COINTEGRATING RANK SELECTION IN MODELS WITH TIME-VARYING VARIANCE. Xu Cheng and Peter C. B. Phillips. January 2009

Transcription

1 COINTEGRATING RANK SELECTION IN MODELS WITH TIME-VARYING VARIANCE By Xu Cheg ad Peter C. B. Phillips Jauary 29 COWLES FOUNDATION DISCUSSION PAPER NO. 688 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 2828 New Have, Coecticut

2 Coitegratig Rak Selectio i Models with Time-Varyig Variace Xu Cheg Departmet of Ecoomics Yale Uiversity ad Peter C. B. Phillips Cowles Foudatio, Yale Uiversity Uiversity of Aucklad & Sigapore Maagemet Uiversity October, 28 Abstract Reduced rak regressio (RRR) models with time varyig heterogeeity are cosidered. Stadard iformatio criteria for selectig coitegratig rak are show to be weakly cosistet i semiparametric RRR models i which the errors have geeral oparametric short memory compoets ad shiftig volatility provided the pealty coe ciet C! ad C =! as! : The AIC criterio is icosistet ad its limit distributio is give. The results exted those i Cheg ad Phillips (28) ad are useful i empirical work where structural breaks or time evolutio i the error variaces is preset. A empirical applicatio to exchage rate data is provided. Keywords: Coitegratig rak, Cosistecy, Heterogeeity, Iformatio criteria, Model selectio, Noparametric, Time varyig variaces, Uit roots. JEL classi catio: C22, C32 Itroductio Much attetio has bee give to ecoometric estimatio ad iferetial procedures for time series with time-varyig variaces or ostatioary volatility. Amog others, Paga ad Schwert (99), Loreta ad Phillips (994), ad Watso (999) Cheg ackowledges the support of a Aderso Fellowship from the Cowles Foudatio for Research i Ecoomics. Phillips ackowledges partial support from the NSF uder Grat No. SES

3 documeted empirical evidece for temporal heterogeeity i the variatio of may macroecoomic ad acial time series. Particular cocer has recetly bee give to the e ect of the presece of heterogeeous ucoditioal variatio ad variace breaks o the validity of uit root tests. Several authors (Hamori ad Tokihisa, 997; Kim et al, 22; Cavaliere, 24; Cavaliere ad Taylor, 27) have show that covetioal uit root tests may su er size distortios ad reduced power whe there is persistet heterogeeity i variatio. Depedig o the speci c patter of the volatility chages, the size distortios ca be large eough to justify the use of more robust iferetial techiques or adaptive estimatio methods to secure gais i e ciecy, such as those developed for autoregressive models (Phillips ad Xu, 26; Xu ad Phillips, 28). The e ect of variace shifts o KPSS tests has also bee studied (Busetti ad Taylor, 23; Cavaliere, 24; Cavaliere ad Taylor, 25). Modi ed uit root tests have bee proposed to deal with various forms of departure from homoskedasticity for ostatioary time series. Kim et al (22) dealt with the case of a sigle abrupt chage i variace by usig a two-stage procedure where the breakpoit together with the pre- ad post-break variaces are estimated i the rst step. Cavaliere ad Taylor (27) developed tests that are robust to multiple abrupt or smooth volatility chages usig simulatio based methods. Ad Beare (27) used kerel methods to remove the heteroskedasticity before applyig stadard semiparametric procedures such as the Phillips-Perro test. Boswijk (26) evaluated the power loss of various uit root tests, derived the asymptotic power evelope agaist a sequece of local alteratives to a uit root uder ostatioary volatility ad gave a adaptive test procedure based o volatility lterig. I cotrast to these uivariate studies i the presece of persistet shifts i volatility, the preset paper deals with multivariate systems ad uses iformatio based methods rather tha Neyma Pearso tests. The focus of attetio is the rak of the coitegratig space i a model with some uit roots. Aalogous to scalar uit root tests, residual based coitegratio tests su er from size distortio uder ostatioary volatility. Alterative methods based o vector autoregressios, such as the Johase (987, 995) trace test, are also ivalidated by time varyig variaces. Some of these methods impose strog parametric assumptios o the form of the model. The iformatio theoretic approach take here uses a semiparametric framework ad is show to be robust to variace chages of a very geeral form. It may be used to cosistetly estimate coitegratig rak i a multivariate time series eviromet or as a scalar uit root test. I both cases, the procedure is robust to persistet shifts i volatility ad is easy to implemet i practical work. The paper is closely related to past work o ecoometric model selectio usig iformatio criteria. The most commo applicatios of these methods ivolve choice of lag legth i (vector) autoregressio, variable choice i regressio, ad coitegratig rak selectio i parametric settigs (Phillips, 996). Cheg ad Phillips (28) show that coitegratig rak selectio by suitable iformatio criteria is cosistet i a more geeral semiparametric framework usig reduced rak regressio (RRR) i a simple 2

4 VAR model with oe lag. I particular, RRR may be implemeted without explicitly takig ito accout weak depedece i the errors. The preset paper stregthes the results i Cheg ad Phillips (28) by showig that these methods remai cosistet whe the errors are weakly depedet ad there are persistet shifts i volatility. More speci cally, iformatio criteria are weakly cosistet for selectig coitegratig rak provided that the pealty term goes to i ity at a rate slower tha the sample size. The approach is quite straightforward for practical implemetatio. Simulatios idicate that uder may forms of heteroskedasticity, the usual BIC criterio for coitegratig rak selectio performs satisfactorily. The mai practical import of the paper, therefore, is that the same coitegratig rak selectio method may be used i empirical work for a wide rage of semiparametric models of coitegratio with shiftig variaces. Aother cotributio of the paper is to provide a limit theory for regressio i multivariable systems with some uit roots, weakly depedet errors ad ostatioary volatility. This limit theory is useful i studyig cases where reduced rak regressios are misspeci ed, possibly through the choice of iappropriate lag legths i the vector autoregressio or igorace of the persistet shifts i variace. The orgaizatio of the paper is as follows. Sectio 2 itroduces the semiparametric heteroskedastic error correctio model (ECM) ad gives assumptios ad estimatio details. Asymptotic results are give i Sectio 3. Sectio 4 brie y reports some simulatio results. A empirical applicatio to exchage rate data is reported i Sectio 5. Sectio 6 cocludes. Proofs ad techical material are i the Appedix. 2 The Semiparametric Heteroskedastic ECM We cosider the semiparametric ECM model X t = X t + u t ; t 2 f; :::; g ; () where X t is a m- vector time series, ad ad are m r full rak matrices. The iteger r is the ukow coitegratig rak parameter. The error term fu t g is weakly depedet ad heterogeeously distributed accordig to u t = D (L) " t = " t = V X D j " t j ; j= t e t ; e t iid (; e ) ; (2) where V () = diag fv () ; ; V m ()g ad V k () ; for k = ; ; m; is a ukow positive scale fuctio. Uder this speci catio, the iovatio term " t has mea zero ad time-varyig variace V t e V t : The series Xt is iitialized at t = by some (possibly radom) quatity X = O p () ; although other iitializatio assumptios 3

5 may be cosidered, as i Phillips (28). Followig covetios i the literature, we eglect the triagular array otatio for fx t g ; fu t g ; ad f" t g : Assumptio below imposes coditios o the liear process u t that facilitate the partial sum limit theory. Assumptio 2 gives coditios o the iovatio variace that are aalogous to those used i Phillips ad Xu (26). The coditios i Assumptio 3 are stadard i the study of reduced rak regressios with some uit roots (Johase, 988, 995; Phillips, 995). Assumptio The lag polyomial D(L) = P j= D jl j satis es that D = I; D () is full rak, ad P j= j jjd jjj < ; where jjjj is some matrix orm. The covariace matrix e is positive with uity diagoal elemets ad E jje t jj 4 < : Assumptio 2 V k () ; for k = ; :::; m; is o-stochastic, measurable ad uiformly bouded o the iterval ( ; ]; with a ite umber of poits of discotiuity, V k () > ad satis es a Lipschitz coditio except at poits of discotiuity. Assumptio 3 (a) The determiatal equatio I m (I m + )L = has roots o or outside the uit circle, i.e. jlj : (b) Set = I m + where ad are m r matrices of full colum rak r, r m: (If r = the = I m ; if r = m the has full rak m) (c) The matrix R = I r + has eigevalues withi the uit circle. Uder (), the time series X t is coitegrated with coitegratio matrix of rak r ; so there are r coitegratig relatios i the true model. As i Cheg ad Phillips (28), we treat () semiparametrically with regard to u t ad to estimate r directly i () by iformatio criteria. The procedure we use here is idetical to that of Cheg ad Phillips (28) ad is straightforward. Model () is estimated by covetioal RRR for all values of r = ; ; ; m just as if u t were a martigale di erece, ad r is chose to optimize the correspodig iformatio criteria as if () were a correctly speci ed parametric framework up to the order parameter r. Thus, o accout is take of the weak depedece structure ad time-varyig variace of u t i the process. Followig Cheg ad Phillips (28), the iformatio criterio used to evaluate coitegratig rak is IC (r) = log b (r) + C 2mr r 2 ; (3) with coe ciet C = log ; log log ; or 2 correspodig to the BIC (Schwarz, 978; Akaike, 977; Rissae, 978), Haa ad Qui (979), ad Akaike (974) pealties, respectively, or eve sample iformatio-based versios (Wei, 992; Phillips ad Ploberger, 996). The BIC versio of (3) was give i Phillips ad McFarlad (997). I (3) the degrees of freedom term 2mr r 2 is calculated to accout for the 2mr elemets of the matrices ad that have to be estimated, adjusted for the r 2 restrictios that are eeded to esure structural ideti catio of i RRR. 4

6 The procedure is ow the same as i Cheg ad Phillips (28). Oly the limit theory di ers because this depeds o the persistet shifts i volatility. For each r = ; ; ; m; we estimate the m r matrices ad by RRR ad, for use i (3), we form the correspodig residual variace matrices b (r) = X t= X t ^^ X t X t ^^ X t ; r = ; :::; m with b () = P t= X tx t: The, r is selected as br = argmi rm IC (r) : De e S = S = X X t Xt; S = t= X X t Xt ; t= X X X t Xt ; ad S = X t Xt: t= t= ad the (Johase, 995) b (r) = js j r i= b i ; (4) where b i ; i r; are the r largest solutios to the determiatal equatio S S S = : (5) S The criterio (3) is the well determied for ay give value of r: 3 Asymptotic Results Assumptio 3 esures that the matrix has full rak. Let? ad? be orthogoal complemets to ad ; so that [;? ] ad [;? ] are osigular ad?? = I m r : As i Cheg ad Phillips (28), we have the Wold represetatio of X t X vt := Xt = R i u t i = R (L) u t = R (L) D (L) " t ; (6) i= ad the partial sum (or geeralized Grager) represetatio X t = C where C =? (??)? : tx u s + R (L) u t + CX ; (7) s= 5

7 For k = ; :::; m; we de e Z k (r) := Z r V k (s) ds V k (s) ds ad k := Z V k (s) ds: (8) The volatility of the k th elemet of " t is characterized by its variace pro le k (r) ; which is equal to r oly whe the iovatio is homogeeous. The variace pro le k (r) is ormalized by the average iovatio variace k so that k (r) is a icreasig homeomorphism o [; ] with k () = ad k () = : Lemma Uder Assumptios -3, (a) [] X =2 u s ) D () B " () ; where B " (r) = s= ad B e () is a Browia motio with variace e. (b) B V () := D () B " () = D () B () =2 e ; Z r V (s) db e (s) ; where = diag ( ; :::; m ) ; B () = (B ( ()) ; :::B m ( m ())) ; ad B () ; :::; B m () are idepedet stadard Browia motios. (c) [] X =2 s= v s ) B V () ; =2? X [] )??? B V () : These limit laws ivolve the variace trasformed Browia motio B V () ; which is Browia motio uder time deformatio. I particular, at time r 2 [; ] ; B k ( k ()) has the same distributio as the stadard Browia motio B k () at time k (r) 2 [; ] : Let v t = G(L)" t ad X t = v t + u t = W (L)" t ; (9) where G (L) = R (L) D (L) by (6) ad W (L) = LG (L) + D(L): De e the average variace of " t by V := Z V (r) e V (r) dr: () The followig results provide some asymptotic limits that are useful i derivig the asymptotic properties of b (r), extedig a correspodig result i Cheg ad Phillips (28). 6

8 Lemma 2 Uder Assumptios -3, where S! p ; S! p ; S! p Z? S? )??? B V BV Z? S )??? B V dbv Z? S )??? B V dbv = wu = X W j V Wj+h ; = j= X h= j= ad w t =? X t =? W (L)" t:??? ; ( ) + wv;???? + X G j V G j+h ; = j= X? W jv D j+h ; wv = X h= j= wu + wv ; X G j V Wj+h ; () j= X? W jv G j+h ; (2) Remarks: Whe the iovatio " t has time-varyig variaces, the asymptotic limit associated with the o-statioary process? X t ivolves the variace trasformed Browia motio B V () : Uder homogeeous iovatios, B V () becomes a m- vector Browia motio with variace 2 D () e D () ad V reduces to 2 e ; where = = = m. As such, the sample variace ad covariace terms ; ; ad the oe sided log ru variace wu ad wv are all simpli ed to momets of X t ad v t ; both of which are ow statioary. Those results uder homogeeous errors were give i Cheg ad Phillips (28). De e e = (3) ad let e? be a m (m r) orthogoal complemet to e such that [e; e? ] is osigular. The followig reproduces Lemma 2 i Cheg ad Phillips (28) uder shiftig variaces. Lemma 3 Uder Assumptios -3, whe the true coitegratio rak is r ; the r largest solutios to (5); deoted by b i with i r ; coverge to the roots of = : (4) The remaiig m r roots, deoted by b i with r + i m; decrease to zero at the rate ad f b i : i = r + ; :::; mg coverge weakly to the roots of Z Z Z G u G u G u dg u? + e? e? e? e?? dg u G u + = ; 7 (5)

9 r dimesioal variace trasformed Brow- where G u (r) = (??)? B V (r) is m ia motio ad = wu + wv : Comparig Theorem 3 with the results uder homogeeous or martigale di erece errors, we see that i all cases the r largest roots of (5) are all positive i the limit ad the m r smallest roots coverge to at the rate : However, uder the preset setup, the determiatal equatio (5) ivolves a variace trasformed Browia motio G u ; which reduces to Browia motio whe the iovatio is homogeeous, as i Cheg ad Phillips (28). Allowig for weak depedece i the errors, equatio (5) ivolves the oe sided log ru variace matrix : A geeral form of the oe sided log ru variace uder weakly depedet heterogeeously distributed errors was rst give i Phillips ad Park (988). Uder Assumptio 2, the compoets of = wu + wv ca be expressed as i (2) usig the average iovatio variace V by meas of Lemma 4 i the Appedix. The mai result ow exteds the correspodig theorem Cheg ad Phillips (28) to allow for shiftig variaces. Theorem Uder Assumptios -3, (a) the criterio IC(r) is weakly cosistet for selectig the rak of coitegratio provided C! at a slower rate tha ; (b) the asymptotic distributio of the AIC criterio (IC(r) with coe ciet C = 2) is give by lim P (^r AIC = r )! " ( rx )# = P i < 2 (r r ) (2m r r ) ; m \ r=r + i=r + ad = P lim P (^r AIC = rjr > r )! ( ( r X m \ r =r+ r \ r =r ( rx i=r + i=r+ i < 2 r r (2m r r) )! i > 2 r r 2m r r )!) ; lim P (^r AIC = rjr < r ) = ;! where r +; :::; m are the ordered roots of the limitig determiatal equatio (5) : This result provides a coveiet basis for cosistet coitegratio rak selectio i most empirical cotexts uder very geeral assumptios o the errors. As i the \ 8

10 homogeeous variace case, BIC, HQ ad other iformatio criteria with C! ad C =! are all cosistet i the presece of weakly depedet errors with time-varyig variace. The iformatio criterio cosistetly selects coitegratig rak uder geeral assumptios o the errors without havig to specify ay parametric model of short memory or heterogeeity. Whe m =, the uit root model correspods to r = ad r = to the statioary model. Ulike some stadard uit root tests, model choice by iformatio criteria is the robust to the presece of permaet shifts i variace. The theorem also applies i the case of models with itercepts ad drifts. AIC is icosistet, asymptotically ever uderestimates coitegratig rak, ad favors more liberally parametrized systems. This outcome is aalogous to the wellkow overestimatio tedecy of AIC i lag legth selectio i autoregressio ad is cosistet with earlier results o coitegratio rak selectio uder homogeous errors. 4 Simulatios This sectio reports some brief simulatios for di eret forms of the variace fuctio V (), di eret settigs for the true coitegratig rak, ad various choices of the pealty coe ciet C : The data geeratig process follows () ad the desig of the reduced rak coe ciet follows Cheg ad Phillips (28). Thus, whe r = we have = ; ad whe r = the reduced rak coe ciet matrix is set to = (; :5) Whe r = 2; two di eret simulatios (desig A ad desig B) were performed, oe with smaller ad oe with larger statioary roots as follows: A : :5 : = ; with statioary roots :2 :4 i I + = f:7; :4g ; B : :5 : = ; with statioary roots :2 :5 i I + = f:9; :45g : Followig Cavaliere ad Taylor (27), we assessed the performace of the iformatio criteria ucotamiated by serial depedece by settig u t = " t : To evaluate the method uder weak depedece, simulatios were also coducted uder the followig AR(), MA(), ARMA(,) formulatios u t = Au t + " t ; u t = " t + B" t ; u t = Au t + " t + B" t ; (6) with coe ciet matrices A = I m ; B = I m ; where j j < ; jj < : The iovatios with time-varyig variace are t " t = V e t ad e t = iid N (; " ) ; (7) 9 :

11 where " = + > : The parameters for these models were set to = = :4 ad = :25: The desig of the variace matrix V () follows that i Cavaliere (24), Cavaliere ad Taylor (27) ad Phillips ad Xu (26). We assume that for ay r 2 ( ; ]; the m m diagoal variace matrix V (r) = g (r) 2 I m ; where g () is a real positive fuctio. Uder this setup, all variables share the same variace pro le, characterized by the variace fuctio g () : Three models for the variace fuctio g () were used:. g (r) 2 = frg; r 2 [; ] ; 2. g (r) 2 = fr< g; r 2 [; ] ; 2 [; =2]; 3. g (r) 2 = r m ; r 2 [; ] : (8) There is a sigle volatility shift from 2 to 2 at time [] i model ad there are two volatility shifts i model 2, which happes at time [] ad [], respectively. I cotrast to the abrupt volatility jumps i these two models, model 3 models the situatio where volatility chages smoothly from 2 to 2 : The parameters i the simulatio are setup as follows. I model, the break date takes values withi the set f:; :5; :9g ; so that early, middle ad late breaks are all ivestigated. I model 2, takes value from f:; :4g ; where a small correspods to the case where the rst jump happes early i the sample ad the secod jump happes late i the sample. I model 3, we allow for both liear tred ad quadratic tred by settig m 2 f; 2g : Without loss of geerality, we set 2 = i all cases. The steepess of the break is measured by the ratio of the post-break ad pre-break stadard deviatio: = = ; which takes values withi the set f:2; 5g for all three models to allow for both positive ( > ) ad egative ( < ) shifts. The performace of AIC ad BIC was ivestigated for sample sizes = ; 4 i all cases icludig 5 additioal observatios to elimiate start-up e ects from the iitializatios X = ad " = : The results are based o 2; replicatios. Tables -3 give simulatio results for desig A where the error u t follows a AR() process. Similar results were obtaied for the other error geeratig schemes i (6) : As is evidet i the tables, BIC geerally performs well uder di eret forms of volatility chages whe the true rak r is or 2, although whe r = ; it may overestimate i some cases uder abrupt volatility shifts, depedig o the patter of the chages. Speci cally, i model, the overestimatio teds to happe whe there is a early egative shift ( = :; = :2) or a late positive ( = :9; = 5) shift, but ot uder early positive shifts or late egative shifts; i model 2, the overestimatio happes whe a very early shift is positive ad a very late shift is egative. I the worst It is show i Cheg ad Phillips (28) that AIC ad BIC geerally have better performace tha other criteria such as Haa-Qui (HQ) or criteria with eve weaker pealties tha HQ such as C = log log log.

12 case, BIC selects the true coitegratio rak r = with a probability aroud 65% whe the sample size is 4: We also observe that the covetioal tedecy of BIC to uderestimate order (here coitegratig rak) is mild whe = ad disappears completely whe = 4: These results are aalogous to those i Phillips of Xu (26), who show that i a stable autoregressive model various t statistics ted to over-reject uder early egative shifts or late positive shifts ad that this tedecy is atteuated whe the error variace dyamics follow a polyomial shape as i model 3. I all cases, BIC performs much more satisfactorily tha AIC, which has a strog tedecy to overestimate order, just as it does i lag legth selectio i autoregressive models. Table Coitegratio rak selectio i desig A whe u t follows a AR() process uder model = 4 = r = r = r = 2 r = r = r = 2 br AIC BIC AIC BIC AIC BIC AIC BIC AIC BIC AIC BIC

13 Table 2 Coitegratio rak selectio i desig A whe u t follows a AR() process uder model 2 = 4 = r = r = r = 2 r = r = r = 2 br AIC BIC AIC BIC AIC BIC AIC BIC AIC BIC AIC BIC Table 3 Coitegratio rak selectio i desig A whe u t follows a AR() process uder model 3 = 4 = r = r = r = 2 r = r = r = 2 m br AIC BIC AIC BIC AIC BIC AIC BIC AIC BIC AIC BIC To show the e ect of variace shifts whe they are ucotamiated by temporal depedece, we performed simulatios for desig A uder idepedet errors with the variace structure speci ed i (8) : To save space, we oly report the results uder model, as show i Table 4. Comparig Tables 4 ad, we d that BIC is geerally more reliable whe the errors have low temporal depedece. Similar results were foud for models 2 ad 3. 2

14 Table 4 Coitegratio rak selectio i desig A whe u t is idepedet uder model = 4 = r = r = r = 2 r = r = r = 2 br AIC BIC AIC BIC AIC BIC AIC BIC AIC BIC AIC BIC The results for desig B, where the statioary roots of the system are closer to uity, are show i Table 5. Just as i Cheg ad Phillips (28), whe the statioary root is large, BIC has a tedecy to uderestimate the rak whe = ad r = 2; thereby choosig more parsimoiously parameterized system i this case. Whe = 4; the uderestimatio is sigi catly atteuated. 3

15 Table 5 Coitegratio rak selectio i desig B whe u t follows a AR() process uder model = 4 = r = r = r = 2 r = r = r = 2 br AIC BIC AIC BIC AIC BIC AIC BIC AIC BIC AIC BIC I summary, the simulatio results show that the BIC criterio for coitegratio rak selectio is robust to weak depedece ad heterogeeity of the errors, geerally co rmig the asymptotic theory. The mai weakess of BIC is that it teds to overestimate whe early egative or late positive volatility shifts happe i a system without coitegratio ad to uderestimate whe the system is statioary but with a root ear uity. The performace of BIC sigi catly improves as the sample size gets larger, the volatility shifts become smoother, or the temporal depedece of the errors is weaker. I all cases, BIC performs much better tha alterative criteria such as AIC ad seems su cietly reliable to recommed for empirical practice 5 Empirical Applicatio This sectio reports the applicatio of model selectio techiques to coitegratig rak estimatio i a dyamic exchage rate system. Usig Johase s trace test, Baillie 4

16 ad Bollerslev (989) foud evidece of oe coitegratio relatio i vector autoregressios of seve daily spot ad seve oe-moth forward rates. They cocluded that these oatig exchage rates follow oe log-ru equilibrium path. However, whe addig a itercept to the model, Diebold et al. (994) foud o support for a coitegratig relatio i these data. I additio to covetioal coitegratio tests, various fractioal coitegratio formulatios have bee cosidered i the same dyamic exchage rate settig, icludig Baillie ad Bollerslev (994), Kim ad Phillips (2), Nielse (24), Hassler et al. (26), ad Nielse ad Shimotsu (27). These papers o fractioal coitegratio geerally agree o the existece of fractioal coitegratio amog the exchage rates of di eret currecies uder the oatig exchage rate regime. Our focus i this applicatio is to apply semiparametric rak selectio methods to ivestigate possible coitegratig relatios amog exchage rates uder both oatig exchage rate regimes (post 973) ad xed exchage rate regimes (uder the Bretto Woods agreemet of ). It is ow a well-established stylized fact that may macro-ecoomic ad acial variables, icludig exchage rates, are characterized by breaks i volatility. So our approach, with its robustess to shiftig variaces icludig both abrupt breaks ad smooth trasitios, seems well suited to this applicatio. Moreover, there is o eed to specify a particular parametric model for variace shifts or weak depedece i our approach, makig it easy to implemet ad robust to a variety of di eret model speci catios. Our data set cocetrates o the same exchage rates as those i the literature cited above. The data comprise log exchage rates for seve currecies: the Caadia Dollar, Frech Frac, Deutsche Mark, Italia Lira, Japaese Ye, Swiss Frac ad British Poud, all relative to the US Dollar. Baillie ad Bollerslev (989,994) ad Diebold et al. (994) used these seve omial exchage rates observed daily from 98 to 985, Kim ad Phillips (2) used quarterly data from 957 to 997, ad Nielse ad Shimotsu (27) applied their estimatio techiques to a data set of mothly averages of oo (EST) buyig rates ruig from Jauary 974 through December 2. Our data set, take from the DRI Ecoomics Database (previously Citibase), is also mothly averages of oo buyig rates ad rus from November 967 to December The data set ca be divided ito two subperiods: the rst period, from November 967 to December 973, correspods to the xed exchage rate regime by the Bretto Woods agreemet ( ); ad the secod period, from Jauary 974 to December 998, correspods to the oatig exchage rate regime before the itroductio of the Euro. Compared with earlier applicatios, our data for the oatig exchage rate period covers a log time spa of 25 years with 3 observatios. We do ot iclude observatios after December 998, as i Nielse ad Shimotsu (27), because sice the the exchage rates betwee some major Europea currecies have bee xed i relatio to the Euro. The log exchage rate data series are plotted i Figure. The time-varyig behavior of the exchage rate volatilities is well characterized 2 Our data, take from the curret versio of the same source as that i Kim ad Phillips (2), starts from November

17 Figure : Log exchage rates by its variace pro le k (r) ; for k = ; ; m; which is icreasig from to ad oly equal to r uder homogeeous errors. We rst estimate the variace pro le of each exchage rate series usig the method of Cavaliere ad Taylor (27). Let fbu t g deote the residuals from the liear regressio of X b t o X b t ; where X b t is the residual of X t after detredig. Detredig X t is ecessary whe we iclude a itercept i () : The estimator of the variace pro le, which is the sample aalogue of (8) liearly iterpolated betwee the observed sample data, ca be writte as b k (r) = P [r] t= bu2 t (r [r]) bu 2 [r]+ P : (9) t= bu2 t Cavaliere ad Taylor (27) show that b k () is a uiformly cosistet estimator for the variace pro le k () : The estimated variace pro les are preseted i Figure 2. The rst two rows are the estimated variace pro les for each currecy (to the US Dollar) i the post-973 oatig exchage rate period, while the last two rows are the correspodig variace pro les estimated by data withi the xed exchage rate period. The 45 o lie correspods to the variace pro le for homogeeous errors. Durig the relatively log time spa after 973, we see that most exchage rates did ot experiece sharp chages i volatility, although multiple shifts ad smooth trasitios do exist i most series. Speci cally, i this period, the Caadia Dollar has the smoothest volatility pro le, followed by 6

18 Frech Frac, Deutsche mark ad Swiss Frac, whose volatility geerally exhibits a smooth icrease at the begiig of the period ad a smooth decrease at the ed of the period. Compared with these currecies, the Lira had sharper positive shifts at both the begiig ad the ed of the period, each followed by a immediate sharp egative shift, ad the British Poud has a abrupt positive shift ear the ed of the period, which is also followed by a egative shift. The variace pro le of the Ye exhibits a positive shift i the begiig ad several small shifts i the middle. These pro les idicate that the major Europea currecies are more closely related to each other ad the Ye ad the Caadia Dollar are relatively idepedet. As we ca see from the last two rows of Figure 2, the volatility shifts uder the xed exchage rate regime is much steeper, partially due to the relatively short time spa. At the ed of the xed exchage rate period, all currecies except the Caadia Dollar had steep icreases i volatility. Iformatio criteria are rst used to reveal the domiat time series characteristic of the exchage rate data with r = sigifyig I () ad r = sigifyig I (). The digs co rm earlier coclusios that omial exchage rates are well characterized as I() processes (c.f., Corbae ad Ouliaris, 988, ad Baillie ad Bollerslev, 989). Table 6 ad 7 3 report results for AIC ad BIC for each currecy uder both exible ad xed exchage rate regimes. Both the theory ad the simulatio results predict that AIC geerally overestimates order, which biases results to statioarity. BIC, which is more reliable, shows almost all series to be I() processes. The oly exceptio is the British Poud i the xed exchage rate period. However, from the simulatio digs i the last sectio, this outcome for the British Poud may well be due to overestimatio resultig from the huge volatility jump of the Poud at the begiig of the period. Next, coitegratig rak amog the seve exchage rates is estimated by AIC ad BIC uder (). The method allows for both weak depedece ad variace heterogeeity as detected i Figure 2. The estimatio results are preseted i Table 8 ad 9. Uder the oatig exchage rate regime, AIC ds 4 coitegratig relatios ad BIC ds o coitegratio i the system. Cosiderig the overestimatio problem associated with AIC ad the small uderestimatio probability of BIC give our large sample size, we coclude that there is o I () =I () coitegratio i the exchage rate dyamic system. Our result is cosistet with that obtaied usig the Johase trace test, where the optimal umber of lags is selected with iformatio criterio (Diebold, et al, 994). Compared with Johase s method, our procedure do ot require a rst step estimatio of the umber of lags i the ECM, is more robust to model speci catio ad is valid i the presece of time-varyig variace. Table 9 shows that coitegratig rak is estimated as 6 by AIC ad by BIC uder the xed exchage rate regime. The di erece i these outcomes is substatial, but we ote that: (i) simulatios show that AIC has a strog tedecy to overestimate 3 AIC() ad BIC() i table -4 are ormalized to for computatioal coveiece, but this ormalizatio does ot a ect estimatio results. 7

19 coitegratig rak, whereas BIC shows oly a small tedecy to uderestimate rak; ad (ii) the empirical results show that the BIC estimate is more sharply determied tha AIC. Takig the more reliable result give by BIC, we coclude that uder the xed exchaged rate regime di eret currecies were tied to oe equilibrium path i the log ru ad that deviatios from this log ru path were temporary. This result is compatible with the ature of the xed exchage rate regime, whereby uder the Bretto Woods agreemet, exchage rates were tied to each other, allowig some adjustmets oly uder special circumstaces. Thus, empirical co rmatio of some log ru equilibrium relatioship amog the exchage rates is to be expected durig the Bretto Woods era. Table 6 Uit root test for idividual series uder oatig exchage rate regime CAN FRA GER ITL JAN SW UK AIC r = r = BIC r = r = AIC br BIC br Table 7 Uit root test for idividual series uder xed exchage rate regime CAN FRA GER ITL JAN SW UK AIC r = r = BIC r = r = AIC br BIC br Table 8 Coitegratio rak estimatio uder oatig exchage rate regime r br AIC BIC Table 9 Coitegratio rak estimatio uder xed exchage rate regime r br AIC BIC

20 6 Coclusio This paper shows that coitegratig rak ca be cosistetly selected by iformatio criteria uder weak coditios o the expasio rate of the pealty coe ciet. I cotrast to traditioal reduced rak ad other coitegratio estimatio methodologies, our method does ot require a full parametric model ad it is robust to both weak depedece ad variace heterogeeity. As a coitegratig rak selector or as a simple uit root test it o ers substatial coveiece to the empirical researcher i the presece of these complicatios. Some further extesios of this semiparametric coitegratig rak selectio approach are possible ad may be useful i empirical research. We metio a few ideas here. First, allowace for stochastic volatility shifts seems importat for practical work, especially i acial ecoometric applicatios. Secod, there is scope for usig BIC to test for a shifts i variace while joitly coductig coitegratig rak estimatio. Fially, models of fractioal coitegratio might be ecompassed by usig a multivariate versio of the exact local Whittle procedure (Shimotsu ad Phillips, 25) to joitly estimate the fractioal di erecig paramaters ad a reduced rak coe ciet matrix, by meas of which coitegratig rak might be assessed as i the much simpler model () used here. 7 Appedix Lemma 4 Uder Assumptio -3, if a t = A(L)" t = P P j= A j" t j ad b t = B(L)" t = j= B j" t j with P j= j jja jjj < ad P j= j jjb jjj < : The X a t b t+h! a:s: t= X A j V Bj+h ; j= X Xt E(a j b t)! t=2 j= X h= j= X A j V Bj+h : Proof of Lemma 4: Usig the fact that " t = V t et ad e t is iid (; e ) ; we have X t= E " t " t = X t= t V Ee t e t t V! Z V (r) e V (r) dr: Sice " t is a martigale di erece sequece, we d, e.g. as i Phillips ad Solo (992), that X X Z a t b t+h! a:s: A j V (r) e V (r) dr Bj+h = X A j V Bj+h : (2) t= j= j= 9

21 Next, ote that X Xt X X h E(a j b t) = E(a t b t+h ) t=2 j= X X h X t j = A j V = + LX j= X j=l+ h= t= j= X Xh t j A j V h= t= h= t= X Xh t j A j V h= t= e V t e V t j j t e V B j+h B j+h j B j+h ; (2) for ay iteger L > chose so that L + L! : For each xed j L; we have where X Xh t j V h= t= t e V j B h j! h = V e V As! ; the sum ivolvig! h satis es X! h = h= = =! j+h = X hx! h Bj+k ; (22) h= h j : k= X h j h j V e V h= X Z h+ [r] j [r] j V e V dr h h= Z [r] j [r] j V e V dr Z V (r) e V (r) dr = V ; (23) uiformly i j; for j L: By (22), (23) ; ad the Toeplitz Lemma, we the have X Xh t j A j V h= t= uiformly i j; for j L: As a result, LX j= X Xh t j A j V h= t= t e V t e V 2 j Bj+h! X A j V Bj+h ; j h= B j+h! X j= h= X A j V Bj+h ; (24)

22 as! ad L! : Let C be a positive costat such that V (r) is uiformly bouded above by CI m for r 2 ( ; ]: The X X X h t j t j A j V e V B j+h j=l+ h= t= X X C 2 k e k ka j k B j+h! ; (25) j=l+ h= as L! sice P j= jja jjj < ; P j= jjb jjj <. It follows from (2) ; (24) ad (25) that : X Xt E(a j b t)! t=2 j= X h= j= X A j V Bj+h : Proof of Lemma : This is a vector geeralizatio of Theorem of Cavaliere ad Taylor (27). Usig the Phillips-Solo device, [r] X =2 [r] X u s = D () V s= s= s Z r es p + o p () ) D () V (s) db e (s) ; (26) where B e () a m-vector Browia motio with variace e : Uder Assumptio, the o p () term i (26) ca be veri ed i the same way as i Cavaliere ad Taylor (27). By Lemma 2 of Cavaliere (24), Z r V (s) db e (s) = Z r V k (s) db (s) = k B ( k (r)) ; for k = ; :::; m: Because V () is diagoal, we have R r V (s) db (s) R. r V m (s) db k (s) C A =2 e B B ( (r)). B k ( m (r)) C A =2 e ; (27) where = diag ( ; :::; m ) ad B k (); for k = ; ; m; are all stadard Browia motios that are idepedet of each other. By (26) ad (27) we obtai [] X =2 s= where B () = (B ( ()) ; ; B m ( m ())) : u s ) B V () := D () B () =2 e ; 2

23 I view of (7) we have? X t =? C tx u s +? R (L) u t +? CX s= ( =? tx )?? u s + X +? R (L) u t ; s= so that the stadardized process =2? X [] ) (??)? B V () : Usig (6) ad the fact that R () = P i= Ri = (I R) =, we have : [] X =2 s= X s ) B V () : (28) Proof of Lemma 2: Writig X t = W (L) " t ad X t = G(L)" t as i (9) ad otig that the lag polyomials W (L) ad G(L) satisfy the coditios of Lemma 4 by virtue of Assumptios ad 3, we have S = S = S = X X t Xt! p t= X j= W j V W j = ; X X t X X t!p G j V G j = ; t= j= X X X t Xt! p G j V Wj+ = : t= Usig Lemma ad Lemma 4, it follows from Park ad Phillips (988) that Z? S? )??? B V BV??? ; ( ) X? (S S ) =? X t X t X t? S = X t= = X t= t=? X t p j= u t p )??? X t Z X t p p )??? Z? B V dbv + wu; (29) B V db V ( ) + wv; (3) 22

24 where wu = lim! wv = lim! X Xt E(? X t u t ) = t=2 j= X Xt t=2 j= Fially, usig (29) ad (3), we obtai E(? X t X t X h= j= X? W jv D j+h ; ) = X h= j= X? W jv G j+h :? S =? (S S ) +? S Z )??? B u dbu???? + wu + wv ; sice ( ) +???? = I (e.g., Johase, 995, p. 39). Proof of Lemma 3: This Lemma follows the proof of Lemma 2 of Cheg ad Phillips (28) by replacig B u (r) with the variace trasformed Browia motio B V (r) : Proof of Theorem : The proof follows i the same way as the proof of Theorem of Cheg ad Phillips (28). 8 Refereces Akaike, H. (973). Iformatio theory ad a extesio of the maximum likelihood priciple. I B. N. Petrov ad F. Csaki (eds.), Secod Iteratioal Symposium o Iformatio Theory. Budapest: Akademiai Kiado. Akaike, H. (977). O etropy maximizatio priciple. I P. R. Krisharah (ed.), Applicatios of Statistics. Amsterdam: North Hollad. Baillie, R. T. ad Bollerslev, T. (989). Commo stochastic treds i a system of exchage rates, Joural of Fiace, 44, Baillie, R. T. ad Bollerslev, T. (994). Coitegratio, fractioal coitegratio, ad exchage rate dyamics, Joural of Fiace, 49, Beare, B. (27). Robustifyig uit root tests to permaet chages i iovatio variace. Workig paper, Yale Uiversity. Boswijk, H. P. (25). Adaptive testig for a uit root with ostatioary volatility. Workig paper 25/7, Uiversity of Amsterdam. Busetti, F. ad A. M. R. Taylor (23). Testig agaist stochastic tred i the presece of variace shifts. Joural of Busiess ad Ecoomic Statistics, 2,

25 Cavaliere, G. (24). Uit root tests uder time-varyig variace shifts, Ecoometric Reviews, 23, Cavaliere, G. ad A.M. R. Taylor (25). Statioarity tests uder time-varyig variaces, Ecoometric Theory, 2, pp Cavaliere, G. ad A.M. R. Taylor (27). Testig for uit roots i time series models with o-statioary volatility. Joural of Ecoometrics, Vol. 4, Chao, J. ad P. C. B. Phillips (999). Model selectio i partially o-statioary vector autoregressive processes with reduced rak structure, Joural of Ecoometrics, Vol. 9, Cheg, X ad P. C. B. Phillips (28). Semiparametric Coitegratig Rak Selectio, The Ecoometrics Joural, forthcomig. Corbae, D ad S. Ouliaris (988), Coitegratio ad Tests of Purchasig Power Parity, The Review of Ecoomics ad Statistics 7, Diebold, F. X., Gardeazabal, J. ad Yilmaz, K. (994). O coitegratio ad exchage rate dyamics, Joural of Fiace, 49, Hamori S. ad A. Tokihasa (997). Testig for a uit root i the presece of a variace shift. Ecoomics Letters, 57, Haa, E. J. ad B. G. Qui (979). The determiatio of the order of a autoregressio,. Joural of the Royal Statistical Society, Series B, 4, 9-95 Hassler, U., Marmol, F., Velasco, C. (26). Residual log-periodogram iferece for log-ru relatioships, Joural of Ecoometrics, 3, Kim, C. S. ad Phillips, P. C. B. (2). Fully modi ed estimatio of fractioal coitegratio models. Workig paper, Yale Uiversity. Johase, S. (988). Statistical Aalysis of Coitegratio Vectors, Joural of Ecoomic Dyamics ad Cotrol 2, Johase, S. (995). Likelihood-Based Iferece i Coitegrated Vector Autoregressive Models, Oxford: Oxford Uiversity Press. Loreta, M. ad P. C. B. Phillips (994). Testig covariace statioarity of heavy tailed time series: A overview of the theory with applicatios to several acial datasets, Joural of Empirical Fiace,, Nielse, M. Ø: (24). Optimal residual-based tests for fractioal coitegratio ad exchage rate dyamics, Joural of Busiess ad Ecoomic Statistics, 22,

26 Nielse, M. Ø:, Shimotsu, K. (27). Determiig the coitegratig rak i ostatioary fractioal systems by the exact local Whittle approach, Joural of Ecoometrics, 4, Paga, A. R. ad G. W. Schwert (99a). Testig for covariace statioarity i stock market data, Ecoomics Letters 33, Phillips, P. C. B. (99). Optimal Iferece i Coitegrated Systems, Ecoometrica 59, Phillips, P. C. B. (995). Fully modi ed least squares ad vector autoregressio, Ecoometrica 63, Phillips, P. C. B. (996). Ecoometric Model Determiatio, Ecoometrica, 64, Phillips, P. C. B. (998). New Tools for Uderstadig Spurious Regressios, Ecoometrica, 66, Phillips, P. C. B. (25). Challeges of Tredig Time Series Ecoometrics, Mathematics ad Computers i Simulatio, 68, Phillips, P. C. B. (28). Uit root model selectio, Joural of the Japa Statistical Society, 38, Phillips, P. C. B. ad J. McFarlad (997). Forward Exchage Market Ubiasedess: the Case of the Australia Dollar sice 984, Joural of Iteratioal Moey ad Fiace, 6, Phillips, P. C. B. ad K.-L. Xu (26), Iferece i autoregressio uder heteroskedasticity, Joural of Time Series Aalysis, 27, Phillips P. C. B. ad W. Ploberger (996). A Asymptotic Theory of Bayesia Iferece for Time Series, Ecoometrica, 64, Phillips, P. C. B. ad V. Solo (992). Asymptotics for Liear Processes, Aals of Statistics 2, 97. Pötscher, B. M. (989). Model selectio uder ostatioarity: Autoregressive models ad stochastic liear regressio models, Aals of Statistics 7, Rissae, J. (978). Modelig by shortest data descriptio, Automatica 4, Schwarz, G. (978). Estimatig the dimesio of a model, Aals of Statistics 6,

27 Shimotsu, K. ad P. C. B. Phillips (25). Exact Local Whittle Estimatio of Fractioal Itegratio, Aals of Statistics, 33, Tsay, R. S. (984). Order selectio i ostatioary autoregressive models, Aals of Statistics, 2, Watso, M. W. (999). Explaiig the icreased variability i log-term iterest rates. Federal Reserve Bak Richmod Ecoomic Quarterly, 85, Wei, C. Z. (992). O predictive least squares priciples, Aals of Statistics 2, 42. Xu, K.-L. ad Phillips, P. C. B. (28). Adaptive estimatio of autoregressive models with time-varyig variaces, Joural of Ecoometrics, 42,

28 Figure 2: Estimated variace pro les of exchage rates over two periods (: ; ad 2: ). CAN: Caadia Dollar; FRA: Frech Frac; GER: Deutsche Mark; ITL: Italia Lira; JAN: Japaese Ye; SW: Swiss Frac; UK: British Poud. 27