sta tio nary.In o rder to determine the number of com mon trends and test fo r the zero-sum restrictio n we use Johansen′s maxim um likelihood appro ach.Rather than directly a nalysing the presence o f com mon trends,Johansen has sugg ested sta rting the analysis with the k o rder V AR m odel:x t=A1x t-1+A2x t-2+…+A k x t-k+X t230J o urnal of Sy stem s Scie nce and Sy stems Engineering V ol.9N o.2　This can be w ritten in error -correction form :△x t =B 1△x t -1+B 2△x t -2+…+B k -1△x t -k +1+B k x t -k +X t (11)w here B i =-I +A 1+A 2+…+A i are p ×p dimensional ma trices of parameters ,a nd Xt is a v ecto r erro r term.If x t is I (1),then △x t is I (0).If so me linear com binatio ns o f x t a re sta-tionary,that is,there a re som e cointeg ra ting relatio nships am ong the va riables in x t ,then the ma trix B k should no t be o f full rank .In o ther w ords ,ex pectation hy po thesis is equiva-lent to a reduced rank o f B k .Use r to denote this reduced rank,and co nsider the facto riza tion B k =T U ′,w here T and U are (p ×r )matrices o f full co lum n rank .Here ,U is the m atrix of cointegra tion v ec-to rs ,as in Equa tion (9),and T is a co rrespo nding ma trix o f erro r co rrection parameters .Maxim um likeliho od estima tion and hypo thesis testing based o n T o bserv atio ns o f x t is o b-tained by reduced ra nk reg ressions.Johansen sugg ests tw o likelihood ratio tests for the number ,r ,of cointeg ratio n v ec-to rs.The first is the maxim al eigenva lue test,λmax .The null hypothesis is r cointeg ratio n v ecto rs,against the alterna tiv e of r +1cointeg ra tion vecto rs.The second test is the tracetest ,λtrace ,w here the null hy po thesis is a t most r cointeg ra tion v ectors ,w ith mo re tha n r v ecto rs under the alternativ e.Earlier in this section,w e sa w that the ex pectation hy po thesis implies r =p -1cointe-g ratio n v ectors ,and tha t the cointeg ratio n space should be spanned by the co lum ns of ma-trix U defined in Equatio n (10).This is exactly the so rt o f hypothesis (o f the form U=U O )which Johansen has show n can be tested by likelihood ratio statistics,using the usual i 2asy mpto tic distributio n.3　Empirical Tests3.1　DataAs the interest rates in China a re not determined by the market,and Londo n Inter-Bank Offered Rate w as often used as an interna tio nal reference ,all interest rates used in the tests are UK spot rates.The 1-mo nth and 3-mo nth spo t ra tes a re o btained fro m DATAST REAM.The 5-yea r,10-y ear and 20-year spo t rates are end-of-month nominal g ross redemptio n yields on zero-coupo n gilts.As no such securities exist in the U K these ra tes hav e been deriv ed by the Bank of Eng land from the theo retical zero coupo n yield curv e,the Sv ensson model yield curv e (Breedon,1995),using UK gov ernm ent bond prices.Mo nthly data rather than daily da ta ,fro m January 1982to April 1997,w as used .This may appear to reduce the info rmational co ntent o f the data.Ho wever,in the co ntex t o f testing fo r unit roo ts,this is not actually the case.Shiller a nd Perro n (1985)show tha t in testing for a unit ro ot the pow er of the Dickey -Fuller (1981)′t -test ′is determined pri-ma rily by the span o f the da ta ,rather than by the num ber of observa tions .They ev en show that the pow er of the test ca n be destro yed by having too high a frequency of obser-va tions ov er a given span.231CHAO J ia n -xiong Cointeg ra tio n T est of Ex pec ta tio n Hy po thesis …232J o urnal of Sy stem s Scie nce and Sy stems Engineering V ol.9N o.2　3.2　Stationarity Test for Interest Rate SpreadsLet x t=(M1,M3,Y5,Y10,Y20)f.M1,M3,Y5,Y10a nd Y20represent the a nnu-alized interest rates with1-m onth,3-month,5-year,10-year and20-y ea r m aturity respec-tiv ely.In the statio nary test,the X t represents interest rate spreads M3-M1,ADF(1) △X t=-0.0698-0.466X t-1-0.195△X t-1-3.35　-5.81 -2.65 DW= 2.0179　2=0.306H0:(d-1)=0is rejected,which means unit roo t o f X t is rejected,X t=M3-M1is statio n-ary.This result sugg ests that expecta tio n hypothesis prev ails a t the sho rt end o f the term structure.The results fo r o ther interest rate spreads a re listed in Table1.Table1　Test f or un it roots in UK interest rate spreadsSp read s DF ADF StationarityM3-M1-8.58-5.81ADF(1)StationaryY5-M1-2.3-2.25ADF(1)Stationary,weakY10-M1-1.92-1.96ADF(1)Stationary,weakY20-M1-1.8-1.82ADF(1)Stationary,v ery w eakY10-Y5-3.55-3.11ADF(2)StationaryY20-Y5-3.55-3.22ADF(1)StationaryY20-Y10-3.86-3.03ADF(1)StationaryFor the period from Janua ry1982to April1997,spreads at the sho rt end of the m aturity spectrum,as w ell as the spreads between interest rates after5-years,are statio nary.The spreads betw een lo ng er term interest ra tes and the one-m onth ra te appear stationa ry, though the ev idence is w eak.The results reg arding the statio narity im plica tions o f the ex-pecta tio n hy po thesis are therefo re generally in fav our of the hy po thesis.3.3　The Johansen TestThe big gest practical challeng e in V AR modelling is to choo se the appropria te lag leng th,o n bala ncing the pow er a nd bias of estimatio n.As there ex ist fiv e va riables in the fiv e-equation V AR mo del,o ne lag o f each va riable in each equa tion will mean5lagg ed pa-ram eters in each equatio n plus the co nsta nt term,fo r a to tal of30pa ram eters in the sy s-tem.Each additio nal lag will add25mo re parameters to the system.U nless the sa mple size is large,estimating that many parameters will co nsume a lot of deg rees of freedo m. The maximum lag of o ne in the V AR model w as cho sen fo r tw o reaso ns:first,there a re 184observ atio ns in the test data set,limiting the number o f lag s co nsumes less degrees o f freedom;second,Table1show s tha t,in m ost cases,one lag in augm ented D-F test for unit roo ts in UK interest rate spreads can reduce the serial co rrela tion pro blem substantial-ly.Under these assum ptions,theλmax v alues and the co nv entio nal critical va lues fo r i2 asym ptotic distributions hav e been show n o n Table2.Table 2　Johansen Maximum Likelihood Procedure (Non -trended case )C ointegrat ion LR Test Based on Maximal Eigenvalue of the Stochastic Mat rix************************************************183obs ervations from 1982M 2to 1997M 4.M aximum lag in V A R = 1.Lis t of v ariables included in the cointeg rating v ecto r:M 1M 3Y 05Y 10Y 20InterceptLis t of eig envalues in d escending o rd er :.39436　.26957　.17384　.077182　.025248　-.0000************************************************NullAlternative Statis tic 95%Critical Value 90%Critical Valu e r =0r =191.769534.400031.6640r 1r =257.484628.138025.5590r 2r =334.947522.002019.7660r 3r =414.699315.672013.7520r 4r =5 4.67969.24307.5250************************************************Table 2is used to determine r ,the num ber of cointeg rating v ectors .As is show n ,the the null hypo thesis of r ≤2,and that of r =5are stro ng ly rejected.The mo st likely rank,r ,is 3o r 4,because 14.6993lies between 15.6720(95%)a nd 13.7520(90%).This result m eans a reduced rank o f B 1,which means the interest rates M 1,M 3,Y 5,Y 10and Y 20a re cointeg ra ted .The nex t step is to test the zero-sum restriction o n 4v ectors.W hen the ra nk o f 4is chosen,the pa ram eter o f matrix T ,U and B 1a re estimated as (In the fo rm o f B 1=T U ′):-0.460.430.27-0.19-0.100.25-0.310.36-0.27-0.110.22-0.20-0.020.00-0.120.17-0.190.23-0.320.060.27-0.290.060.07-0.24=- 1.69-0.60 1.91-0.600.45-0.71 1.83-0.880.680.690.38- 1.700.39-0.500.36- 1.540.700.41 1.46-0.600.33-0.350.020.00-0.02-0.040.08-0.220.28-0.150.04-0.060.080.02-0.120.000.00-0.050.12-0.02As can be seen,the sum of the pa ram eters in each ro w of the m atrix U ′appro aches zero.Likelihood Ratio test of zero-sum restrictio ns show s i 2(8)=0.87416,which m eans the ze-ro -sum restriction can no t be rejected .There are four cointeg rated v ectors in the system o f fiv e va riables.Therefo re,these interest rates are driv en by o ne comm on stochastic trend.The Johansen test suppo rts fo r the ex pectation hypothesis.Since the spread between 20-y ear a nd one-m onth rates show s v ery w eak sta tio narity,even under the co ndition of the smo oth yield curv e,it may look entirely impla usible to re-ga rd long -da ted bonds as close substitutes fo r shor t-da ted Trea sury B ills.Nev er theless,it is alw ays possible fo r an inv esto r to substitute a bo nd with bo nds of adjacent ma turity .233CHAO J ia n -xiong Cointeg ra tio n T est of Ex pec ta tio n Hy po thesis …234J o urnal of Sy stem s Scie nce and Sy stems Engineering V ol.9N o.2　Such a substitution can transit fro m the sho rt end to the long end of the term structure and vice versa.Therefo re,the evidence of cointeg ration in the multiva ria te sy stem is stro ng e-nough to v alidate the ex pecta tion hypothesis fo r the UK interest rate term structure.3.4　DiscussionThe finding s from this a nalysis are co nsistent with findings fro m studies by Mills (1991)and previous studies by MacDonald and Speig ht(1988)on UK data.How ev er,the em pirical study in this pa per disting uishes from other′s wo rks in tw o important aspects:(1)Test a t the sho rt end o f the term structureMacDona ld and Speigh t ex amined UK quarterly data from1963to1987,and find evi-dence co nsistent with the ex pectation hy po thesis.By adopting the sam e m ethodo log y, Mills examined further the expectatio n hy pothesis o f the term structure using UK interest ra tes,but ex tending the analy sis to ex amine a m uch lo ng er period fro m1870to1988. Their research w orks,how ev er,are confined to th e testing of the ex pectation hy po thesis a t the lo ng end o f term structure.As sho rt-term instruments in the UK hav e,until recent-ly,been limited to three-m onths duration,inv estig atio n o f the hy po thesis at the short end o f the term structure is not po ssible at the tim e w hen Mills and o thers conduct their re-searches.Since the choice of interest rates w as made prima rily on the g rounds of da ta av ailabili-ty,this paper concentrates on the period fro m1982to1997,making tw o series of shor t-term interest ra tes available,i.e.the one-m onth and th ree-mo nth no minal ra tes,thus fills the vaca ncy o f study at the shor t end of the term structure.(2)AccuracyIn Mills′model,the sho rt ra te r fo r the pre-w ar period is the3-mo nth Prim e Bank B ill ra te,w hile the long rate R∞is the yield o n Co nso ls.Fo r the post-w ar perio d,the short ra te r is th e yield o n91day Treasury B ill,the lo ng rates a re the yields on5and20 y ea r gilts a nd yield o n3.5%War Loan.The problem is tha t the Prime Bank Bill ra te is no t a risk-free rate.Fur thermore,the3.5%War Loa n is rarely traded and its yield is de-termined from quoted prices rather than prices determined in a n activ e market.Such yield should thus no t reflect the w o rking of the ex pecta tion hypothesis o f term structure.E v en the co upon-bearing bond yield o n5-year gilts does no t satisfy the co nditio n of n→∞in the m odel.These facto rs cast som e doubt on the accuracy of the tests.All the interest rates used in the empirical test of this paper are zero-coupo n yields o n g overnment bonds,which in theory will provide mo re accurate estima tion of term struc-ture tha n any oth er yields.Thus the results in this paper are m ore reliable.The results are different from tho se obtained by Mankiw and Summer on US da ta (Mankiw a nd Summ er,1984).The ex pectatio n hy po thesis has been rejected from their analysis o f U S interest rate data from the1960s to the early1980s and this may be simply a reflection of this particular,possibly a ty pical,sam ple.Studies of the ex pecta tion hy-poth esis ov er different time period befo re1979o n UK data by Mills also rev eal different results.During the period befo re W W I the hypothesis w as rejected.This should be nosurprise ,because the dominant sho rt -term fina ncial instrument then w as the com mercial bill ,w hose draw ers w ere insensitiv e to chang es in either ex pected o r real interest ra tes and w ho could no t m ov e freely from o ne market to ls ′evidence rejecting the ex-pecta tio n hypothesis fo r data prio r to 1979is w eaker than those prio r to W W I ,though still reflect the seg menta tion of the gilt -edg ed security ma rket in that period .The inv estig atio n o f the ex pecta tion hypothesis cov ering the mo st recent fifteen years has mo re practical sig-nifica nce than purely theoretical interest.The dominance of relationship betw een long and short rates by the ex pecta tio n hypo thesis signals less segm entatio n,o r better efficiency,o f UK g ov ernment bo nd market over the last decade .4　ConclusionsIn this paper the theo ries o f the term structure of interest ra te hav e been investiga ted using the recent UK Trea sury bill rates and the zero -coupo n bond yields co nstructed by the Bank o f Eng land .The basic purpo se w as to test the ex pectatio n hy po thesis by ex ploiting the re-cent literature o n cointeg rated series.Thus the ex pectatio n hypo thesis w as tested in a bi-va riate auto reg ressio n fo rm using the spread betw een long a nd shor t interest rates.It w as found that all interest rate spreads a re stationa ry ,which sig nals suppor t fo r the ex pecta-tion hypo thesis.Further ev idence to support for the hypoth esis w as obtained from co n-ducting Johansen test in a multiv ariate v ecto r auto reg ressive mo del.The test results a re sufficient to justify po stula ting tha t lo ng and sho rt interest ra tes in the U K are determined by the ex pectatio n model of the term structure .Since all the interest ra tes used in this analysis are risk-free zero -coupo n yields,the test results are m ore accurate than those from using o ther data.The validity o f the expectatio n hypo thesis sug gests that the Bank o f Eng land can hav e a direct effect o n long interest rates by opera ting a t the sho rt end o f the ma rket witho ut altering the na ture o f the term structure itself.References:[1]　Breed on F.Bond prices and mark et expectations of in flation.Bank of England Quarterly Bulletin ,M ay,1995,160～165.[2]　Cam pbell J Y,Per ron P.Pitfalls and opportunities :w hat macroeconomists should k now ab ou t unit roo ts.NBE RM acroeconomics An nual,1991,6:141～201.[3]　Campbell J Y,Shiller R J .Cointegration and tes t of pres ent-value models.Journal of Political Economy,1987,95:1063～88.[4]　Dick ey D A ,Fuller W A .Likelihood ratio s tatistics for autoregressive time series with a unit roo t 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