LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS. PETER C. B. PHILLIPS and TASSOS MAGDALINOS COWLES FOUNDATION PAPER NO. 1244

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1 LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS BY PETER C. B. PHILLIPS ad TASSOS MAGDALINOS COWLES FOUNDATION PAPER NO COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box New Have, Coecticut

2 Ecoometric Theory, 24, 2008, Prited i the Uited States of America+ doi: s LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS PETER C.B. PHILLIPS Cowles Foudatio for Research i Ecoomics, Yale Uiversity Uiversity of Aucklad ad Uiversity of York TASSOS MAGDALINOS Uiversity of Nottigham A limit theory is developed for multivariate regressio i a explosive coitegrated system+ The asymptotic behavior of the least squares estimator of the coitegratig coefficiets is foud to deped upo the precise relatioship betwee the explosive regressors+ Whe the eigevalues of the autoregressive matrix Q are distict, the cetered least squares estimator has a expoetial Q rate of covergece ad a mixed ormal limit distributio+ No cetral limit theory is applicable here, ad Gaussia iovatios are assumed+ O the other had, whe some regressors exhibit commo explosive behavior, a differet mixed ormal limitig distributio is derived with rate of covergece reduced to M+ I the latter case, mixed ormality applies without ay distributioal assumptios o the iovatio errors by virtue of a Lideberg type cetral limit theorem+ Covetioal statistical iferece procedures are valid i this case, the statioary covergece rate domiatig the behavior of the least squares estimator+ 1. INTRODUCTION Autoregressios with a explosive root 6u6 1 came to promiece after the early work of White ~1958! ad Aderso ~1959!+ Assumig Gaussia iovatio errors, these authors derived a Cauchy limit theory for the cetered least squares estimator with rate of covergece u + The theory was geeralized by Mijheer ~2002! to o-gaussia explosive processes geerated by iovatios satisfyig a stability property+ I each of these works, o cetral limit theory applies, ad the asymptotic distributio of the least squares estimator is characterized by the distributioal assumptios imposed o the iovatios+ Phillips thaks the NSF for research support uder grat SES Address correspodece to Peter C+B+ Phillips, Departmet of Ecoomics, Yale Uiversity, P+O+ Box , New Have, CT , USA; peter+phillips@yale+edu Cambridge Uiversity Press $

3 Z Z 866 PETER C.B. PHILLIPS AND TASSOS MAGDALINOS I this paper, we cosider a explosively coitegrated system y t Ax t «t, x t Qx u t, Q I K C, C diag~c 1,+++,c K!, c i ~,2! ~0,! i, (1) (2) where A is a m K matrix of coitegratig coefficiets, x t is a K-vector of explosive autoregressios iitialized at x 0 0, ad v t ~«t, u t! is a sequece of idepedet, idetically distributed ~0, S! radom vectors with absolutely cotiuous desity, where S is a positive defiite matrix partitioed coformably with v t as S diag~s ««, S uu!+ We deote by u i 1 c i the ith diagoal elemet of Q ad by 7Q7 max 1iK 6u i 6 the spectral orm of Q+ The asymptotic behavior of the least squares estimator A y t x t 1 x t x t is foud to deped o the relatioship betwee the regressors i ~2!, i+e+, o the precise form of the matrix Q+ As Theorem 2+1, which follows, shows, the rak of the limit matrix of the ormalized sample secod momets, ad hece the order of magitude of ~ x t x t! 1, is determied exclusively by Q+ Whe Q yields a osigular limit i Theorem 2+1, AZ A is foud to have a Q rate of covergece ad a mixed ormal limitig distributio, uder the assumptio of Gaussia iovatios ~cf+ Aderso, 1959!+ But whe the limit momet matrix of Theorem 2+1 is sigular, A A has a degeerate mixed ormal limitig distributio with covergece rate reduced to The asymptotics i the sigular case are obtaied by rotatig the regressio coordiates i a way that the sigularity is elimiated ad cetral limit theory applies+ Cosequetly, the mixed ormal limit theory i the sigular case applies without ay distributioal assumptios o the iovatio errors+ Explosive systems are useful i modelig periods of extreme behavior i ecoomic ad fiacial variables+ Ecoomic growth amog the Asia dragos durig the 1980s ad recet growth i Chia provide examples of mildly explosive growth i macroecoomic variables+ Hyperiflatio i Germay i the 1920s ad Yugoslavia i the 1990s are examples of some of the may historical istaces of explosive behavior i prices+ Fiacial bubbles i asset prices are aother example, the recet rise ad subsequet fall i price of Iteret stocks i the NASDAQ market creatig ad destroyig some $8 trillio of shareholder wealth+ To the extet that periods of explosive movemet i such variables ifluece ecoomic decisios or cotamiate other variables, we may expect models of explosive coitegratio such as ~1! ad ~2! to be relevat i relatig these variables+ Whe there is a sigle source of the extreme movemet, the such a system may also have explosively coitegrated regressors, ad the degeeracy described earlier may occur+

4 Z 2. RESULTS LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS 867 We develop a limit theory for the cetered least squares estimator A A «t x t 1 x t x t + It turs out that the asymptotic order of AZ A depeds o the rak of the limit of the ormalized sample momet matrix x t x t + The latter ca be derived, usig a similar method to Aderso ~1959!, i terms of the radom vector X Q Q j u j, j1 (3) where the series coverges almost surely by virtue of the martigale covergece theorem+ THEOREM 2+1+ The sample momet matrix of the explosive process (2) satisfies Q x t x t Q r a+s+ Q j X Q X Q Q j as r, (4) j0 where X Q is the radom vector defied i (3). Note that the almost sure limit j0 Q j X Q X Q Q j of the ormalized sample momet matrix is ot always osigular+ Deote the ith elemet of the radom vector X Q by X! Q ad defie the ~i matrices M Q : u i u j u i u j 1 : i, j $1,+++,K% ad X Q : diag~x Q ~1!,+++,X Q ~K!!+ Because u 1 admits a absolutely cotiuous desity, X Q ~i! 0 almost surely for each i+ Thus, the idetity j0 Q j X Q X Q Q j X Q M Q X Q implies that j0 Q j X Q X Q Q j is osigular wheever the matrix M Q is osigular, i+e+, if ad oly if c i c j for all i j ~cf+ Lemma 4+3!+ O the other had, whe ay two localizig coefficiets c i, c j are the same, the matrix M Q will have two idetical colums ad will, therefore, be sigular+ We begi by discussig the osigular asymptotic momet matrix case+

5 868 PETER C.B. PHILLIPS AND TASSOS MAGDALINOS THEOREM 2+2+ For the explosive coitegrated system geerated by (1) ad (2) with v t d N~0, S! ad c i c j for all i j, the followig limit theory applies as r : vec@~ AZ A!Q # MN0, j 1 Q j X Q X Q Q S ««. Remarks 2.1. j0 ~i! The assumptio of Gaussia iovatios is essetial to obtai a mixed ormal limitig distributio for the least squares estimator+ This is because, despite beig asymptotically equivalet to a martigale array ~see ~34! i Sect+ 4!, the sample covariace does ot satisfy the requiremet of uiform asymptotic egligibility or the Lideberg coditio ~cf+ Hall ad Heyde, 1980, Sect+ 3+2!+ As a result, o cetral limit theory applies i geeral, ad mixed ormality requires Gaussia iovatios, as i the AR~1! case of Aderso ~1959!+ ~ii! Whe v t d N~0, S!, X Q d N~0, j1 Q j S uu Q j!+ ~iii! I the simplest case of a two-equatio system, K 1, ad so x t, A a, ad Q u are scalar+ Lettig Z be a N~0,1! variate, the previous remark yields ~u 2 1! 102 X Q d N~0, S uu! d S 102 uu Z, u Q j X Q X Q Q j 2 u 2j X 2 Q d j0 j0 ~u 2 1! S 2 uu Z 2 + Thus, Theorem 2+2 reduces to u ~ a[ a! MN0, S ««~u 2 1! S uu 2 Z 2 u 2 where Y ad Z are idepedet N~0,1! variates, or u S uu d u 2 1 S ««u S uu 102 Y Z, u 2 1 ~ a[ a! S ««C, where C is a stadard Cauchy variate+ I the geeral case, the exact form of the limitig distributio of Theorem 2+2 ca be obtaied by usig a matrix quotiet argumet, as i Phillips ~1985!+ We ow tur to the discussio of the limit theory i the case of two or more equal localizig coefficiets+ We have see that this case gives rise to a sigular limit matrix for the sample variace, reflectig the fact that the regressors x t are themselves explosively coitegrated+ Because the mixig radom matrix j0 Q j X Q X Q Q j is sigular, the limit theory of Theorem 2+2 does ot apply+ The asymptotic behavior of the least squares estimator ca be determied by a rotatio of coordiates to isolate the explosive ad oexplosive behavior, a

6 LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS 869 method used by Park ad Phillips ~1988, 1989! i the settig of coitegrated processes+ Here, however, the rotatio is radom ad is determied by the limit vector X Q + We start by groupig together the repeated diagoal elemets of Q+ This ca be doe without loss of geerality by premultiplyig ~2! by a appropriate permutatio matrix ~i+e+, a square matrix cosistig of zeros ad oes that cotais exactly oe elemet 1 i each row ad each colum!+ If there are p groups of repeated diagoal elemets of Q the autoregressive matrix ca be rearraged as F diag~f 1, F 2!, F 1 diag~u 1 I r1,+++,u p I rp!, F 2 diag~w 1,+++,w Kr! p r r i, (5) i1 where all w i ad u i are diagoal elemets of Q with w s u l for all s, l ad w i w j, u i u j for all i j+ This effectively rearrages the system of equatios i ~2! ito a system of the form x1t J t u1 x1 u1t J J x pt u p x p x p1, F 2 x p1, (6) u pt u p1, t, where x it R r i icludes the regressors i ~2! that cotai the repeated root u i for each i $1,+++,p% ad x p1, t R Kr icludes the regressors that cotai all distict diagoal elemets of Q+ Lettig xi t ~x 1t,+++,x pt, x p1, t! ad ui t ~u 1t,+++,u pt, u p1, t!, ~6! ca be obtaied from ~2! as follows+ Cosider the K K permutatio matrix P that trasforms x t ito xi t : Px t xi t + The, usig orthogoality of permutatio matrices, ~2! yields xi t PQx ui t PQP Px ui t FxI ui t, (7) where F PQP has the explicit form give i ~5! ad, by orthogoality of P, satisfies the useful idetity F j PQ j P for all j N+ (8) Similarly, we ca write ~1! i terms of xi t as y t AP xi t «t CxI t «t, (9) where C AP + Because ZC C ~ AZ A!P, the asymptotic behavior of AZ is completely determied by that of ZC + I what follows, we show that oly the

7 870 PETER C.B. PHILLIPS AND TASSOS MAGDALINOS first r rows of the permutatio matrix P will cotribute to the limitig distributio of M ~ AZ A!+ It is therefore coveiet to partitio P as P 1 rk P P 2 ~Kr!K, where, by the orthogoality of P, P 1 ad P 2 satisfy P 1 P 1 I r, P 2 P 2 I Kr, P 1 P 2 0, P 1 P 1 P 2 P 2 I K + (10) I particular, the first two equalities i ~10! imply that rak~p 1! r ad rak~p 2! K r+ Coformably, we partitio F j as F j diag~f 1 j, F 2 j!+ The partitioed form of P together with ~8! the gives rise to the idetities F 1 j P 1 Q j P 1, F 2 j P 2 Q j P 2, P 1 Q j P 2 0, (11) Q j P 1 F 1 j P 1 P 2 F 2 j P 2 (12) for all j N+ The limit theory for the coitegrated system ~9! ad ~7! is derived by rotatig the regressio space i a directio orthogoal to X F : P 1 X Q ~P 1 Q j P 1!P 1 u j F j 1 P 1 u j, j1 j1 where the last equality is obtaied usig ~11!+ Correspodig to the partitio of P 1 x t, defie X F ~X F1,+++,X Fp!, X Fi R r i, ad HFi X Fi 0~X Fi X Fi! 102 for each i $1,+++,p%+ We cosider a r i ~r i 1! orthogoal complemet H 4i to each H Fi satisfyig H 4i H Fi 0 ad H 4i H 4i I ri 1 almost surely for all i $1,+++,p%+ The H H 41 0 J H 42 J 0 0 J J J J J 0 0 J H 4p 0 H F1 0 J H F2 J 0 0 J J J J J 0 0 J H Fp J 0 I Kr (13)

8 I is a K K orthogoal matrix that ca be partitioed as H 4 HF 0 4 H U U, U 4 F ~Kr!~rp!, U F 0 r~kr! 0 ~Kr!p I Kr H 4 diag~h 41,+++,H 4p!, H F diag~h F1,+++,H Fp!+ (14) By costructio, the orthogoal complemet matrix H 4 satisfies H 4 X F H 4 H F 0 ad H 4 H 4 I rp almost surely+ Although H 4 is ot uique, its outer product is uiquely defied by the relatio H 4 H 4 I r H F H F a+s+ (15) ~see, e+g+, Abadir ad Magus, 2005, 8+67!+ Moreover, ~14! implies a similar relatioship betwee U F ad U 4, amely, U 4 U F 0, U 4 U 4 I rp ad U 4 U 4 I K U F U F a+s+ (16) ad Applyig the orthogoal trasformatio H to the explosive regressor yields z t HxI 1t, z 2t #, z 1t H 4 P 1 x t R rp, z 2t U F Px t R Krp, (17) ZC C LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS 871 «t xi t H H x t I «t z t z t z t 1 x t H With this rotatio, the limit matrices of both 1 H, H+ (18) z t z t ad ~ z t z t! 1 are well defied after appropriate ormalizatio+ To see this, first observe that, i view of the idetities ~H 4 F 1 1 H 4! j H 4 F 1 j H 4, H 4 F 1 i H F 0 j N, i Z, (19) z 1t satisfies the reverse autoregressio z 1t ~H 4 F 1 1 H 4!z 1 H 4 F 1 1 P 1 u, (20) which, upo recursio, yields for each t t z 1t ~H 4 F 1 1 H 4! t z 1 H 4 F j 1 P 1 u tj + (21) j1

9 872 PETER C.B. PHILLIPS AND TASSOS MAGDALINOS Proofs of ~19! ad ~21! are give i Sectio 4+ Usig ~10! ad ~12! the secod term of ~21! ca be writte as t H 4 F j 1 P 1 u tj H 4 P 1 ~P 1 F j 1 P 1!u tj j1 t j1 t H 4 P 1 ~Q j P 2 F j 2 P 2!u tj j1 t H 4 P 1 Q j u tj j1 r a+s+ H 4 P 1 Q j u tj j1 as r by the martigale covergece theorem+ For the first term of ~21!, usig ~19! ~15!, ~10!, ad ~12! we obtai ~H 4 F 1 1 H 4! t z 1 H 4 F 1 ~t! H 4 H 4 P 1 x H 4 F 1 ~t! ~I r H F H F!P 1 x H 4 F 1 ~t! P 1 x H 4 F 1 t P 1 ~P 1 F 1 P 1!x H 4 F 1 t P 1 ~Q P 2 F 2 P 2!x H 4 F 1 t P 1 Q x H 4 F 1 t ~H 4 H 4 H F H F!P 1 Q x ~H 4 F 1 t H 4!H 4 P 1 Q x r a+s+ ~H 4 F 1 t H 4!H 4 P 1 X Q 0 because X F P 1 X Q + Thus, ~21! implies that z 1t is a R rp -valued statioary ergodic process with the followig liear process represetatio: z 1t H 4 P 1 z a+s+ z Q j u tj + (22) j1 The ergodic theorem the yields, as r, 1 z 1t z 1t H 4 P 1 1 z z P 1 H 4 r a+s+ H 4 P 1 E~z 1 z 1!P 1 H 4 H 4 P 1 Q j S uu Q j P 1 H 4 0, (23) j1

10 LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS 873 where positive defiiteess follows because S uu 0, P 1 has full row rak equal to r, ad H 4 has full colum rak equal to r p+ Thus, i the directio of H 4, the sample variace has the usual 1 ormalizatio that applies uder statioarity+ By stadard iversio of a partitioed matrix ~e+g+, Abadir ad Magus, 2005, 5+18! we obtai 1 z t z t z 1t z 1t z 2t z 1t z 1t z 2t z 2t z 2t 1 Z 1 Z 1 Z 1 1 Z 2 Z 2 Z 1 Z 2 Z 2 ~Z 1 Q 2 Z 1! 1 P 2 ~Z 1 Q 2 Z 1! 1 ~Z 1 Q 2 Z 1! 1 P 2 ~Z 2 Z 2! 1 P 2 ~Z 1 Q 2 Z 1! 1, (24) P 2 where Z 11, z 12,+++,z 1 # R ~rp!, Z 21 R ~Krp!, P 2 ~Z 2 Z 2! 1 Z 2 Z 1, ad Q 2 I Z 2 ~Z 2 Z 2! 1 Z 2 +, z 22 Lemma 4+4 implies that 7Z 2 Z 2 7 O p ~7Q7 2!,7P 2 7 O p ~7Q7!, ad ~ 1 Z 1 Q 2 Z 1! 1 ~ 1 Z 1 Z 1! 1 O p ~ 1!+,+++,z 2 # Thus, i view of ~23!, the large-sample behavior of the sample momet matrix after rotatio of the regressio space is give by 1 z t z t 1 1 Z 1 Q 2 Z 1 O p ~7Q7! O p ~7Q7!! O p ~7Q z 1t z 1t O p ~ 1! O p ~7Q7! O p ~7Q7! p H 4 P 1 r j1 Q j S uu Q j P 1 H 4 1 O p ~7Q7 2! (25) 0 Now that we have established a osigular limit for the sample momet matrix i the ew regressio coordiates the limit theory for the coefficiet

11 874 PETER C.B. PHILLIPS AND TASSOS MAGDALINOS matrix C i ~9! is drive by the sample covariace 102 ~z 1t «t!, which has a mixed ormal asymptotic distributio 1 ~z 1t «t! MN0,H 4 P 1 Q j S uu Q j P 1 H M j1 4 S ««(26) by virtue of a martigale cetral limit theorem+ The proof of ~26! is give i Sectio 4+ For the least squares estimator of C, combiig ~18!, ~14!, ad ~25! yields M ~ ZC C! 1 M «t z 1t 1 1 z 1t z 1t H 4, 0 o p ~1!+ It is ow straightforward to derive a limit theory for the origial coitegrated system ~1! ad ~2! by usig the relatioship AZ A ~ ZC C!P, so that M ~ AZ A! 1 Mvec~ Z M «t z 1t A A! P 1 H z 1t z 1t 1H 4 P 1 o p ~1!, 1 z 1t z 1t I m 1 ~z 1t «t! o p ~1!, M ad the limit distributio of the least squares estimator follows as a cosequece of ~23! ad ~26!+ THEOREM 2+3+ For the explosive coitegrated system geerated by (1) ad (2) with c i c j for some i j the followig limit theory applies as r : Mvec~ AZ A! MN0, P 1 H 4H 4 P 1 Remarks 2.2. j1 Q j S uu Q j P 1 H 4 1 H 4 P 1 S ««. ~i! The limit distributio of the least squares estimator is mixed Gaussia ad sigular, because rak~h 4! r p ad rak~p 1! r implies that P 1 H 4H 4 P 1 j1 Q j S uu Q j P 1 H 4 1 H 4 P 1 is a sigular matrix of rak r p+ Moreover, a M covergece rate applies, which is much slower tha the usual Q rate for explosive processes appearig i Theorem 2+2+ This reductio i the covergece rate results from the fact that some regressors i certai directios are explosively coitegrated with a commo explosive form, whereas the com-

12 [ plemetary set of regressors behave like statioary variates+ These variates slow dow the covergece rate, ad stadard limit theory applies+ ~ii! Ulike Theorem 2+2, Theorem 2+3 does ot require ay distributioal assumptios o the iovatios v t + The limitig distributio of Theorem 2+3 is valid for o-gaussia iovatios as a cosequece of the cetral limit theorem applyig for the sample covariace i ~26!+ ~iii! I the polar case where all localizig coefficiets are equal, Q ui K, r K, p 1, ad P 1 I K, ad so Theorem 2+3 reduces to LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS 875 u 2 1 vec~ AZ A! MN~0, H 4 ~H 4 S uu H 4! 1 H 4 S ««!+ ~iv! A iterestig feature of the limit distributio of Theorem 2+3 isthe relatioship betwee the rak of the limitig covariace matrix ad the order of coitegratio betwee the explosive regressors+ As oted i Remark 2+2~i!, the rak of the limitig covariace matrix is give by ~r p!m r i pm, p i1 where p is the umber of repeated roots of Q ad r i is the umber of times that the repeated root u i appears i Q+ Hece, the limitig covariace matrix assumes its maximum rak, ~K 1!m, whe all diagoal elemets of Q are equal+ O the other had, the iequality r 2p implies that the miimum rak, m, occurs whe r 2 ad p 1, i+e+, whe Q has exactly two equal diagoal elemets+ The rak of the limitig distributio of Theorem 2+3 reflects the fact that the orthogoal trasformatio H removes the sigularity i ~4! by cacelig out the effect of the regressors i ~2! that are ot coitegrated+ The M limit theory of Theorem 2+3 is drive exclusively from the coitegrated part of x t, i+e+, the regressors i ~2! that cotai repeated explosive roots+ ~v! I view of Theorem 2+3, the limit behavior of 1 1 x t x t z t z t HP 1 P H 1 r p P 1 H 4H 4 P 1 j1 Q j S uu Q j P 1 H 4 1 H 4 P 1, ad the fact that ZS ««1 «t «[ t r p S ««, where «[ t y t AZ x t, we obtai covetioal asymptotic chi-squared distributios uder the ull hypothesis for regressio Wald tests such as

13 Z 876 PETER C.B. PHILLIPS AND TASSOS MAGDALINOS W g~ Z A! G A 1 x t x t ZS ««1 G A g~ AZ!, G A ]g ] vec A, for some aalytic restrictios of the form H 0 : g~a! 0+ ~vi! Note that the matrix P 1 H 4H 4 P 1 j1 Q j S uu Q j P 1 H 4 1 H 4 P 1 is ivariat to the coordiate system defiig H 4, ad so the limit theory of Theorem 2+3 is also ivariat to the choice of coordiates+ We ow provide a discussio of the asymptotic behavior of AZ A i the directio of X Q + Recallig the partitioed form of P ad ~14!, the vector HPX U 4 PX Q Q Q U F PX H 4 P 1 X Q Q 0 Q U F PX ~rp!1 U F PX cacels out the effect of ~Z 1 Z 1 0! 1 o the variace matrix i ~24! ad produces a typical explosive limit theory for A + More specifically, lettig D : U F FU F, ~8!, ~18!, ad ~24! yield ~ AZ A!Q X Q ~ ZC C!F PX Q ~ ZC C!H ~HF H!HPX Q ~ ZC C!H diag~u 4 F U 4,U F F U F!HPX Q 0 «t z t z t z t 1 Q ~rp!1 D U F PX «t z 2 Z 2! 1 D P 2 ~Z 1 Q 2 Z 1! 1 P 2 D #U F PX Q «t z 1t ~Z 1 Q 2 Z 1! 1 P 2 D U F PX Q + (27) From the aalysis precedig Theorem 2+3, we kow that 7P 2 7 O p ~7Q7!, ~Z 1 Q 2 Z 1! 1 O p ~ 1!, ad «t z 1t O p ~ 102!+ Thus, because D is a diagoal matrix cosistig of all distict diagoal elemets of Q, 7D7 7Q7, ad the last term i ~27! has asymptotic order O p ~ 102!+ O the other had, usig ~16! ad the fact that U 4 F U F H 4 F 1 H F 0, we ca write

14 «t z 2t D «t x t P U F U F F U F «t x t P ~I K U 4 U 4!F U F «t x t ~P F P!P U F «t x t Q P U F, (28) so that «t z 2t O p ~7Q7!, ad the secod term i ~27! has asymptotic order O p ~ 1!+ Thus, Lemma 4+4~i! ad ~28! yield ~ Z A A!Q X Q «t z 2t D ~D Z 2 Z 2 D! 1 U F PX Q O p ~ 102! «t x t Q P U FU F PQ x t x t Q P U F 1 U F PX Q + The asymptotic behavior of AZ A i the directio of X Q is determied by a argumet idetical to the osigular case of Theorem 2+2+ For Gaussia iovatios u t, Lemma 4+2, ~33!, ad ~34! imply that Q x t x t Q ad «t x t Q coverge joitly i distributio, leadig to a mixed ormal limit, stated formally as follows+ THEOREM 2+4+ For the explosive coitegrated system geerated by (1) ad (2) with v t d N~0, S! ad c i c j for some i j the followig limit theory applies o the directio of X Q : ~ AZ A!Q X Q MN~0, X Q P U F V 1 U F PX Q S ««!, where V U F P Q j X Q X Q Q j P U F. j0 Remarks 2.3. LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS 877 ~i! The limit theory for the least squares estimator i the directio of X Q is mixed Gaussia with full rak covariace matrix of order m ad the usual explosive rate of covergece+ As i the osigular case, the

15 Z 878 PETER C.B. PHILLIPS AND TASSOS MAGDALINOS assumptio of Gaussia iovatios is essetial for the reasos explaied i Remark 2+1~i!+ ~ii! Rotatio of the regressio space i the directio of X Q determies the limit theory i the explosive directio resolvig the sigularity of the limitig momet matrix j0 Q j X Q X Q Q j + ~iii! I the polar case of equal localizig coefficiets, we have Q ui K with P I K + Thus, Theorem 2+4 reduces to u 1 Mu 2 1 ~ AZ A!X Q MN~0, S ««!+ 3. DISCUSSION This paper provides a limit theory for explosively coitegrated systems+ Both the ormalizatio ad the limit distributio of the cetered least squares estimate A A are foud to vary accordig to whether the regressors cotai commo explosive roots+ Whe all the explosive roots are distict, the Q expoetial rate of covergece ad a full rak mixed ormal limitig distributio apply uder the assumptio of Gaussia iovatios+ O the other had, repeated explosive roots give rise to a degeeracy i the regressio limit theory+ This degeeracy is resolved aalytically by a appropriate orthogoal rotatio of the regressio coordiates+ The resultig limit theory is mixed ormal ad of reduced rak+ The rak of the limit distributio depeds o the umber of repeated roots but is ivariat to both the choice of coordiates ad the distributio of the iovatios+ Thus, i the case where some explosive roots are commo, a ivariace priciple holds+ The authors have show that similar results to those give here hold for mildly explosive coitegrated systems with roots that approach uity at rates slower tha 1 + I particular, Magdalios ad Phillips ~2006! cosider models such as ~1! ad ~2! with mildly explosive roots of the form Q I K C a, a ~0,1!, C diag~c 1,+++,c K! 0+ For such systems, a mixed ormal asymptotic distributio is derived for the least squares estimator with the mildly explosive rate of covergece a Q whe C has distict diagoal elemets ad with the moderately statioary rate ~1a!02 whe C has repeated roots, correspodig to Theorems 2+2 ad 2+3, respectively+ A attractive feature of mildly explosive systems is that cetral limit theory applies i both cases ad asymptotic mixed ormality is valid without distributioal assumptios o the iovatios eve whe C has distict diagoal elemets+ Such systems may also be more realistic for practical work+

16 4. PROOFS LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS 879 This sectio cotais some techical lemmas ad also proofs of various statemets ad results i the paper+ Throughout, we use the otatio k : {02}, k X k : Q j u j, (29) j1 F t : s~v 1,+++,v t! for the atural filtratio of the iovatios, ad we let C be a boudig costat i ~0,! that may assume differet values+ The precedig choice for k is made for the sake of simplicity, ad the results hold for ay iteger-valued sequece k satisfyig 1 7Q7 2k ad 7Q7 ~k! k 102 r 0 as r + LEMMA 4+1+ For k ad X k as defied i (29), we have t max Q j u j jk a+s+ o 1 as r. 1 M Proof. Usig Doob s iequality for martigales we obtai, for each d 0, t P max Q j u j d E Q j u j 2 k 1t 1 k 1t jk 1 M 1 d 2 1 E7u 17 2 d 2 1 jk 1 C 7Q7 2k 1 7Q7 2j jk 1 C 7Q7 + 1 LEMMA 4+2+ For k ad X k as defied i (29), we have, as r, ~Q I m! ~x t «t! tk 1 ~Q t I m!~x k «t! o a+s+ ~1!. Proof. The lemma will follow by showig ~30! ad ~31!+ k ~Q t I m!@~q t x t! «t # o a+s+~1! (30)

17 880 PETER C.B. PHILLIPS AND TASSOS MAGDALINOS tk 1 ~Q t I m! t jk 1 Q j u j «t o a+s+~1!+ (31) For ~30!, because 7Q t x t 7 7Q t x t X Q 7 7X Q 7 ad 7Q t x t X Q 7 o a+s+ ~1!, there exists a costat C ~0,! such that 7Q t x t 7 C 7X Q 7 t 1 a+s+ (32) with 7X Q 7 almost surely by the martigale covergece theorem+ Thus, by ergodicity, k ~Q t I m!@~q t x t! «t # k ~C 7X Q 7!7Q7 7Q7 t 7«t 7 ~C 7X Q 7!7Q7 k 202 7Q7 O a+s+ ~7Q7 ~k! k 102! o a+s+ ~1!, k «t 7 showig ~30!+ For ~31!, Lemma 4+1 ad the ergodic theorem yield tk 1 k ~Q t I m! max k 1t t t jk 1 max k 1t t jk 1 jk 1 CM max k 1t Q j u j «t Q j u j tk 1 Q j u j t jk 1 tk 1 Q j u j 1 7Q7 t 7«t 7 7Q7 2~t!102 tk 1 Proof of Theorem 2.1. By ~32! we obtai, almost surely, 7«t 7102 tk 1 7«t 7102 o a+s+ ~1!+ k Q ~t! Q t x t x t Q t Q ~t! ~C 7X Q7! 2 7Q7 2~t! O a+s+ ~7Q7 2~k!!+

18 Thus, Lemma 4+1 yields Q x t x t Q tk 1 tk 1 Q ~t! Q t x t x t Q t Q ~t! o a+s+ ~1! Q ~t! X k X k Q ~t! o a+s+ ~1! k 1 Q j X k X k Q j o a+s+ ~1!, (33) j0 ad Theorem 2+1 follows immediately by the martigale covergece theorem+ Proof of Theorem 2.2. By ~33! ad Lemma 4+2, vec@~ AZ A!Q # 1 Q x t x t Q k 1 j1 Q j X k X k Q j0 I m~q I m! ~x t «t! I m tk 1 The assumptio of Gaussia errors yields, coditioal o F k, tk 1 ~Q t X k «t! o a+s+ ~1!+ ~Q t X k «t! d N0, j k 1 Q j X k X k Q S ««, (34) j0 which leads to vec@~ Z LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS 881 k 1 A A!Q # d j0 MN0, j102 Q j X k X k Q I mn~0, I K S ««! j0 j 1 Q j X Q X Q Q S ««, as r, because X k r a+s+ X Q + LEMMA 4+3+ The determiat of the matrix 1 W : ~s! w i w j 1 : i, j $1,+++,s%

19 882 PETER C.B. PHILLIPS AND TASSOS MAGDALINOS is give by 1 6W ~s! 6 ~w 2 1 1! +++~w 2 s 1! s1 ~w j w j1! 2 ~w j w j2! 2 +++~w j w s! 2 j1 ~w j w j1 1! 2 ~w j w j2 1! 2 +++~w j w s 1!. 2 Cosequetly, the matrix M W : w i w j w i w j 1 : i, j $1,+++,s% is osigular if ad oly if w i w j for all i j. Proof. We use iductio+ The result is immediate for s 2+ If we assume the result for s 1 ad partitio W ~s! as W ~s1! W ~s! w w 1 w s 2 1 w : 1 w 1 w s 1,+++, 1 we have ~e+g+, Abadir ad Magus, 2005, 5+29! ~w s 2 1!6W ~s! 6 6W ~s1! ~w s 2 1!ww 6+, w s1 w s 1 Because the matrix o the right is equal to diag w 1 w s w 1 w s 1,+++, w s1 w s w s1 w s diag 1W w ~s1! 1 w s w 1 w s 1,+++, w s1 w s ad 6W ~s1! 6 is kow from the iductio hypothesis, we obtai 6W ~s! 6 1 w s 2 1 ~w 1 w s! 2 ~w 1 w s 1! ~w s1 w s! 2 ~w s1 w s 1! 2 6W ~s1! 6 1 s1 ~w i w s! 2 ~w 2 1 1! +++~w 2 s 1! i1 ~w i w s 1! 2 s2 ~w j w j1! 2 +++~w j w s1! 2 j1 ~w j w j1 1! 2 +++~w j w s1 1! 2 1 s1 ~w j w j1! 2 +++~w j w s! 2 ~w 2 1 1! +++~w 2 s 1! j1 ~w j w j1 1! 2 +++~w j w s 1! 2 w s1 w s 1

20 LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS 883 as required+ Hece, W ~s! is osigular if ad oly if w i w j for all i j+ The idetity M W diag~w 1,+++,w s!w ~s! diag~w 1,+++,w s! implies that osigularity of M W is equivalet to osigularity of W ~s! + LEMMA 4+4+ Let D U F FU F. The followig results hold as r : (i) D Z 2 Z 2 D r a+s+ U F P~ j0 Q j X Q X Q Q j!p U F 0a+s+, (ii) 7Z 2 Z 1 7 O p ~7Q7!, (iii) ~Z 1 Q 2 Z 1 0! 1 ~Z 1 Z 1 0! 1 O p ~ 1!. Proof. For part ~i!, first ote that U F F j U 4 H F F 1 j H 4 0 for all j N+ Thus, usig ~17! ad ~16! we obtai D j z 2t U F F j U F U F Px t U F F j ~I K U 4 U 4!Px t U F F j Px t ~U F F j U 4!U 4 Px t U F P~P F j P!x t U F PQ j x t, (35) for all j N, ad similarly D j U F PX Q U F PQ j X Q + (36) The limit matrix of part ~i! ow follows immediately from ~35! ad Theorem 2+1: D Z 2 Z 2 D U F F U F z 2t z 2t U F F U F U F PQ x t x t Q P U F r a+s+ U F P Q j X Q X Q Q j P U F + j0 To establish the osigularity of the limit matrix, ote that, by ~5!, D U F FU F diag~u 1,+++,u p, w 1,+++,w Kr! cosists of all distict diagoal elemets of Q+ The, deotig by d i the ith diagoal elemet of D, Lemma 4+3 implies that the matrix M D : d i d j d i d j 1 : i, j $1,+++,Krp%

21 884 PETER C.B. PHILLIPS AND TASSOS MAGDALINOS ~i is osigular+ Deotig by S! F the ith elemet of the vector S F U F PX Q ad lettig SX F : diag~s ~1! F,+++,S ~Krp! F!, ~36! gives U F P Q j X Q X Q Q j P U F D j S F S F D j j0 j0 SX F M D SX F + The last matrix is osigular almost surely because M D is osigular ad S F ~i! 0 almost surely for each i by absolute cotiuity of u t + For part ~ii!, usig a matrix Cauchy Schwarz iequality ~e+g+, Abadir ad Magus, 2005, 12+5! we obtai 7z 1t 7 2 7H 4 z 7 2 tr~h 4 H 4 z 4 H 4!# 7 2 tr~z z!# 4 H 4!# 102 7z 7 2 ~r p! 102 7z 7 2 because H 4 H 4 I rp + Also, usig ~35! ad a stadard trace iequality we ca write E7z 2t 7 2 2t z 2t!# F U F!tr~ xi t xi t!# KE7xI t 7 2 KE7uI F F72t + Thus, E7vec~Z 2 Z 1!7 E~7z 1t 77z 2t 7! ~E7z 1t 7 2! 102 ~E7z 2t 7 2! 102 ~r p! 104 ~E7z 1 7 2! 102 ~E7z 2t 7 2! 102 KE7 I ~r p! 104 u 17 2 E7z F F7 t O~7F7! ad the result follows because 7F7 7Q7+ For part ~iii!, ote that part ~i! implies that 7Z 2 Z 2 7 O p ~7Q7 2!+ Thus, 1 7Z 1 Z 2 ~Z 2 Z 2! 1 Z 2 Z Z 1 Z Z 2 Z O p 1 by part ~ii!, ad the result follows from the defiitio of Q 2 +

22 LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS 885 Proof of (19). Because F 1 i diag~u 1 i I r1,+++,u p i I rp!, ~14! gives, for all i Z, H 4 F i 1 H F diag~u i 1 H 41 H F1,+++,u i p H 4p H Fp! 0+ O the other had, ~15! ad the precedig idetity for i 1 yield ~H 4 F 1 1 H 4! 2 H 4 F 1 1 H 4 H 4 F 1 1 H 4 H 4 F 1 1 ~I r H F H F!F 1 1 H 4 H 4 F 1 2 H 4 ~H 4 F 1 1 H F!H F F 1 1 H 4 H 4 F 1 2 H 4, ad so we have proved the idetity ~H 4 F 1 1 H 4! j H 4 F j 1 H 4 for j 2+ The geeral case follows by straightforward iductio+ Proof of (21). Usig ~10! ad ~11! the defiitio of z 1t yields z 1t H 4 P 1 x t H 4 P 1 ~Q 1 x Q 1 u! H 4 P 1 Q 1 x H 4 P 1 Q 1 u H 4 P 1 Q 1 ~P 1 P 1 P 2 P 2!x H 4 P 1 Q 1 ~P 1 P 1 P 2 P 2!u H 4 ~P 1 Q 1 P 1!P 1 x H 4 ~P 1 Q 1 P 1!P 1 u H 4 F 1 1 P 1 x H 4 F 1 1 P 1 u + The secod term has the form that appears i ~21!+ For the first term, usig the fact that H 4 F 1 H F 0, we ca write H 4 F 1 1 P 1 x H 4 F 1 1 ~H 4 H 4 H F H F!P 1 x ~H 4 F 1 1 H 4!H 4 P 1 x ~H 4 F 1 1 H 4!z 1, as required+ Proof of (26). Recallig the otatio z j1 Q j u tj, ~22! yields the followig expressio for the sample covariace: 1 ~z 1t «t! 1 ~H 4 P 1 z «t! M M ~H 4 P 1 I m! j t, (37)

23 886 PETER C.B. PHILLIPS AND TASSOS MAGDALINOS where j t : 102 ~z «t! is a martigale differece array with respect to F, because z is s~v,v t2,+++! measurable+ The coditioal variace of j t is give by E Ft ~j t j t! 1 1 Ft ~z z! «t «t z! «t «t # j Q j S uu Q 1 r a+s+ j Q j S uu Q S ««, j1 «t «t by the ergodic theorem+ Thus, provided that the Lideberg coditio E Ft ~7j t 7 2 1$7j t 7 d%! r p 0 d 0 (38) holds, Corollary 3+1 of Hall ad Heyde ~1980! ad the Cramér Wold theorem imply that j t N0, j Q j S uu Q S ««+ (39) j1 The proof of ~38! is give ext+ The proof of ~26! follows from ~37! ad ~39!+ Proof of (38). The Lideberg coditio ~38! is equivalet to 1 7«t 7 2 E Ft ~7z 7 2 1$7z 77«t 7 d 102 %! o p ~1!+ (40) Applyig the iequality 1$7z 77«t 7 d 102 % 1$7z 7 d % 1$7«t 7 d % to ~40!, we deduce that ~38! will follow if the followig terms, S 1 1 7«t 7 2 E~7z 7 2 1$7z 7 d %!, S 2 1 7«t 7 2 1$7«t 7 d %E7z 7 2,

24 LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS 887 are o p ~1!+ The term S 1 r 0iL 1 because, usig the fact that z is a strictly statioary sequece with E7z 1 7 2, ES 1 max E~7z 7 2 1$7z 7 d %!E7« t E~7z $7z 1 7 d %!E7«1 7 2 r 0+ The term S 2 also teds to 0 i L 1 because ES 2 E7z E~7«t 7 2 1$7«t 7 d %! E7z E~7« $7«1 7 d %! r 0, by itegrability of 7« Thus, ~40! ad ~38! follow+ REFERENCES Abadir, K+M+ &J+R+ Magus ~2005! Matrix Algebra+ Ecoometric Exercises, vol+ 1+ Cambridge Uiversity Press+ Aderso, T+W+ ~1959! O asymptotic distributios of estimates of parameters of stochastic differece equatios+ Aals of Mathematical Statistics 30, Hall, P+ &C+C+ Heyde ~1980! Martigale Limit Theory ad Its Applicatio+ Academic Press+ Magdalios, T+ &P+C+B+ Phillips ~2006! Limit Theory for Coitegrated Systems with Moderately Itegrated ad Moderately Explosive Regressors+ Workig paper, Yale Uiversity+ Mijheer, J+ ~2002! Asymptotic iferece for AR~1! processes with ~oormal! stable iovatios, part V: The explosive case+ Joural of Mathematical Scieces 111, Park, J+Y+ &P+C+B+ Phillips ~1988! Statistical iferece i regressios with itegrated processes, part 1+ Ecoometric Theory 4, Park, J+Y+ &P+C+B+ Phillips ~1989! Statistical iferece i regressios with itegrated processes, part 2+ Ecoometric Theory 5, Phillips, P+C+B+ ~1985! The distributio of matrix quotiets+ Joural of Multivariate Aalysis 16, White, J+S+ ~1958! The limitig distributio of the serial correlatio coefficiet i the explosive case+ Aals of Mathematical Statistics 29,