Limit Theory for VARs with Mixed Roots Near Unity

Transcription

1 Limit Theory for VARs with Mixed Roots Near Uity Peter C. B. Phillips Yale Uiversity, Uiversity of Aucklad, Sigapore Maagemet Uiversity & Uiversity of Southampto Ji Hyug Lee Yale Uiversity August, Abstract Limit theory is developed for ostatioary vector autoregressio (VAR) with mixed roots i the viciity of uity ivolvig persistet ad explosive compoets. Statistical tests for commo roots are examied ad model selectio approaches for discrimiatig roots are explored. The results are useful i empirical testig for multiple maifestatios of ostatioarity i particular for distiguishig mildly explosive roots from roots that are local to uity ad for testig commoality i persistece. Keywords: Commo roots, Local to uity, Mildly explosive, Mixed roots, Model selectio, Persistece, Tests of commo roots. JEL classi catio: C Itroductio Ama Ullah s cotributios cover a wide spectrum of ecoometrics with sustaied scieti c work over the last four decades i ite sample theory, oparametric estimatio, spatial ecoometrics, pael data modelig, acial ecoometrics, time series ad applied This paper is based o the rst part of a Yale take home examiatio i /. The authors thak the Editor ad two referees for helpful ad costructive commets o the origial versio. Phillips ackowledges support from the NSF uder Grat No. SES

2 ecoometrics. His advaced textbook o Noparametric Ecoometrics (999, with Adria Paga) has bee particularly i uetial, helpig to educate a geeratio of ecoometricias i oparametric methods ad providig a accessible referece for applied researchers. His moograph o Fiite Sample Ecoometrics (4) ecapsulates may of his ow cotributios to this subject ad touches some of the wider reaches of this di cult ad vitally importat eld. Oe eld of ecoometrics that his work has less frequetly touched is ostatioary time series ad uit root limit theory. Sice the mid 98s models with autoregressive roots i the viciity of uity have attracted much attetio. These models are particularly useful i empirical work with ostatioary series whe it may be too restrictive to isist o the presece of roots precisely at uity or where mildly itegrated or mildly explosive behavior may be more relevat tha uit roots. Whe multiple time series are cosidered, it may be useful to allow simultaeously for various types of behavior i the idividual series: some roots that are local to uity (of the form + c for xed c ad give sample size ) ad others that are mildly itegrated (of the form + b k for xed b < ad a sequece k! slower tha ) or mildly explosive (of the form + b k for xed b > xed ad a similar sequece k! slower tha ). Roots of the latter form lie i a wide ad thereby accommodate some iterestig alteratives viciity of uity of radius O k where the behavior of the process, icludig its limit behavior, di er from that of the radom waderig character associated with uit root processes. The mathematical form of these roots ivolves the localizig coe ciet b ad a rate sequece k which it is ofte coveiet to write i the expoet form k for some parameter (; ) : Limit theory for regressors with roots local to uity developed early i the literature of this eld (Phillips, 987b; Cha ad Wei, 987). More recet work has cosidered mildly itegrated ad mildly explosive cases (Phillips ad Magdalios, 7a, 7b; [PM7a&b]). The latter theory has proved particularly relevat i studyig data durig periods of - acial exuberace (Phillips, Wu ad Yu, ; Phillips ad Yu, ). I such cases, exuberace ca be modeled i terms of a mildly explosive process with a autoregressive root + b k for which b > ad k + k! : It is especially iterestig i practical work to study trasitios betwee ormal market behavior, which ca be represeted i terms of a uit root or ear uit root model, ad exuberat behavior. The emergece of market exuberace or a asset price bubble may the be modeled as a structural break i which the (log ru) autoregressive coe ciet of the model shifts from beig ear to uity to mildly explosive. Datig such a trasitio amouts to date stampig the emer-

3 gece of exuberace. A similar trasitio from a exuberat to a mildly itegrated or mea revertig process captures the collapse of a asset price bubble ad correspodigly eables the date stampig of bubble termiatio. Phillips, Wu ad Yu () ad Phillips ad Yu () showed how to perform tests of these hypotheses ad costruct date stampig algorithms that were empirically implemeted to characterize the 99s Nasdaq bubble ad the evets leadig up to ad followig the recet global acial crisis. The work i those papers dealt with the special case where the ormal period model was a strict uit root process. The methods of the preset paper allow for these methods to be exteded to the wider class of local to uity processes (for ormal periods) ad eable tests to be developed to distiguish such roots from mildly explosive ad mildly itegrated roots, thereby wideig the rage of potetial empirical applicatios. I particular, the preset paper cosiders time series models with mixed ad commo roots i the viciity of uity. To simplify expositio, we work with a bivariate model ad aalyze a case of primary iterest where there is oe local to uit root ad oe mildly explosive root. Models of this type may be aticipated whe there are maifestatios of ostatioarity i the data but somewhat di eret idividual characteristics i the two series. Or it may be that the behavior is commo across the series for istace i several asset prices arisig from a sigle source of persistece or exuberace. We may be particularly iterested empirically i testig commoality i persistece or log ru behavior across series, which occurs whe the autoregressive roots have the same value. The methods of the curret paper eable empirical researchers to coduct such tests. The remaider of the paper is orgaized as follows. Sectio cosiders mixed VARs whose variates have mixed degrees of persistece that allow for a local to uit root ad a mildly explosive root. A limit theory for least squares regressio ad associated Wald tests for commoality i the autoregressive coe ciets is developed. Sice the ull hypothesis is composite ad ivolves the ukow (local to uity) localizig coe ciet, stadard Wald tests have limit distributios that are parameter depedet ad do ot have uiform size. Modi ed Wald statistics for testig commoality i log ru behavior are developed ad show to produce cosistet tests. I particular, this modi catio esures that the tests are completely cosistet i the sese that size goes to zero ad power to uity asymptotically. Sectio 3 cosiders a model selectio approach ad shows that the BIC criterio ca also distiguish persistet ad mildly explosive behavior. Sectio 4 cocludes. A techical Appedix icludes subsidiary lemmas ad proofs of the mai results. 3

4 Mixed Variate VARs For simplicity of expositio, we cosider the bivariate VAR() model X t R X t + u t ; t ; :::; ; (.) R ; + c ; + b ; b > ; (.) k which we write i compoet form as X t X t X t X t + u t u t ; t ; :::; (.3) with iitializatio X o p k ; ad martigale di erece iovatios u t satisfyig Assumptio below. Our results may be exteded to systems with weakly depedet errors u t uder coditios like those i the liear process framework of Magdalios ad Phillips (9), but all the key ideas follow as i the simpler VAR() model studied here so we do ot provide details. The coe ciet + c is local to uity, + b k is a mildly explosive coe ciet with b > ad the sequece k satis es k + k! as! : The power rate k for (; ) satis es this latter coditio as well as coditios we use later i the paper to develop cosistet test procedures that ivolve slowly varyig fuctios L! : I particular L! for all such fuctios L : Although! as! (so both coe ciets are i the viciity of uity), k! b > ad so is further from uity tha for all ite c as! : I order to distiguish the mildly explosive behavior iduced by from the persistece iduced by, statistical tests eed to di eretiate from for all ite c as! : As we will show, cosistet tests ca be costructed to discrimiate betwee such localizig coe ciets. The fact that cosistet tests of hypotheses ivolvig localizig coe ciets is possible is relevat to practical work where there is substatial iterest i idetifyig exuberace i asset price data. It is also of theoretical iterest because it is well kow that the localizig coe ciet c caot be cosistetly estimated i local to uity speci catios. By cotrast, the localizig coe ciet b i mildly itegrated ad mildly explosive speci - catios is cosistetly estimable, ad it is this feature of the model that makes possible cosistet testig of di ereces i localizig behavior. The diagoal form of R i (.) coforms with stadard practice i the stochastically 4

5 ostatioary literature. The presece of o-zero o diagoal elemets i R iduces higher order stochastic treds or explosive mechaisms i the time series, at least uless those coe ciets are local to zero or egligible. Hece, o-zero o diagoal elemets i R result i ampli ed feedback across series i ostatioary autoregressios. It is therefore covetioal practice to retai a diagoal form R i developig a limit theory, as we do here. Of course, if the cotext ad characteristics of the series suggest the presece of such feedbacks the they may be icluded ad their e ects o the limit theory ca be aalyzed. Assumptio. The errors fu t g i (.) form a martigale di erece sequece with respect to the atural ltratio F t (u t ; u t ; :::) satisfyig E Ft u t u t ad EFt ku t k a:s: for all t (.4) for some > ad positive de ite matrix ; sup t E ku t k 4 < ; ad max E ku t k fku t k > g! as! (.5) t for ay sequece ( ) N such that!, ad where is the spectral orm of the matrix M: o kmk max i i : i is a eigevalue of M M As expected from the di ereces i the coe ciets ad i (.3), the time series compoets X t ad X t have di eret orders of magitude as! : These di ereces traslate ito di eret rates of covergece of the sample momets of X t ad the least squares regressio compoets. To accommodate these di ereces we employ the (asymptotically equivalet) ormalizig matrices D : k ad F : ( ) The urestricted least squares regressio estimate of R i (.) is writte i stadard otatio as ^R X X X X : This estimate is cosistet ad has a limit distributio that is obtaied from a combiatio of fuctioal limit theory that applies to : 5

6 the persistet compoets ad cetral limit theory that applies to the mildly explosive compoets, as detailed i the followig result. Theorem. As! ; ^R R F ) R J c(r)db(r) R J c(r) dr Y (b) X (b) : ; (.6) where J c (r) R r ec(r s) db (s); which is a Orstei-Uhlebeck (O-U) process, B (r) (B (r) ; B (r)) is bivariate Browia motio with variace matrix ; X(b) (X (b) ; X (b)) N(; b ); Y (b) d X(b); ad X(b) ad Y (b) are idepedet. The two colum compoets R J c(r)db(r) of the limitig matric variate are idepedet. R J c(r) dr ad Y (b) X (b) Remarks. The two colums of ^R R coverge at di eret rates, the rst at the usual O () rate for ear itegrated regressios ad the secod at the mildly explosive rate O (k ) O k e bk : I particular, writig ( ij ) ; we have ( ) (br r ) ) (br r ) ) R J c(r)db (r) R J ; (.7) c(r) dr R J c(r)db (r) R J ; (.8) c(r) dr ( ) (br r ) ) Y (b) X (b) ; ( ) (br r ) ) Y (b) X (b) : (.9). The process J c (r) R r ec(r s) db (s) that appears i the limit variate ivolves compoet B (r) of B(r); so that the limit variate R J c(r)db (r) R J c(r) dr has a stadard local uit root distributio that is idepedet of but is depedet o c: 3. The limit variate Y (b) X (b) (b) Y (b) : Y (b) X (b) X is idepedet of b ad we ca therefore write Y (b) X (b) : Y X ; where Y N (; ) ; X (X ; X ) N (; ) ; ad X ad Y are idepedet. 6

7 As idicated earlier, we may be iterested i testig commoality of persistece characteristics i the compoet series X t ad X t : I the preset case, settig R (r ij ) ad uder a maitaied hypothesis that R is diagoal with roots local to uity, commoality amouts to testig the hypothesis H : r r + c for some ite c ( ; ) : The ull ca be writte as H : a vec (R ) where a [; ; ; ] without explicitly specifyig a commo persistece parameter r + c: H may also be subsumed i a block test of R r I for some r + c ; which we ca write i the form HA : A vec (R ) where we use row vectorizatio i the vec operator ad A 6 4 The stadard Wald test of H uses the statistic W : 4 a a a : a o vec ^R a ^ X X a ; ad the correspodig block test of H A has the form W A A vec ^R A ^ o X X A A vec ^R A ^ X o X A A vec ^R A vec ^R ; where ^ P t ^u t^u t is a cosistet estimator of based o the least squares residuals ^u t X t ^R X t : b k Uder (.3) the coe ciets r ad r ; so that r r c o(); which is local to zero. Hece the model (.) actually correspods to a local alterative to the ull H : Theorem. Uder the ull hypothesis H : R r I with r + c, as! b k W ) a (a ) R ; (.) J c(r)j c (r) a 7

8 ad Z ) W A ) A A ( J c (r)j c (r) dr A! A ; (.) where J c (r) R r ec(r s) db(s); vec () ad R dbj c R J cj c : Uder the alterative H : R diag ( ; ) k b W ; W A R f + o p ()g O p : (.) J c(r) k dr Remarks 4. The ull limit distributios (.) ad (.) are parameter depedet. The depedece ivolves the localizig coe ciet c ad the variace matrix : Whe c ; Z Z Z Z dbb BB dv V V V : V where V BM (I ) is stadard vector Browia motio. The limit distributio of the Wald statistic is the W ) a a V R V V a (b V ) R b I V V b ; (.3) where V vec ( V ) ad b a (a ( ) a ) lies o the uit sphere b b : Thus, eve i the case of a commo uit root, the ull limit distributio of the test depeds o ; although this matrix is cosistetly estimable by the residual momet matrix ^: I the geeral case, the limit distributios (.) ad (.) both have uisace parameters (c; ) : 5. The parameter c is ot cosistetly estimable ad it is therefore ot possible to costruct a stadard test of the composite H : However, modi ed tests are available to distiguish H from alteratives that ivolve a mildly explosive compoet. For istace, for some (possibly slowly varyig) sequece L! ; the statistic W L 8

9 W L! p uder H for all ite c: The, uder the alterative hypothesis H ; W L O p kl which diverges for all sequeces L! such that k L! : I particular, if k O ( ) for some (; ) ad L is slowly varyig at i ity, the W L O ( ) p L! as! ad tests based o the statistic W L with ay xed critical value are cosistet ad have zero size asymptotically. Similar remarks apply to the block test based o WL A W A L : 6. I view of (.), W ; W A O p ad the Wald statistics diverge, as do the k scaled statistics W L ad WL A. So there is discrimiatory power uder the local alterative H : r + c ; r + b k : 3 Model Selectio Aother approach to testig for commo roots i (.) is to apply model selectio methods. This ivolves estimatig (.) i the restricted case uder the ull of a commo root ad uder the alterative of urestricted roots. Estimatig (.) uder the restrictio R r I gives the pooled least squares estimator ^r ( P t X tx t ) P t X t X t of the commo root r : We have the followig limit theory for ^r uder the ull hypothesis ad alterative. Lemma 3. (i) Uder the ull R r I with r + c ; ^r has the limit distributio Z (^r r ) ) Z J c (r) db J c (r) J c (r)dr ; (3.) ad the residual momet matrix ~ P t ~u t~u t! p ; where ~u t X t ^r X t ; has the form ~ X u t u t + O : (3.) t (ii) Uder the alterative hypothesis where R diag ( ; ) ; ^r has the limit distributio k (^r ) ) b Y (b) X (b) ; (3.3) For example, asymptotic critical values might be computed for the limit distributio (.3) with I ad b a (a a) 9

10 where Y (b) d X (b) N ; b ad Y (b) ad X (b) are idepedet. The residual momet matrix ~ of the restricted regressio has the followig asymptotic behavior uder the alterative hypothesis: ~ X t u t u t + b k P t X t f + o p ()g : (3.4) Sice P t u tu t! p, it follows from (3.) that ~ is cosistet for uder the ull. However, from (3.4) ad the fact that P t X t ) R J c ; it is apparet that ~ is cosistet for whe o k but is icosistet whe k O () ad, i particular, whe k o : These results eable us to determie coditios for the cosistecy of model selectio criteria such as the Schwarz criterio (BIC). are: For the model (.), the restricted regressio ad urestricted regressio BIC criteria BIC r log ~ + log ; BIC u log ^ + 4 log : Whe the ull holds ad R r I it is evidet that BIC r log ~ + log log + log + O p ; (3.5) whereas for the urestricted regressio BIC u log ^ + 4 log log + 4 log + O p (3.6) sice ^ + O p aalogous to the proof of (3.). I view of (3.5) ad (3.6), BIC r < BIC u up to a term of O p : The restricted model will therefore be correctly chose with probability approachig uity uder the ull. Whe the alterative holds, (3.6) cotiues to apply for the urestricted regressio. But uder the alterative for the restricted regressio we have from (3.4) log ~ log + b P t X t k f + o p ()g log + log I + b P t X t f + o p ()g k

11 where : log + b tr ( k log jj + b k tr ( P t X t P t X t log jj + b P t X t k f + o p ()g ; : : The ) ) f + o p ()g f + o p ()g BIC r log ~ + log log + b P t X t f + o p ()g + log : : k It follows that BIC r > BIC u uder the alterative as! wheever b P t X t k : > 3 log ; which iequality holds with probability approachig uity provided! as! k log because P t X t ) R J c > with probability oe. Hece, uder the alterative, the urestricted model will be chose with probability approachig uity as! provided k goes to i ity slower tha (log ) k(log ) ; that is provided! : It follows that model selectio by BIC is cosistet ad as! the criterio will successfully distiguish roots i the viciity of uity provided oe of the roots + b k is mildly explosive ad su cietly di eret from local to uity i the sese that k! slower tha O where L L is a slowly varyig fuctio that diverges at least as fast as L (log ) ; i.e., lim if! >. I this respect, the discrimiatory capability of (log ) model selectio is aalogous to that of classical Wald testig. 4 Coclusio Model selectio by BIC is well kow to be blid to local alteratives i geeral (see Ploberger ad Phillips, 3; ad Leeb ad Poetscher, 5). For istace, i the curret set up, BIC caot cosistetly distiguish betwee a model with a uit root ( ) ad models with roots local to uity ( + c ), just as localizig coe ciets such as

12 the parameter c are ot cosistetly estimable. O the other had, as show here, BIC ad classical tests ca successfully distiguish roots i the immediate locality of uity like from roots that are i the wider viciity of uity like ; which opes the door to distiguishig mildly explosive behavior i data. We expect these model selectio results to be geeralizable to models with weakly depedet iovatios, aalogous to the digs i Phillips (8) o uit root discrimiatio ad Cheg ad Phillips (9) for coitegratig rak determiatio. Tests of this type will be useful i empirical work where it is of iterest to di eretiate betwee the behavioral time series character of acial data such as asset prices ad the fudametals that are believed to determie prices, like divideds ad earigs. I such cases, the primary maitaied hypothesis is that the series have roots that are local to uity (without beig speci c about the localizig coe ciet) ad the alterative is that oe or other of the series may be mildly explosive at least over subperiods of data (see Phillips, Wu ad Yu, ; Phillips ad Yu, ). O the other had, if the primary maitaied hypothesis is that both series may be mildly explosive ad the ull hypothesis is commoality i the roots, the problems of bias ad icosistecy may arise i testig ad model selectio. Recet work by Nielse (9) ad Phillips ad Magdalios () provide a limit theory for least squares regressio i the case of purely explosive commo roots ad show that least squares regressio is icosistet. That work may be exteded to the case of commo mildly explosive roots ad will be explored i later work. 5 Appedix 5. Prelimiary Lemmas We start with some lemmas that assist i the asymptotic developmet. These results rely o existig limit theory so we oly sketch the mai details here for coveiece. We repeatedly use the fact that k ( ) b+o( k ) ad exp( b k ) f + o()g o(): The rst result is from PM7a. See also Phillips ad Magdalios (8) ad Magdalios ad Phillips (9) for related results o systems with explosive ad mildly explosive processes.

13 Lemma 5. (PM7a) De e X (b) Y (b) X (b) X (b) Y (b) Y (b) : p k : p k X j j u j ; X ( j) u j : j The, as! ; X (b))x(b) (X (b) ; X (b)) N(; b ); ad Y (b))y (b) (Y (b) ; Y (b)) ; where Y (b) d X(b); ad X(b) ad Y (b) are idepedet. Lemma 5. De e S (r) : p P brc j u j ad X(r) c X brc p brc X p j u brc j ; X (b) X p k p k j X j u j j : The, as! ; (i) S (r) p P brc j u j p P brc j u j ) B (r) B (r) B(r) BM(); (ii) X c (r))j c(r) R r ec(r s) db (s) ad P j X t u t ) R J c(r)db(r); (iii) X (b))x (b); where X (b) N ; b ; (iv) J c (r) ad X (b) are idepedet. (v) For all s; r > the followig joit covergece applies: Xbrc X bsc p ; p k bsc ) [J c (r); X (b)] ; as! : Proof. Result (i) is stadard, (ii) is from Phillips (987b), ad (iii) is from lemma 5.. To prove (iv), it su ces to show that B (r) ad X (b) are idepedet, sice J c (r) is a 3

14 fuctioal of fb (s)g sr : Note that the covariace E (S ()X (b)) E 4@ p p k X X j k k u j A p k X k p k p k f + o()g b u k k! 3 5 r k f + o()g o(); as! : Idepedece of the limit processes J c (r) ad X (b) follows. To prove (v), rst observe that for ay (iteger sequece) L! such that L X (b): Note that X (b) E X u j p k j jl + X L p k L sice L + b L k X L pk L k + p P u j k jl + j X jl + j k L ad k k L+! ; we have o(); +L X L pk L + b k L k k exp( b L k ) + o() o(): Hece, ) X (b) by lemma 5.. Now let L bsc for ay s > ad the X bsc p ) k bsc X (b): Joit covergece ad (v) follow from margial covergece ad asymptotic idepedece of the compoets. Lemma 5.3 As! ; P (i) k t X t ) (X (b)) b ; (ii) P t X t ) R J c(r) dr; (iii) k P t X t X t o p (): Proof. (i) follows from PM7a ad (ii) is stadard h (Phillips, i 987a&b). For (iii), it is Xbrc coveiet to take a probability space where p ;! p [J c (r); X (b)] : The, X bsc p k bsc ) 4

15 for ay sequece L! such that L k X X t X t t! ; we have ( X L p k X (b) p k + L p k t X + tl + XL t X tl + J c t ) Xt p Xt p k t t Xt p Xt p k t t f + o p ()g t L Now P t J c E X (b) p k X (b) p k X t X tl + X t t J c t t J c t + o p (): t t has zero mea ad variace! t J c t X X t s M L L f + o p ()g + O p p k for some ite costats M ad M : It follows that X! t Var p k J c t tl t s E J c J c t+s M k b ; k O k O k o(); P leadig to p t k tl J c t P o p (); which implies that k t X t X t o p () ad this also holds i the origial probability space, givig the required result. Lemma 5.4 As! ; R (i) D X X D ) J c(r) dr h R (ii) u X D ) J c(r)db(r) X (b)y (b) (X (b)) b 5 i : ;

16 Proof. Usig lemma 5.3 D X X D D )! X X t Xt t P t X t k D P k t X t X t P t X t X t R J c(r) dr (X (b)) b ; k P t X t givig (i). Result (ii) follows directly from lemmas 5. ad 5.3 as u X D ) h h P t P t X t u t k Xt p p ut P t X t p k h R J c(r)db(r) X (b)y (b) u t i P t Xt i : pk t u t t Joit covergece follows from the idepedece betwee B(r) ad (X (b); Y (b)): i 5. Proofs of the Mai Results Proof of Theorem.. we have ^R Usig Lemma 5.4, cotiuous mappig ad joit covergece, R D u X D D X X D R ) J c(r)db(r) R J c(r) dr Y (b) X (b)b : Sice ( ) b k ( + o()) the equivalet result ^R R F ) R J c(r)db(r) R J c(r) dr Y (b) X (b) ; holds as stated. Proof of Theorem.. We rst prove (.) ad (.) for the statistic W : Uder the ull we have by stadard theory ^R R ) Z Z Z dbjc J c Jc : ; X X ) J c Jc (5.) 6

17 ^ P t ^u t^u t! p ; ad (.) follows directly for W ad (.) for W A. Uder the alterative from theorem. with correct ceterig we have o a vec ^R R F (br r ) ( ) (br r ) )a vec; whereas uder (.) with b > ; the ull cetred liear combiatio behaves as a vec ^R (br br ) (br r ) (br r ) + (r r ) (br r ) ( ) (br r ) ( ) b + c k (br r ) + c b k + o p () (br r ) + O p ( k )! ; as! ; i view of (.7) - (.9) ad sice ( ) settig d P P t X t t X t d that 8 X X >< d Xt Xt >: ad t X t Xt t t k k k ( ) O( exp(b )) o(): Next, k ( P t X t X t ) ad usig Lemma 4.3 we P k t X t X t P t X t k P t X t X Xt f o p ()g ; (5.) 9 > >; d k X X X t k t Xt f o p ()g t Z! ) J c (r) X (b) dr : b 7

18 It follows that P X P X t X t t X t X t P d t X P t X t t X t 3 Sice ^! p ; we have 4 4 P t X t P t X t P X t P t X t t X t P t X t + op () o p () o p () o p () P t X t P X t P t X t t X t P t X t 3 5 ) 4 5 f + o p ()g(5.3) R J c(r) dr a o ^ X X a 8 P < 39 a t X t : ( + o p ()) 4 + op () o p () 5 o p () o p () ; a 8 < R 39 ) a : 4 J c(r) dr 5 ; a Z J c (r) dr : 3 5 : It follows that W a o vec ^R a ^ X X o (br r ) + O p ( k ) R J c(r) dr + op () O p k a ; givig the stated result. The proof of (.) for the statistic W A uder the alterative follows the same lies but ivolves more complex calculatios to cope with di eret orders of magitude i the compoets. First cosider the behavior of the cetred elemets uder the alterative. By 8

19 (.7) - (.9) we have o A vec ^R R F 6 4 (br r ) ( ) (br 3 r ) ( ) (br r ) 7 5 (br r ) )A vec: O the other had uder (.) with b > ; the ull-cetred liear combiatios behave as follows. First, a vec ^R (br br ) (br r ) (br r ) + (r r ) (br r ) ( ) (br r ) ( ) b + c k (br r ) + O p ( k )! ; as! ; as for W. Secod a vec ^R br ( ) br ( )! k O p exp(b o p () ; k ) ad third a 3 vec ^R br ) a 3vec; as! : Also, as i (5.3) X X 4 P P t X t t X t P X t P t X t t X t P t X t X t P P t X t t X t P t X t 3 5 f + o p ()g : We ow evaluate each of the compoets of the matrix ^ A o X X A 3 a 6 4 a 7 ^ X 5 X ^ X X ^ X X ^ X X [a ; a ; a 3 ] : a 3 9

20 Usig lemma 4.3 we d a ^ X X ^ X X a ^ X X ^ X X ^ P t X t ^ P f + o p ()g ; t X t P t + ^ X t X t P P + ^ P t X t t X t t X t f + o p ()g a ^ X X ^ X X ^ X X ^ X X a P t ^ X t X t P P + ^ P t X t t X t t X t f + o p ()g P t ^ X t X t P P f + o p ()g ; t X t t X t a a 3 ^ X X ^ X X ^ X X ^ X X a ^ P f + o p ()g ; t X t ^ X X ^ X X ^ X X ^ X X a 3 ^ P f + o p ()g ; t X t a ^ X X ^ X X a 3 ^ X X ^ X X ^ P t X t P t ^ X t X t P P f + o p ()g t X t t X t P t ^ X t X t P P f + o p ()g ; t X t t X t a ^ X X ^ X X P t ^ X X ^ X X a 3 ^ X t P X t P f + o p ()g ; t X t t X t

21 ad Hece a 3 ^ X X ^ X X ^ X X ^ X X a 3 ^ P f + o p ()g : t X t ^ A o X X A ^ P t X t 6 4 ^ P t X t X t P P t X t t X t ^ P t X t ^ P t X t X t P P t X t t X t ^ P t X t X t P P t X t t X t ^ P t X t f + o p ()g : Set K diag(; ; k ) ad observe that ^ K A o X X AK ^ P o p () o p () t X t 6 ^ P o p () 4 t X t k ^ P t X t f + o p ()g sice P t X t X t P P t X t t X t o p () ; k P t X t X t P P t X t t X t k P t X t X t P t X t k P t X t o p () ;

22 by lemma 4.3(iii). We deduce that W A A vec ^R A ^ o X X A A vec ^R A vec ^R K K ^ A o X X AK K A vec ^R 3 P 6 t X t 7 ^R K A vec 6 4 P t X t k P t X t K A vec ^R f + o p ()g : (5.4) 7 5 Next A vec ^R K A vec ^R R K + A vec (R ) K 3 3 (br r ) (br r ) (r r ) (br r ) k (br r ) (br r ) + O p O p k k (br r ) k 3 c k b ; (5.5) from Theorem. ad (.7) - (.9). It ow follows from (5.4) ad (5.5) that W A 6 4 b + O p () ; o p () ; O p () 6 k 4 k b k b + O p () o p () O p () Z givig the stated result P t X t J c (r) dr f + o p ()g ; P t X t k P t X t 3 7 5

23 Proof of Lemma 3.. Part (i) follows by stadard methods i view of Lemmas Also ~u t X t ^r X t u t (^r r ) X t, ad so we have X X ~ u t u t + (^r r ) t t t X t u t + u t Xt X + (^r r ) X t Xt X u t u t + O p ; (5.6) as stated. For part (ii) to obtai the limit distributio uder the alterative, write ^r as ^r X X t X t + t P t X t P t X t +! X X t X t t P t X t + P t X t P + t X t P t X t + P t X t P t X t + P t u tx t + X Xt + t P t + u tx t P t X t X t X t! t + P t u tx t + P t X t P t X t + P t u tx t P t X t + P t X t P t X t P t + X t P t X t P t X t P t + u tx t + P P t X t t u tx t P t X t P t + X t P : t X t The, usig Lemma 5.3 ^r ( ) P t X t P t X t P t f + o p ()g + u tx t + P t u tx t P t X t f + o p ()g k + k P k t u tx t + c b k k k P t X t P t u tx t P t X t P f + o p ()g k t X t P k k t u tx t f + o p ()g ; k P t X t f + o p ()g 3

24 ad i view of Lemma 5. k (^r ) ) X (b) Y (b) X (b) b b Y (b) X (b) ; givig the stated result (3.3). To prove (3.4), rst ote that b ^r (^r ) + ( ) k c + O p k : The restricted regressio residuals are ~u t X t ^r X t u t (^r I R ) X t u t u t (^r ) X t + u t + b k X t ( ) X t f + o p ()g : ^r ^r X t Let P t u tu t ad the! p ad ~ + b k + b k + b k ( X t X t X t h i ) u t + u t X t h i X t X t P t X t f + o p ()g ; sice P t X t u t O p () by Lemma 5.(ii) ad k! : 6 Refereces Cha, N. H. ad C. Z. Wei (987). Asymptotic Iferece for Nearly Nostatioary AR() Processes, Aals of Statistics 5, Cheg, X ad P. C. B. Phillips (9). Semiparametric coitegratig rak selectio, 4

25 The Ecoometrics Joural,, pp. S83-S4. Hall, P. ad C.C. Heyde (98). Martigale Limit Theory ad its Applicatio. Academic Press. Lai, T.L. ad C.Z. Wei (98). Least Squares Estimates i Stochastic Regressio Models with Applicatios to Ideti catio ad Cotrol of Dyamic Systems. Aals of Statistics,, Leeb H. ad B. Pötscher, Model selectio ad iferece: facts ad ctio, Ecoometric Theory,, pp Magdalios, T. ad P. C. B Phillips (9). Limit Theory for Coitegrated Systems with Moderately Itegrated ad Moderately Explosive Regressors. Ecoometric Theory, 5, Nielse, B. (9). Sigular vector autoregressios with determiistic terms: Strog cosistecy ad lag order determiatio. Uiversity of Oxford workig paper. Paga, A. ad A. Ullah (999). Noparametric Ecooometrics. Cambridge Uiversity Press. Ploberger, W. ad P. C. B. Phillips (3). Empirical limits for time series ecoometric models Ecoometrica, 7, Phillips, P. C. B. (987a). Time Series Regressio with a Uit Root, Ecoometrica, 55, 77 3 Phillips, P. C. B. (987b). Towards a Ui ed Asymptotic Theory for Autoregressio, Biometrika 74, Phillips, P. C. B. (8). Uit root model selectio, Joural of the Japa Statistical Society, 38, Phillips, P. C. B., T. Magdalios (7a), Limit Theory for Moderate Deviatios from a Uit Root, Joural of Ecoometrics 36, 5-3. Phillips, P. C. B., T. Magdalios (7b), Limit Theory for Moderate Deviatios from a Uit Root Uder Weak Depedece, i G. D. A. Phillips ad E. Tzavalis (Eds.) 5

26 The Re emet of Ecoometric Estimatio ad Test Procedures: Fiite Sample ad Asymptotic Aalysis. Cambridge: Cambridge Uiversity Press, pp.3-6. Phillips, P. C. B. ad T. Magdalios (8). Limit theory for explosively coitegrated systems, Ecoometric Theory, 4, Phillips, P. C. B., T. Magdalios (), Icosistet VAR Regressio with Commo Explosive Roots, Workig Paper, Yale Uiversity. Phillips P. C. B., Y. Wu ad J. Yu (). Explosive behavior i the 99s Nasdaq: Whe did exuberace escalate asset values?, Iteratioal Ecoomic Review, 5, pp. -6. Phillips P. C. B. ad J. Yu (). Datig the Timelie of Fiacial Bubbles durig the Subprime Crisis, Quatitative Ecoomics,, pp Ullah, A. (4). Fiite Sample Ecoometrics. Oxford: Oxford Uiversity Press. 6