LIMIT THEORY FOR EXPLOSIVELY COINTEGRATED SYSTEMS BY PETER C. B. PHILLIPS ad TASSOS MAGDALINOS COWLES FOUNDATION PAPER NO. 1244 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Have, Coecticut 06520-8281 2008 http://cowles.eco.yale.edu/
Ecoometric Theory, 24, 2008, 865 887+ Prited i the Uited States of America+ doi:10+10170s0266466608080353 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-04-142254+ Address correspodece to Peter C+B+ Phillips, Departmet of Ecoomics, Yale Uiversity, P+O+ Box 208268, New Have, CT 06520-8268, USA; e-mail: peter+phillips@yale+edu+ 2008 Cambridge Uiversity Press 0266-4666008 $15+00 865
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 102 + 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+
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+
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 1 102 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
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
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 0 0 0 H 42 J 0 0 J J J J J 0 0 J H 4p 0 H F1 0 J 0 0 0 H F2 J 0 0 J J J J J 0 0 J H Fp 0 0 0 J 0 I Kr (13)
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 t @z 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
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
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 1 @z 11, z 12,+++,z 1 # R ~rp!, Z 2 @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 ~7Q7 2 1 1 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! 0 0 + (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
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 4 1 1 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-
[ 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
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 2t @~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
«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
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+
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)
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 2102 7«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!!+
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%
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! 2 +++ ~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
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%
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 z! @tr~h 4 H 4!# 102 @7z 7 2 tr~z z!# 102 @tr~h 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 E @tr~z 2t z 2t!# E @tr~u F U F!tr~ xi t xi t!# KE7xI t 7 2 KE7uI 17 2 7F7 2 1 7F72t + 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 1 7 2 102 7F7 2 1 7F7 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 1 7 1 7Z 1 Z 2 7 2 7Z 2 Z 2 7 1 O p 1 by part ~ii!, ad the result follows from the defiitio of Q 2 +
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)
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 j1 @E Ft ~z z! «t «t # @E~z 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 102 104 % 1$7«t 7 d 102 104 % 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 102 104 %!, S 2 1 7«t 7 2 1$7«t 7 d 102 104 %E7z 7 2,
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 102 104 %!E7«1 7 2 1t E~7z 1 7 2 1$7z 1 7 d 102 104 %!E7«1 7 2 r 0+ The term S 2 also teds to 0 i L 1 because ES 2 E7z 1 7 1 2 E~7«t 7 2 1$7«t 7 d 102 104 %! E7z 1 7 2 E~7«1 7 2 1$7«1 7 d 102 104 %! r 0, by itegrability of 7«1 7 2 + 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, 676 687+ 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, 3854 3856+ Park, J+Y+ &P+C+B+ Phillips ~1988! Statistical iferece i regressios with itegrated processes, part 1+ Ecoometric Theory 4, 468 497+ Park, J+Y+ &P+C+B+ Phillips ~1989! Statistical iferece i regressios with itegrated processes, part 2+ Ecoometric Theory 5, 95 131+ Phillips, P+C+B+ ~1985! The distributio of matrix quotiets+ Joural of Multivariate Aalysis 16, 157 161+ White, J+S+ ~1958! The limitig distributio of the serial correlatio coefficiet i the explosive case+ Aals of Mathematical Statistics 29, 1188 1197+