Common trends and cycles in I(2) VAR systems

ARTICLE IN PRESS

Journal of Econometrics 132 (2006) 143–168 www.elsevier.com/locate/econbase

Common trends and cycles in I(2) VAR systems$ Paolo Paruolo Department of Economics, University of Insubria, Via Ravasi 2, 21100 Varese, Italy Available online 10 March 2005

Abstract This paper analyzes common cycles in I(2) vector autoregressive (VAR) systems. We consider different choices of stationary variables extracted from a VAR, including deviations from equilibria. This extension is based on the equilibrium dynamics representation of the system, introduced in this paper. Inference on the number of common features is addressed via reduced rank regression, as well as estimation of the cofeature relations and testing. An application to Australian prices is presented; it is found that the deviation from one equilibrium relation is an innovation process, whereas no common cycles can be obtained for the acceleration rates. r 2005 Elsevier B.V. All rights reserved. JEL classification: C32; C51; C52 Keywords: Common features; Cointegration; Common cycles; I(2); Reduced rank regression

1. Introduction The notion of common factors is a classical idea in statistics; during the last two decades it has received new momentum in econometrics, thanks to the introduction of concepts like cointegration, see Engle and Granger (1987), and of common features, see Engle and Kozicki (1993) and Vahid and Engle (1993). As for the case

$

Invited paper at the conference ‘Common features in Rio’, Rio de Janeiro 28–31 July 2002.

Tel.: +39 0332 215500; fax:+39 0332 215509.

E-mail address: [email protected]. 0304-4076/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jeconom.2005.01.026

ARTICLE IN PRESS 144

P. Paruolo / Journal of Econometrics 132 (2006) 143–168

of cointegration, common cycles (CC)—defined as non-innovation common features—imply restrictions on the dynamic representation of the system. The interplay between common trends and common cycles in processes integrated of order 1, I(1), has been considered in Vahid and Engle (1993). I(1) systems have also been the focus of much of the ensuing literature; see Kugler and Neusser (1993), Vahid and Engle (1997), Cubadda and Hecq (2001), Paruolo (2003) inter alia. No contributions in the literature has investigated the interplay between common trends and cycles for systems integrated of order 2, I(2); this is focus of the present paper. I(2) systems have been analyzed in Johansen (1992, 1995a, 1997), Stock and Watson (1993), and Boswijk (2000) inter alia; see Haldrup (1998) for a survey and references prior to 1998. In this paper it is shown how the I(2) equilibrium correction form is the basis for the analysis of common cycles in the second differences of the process. An additional representation is introduced, called the ‘equilibrium dynamics’ form, which is the basis for the analysis of common cycles in deviations from equilibrium. The application of CC both to second differences and to equilibrium relations sheds light on different features of a system. Extensions of the notion of CC to unpredictable linear combinations of several lags of the process are also considered. As for I(1) systems, the notion of CC is directly related to rank deficiency of some autoregressive coefficient matrices; this provides a unified framework for inference. When the cointegration parameters are known, the Gaussian likelihood analysis consists of reduced rank regression (RRR), see Anderson (1951). The same locally asymptotically normal (LAN) results apply once the cointegration parameters have been substituted with their maximum likelihood (ML) estimates or the two stage I(2) estimates of Johansen (1995a), thanks to the superconsistency of estimates of cointegration parameters. This property does not extend to other permanenttransitory transformations involving estimated parameters that converge at the standard T 1=2 rate, as in the I(2) analog of the transformation of Gonzalo and Granger (1995). The RRR framework provides a common format for testing, estimation and specification search on the cofeature vectors. These techniques are illustrated on the Australian prices data set analyzed in Banerjee et al. (2001); it is found that the newly introduced equilibrium dynamics form supports the presence of a single cofeature vector, while none can be obtained for the equilibrium correction form. The rest of the paper is organized as follows: Section 2 presents notation and definitions. Section 3 reports various representations of I(2) systems. CC are treated in Section 4. Section 5 discusses unpredictable combinations involving variables at different lags. Section 6 addresses inference through RRR techniques. Section 7 contains the application to Australian prices. Section 8 concludes. Proofs are reported in the appendix. In the following a:¼b and b¼:a indicate that a is defined by b; ða: bÞ indicates the matrix obtained by horizontally concatenating a and b. ei indicates the i-th column of the identity matrix. For any full column rank matrix H, colðHÞ is the linear span ¯ indicates HðH 0 HÞ
1 and H ? indicates a basis of col? ðHÞ; the of the columns of H, H ¯ 0 ¼ HH ¯ 0 is the orthogonal projector orthogonal complement of colðHÞ: PH ¼ HH

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

145

matrix onto colðHÞ: vec is pthe column stacking operator and A  B:¼½aij B defines d the Kronecker product. ! and ! indicate convergence in probability and in distribution respectively. All processes W t are understood to be multivariate, i.e. of dimension q  1; W t ¼ ðW 1t : . . . : W qt Þ0 : Individual time series, or linear combinations thereof, are called components of the process.

2. Notation and definitions In this section we introduce general notation and definitions. We follow Johansen (1996, Chapter 3), for the definition of (co)integration and Engle and Kozicki (1993), Vahid and Engle (1993) for the general definition of non-innovation common features. P We consider VAR(k), kX2; systems of the type X t ¼ ki¼1 Ai X t
i þ e
t ; where X t and e
t ¼ m1 t þ m0 þ md d t þ
t are p  1: t, 1, d t are the deterministic components, d t :¼ðd 1;t : . . . d r
1;t Þ0 is a vector of demeaned seasonal dummies, i.e. of the form d i;t ¼ 1ðt mod r ¼ iÞ
1=r; 1ðÞ is the indicator function and r is the number of seasons. L and D:¼1
L are the lag and difference operators, where negative powers of D indicate summation.
t is assumed to be an innovation process w.r.t. F t
1 ; the sigmafield generated by X t
i ; iX1; more specifically
t is assumed to be a martingale difference sequence w.r.t. F t ; Eð
t jF t
1 Þ ¼ 0; with second moments Eð
t
0t jF t
1 Þ ¼ O; where 0ox0 Oxo1 for all x 2 Rp : All innovation processes in this paper are understood to be linear combination of
t :1 We assume, ‘Assumption 1’, that, apart fromProots at z ¼ 1; all other characteristic roots of the AR polynomial AðzÞ:¼I
ki¼1 Ai zi are outside the unit circle, i.e. of the stationary type; if all roots are outside the unit circle then AðzÞ is called stable. Stationary processes derived P from X ti
EðX t Þ have a moving average representation C W ðLÞ
t ; where C W ðzÞ:¼ 1 i¼0 C W ;i z is summable for jzjo1 þ K and K40; i.e. they are linear processes. The first coefficient matrix C W ;0 is assumed to have full row-rank, but not necessarily to be equal to the identity matrix. When a linear process has sum of coefficients C W ð1Þ different from the 0 matrix, then the linear process is called integrated of order 0, Ið0Þ; see Johansen (1996). In the following, an Ið0Þ process W t is said to have Ið0Þ rank q if rankðC W ð1ÞÞ ¼ q: The row-rank deficiency of C W ð1Þ is associated with the presence of cointegration in D
1 W t ; see Johansen (1996, Chapter 3). A process X t is said to be integrated of order d, IðdÞ; if Dd X t
EðDd X t Þ is Ið0Þ; d ¼ 1; 2; . . . : Ið0Þ processes W P t
EðW t Þ ¼ C W ðLÞ
t are in general autocorrelated, with j-th 0 autocovariance gj :¼ 1 i¼0 C W ;i OC W ;iþj ; a feature associated with predictability and the presence of cycles. Innovation processes are a special case of I(0) processes without this feature; they correspond to C W ðLÞ ¼ C W ;0 ; with 0 autocovariances because C W ;i ¼ 0 for i ¼ 1; 2; . . . in gj : With a slight abuse of language, we refer to any linear process that is not an innovation process as a cycle. 1

In the statistical analysis
t is assumed i.i.d. Nð0; OÞ: The assumption kX2 is common to the literature on I(2) systems. Other deterministic terms could also be incorporated.

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

146

Consider an Ið0Þ process W t with rank q40; if there exist some non-zero vector bi such that b0i ðW t
EðW t ÞÞ is an innovation process, then the system is said to present non-innovation common features, or common cycles, CC, and bi is called a cofeature vector. Let ‘ be the maximum number of linearly independent cofeature vectors b1 ; . . . ; b‘ ; the matrix b:¼ðb1 : . . . : b‘ Þ is called the CC cofeature matrix, and the systems is said to have cofeature rank ‘: Equivalently, W t is said to present q
‘ common I(0) cycles. Implicit in this notation is the fact that the maximum number of I(0) cycles is given by the I(0) rank. This result is parallel to Theorem 1 in Vahid and Engle (1993) for I(1) systems, although it applies more generally also to I(2) systems. Proposition 1 (Upper bound on cofeature rank). A p  1 Ið0Þ process W t with Ið0Þ rank qpp presents at most q linearly independent innovation processes; hence the cofeature rank ‘ is bounded by q, ‘pq: When qop; the remaining p
q components of an I(0) process with rank q are integrated of negative order, which are cyclic by definition. Hence nothing can be said about the commonality in the remaining p
q directions. This point is further discussed in Section 5. Before closing this section, we introduce a motivating example. Example 2. Consider a bivariate system X t :¼ðX 1t : X 2t Þ0 run by i.i.d. innovations Zt :¼ðZ1t : Z2t Þ0 ; defined by the equations X 1t ¼ DX 2t þ Z1t ; D2 X 2t ¼ ct ;

where

ct ¼ Rct
1 þ Z2t ;

jRjo1.

Here ct represents a cycle for Ra0: Observe that the system is Ið2Þ because X 2t needs second differences to become Ið0Þ: From the first equation one sees that DX 1t ¼ D2 X 2t þ DZ1t ¼ ct þ DZ1t is affected by the cycle ct ; which is common to the 2 variables in the system. One thus wishes to adopt a notion of common features that, when applied to this system, would indicate the presence of a common cycle.

3. Common trends In this section we review three representations for I(2) systems: the common trends representation, the equilibrium correction formulation and the equilibrium dynamics form. The connection between the first two representations is treated in Johansen’s representation theorem, see Johansen (1992) or (1996, Theorem 4.6). The last representation can be found in the proof of Johansen’s representation theorem, and it is used here in order to discuss common features in deviations from equilibrium. It is convenient to rewrite AðLÞ as AðLÞ ¼
PL
GDL þ UðLÞD2 ; i.e. consider D2 X t ¼ PX t
1 þ GDX t
1 þ gV t þ e
t

(1)

for kX2; where notation is given in Table 1. We next list the conditions for (1) to be I(2).

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

147

Table 1 Notation Symbol

Definition

Symbol

Definition

Y 1t :¼ Y 2t :¼ g:¼ U:¼

b01 DX t b02 D2 X t ðU1 : . . . : Uk
2 Þ Pk
2 i¼1 Ui ð1: d 0t Þ0 ðm0 : md Þ ðmy0 : md Þ Dðm0
z1 b00 Þ Dm D
t ¯ 0 my
DUk
2 bb

Y 0t :¼ Y t :¼ V t :¼ y:¼

b0 X t þ db02 DX t þ b00 t ðY 00t : Y 01t : Y 02t Þ0

C :¼ U zt :¼ C :¼

ðY 00t
1 : DX 0t
1 ðb: b1 Þ: V 0t Þ0 ðA1 : . . . : Ak
2 : DðUk
3
Uk
2 Þ: Ak
1;1 :
Ak
1;2 Þ ðY 0t
1 : . . . : Y 0t
kþ2 : Y 00;t
kþ1 : Y 01;t
kþ1 : b0 DX 0t
kþ1 Þ0 ¯ z1 þ b: ¯ z2 þ b¯ 1 : gÞ Dða þ b:

ðmz0 : md Þ

U t :¼

ðY 00t
1 : DX 0t
1 ðb: b1 Þ: V 0t Þ0

Dt :¼ m:¼ my :¼ my0 :¼ m :¼
t :¼ mz0 :¼ mz :¼

0

0

D:¼ z:¼ C:¼ U t :¼ z

ðD2 X 0t
1 : . . . : D2 X 0t
kþ2 Þ0 ¯ a0 ðG
PÞ þ I
U ðG
PÞb¯ ðb þ b2 d0 : b1 : b2 Þ0 ðz1 : z2 Þ ða: z1 : z2 : gÞ

I(2) conditions: (a) P ¼ ab0 ; where a and b are p  p0 matrices of full rank p0 op; (b) Pa? GPb? ¼ a1 b01 where a1 and b1 are p  p1 matrices of full rank p1 op
p0 ; or, equivalently, a0? Gb? ¼ xZ0 where x ¼ a0? a1 and Z ¼ b0? b1 are p
p0  p1 matrices of full rank p1 op
p0 ; (c) a02 yb2 has full rank p2 :¼ p
p0
p1 ; where a2 :¼ða: a1 Þ? ; b2 :¼ðb: b1 Þ? ; (d) m1 ¼ ab00 ; with b00 a p0  1 vector; ¯ 0 ; with Z0 a p1  1 vector. (e) a0? m0 ¼ xZ00 þ a0? Gbb 0 0 Under Assumption 1, Johansen’s representation theorem, see Rahbek et al. (1999), establishes that necessary and sufficient conditions for D2 X t ; Y 0;t :¼b0 X t þ db02 DX t þ b00 t; Y 1;t :¼b01 DX t to be stationary, apart from deterministic components and initial values, are conditions (a)–(e). Under the I(2) conditions the following common trends representation holds X t ¼ C2

t X s X s¼1 i¼1


i þ C 1

t X


i þ C 0 ðLÞ
t þ m0 þ m1 t þ mðLÞd t ,

(2)

i¼1

P P where the I(2) component is C 2 ts¼1 si¼1
i ; with C 2 ¼ b2 ða02 yb2 Þ
1 a02 P of rank p2 ; hence there are p2 common I(2) trends. The I(1) component is C 1 ti¼1
i ; for remaining definitions see Rahbek et al. (1999). The I(2) common trends representation implies that D2 X t is an I(0) process with rank p2 : Taking in fact second differences in (2) one finds D2 X t ¼:C n ðLÞ
t þ mnt ¼ C 2
t þ C 1 D
t þ C 0 ðLÞD2
t þ mnt ,

(3)

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

148

where mnt :¼mðLÞD2 d t : This equation shows that C n ð1Þ ¼ C 2 ; which is of rank p2 ; and hence D2 X t is an I(0) process with rank p2 : If the I(2) conditions hold, then system (1) can be rewritten in many equilibrium correction forms. The following one is taken from Paruolo and Rahbek (1999): D2 X t ¼ a½Y 0;t
1  þ ðz1 : z2 Þ½ðb: b1 Þ0 DX t
1  þ gV t þ mDt þ
t ¼ CU t þ mDt þ
t ,

ð4Þ

where the constraint m1 ¼ ab00 has been imposed see also Table 1. The terms in square brackets in (4) are I(0) by Johansen’s representation theorem; these equations emphasize the correction of the variables D2 X t towards equilibrium. A final representation is the one that defines the dynamics of the stationary cointegration relations themselves. We consider Y 0t :¼b0 X t þ db02 DX t þ b00 t; Y 1t :¼b01 DX t ; Y 2t :¼b02 D2 X t as the stationary variables of interest, where Y t :¼ðY 00t : Y 01t : Y 02t Þ0 is p  1: Other choices are possible, see Corollary 4. Theorem 3 (Equilibrium dynamics representation). Let the Ið2Þ conditions hold and let Y t :¼ðY 00;t : Y 01;t : Y 02;t Þ0 ; then Y t follows a VARðkÞ; A ðLÞY t ¼ my Dt þ
t ,

(5) Pk

where the AR polynomial A ðLÞ:¼I
i¼1 Ai Li is stable, and can be inverted to give Y t ¼ C  ðLÞðmy Dt þ
t Þ; with C  ðLÞ
t ¼ C Y ðLÞ
t is a Ið0Þ process with rank p, where C Y ;0 ¼ D; a full rank matrix. Let the AR matrices Ai be partitioned column-wise conformably with Y t ; i.e. let Ai;j be the p  pj block that multiplies Y j;t
i ; j ¼ 0; 1; 2: The last AR matrices Ai in (5) are constrained as follows: Ak;0 d ¼
Ak
1;2 ;

ðAk;1 : Ak;2 Þ ¼ 0.

(6)

The constraints (6) can be incorporated in (5) by substituting Y t
k ; Y 2;t
kþ1 with Db0 X t
kþ1 in the r.h.s. of (5), or, alternatively using the same r.h.s. variables as (4), obtaining, respectively, Y t ¼ Cz U zt þ mz Dt þ
t , ¼C



U t



þ m Dt þ


t .

ð7Þ ð8Þ

Moreover C ¼ Cz A; for A square and non-singular, and hence rankðC Þ ¼ rankðCz Þ and colðCz Þ ¼ colðC Þ: (7) is called the ‘equilibrium dynamics form’ and (8) is called the ‘mixed form’.2 Eq. (5) or (7) describes the dynamics of the equilibrium relations. Note also that the I(0) rank of process Y t is equal to the dimension p of the process X t (and Y t ). Thus the transformation from D2 X t to Y t allows to express all cycles as I(0) cycles. Observe also that C in (4) and C (respectively, Cz ) in (7) (resp. (8)) are not similar and possibly have different ranks. 2

Note that when k ¼ 2; the mixed form (8) and the equilibrium dynamics representation (7) coincide.

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

149

In the following corollary we show that the properties of the equilibrium dynamics for Y t discussed in Theorem 3 are carried over to a different choice of the Y 1t component. Let t1 be any p  p1 matrix such that colðb: b1 Þ ¼ colðb: t1 Þ and define G t :¼ðY 00t : DX 0t t1 : Y 02t Þ0 : Corollary 4. Under the same assumptions of Theorem 3, G t follows a VARðkÞ of the type B ðLÞY t ¼ Kmy Dt þ K
t ; where B ðLÞ is stable, K is a square matrix of full rank defined in the appendix and the AR coefficients of B ðLÞ satisfy the following constraints, equivalent to (6): ¯ ¼ 0; Bk;0 d þ Bk
1;2
2Bk
1;1 t01 bd

ðBk;1 : Bk;2 Þ ¼ 0.

The choice G t is hence equivalent to Y t : For ease of exposition, in the following we will only discuss the choice Y t ; simply noting that G t enjoys the same properties.

4. Common cycles This section discusses the application of common features to I(2) systems, as defined in Section 2. The notion is applied both to D2 X t and Y t ; where Y t has been defined in Section 3, see (5). In the following we will indicate with W t either D2 X t or Y t: A matrix b, of dimension p  ‘ and rank ‘; defines unpredictable linear combinations for W t if b0 ðW t
EðW t ÞÞ ¼ g0
t is an innovation process, where W t
EðW t Þ is a p  1 I(0) process of rank q. We say that b is a CC cofeature matrix for W t with cofeature rank ‘ when ‘ is chosen to be maximal. The above definition of CC allows to decompose the time series W t into cyclical and idiosyncratic components, analogously to the permanent-transitory decompositions discussed in Gonzalo and Granger (1995). Let b be the cofeature matrix of W t : Using the orthogonal projection identity I p ¼ Pb þ Pb? ; one can define a first decomposition ¯ t þ b¯ ? ct , W t ¼ bZ

(9)

ct :¼b0? W t

is the common cyclical component of dimension p
‘ and I(0) where rank q
‘; while Zt :¼b0 W t ¼ b0 EðW t Þ þ g0
t is the idiosyncratic noise, i.e. a white noise component of dimension and rank equal to ‘: A second decomposition can be obtained using non-orthogonal projections; let a be of the same dimension of b with b0 a of full rank. One has I n ¼ aðb0 aÞ
1 b0 þ b? ða0? b? Þ
1 a0? ¼: ab b0 þ b?a? a0? ; and hence W t ¼ ab Zt þ b?a? cat ? , a ct ? :¼a0? W t

(10)

; ab :¼aðb0 aÞ
1 ; b?a? :¼b? ða0? b? Þ
1 : Note that both decompositions where (9) and (10) contain the same idiosyncratic component Zt with different loading matrices, which correspond to a different definition of the cyclical component, ct and cat ? : The cyclical parts are autocorrelated and present no CC.

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

150

The cyclical (ct or cat ? ) and idiosyncratic components (Zt ) are in general correlated, except for the
1 choice a? :¼O
1 g? : This particular decomposition has the property O g? that CovðZt ; ct Þ ¼ g0 OO
1 g? ¼ 0; i.e. the cyclic and idiosyncratic parts are uncorrelated. As stated in Proposition 1, the cofeature rank ‘ is bounded by the I(0) rank q, i.e. ‘pq: As noted in Eq. (3), D2 X t is an I(0) process of rank p2 ; while Y t is an I(0) processes of rank p, see Theorem 3. Hence the upper bound ‘pq is more restrictive for the choice W t ¼ D2 X t than W t ¼ Y t because p2 pp: The existence of a cofeature matrix b for D2 X t or Y t is associated to a rank reduction of the coefficient matrices in the equilibrium correction or equilibrium dynamics representations respectively, as shown in the following proposition. Proposition 5 (Cofeatures and reduced rank). W t presents common feature with cofeature rank ‘ if and only if CðÞ is of reduced rank, where CðÞ ¼ C in (4) for the choice W t ¼ D2 X t ; and CðÞ ¼ C in (8) for the choice W t ¼ Y t : The reduced rank condition rankðCðÞ Þ ¼ p
‘ can be written CðÞ ¼ jt0 ; with j and t of full column rank s:¼p
‘: In this case the cofeature matrix b can be chosen equal to j? : We next discuss the two choices for W t in more detail. Consider first the choice W t ¼ D2 X t : The following representation theorem for cofeatures in I(2) systems parallels Proposition 1 of Vahid and Engle (1993) for I(1) systems. Theorem 6 (Common cycles representation). Under the I(2) conditions, there exist a cofeature matrix b such that b0 ðD2 X t
EðD2 X t ÞÞ ¼ b0
t if and only if in (2) or (3) one has b0 C 0;i ¼ 0;

i ¼ 0; 1; 2; . . . ,

b0 C 1 ¼ 0.

(11) (12)

When (11) holds, the second condition (12) holds if and only if any of the following equations holds: b0 C 2 ¼ b0 , b ¼ a2 c ? ;

(13) a02 ðI
yÞb2 ¼ cd 0 ,

(14)

where c and d are p2  p2
‘ matrices of rank p2
‘: Theorem 6 shows that, when it exists, the cofeature matrix b for D2 X t must be of the form b ¼ a2 u for an appropriate matrix u, b0
t ¼ u0 a02
t : Recall also from (2) that a02
t are the second increments of the p2 common I(2) trends. Hence the interpretation of the CC cofeature linear combinations b0 ðD2 X t
EðD2 X t ÞÞ is that of observable second increments to the common I(2) trends. We next consider the choice W t ¼ Y t : If the cofeature matrix b selects elements of Y 0t or Y 1t ; then the cofeature relations imply that certain deviations from equilibrium are innovation processes, as expected by some economic models with rational expectations. If the cofeature matrix b selects elements from Y 2t ; the interpretation is similar to the one given for the choice W t ¼ D2 X t : It would

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

151

thus be useful in this context to test exclusion restrictions on b0 W t similar to the ones of a system of structural equations. We refer to this possibility as specification-test on b. A possible disadvantage of the choice W t ¼ Y t is that the components of Y t are themselves linear combinations of X t ; DX t ; D2 X t ; so that the interpretation of the cofeature matrix is possible only after identification of the components of Y t ; and after specification-testing on b itself. This problem, however, is solved by careful modelling of the cointegration properties of a system and by appropriate specification testing on b, see Section 6 below, where we discuss model specification. We next apply the previous definitions to the example of Section 2. Example 7 (Example 2 continued). Observe that b ¼ ð1: 0Þ0 ; d ¼
1; b2 ¼ ð0: 1Þ0 ; Y 0t ¼ X 1t
DX 2t ; Y 2t ¼ D2 X 2t ; Y t ¼ ðY 0t : Y 2t Þ0 : The equilibrium dynamics representation is ! ! ! ! Z1t 0 0 X 1t
DX 2t X 1t
1
DX 2t
1 ¼ þ . Z2t 0 R D2 X 2t D2 X 2t
1 Observe that the AR matrix A1 is of deficient rank, so that b ¼ ð1: 0Þ0 is a cofeature vector. The cofeature relation is X 1t
DX 2t ¼ Z1t ; hence common features applied to Y t correctly signal the presence of a common cycle. The equilibrium correction representation is 0 1 ! ! ! X 1t
1
DX 2t
1 2
1
1 R B
1t D X 1t C DX 1t
1 ¼ , @ Aþ 0 0 R
2t D2 X 2t D2 X 2t
1 where
1t :¼Z1t þ Z2t ;
2t :¼Z2t : Note that the regression matrix on the r.h.s. is of full rank 2 for any Ra0; so that there is no cofeature vector for D2 X t : In this case common features applied to D2 X t would not signal the presence of a common cycle. The reason is that when W t ¼ D2 X t the type of cofeature relation is the one of observable second increments to the Ið2Þ trends, which is not the case here. The conclusion is that for some systems there may exist cofeatures in the equilibrium correction formulation, W t ¼ D2 X t ; and for some other systems there may exist cofeatures in the equilibrium dynamics representation, W t ¼ Y t : Both definitions may hence turn out to be important. Ultimately which option is relevant remains an empirical question. In the following section we consider some more general notions of unpredictable linear combinations.

5. Unpredictable combinations In this section we discuss unpredictable linear combinations of possibly larger sets of variables, which may include different lags of X t ; we call this case UP,

ARTICLE IN PRESS 152

P. Paruolo / Journal of Econometrics 132 (2006) 143–168

unpredictable polynomial combinations.3 This notion offers some relief to the lack of invariance of CC to changes in the timing of the variables. This phenomenon was first observed by Ericsson in his comments to Engle and Kozicki (1993), and it applies as well to the results in previous sections. Unlike CC, however, there is no direct decomposition in idiosyncratic and cyclic components associated with UP, of the type discussed in Section 4. Unpredictable combinations of X t and its lags are motivated e.g. by present value models. Consider in fact a present value model of the form X 1t ¼ j0 ð1
P 0 i j1 Þ 1 j i¼0 1 Et ðX 2tþi Þ for the bivariate time series X t :¼ðX 1t : X 2t Þ ; see Campbell and 4 Shiller (1987). Special cases of this model are obtained in the expectation theory of interest rates, where X 1t represents the long term yield and X 2t the one-period interest rate, the present value model for stock prices, where X 1t is the stock price and X 2t the dividend, and (with modifications) to the permanent income theory of consumption. This model implies that the linear combination X 1t
j
1 1 ðX 1t
1
j0 ð1
j1 ÞX 2t
1 Þ is an innovation process with respect to past history F t of X t : This innovation process ! X 1t
1 1 :
j1 DX 1t
ð 1
j0 Þ (15) X 2t
1 j1 involves DX t and X t
1 ; or X t and X t
1 ; i.e. X t at different lags. The relation (15) is compatible with X t being I(1) or I(2); in the latter case, (15) represents a multicointegrating relation. The unpredictable combination (15) is similar to a CC cofeature relation defined in Section 4, although applied to two lags of X t : We hence consider a polynomial matrix bðLÞ in the lag operator; a set of ‘ linear combinations bðLÞ0 X t is defined to be an UP linear combinations if it is an innovation process with respect to F t and ‘ is maximal. Observe that CC discussed in Section 4 corresponds to the special case bðLÞ0 X t ¼ b0 W t :5 The notion of UP combinations provides a partial answer to the lack of invariance of CC with respect to timing of the variables, as illustrated by the following example. Example 8. Consider the stationary bivariate system W t :¼ðW 1t : W 2t Þ0 with innovations Zt W 1t ¼ R1 W 2t
j þ Z1t ; W 2t ¼ R2 W 2t
2 þ Z2t ;

where

jR2 jo1;

R1 a0;

R2 a0,

and the lag j in the first equation can be either 1 or 2. When j ¼ 2; the system W t presents a common cycle in the definition of Section 4, because R2 W 1t
R1 W 2t ¼ 3

The transformation Y t already involves lags of X t : The unpredictable linear combinations may involve other lags not included in Y t : 4 This reference was suggested by one of the referees; it assumes X t is I(1) and discusses the cointegration implications of (15) in this case. 5 The UP notion contains also the special case of ‘polynomial serial correlation common feature’ defined as any polynomial linear combination of DX t that is an innovation process, see Cubadda and Hecq (2001).

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

153

R2 Z1t
R1 Z2t is an innovation process. Changing the lag to j ¼ 1 yields no common cycles for W t in the definition of Section 4. However, if one considers UP in the extended vector ðW 0t : W 0t
1 Þ0 ; one finds that W 1t
R1 W 2t
1 ¼ Z1t is an innovation process. Hence, looking for UP in the extended vector ðW 0t : W 0t
1 Þ0 for j ¼ 1 yields the same number of unpredictable linear combinations as when considering CC for j ¼ 2; although the weights of the linear combination are different in the two cases. This shows that UP goes some way in preserving the common cycle properties of a system when changing the timing of the variables. The previous example shows that employing the definition of UP relations, the system in the example contains a common cycle both for j ¼ 1 and 2, although the linear combination involving W 1t and W 2t
h has different weights in the two cases. Observe that not all UP combination can be interpreted as cofeature combination in the CC sense. Some UP combinations may for instance involve the same variable at different lags; for instance W 1t
jW 1t
1 could be an innovation. This occurrence would signal that W 1t has autonomous dynamics, i.e. it is not Granger-caused by the other variables in the system. Obviously the interpretation in this case would be very different from the case of CC. Finding some UP relations in a system does not tell whether these conform to the predictions of an economic model as in (15), or they correspond to asynchronous cycles in different variables as in Example 8, or to autonomous dynamics of the above type, or even some other case. Just as in cointegration analysis, one can investigate these issue e.g. by hypothesis testing on the UP relations; this specification analysis is discussed in Section 6. In the rest of this section we analyze some representation properties of UP combinations. In the following, we let W t be an I(0) process with rank q, which is taken to be either D2 X t or Y t as above. We will also indicate by Rt an h  1 vector of additional stationary variables, constructed from lags of X t : Let Z t :¼ðW 0t : R0t Þ0 : A matrix b, of dimension ðp þ hÞ  ‘ and rank ‘; is defined to be an UP matrix for Zt if b0 ðZ t
EðZ t ÞÞ is an innovation process. We say that b is a UP matrix for Z t with UP rank ‘ when ‘ is chosen to be maximal. If b:¼ðb01 : b02 Þ0 is partitioned conformably with Z t :¼ðW 0t : R0t Þ0 ; choosing b2 ¼ 0 delivers the CC definition given in Section 4. A consequence of the definition is that in UP combinations the contemporaneous variables W t are always involved, in the sense of the following proposition. Proposition 9. If b:¼ðb01 : b02 Þ0 is a ðp þ hÞ  ‘ UP matrix for Zt :¼ðW 0t : R0t Þ0 ; where Rt depends on lagged X t s and b is partitioned conformably with Z t ; then b1 has full column rank ‘: The inclusion of additional variables Rt is meant to be minimal. In this sense it would be interesting to investigate what set of additional variables Rt makes the choices ðD2 X 0t : R0t Þ0 and ðY 0t : R0t Þ0 equivalent for UP purposes. This is reported in the following proposition. Proposition 10. The UP properties of U 1t :¼ðD2 X 0t : R0t Þ0 and U 2t :¼ðY 0t : R0t Þ0 are identical when Rt :¼ðY 00;t
1 : DX 0t
1 ðb: b1 ÞÞ0 :

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

154

In the next proposition we state the necessary and sufficient conditions in order to have UP relations; this proposition extends Proposition 5. In the following, we indicate W t with Z 0t ; and we let Z 2t :¼ðR0t : d 0t Þ0 ; in order to simplify the notation of later statements. We define
nt :¼C W ;0
t ; where C W ;0 ¼ I for W t ¼ D2 X t and C W ;0 ¼ D for W t ¼ Y t ; see Theorem 3. The covariance matrix of
nt is indicated by On :¼C W ;0 OC 0W ;0 : Similarly we let mn0 indicate m0 ; my0 ; mz0 or m0 : Proposition 11. Let Z 2t :¼ðR0t : d 0t Þ0 ; and assume that Z 0t :¼W t ; Z 1t and Rt be variables generated from a stationary VAR with innovations
t ; where Z 0t satisfies Z 0t ¼ BZ 1t þ FZ2t þ mn0 þ
nt ,

(16)

and Z 1t ; Rt depend on lagged
t s: Partition also F:¼ðF1 : F2 Þ conformably with Z 2t :¼ðR0t : d 0t Þ0 : Then a necessary and sufficient condition for b to be an UP matrix for ðW 0t : R0t Þ0 ¼ ðZ00t : R0t Þ0 is that B is of reduced rank, B ¼ jt0 ; with j and t of full column rank. In this case the UP matrix has representation b0 ¼ ðj0? : j0? F1 Þ and b0 ðW 0t : R0t Þ0 ¼ j0? W t þ j0? F1 Rt ¼ j0? ðF2 d t þ mn0 Þ þ
nt . In the following we use the characterization given in Proposition 11, simply stating the reduced rank restrictions implied by different choices of variables in Z 0t ; Z1t ; Z 2t :

6. Estimation and testing This section describes inference on I(2) VAR systems with common trends and cycles. The cointegration analysis of I(2) systems has been extensively discussed in the literature; hence it is not described here for space constraints. We refer to Johansen (1995a, 1997), Rahbek et al. (1999), Boswijk (2000), Paruolo (2000). This section concentrates on the analysis of cofeatures after the cointegration analysis has been performed, fixing the cointegration parameters b; b1 ; d to their maximum likelihood estimates or the two stage I(2) estimates of Johansen (1995a) see Rahbek et al. (1999). These estimators of the cointegration parameters are superconsistent, and using the estimates in place of the parameters does not change the limit distributions of the common feature statistics described below. In the rest of this section we simply do not distinguish b; b1 ; d and their estimated values, and we assume that
t is i.i.d. Gaussian. In Section 6.1 we review the statistical analysis of CC and UP through RRR; in Section 6.2 we discuss specification analysis. 6.1. RRR analysis For any given model, see Table 2, the analysis of common features may be organized by first determining the cofeature rank ‘: The cofeature matrix b can then be estimated, for the selected cofeature rank ‘; possibly testing restrictions on b. In some cases, economic theory may suggest the specific value of the cofeature matrix b;

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

155

Table 2 Rank restrictions in the regression format of (17) using the notation RRRðZ0t ; Z1t jZ2t ; 1Þ b01 aðÞ

Case

b01 zðÞ

b01 gðÞ

Z1t 0

(a)

CC

0

0

0

(b)

UP

*

0

0

(c)

UP

0

*

0

(d)

UP

0

0

*

Z2t ; 1 1

Y 0;t
1 B ðb: b Þ0 DX C t
1 A @ 1 Vt ! ðb: b1 Þ0 DX t
1

Y 0;t
1

*

*

0

Vt

!

Dt

Vt !

ðb: b1 Þ0 DX t
1 UP

Y 0;t
1

ðb: b1 Þ0 DX t
1

Y 0;t
1 Vt

(e)

Dt

!

Vt 0

!

!

Dt

Dt

1 Y 0;t
1 B ðb: b Þ0 DX C t
1 A @ 1 Dt

The dependent variables Z0t is either D2 X t for the equilibrium correction form (4) or Y t for the equilibrium dynamics mixed form (8). aðÞ indicates either a or a ; similarly for z; g: * indicates unrestricted entries.

in this case it would be of interest to test that a certain vector is a cofeature vector. Finally one may analyze the cofeature relations b0 W t ¼ ut or b0 Zt ¼ ut as a system of simultaneous equations, where ut are ‘ linear combinations of the innovations
t : All these hypotheses are considered in this subsection; we consider either likelihood ratio tests, labelled Qi ; or Wald-type tests, indicated by J i : All test statistics are asymptotically w2 distributed with degrees of freedom equal to the number of constraints; moreover all tests are consistent under fixed alternatives. Proofs of these statements can be found in the working paper version of this paper at http:// ideas.repec.org; see also Paruolo (2003). We first treat the case of unknown cofeature matrix b. Several cases of CC and UP have been presented in Sections 4 and 5, see Table 2. As stated in Propositions 5 and 11, they can all be put in the regression format Z 0t ¼ BZ 1t þ FZ2t þ mn0 þ
nt

(17)

where the cofeature restriction is HðsÞ:

B ¼ jt0 n

(18) n

n n0

and j; t; F; m0 and O :¼Eð
t
t Þ are unrestricted. s indicates the number of columns in j; t; where j is p  s and t is j  s: Because when jop there always exists a cofeature matrix of rank p
j; we exclude these trivial cases by assuming jXp; i.e. there are more regressors than dependent variables.6 We indicate as the ‘HðsÞ model’ the regression model (17) under the reduced rank restriction (18). 6

Most of the derivations are unaffected by this assumption.

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

156

The HðsÞ model is analyzed by reduced rank regression, indicated in the following with the shorthand RRRðZ

0t ; Z 1t jZ 2t ; 1Þ: The P Gaussian log-likelihood function is proportional to
Tðln
On
þ trðOn
1 T
1 Tt¼1
nt
tn0 ÞÞ; which is maximized by considering the following eigenvalue problem: jlS 11
S10 S
1 00 S 01 j ¼ 0

(19)

4lp and associated eigenvectors with eigenvalues l1 4    li 4   P PT vi ; where T 0
1
1 S ij :¼M ij
M i2 M
1 M ; M :¼T ðZ
m ÞðZ
m Þ ; m :¼T 2j ij it i jt j i 22 t¼1 t¼1 Z it ; i, j ¼ 0; 1, 2, see e.g. Johansen (1996). The LR test statistic for hypothesis (18) of HðsÞ versus HðpÞ about the rank of B is given by Q1 ðsÞ:¼
T

p X

lnð1
li Þ.

i¼sþ1

This test is asymptotically w2 with degrees of freedom df Q1 ðsÞ ¼ ðj
sÞðp
sÞ: Eq. (19) provides also the maximum likelihood estimates for given dimension s. In particular bt ¼ ðv1 : . . . : vs Þ and b ¼ S 01btðbt0 S 11btÞ
1 ; j

bB ¼ j bbt0 ¼ S 01btðbt0 S11btÞ
1bt0 ,

b ¼ ðM 02
bBM 12 ÞM
1 ; F 22

b n ¼ S00
S 01btðbt0 S11btÞ
1bt0 S 10 , O

ð20Þ

t ¼ diagðl1 ; . . . ; ls Þ ¼: L1 : where bt is normalized by bt0 S 11bt ¼ I s ; bt0 S 10 S
1 00 S 01b In order to identify parameters, it is convenient to normalize bt by the justidentifying restrictions btc :¼btðc0btÞ
1 ; where c is a known matrix of the same dimensions of t; such that c0 t is a square non-singular matrix, see Johansen (1996, b obtained by substituting btc in place Section 5.2) or Paruolo (1997). The choice of j b :¼bBc; which satisfies c j bbt0c ¼ bB: of bt in (20) is given by c j b ?a? :¼b b ? Þ
1 also In the following we use the just-identifying normalization j j? ða0? j for the estimator of j? : We note that j? is estimated unrestrictedly as the matrix of eigenvectors associated with the last p
s eigenvalues of the dual problem to (19) jlS 00
S01 S
1 11 S 10 j ¼ 0,

(21)

b? ¼ which has the same li eigenvalues of (19) and eigenvectors u1 ; . . . ; up ; one has j b ? is normalized by ðusþ1 : . . . : up Þ; see Johansen (1996, Theorem 8.5). Here j b 0? S 00 j b 0? S01 S
1 b ? ¼ I p
s ; j b ? ¼ diagðlsþ1 ; . . . ; lp Þ ¼: L2 : The corresponding j 11 S 10 j b ?a? :¼b b ? Þ
1 ; where a? is a known, full column just-identified estimator is j j? ða0? j rank matrix of the same dimensions of j? ; and it is assumed that a0? j? is of full rank. b ?a? ¼ a0? j?a? ¼ I ‘ ; so that Due to the just-identifying restrictions, one has a0? j 0 a? ðb j?a?
j?a? Þ ¼ 0; ðb j?a?
j?a? Þ ¼ Pa ðb j?a?
j?a? Þ; see Paruolo (1997), and one only needs to report the limit distribution for a¯ 0 ðb j?a?
j?a? Þ: This is given in the following proposition.

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

157

Proposition 12. Let the I(2) assumption hold. Under (18) one has d

T 1=2 ðvecð¯a0 ðb j?a?
j?a? ÞÞÞ ! Nð0; ðj0?a? On j?a?  ða0 BS11 B0 aÞ
1 ÞÞ.

(22)

A consistent estimator of the asymptotic covariance matrix is obtained substituting parameters with the corresponding ML estimators and S11 with S11 : Consider exclusion restrictions for all columns of j? simultaneously of the type H0 :

j? ¼ Hf,

(23)

where H is p  h; hX‘: Under restriction (23), the likelihood function is maximized by solving jln H 0 S00 H
H 0 S 01 S
1 11 S 10 Hj ¼ 0, with eigenvalues ln1 4    4lnh and corresponding eigenvectors vni ; see e.g. Paruolo (1997), Appendix C, or Johansen (1996, Theorems 8.4 and 8.5). The corresponding LR test statistic of (23) in HðsÞ is given by ! p h X X n Q2 :¼T lnð1
li Þ
lnð1
li Þ , i¼sþ1

i¼h
pþsþ1

b ? ¼ Hðvnh
pþsþ1 : . . . : vnh Þ: This test is and the restricted estimate of j? is j 2 asymptotically w distributed with degrees of freedom df Q2 :¼2ps
s2
2pðp
hÞ
ðp
s
hÞð2h
p þ sÞ: Hypothesis (23) is a special case of K 0 vecð¯a0 j?a Þ ¼ j; where K has h columns. The Wald test associated with the latter is given by n

b j b ?a? Þ
jÞ0 ððb b ?a? Þ
1  ða0bBS 11bB0 aÞÞ J 1 :¼TðK 0 vecð¯a0 j j0?a? O b ?a? Þ
jÞ ðK 0 vecð¯a0 j

ð24Þ

which is asymptotically w2 ðhÞ: Consider now the case where b is (partly) known. Let K be a known p  h matrix of rank hp‘; and consider the hypothesis that K is a submatrix of b, b ¼ ðK: bn2 Þ; i.e. H0 :

K 0 B ¼ 0.

(25)

A Wald test of (25) can be based on the unrestricted maximum likelihood estimates b n :¼S 00:1 :¼S 00
S 01 S
1 S 10 ; and equals bB:¼S01 S
1 ; O 11 11 J 2 :¼T trððK 0 S 00:1 KÞ
1 K 0bBS11bB0 KÞ ¼ T trððK 0 S 00:1 KÞ
1 K 0 S 01 S
1 11 S 10 KÞ

(26)

which is asymptotically w2 ðhjÞ: The corresponding LR test of (25) in HðsÞ; labelled Q3 ; is found by solving the eigenvalue problem jl K 0? S00:K K ?
K 0? S 01:K S
1 11:K S 10:K K ? j ¼ 0,

(27)

with eigenvalues l1 4    4lh and corresponding eigenvectors vi ; where S ij:K :¼S ij
S i0 KðK 0 S00 KÞ
1 K 0 S0j ; i; j ¼ 0; 1; see Johansen (1996, Theorems 8.2 and 8.5). The test

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

158

of (25) in HðsÞ is given by Q3 :¼T ¼T

s X

lnð1


li Þ




s X

! lnð1
li Þ

i¼1

i¼1

p X

h
pþs X

lnð1
li Þ


i¼sþ1

i¼sþ1

! 0 jK S Kj 00:1 lnð1
li Þ
ln , jK 0 S00 Kj

ð28Þ

b? ¼ where S 00:1 :¼S 00
S 01 S
1 11 S 10 : The restricted estimate under (25) is j b ?a? : The Q3 test is ðK: K ? ðvsþ1 : . . . : vh
pþs ÞÞ; which again can be identified via j asymptotically w2 ; with degrees of freedom df Q3 :¼sh: The tests Q1 ðsÞ and Q3 can be combined to obtain the LR test of (25) in HðpÞ; Q4 :¼Q1 ðsÞ þ Q3 : Again d Q4 ! w2 ðdf Q4 Þ; with df Q4 ¼ df Q1 ðsÞ þ df Q3 : Finally, observe that j0? Z 0t ¼ ut ; where ut contains ‘ linear combinations of
t ; defines a system of ‘ simultaneous equations. Homogeneous separable restrictions on each equation can be written in the form j? ¼ ðH 1 f1 : . . . : H ‘ f‘ Þ, see Johansen (1995b) for the discussion of identification in this case. We just mention here that the algorithm for the maximization of the likelihood proposed there, see also Johansen (1996, Theorem 7.4), can be applied to the estimation of j? in the dual problem (21), interchanging b and j? ; the subscripts 0 and 1, and choosing the smallest eigenvalues instead of the largest ones. 6.2. Specification analysis A list of different UP and CC cases is given in Table 2, using the format of Eq. (16). We observe that case (d) for W t ¼ D2 X t corresponds to the conditions for b01 X t to be weakly exogenous for the cointegrating parameters b; b1 ; d; see Paruolo and Rahbek (1999). In particular, these conditions can be written as b01 ða: zÞ ¼ 0; which simply states that the equations of b01 X t in the equilibrium correction formulation (4) have zero adjustment coefficients. This situation may be described as ‘no levels- and difference-feedback’ in the equations of b01 D2 X t : Cases (b), (c), (e) are similar to the definition of ‘weak form of common features’ proposed in Hecq et al. (2005) for I(1) systems. The idea is that some elements in D2 X t inherit the cyclic part included in deviations from equilibrium in Y 0;t
1 and/or ðb: b1 Þ0 DX t
1 : Several models given in Table 2 are nested. This suggest the possibility to ‘test down’ for cofeatures from the most general to the most specific model. The sequence starts from models characterized by the less stringent restrictions, represented by case (e). Rejection of the reduced rank restrictions in this model implies also rejection for any nested submodel. Hence, finding that model (e) does not support the presence of cofeatures implies that no submodel (cases (a), (b), (c)) presents cofeatures. When the presence of UP is supported in a model, like model (e), one can continue testing more restricted submodels. Cases (b) and (c) are nested within model (e), but mutually non-nested. Both submodels can be investigated. If both submodels do not

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

159

support cofeatures, then one returns to the first nesting model that supports cofeatures. Eventually the sequence may reach the CC model (a). Obviously, the significance level of the individual tests in the testing-down procedure must be chosen in order to guarantee a given overall size, e.g. by use of Bonferroni-type inequalities. Hence a typically small nominal size must be chosen for each component test. One can also arrange the specification search starting from the most restricted model; we call this strategy the ‘testing up’ procedure. In this case the most restricted model is the CC model, case (a). If this model does not support cofeatures, one considers less stringent models, like models (b) and (c). Eventually the specification search may reach the least stringent model (e). Note that, in all cases, the models with restrictions are compared with a baseline reference model, which is the unrestricted equilibrium correction formulation (4) or the equilibrium dynamics mixed form (8). Hence also the ‘testing up’ procedure is in line with the general-to-specific framework, see Paruolo (2001) and reference therein. In this procedure, the sizes of the tests in the sequence are fixed at the overall nominal level; the overall procedure can be shown to have the asymptotic nominal size of each component test, if each test has probability of rejection that converges to 1 under a fixed alternative, which is the case for the tests in Section 6.1. For further details on this type of procedure we refer to Paruolo (2001) and reference therein. 7. An application to Australian prices In this section we present an application to the Australian prices data set analyzed by Banerjee et al. (2001) and Omtzigt and Paruolo (2005), inter alia.7 We refer to the latter paper for the analysis of the initial specification and the cointegration analysis. The remaining computations of the CC analysis reported here were performed in Gauss 3.6 and PcGive 10.0. The data set consists of three Australian macroeconomic time series: the consumer price deflator at factor cost (lpfc), unit labor costs in the non-farm sector (lulc) and import prices (lpm). All three variables are quarterly data, measured in natural logs, and run from 1970Q1 to 1995Q2 for a total of 102 observations. The time series are graphed in Fig. 1. The common trend analysis selected p0 ¼ 1; p1 ¼ 1; the ML estimates of the cointegration parameters turned out to be 0 b b1 X t ¼
0:28lpfct
0:72lulct þ lpmt , 0 0 0 Y 0t ¼ b b Xt þ b db b2 DX t þ b b0 t ¼ lpfct
0:74lulct
0:26lpmt þ 2:68    Dlpfct þ Dlulct þ Dlpmt þ 0:0013t.

ð29Þ

We next fixed the cointegration parameters b; b1 ; d at these estimated values. The equilibrium corrections shows how the original variables adjust to various 7

The data set is available at the data archive of the Journal of Applied Econometrics: http:// qed.econ.queensu.ca/jae.

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

160

∆ lpfc

lpfc 0.04

4

0.02 3 1970

1975

1980

1985

1990

1995

lulc

1970

1975

1980

1990

1995

1990

1995

1990

1995

∆ lulc

0.10

6

1985

0.05 0.00

5 1970

1975

1980

1985

1990

1995

1970

1975

1980

1985

∆ lpm

lpm 0.1

4

0.0 3 1970

1975

1980

1985

1990

1995

1970

1975

1980

1985

Fig. 1. Data in levels and first differences.

Table 3 Restricted estimates of the adjustment coefficients a; z in the equilibrium correction form (4)

D2 lpfc D2 lulc D2 lpm

b a

bz1

bz2


0.0596 0 0


0.329 1.256 0.639

0 0.179
0.866

The zero restrictions, for fixed cointegration coefficients, gave a LR test of 2.5709 with a w2 ð3Þ p-value of 0.4626. All remaining coefficients are significant at conventional confidence levels.

disequilibria. The ML estimates of the adjustment coefficients a and z are reported in Table 3, after deleting insignificant coefficients. They show0 a significant adjustment to the growth rate of the autonomous price component b b1 DX t
1 ; both for D2 lulct 2 and D lpmt : We interpret this finding as evidence that the I(1) autonomous price 0 component b b1 X t contains international trends, which also influence the labor market. Note also that the adjustment to the multicointegrating relation Y 0;t
1 is significant only in the equation for D2 lpfc, suggesting that Y 0t measures (deviations from) an internal price equilibrium. Using this specification we also were not able to reject the null of absence of structural breaks. We performed the CC analysis in model (a) in Table 2, both for the equilibrium correction form and for the equilibrium dynamics form. We employed the Q1 test statistic to investigate the presence of cofeature vectors, taking Z 0t either equal to D2 X t or Y t ; Z1t equal to ðY 0;t
1 : Y 1;t
1 : b0 DX t
1 Þ0 and Z 2t ¼ ðd 0t : d tn0 Þ0 ; where d nt indicate

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

161

Table 4 Test statistics Q1 ðsÞ for the equilibrium correction form (4) for D2 X t and for the equilibrium dynamics in mixed form (8) for Y t Specification

s

0

1

2

(4)

Q1 ðsÞ df p-value w2 ðdf Þ

250.28 9 0.0000

116.33 4 0.0000

33.00 1 0.0000

(8)

Q1 ðsÞ df p-value w2 ðdf Þ

562.08 9 0.0000

43.00 4 0.0000

0.3538 1 0.5519

df indicates the number of degrees of freedom.

intervention dummies defined in Omtzigt and Paruolo (2005). Table 4 reports the result of the test statistics. It can be seen that the tests reject the presence of cofeatures in the equilibrium correction form (4) for D2 X t ; while they indicate the presence of a single cofeature vector for the equilibrium dynamics in mixed form (8) for Y t : The corresponding estimate of j? in Y t for s ¼ 2; i.e. ‘ ¼ 1; is reported in Table 5. We normalized the estimate on the coefficient to Y 1t ; by choosing a? ¼ e2 and a ¼ ðe1 : e3 Þ in (22), where ei is a unit vector with all zeros and a 1 in position i. Table 5 reports also asymptotic standard errors based on Proposition 12, and the corresponding asymptotically normal t-ratios. These estimates suggest the vector ð0: 1: 0Þ0 as a candidate cofeature vector, i.e. that Y 1t
EðY 1t Þ is an innovation process in this system. In order to test the hypothesis j? ¼ ð0: 1: 0Þ0 we employed the LR test Q2 of (23) in HðsÞ; specifying H ¼ ð0: 1: 0Þ0 :8 We obtained Q2 ¼ 2:0358; with a p-value of 0:3614 when compared with a w2 ð2Þ: The corresponding Wald test J 1 in (24) was equal to 1:8453 with a p-value of 0:3975 when compared with a w2 ð2Þ: Hence, both tests support the hypothesis j? ¼ ð0: 1: 0Þ0 : The same conclusion for the equilibrium dynamics can be derived by testing that single equations of the system Y t :¼ðY 0t : Y 1t : Y 2t Þ0 have all coefficients to stochastic regressors equal to 0 in HðpÞ: This hypothesis is of the type (25) with K ¼ ei ; the associated Wald test statistic J 2 in (26) is reported in Table 6. We also calculated the LR test of (25) in HðsÞ; i.e. the test Q4 : Both tests confirm that there exists a CC cofeature vector b ¼ ð0: 1: 0Þ0 in the system for Y t ; i.e. that all coefficients of lagged variables in the equation for Y 1t can be restricted to 0. Hence Y 1t
EðY 1t Þ is an innovation process, i.e. the autonomous I(1) component is one of the I(1) trends in the system. The fit of the restricted equilibrium dynamics b 1t Þ: This is graphed in Fig. 2, along with the estimated innovation process Y 1t
EðY analysis also suggests that Y t contains 3
1 ¼ 2 common I(0) cycles, which can be represented e.g. by the equations for Y 0t and Y 2t : Recall in fact that the equilibrium dynamics form has I(0) rank equal to p ¼ 3: 8

The same test can also be obtained as a special case of Q3 :

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

162

Table 5 Estimates of the cofeature vector j? for the equilibrium dynamics in mixed form (8) for Y t Equations

Y 0t

Y 1t

Y 2t

b 0?a? j Standard errors t-ratios


0:0092

1

0:1026

0:0593
0:1551

— —

0:0756 1:3581

Table 6 Test statistics J 2 ; Q4 and w2 ð3Þ p-values in brackets of hypothesis (25) with K ¼ ei ; i ¼ 1; 2; 3; corresponding to the zero coefficients in the equations indicated at the top of each column Eq.

Test statistic

Equation

J2 Q4

D2 lpfc 50.162 [0.0000] 45.949 [0.0000]

D2 lulc 165.74 [0.0000] 107.40 [0.0000]

D2 lpm 111.81 [0.0000] 83.297 [0.0000]

J2 Q4

Y 0t 31.243 [0.0000] 30.990 [0.0000]

Y 1t 2.0798 [0.5560] 2.3896 [0.4956]

Y 2t 111.18 [0.0000] 82.979 [0.0000]

(4)

(8)

Y0t

Y1t

Fitted

Fitted

0.10

-0.8

0.05

-1.0

0.00

-1.2

-0.05 1970

1975

1980

1985

1990

1995

1970

1975

^ Y1t −E(Y 1t) (scaled)

3

1980

1985

1990

1995

Fitted

Y2t

0.1

2 1

0.0

0 -1 1970

-0.1 1975

1980

1985

1990

1995

1970

1975

1980

1985

1990

1995

Fig. 2. Fit of the equilibrium dynamics Y t in mixed form, where the coefficients of Y 0;t
1 ; Y 1;t
1 and Db0 X t
1 are constrained to 0 in the equation for Y 1t ; i.e. (25) holds with K ¼ e2 : The lower left panel b 1t Þ; which is an innovation process. reports Y 1t
EðY

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

163

Moreover the finding that Y 1t is an innovation process allows to better interpret the adjustment coefficients both in the equilibrium correction and in the equilibrium dynamics forms. In fact, the adjustment to Y 1;t
1 can be interpreted as reaction to the unpredictable autonomous I(1) component in inflation. Summarizing, the application to Australian prices shows the relevance of the equilibrium dynamics form in cofeature analysis. Only trivial cases of cofeatures could be obtained for the equilibrium correction form.

8. Conclusions In this paper we have discussed various applications of the notion of common features that can possibly arise in I(2) systems. For each case we have discussed how to address inference both for known and unknown cofeature vectors, using RRR. As in the I(1) case, the cointegration analysis needs to precede the analysis of common features. After fixing the cointegration parameters, all subsequent inference is LAN. The notions of cofeatures introduced in the paper have been found to have empirical relevance in the data set of Australian inflation analyzed in Banerjee et al. (2001) inter alia. For these data, the equilibrium dynamics form supports the presence of a single cofeature vector, while only trivial cases of cofeatures can be obtained for the equilibrium correction form.

Acknowledgements Partial financial support from Italian MIUR grants FAR and Cofin2002 is gratefully acknowledged. I wish to thank the participants of the conference ‘Common features in Rio’, Alain Hecq, Søren Johansen, the guest editors and two anonymous referees for useful comments on two earlier versions of this paper. The usual disclaimer applies.

Appendix A Proof of Proposition 1. Let W t ¼ C W ðLÞ
t : From the definition of cofeature rank, b0 W t ¼ b0 C W ;0
t (i.e. b0 C W ;i ¼ 0; iX1) and V :¼b0 C W ;0 OC 0W ;0 b is positive definite. Because b0 C W ;i ¼ 0; iX1; one has b0 C W ;0 ¼ b0 C W ð1Þ; and V ¼ b0 C W ð1ÞOC 0W ð1Þb: Because C W ;0 is assumed to be of full rank, ‘ ¼ rankðb0 C W ;0 Þ ¼ rankðb0 C W ð1ÞÞp rankðC W ð1ÞÞ: Alternatively, in order for V to be positive definite, ‘:¼rankðbÞ must be less or equal to rankðC W ð1ÞÞ ¼: q; where O is of full rank by assumption. & P 0 0 i Proof of Theorem 3. Let ut :¼md d t þ
t ; UðLÞ:¼I
k
2 i¼1 Ui L ; D:¼ðb þ b2 d : b1 : b2 Þ : By the proof of Johansen’s representation theorem, see Johansen (1992, 1996), the

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

164

process Y t follows an I(0) VAR(k) process, with stable AR polynomial A ðLÞ under the I(2) assumptions. These references also describe how to transform Y t back to the autoregressive form, and hence to the equilibrium correction form (5). Tedious calculations show that the AR matrices are given by ¯ z2 þ ðI þ U1 Þb¯ 1 :
ðb¯ þ z1 Þd þ U1 ðb¯ 2
bdÞÞ, ¯ A1 ¼ Dðz1 þ a þ ð2I þ U1 Þb: ¯ ¯ ðU2
U1 Þb¯ 1 : U2 ðb¯ 2
bdÞ ¯
U1 bdÞ, A ¼ Dð
z1
ðI
U2 þ 2U1 Þb: 2

¯ ¯ ðUi
Ui
1 Þb¯ 1 : Ui ðb¯ 2
bdÞ ¯
Ui
1 bdÞ, Ai ¼ DððUi
2Ui
1 þ Ui
2 Þb: i ¼ 3; . . . ; k
2; ¯ ¯
Uk
2 b¯ 1 :
Uk
2 bdÞ, Ak
1 ¼ Dðð
2Uk
2 þ Uk
3 Þb: ¯ 0: 0Þ. A ¼ DðUk
2 b: k

These expressions imply restrictions (6). In order to impose them, observe that Ak
1;0 Y 0;t
kþ1 þ Ak
1;2 Y 2;t
kþ1 þ Ak;0 Y 0;t
k equals ¯ 2;t
kþ1 þ DUk
2 bY ¯ 0;t
k ¯ 0;t
kþ1
DUk
2 bdY Dð
2Uk
2 þ Uk
3 ÞbY ¯ 0;t
kþ1
Y 0;t
k þ dY 2;t
kþ1 Þ þ Dð
Uk
2 þ Uk
3 ÞbY ¯ 0;t
kþ1 ¼
DUk
2 bðY 0 ¯ ¯ 0;t
kþ1 , ¼
DUk
2 bðDb X t
kþ1 þ b00 Þ þ Dð
Uk
2 þ Uk
3 ÞbY

so that one can simply substitute Y t
k ; Y 2;t
kþ1 with Db0 X t
kþ1 ; changing the coefficients of the constant and of Y 0;t
kþ1 : We finally show how the mixed form can be obtained. Let unt :¼gV t þ md d t þ
t : Insert I ¼ D
1 D before D2 X t in the l.h.s. of (4); one finds 0 0 1 0 1 Y 0;t
1 b DX t þ db02 D2 X t B C B 0 C ¯ b¯ 1 D: b¯ 2
bdDÞ ¯ b01 DX t ðbD: @ A ¼ ða: z1 : z2 Þ@ b DX t
1 A þ m0 þ unt . b02 D2 X t

b01 DX t
1

Adding b0 to the top block of variables on the l.h.s. and rearranging 0 1 0 1 Y 0;t
1 Y 0t 0 C C ¯ b¯ 1 : b¯ 2
bdÞ ¯ z1 þ b: ¯ B ¯ z2 þ b¯ 1 ÞB ðb: @ Y 1t A ¼ ða þ b: @ b DX t
1 A þ m0 þ un t

b01 DX t
1

Y 2t

Pre-multiplication by D:¼ðb þ b2 d0 : b1 : b2 Þ0 gives 0 1 Y 0;t
1 0 C ¯ z1 þ b: ¯ z2 þ b¯ 1 ÞB Y t ¼ Dða þ b: @ b DX t
1 A þ Dm0 þ Dunt , b01 DX t
1

which gives (8). The proof that C ¼ Cz A; for A square and non-singular is similar to the proof of Proposition 10 below and it is therefore omitted. &

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

165

Proof of Corollary 4. Given that colðb: b1 Þ ¼ colðb: t1 Þ; one has t1 ¼ bf 0 þ b1 h0 ; 0 0 where f 0 :¼b¯ t1 ; h0 ¼ b¯ 1 t1 and h must be invertible. Let a:¼h
1 f and define 0 1 0 1 I p0 I p0 B C B C DðLÞ:¼@ aD I p1
ad A; D
1 ðLÞ ¼ @
aD I p1 ad A. I p2 I p2 Let H:¼diagðI p0 ; h; I p2 Þ and note that G t ¼ HDðLÞY t : Let also vt :¼my Dt þ
t ; inserting D
1 ðLÞH
1 HDðLÞ between A ðLÞ and Y t in A ðLÞY t ¼ vt ; one obtains B ðLÞGt ¼ ut where B ðLÞ:¼ KA ðLÞD
1 ðLÞH
1 ; ut :¼Kvt ; K:¼HDð0Þ: The AR polynomial B ðLÞ:¼ KA ðLÞD
1 ðLÞH
1 is of the same degree k as A ðLÞ and has the same roots, because DðLÞ is a unimodular matrix. The constraints on the coefficients of B ðLÞ are obtained from (6) substituting A ðLÞ into the definition of B ðLÞ: & Proof of Theorem 6. Let mnt :¼mðLÞD2 d t : Taking second differences in (2) one finds that D2 X t
mnt equals C 2
t þ C 1 D
t þ C 0 ðLÞD2
t , ¼ C 2
t þ C 1 ð
t

t
1 Þ þ

1 X

C 0;i Li
t
2

i¼0

1 X

C 0;i Liþ1
t þ

i¼0

1 X

C 0;i Liþ2
t

i¼0

¼ ðC 2 þ C 1 þ C 0;0 Þ
t þ ð
C 1 þ C 0;1
2C 0;0 Þ
t
1 1 X þ ðC 0;i
2C 0;i
1 þ C 0;i
2 Þ
t
i i¼2

¼:
t þ C n1
t
1 þ

1 X

C ni
t
i ¼: C n ðLÞ
t ,

ð30Þ

i¼2

where in the last line we have used the normalization of the process C n ð0Þ ¼ I; i.e. C 2 þ C 1 þ C 0;0 ¼ I.

(31) 0

2

n

0

There exist a cofeature matrix b such that b ðD X t
mt Þ ¼ b
t if and only if all the coefficient matrices to the lagged
t in (30) cancel when pre-multiplied by b0 ; i.e. iff b0 C ni ¼ 0; i ¼ 1; 2; . . . : Let ai :¼b0 C 0;i : The condition b0 C ni ¼ 0; for iX2 is aj
2ajþ1 þ ajþ2 ¼ 0;

j ¼ 0; 1; . . . .

This is a difference equation with solution aj ¼ a0 þ jða1
a0 Þ: From the summability of C 0 ðzÞ for jzjo1 þ K and K40; it follows that a0 ¼ a1
a0 ¼ 0; i.e. b0 C 0;i ¼ 0 for all iX0; which is condition (11). The condition b0 C n1 ¼ 0 gives b0 ð
C 1 þ C 0;1
2C 0;0 Þ ¼ 0; where b0 C 0;1 ¼ b0 C 0;0 ¼ 0 by (11). Hence one finds b0 C 1 ¼ 0; condition (12). From (31) one has C 1 ¼ I
C 2
C 0;0 ; so that b0 C 1 ¼ b0 ðI
C 2
C 0;0 Þ ¼ b0 ðI
C 2 Þ; where the last equality follows from (11). This proves the equivalence between (12) and (13). Assume (13) holds, b0 C 2 ¼ b0 : From the definition of C 2 ; see (2), it follows that b 2 colða2 Þ; i.e. that b ¼ a2 u for some u. Substituting into b0 C 2 ¼ b0 one finds u0 ða02 b2 ða02 yb2 Þ
1
I p2 Þa02 ¼ 0; which holds iff u0 ða02 b2
a02 yb2 Þ ¼ 0; i.e. if c belongs to A:¼col? ða02 ðI
yÞb2 Þ: In order for A not to contain only the zero

ARTICLE IN PRESS 166

P. Paruolo / Journal of Econometrics 132 (2006) 143–168

vector, a02 ðI
yÞb2 must be of deficient rank, i.e. a02 ðI
yÞb2 ¼ cd 0 for some full column rank p2  p2
‘ matrices c and d. Hence u ¼ c? : The converse statement is direct. This completes the proof. & Proof of Proposition 5. If CðÞ ¼ jt0 then j0? ðW t
EðW t ÞÞ is an innovation process. Conversely assume W t has cofeature matrix b, i.e. b0 ðW t
EðW t ÞÞ is an innovation process. From (4) and (5) one finds that b0 ðW t
EðW t ÞÞ contains b0 CU t or b0 CU t in addition to an innovation process. Hence b0 C ¼ 0; i.e. b 2 col? ðCÞ: In order for b to be different from the zero vector one must have rankðCÞ ¼ p
‘; i.e. C ¼ jt0 : This completes the proof. & Proof of Proposition 9. Let W t ¼ C W ðLÞ
t : Because b is a cofeature matrix, one has ðb01 : b02 ÞðW 0t : R0t Þ0 ¼ b01 C W ;0
t ; given that Rt does not depend on
t : Moreover V ¼ Varðb0 Z t Þ ¼ b01 C W ;0 OC 0W ;0 b1 is of full rank ‘: This holds only if b1 has full column rank ‘: This completes the proof. & Proof of Proposition 10. We wish to show U 2t can be obtained linearly from U 1t and vice versa. To this end simply observe that 1 0 01 0 0 1 0 b þ db02 I p0 I p0 Y 0t b0 0 1 B b0 DX C B C B b0 I p1 C D2 X t C B 1 t C 1 C B C B C B 0 2 B C B C 0 CB B b2 D X t C B Y 0;t
1 C C B b2 C B CB B C B B C, ¼ þ C CB 0 B C B I p0 C@ b DX t
1 C B Y 0;t
1 C B B C B A C C B 0 C B C b0 DX t
1 B b DX t
1 C B B I A @ p 0 1 A A @ @ b01 DX t
1 I p1 where we have omitted zeros for readability. Conversely 1 0 1 0¯ ¯ 0 ¯ ¯ 0 b b1 b¯ 2
bd
b¯ D2 X t
bb 0 C B C B B C B B Y 0;t
1 C B I p0 C B C B B C¼B CþB B 0 C B B b DX t
1 C B A @ A @ @ 0 b1 DX t
1 1 0 Y 0t C B 0 B b1 DX t C C B C B 0 2 B b2 D X t C C B B C. B Y 0;t
1 C C B C B 0 B b DX t
1 C A @


b¯


b¯ 1

I p0 I p1

1 C C C C C A

b01 DX t
1

This completes the proof.

&

Proof of Proposition 11. Sufficiency is proved by substituting B ¼ jt0 in (16) and premultiplication by j0? : In order to prove necessity, assume b:¼ðb01 : b02 Þ0 is the cofeature

ARTICLE IN PRESS P. Paruolo / Journal of Econometrics 132 (2006) 143–168

167

matrix with ‘40 columns, and b01 Z0t þ b02 Z 2t ¼ b01 ut be the cofeature relations. Premultiplication of (16) by b01 gives b01 Z0t ¼ b01 BZ 1t þ b01 FZ 2t þ b01 ut ; which, substituted back implies
b01 BZ 1t þ ðb02
b01 FÞZ 2t ¼ 0. In order for this to be identically zero, both coefficients of Z 1t and Z 2t must be zero. This shows that b1 2 col? ðBÞ and that b02 ¼ b01 F: Since ‘40 was assumed, B must be of deficient rank, B ¼ jt0 ; and b1 ¼ j? : &

References Anderson, T.W., 1951. Estimating linear restrictions on regression coefficients for multivariate normal distributions. Annals of Mathematical Statistics 22, 327–351 (correction, Annals of Statistics 8, 1980, 1400). Banerjee, A., Cockeress, L., Russell, B., 2001. An I(2) analysis of inflation and the markup. Journal of Applied Econometrics 16, 221–240. Boswijk, P., 2000. Mixed normality and ancillarity in I(2) systems. Econometric Theory 16, 878–904. Campbell, J.Y., Shiller, R.J., 1987. Cointegration and tests of present value models. Journal of Political Economy 95, 1062–1087. Cubadda, G., Hecq, A., 2001. On non-contemporaneous short-run co-movements. Economics Letters 73, 389–397. Engle, R.F., Granger, C.W.J., 1987. Co-integration and error correction: representation, estimation and testing. Econometrica 55, 251–276. Engle, R.F., Kozicki, S., 1993. Testing for common features. Journal of Business and Economic Statistics 11, 369–380. Gonzalo, J., Granger, C.W.J., 1995. Estimation of common long-memory components in cointegrated systems. Journal of Business and Economic Statistics 13, 27–36. Haldrup, N., 1998. An econometric analysis of I(2) variables. Journal of Economic Surveys 12, 595–650. Hecq, A., Palm, F.C., Urbain, J.P., 2005. Common cyclical features analysis in VAR models with cointegration. Journal of Econometrics, this issue. doi: 10.1016/j.jeconom.2005.01.025 Johansen, S., 1992. A representation of vector autoregressive processes integrated of order 2. Econometric Theory 8, 188–202. Johansen, S., 1995a. A statistical analysis of cointegration for I(2) variables. Econometric Theory 11, 25–59. Johansen, S., 1995b. Identifying restrictions of linear equations with application to simultaneous equations and cointegration. Journal of Econometrics 69, 111–132. Johansen, S., 1996. Likelihood–based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press, Oxford (Second corrected printing). Johansen, S., 1997. A likelihood analysis of the I(2) model. Scandinavian Journal of Statistics 24, 433–462. Kugler, P., Neusser, K., 1993. International interest rates equalization: a multivariate time series approach. Journal of Applied Econometrics 8, 163–174. Omtzigt, P., Paruolo, P., 2005. Impact factors. Journal of Econometrics, in press. Paruolo, P., 1997. Asymptotic inference on the moving average impact matrix in cointegrated I(1) VAR systems. Econometric Theory 13, 79–118. Paruolo, P., 2000. Asymptotic efficiency of the two stage estimator in I(2) systems. Econometric Theory 16, 524–550. Paruolo, P., 2001. The power of lambda max. Oxford Bulletin of Economics and Statistics 63, 395–403. Paruolo, P., 2003. Common dynamics in I(1) systems. WP 2003/35. Available at http://eco.uninsubria.it/ dipeco/Quaderni

ARTICLE IN PRESS 168

P. Paruolo / Journal of Econometrics 132 (2006) 143–168

Paruolo, P., Rahbek, A., 1999. Weak exogeneity in I(2) VAR systems. Journal of Econometrics 93, 281–308. Rahbek, A., Kongsted, H.C., Jørgensen, C., 1999. Trend-stationarity in the I(2) cointegration model. Journal of Econometrics 90, 265–289. Stock, J., Watson, M., 1993. A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 61, 783–820. Vahid, F., Engle, R.F., 1993. Common trends and common cycles. Journal of Applied Econometrics 8, 341–360. Vahid, F., Engle, R.F., 1997. Codependent cycles. Journal of Econometrics 80, 199–221.