Cointegration in functional autoregressive processes

Transcription

1 Dipartimento di Scienze Statistice Sezione di Statistica Economica ed Econometria Massimo Franci Paolo Paruolo Cointegration in functional autoregressive processes DSS Empirical Economics and Econometrics Working Papers Series DSS-E3 WP 17/5

2 Dipartimento di Scienze Statistice Sezione di Statistica Economica ed Econometria Sapienza Università di Roma P.le A. Moro Roma - Italia ttp://

3 Cointegration in functional autoregressive processes MASSIMO FRANCHI AND PAOLO PARUOLO DECEMBER, 17 Abstract. Tis paper derives a generalization of te Granger-Joansen Representation Teorem valid for H-valued autoregressive (AR) processes, were H is an infinite dimensional separable Hilbert space, under te assumption tat 1 is an eigenvalue of finite type of te AR operator function and tat no oter non-zero eigenvalue lies witin or on te unit circle. A necessary and sufficient condition for integration of order d = 1,,... is given in terms of te decomposition of te space H into te direct sum of d + 1 closed subspaces τ, =,..., d, eac one associated wit components of te process integrated of order. Tese results mirror te ones recently obtained in te finite dimensional case, wit te only difference tat te number of cointegrating relations of order is infinite. 1. Introduction Te teory and applications of time series tat take values in infinite dimensional separable Hilbert spaces, or H-valued processes, are recently gaining increasingly attention in econometrics. Tey allow to represent directly te dynamics of infinite-dimensional objects, suc as bounded continuous function on a compact. An important early contribution to te literature of functional time series is Bosq (), were a teoretical treatment of linear processes in Banac and Hilbert spaces is developed. Tere, empasis is given to te derivations of laws of large numbers and central limit teorems tat allow to discuss estimation and inference for H-valued autoregressive (AR) models. Empirical applications of functional time series analysis include studies on te term structure of interest rates, Kargin and Onatski (8), and on intraday volatility, Hörmann et al. (13) and Gabrys et al. (13). Te recent monograps by Horvát and Kokoszka (1) and Kokoszka and Reimerr (17) and te review in Hörmann and Kokoszka (1) report additional examples. Key words and prases. Functional autoregressive process, Unit roots, Cointegration, Common Trends, Granger- Joansen Representation Teorem. M. Franci, Sapienza University of Rome, P.le A. Moro 5, 185 Rome, Italy; massimo.franci@uniroma1.it. P. Paruolo, European Commission, Joint Researc Centre (JRC), Via E.Fermi 749, I-37 Ispra (VA), Italy; paolo.paruolo@ec.europa.eu, corresponding autor. Te ideas in te present paper were conceived wile te first autor was visiting te Department of Economics, Indiana University, in January 17; te ospitality of Yoosoon Cang and Joon Park is gratefully acknowledged. 1

4 In a recent paper, Cang et al. (16) present statistical tools for functional time series tat are integrated of order one, I(1), and find evidence of unit root persistence and cointegration in te time series of te several coordinates of te cross-sectional distributions of individual earnings and te intra-mont distributions of stock returns. 1 Te framework proposed by Cang et al. (16) as (by construction) a finite number of I(1) stocastic trends and an infinite dimensional cointegrating space. Te teory is developed starting from te infinite moving average representation of te first differences of te process; te potential unit roots are identified and tested troug a functional principal components analysis. In a different vein, Hu and Park (16) start from an AR process of order one wit compact AR operator and provide an I(1) condition tat extends te Granger-Joansen Representation Teorem, see Teorem 4. in Joansen (1996), to I(1) H-valued AR(1) processes. Te corresponding common trends representation, or functional Beveridge-Nelson decomposition, displays a finite number of I(1) stocastic trends and an infinite dimensional cointegrating space. Tey furter propose an estimator for te functional autoregressive operator wic builds on Cang et al. (16). Beare et al. (17) consider an H-valued AR(s), s > 1, process wit compact AR operators and provide a reformulation of te Joansen I(1) condition tat extends te Granger-Joansen Representation Teorem to tis more general setup. Again ere, te number of I(1) stocastic trends is finite and te dimension of te cointegrating space is infinite. Finally, Cang et al. (16) consider an error correction form wit compact error correction operator and sow tat in tis case te number of I(1) stocastic trends is infinite and te dimension of te cointegrating space is finite. Moreover, te Granger-Joansen Representation Teorem continues to olds in a form similar to te finite dimensional case. Te present paper provides an extension of te representation results in Beare et al. (17) and Hu and Park (16) for H-valued AR processes in te generic I(d), d = 1,,... case, under te assumption tat 1 is an eigenvalue of finite type of te AR operator function and tat no oter non-zero eigenvalue of te AR operator lies witin or on te unit circle. It is found tat te conditions and properties in te H-valued cointegrated AR processes coincide wit tose te finite dimensional AR processes, except for te fact tat te number of cointegrating relations of order is infinite for H-valued cointegrated AR processes. Te assumption tat 1 is an eigenvalue of finite type of te AR operator function means tat te inverse of te AR operator function as an isolated pole at 1, as in te finite dimensional case. Te assumptions in Hu and Park (16) and Beare et al. (17) imply tat 1 is an eigenvalue of finite type (but not viceversa), and ence te present analysis applies to tose setups as special cases. 1 Beare (17) argues tat te nonnegativity property of densities is not compatible wit te postulated unit root nonstationarity. Neverteless, wile concurring tat te framework in Cang et al. (16) is useful for generic H-valued processes.

5 A necessary and sufficient condition for I(d), d = 1,,..., in terms of te decomposition of te space H into te direct sum of d + 1 closed subspaces, tat are defined recursively from te AR operators, H = τ τ 1 τ d, were τ is closed and infinite dimensional and τ, is closed and finite dimensional for = 1,..., d. Te infinite dimensional subspace τ τ 1 τ d 1 is te cointegrating space of x t wile te finite dimensional subspace τ d is te attractor space of te I(d) stocastic trends. Hence an H-valued cointegrated AR process x t as a finite number of I(d) stocastic trends and infinitely many cointegrating relations. Te properties of v, x t vary wit v in H: for v τ, wic is infinite dimensional, one can combine v, x t wit differences n x t for n = 1,..., d 1 to find I() polynomial cointegrating relations, for v τ 1, wic is finite dimensional (and can as well be equal to ), one can combine v, x t wit wit differences n x t for n = 1,..., d and find at most I(1) polynomial cointegrating relations, and so on up to v τ d 1, wic is finite dimensional (and possibly of dimension), for wic v, x t I(d 1) does not allow for polynomial cointegration. Wen v τ d, wic is finite dimensional and different from, one as v, x t I(d), i.e. tere is no cointegration. For any v in te cointegrating space, te explicit expression of te coefficients of te polynomial cointegrating relations is provided in terms of operators tat are defined recursively from te AR operators togeter wit te sequence of τ. Te present results sow tat te infinite dimensionality of te space does not introduce additional elements in te analysis, apart from te fact tat te number of I() cointegrating relations is infinite. Tat is, te conditions and properties of H-valued cointegrated AR process coincide wit tose tat apply in te finite dimensional case. Te present derivations parallel te development of te representation teory for finite dimensional autoregressive processes developed by Joansen (1996) for te I(1) and I() cases and extended in Franci and Paruolo (16) to I(d) processes. Te present treatment makes extensive use of projection matrices in te ambient space, as well as of Moore-Penrose generalised inverses. Tese tools ave direct counterparts bot in te finite dimensional case in Franci and Paruolo (16) and in te present infinite dimensional space. Te rest of te paper is organized as follows: te remaining part of tis introduction reports notation and preliminaries; Section presents basic definitions and concepts and Section 3 reports an existence result. Section 4 provides a caracterization of I(1) and I() cointegrated H-valued AR processes, Section 5 extends te result to te general I(d), d = 1,,..., case and Section 6 concludes. Appendix A reviews notions and results on separable Hilbert spaces and on operators acting on tem, Appendix B presents basic facts on H-valued random variables and Appendix C contains proofs. 3 Notation and preliminaries. In te present paper H is an infinite dimensional separable Hilbert space wit inner product, and norm, were separable means tat H admits a countable

6 4 ortonormal basis. A random variable tat takes values in H is said to be an H-valued random variable and a sequence of H-valued random variables is called an H-valued stocastic process. Let B(H) be te (Banac) space of bounded linear operators from H to H and consider A B(H); te closed subspace {x H : Ax = }, written Ker A, is called te kernel of A and te subspace {Ax : x H}, written Im A, is called te image of A. Te dimension of Im A, written rank A, is called te rank of A. ortogonal complement of S, written S, and P S P S x = x for all x S and Ker P S = S. Te set {v H : v, y = for all y S H} is called te denotes te ortogonal projection on S, i.e. Let z C and < ρ R; te set {z C : z z < ρ} is called te open disc centered in z wit radius ρ, written D(z, ρ). If a power series n= A n(z z ) n, A n B(H), is absolutely convergent for all z D(z, ρ), i.e. tat n= A n z z n < for all z D(z, ρ), ten n= A n(z z ) n converges in te operator norm to A(z) B(H), i.e. tat N n= A n(z z ) n A(z) as N for all z D(z, ρ). In tis case, te operator function A(z) = n= A n(z z ) n is said to be absolutely convergent on D(z, ρ). Appendix A reviews notions and results on separable Hilbert spaces and on operators acting on tem and Appendix B presents te definitions of expectation, covariance and cross-covariance for H-valued random variables.. Basic definitions and concepts Tis section introduces stocastic processes tat take values in a separable Hilbert space H and presents te notions of weak stationarity, wite noise, linear process, integration and cointegration. Te definitions of weak stationarity and wite noise are taken from Bosq () wile tose of linear process, integration and cointegration are adapted from Joansen (1996) and are similar to tose employed in Beare et al. (17). Definition.1 (Weak stationarity). An H-valued stocastic process {ε t, t Z} is said to be weakly stationary if i) E( ε t ) <, ii) te expectation of ε t does not depend on t and iii) te crosscovariance function of ε t and ε s is suc tat c εt,ε s (, x) = c εt+u,ε s+u (, x) for all, x H and all s, t, u Z. Te basic notion of H-valued wite noise is introduced next. Definition. (Wite noise). An H-valued stocastic process {ε t, t Z} is said to be a wite noise if i) < E( ε t ) <, ii) te expectation of ε t is equal to, iii) te covariance operator of ε t does not depend on t, and iv) te cross-covariance function of ε t and ε s, c εt,ε s (, x), is equal to for all, x H and all s t, s, t Z. It is evident from te definition tat any wite noise is weakly stationary. Te same property applies to linear combinations of a wite noise wit suitable weigts, wic defines te class of linear processes.

7 Definition.3 (Linear process). An H-valued stocastic process {u t, t Z} is said to be a linear process if u t = B n ε t n, B = I, n= were {ε t, t Z} is wite noise, B n : H H is a bounded linear operator and B(z) = n= B nz n is absolutely convergent on D(, ρ) for some ρ > 1. As discussed in Section 7.1 in Bosq (), existence and weak stationarity of u t = n= B nε t n are guaranteed if n= B n <. Observe tat te requirement tat B(z) is absolutely convergent D(, ρ) for some ρ > 1 is stronger. In fact, n= B n z n < for all z D(, ρ), ρ > 1, implies tat n= B n is finite, so tat n= B n is finite and u t in Definition.3 is a well defined and weakly stationary process. Moreover, te absolute convergence of B(z) implies tat n= B nz n = B(z) B(H) for all z D(, ρ), ρ > 1, so tat B(1) = n= B n is a well defined bounded linear operator. Tis is needed for te notions of integration and cointegration in Definition.4 below. Finally note tat te series obtained by termwise k times differentiation, n=k n(n 1) (n k+ 1)B n z n k, is absolutely convergent on D(, ρ), ρ > 1, and coincides wit te k-t derivative of B(z) for eac z D(, ρ). Hence an absolutely convergent operator function is infinitely differentiable on D(, ρ). In te following := 1 L is te difference operator at frequency. For simplicity, only te case of integration and cointegration at frequency is considered in te following; similar definitions old for any oter root on te unit circle. Definition.4 (Integrated and cointegrated processes at frequency ). A linear process u t = B(L)ε t is said to be integrated of order, written u t I(), if B(1) = n= B n. An H- valued stocastic process {x t, t =, 1,... } is said to be integrated of order d (at frequency zero), written x t I(d), if d x t = B(L)ε t I(). An I(d) process x t is said to be cointegrated (at frequency zero) if B(1) is non-invertible; in tis case any non-zero vector v (Im B(1)) is suc tat v, x t I(d j) for some j > and any non-zero vector v Im B(1) is suc tat v, x t I(d). (Im B(1)) is called te cointegrating space of x t and Im B(1) is called te attractor space of te I(d) trends. Observe tat I()-ness implies weak stationarity but not viceversa and tat a wite noise is necessarily I(). Also note tat te notions of integration and cointegration in Definition.4 are invariant to bounded linear invertible transformations of te process. Tat is, if x t I(d) and A is a bounded linear invertible transformation ten Ax t I(d). Remark tat te same invariance property olds replacing A wit A(z) = n= A nz n, A n B(H), invertible and absolutely convergent for all z D(, ρ), ρ > 1. Te next definition introduces te class of processes tat is studied in te present paper. 5

8 6 Definition.5 (Cointegrated H-valued AR at frequency ). Consider an H-valued AR process of order s x t = A 1x t A sx t s + ε t, were A n : H H, n = 1,..., s, is a bounded linear operator and {ε t, t Z} is wite noise as in Definition., and define te AR operator function A(z) = I A 1 z A sz s, z C. An H-valued AR process A(L)x t = ε t is said to be cointegrated (at frequency zero) if i) A(1), ii) A(z) as an eigenvalue of finite type at z = 1, and iii) A(z) is invertible in te punctured disc D(, ρ) \ {1} for some ρ > 1. Remark.6. Beare et al. (17) consider H-valued AR processes Φ(L)x t = ε t, Φ(z) = I Φ 1 z Φ k z s, for wic wenever Φ(z) is non-invertible one as eiter z = 1 or z > 1, and were Φ n are compact if s > 1. Hu and Park (16) consider H-valued AR processes A(L)f t = ε t, A(z) = I Az, suc tat A is compact and 1 is an isolated eigenvalue of A. Te present setup contains te ones considered in Beare et al. (17) and Hu and Park (16) as special cases. In fact, because te sum of compact operators is compact, see Teorem 16.1 in Capter II in Goberg et al. (3), and I B is Fredolm of index if B B(H) is compact, see Teorem 4. in Capter XV in Goberg et al. (3), Φ(1) in Beare et al. (17) and A(1) in Hu and Park (16) are Fredolm of index and 1 is isolated, i.e. 1 is an eigenvalue of finite type. For tis reason bot te processes considered by Beare et al. (17) and Hu and Park (16) are cointegrated H-valued AR process in te sense of Definition.5. Hu and Park (16) also discuss estimators and asymptotic results for H-valued AR(1) models wit a compact operator. Remark.7. Cang et al. (16), see teir Assumption.1, study processes tat satisfy w t = Φ(L)ε t = n= Φ nε t n, were n= n Φ n < and Im Φ(1) is finite dimensional. As sown below, a cointegrated H-valued AR process necessarily meets tese requirements. Hence teir asymptotic analysis applies and teir test can be employed in te present setup as well. Remark.8. As sown in Capter XI in Goberg et al. (199), te assumption tat 1 is an eigenvalue of finite type of A(z) implies tat A(1) is Fredolm of index, wic means tat dim Ker A(1) and codim Im A(1) = dim(h/ Im A(1)) are finite and equal. Because Ker A(1) is finite dimensional, it is complemented, see Teorem 5.7 in Capter XI in Goberg et al. (3), and ence its ortogonal complement (Ker A(1)) is closed. Moreover, because H/ Im A(1) is finite dimensional, Im A(1) is closed, see Corollary.3 in Capter XI in Goberg et al. (199). Tis sows tat te subspaces Im A(1) and (Ker A(1)) are closed and infinite dimensional wile teir ortogonal complements (Im A(1)) and Ker A(1) are closed and finite dimensional. Hence (Ker A(1)) = Im A(1), were A(1) is te adjoint of A(1).

9 7 3. Existence results for cointegrated H-valued AR processes Tis section presents an existence result in Teorem 3.1 for cointegrated H-valued AR processes A(L)x t = ε t. Tis result is a direct consequence of Teorem A.1 in Appendix A. Teorem 3.1 sows tat x t I(d) for some d = 1,,... as necessarily finitely many I(d) trends, finitely many I(j), 1 j d 1, cointegrating relations and infinitely many I() cointegrating relations. Apart from te fact tat te number of I() cointegrating relations is infinite, tis mirrors te finite dimensional case in Franci and Paruolo (17). Teorem 3.1 (Existence results for cointegrated H-valued AR processes). Let A(L)x t = ε t be a cointegrated H-valued AR process and assume tat x t is I(d) for some d = 1,,.... Ten tere exist integers = j < j 1 < j < < j w = d and projections P, P 1,..., P w suc tat H = Im P Im P 1 Im P w, were Im P, Im P 1,..., Im P w are closed, dim Im P = and < dim Im P <, = 1,..., w. In tis case, x t = C s d,t + C 1 s d 1,t + + C d 1 s 1,t + C (L)ε t + v, Im C = Im P w, were s,t = t i=1 s 1,i I(), s,t = ε t, C (L)ε t is a linear process, and v collects initial values. Tis sows tat Im P Im P 1 Im P w 1 is te cointegrating space of x t and Im P w is te attractor space of te I(d) trends. Moreover, tere exists an operator function G(z) = n= G n(1 z) n, G = I, invertible and absolutely convergent for all z D(, ρ), ρ > 1, suc tat is I(j ) for all v Im P. d j 1 v, x t + v, G n n x t n=1 Remark 3.. In te I(1) case, one as w = 1, = j < j 1 = 1 in Teorem 3.1 and t x t = C ε i + C (L)ε t + v, Im C = Im P 1, H = Im P Im P 1, i=1 were C (L)ε t is a linear process, v collects initial values, Im P, =, 1 is closed, dim Im P = and < dim Im P 1 <. Because Im C = Im P 1 one as tat Im P is te cointegrating space of x t and Im P 1 is te attractor space of te I(1) trends. Moreover, v, x t I() for all v Im P, v, x t I(1) for all v Im P 1. Observe tat te number of I(1) trends in x t is equal to dim Im P 1 and te number of I() cointegrating relations is equal to dim Im P, so tat x t as finitely many I(1) trends and infinitely many I() cointegrating relations. Tis fact is also documented in Hu and Park (16) and in Beare et al. (17). Apart from te fact tat te number of I() cointegrating relations is infinite, tis ecoes te finite dimensional case, see Teorem 4. in Joansen (1996).

10 8 Remark 3.3. In te I() case, one as eiter w = 1 or w =. If w = 1, = j < j 1 = and t s t x t = C ε i + C 1 ε i + C (L)ε t + v, Im C = Im P 1, H = Im P Im P 1, s=1 i=1 i=1 were C (L)ε t is a linear process, v collects initial values, Im P, =, 1 is closed, dim Im P = and < dim Im P 1 <. Because Im C = Im P 1 one as tat Im P is te cointegrating space of x t and Im P 1 is te attractor space of te I() trends. Moreover, v, x t + v, G 1 x t I() for all v Im P, v, x t I() for all v Im P 1. Observe tat te number of I() trends in x t is equal to dim Im P 1 and te number of I() cointegrating relations is equal to dim Im P, so tat x t as finitely many I() trends and infinitely many I() cointegrating relations. Futer observe tat v is eiter not cointegrating (wen v Im P 1 ) or it allows for polynomial cointegration (wen v Im P ) of order. In te I() case wen w =, tis does not old; in fact, one as tat = j < j 1 = 1 < j = and t s t x t = C ε i + C 1 ε i + C (L)ε t + v, Im C = Im P, H = Im P Im P 1 Im P, s=1 i=1 i=1 were C (L)ε t is a linear process, v collects initial values, Im P, =, 1, is closed, dim Im P = and < dim Im P <, = 1,. Because Im C = Im P one as tat Im P Im P 1 is te cointegrating space of x t and Im P is te attractor space of te I() trends. Moreover, v, x t + v, G 1 x t I() for all v Im P, v, x t I(1) for all v Im P 1 and v, x t I() for all v Im P, so tat tere are cointegrating vectors tat allow for polynomial cointegration (wen v Im P ) of order and oters tat don t (wen v Im P 1 ). Observe tat te number of I() trends in x t is equal to dim Im P, te number of I(1) cointegrating relations is equal to dim Im P 1 and te number of I() cointegrating relations is equal to dim Im P. Hence x t as finitely many I() trends, finitely many I(1) cointegrating relations and infinitely many I() cointegrating relations. Apart from te fact tat te number of I() cointegrating relations is infinite, tis mimics te finite dimensional case, see Teorem 4.6 in Joansen (1996). Te same structure applies in general: from Teorem 3.1 one as tat te number of I(d) trends is equal to dim Im P d and te number of I(j ) cointegrating relations is equal to dim Im P, =,..., w 1. Because dim Im P = and < dim Im P <, = 1,..., w, x t as finitely many I(d) trends, finitely many I(j ), = 1,..., w 1, cointegrating relations and infinitely many I() cointegrating relations. As in te I(1) and I() cases above, te finite and te infinite dimensional cases are similar, apart from te fact tat te number of I() cointegrating relations is infinite, see Teorem 3.3 in Franci and Paruolo (17). Finally observe tat Teorem 3.1 provides information about te existence of te projections P, te dimensions of teir images, te orders of integration j, and te G n operators wic are

11 relevant to describe te properties of a generic I(d) cointegrated H-valued AR process. However, it is completely silent about ow tose relevant quantities are related to te structure of te AR operators and ow one can construct tem in practice. Teir form is described in Section 4 for te I(1) and I() cases and in Section 5 for te generic I(d) case. 4. A caracterization of I(1) and I() cointegrated H-valued AR processes Tis section presents a caracterization of I(1) and I() cointegrated H-valued AR processes A(L)x t = ε t. Te I(1) case parallels te results in Beare et al. (17), wile te results for te I() case are novel for te infinite dimensional case. All te relevant quantities are expressed in terms of te coefficients of te expansion of te AR operator function A(z) = I s n=1 A nz n around 1, { A(z) = A (1 z) I s n=1, A = A n for = ( 1) +1 s ). (4.1) n= A n+ for = 1,,... = In Teorem 4.1 below a necessary and sufficient condition for x t I(1) is given in terms of te decomposition of te space into te sum of two closed subspaces, H = τ τ 1, tat are defined using A and A 1, see (4.) and (4.3) below. Te infinite dimensional cointegrating space coincides wit τ and te finite dimensional attractor space of te I(1) trends wit τ 1. In Remark 4. te ( n+ equivalence wit te condition in Beare et al. (17) is proved. In Teorem 4.4 a necessary and sufficient condition for x t I() is given in terms of te decomposition of te space into te sum of tree closed subspaces, H = τ τ 1 τ, were τ is defined in (4.4) below using A, A 1 and A. Te infinite dimensional cointegrating space coincides wit τ τ 1 and τ is te finite dimensional attractor space of te I() trends. In τ, wic is infinite dimensional, one finds te cointegrating vectors tat allow for polynomial cointegration and in τ 1, wic is finite dimensional (and can as well be equal to ), tose tat don t allow for polynomial cointegration. Te following notation is employed: consider A in (4.1) and define ζ = Im A, ζ = (Im A ), τ = (Ker A ), τ = Ker A. (4.) As sown in Remark.8, te assumption tat 1 is an eigenvalue of finite type of A(z) implies tat te subspaces ζ and τ are closed and infinite dimensional, so tat τ = (Ker A ) = Im A, wile teir ortogonal complements ζ and τ are closed and finite dimensional. In te following, P x indicates te ortogonal projection on x. Teorem 4.1 (A caracterization of I(1) cointegrated H-valued AR processes). Consider a cointegrated H-valued AR process A(L)x t = ε t, were A(z) is written as in (4.1). Let ζ and τ be as in (4.) and define ζ 1 = Im P ζ A 1 P τ, τ 1 = (Ker P ζ A 1 P τ ). (4.3) 9 Ten x t is I(1) if and only if H = τ τ 1,

12 1 called te I(1) condition, were τ and τ 1 are closed, dim τ = and < dim τ 1 <. In tis case, t x t = C ε i + C (L)ε t + v, Im C = τ 1, i=1 were C (L)ε t is a linear process and v collects initial values. Tis sows tat τ is te cointegrating space of x t and τ 1 is te attractor space of te I(1) trends. Moreover, v, x t I() for all v τ and v, x t I(1) for all v τ 1. Finally, te I(1) condition can be equivalently stated as H = ζ ζ 1, were ζ and ζ 1 are closed, dim ζ = and < dim ζ 1 <. Remark 4.. As sown next, te I(1) condition in Teorem 4.1 is equivalent to te I(1) condition in Beare et al. (17), H = ζ A 1 τ, see teir equation (4.15). First assume tat H = ζ A 1 τ ; ten I = P ζ + P A1 τ, so tat I P ζ = P ζ = P A1 τ. Tis implies tat A 1τ = P ζ A 1τ = Im P ζ A 1 P τ = ζ 1. Tis sows tat H = ζ ζ 1, i.e. te I(1) condition in Teorem 4.1 olds. Conversely, assume tat H = ζ ζ 1. Because ζ 1 = Im P ζ A 1 P τ, one as tat ζ = ζ 1 Im A 1 P τ and ence dim ζ dim A 1τ ; because dim A 1τ dim τ = dim ζ, one tus as ζ 1 = Im A 1 P τ = A 1 τ. Tis sows tat H = ζ A 1 τ, i.e. te I(1) condition in Beare et al. (17) olds. Remark 4.3. In te finite dimensional case H = R p, Franci and Paruolo (16) sow tat te I(1) condition in Teorem 4. in Joansen (1996) can be equivalently stated as R p = ζ ζ 1 = τ τ 1, were ζ = span(α ), τ = span(β ), =, 1, and te bases α, β are defined by te rank factorizations A = α β and P α A 1P β = α 1 β 1, i.e. tey are full column rank matrices tat respectively span te column space and te row space of te corresponding matrix. Apart from te fact tat dim ζ = dim τ = rank A is finite wen H = R p, tis mirrors wat appens in te present infinite dimensional case. Te I() case is presented next. Teorem 4.4 (A caracterization of I() cointegrated H-valued AR processes). Consider a cointegrated H-valued AR process A(L)x t = ε t, were A(z) is written as in (4.1). Let ζ and τ be as in (4.), let ζ 1 and τ 1 as in (4.3) and let Z = ζ ζ 1 and T = τ τ 1 ; furter define ζ = Im P Z A,1 P T, τ = (Ker P Z A,1 P T ), A,1 = A A 1 Q A 1, Q = A +, were Q is te generalized inverse of A. Ten x t is I() if and only if H = τ τ 1 τ, (4.4)

13 11 called te I() condition, were τ, τ 1 and τ are closed, dim τ =, dim τ 1 <, and < dim τ <. In tis case, t s t x t = C ε i + C 1 ε i + C (L)ε t + v, Im C = τ, s=1 i=1 i=1 were C (L)ε t is a linear process and v collects initial values. Tis sows tat τ τ 1 is te cointegrating space of x t and τ is te attractor space of te I() trends. Moreover, v, x t + v, Q A 1 x t I() for all v τ, v, x t I(1) for all v τ 1 and v, x t I() for all v τ. Finally, te I() condition can be equivalently stated as H = ζ ζ 1 ζ, were ζ, ζ 1 and ζ are closed, dim ζ =, dim ζ 1 <, and < dim ζ <. Remark 4.5. As sown in Teorem 3 in Capter 9 in Ben-Israel and Greville (3), if A B(H) and Im A is closed, ten its generalized inverse A + exists and it is te unique solution of te system AA + A = A, A + AA + = A +, (AA + ) = AA +, (A + A) = A + A. Te assumption tat 1 is an eigenvalue of finite type of A(z) implies tat ζ = Im A is closed, see Remark.8; ence Q exists and it is unique. Remark 4.6. In te finite dimensional case H = R p, Franci and Paruolo (16) sow tat te I() condition in Teorem 4.6 in Joansen (1996) can be equivalently stated as R p = ζ ζ 1 ζ = τ τ 1 τ, were ζ = span(α ), τ = span(β ), =, 1,, and te bases α, β are defined by te rank factorizations A = α β, P α A 1P β = α 1 β 1 and P a (A A 1 Q A 1 )P b = α β, were a = (α, α 1 ), b = (β, β 1 ) and Q = β (β β ) 1 (α α ) 1 α. Again ere, apart from te fact tat dim ζ = dim τ = rank A is finite wen H = R p, tis is exactly wat appens in te infinite dimensional case. Summing up, Teorem 4.1 sows tat d = 1 if and only if circumstances old: te first, τ H (or equivalently ζ H), establises tat d > and te second, τ 1 = τ (or equivalently ζ 1 = ζ ), establises tat d = 1. In τ, wic is infinite dimensional, one finds te I() cointegrating components and in τ 1, wic is finite dimensional and different from, tose tat are not cointegrating. Te combinations of tese circumstances makes up te I(1) conditions. Similarly, Teorem 4.4 sows tat d = if and only if 3 circumstances old: te first, τ H (or equivalently ζ H), establises tat d >, te second, τ 1 τ (or equivalently ζ 1 ζ ), establises tat d > 1 and te tird, τ = (τ τ 1 ) (or equivalently ζ = (ζ ζ 1 ) ) establises tat d =. In τ, wic is infinite dimensional, one finds te cointegrating components tat allow for polynomial cointegration of order, in τ 1, wic is finite dimensional (and can as well be equal to ), tose tat are I(1) and don t allow for polynomial cointegration, and in τ, wic is finite dimensional (and different from ), tose tat are not cointegrating. Te combinations of tese circumstances makes up te I() conditions. As sown in te next section, tis construction is true in te general I(d) case.

14 1 5. Te general result Tis section extends te results in Section 4 to te general I(d), d = 1,,..., case. Teorem 5.4 provides a necessary and sufficient condition for an H-valued AR processes A(L)x t = ε t to be I(d) and it is sows tat under tis condition te space is decomposed into te sum of d + 1 closed subspaces, H = τ τ 1 τ d, tat are defined in terms of A, A 1,..., A d in (4.1), see Definition 5.1 below. Te infinite dimensional cointegrating space coincides wit τ τ 1 τ d 1 and τ d is te finite dimensional attractor space of te I(d) trends. In τ, wic is infinite dimensional, one finds te cointegrating vectors tat allow for polynomial cointegration of order, in τ, = 1,..., d, wic is finite dimensional and can as well be equal to, tose tat allow for polynomial cointegration of order and in τ d 1, wic is finite dimensional and can as well be equal to, tose tat are I(d 1) and don t allow for polynomial cointegration. Finally, in τ d, wic is finite dimensional and different from, tose tat are I(d) and don t allow for cointegration. Definition 5.1 (ζ, τ subspaces and Q, A,n operators). Consider a cointegrated H-valued AR process A(L)x t = ε t, were A(z) is written as in (4.1). Let ζ = Im A, τ = (Ker A ), Q = A +, were Q is te generalized inverse of A, and for = 1,,... define S = P Z A,1 P T, ζ = Im S, τ = (Ker S ), Q = S +, were Q is te generalized inverse of S, and A,n = { Z = ζ ζ 1, T = τ τ 1, A n for = 1 A 1,n+1 + A 1,1 j= Q ja j+1,n for =, 3,..., n = 1,,.... Remark 5.. As sown in Remark.8, te assumption tat 1 is an eigenvalue of finite type of A(z) implies tat te subspaces ζ = Im A and τ = (Ker A ) = Im A are closed and infinite dimensional wile teir ortogonal complements ζ = (Im A ) and τ = Ker A = (Im A ) are closed and finite dimensional. Tis implies tat te subspaces ζ, τ, = 1,,..., are closed and finite dimensional. Because ζ, =, 1,..., is closed, te corresponding generalized inverse Q, =, 1,..., exists and it is unique, see Remark 4.5. Remark 5.3. By construction, for s, ζ s is ortogonal to ζ and τ s is ortogonal to τ ; moreover, it is possible tat ζ, τ,, are equal to. Teorem 5.4 (A caracterization of I(d) cointegrated H-valued AR processes). Consider a cointegrated H-valued AR process A(L)x t = ε t, were A(z) is written as in (4.1), and let ζ, τ, Q, and A,n be as in Definition 5.1. Ten x t is I(d) if and only if H = τ τ 1 τ d,

15 were τ, τ 1,..., τ d are closed, dim τ =, dim τ < for any, d, and < dim τ d <. In tis case, x t = C s d,t + C 1 s d 1,t + + C d 1 s 1,t + C (L)ε t + v, Im C = τ d, were s,t = t i=1 s 1,i I(), s,t = ε t, C (L)ε t is a linear process, and v collects initial values. Tis sows tat τ τ 1 τ d 1 is te cointegrating space of x t and τ d is te attractor space of te I(d) trends. Moreover, d 1 v, x t + v, Q A +1,n n x t I() for all v τ, =,..., d, n=1 v, x t I(d 1) for all v τ d 1, and v, x t I(d) for all v τ d. Finally, te I(d) condition can be equivalently stated as H = ζ ζ 1 ζ d, were ζ, ζ 1,..., ζ d are closed, dim ζ =, dim ζ < for any, d, and < dim ζ d <. Remark 5.5. In te finite dimensional case H = R p, Franci and Paruolo (16) sow tat d = 1,,... if and only if R p = ζ ζ d = τ τ d, were ζ = span(α ), τ = span(β ), =, 1,..., and te bases α, β are defined by te rank factorizations P a A,1 P b = α β, were a = (α,..., α 1 ), b = (β,..., β 1 ), Q = β (β β ) 1 (α α ) 1 ζ and A,1 is as in Definition 5.1. Again ere, apart from te fact tat dim ζ = dim τ = rank A is finite wen H = R p, tis mirrors wat appens in te infinite dimensional case. Hence te infinite dimensionality of te space does not introduce additional elements in te I(d) analysis. Summing up, Teorem 5.4 sows tat d = 1,,... if and only if d + 1 conditions old: te first, τ H (or equivalently ζ H), establises tat d > and for = 1,..., d 1, τ (τ τ 1 ) (or equivalently ζ (ζ ζ 1 ) ), establises tat d > and te last one, τ d = (τ τ d 1 ) (or equivalently ζ d = (ζ ζ d 1 ) ), establises te value of d = 1,,.... Te infinite dimensional cointegrating space coincides wit τ τ 1 τ d 1 and τ d is te finite dimensional attractor space of te I(d) trends. In τ, wic is infinite dimensional, one finds te cointegrating vectors tat allow for polynomial cointegration of order, in τ, = 1,..., d, wic is finite dimensional and can as well be equal to, tose tat allow for polynomial cointegration of order and in τ d 1, wic is finite dimensional and can as well be equal to, tose tat are I(d 1) and don t allow for polynomial cointegration. 13 Te coefficients of te polynomial cointegrating relations are Q A +1,n, wic are calculated recursively as in Definition Conclusion Te present paper caracterizes te cointegration properties of an H-valued AR process A(L)x t = ε t under te assumptions tat i) A(1), ii) A(z) as an eigenvalue of finite type at z = 1, and iii) A(z) is invertible in te punctured disc D(, ρ) \ {1} for some ρ > 1.

16 14 A necessary and sufficient condition for x t I(d), d = 1,,..., is given and it is sown tat under tis condition te space is decomposed into te sum of d + 1 closed subspaces tat are defined recursively from te AR operators, H = τ τ 1 τ d, were τ is closed and infinite dimensional and τ, = 1,..., d is closed and finite dimensional. Te infinite dimensional subspace τ τ 1 τ d 1 is te cointegrating space of x t wile te finite dimensional subspace τ d is te attractor space of te I(d) stocastic trends. Hence an H-valued cointegrated AR process as a finite number of I(d) trends and infinitely many cointegrating relations. Te properties of v, x t vary wit v in te cointegrating space: for v τ, wic is infinite dimensional, one can combine v, x t wit differences of te process and find I() polynomial cointegrating relations, for v τ 1, wic is finite dimensional and can as well be equal to, one can combine v, x t wit differences and find at most I(1) polynomial cointegrating relations, and so on up to v τ d 1, wic is finite dimensional and can as well be equal to, for wic v, x t I(d 1) does not allow for polynomial cointegration. For any v in te cointegrating space, te explicit expression Q A +1,n of te coefficients of te polynomial cointegrating relations is provided in terms of operators tat are defined by te same recursion of te τ. Te present results sow tat te infinite dimensionality of te space does not introduce additional elements in te analysis, under te assumption tat 1 is an eigenvalue of finite type of te AR operator function and tat no oter non-zero eigenvalue of te AR operator lies witin or on te unit circle. Tat is, apart from te fact tat te number of I() cointegrating relations is infinite, te conditions and properties of H-valued cointegrated AR process coincide wit tose tat apply in te finite dimensional case.

17 15 Appendix A. Separable Hilbert spaces and operators acting on tem Tis section reviews definitions (typewritten in italics) and basic facts on separable Hilbert spaces and on operators tat act on tem. Te material is based on Capter I, II and XV in Goberg et al. (3) and Capter XI in Goberg et al. (199). Tis section also introduces te basic system (A.), wic is central in te proofs in Appendix C, and a useful intermediate result in Lemma A.. Let H be an Hilbert space; H is called separable if tere exist vectors v 1, v,... wic span a subspace dense in H, i.e. every vector in H is te limit of a sequence of vectors in span(v 1, v,... ). It can be sown tat (only) separable Hilbert spaces ave countable ortonormal bases and tat a closed subspace of a separable Hilbert space is separable. A separable Hilbert space H is said to be te direct sum of subspaces M and N, written H = M N, if every vector v H as a unique representation of te form v = x + y, were x M and y N. Te dimension of N, written codim M, is called te codimension of M and M is said to be complemented if N is closed. Let H be a separable Hilbert space wit inner product, and norm ; a function A wic maps H into H, A : H H, is called a linear operator if for all x, y H and c C, A(x + y) = Ax + Ay and A(cx) = cax, were Az and A(z) bot indicate te action of A on z. A linear operator A : H H is called bounded if sup x =1 Ax < and its norm A is given by sup x =1 Ax. Te set of bounded linear operators wic map H into H is denoted by B(H). An operator A B(H) is said to be invertible if tere exists an operator B B(H) suc tat BAx = ABx = x for every x H; in tis case B is called te inverse of A, written A 1. A B(H) is said to be a projection if A = A and it can be sown tat if A is a projection, Ker A is complemented, i.e. H = Im A Ker A and Im A = Ker(I A) is closed. An operator A B(H) is said to be Fredolm of index n(a) d(a) if te numbers n(a) = dim Ker A and d(a) = codim Im A are finite. It can be sown tat if H is finite dimensional, any operator is Fredolm of index. Let D(u, ρ) = {z C : z u < ρ} be te open disc centered in u wit radius ρ > and let F : D(u, ρ) B(H) be an absolutely convergent operator function; a point z D(u, ρ) is said to be an eigenvalue of finite type of F (z) if F (z ) is Fredolm, F (z )x = for some non-zero x H and F (z) is invertible for all z in some punctured disc D(z, ρ) \ {z }. It can be sown tat if z is an eigenvalue of finite type ten F (z ) is Fredolm of index.

18 16 Let T, W : D(u, ρ) B(H) be absolutely convergent operator functions; T (z) is said to be locally equivalent at z to W (z) if i) T (z) = E(z)W (z)g(z) in some disc D(z, ρ) and ii) E(z) and G(z) are invertible and absolutely convergent on D(z, ρ). Teorem A.1. Let T : D(u, ρ) B(H) be an absolutely convergent operator function. Assume tat tere exists z D(u, ρ) suc tat T (z ) is Fredolm of index and non-invertible. Ten T (z) is locally equivalent at z to s W (z) = W + W 1 (z z ) m W s (z z ) ms, W W j = δ j W, W = I, were m 1 m m s are positive integers, δ j is te Kronecker delta, W, W 1,..., W s are mutually disjoint projections tat decompose te identity, W is Fredolm of index and W 1,..., W s ave rank one. Tat is, tere exists ρ > suc tat T (z) = E(z)W (z)g(z) for all z D(z, ρ), were E(z) and G(z) are invertible and absolutely convergent on D(z, ρ). Hence T (z) 1 = G(z) 1 W (z) 1 E(z) 1 as a pole of order d = m s at z and it admits representation T (z) 1 = U n (z z ) n d, z D(z, ρ) \ {z }, (A.1) n= were U,..., U d 1 are operators of finite rank and U d is Fredolm of index. = Proof. See Teorem 8.1, Corollary 8.4 and eq.() in section XI.9 in Goberg et al. (199). Consistently wit te terminology employed in te finite dimensional case, see Goberg et al. (1993), te operator function W (z), te positive integers m 1 m m s and te operator functions E(z), G(z) are respectively called te local Smit factorization, te partial multiplicities and extended canonical system of root functions of T (z) at z. Expanding T (z) around z as T (z) = n= T n(z z ) n and considering (A.1), one writes te identity T (z)t (z) 1 = I = T (z) 1 T (z) as te following linear systems in te T n, U n operators T U = = U T T U 1 + T 1 U = = U T 1 + U 1 T. T U d T d 1 U = = U T d U d 1 T (A.) T U d + T 1 U d T d U = I = U T d + U 1 T d U d T T U d+1 + T 1 U d + + T d+1 U = = U T d+1 + U 1 T d + + U d+1 T. In te following, equations in te system (A.) are indexed according to te igest value of te subscript of U n ; for instance te identity appears in equation d, wic is te order of te pole.

19 17 Te equations tat derive from T (z)t (z) 1 = I are called left versions and tose tat derive from I = T (z) 1 T (z) are called rigt versions; for instance T U d + T 1 U d T d U is te left version of equation d. Lemma A.. Let T (z) be as in Teorem A.1 and define ζ = Im T and τ = (Ker T ). Ten U in (A.1) satisfies U = P τ U = U P ζ = P τ U P ζ. Proof. Te left version of equation, T U =, implies tat Im U Ker T. Let τ = (Ker T ), so tat τ = Ker T, and define te associated ortogonal projections P τ and P τ. Clearly P τ and P τ are disjoint and decompose te identity. Hence U = P τ U + P τ U = P τ U. Similarly, from te rigt version of equation, U T =, one as Im T Ker U. Defining ζ = Im T, so tat ζ = (Im T ), and te associated ortogonal projections P ζ and P ζ, one as U = U P ζ + U P ζ = U P ζ and ence te statement. Appendix B. Random variables in separable Hilbert spaces Te following definitions are taken from Bosq (). Let H be a separable Hilbert space wit inner product,, norm and Borel σ-algebra σ(h) and let (Ω, A, P ) be a probability space. A function Z : Ω H is called an H-valued random variable on (Ω, A, P ) if it is measurable, i.e. for every subset S σ(h), {ω : Z(ω) S} A. For a C-valued random variable U on (Ω, A, P ), define E(U) = Ω U(ω)dP (ω); te expectation of an H-valued random variable Z, written µ Z, is defined as te unique element of H suc tat E(, Z ) =, µ Z for all H. It can be sown tat te existence of µ Z is guaranteed by te condition E( Z ) <. Te covariance function of an H-valued random variable Z is defined as c Z (, x) = E(, Z µ Z x, Z µ Z ),, x H. It is immediate to see tat c Z (, x) = E(, Z x, Z ), µ Z x, µ Z = E(, x, Z Z ), µ Z x, µ Z. If E( x, Z Z ) <, te expectation of te H-valued random variable x, Z Z exists and it is te unique element of H suc tat E(, x, Z Z ) =, µ x,z Z for all H. One tus as c Z (, x) =, µ x,z Z, µ Z x, µ Z,, x H. Because x, Z Z = x, Z Z x Z, te existence of te covariance function of Z is guaranteed by te condition E( Z ) <. Define te operator C Z : H H tat maps x into µ x,z Z and rewrite te covariance function as c Z (, x) =, C Z (x), µ Z x, µ Z,, x H. As C z is completely determined by te covariance function, it is called te covariance operator of Z. Similarly, te cross-covariance function of two H-valued random variables Z and U is defined

20 18 as c Z,U (, x) = E(, Z µ Z x, U µ U ),, x H. Tis completely determines te cross-covariance operators of Z and U, C Z,U and C U,Z, respectively defined as te mappings x µ x,z U and x µ x,u Z. Appendix C. Proofs Proof of Teorem 3.1. Te result is a direct consequence of Teorem A.1 in Appendix A. By definition, a cointegrated H-valued AR process A(L)x t = ε t is suc tat A(1), A(z) as an eigenvalue of finite type at z = 1 and A(z) is invertible in some punctured disc D(, ρ) \ {1}, ρ > 1. Hence A(1) is Fredolm of index. Because A : C B(H), one can apply Teorem A.1. Tis states tat tere exists ρ > suc tat A(z) = E(z)W (z)g(z) for all z D(1, ρ), (C.1) were E(z) and G(z) are invertible and absolutely convergent on D(1, ρ) and W (z) = W + W 1 (1 z) m W s (1 z) ms, W W j = δ j W, s W = I, were te positive integers m 1 m m s are te partial multiplicities of A(z) at 1, W, W 1,..., W s are mutually disjoint projections tat decompose te identity, W is Fredolm of index and W 1,..., W s ave rank one. Given tat G = G(1) is invertible, one can normalize it to be equal to I, in fact A(z) = E (z)w (z)g (z), were E (z) = E(z)G, W (z) = G 1 W (z)g, and G (z) = G 1 G(z) sare te same properties of E(z), W (z) and G(z) in (C.1). In te following one can set G(1) = I. Moreover, A(z) is invertible in te punctured disc D(, ρ) \ {1}, ρ > 1, and because A(z) 1 = G(z) 1 W (z) 1 E(z) 1, tis implies tat E(z) and G(z) are invertible and absolutely convergent for all z D(, ρ), ρ > 1. Let w be te number of distinct partial multiplicities and organize tem as in m 1 = = m q1 < m q1 +1 = = m q1 +q }{{} < < m }{{} w 1 i=1 q i+1 = = m s }{{} =j 1 =j = =j w, were q is te number of partial multiplicities tat are equal to te given value j. Tis leads to define te projections P as te sum of te projections W n tat load te same partial multiplicity into W (z), i.e. P 1 = W W q1, P = W q W q1 +q,..., P w = W w 1 i=1 q i W s, so tat W (z) = w = P (1 z) j, were j =, P = W and P = j n=1 W n+ 1 i=1 q, = 1,..., w. i Observe tat P P j = δ j P and w = P = I, were P as infinite rank q = dim Im P = and P 1,..., P w ave finite rank q. Tis leads to te direct sum decomposition H = Im P Im P 1 Im P w, (C.)

21 were Im P is closed because P is a projection. Substituting (C.1) in A(L)x t = ε t and rearraging one as W (L)y t = u t, y t = G(L)x t, u t = E(L) 1 ε t, and using W (z) = w = P (1 z) j and w = P = I one finds P y t + j 1 P 1 y t + + jw P w y t = P u t + P 1 u t + + P w u t, were P y t and P u t belong to Im P. As a consequence of te direct sum decomposition in (C.), one as j P y t = P u t and ence P y t = P u j,t + P v,, were u j,t I(j ) is te j -t cumulation of u t I() and v, collects initial values. w = P v,, one as y t = P u j,t + P 1 u j1,t + + P w u jw,t + v, u,t = 19 As y t = w = P y t = w = P u j,t + t u 1,i I(), i=1 u,t = u t I(), were v = w = P v, collects initial values. Tis sows tat y t = G(L)x t I(j w ), q = dim Im P is te number of I(j ) processes in y t and v, y t I(j ) if and only if v Im P. Because G(1) = I, x t = G(L) 1 y t = y t + F 1 y t +..., were G(z) 1 = F (z) = n= F n(1 z) n, one as x t = (P u j,t + P 1 u j1,t + + P w u jw,t + v ) + F 1 (P u j,t + P 1 u j1,t + + P w u jw,t + v ) +..., wic sows tat x t is I(j w ), v, x t I(j w ) if and only if v Im P w and ence Im P Im P 1 Im P w 1 is te cointegrating space of x t. Finally, let n = j w j and expand G(z) around 1 as G(z) = n 1 n= G n(1 z) n + (1 z) n G n (z), were G = I and G n (z) is absolutely convergent on D(, ρ), ρ > 1. Ten x t + G 1 x t + + G n 1 n 1 x t = P u j,t + P 1 u j1,t + + P w u jw,t + x j,t + v, were x j,t = G n (L) n x t = G n n x t + = G n n P w u jw,t + = G n P w u j,t +... is at most integrated of order j w n = j, so tat P ( xt + G 1 x t + + G n 1 n 1 x t ) = P (u j,t + x j,t) + P v. Because u j,t +x j,t = (I G n P w )u j,t +..., one as tat v, (u j,t +x j,t)x for any v Im P and any x Im P Im P w 1. Tis sows tat v, x t + n 1 n=1 v, G n n x t is I(j ) for all v Im P and completes te proof. Proof of Teorem 4.1. Replacing T wit A and U wit C in system (A.), for d = 1 one as A C = = C A A C 1 + A 1 C = I = C A 1 + C 1 A A C + A 1 C 1 + A C = = C A + C 1 A 1 + C A.

22 Because P ζ A = and C = P τ C, see Lemma A., te left version of equation 1 implies tat (P ζ A 1 P τ )C = P ζ. Tis sows tat if x Im P ζ ten x Im P ζ A 1 P τ, i.e. Im P ζ Im P ζ A 1 P τ ; because te reverse inclusion is clearly satisfied, one as tat Im P ζ A 1 P τ = Im P ζ, i.e. ζ 1 = ζ. Similarly, because A P τ = and C = C P ζ, see Lemma A., te rigt version of equation 1 implies tat C (P ζ A 1 P τ ) = P τ. Hence if x Ker P ζ A 1 P τ ten x Ker P τ, i.e. Ker P ζ A 1 P τ Ker P τ ; because te reverse inclusion is clearly satisfied, one as tat Ker P ζ A 1 P τ = Ker P τ, so tat (Ker P ζ A 1 P τ ) = (Ker P τ ), i.e. τ 1 = τ. Tis sows tat if d = 1 ten ζ 1 = ζ, τ 1 = τ, C = P τ1 C P ζ1, Im C = τ 1 and Ker C = ζ. Finally note tat if d > 1, because te identity is in equation d, te same analysis leads to (P ζ A 1 P τ )C = and C (P ζ A 1 P τ ) =. In tis case tere exist a nonzero x Im C suc tat x Ker P ζ A 1 P τ and a nonzero y Im P ζ A 1 P τ suc tat y Ker C. i.e. τ 1 τ and ζ 1 ζ. Tis sows tat τ 1 = τ (or any of ζ 1 = ζ, Im C = τ 1 and Ker C = ζ ) is a necessary and sufficient condition for d = 1. Because x t = C(L)ε t I() and Im C = τ 1, τ 1 is te attractor space of te I(1) trends, i.e. v, x t I(1) for all v τ 1, and τ1 = τ is te cointegrating space of x t. In order to prove tat v, x t I() for all v τ, write P τ x t = P τ C ε t + P τ C 1 ε t + C (L) ε t, were ere and in te following C (z), C (z), C (z) represent remaining terms. Because P τ C =, P τ x t = P τ C 1 ε t + C (L) ε t is a linear process. Next consider te generalized inverse of A, Q = A +, and note tat Q A = P τ, see Lemma C.1 below. From te left version of equation 1 one as Q A C 1 +Q A 1 C = Q and tus P τ C 1 = Q (I A 1 C ); tis implies tat v, P τ C 1 x for any v τ and any x ζ, i.e. v, x t I() for all v τ. Proof of Teorem 4.4. Replacing T wit A and U wit C in system (A.), for d = one as A C = = C A A C 1 + A 1 C = = C A 1 + C 1 A A C + A 1 C 1 + A C = I = C A + C 1 A 1 + C A A C 3 + A 1 C + A C 1 + A 3 C 1 = = C A 3 + C 1 A + C A 1 + C 3 A. From te proof of Teorem 4.1 one as tat if d > 1 te left versions of equations and 1 lead to A C = and (P ζ A 1 P τ )C = and te rigt versions to C A = and C (P ζ A 1 P τ ) =. Hence Im C (Ker A Ker P ζ A 1 P τ ) = (τ τ 1 ) = T and Ker C (Im A Im P ζ A 1 P τ ) = (ζ ζ 1 ) = Z. Hence C = P T C = C P Z = P T C P Z. Because P ζ A =, te left version of equation implies tat P ζ A 1 C 1 +P ζ A C = P ζ. Inserting I = P τ +P τ in te first term one as P ζ A 1 C 1 = P ζ A 1 P τ C 1 +P ζ A 1 P τ C 1 and ence P Z A 1 C 1 = P Z A 1 P τ C 1, because Im P ζ A 1 P τ = ζ 1. Because Q A = P τ, see Lemma C.1 below, te left version of equation 1 implies P τ C 1 = Q A 1 C ; ence one as P ζ A 1 C 1 = P ζ A 1 Q A 1 C + P ζ A 1 P τ C 1 and