Gianluca Cubadda

Alain Hecq

Franz C Palm
email: f.palm@ke.unimaas.nl.

Studying co-movements in large multivariate data prior to multivariate modelling

Keywords: Interactions, multiple time series, co-movements, ARIMA, cointegration, common cycles, dynamic panel data. JEL: C32

For non-stationary vector autoregressive models (VAR hereafter, or VAR with moving average, VARMA hereafter), we show that the presence of common cyclical features or cointegration leads to a reduction of the order of the implied univariate autoregressive-moving average (ARIMA hereafter) models. This …nding can explain why we identify parsimonious univariate ARIMA models in applied research although VAR models of typical order and dimension used in macroeconometrics imply nonparsimonious univariate ARIMA representations.

Next, we develop a strategy for studying interactions between variables prior to possibly modelling them in a multivariate setting. Indeed, the similarity of the autoregressive roots will be informative about the presence of co-movements in a set of multiple time series. Our results justify both the use of a panel setup with homogeneous autoregression and heterogeneous cross-correlated vector moving average errors and a factor structure, and the use of cross-sectional aggregates of ARIMA series to estimate the homogeneous autoregression.

Introduction

The analysis of interactions between individuals, groups of agents, organizations, regions or countries is a key objective of scienti…c inquiry. Huge collections of data (national accounts for virtually all countries, unemployment data for hundreds of types of jobs, prices for thousands of products, assets and services, purchases by individual consumers, environmental data on CO2 emissions and climate change, on health care, education,. . . ) are almost instantaneously available at relatively low cost. The IMF and the World Bank publish statistical information for 150 countries. In the European Union, starting for the six founding countries, Eurostat now collects and analyzes hundreds of variables for the 27 member countries. The availability of sets of data for a large number of individual entities and the importance of understanding individual behavior and interaction between individuals have created a demand for methods to analyze these types of phenomena using large sets of data.

To capture interactions between a large set of time series requires imposing structure on the models used in the analysis. Examples of structure often imposed are the dynamic factor models put forward by [START_REF] Forni | The Generalized Dynamic Factor Model: Identi…cation and Estimation[END_REF] and by [START_REF] Stock | Macroeconomic Forecasting Using Di¤usion Indexes[END_REF]. Another example is the approach of using the common factor structure adopted in the recent literature on dynamic macro-panels to account for cross-sectional correlation (e.g. [START_REF] Bai | A PANIC Attack on Unit Roots and Cointegration[END_REF]. In these approaches, the common factors usually account for the impact of changes in the environment that is largely exogenous and in which individual entities have to take decisions and to interact. Individual speci…cities are often accounted for by including individual e¤ects.

Alternatively, partial systems are built for country-speci…c analyses or for a speci…c variable such as for instance GDP for a subset of countries or regions. An often adopted framework to analyze a limited number of time series is the vector autoregressive model (VAR hereafter). A VAR, allowing all variables to be endogenous, can only be implemented for small systems, with typically …ve or six variables. Still, in such cases, the number of parameters is large. To deal with the dimensionality problem in VARs, di¤erent approaches have been adopted such as inter alia Bayesian analysis, simulation-based techniques, separability assumptions (as for instance used in panel data models when they are accompanied by the homogeneity assumption of the autoregressive parameters across units), automatic selection by deleting non-signi…cant coe¢ cients and reduced-rank regressions.

In this paper we take a di¤erent route and make two new contributions to the literature. Instead of designing a system which captures interactions between entities, we …rst address the question what the implications are of the presence of common factors in a possibly large VAR system on the marginal processes for individual entities. The common factor structure could arise from the presence of common cyclical or other features (see e.g. [START_REF] Engle | Testing for Common Features (with comments)[END_REF]. For non-stationary series, a common factor structure could result from the presence of cointegration, that is from the occurrence of common stochastic trends.

The …rst main contribution of the paper is that the presence of common factors or common features in a VAR leads to a reduction of the order of the implied univariate autoregressive-moving average (ARIMA hereafter) schemes for the individual series. In a way, individual series keep a print of the system as a whole. Second, we propose a strategy, that allows to identify the print of common features from individual data, that is to study aspects of interactions between individual variables without or prior to modelling these interactions

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in a multivariate framework.

Before implementing this strategy, we derive the implications of the presence of a common factor structure for the univariate processes implied by a VAR, both in absence of and under exogeneity restrictions. More precisely, in Section 2, we extend results by [START_REF] Zellner | Time Series Analysis and Simultaneaous Equation Econometric Models[END_REF] by showing that VARs with reduced rank restrictions coming from short-or long-run co-movements (see inter alia Vahid andEngle, 1993, and[START_REF] Ahn | Inference of Vector Autoregressive Models with Cointegration and Scalar Components[END_REF]) o¤er an alternative explanation to the well known paradox in time series: on the one hand, small multivariate systems imply non-parsimonious individual ARIMA processes whereas on the other hand empirically selected and estimated univariate models are generally of low order. Section 3 proposes and evaluates an estimation strategy based upon the common univariate autoregressive parameters of series generated by the same VAR system. Indeed, a homogeneous panel framework with common autoregressive parameters is the by-product of a large VAR. We propose to use cross-sectional aggregates to estimate these common autoregressive parameters, and we evaluate this estimation strategy in a small Monte Carlo experiment. Section 4 focuses on the link between the growth rates of GDP among Latin American economies. The occurrence of common autoregressive parts for the individual series is used as an indication that these series exhibit the same common features and therefore are subject to the same co-movements. This indication is then formally tested and found to be supported by the information in the data. A …nal section concludes and provides suggestions for further research.

Multivariate and implied univariate time series models

The aim of this section is to present new results on the relations between univariate and multivariate time series models. In particular, Subsection 2.1 brie ‡y reviews the paradox, namely that VARs of typical order and dimension used in macroeconometrics imply non-parsimonious univariate ARMA representations, which are rarely observed in empirical applications. Subsection 2.2 shows how VARs with certain reduced rank structures can solve this paradox, and Subsection 2.3 specializes the results to the case of block-diagonal and block-triangular VARs. Moreover, Subsection 2.4 explores the implications of short-run co-movements for the vector moving average (VMA hereafter) component of the so-called …nal equation representation of a VAR model [START_REF] Zellner | Time Series Analysis and Simultaneaous Equation Econometric Models[END_REF]. The …ndings may explain why MA orders of univariate models are often found to be smaller than those theoretically implied by multivariate models. Finally, Subsection 2.5 illustrates concepts and methods with an empirical analysis of the relationship between quarterly growth rates of the industrial production indexes in the US and Canada.

The paradox

The literature on the link between multivariate time series and the behavior of univariate variables stresses the fact that univariate ARIMA analyses provide tools for the diagnostic testing of VAR (or VARMA) models. Important contributions in this area are the monograph by [START_REF] Quenouille | The Analysis of Multiple Time-Series[END_REF] and papers by [START_REF] Zellner | Time Series Analysis and Simultaneaous Equation Econometric Models[END_REF][START_REF] Zellner | Time Series and Structural analysis of Monetary Models of the US Economy[END_REF], 2004), [START_REF] Palm | Large-Sample Estimation and Testing Procedures for Dynamic Equation Systems[END_REF], [START_REF] Palm | On Univariate Time Series Methods and Simultaneous Equation Econometric Models[END_REF] or [START_REF] Maravall | Encompassing Univariate Models in Multrivariate Time Series: a Case Study[END_REF]. Some textbooks devote a few pages to that issue (e.g. Franses (1998, 198-199) or Lütkepohl (2005, 494-495)).

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More formally, let us …rst consider the stationary1 multivariate VARMA process for an n vector of time series z t = (z 1t ; : : : ; z nt ) 0 , and without deterministic terms for simplicity:

(L)z t = (L)" t ; t = 1; : : : ; T;

where (L) and (L) are …nite order polynomial coe¢ cient matrices with the usual lag operator L such that

Lz t =
(L)z t = " t : (2)
Following [START_REF] Zellner | Time Series Analysis and Simultaneaous Equation Econometric Models[END_REF], the univariate representation of elements of z t can be obtained by premultiplying both sides of ( 2) by (L) adj ; the adjoint matrix (or the adjugate) associated with (L); in order to obtain the "…nal equations" (FEs henceforth):

det[ (L)]z t = (L) adj " t ; (3)
where the determinant det[ (L)] is a scalar …nite order polynomial in L: This means that each series is a …nite order ARMA(p ; q ), with the same lag structure and the same coe¢ cients for the autoregressive part for every series, although the multivariate system is a …nite order VAR(p). For instance for the ith element of z t we have

det[ (L)]z it = i (L) adj " t = i (L)u it ;
i (L) adj denoting the ith row of the matrix (L) adj ; i (L) is a scalar polynomial in L and u it is a scalar innovation with respect to the past of z it : This recognition and the compatibility of these p and q with n and p of the VAR(p) is the …rst step of an approach that has been developed in Zellner andPalm (1974, 1975) as a general modelling strategy called SEMTSA (Structural Economic Modelling Time Series Analysis). In the …rst stage of this approach, information from univariate schemes is used to restrict the dynamics of the structural model. However, given the conditions associated with that …rst diagnostic checking, one often faces a paradoxical situation: a small order VAR with few series already generates univariate ARMA with large p and q , an implication that is rejected when tested on economic data where one usually …nds quite parsimonious models with low order autoregressive and moving average polynomials.

Indeed, an n dimensional VAR(p) would imply at most individual ARMA(np; (n 1)p) processes. This well known result is simply due to the fact that det[ (L)] contains by construction up to L np terms and the adjoint matrix is a collection (n 1) (n 1) cofactor matrices, each of the matrix elements can contain the terms 1; L; ::L p : But whatever the simplicity of that result, it leads to implausible univariate models for

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most economic series. 2 This is illustrated in the following numerical example. the determinant of which is 0:48L 4 0:189 L3 + 0:571L 2 0:66L + 1 and the adjoint matrix such that (L) adj = " 1 0:76L + 0:8L 2 0:21L + 0:8L 2 0:7L + 0:3L setting. The next subsection provides an alternative explanation for solving that paradox thanks to the use of short-run co-movements and reduced rank models.

Solving the paradox under common features

We introduce a new framework aimed at understanding the gap between implied individual ARMA processes for VARs and the estimated univariate models that we get in empirical studies. More speci…cally, we investigate the implications of the presence of common features such as common cyclical features and common trends in a multivariate dynamic model for the order of the marginal processes of the individual series.

The strongest form of common cyclical features is the notion of serial correlation common feature (henceforth, SCCF) proposed by [START_REF] Engle | Testing for Common Features (with comments)[END_REF] and [START_REF] Vahid | Common Trends and Common Cycles[END_REF]. 3 In this framework, series [START_REF] Cubadda | On Non-Contemporaneous Short-Run Comovements[END_REF]. In particular, these latter authors assume that there exists an n s polynomial matrix such that 0 (L)z t ( 0 + 1 L) 0 z t = 0 0 " t . This model is a polynomial SCCF model of order one, which is denoted PSCCF(1). 4 The notion of PSCCF can be generalized to account for adjustment delays of m periods instead of one lag as in the PSCCF [START_REF] Ahn | Inference of Vector Autoregressive Models with Cointegration and Scalar Components[END_REF]. In this case the polynomial matrix (L) is of order m, where m p, a model denoted PSCCF(m) (see [START_REF] Cubadda | On Non-Contemporaneous Short-Run Comovements[END_REF] for details).

z
For simplicity reasons, we start with the most parsimonious model, in which there exists a SCCF matrix.

We …rst illustrate the consequences of the presence of one SCCF relationship in a bivariate example with p = 2 before generalizing to the n dimensional case for any polynomial order p and other forms of non contemporaneous co-movements (e.g. PSCCF):

Example 2 Let us consider the polynomial matrix (L) of the Example 1 but now there exists a SCCF relationship with a cofeature vector 0 = (1 : 1) for

(L) = " 1 0 0 1 # " 0:1 0:2 0:1 0:2 # L " 0:2 0:6 0:2 0:6 # L 2 : (4)
Because the second and the third coe¢ cient matrices 1 and 2 are of reduced rank with the same left null space, it follows that

0 (L) = 0
. The determinant of (L) in ( 4) is 0:4L 2 0:1L + 1 and the adjoint matrix is

(L) adj = " 0:2L + 0:6L 2 + 1 0:2L + 0:6L 2 0:1L 0:2L 2 0:1L 0:2L 2 + 1 # :
This implies that the two series are at most ARMA(2,2) and not ARMA(4,2) as in Example 1.

More generally, Table 1 summarizes the reduction of the individual ARMA orders due to common feature restrictions. Basically, Table 1 documents that multivariate systems with additional commonality deliver more parsimonious, and therefore empirically more plausible, ARMA models than those implied by unrestricted VAR models. For instance, in a small VAR system with four variables and two lags, implied models are at most ARMA [START_REF] Engle | Testing for Common Features (with comments)[END_REF][START_REF] Cubadda | A Unifying Framework for Analyzing Common Cyclical Features in Cointegrated Time Series[END_REF] while they would reduce to ARMA(2,2) models in the presence of a unique common cycle, that is, with s = 3: This application to Quenouille's results, who already stressed the impact of rank de…ciency, is one of the diagnostic tools for data-admissibility that we want to formalize and emphasize.

Note also that additional zero coe¢ cient restrictions on the common factors and/or the loadings may reduce

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these implied maximal orders. This means that the ARMA(2,2) obtained under short-run co-movements can be more parsimonious with additional exclusion restrictions.

Let us formally prove the results that are reported in Table 1.

Q(L)x t = e t ;
where

x t = M z t , e t = M " t , Q(L) = M (L)M 1 , M 0 [ : ? ],
and ? is the orthogonal complement.

Given that x t is a non-singular linear transformation of z t , the maximum AR and MA orders of the univariate representation of elements of z t must be the same as those of elements of x t . Since M 1 = [ : ? ],

where = ( 0 ) 1 , and ? = ? ( 0 ? ? ) 1 , we have

Q(L) = " I s 0 s (n s) 0 ? (L) 0 ? (L) ? # ; from which it easily follows that det[Q(L)] = det[ 0 ? (L) ?
] is a polynomial of order (n s)p. Hence, the univariate AR order of each element of z t is, at most, (n s)p. To prove (ii), let P (L) denote a submatrix of Q(L) that is formed by deleting one of the …rst s rows and one of the …rst s columns of Q(L). We can partition P (L) as follows

P (L) = " P 11 P 12 P 21 (L) P 22 (L) # : (5)
Now, P 11 is a (s 1) (s 1) identity matrix, P 12 is a (s 1) (n s) matrix of zeros, P 21 (L) is a (n s) (s 1) polynomial matrix of order p, and P 22 (L) is a (n s) (n s) polynomial matrix of order p.

Hence, det[P (L)] = det[P 11 ] det[P 22 (L)], which tells us that det[P (L)] is of order (n s)p. Since cofactors associated with the blocks of Q(L) di¤ erent from P 11 are polynomials of degree not larger than (n s)p, we conclude that the univariate MA order of each element of z t is, at most, (n s)p.

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Proposition 4 Stationary VAR with s PSCCF [START_REF] Ahn | Inference of Vector Autoregressive Models with Cointegration and Scalar Components[END_REF]. In an n stationary VAR(p) with s PSCCF(1) the individual ARMA processes have: (i) AR orders not larger than (n s)p + s;(ii) MA orders not larger than (n s)p + (s 1).

Proof. Along the same lines of the previous proof, let us consider

R(L) = M 0 (L)M 1 0 , M 0 0 [ 0 : 0? ],
and

M 1 0 [ 0 : 0? ]. Then, we have R(L) = " 0 (L) 0 0 1 0? L 0 0? (L) 0 0 0? (L) 0? # : Hence, det[R(L)] = det[ 0 (L) 0 ] det[ 0 0? (L) 0? 0 0? (L) 0 ( 0 (L) 0 ) 1 0 1 0? L]: Writing ( 0 (L) 0 ) 1 = ( 0 (L) 0 ) adj = det[ 0 (L) 0
] and substituting it above we get

det[R(L)]det[ 0 (L) 0 ] n s 1 | {z } s(n s 1) = det 8 > < > : 0 0? (L) 0? | {z } p det[ 0 (L) 0 ] | {z } s 0 0? (L) 0 | {z } p ( 0 (L) 0 ) adj | {z } s 1 0 1 0? L | {z } 1 9 > = > ;
from which follows that the degree of the polynomial of det[R(L)] is, at most, equal to (n s)p+s. Regarding the order of the MA component, let us denote G(L) a submatrix of R(L) that is formed by deleting one of the …rst s rows and one of the …rst s columns of R(L). We can partition G(L) as follows

G(L) = " G 11 (L) G 12 L G 21 (L) G 22 (L) # ; where G 11 (L) is a (s 1) (s 1) polynomial matrix of order 1, G 12 is a (s 1) (n s) matrix, G 21 (L)
is a (n s) (s 1) polynomial matrix of order p, and G 22 (L) is a (n s) (n s) polynomial matrix of order p. Hence, following a similar reasoning as above, we conclude that the individual MA order is, at most, (n s)p + (s 1).

These results can be easily generalized for the PSCCF(m) case as reported in Table 1. Also note that we do not consider a mixed model that can have both SCCF and PSCCF relationships but results can be trivially deduced from results beneath.

The above propositions can be extended to the case of an I(1) VAR. Let us consider the vector error

correction model (VECM henceforth) of variables z t (L) z t = 0 z t 1 + " t ; (6)
where = (1 L), (L) = I n P p 1 i=1 i L i , i = P p j=i+1 j for i = 1; 2; :::; p 1, and are fullcolumn rank n r (r < n) matrices such that (1) = 0 and 0 ? (1) ? has full rank. The process z t is cointegrated of order (1,1), denoted by CI(1,1), the columns of span the cointegrating space, the elements of are the corresponding adjustment coe¢ cients, see e.g. [START_REF] Johansen | Likelihood-Based Inference in Cointegrated Vector Autoregressive Models[END_REF].

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A well known implication of cointegration is that (n r 1) unit roots are common to the individual AR and the MA polynomial. Let us formalize this result in order to later extend it to di¤erent forms of common cyclical features. To do so, consider again equation ( 2) and apply the transformation

B(L)y t = u t ;
where y t = P z t , u t = P " t , B(L) = P (L)P 1 , and P 0 [ : ? ]. Again, the maximum AR and MA orders of the univariate representation of elements of z t are the same as those of elements of y t . Since 0 z t is I(0) and 0 ? z t is I(1), we can partition B(L) as follows

B(L) = " B 11 (L) B 12 (L) B 21 (L) B 22 (L) # :
We have that

det[B(L)] = n r det[B 22 (L)] det[B 11 (L) B 12 (L)B 22 (L) 1 B 21 (L)] n r det[B 22 (L)] det[B (L)]:
Since B 22 (1) 6 = 0, and B (1) = B 11 (1) 6 = 0, there are (n r) unit roots in det[B(L)]. However, since 0 ? z t is I(1), there must be (n r 1) unit roots in the univariate MA polynomials of elements of ? z t . Hence, the AR and MA orders of elements of z t respectively are, at most, n(p 1) + r + 1 and (n 1)(p 1) + r.

For the VECM in [START_REF] Cubadda | A Unifying Framework for Analyzing Common Cyclical Features in Cointegrated Time Series[END_REF], the existence of s SCCF's implies that there exists an n s full-rank matrix such that 0 z t = 0 " t or equivalently 0 (L) = 0 :

As shown by [START_REF] Vahid | Common Trends and Common Cycles[END_REF], the presence of s SCCF's is equivalent to the existence of (n s) common and synchronous cycles 5 .

The existence of s PSCCF(1) relationships requires, instead, that there exists an n s full-rank matrix

0 such that 0 0 z t + 0 1 z t 1 = 0 0 " t where 1 = 0 1 0 or equivalently (L) 0 (L) = 0 0 :
The presence of the PSCCF(1) has an interesting implication for the cycles of series z t . Indeed, Cubadda and Hecq (2001) proved that the same PSCCF(1) relationships cancel the dependence from the past of both the …rst di¤erences and cycles of series z t .

Finally, the existence of s weak form SCCF (WF henceforth) implies that there exists an n s full-rank matrix ~ such that ~ 0 z t ~ 0 0 z t 1 = ~ 0 " t [START_REF] Hecq | Common Cyclical Features Analysis in VAR Models with Cointegration[END_REF] or equivalently

~ (L) 0 (L) = ~ 0 ; A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT where ~ (L) = ~ + ~ 1 L; ~ 1 = ( 0 + I n ) ~ .
Interestingly, the same WF relationships cancel the dependence from the past of both the levels and cycles of series z t [START_REF] Cubadda | A Unifying Framework for Analyzing Common Cyclical Features in Cointegrated Time Series[END_REF]. Hence, we can interpret the WF as an analogous property to the PSCCF(1) that applies to the levels rather than to the di¤erences of series z t . Proof. By construction of M , and keeping in mind that 0 (L) = 0 , we notice that

det[Q(L)] = det " I s 0 s (n s) 0 ? (L) 0 ? (L) ? # = s det[ 0 ? (L) ? ]
is a polynomial of order (n s)p + s. Since det[Q(L)] is a polynomial of the same order as det[ (L)],

we conclude that the univariate AR order of each element of z t is, at most, (n s)p + s. Note that this result means that the AR part is of order (ii) MA orders not larger than (n s)(p 1) + r + s 1.

Proof. Since it is similar to that of Proposition 4, it is not reported to save space.

#

Hence, the proof is entirely analogous to that of Proposition 4 with the H(L) matrix in place of R(L).

Notice that the results for the VAR in the presence of common features can be extended in a fairly trivial way for the VARMA model. In the presence of common features in a VARMA(p; q) model, the orders reported in Tables 1 and2 hold true as well for the AR part whereas the order of the MA parts has to be augmented by q (at most). If common feature restrictions a¤ect the VAR part only, the MA part of the univariate processes is expected to be increased by q. The proofs are similar to those reported for the VAR(p) model except for the presence of the q extra lags in the MA parts. If the common features restrictions a¤ect the MA part of the VARMA model, the increase of the MA part of the implied univariate models might be smaller than q.

Block-diagonal and block-triangular VARs and the separation hypothesis

From the results of the previous subsection, there naturally arises the question about block-diagonal and block-triangular systems, namely about a situation in which we can disentangle groups of variables having many within co-movements but being roughly independent of other sets of variables. Indeed, intuition suggests that additional restrictions due to block-diagonality or block-triangularity should induce further simpli…cations of the individual ARMA structures. A block-triangular structure is required for the absence of Granger-causation between subvectors of z t accordingly partitioned and for strong exogeneity.

For the sake of simplicity, we consider two groups of variables and SCCF restrictions only. The …rst model under analysis is a block-diagonal stationary VAR(p) with SCCF within each block of variables n 1 and n 2 respectively; with n 1 + n 2 = n such that

(L) = " 11 (L) 0 n1 n2 0 n2 n1 22 (L) # ;
where there exist two full column rank matrices with ranks s 1 and s 2 respectively, with s 1 + s 2 = s such that 0 1 11 (L) = 0 1 and 0 2 22 (L) = 0 2 : This is a system with separability in common features [START_REF] Hecq | Separation, weak exogeneity and P-T decomposition in cointegrated VAR systems with common features[END_REF] with a block-diagonal co-feature matrix such that

= " 1 0 n1 s2 0 n2 s1 2 # :
In view of Table 1, the maximum orders for the implied univariate ARMA processes are (n 1 + n 2 s 1 s 2 )p for both the AR and MA components, assuming the same polynomial order p for each group: With di¤erent polynomial orders p 1 and p 2 ; these orders become (n 1 s 1 )p 1 + (n 2 s 2 )p 2 respectively: However, it is clear that these orders can be further reduced due to the presence of blocks of zeros. Because this leads to

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the presence of common roots between the AR and MA parts, the maximum orders of both AR and MA components are (n 1 s 1 )p 1 for the …rst block and (n 2 s 2 )p 2 for the second group of series.

Due to separation in the form of block-diagonality of (L); the implied univariate ARMA-processes need not to have identical AR polynomials. In fact, the model allows for inter-block AR parameter heterogeneity and intra-block AR parameter homogeneity. Note that this form of separation is often used when studying data for a set of countries using a panel-data framework. In some panel-data studies, a single variable is analyzed for a set of countries which can be clustered in groups with intergroup block-diagonality of (L) combined with intragroup homogeneity so that the implied univariate ARMA models for a given group have identical AR polynomials.

The second model is a block-triangular VAR such that

(L) = " 11 (L) 12 (L) 0 n2 n1 22 (L) # ; with 0 1 [ 11 (L) : 12 (L)] = 0 1 and 0 2 22 (L) = 0 2 :
The situation is similar as before because ? is still block-diagonal. However fewer cancellations of common roots are observed and the implied models are (n 1 s 1 )p 1 + (n 2 s 2 )p 2 for block 1 and (n 2 s 2 )p 2 in the block 2.

In this case, the AR polynomials of the two blocks could have factors in common which are identical to the AR polynomial of the second block. Notice also that under the additional assumption of appropriate block-diagonality of the contemporaneous covariance matrix ; the n 2 1 subvector x t of z t = (y 0 t ; x 0 t ) 0 is strongly exogenous. Multiplying the subsystem for y t by the adjoint matrix of 11 (L); one obtains the set of transfer functions (TF henceforth) for y t ; that is a set of dynamic equations with the same scalar AR polynomial, and vector moving averages in x t and in " 1t ; with " it being the subvector of " t corresponding to the partitioning of z t as (y 0 t ; x 0 t ) 0 : These TF equations are conditional models which can be used to study the dynamics of y t or each of its components given x t and its past and the past of y t .

Obviously, similar results can be obtained for the general case of block-diagonal or block-triangular systems with k blocks.

Interpretation in terms of a VMA index model

This subsection further investigates the consequences of the presence of short-run co-movements for the VMA part of FEs. We consequently look at the adjoint matrix of (L) and emphasize that the VMA representation follows a sort of multivariate index model (see [START_REF] Reinsel | Some Results on Multivariate Autioregressive Index Models[END_REF]. In order to introduce the problem, let us look more closely at the adjoint matrix (L) adj ; we have computed in Example 2 for s = 1 SCCF relationship; i.e. We can also write the previous expression in terms of coe¢ cient matrices and we observe the presence of a common right factor, namely of common right null spaces such that

(L) adj " t = " " 1t " 2t # + " : 2
0:1 # h 1 1 i " " 1t 1 " 2t 1 # + " 0:6 0:2 # h 1 1 i " " 1t 2 " 2t 2 #
with the obvious numerical observation that (L) adj ? = ? , and 0 ? = (1 : 1). We now show that the VMA component of (3) has a structure that is analogous to the multivariate index model of [START_REF] Reinsel | Some Results on Multivariate Autioregressive Index Models[END_REF], and this leads to the following general proposition: Proposition 8 In a stationary VAR(p), the existence of s SCCF vectors implies that in the FE representation the VMA coe¢ cient matrices associated with degrees strictly larger than (n s 1)p have a common right null space that is spanned by ? : Hence, post-multiplying the adjoint matrix (L) adj by ? reduces the order of the VMA component to a degree of at most (n s 1)p instead of (n s)p.

Proof. We know from Proposition 3 that det[ (L)] and (L) adj are both polynomials of order (n s)p.

Moreover, under SCCF we have that

(L) = I n p X j=1 ? 0 j L j ;
where j = ? from which it follows that (L) adj ? is a polynomial of order (n s 1)p.

Corollary 9

In the particular case with n 1 = s, the VMA component of the FEs follows an index model as in [START_REF] Reinsel | Some Results on Multivariate Autioregressive Index Models[END_REF].

The last result is more a mathematical curiosity than a device to be used in an empirical analysis. We might think to use it for fully e¢ cient estimation of the FEs for instance. But this shows that there exists a factor structure in the VMA component. In empirical investigations, this result can partially explain why the MA order is often found to be smaller than what it should be according to the theoretical implied models in Tables 1 and2. Consider, for instance, the case with w t = " t + H 1 " t 1 , where w t = (L) adj " t . The …rst autocovariance of w t = det[ (L)]z t is then given by E(w t 1 w 0 t ) = H 1 and, since ? spans the right null space of H 1 , we have E(w t 1 w 0 t ) = 0 when = ? 0 ? , where is a symmetric semi-positive de…nite matrix. Of course, under the condition = ? 0 ? the VAR innovations would be perfectly collinear and this cannot occur in practice. However, it might well be that ? 0

? , and we label this case as a near coincidental situation. . 06 1965 1970 1975 1980 1985 1990 1995 2000 Canadian growth American growth 2.5 An example: Industrial output growth in Canada and the US [START_REF] Engle | Testing for Common Features (with comments)[END_REF] …nd that there exists a SCCF relationship between the Canadian and the US quarterly growth rates of output (seasonally adjusted series). They have considered a sample from the late 1950s to the late 1990s. We select the same countries and we use the seasonally adjusted industrial production indexes from OECD main indicators over the period 1960:Q1-2004:Q3, namely we have 175 observations. Figure 1 plots these series. The model selection criteria LR, AIC, HQ and SBC lead to selecting p = 2 for the log-levels of the bivariate processes. We reject the presence of cointegration at usual signi…cance levels using Johansen's trace test. Consequently the analysis is performed in …rst di¤erences, namely with quarterly growth rates, with one lag only. In order to check whether the coe¢ cient matrix is of reduced rank we compute a SCCF test statistics using a canonical correlations approach (e.g. [START_REF] Vahid | Common Trends and Common Cycles[END_REF] between z t = ( ln U SA t : ln CA t ) 0 and z t 1 : The results are as follows: p value = 0:31 is the p-value associated with the null hypothesis

The estimation by OLS of the VAR(1) in …rst di¤erences delivers (standard errors in brackets)

( 2 (1)
) that a linear combination of z t is orthogonal to the past of z t . Information criteria also lead to where for both equations, the null of no disturbance autocorrelation is not rejected using LM tests.

One part of our story is con…rmed, i.e., the implied individual ARMA models have smaller AR orders than those that we should get in absence of common feature restrictions and the two series exhibit one signi…cant SCCF relationship. Moreover, the AR roots are similar for both countries. What might look as evidence against our prediction is the absence of a MA component because both growth rates should be ARMA(1,1). 6 However, since ^ is close to (1 : 1) 0 , the variances of VAR residuals are similar, and the correlation of VAR residuals is around 0.65, the results presented in Subsection 2.4 may explain why the MA(1) components are almost negligible.

Estimation procedures for implied univariate models and modelling considerations

As mentioned above, the …nding that a set of series have identical autoregressive polynomials is an indication that these series have been generated by a VAR with non-(block)-diagonal and non-(block)-triangular matrix (L) and that beyond having this common univariate autoregressive polynomial, they share other common features. 7 This indication for the existence of common features calls for testing whether the moving average part of the set of variables z it with identical autoregressive polynomials exhibits the multivariate index structure implied by the presence of common features (see Subsection 2.4). A test of the multivariate index structure could be carried out using the residuals of det[ (L)]z it . The properties of such a residual-based test would be a¤ected by the properties of the estimator of the univariate autoregressive polynomials i (L):

It is therefore important to estimate these polynomials as accurately as possible.

We address the problem of estimation and testing of univariate ARMA models implied by a VAR(p) model with cofeature restrictions. Univariate ARMA models can be estimated in a straightforward way by the maximum likelihood method, that is we identify and estimate by ML for each series individually the

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT parsimonious empirical ARMA(p i ; q i ) such that z it = ^ i + pi j=1 ^ ij z it j + qi k=1
^ ik ûit k + u it ; i = 1 : : : n; t = 1 : : : T;

where ^ i ; ^ ij and ^ ik are estimated scalar parameters for series i; i 2 f1; 2; :::; ng; p i and q i are the lag orders of the ARMA model for the ith series and they might empirically di¤er from series to series. This …rst stage of the analysis is helpful because we can get a …rst idea about a maximum AR order p as well as of the number of series that might share co-movements. 8 In doing so, we obtain a rough indication, a sort of upper bound say, regarding the possible number of common feature vectors, with max(p i ) = (n s)p: Also, if for a series the AR order di¤ers much from that for other series, then probably it does not share a common cycle with these other variables. Indeed, in absence of cancellation of common roots in the ARMA representations, the ij 's should be the same for all z it 's.

For sets of univariate ARMA models derived from a stationary multivariate ARMA model, [START_REF] Wallis | Multiple Time Series Analysis and the Final Form of Econometric Models[END_REF] has considered maximum likelihood estimation, whereas [START_REF] Palm | Large-Sample Estimation and Testing Procedures for Dynamic Equation Systems[END_REF] have considered both maximum likelihood and e¢ cient two-step estimation. For (di¤erence)-stationary series, the asymptotic properties of univariate and multivariate estimation and test procedures are standard and known. For integrated processes with or without cointegration present, when the unit root is not imposed, the asymptotic procedures generally are non-standard. Whether standard asymptotics hold or not, likelihood and e¢ cient two-step methods for systems may be cumbersome to implement when the set of variables is large. Therefore, for empirical work, it will be useful to have tools at the disposal which can be easily implemented and lead to reliable inference. Satisfying this need is the objective of this section.

Estimation of the common autoregressive polynomial using aggregates of individual series

It has been emphasized in Section 2 that the series implied by the VAR must have the same AR coe¢ cients.

Consequently we investigate whether imposing this restriction using an estimator of the common AR part 9such that

z it = ^ i + pi j=1 ^ j z it j + qi k=1 ^ ik ûit k + ûit ; (8)
where ^ j is the jth lag order common coe¢ cient to all n that should be preferred to estimating ARMA models for the individual series: In order to avoid new notation, we use the same symbols as in [START_REF] Cubadda | Macro-panels and Reality[END_REF] for the estimated intercept and moving average parameters.

The intuition underlying the use of ( 8) is twofold. First, under the null hypothesis, this is a correct way to impose the commonality observed in di¤erent series. Secondly, under the alternative namely when including a set of n 1 variables that does not belong to the same multivariate process then, erroneously imposing a

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

common AR root for the n n 1 variables induces autocorrelation in the error terms. As a result, the detected order of the MA part can be higher than those observed for individual components. But before tackling this issue, we need to get estimates of the common AR component.

After having identi…ed and estimated the parsimonious empirical ARMA(p i ; q i ) by ML for each series individually; one solution is to compute the average of n AR parameters such that ^ mg j = n 1 P n i=1 ^ ij ; j = 1::: max(p i ): However proceeding this way the homogeneity we have under the null is ignored. Moreover the estimation of n equations by ML induces a lot of variability in the estimation of that average. Instead, because the implied model for det[ (L)] z t is at most a MA[(n s)p)] model in the presence of SCCF vectors for instance, we further use this observation and estimate an ARMA model for the average of the n series in the ML estimation of

z t = ^ p j=1 ^ av j z t j + q k=1 ^ k ^ t k + ^ t ; (9)
with z t = n 1 n i=1 z it the simple average of n series, t being the innovation of the univariate ARMA(p; q) for z t and k being the k-th lag parameter of the moving average part of z t : In this second setting we might use a LM test for no autocorrelation or graphical tools (i.e. ACF, PACF) on ^ t to check the white noise hypothesis. A rejection of that null is a sign of misspeci…cation, namely that we have likely included in the analysis a series that is not implied by the same system and consequently does not have the same …nal equation or does not have features in common with the other series in the system and therefore has higher degree AR and MA polynomials. [START_REF] Cubadda | Macro-panels and Reality[END_REF] have shown by simulations that estimating a parsimonious ARMA model on aggregates is the preferred strategy for obtaining a common AR coe¢ cient. Indeed, this procedure not only imposes the common AR parameters but also reduces the MA parts that might be annihilated by linearly combining the series (see Section 3). For instance, using the averages for Canada and the US with

z t = 1
2 ( ln U SA t + ln CA t ), the best model (using SBC) for the aggregates is the AR(1) z t = 0:016 where t-ratios are in brackets. LM autocorrelation tests on the residuals do not reject the null at any common signi…cance level.

Moreover, aggregation can also yield an additional "virtuous" property for such models. Indeed, crosssectional aggregation implies a left multiplication of the adjoint by a vector, and this operation can reduce in some circumstances the order of the MA component of the aggregate. From equation (3), we see that the ARMA model for the aggregate is the following det[ (L)] z t = 0 (L) adj " t ;

where = 1 n , and is an n vector of ones. Moreover, we know from Proposition 8 that the coe¢ cient matrices of (L) adj associated with degrees strictly larger than (n s 1)p have a reduced-rank structure

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

with a common right null space that is spanned by ? . Hence, it is possible that the polynomial matrix 0 (L) adj is, for instance, of order (n s)p 1 instead of (n s)p. This would require that the vector lies in the left null space of the coe¢ cient matrix of (L) adj associated with degree (n s)p , which is equal to p? G 0 , where G is a full rank s s matrix, because of the relation

det[ (L)] | {z } (n s)p I n = (L) (L) adj = 0 @ I n p X j=1 ? 0 j L j 1 A (L) adj | {z } (n s)p :
By similar reasoning, the order of the MA component of model ( 9) can be reduced till (n s 1)p. This result explains why in empirical work it might occur that the MA order of the aggregate is even smaller than those of the individual series. Thus, we may interpret this occurrence as an evidence of the presence of co-movements, although the opposite conclusion is not valid.

A Monte Carlo experiment on the estimator using aggregates

Using the cross-sectional average is …ne under the null that the FEs are from the same initial VAR model. In this subsection, we evaluate by simulation whether this is also true when we "erroneously substitute" to the group of variables having common cycles, additional series from another group. Next we look at alternative weighted cross-sectional averages of series having di¤erent standard deviations.

In both situations we look at the FEs of a VAR(1) with a reduced rank structure

z t = + 1 z t 1 + " t = + ? C 0 1 z t 1 + " t : (10)
In order to keep …xed the AR and the MA orders in the n implied models, we follow [START_REF] Cubadda | Macro-panels and Reality[END_REF] and impose that s = n 1 SCCF relationships, leading to n implied ARMA(1; 1) models for any n. The associated common feature matrix assumes full short-run convergence between economies (or variables).

Indeed, one of the issues we want to address is to select a core of countries (or variables) with the maximum number of co-movements. The cofeature matrix has thus the following shape, for instance for n = 5;

= 0 B B B B B B B @ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 C C C C C C C A ;
which leads to (up to a normalization) ? = (1 : 1 : 1 : 1 : 1) 0 : is a n dimensional vector of intercepts that we generate from a uniform distribution on (0, In particular, we now use in DGP [START_REF] Forni | The Generalized Dynamic Factor Model: Identi…cation and Estimation[END_REF] a covariance matrix of the VAR " such that its diagonal elements are (1 2 ; 2 2 ; :::; n 2 ) and the correlation is equal to 0.7 for any pair of errors. This implies a contemporaneous covariance matrix E(w t w 0 t ) = " + 1 " Since weighted averages are more appropriate than simple averages when series have di¤erent variances, we compare the performances of four types of aggregates:

1. Av 1 is a simple average of the series; 2. Av 2 is a weighted average where the weights are proportional to the inverse of the unconditional standard deviations of the series; 3. Av 3 is a weighted average where the weights are proportional to the inverse of the conditional standard deviations of the series ui obtained through the estimated ARMA(1,1) models; 4. Av 4 is a weighted average where the weights are proportional to the inverse of the standard deviations q 2 ui (1 + 2 i1 ) of the MA(1) innovations of the series obtained through the estimated ARMA(1,1) models.

Table 4 reports the empirical bias and RMSE of the estimators of the common AR parameter for the above aggregates. It emerges that the estimator based on Av 4 has, overall, the best performance. The intuition behind this …nding is that, since the MA component is noise in this estimation problem, the Av 4 weighted average gives more weight to less noisy series. However, the di¤erences in terms of performance of the various estimators are not large, and they tend to disappear as both n and T become large.

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Analysis of per capita real GDP in Latin America

We study the …rst di¤erences of the log levels of the per capita real gross domestic product (namely the growth rates) of nine Latin American economies. We have used annual data from 1950 to 2002, i.e. 53 observations. The series are derived from the Total Economy Database. 10 We have used on purpose the same series as in [START_REF] Hecq | Common Cyclical Features Analysis in VAR Models with Cointegration[END_REF] because it emerged there that it was not obvious to determine a priori the variables to be included in a common trend, common cycle analysis. Table 5 reports the list of these countries as well as the univariate ARMA models and outliers we have identi…ed using the software program Tramo/Seats implemented in Eviews 6 (see [START_REF] Gómez | Programs TRAMO (Time series Regression with Arima noise, Missing observations, and Outliers) and SEATS (Signal Extraction in Arima Time Series)[END_REF]. This illustrates the use of a well known alternative to ACF and PACF analyses with the advantage to determine the presence of aberrant values. We also report the estimates of the signi…cant coe¢ cients at a 10% level.

As far as short-run co-movements are concerned, we can …rst form a group of countries that includes Brazil, Columbia and Guatemala for which an AR(1) was found. 11 With these three countries, p = 2 for the log levels was su¢ cient to capture the dynamics using a VAR. Moreover we do not conclude to the existence of long-run cointegration relationships between the log of per capita real GDP using Johansen's trace tests. Consequently, the VAR(1) for …rst di¤erences implies ARMA(3,2) models for growth rates. This is not compatible with the parsimonious AR(1) we identify from the data. We interpret this as a sign of the where z t is the weighted average Av 3 with estimated weights, which is obviously equal to Av 4 in this case, for which the residuals êt show no sign of autocorrelation. In Table 5, we report the orders of the selected processes, the point estimates for the coe¢ cients (between parentheses) and the years for which outliers have been detected and accounted for.

From these results we conjecture that there exist strong short-run co-movements characterized by a unique common cycle for that group of three countries. Both the individual ARMA orders and the absence of any autocorrelation in the residuals of the aggregated estimated equation support this conclusion. To have an alternative look at this issue (it is possible in this case because n is small), we use SCCF test statistics for the number of cofeatures using a canonical correlation approach as well as the value of information criteria. We estimate the VAR after correcting for the outliers detected at the univariate level. The p value associated with s = 2 is 0.2 and information criteria also select s = 2 (see [START_REF] Hecq | Common Cyclical Features Analysis in VAR Models with Cointegration[END_REF][START_REF] Hecq | Cointegration and Common Cyclical Features in VAR Models: Comparing Small Sample Performances of the 2-Step and Iterative Approaches[END_REF].

Conclusion

This paper has derived the implications for the univariate processes of the presence of common factors or trends resulting from common cyclical features or cointegration in multivariate time series. Exploiting the information about common dynamics contained in individual series and exhibited in individual ARIMA estimations, a strategy has been proposed to study the presence of co-movements in large sets of series without the need to develop and jointly estimate a large complex multivariate model for these series. This

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

strategy is based on the theoretical results for the orders of the FEs for VAR models with co-features. The strategy is shown to yield sensible results in an application involving GDP data for nine countries.

Many further developments are currently under investigations such as, inter alia, the development of panel unit root tests and the extension to seasonal models. Moreover, the small sample and asymptotic properties of the proposed methods must be more deeply evaluated.

The tools we introduce can be extended in several directions. For instance, they allow to study comovements between variables or convergence among economies, forecast series, and build business cycle indices without requiring a full parametric system with many variables. The advantages of our approach are:

1) its feasibility when it is not possible to jointly analyze a complete system or when we prefer to work using a sub-system, 2) its usefulness to detect sets of variables that are likely to be generated by seemingly unrelated subsystems for subsets of variables that have some features in common, 3) the accuracy of forecasts, 4) the ease of its implementation when a large number of jointly dependent variables has to be studied in complex situations, 5) the potential empirical applications in many …elds.

Finally, the insights obtained from the analyses of single series and subset of series with identical AR parts, and from testing for short-run and long-run co-feature restrictions can be incorporated in the multivariate model for all series to be modeled.

Example 1

 1 Let us consider the bivariate VAR(2), (L)z t = (I 1 L 2 L 2 )z t = " t with the following numerical values

Proposition 6

 6 (n s)(p 1) + n: The polynomial degree of the moving average component is obtained along the same line as in Proposition 3 but with P 11 instead of P 11 in (5). Hence, det[P (L)] = det[P 11 ] det[P 22 (L)], which tells us that det[P (L)] is of order (n s)p + (s 1): Since the presence of cointegration implies that (n r 1) common unit roots cancel out from the individual AR and the MA polynomials, the result follows. Cointegrated VAR with s PSCCF(1). In a CI(1,1) VAR(p) with s PSCCF(1) the individual ARMA processes of the …rst di¤ erences have: (i) AR orders not larger than (n s)(p 1) + s + r;

Proposition 7 e 0 1 e ? ( e 0 ?e ? ) 1

 7101 Cointegrated VAR with s WF. In a CI(1,1) VAR(p) with s WF the ARMA processes of the …rst di¤ erences have: (i) AR orders not larger than (n s)(p 1) + r; (ii) MA orders not larger than (n s)(p 1) + r.A C C E P T E D M A N U S C R I P TACCEPTED MANUSCRIPTProof. Let us consider H(L) = f M (L) f M 1 , f M 0 [ e : e ? ],andH(L) =" e 0 (L) e ( e 0 e ) 1 L e 0 ? (L) e ( e 0 e ) 1 e 0 ? (L) e ? ( e 0 ? e ? ) 1

0j:

  Hence we can write det[ (L)]I n = (L) adj (L) = (L) adj [ (L) adj ? ]

Figure 1 :

 1 Figure 1: Quarterly growth rates of industrial production indexes (industrial sector)

t 1 ln CA t 1 #

 11 :

A(

  C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPTselect s = 1[START_REF] Hecq | Cointegration and Common Cyclical Features in VAR Models: Comparing Small Sample Performances of the 2-Step and Iterative Approaches[END_REF]: The estimated common cyclical feature relationship is ln U SA t 1:05 ln CA t :Without any constraints on the short-run, both implied series should follow ARMA(2,1) processes. However, the sample ACF and PACF indicate (…gures not reported in the paper) that the orders are probably shorter and that AR(1) models for industrial production in both countries are more appropriate. For univariate AR(1) models, the estimated equations have quite similar AR roots d ln U SA t = 0:003

z t 1

 1 + êt ;

3 . 2 . 2

 322 1): In order to have the same AR coe¢ cient whatever the odd number of individuals we choose C This guarantees both the stationarity of the multivariate process and the common value of the AR root ( i1 = 0:5; i = 1:::n) for every series for any n. To show this we can simply use the property of partitioned matrices and compute A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT Experiment 2: Simple vs. weighted averages In this second experiment, we extend the Monte Carlo analysis of Cubadda et al. (2008) to allow for heterogenous variances of the individual series.

0 1 .

 1 We use M = 2000 replications of n = 5; 11; 23 individuals with T = 50; 100; 250 observations after having discarded 50 points as starting values.

A( 3 z t 1

 31 C C E P T E D M A N U S C R I P TACCEPTED MANUSCRIPTpresence of short-run co-movements. Indeed, ARMA(1,1) and AR(1) are compatible with s = 2 short-run cofeature relationships. If we identify an ARMA model for the aggregate of the three countries using Tramo, the most parsimonious model is the AR(1) without outliers z t = 0:64 + êt ;

  z t 1 : The sequence of " t is a multivariate white noise process with each "

t N (0; ): The VARMA model in (1) encompasses several useful speci…cations. For example, if (L) I we have the vector moving average representation or VMA; if (L) I we have a VAR(p). Let us concentrate the analysis on VAR models of order p, denoted VAR(p), one of the cornerstone speci…cation in empirical macroeconometrics:

Table 1

 1 Maximum ARMA orders of univariate series generated by an n-dimensional I(0) VAR(p) with s cofeature restrictions

	Models	AR order	MA order
	VAR(p)	np	(n 1)p
	SCCF	(n s)p	(n s)p
	PSCCF(1)

(n s)p + s (n s)p + (s 1) PSCCF(m) (n s)p + sm (n s)p + (s 1)m Proposition 3 Stationary VAR with s SCCF. In an n-dimensional stationary VAR(p) with s SCCF, the individual ARMA processes have: (i) AR orders not larger than (n s)p; (ii) MA orders not larger than (n s)p. Proof. Let us rewrite equation (2) as follows

Table 2

 2 reports the largest ARIMA orders of individual series that are generated by a VECM with common feature restrictions. Let us formally prove these results.

			Table 2
		Maximum ARMA orders of univariate series generated by an
		n-dimensional CI(1,1) VAR(p) with s cofeature restrictions
	Models	AR order	Integration order	MA order
	VAR(p)	n(p 1) + r	1	(n 1)(p 1) + r
	SCCF	(n s)(p 1) + r	1	(n s)(p 1) + r

PSCCF(1) (n s)(p 1) + r + s 1 (n s)(p 1) + r + s 1 WF (n s)(p 1) + r 1 (n s)(p 1) + r Proposition 5 Cointegrated VAR with s SCCF. In a CI(1,1) VAR(p) with s SCCF the individual ARMA processes of the …rst di¤ erences have: (i) AR orders not larger than (n s)(p 1) + r; (ii) MA orders not larger than (n s)(p 1) + r.

Table 4

 4 Various estimators of a common autoregressive parameter

			Bias			RMSE
	n	T = 50 T = 100 T = 250 T = 50 T = 100 T = 250
	5	Av 1 -0.072	-0.028	-0.007	0.205	0.128	0.069
		Av 2 -0.054	-0.012	0.011	0.199	0.127	0.069
		Av 3 -0.046	-0.006	0.018	0.190	0.128	0.073
		Av 4 -0.040	-0.001	0.019	0.183	0.122	0.073
	11 Av 1 -0.065	-0.033	-0.011	0.185	0.120	0.068
		Av 2 -0.054	-0.022	-0.001	0.182	0.115	0.067
		Av 3 -0.051	-0.017	0.003	0.186	0.115	0.067
		Av 4 -0.046	-0.015	0.005	0.177	0.113	0.067
	23 Av 1 -0.079	-0.033	-0.010	0.211	0.124	0.067
		Av 2 -0.077	-0.029	-0.006	0.216	0.123	0.066
		Av 3 -0.073	-0.027	-0.003	0.214	0.122	0.066
		Av 4 -0.067	-0.025	-0.003	0.201	0.121	0.065

Note: A parsimonious empirical ARMA model is chosen for the aggregates using SBC.

Table 5

 5 Identi…ed ARMA(p; q) models

	Identi…ed components: AR	MA	Outliers
	Brazil	1 (0.51) 0	1981
	Chile	0	1 (0.33) 1982,1975
	Columbia	1 (0.39) 0	1999
	Mexico	0	1 (0.44) 1995
	Peru	0	1 (0.58) 1983
	Venezuela	0	0	-
	Argentina	0	1 (0.23) -
	Ecuador	0	0	1987,1999
	Guatemala	1 (0.57) 0	-

The generalisation to non-stationarity processes will be considered later in this section. .

This result generalizes to individual ARMA(np; (n 1)p + q) for VARMA(p; q) processes (see e.g.[START_REF] Lütkepohl | New Introduction to Multiple Time Series Analysis[END_REF]).

This model is also known as a scalar component model of order zero or SCM(0,0) using the terminology of[START_REF] Tiao | Model Speci…cation in Multivariate Time Series (with comments)[END_REF] or white noise direction codependence by[START_REF] Gouriéroux | Detecting a Long-run Relationship[END_REF].

Notice that a SCCF relationship involving, for instance, z 1t and z 2t 1 , where zt = (z 0 1t ; z 0 2t ) 0 , is a particular case of a PSCCF[START_REF] Ahn | Inference of Vector Autoregressive Models with Cointegration and Scalar Components[END_REF].

In this framework, cycles are the remainder of a vector I(1) process after subtracting the (possibly common) random-walk components.

Using univariate maximum likelihood estimation, the MA(1) component has p-values of 0.16 and 0.32 for respectively the US and Canada.

Of course variables z it with di¤erent univariate autoregressive polynomials could be generated by di¤erent sub-systems of a multivariate VAR with a corresponding block-diagonal matrix (L):

Note that even though we do not identify the VAR order, we may nonetheless have an idea about p. For instance, with annual data p is typically at most 4 but p = 2 is the most common choice in empirical applications, whereas for quarterly data p is rarely found to be larger than

9 Note that the common AR operator can be obtained by factorizing the AR-polynomials for the individual series into the product of the individual operators and then cancelling common factors(Lütkepohl, 2005, 496).

0 University of Groningen and The Conference Board, GGDC Total Economy Database www.eco.rug.nl/ggdc. The variables are expressed in 1990 US dollars and converted at "Geary-Khamis" purchasing power parities.1 1 Note that when looking for a set of countries sharing common cycles, white noise processes do not a¤ect the order of the univariate ARMA processes of the remaining variables. This is because having one more white noise series included leads to consider the degree (n + 1 (s + 1))p = (n s)p for the implied AR and MA parts for instance: But this simpli…es the search procedure.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT det[I

1 L] when considering the (n 1) (n 1) upper-left block of (I 1 L). We use M = 2000 replications of T + 50 observations before dropping the …rst 50 points to initialize the random sequence.

Experiment 1: Impact of adding series not generated from a VAR

In this …rst experiment we successively consider n = 5; 11; 23 series generated by the VAR (10) for T = 50; 100; 250. However, for the last n 2 = (n 1)=2 series, we substitute series generated by the following diagonal VAR:

where 1 = diag( ) + U( 0:125; 0:125), and = 0; 0:25; 0:5. These are heterogeneous AR(1) processes such that only when = 0:5 the coe¢ cient comes from an uniform distribution centered around the same AR parameter of the FEs associated with [START_REF] Forni | The Generalized Dynamic Factor Model: Identi…cation and Estimation[END_REF]. For the covariance matrix of the VAR errors we use in this …rst experiment " = I: This implies cross-correlated errors w t = (L) adj " t in the FEs because their

This corresponds to a correlation between the disturbances of the implied equations of = 0:55 for all pairs. We estimate the common AR parameter using the parsimonious ARMA model on aggregates. Indeed, [START_REF] Cubadda | Macro-panels and Reality[END_REF] have found that this estimator has the best properties among various methods when the DGP is given by [START_REF] Forni | The Generalized Dynamic Factor Model: Identi…cation and Estimation[END_REF]. Table 3 reports both the empirical bias and RMSE of this estimator when we generate n 1 = n 2 n 1 series from system (10) and n 2 dimensional series from system [START_REF] Franses | Time Series Models for Business and Economic Forecasting[END_REF]. It emerges that the bias signi…cantly increases if we include series with AR coe¢ cients more distant from the …nal equation common parameters. However, the di¤erences in terms of both bias and RMSE decrease as n and T increase.

Table 3

Estimation of a common autoregressive parameter i1 = 0:5 in a mixed system