Cointegration and error correction mechanisms for singular stochastic vectors

Barigozzi, Matteo; Lippi, Marco; Luciani, Matteo Article Cointegration and error correction mechanisms for singular stochastic vectors Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Barigozzi, Matteo; Lippi, Marco; Luciani, Matteo (2020) : Cointegration and error correction mechanisms for singular stochastic vectors, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 8, Iss. 1, pp. 1-23, https://doi.org/10.3390/econometrics8010003 This Version is available at: https://hdl.handle.net/10419/247552 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ econometrics Article Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors Matteo Barigozzi 1, Marco Lippi 2,* and Matteo Luciani 3 1Università di Bologna, Department of Economics, 40126 Bologna, Italy; [email protected] 2Einaudi Institute for Economics and Finance, 00187 Roma, Italy 3Federal Reserve Board of Governors, Washington, DC 20551, USA; [email protected] *Correspondence: [email protected] Received: 28 March 2018; Accepted: 7 January 2020; Published: 4 February 2020   Abstract: Large-dimensional dynamic factor models and dynamic stochastic general equilibrium models, both widely used in empirical macroeconomics, deal with singular stochastic vectors, i.e., vectors of dimension r which are driven by a q -dimensional white noise, with q<r . The present paper studies cointegration and error correction representations for an I( 1 ) singular stochastic vector yt . It is easily seen that yt is necessarily cointegrated with cointegrating rank c≥r−q . Our contributions are: (i) we generalize Johansen’s proof of the Granger representation theorem to I( 1 ) singular vectors under the assumption that yt has rational spectral density; (ii) using recent results on singular vectors by Anderson and Deistler, we prove that for generic values of the parameters the autoregressive representation of yt has a finite-degree polynomial. The relationship between the cointegration of the factors and the cointegration of the observable variables in a large-dimensional factor model is also discussed. Keywords: singular stochastic vectors; cointegration for singular vectors; Granger representation theorem; large-dimensional dynamic factor models) JEL Classification: C0; C01; E0 1. Introduction An r -dimensional stochastic vector yt such that yt=A0ut+A1ut−1+· · · , where the matrices Aj are r×q and ut is a q -dimensional white noise, with q<r , is said to be singular. Singular stochastic vectors have been systematically analyzed in a number of papers starting with (Anderson and Deistler 2008a,2008b) . A motivation for studying the consequences of singularity, as argued by these authors, is that the factors’ vector in large-dimensional dynamic factor models (DFM), such as those introduced in Forni et al. (2000); Forni and Lippi (2001), (Stock and Watson 2002a,2002b) , is typically singular. Singularity is also an important feature of dynamic stochastic general equilibrium models (DSGE), see e.g., Sargent (1989), Canova (2007), pp. 230–2. Singularity as it arises in DFMs is presented in some detail below. DFMs are based on the idea that all the observed variables in an economic system are driven by a few common (macroeconomic) shocks and by idiosyncratic components which may result from measurement errors and sectoral or regional shocks. Formally, each variable in the n -dimensional dataset xit , i= 1, 2, . . . , n , t= 1, 2, . . . , T , is decomposed into the sum of a common component χit , and an idiosyncratic component eit : xit =χit +eit , where χit and ejs are orthogonal for all i , j , t , s . In the standard version of the DFM the common components are linear combinations of an r -dimensional vector of common factors Ft= (F1tF2t· · · Frt)0, χit =λi1F1t+λi2F2t+· · · +λirFrt =λiFt. (1) Econometrics 2020,8, 3; doi:10.3390/econometrics8010003 www.mdpi.com/journal/econometrics Econometrics 2020,8, 3 2 of 23 Now suppose that the observable variables xit and the common factors Ftare I(1)and that (1−L)Ft=C(L)ut, (2) where ut is a nonsingular q -dimensional white-noise vector 1 , the common shocks. A number of papers analyzing macroeconomic databases find strong empirical support for the assumption that the vector Ft is singular, i.e., that q<r . See, for US datasets, Giannone et al. (2005); Amengual and Watson (2007); Forni and Gambetti (2010), Luciani (2015). For a Euro-area dataset, see Barigozzi et al. (2014). Such results can be easily understood observing that usually the static Equation (1) is just a convenient representation derived from a “primitive” set of dynamic equations linking the common components χit to the common shocks ut . As a simple example, suppose that the variables xit are driven by a common one-dimensional cyclical process ft , such that ( 1 −αL)ft=ut , where ut is scalar white noise, and that the variables xit load ftdynamically: xit =ai0ft+ai1ft−1+eit. (3) In this case we can set F1t=ft , F2t=ft−1=F1,t−1 , λi1=ai0 , λi2=ai1 , so that Equations (1) and (2) take the form xit =λi1F1t+λi2F2t+eit and F1t F2t!= (1−αL)−1 L(1−αL)−1!ut, respectively. Here r= 2 and q= 1 so that Ft is singular. For a general analysis of the relationship between representation (1) and “deeper” dynamic representations like (3) , see e.g., Forni et al. (2009); Stock and Watson (2016). Now suppose that the factors Ft have been estimated. Obtaining ut and the impulse-response functions of the variables xit with respect to ut (or structural shocks obtained by a linear transformation of ut ) requires the estimation of a VAR for the singular I( 1 ) vector Ft . On the other hand, the latter is necessarily cointegrated with cointegration rank c at least equal to r−q (the rank of the spectral density of (1−L)Ftdoes not exceed qat all frequencies and, therefore, at frequency zero). Singular vectors of factors in an I( 1 ) DFM and I( 1 ) singular vectors in DSGE models provide strong motivation for studying singular I( 1 ) vectors in a general time-series context. The main contributions of the paper are: (I) A generalization of Johansen’s proof of the Granger Representation Theorem (from MA to AR), this is Proposition 2. Consider an I( 1 ) singular vector yt , with dimension r , rank q<r , and cointegrating rank c≥r−q . Assuming that ( 1 −L)yt has an ARMA structure, S(L)( 1 − L)yt=B(L)ut and that some simple additional conditions hold, yt has a representation as a vector error correction mechanism (VECM) with cerror correction terms: A(L)yt=A∗(L)(1−L)yt+α(β0yt−1−w) = B(0)ut, (4) where α and β are both r×c and full rank, β0yt−w is I( 0 ) , A(L) and A∗(L) are r×r rational matrices in L . Under the additional assumption that unity is the only zero of B(L) , i.e., if z6= 1 then B(z)is full rank, A(L)and A∗(L)are finite-degree matrix polynomials. (II) Assuming that the parameters of S(L) and B(L) may vary in an open subset of Rλ , see Section 3.2 for the definition of λ , in Proposition 3we show that all the assumptions used to obtain (4) , and also the assumption that unity is the only possible zero of B(L) , hold for generic values of 1Usually orthonormality is assumed. This is convenient but not necessary in the present paper. Econometrics 2020,8, 3 3 of 23 the parameters. This implies that the matrices A(L) and A∗(L) are generically of finite degree, which is obviously not the case for nonsingular vectors.2 The paper is organized as follows. Section 2is preliminary. We firstly recall recent results for stationary singular stochastic vectors with rational spectral density, see (Anderson and Deistler 2008a,2008b). Secondly, we discuss cointegration and the cointegrating rank for I( 1 ) singular stochastic vectors. In Section 3we prove our main results. We also obtain the permanent-transitory shock representation in the singular case: yt is driven by r−c permanent shocks, i.e., r minus the cointegrating rank, the usual result. However, the number of transitory shocks is c−(r−q) , not c as in the nonsingular case. Section 3also contains an exercise carried out with simulated singular I( 1 ) vectors. We compare the results obtained by estimating an unrestricted VAR in the levels and a VECM. Though limited to a simple example, the results confirm what has been found for nonsingular vectors, that under cointegration the long-run features of impulse-response functions are better estimated using a VECM rather than an unrestricted VAR in the levels (Phillips 1998). In Section 4we analyse cointegration of the observable variables xit in a DFM. Our results on cointegration of the singular vector Ft have the implication that p -dimensional subvectors of the n -dimensional common-component vector χt , with p>r−c , are cointegrated. As a consequence, stationarity of the idiosyncratic components would imply that all p -dimensional subvectors of the n -dimensional dataset xt are cointegrated if p>r−c . For example, if q= 3 and d= 1, then all 3-dimensional subvectors in the dataset are cointegrated, a kind of regularity that we do not observe in actual large macroeconomic datasets. This suggests that an estimation strategy robust to the assumption that the idiosyncratic components can be I( 1 ) has to be preferred (for this aspect we refer to Barigozzi et al. 2019). Section 5concludes. Some proofs, a discussion of some non-uniqueness problems arising with singularity and details on the simulations are collected in the Appendix. 2. Stationary and I(1)Singular Vectors 2.1. Stationary Singular Vectors As in this paper we only consider representation issues it is convenient to assume that all stochastic processes are defined for t∈Z . Accordingly, the lag operator L is defined as Lyt=yt−1 for t∈Z (Bauer and Wagner (2012) also study I(1)and cointegrated processes for t∈Z). We start by introducing results on singular vectors with an ARMA structure from (Anderson and Deistler 2008a,2008b). Some preliminary definitions are needed. Definition 1. (Zeros and Poles) (A) When considering matrices V(z) whose entries are rational functions of z∈C we always assume that numerator and denominator of each entry have no common roots. If V(z) is an r×q matrix of rational functions, we say that z∗is a pole of V(z)if it is a pole of some entry of V(z). (B) Suppose that V(z) is an r×q matrix whose entries are polynomial functions of z∈C , with q≤r . We say that z∗∈C is a zero of V(z) if rank (V(z∗)) <q , and that V(z) is zeroless if it has no zeros, i.e., rank(V(z)) = q for all z ∈C. 2 To our knowledge, the present paper is the first to study cointegration and error correction representations for I( 1 ) singular vectors, the factors of I( 1 ) dynamic factor models in particular. An error correction model in the DFM framework is studied in (Banerjee et al. 2014,2017) . However, their focus is on the relationship between the observable variables and the factors. Their error correction term is a linear combination of the variables xit and the factors Ft , which is stationary if the idiosyncratic components are stationary (so that the x ’s and the factors are cointegrated). Because of this and other differences their results are not directly comparable to those in the present paper. Econometrics 2020,8, 3 4 of 23 With a minor abuse of language, we may speak of zeros and poles of the corresponding matrix V(L) . When a r×r polynomial matrix S(L) has all its zeros outside the unit circle we say that S(L) is stable. All the stationary vector processes considered have an ARMA structure. Precisely, the r-dimensional process ythas an ARMA structure with rank q,q≤r, if there exist (i) a non-singular q-dimensional white-noise process ut, (ii) an r×rstable polynomial matrix S(z), with S(0) = Ir, (iii) an r×q matrix B(z) whose rank is q for all z with the exception of a finite subset of C , such that yt=V(L)ut, (5) where V(L) = S(L)−1B(L). Suppose that yt has also the representation yt=˜ S(L)−1˜ B(L)˜ut , where ˜ut is a ˜ q -dimensional nonsingular white noise. Denoting by Σy(θ)the spectral density of yt, Σy(θ) = (2π)−1V(e−iθ)ΣuV0(eiθ), so that the rank of Σy(θ) is q for all θ , with the exception of a finite subset of [−π , π] . As the spectral density is independent of the ARMA representation, q=˜ q and ˜ B(z) has rank q except for a finite subset of C. Remark 1. Let us recall that the equation S(L)ζt=B(L)ut, in the unknown vector process ζt , where S(L) is stable, has only one stationary solution, and this is yt= S(L)−1B(L)ut .Thus the ARMA process yt can also be defined as the stationary solution of S(L)ζt=B(L)ut . Definition 2. (Genericity) Suppose that a statement Q depends on p∈ A , where A is an open subset of Rλ . We say that Q holds generically in A , or that Q holds for generic values of p∈ A , if the subset N of A where it does not hold is nowhere dense in A, i.e., the closure of Nin Ahas no internal points. For example, assuming that p∈ A =R , the statement “The roots of the polynomial x2+px+ 1 are distinct” holds generically in A. Definition 3. (Rational reduced-rank family of filters) Assume that r>q and let G be a set of ordered couples (S(L),B(L)), where: (i) B(L)is an r ×q polynomial matrix of degree s1≥0. (ii) S(L)is an r ×r polynomial matrix of degree s2≥0.S(0) = Ir. (iii) Denoting by p the vector containing the λ=rq(s1+ 1 ) + r2s2 coefficients of the entries of B(L) and S(L) , we assume that p∈Π , where Π is an open subset of Rλ such that for p∈Π ,(1) S(z) is stable, (2) rank(B(z)) = q with the exception of a finite subset of C. We say that Gis a rational reduced-rank family of filters with parameter set Π. The notation Sp(L) , Bp(L) , though more rigorous, would be heavy and not really necessary. We use it only in Appendix A.1. Proposition 1. Assume that r >q. (I) Suppose that V(L) is an r×q matrix polynomial in L . If V(z) is zeroless then V(L) has an r×r finite-degree stable left inverse, i.e., there exists a finite-degree polynomial r×r matrix W(L) such that: Econometrics 2020,8, 3 5 of 23 (a) W( 0 ) = Ir ,(b) det(W(z)) = 0implies |z|> 1,(c) W(L)V(L) = V( 0 ) . Let yt be the stationary solution of S(L)ζt=B(L)ut and suppose that B(L) is zeroless. Then yt has a finite vector autoregressive representation (VAR) A(L)yt=B( 0 )ut , where A(L) = N(L)S(L) and N(L) is a finite-degree left inverse of B(L). (II) Assume that yt is the stationary solution of S(L)ζt=B(L)ut , where (S(L) , B(L)) belongs to a rational reduced-rank family of filters with parameter set Π . For generic values of the parameters in Π , B(L) is zeroless so that ythas a finite VAR representation. For statement (I) see Anderson and Deistler (2008a), Theorem 3. Statement (II) is a modified version of their Theorem 2, see for a proof Forni et al. (2009), p. 1327. 2.2. Fundamentalness Assume that the r -dimensional vector yt has an ARMA structure, rank q and the moving average representation (5) . If rank(B(z)) = q for |z|< 1, then ut belongs to the space spanned by yt−k , with k≥ 0, and representation (5) , as well as ut , is called fundamental (for these definitions and results see e.g., Rozanov (1967), pp. 43–7). Note that if (5) is fundamental rank(B( 0 )) = q . Note also that when q=r, the condition that rank(B(z)) = qfor |z|<1 becomes det(B(z)) 6=0 for |z|<1. Remark 2. Note that in Proposition 1, part (II), we do not assume that ut is fundamental for yt . However, Proposition 1, (II), states that for generic values of p∈Π the matrix B(L) is zeroless and therefore ut is fundamental for yt. 2.3. I(1)Singular Vectors To analyze cointegration and the autoregressive representations of singular non-stationary vectors let us first recall the definitions of I( 0 ) , I( 1 ) and cointegrated vectors. This requires some preliminary definitions and results. We denote by L2(Ω , F , P) the space of the square-integrable functions on the probability space (Ω , F , P) . Let zt= (z1tz2t· · · zrt)0 , zht ∈L2(Ω , F , P) , be an r -dimensional stochastic process and consider the difference equation (1−L)ζt=zt, (6) in the unknown r-dimensional process ζ ζ ζt. A solution of (6) is ˜ ψt=       z1+z2+· · · +zt, for t>0 0, for t=0 −(z0+z−1· · · +zt+1), for t<0, see e.g., Gregoir (1999), p. 439, Franchi and Paruolo (2019). All the solutions of (6) are ψt=˜ ψt+φt , where φt= (φ1tφ2t· · · φrt)0 , φht ∈L2(Ω , F , P) , is a solution of the homogeneous equation ( 1 −L)ζt= 0 , so that φt=K , for some r -dimensional stochastic vector K , for all t∈Z . We say that the process φt=K is a constant stochastic process. Obviously a constant stochastic process φt=K is weakly stationary. Its spectral measure has the jump ΣK at frequency zero. Thus φt has a spectral density (has an absolutely continuous spectral measure) if and only if ΣK=0 , i.e., if and only if φt(ω) = k , where k∈Rr, for ωalmost everywhere in Ω. Definition 4. (I(0), I(1) and Cointegrated vectors) I(0). An r-dimensional ARMA ytwith spectral density Σy(θ)is I(0)if Σy(0)6=0. I(1). The r -dimensional vector stochastic process yt is I( 1 ) if it is a solution ( 1 −L)ζt=zt where zt is an r-dimensional I(0)process. The rank of ytis defined as the rank of zt. Econometrics 2020,8, 3 6 of 23 Cointegration. Assume that the r -dimensional stochastic vector yt is I( 1 ) and denote by Σ∆y(θ) the spectral density of ( 1 −L)yt . The vector yt is cointegrated with cointegrating rank c , with 0 <c<r , if rank(Σ Σ Σ∆y(0)) = r−c. If q is the rank of yt and r≥q , then c=r−q+d , where q>d> 0. Thus in the singular case, r>q,ytis necessarily cointegrated with cointegrating rank at least equal to r−q. If yt is I( 1 ) and cointegrated with cointegrating rank c , there exist c linearly independent r× 1 vectors cj , j= 1, . . . , c , such that the spectral density of c0 j( 1 −L)yt vanishes at frequency zero. The vectors cj are called cointegrating vectors and the set cj , j= 1, . . . , c , a complete set of cointegrating vectors. Of course a complete set of cointegrating vectors cj , j= 1, . . . , c , can be replaced by the set dj , j=1, . . . , c, where the vectors djare cindependent linear combinations of the vectors cj. Lemma 1. (I) Assume that yt has an ARMA structure and has the rational representation (5) : yt=V(L)ut . Then ytis I(0)if and only if V(1)6=0. (II) Assume (1−L)ythas an ARMA structure and has the rational representation (1−L)yt=V(L)ut. (7) The process ytis I(1)if and only if V(1)6=0. (III) If yt is I( 1 ) , cointegrated and has representation (7) , the cointegrating rank of yt is c if and only if the rank of V(1)is r −c. Moreover cis a cointegrating vector for ytif and only if c0V(1) = 0. (IV) Assume that yt is I( 1 ) . c is a cointegrating vector for yt if and only if a scalar stochastic variable w∈L2(Ω,F,P)can be determined such that c0yt−w is stationary with an ARMA structure. Proof. (I) is an immediate consequence of Σy( 0 )=( 2 π)−1V( 1 )ΓuV( 1 )0 , where Γu is the nonsingular covariance matrix of ut . (II) and (III) are obtained in the same way from Σ∆y( 0 ) = ( 2 π)−1V( 1 )ΓuV( 1 )0 . (IV) The process ytsolves (6) with zt=V(L)ut, so that, defining µt=       u1+u2+· · · +ut, for t>0 0, for t=0 −(u0+u−1· · · +ut+1), for t<0, (8) we have yt=V(L)µt+K=V(1) + (1−L)V(L)−V(1) 1−Lµt+K=V(1)µt+V∗(L)ut+K, where (i) the entries of V∗(L)=(V(L)−V( 1 ))/( 1 −L) are rational functions of L with no poles of modulus less or equal to unity, (ii) Kis a constant r-dimensional stochastic process. We have: c0yt=c0V(1)µt+c0V∗(L)ut+c0K. (9) If cis a cointegrating vector of ytwe have c0V(1) = 0, so that c0yt=c0V∗(L)ut+c0K. Setting w=c0K , the process c0yt−w=c0V∗(L)ut has the desired properties. Note that w has the equivalent definition w=c0y0−c0V∗(L)u0 . Conversely, suppose that w is such that c0yt−w has an ARMA structure. By (9), c0yt−w=c0V(1)µt+c0V∗(L)ut+c0K−w, Econometrics 2020,8, 3 7 of 23 so that qE(c0yt−w)2+qE(c0V∗(L)ut)2+qE(c0K−w)2≥qc0V(1)ΣµtV0(1)c. The three terms on the left-hand side are finite and independent of t . As Σµt=|t|Σu and Σu is positive definite, the right-hand side diverges for |t| → ∞unless c0V(1) = 0. Lemma 1shows that our definitions of I( 0 ) and I( 1 ) processes are equivalent to Definitions 3.2, and 3.3 in Johansen (1995), p. 35, with two minor differences: (i) our assumption of rational spectral density, (ii) the time span of the stochastic processes is t= 0, 1, . . . in Johansen’s book, t∈Z in the present paper. Also, under the assumption that ( 1 −L)yt has an ARMA structure, our definition of cointegration is equivalent to that in Johansen (1995), p. 37. 3. Representation Theory for Singular I(1)Vectors In Section 3.1 we prove our generalization to singular vectors of the Granger representation theorem (from MA to AR). We closely follow the proof in Johansen (1995), Theorem 4.5, p. 55–57. In Section 3.2 we show that, under a suitable parameterization, the matrix of the autoregressive representation is generically of finite degree. 3.1. The Granger Representation Theorem (MA to AR) Suppose that r≥q , c> 0 and r>c≥r−q . Let B(L) be an r×q polynomial matrix of degree s1≥0 and S(L)an r×rpolynomial matrix of degree s2≥0 with S(0) = Ir. Assumption 1. S(L)is stable. Assumption 2. If z∗is a zero of B(z)(i.e. rank(B(z∗)) <q) then either z∗=1or |z∗|>1. Assumption 2implies that the rank of B(0)is q. The next is a stronger version of Assumption 2: Assumption 3. If z∗is a zero of B(z)then z∗=1. Assumption 4. rank(B(1)) = r−c. Under Assumption 1, let ytbe a solution of the equation (1−L)ζt=S(L)−1B(L)ut. (10) We have yt=S(L)−1B(L)µt+K, (11) where µt is defined in (8) and K is a constant stochastic process. By Assumption 4, S( 1 )−1B( 1 )6=0 , so that ytis I(1)with cointegrating rank c, see Lemma 1, (II) and (III). Consider the finite Taylor expansion of B(z)around z=1: B(z) = B(1)−(1−z)B0(1) + (1−z)2B00(1) + · · · . Assumption 4implies that B(1) = ξη0, where ξ ξ ξ is r×(r−c) of rank r−c , η is q×(r−c) of rank r−c , see Lancaster and Tismenetsky (1985, p. 97, Proposition 3). The Taylor expansion above can be rewritten as B(z) = ξη η η0+ (1−z)B∗+ (1−z)2E(z), (12) Econometrics 2020,8, 3 8 of 23 where B∗=−B0(1)and E(z)is a polynomial matrix. Let ξ ξ ξ⊥ be an r×c matrix whose columns are orthogonal to all columns of ξ : (i) the columns of ξ⊥ are a complete set of cointegrating vectors for B(L)ut , (ii) the columns of the matrix S0( 1 )ξ⊥ are a complete set of cointegrating vectors for yt. Regarding (i), using (11) and (12), we have ξ0 ⊥S(L)yt=ξ0 ⊥B(L)µt+ξ0 ⊥S(1)K= (ξ0 ⊥B∗+ (1−L)ξ0 ⊥E(L))ut+ξ0 ⊥S(1)K, (13) so that ξ0 ⊥S(L)yt−ξ0 ⊥S(1)Khas an ARMA structure. Regarding (ii), see the proof of Proposition 2. Assumption 5. rank " ξ ξ ξ0 ⊥B∗ η η η0!#=rank " ξ ξ ξ0 ⊥B∗ ξ ξ ξ0ξ ξ ξη η η0!#=q. Define S∗(L) = S(L)−S(1) 1−L. Assumption 6. ξ ξ ξ0 ⊥(B∗−S∗(1)S(1)−1ξη0)6=0. Remark 3. Let yt be a solution of (10) so that ( 1 −L)yt is stationary and S(L)[( 1 −L)yt] = B(L)ut . Assumption 2, and therefore 3, implies that utis fundamental for (1−L)yt, see Section 2.2. We are now ready for our main representation result. Proposition 2. (I) Weak form. Suppose that Assumptions 1,2,4,5and 6hold and let yt be a solution of the difference Equation (10) , so that yt=S(L)−1B(L)µt+K , with µt defined in (8) and K a constant stochastic process. Set β=S( 1 )0ξ⊥ . Then a c -dimensional stochastic vector w can be determined such that (i) β0yt−w is I(0),(ii) ythas the error correction representation A(L)yt=A∗(L)(1−L)yt+α(β0yt−1−w) = B(0)ut, (14) where A(L) is a rational r×r matrix with no poles in or on the unit circle, A( 1 ) = Ir , A∗(L) = (A(L)− A(1)L)(1−L)−1,αis r ×c and full rank, αβ0=A(1). (II) Strong form. Under Assumptions 1,3,4,5and 6, statement (I) holds with an r×r stable, finite-degree matrix polynomial A(L). Proof. Multiply both sides of ( 1 −L)S(L)yt=B(L)ut by the r×r invertible matrix Ξ= ξ0 ⊥ ξ0! . We obtain (1−L)ΞS(L)yt=Ξ Ξ ΞB(L)ut =( 0c×q ξ0ξη0!+ (1−L) ξ0 ⊥B∗ ξ0B∗!+ (1−L)2 ξ0 ⊥E(L) ξ0E(L)!)ut = (1−L)Ic0 0 Ir−c!( ξ0 ⊥B∗ ξ0ξη0!+ (1−L) ξ0 ⊥E(L) ξ0B∗!+ (1−L)2 0c×q ξ0E(L)!)ut. (15) Taking the first crows in (15), (1−L)ξ0 ⊥S(L)yt= (1−L)ξ0 ⊥B∗+ (1−L)ξ0 ⊥E(L)ut. This implies that ξ0 ⊥S(L)yt=ξ0 ⊥B∗+ (1−L)ξ0 ⊥E(L)ut+w, (16) Econometrics 2020,8, 3 15 of 23 Now, (23) implies that λpΣ(p) ∆x(0)≥λpΣ(p) ∆χ(0)+λ(p)Σ(p) ∆e(0), (24) where λp(A) denotes the smallest eigenvalue of the hermitian matrix A ; this is one of the Weyl’s inequalities, see Franklin (2000), p. 157, Theorem 1. Because the spectral density matrices are non-negative definite, the right hand side in (24) vanishes if and only if both terms on the right hand side vanish, i.e., the spectral density of ∆x(p) t is singular at zero if and only if the spectral densities of ∆χ(p) tand ∆e(p) tare singular at zero. By definition 4, (i) is proved. Without loss of generality we can assume that S(L) = Ir. By substituting (21) in (22), we obtain xt=Λ[(G1(L)v1t+G2(L)v2t+Tt)+K]+et, (25) where on the right hand side the only non-stationary terms are Tt and possibly et . By recalling that Tt=ξ∑t s=1v2s where ξ is of dimension r×(q−d) and rank q−d , and by defining Gt= Λ[G1(L)v1t+G2(L)v2t+K]and Tt=∑t s=1v2s, we can rewrite (25) as xt=ΛξTt+Gt+et. For x(p) t: x(p) t=χ χ χ(p) t+e(p) t=Λ(p)ξTt+G(p) t+e(p) t, where Λ(p) and G(p) t have an obvious definition. Of course cointegration of the common components χ χ χ(p) t is equivalent to cointegration of Λ(p)ξTt , which in turn is equivalent to rank (Λ(p)ξ)<p . Statement (ii) follows from rank Λ(p)ξ≤min rank(Λ(p)), rank(ξ). The first part of (iii) is obvious. Assume now that p>q−d . If cp χ+cp e=dim(Vχ) + dim(Ve) = p−(q−d) + cp e>p , i.e., if cp e>q−d , then the intersection between Vχ and Ve is non-trivial, so that x(p) tis cointegrated. 5. Summary and Conclusions The paper studies representation theory for singular I( 1 ) stochastic vectors, the factors of an I( 1 ) Dynamic Factor Model in particular. Singular I( 1 ) vectors are cointegrated, with a cointegrating rank cequal to r−q, the dimension of ytminus its rank, plus d, with 0 ≤d<q. If ( 1 −L)yt has rational spectral density, under assumptions that generalize to the singular case those in Johansen (1995), we show that yt has an error correction representation with c error terms, thus generalizing the Granger representation theorem (from MA to AR) to the singular case. Important consequences of singularity are that generically: (i) the autoregressive matrix polynomial of the error correction representation is of finite degree, (ii) the white noise vector driving ( 1 −L)yt is fundamental. We find that yt is driven by r−c permanent shocks and d=c−(r−q) transitory shocks, not c as in the nonsingular case. Using simulated data generated by a simple singular VECM, confirms previous results, obtained for nonsingular vectors, showing that under cointegration the long-run features of impulse-response functions are better estimated using a VECM rather than a VAR in the levels. In Section 4we argue that stationarity of the idiosyncratic components in a DFM produce an amount of cointegration for the observable variables xit that is not observed in the datasets that are standard in applied Dynamic Factor Model literature. Thus the idiosyncratic vector in those datasets is likely to be I( 1 ) , so that an estimation strategy robust to the assumption that some of the idiosyncratic variables eit are I(1)should be preferred. Econometrics 2020,8, 3 16 of 23 The results in this paper are the basis for estimation of I( 1 ) Dynamic Factor Models with cointegrated factors, which is developed in the companion paper (Barigozzi et al. 2019). Author Contributions: All authors contributed equally to the paper. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: Dietmar Bauer, Manfred Deistler, Massimo Franchi, Martin Wagner, three anonymous referees and the Editors of this Special Issue gave important suggestions for improvements. We also thank the participants to the Workshop on Estimation and Inference Theory for Cointegrated Processes in the State Space Representation, Technische Universität Dortmund, January 2016. Part of this paper was written while Matteo Luciani was chargé de recherches F.R.S.- F.N.R.S., and he gratefully acknowledges their financial support. Of course we are responsible for any remaining errors. Disclaimer : The views expressed in this paper are those of the authors and do not necessarily reflect those of the Board of Governors or the Federal Reserve System. Conflicts of Interest: The authors declare no conflict of interest. Appendix A. Proofs Appendix A.1. Assumption 3Holds Generically Proving that Assumption 3holds generically is equivalent to proving that M(z) is generically zeroless, see the argument below Equation (20). We need some preliminary results. Lemma A1, though quite easy, is not completely standard and is therefore carefully stated and proved below. Regarding notation, to avoid possible misunderstandings, let us recall that vectors and matrices are always denoted by boldface symbols, while light symbols denote scalars, see Lemmas A1 and A2 in particular. Lemma A1. Let Aj , j= 1, . . . , s , be scalar polynomials defined on Rλ , let p∈Rλ and Q(p) be the statement Aj(p) = 0, for j=1, . . . , s, for example the statement that all the q×q minors of M( 1 ) vanish, i.e., that rank(M( 1 )) <q . Let Π be an open subset of Rλ. If Q is false for one point p∗∈Rλ, then Q is generically false in Π. Proof. Let N be the closure in Π (in the topology of Π ) of the subset of Π where Q is true. Suppose that Q is not generically false in Π . Then the interior of N in Π , call it N◦ , is not empty. As Π is open, N◦ is open both in the topology of Π and of Rλ . On the other hand a polynomial function defined on Rλ vanishes on an open set if and only if it vanishes on the whole Rλ , which contradicts the existence of a point in Rλwhere Qis false. Lemma A2. Consider the scalar polynomials A(z) = a0zn+a1zn−1+· · · +an,B(z) = b0zm+b1zm−1+· · · +am, with a06= 0and b06= 0, and let αi , i= 1, . . . , n and βj , j= 1, . . . , m , be the roots of A and B , respectively. Then: (i) am 0bn 0∏ i,j (αi−βj) = R(a0,a1, . . . , an;b0,b1, . . . , bm), where R is a polynomial function which is called the resultant of A and B . (ii) The resultant vanishes if and only if A and B have a common root. (iii) Suppose that the coefficients ai and bj are polynomial functions of p∈Π , where Π is an open subset of Rλ . If there exists a point p∗∈Rλ such that a0(p∗)6= 0, b0(p∗)6= 0, and R(p∗)6=0, then generically in Πthe polynomials A and B have no common roots. Econometrics 2020,8, 3 17 of 23 Proof. For (i) and (ii) see van der Waerden (1953, pp. 83-8). Statement (iii) is an obvious consequence of (ii) and Lemma A1. Lemma A3. Recall that a zero of M(z) is a complex number z∗ such that rank(M(z∗)) <q . If M(z) has two q×q submatrices whose determinants have no common roots, then M(z)is zeroless. Proof. If z∗is a zero of M(z), then z∗is a zero of all the q×qsubmatrices of M(z). For the statement and proof of our last result it is convenient to make explicit the dependence of the matrix M(z) and its submatrices on the vector p . Thus we use Mp(z) , etc. The parameters of the matrix S(L) play no role here. Hence, with no loss of generality, we assume s2= 0, so that λ= (r−c)(r+q) + rq(s1+ 2 ) . Lemmas A2–A4 below imply that Assumption 3holds generically in Π. Lemma A4. Let Mp 1(z) , Mp 2(z) , . . . be all the q×q submatrices of Mp(z) and let Lp i be the leading coefficient of det Mp i(z)and Rp ij is the resultant of det Mp i(z)and det Mp j(z). There exist i, j, p∗∈Rλsuch that Lp∗ iLp∗ j6=0 and Rp∗ ij 6=0. Proof. Assume that r=q+1. To each p∈Πthere corresponds the matrix Mp(z) = ξ0 ⊥B∗ ξ0ξη0!+ (1−z) ξ0 ⊥E(z) ξ0B∗!+ (1−z)2 0c×q ξ0E(z)!. Of course, the definition of Mp(z) makes sense for all p∈Rλ , see Equation (19) . Let Mp 1(z) and Mp 2(z) be the matrices obtained from Mp(z) by removing the first and the last row respectively. We have: degree[det(Mp 1(z))] ≤(q−d)(s1+2) + d(s1+1) = d1, degree[det(Mp 2(z))] ≤(q−d−1)(s1+2) + (d+1)(s1+1) = d2. We will construct a point p∗∈Rλ such that: (A) the coefficient of zd1 in det(Mp∗ 1(z)) and the coefficient of zd2 in det(Mp∗ 2(z)) (the leading coefficients) do not vanish, (B) the resultant of det(Mp∗ 1(z)) and det(Mp∗ 2(z)) does not vanish. Let us firstly define a family of matrices, denoted by M(z) , obtained by specifying η η η , ξ , ξ0 ⊥ , B∗ and E(z)in the following way: η0=0(q−d)×dIq−d,ξ= Iq−d 0c×(q−d)!,ξ0 ⊥= K H!, B∗=H00(q+1)×(q−d),E(z) =    E1(z) E2(z) E3(z)  , Econometrics 2020,8, 3 18 of 23 where: K=01×(q−d)101×d,H=0d×(q+1−d)Id, E1(z) =          k1(z)h1(z)· · · 0 0(q−d)×d ...... ...hq−d−1(z) 0· · · kq−d(z)          ,E2(z) = e(z)01×(q−1), E3(z) =     f1(z)g1(z)· · · 0 ......0d×(q−d−1) 0· · · fd(z)gd(z)     , the entries e,ki,hi,fiand gibeing scalar polynomials of degree s1. We denote by q1 the vector including the coefficients of the polynomials fi , i= 1, . . . , d and ki , i= 1, . . . , (q−d) , a total of q(s1+ 1 ) coefficients, by q2 the vector including the coefficients of the polynomials e , gi , i= 1, . . . , d and hi , i= 1, . . . , (q−d− 1 ) , a total of q(s1+ 1 ) coefficients, by q0 the vector including the zeros and the ones in the definition of ξ , η , B∗ , E , and define q= (q0q1q2) , which is a λ -dimensional parameter vector. We put no restriction on q1 and q2 , so that both can take any value in Rν, with ν=q(s1+1). Note that qdoes not necessarily belong to Π. We have: Mq(z) =    01×d01×(q−d) Id0d×(q−d) 0(q−d)×dIq−d   + (1−z)   E2(z) E3(z) 0(q−d)×q   + (1−z)2   01×q 0d×q E1(z)  . (A1) The matrix Mq(z) has zero entries except for the diagonal joining the positions ( 1, 1 ) and (q , q) , and the diagonal joining ( 2, 1 ) and (q+ 1, q) . The matrices Mq 1(z) and Mq 2(z) are upper- and lower-triangular, respectively, and det(Mq 1(z)) = [1+ (1−z)f1(z)] · · · [(1+ (1−z)fd(z)] ×[1+ (1−z)2k1(z)] · · · [1+ (1−z)2kq−d(z)] = Lq 1,d1zd1+· · · +Lq 1,0 det(Mq 2(z)) = (1−z)2q−d−1e(z)[g1(z)· · · gd(z)][h1(z)· · · hq−d−1(z)] = Lq 2,d2zd1+· · · +Lq 2,0. Note that det(Mq 1(z)) does not depend on q2 , while det(Mq 2(z)) does not depend on q1 . Thus we use the notation δq1 1(z) = det(Mq 1(z)),δq2 2(z) = det(Mq 2(z)),Mq1 1,d1=Lq 1,d1,Mq2 2,d2=Lq 2,d2. Now: (i) Let q∗ 2∈Rν be such that none of the leading coefficients of the polynomials e , gi and hi vanishes. Of course Mq∗ 2 2,d2=d26=0. (ii) Let ˇ z be a root of δq∗ 2 2(z) . If ˇ z= 1 then ˇ z is not a root of δq1 1(z) for all q1∈Rν . Suppose that ˇ z is a root of gj(z) , for some j . As the parameters of the polynomials fi and ki are free to vary in Rν , then, generically in Rν , δq1 1(ˇ z)6= 0. Iterating for all roots of δq∗ 2 2(z) , generically in Rν , δq1 1(z) and δq∗ 2 2(z) have no roots in common. Moreover, generically in Rν , Mq1 1,d1=d16= 0. Thus, there exists q∗ 1such that (a) Mq∗ 1 1,d1=d16=0, (b) δq∗ 1 1(z)and δq∗ 2 2(z)have no roots in common. (iii) Now let p∗= (q0q∗ 1q∗ 2), so that det(Mp∗ 1(z)) = δq∗ 1 1(z)), det(Mp∗ 2(z)) = δq∗ 2 2(z). Econometrics 2020,8, 3 19 of 23 Using (i) and (ii), (A) the leading coefficients of det(Mp∗ 1(z)) and det(Mp∗ 2(z)) do not vanish, (B) det(Mp∗ 1(z)) and det(Mp∗ 2(z)) have no root in common so that their resultant does not vanish. This proves the proposition for r=q+1. Generalizing this result to r>q+ 1 is easy. Let us define the family N(z) in the following way: (a) specify η0,ξ,E1(z)and E3(z)as in the definition of M(z), (b) then let K=0(r−q)×(r−d−1)01×(r−q−1)100(r−q)×d,H=0d×(r−d)Id, ξ0 ⊥= K H!,D=H0Ir×(q−d),E2(z) = 0(r−q)×q e(z)01×(q−1)!. We have: N(z) =    0(r−q)×d0(r−q)×(q−d) Id0d×(q−d) 0(q−d)×dIq−d   + (1−z)   E2(z) E3(z) 0(q−d)×q   + (1−z)2   0(r−q)×q 0d×q E1(z)  . It is easy to see that the (q+1)×qlower submatrix of N(z)is identical to the matrix Mq(z)in (A1). Appendix A.2. if R >Q and C ≤Q, Assumptions 5and 6Do Not Imply That etIs a Non-Cointegrated I(0)Process. Let r=3, q=2, S(L) = I3, ξ=   1 0 0   ,η= 0 1!,ξ⊥=   0 0 1 0 0 1   ,B∗=   a b 1 0 1 0   . In this case c=2 and d=1, so that c=q(see Remark 6). We have ξ0 ⊥B∗ η0!=   1 0 1 0 0 1   . We see that Assumptions 5and 6hold. However, rank(ξ0 ⊥B∗) = 1, so that et , though being I( 0 ) , is not a non-cointegrated I( 0 ) process. On the other hand, if the ( 3, 2 ) entry of B∗ is 1 instead of 0, etis non-cointegrated. Appendix B. Non Uniqueness In Proposition 3we prove that a singular I( 1 ) vector with cointegrating rank c has a finite error correction representation with c error terms. On the other hand, as we have seen in Remark 5, when c=r−q the singular vector yt has also an autoregressive representation in the differences, i.e., a representation with zero error terms. In Appendix B.1 we give an example hinting that yt has error correction representations with any number of error terms between d and c . However, in Appendix B.2 we show that all such representations produce the same impulse-response functions. Appendix B.1. Alternative Representations with Different Numbers of Error Terms Let S(L) = Irand consider the following example, with r=3, q=2, c=2, so that d=1: Econometrics 2020,8, 3 20 of 23 ξ0=111 η0=1 2 ξ0 ⊥= 1−1 0 0 1 −1! We have, (1−L) ξ0 ⊥ ξ0!yt=   1−L0 0 0 1 −L0 0 0 1           b∗ 11 −b∗ 21 b∗ 12 −b∗ 22 b∗ 21 −b∗ 31 b∗ 22 −b∗ 32 3 6   + (1−L)ˆ E(L)     ut, where ( 1 −L)ˆ E(L) gathers the second and third terms in M(L) . If the assumptions of Proposition 2 hold, we obtain an error correction representation with error terms ξ0 ⊥yt= y1t−y2t y2t−y3t!. However, we also have (1−L) ξ0 ⊥ ξ0!yt=   1−L0 0 0 1 0 0 0 1    ×        b∗ 11 −b∗ 21 b∗ 12 −b∗ 22 (1−L)(b∗ 21 −b∗ 31) (1−L)(b∗ 22 −b∗ 32) 3 6   + (1−L)ˇ E(L)     ut=   1−L0 0 0 1 0 0 0 1   ˇ M(L)ut. Under suitable assumptions on the coefficients b∗ ij and ˇ E(L) , assuming in particular that the matrix b∗ 11 −b∗ 21 b∗ 12 −b∗ 22 3 6 ! is nonsingular, the matrix ˇ M(L) is zeroless and has therefore a finite-degree left inverse. Proceeding as in Proposition 2, we obtain an alternative error correction representation with just one error term, namely y1t−y2t. This example should be sufficient to convey the idea that yt admits error correction representations with a minimum dand a maximum c=r−q+dof error terms. The problem of error correction representations, with different numbers of error terms, has been recently addressed in Deistler and Wagner (2017). An implication of their main result (see Theorem 1, p. 41) is that if ythas the error correction representation ˜ A(L)yt=˜ A∗(L)(1−L)yt+˜ A(1)yt−1=˜ B˜ut, and rank(˜ A( 1 )) <c (the number of error terms is not the maximum), then ˜ A(L) and ˜ B are not left coprime. The consequences of Deistler and Wagner’s paper have not yet been developed. In Propositions 2 and 3we have only considered representations with c error terms. On non-uniqueness of autoregressive representations for singular vectors with rational spectral density see also Chen et al. (2011); Anderson et al. (2012); Forni et al. (2015). Econometrics 2020,8, 3 21 of 23 Appendix B.2. Uniqueness of Impulse-Response Functions Suppose that the assumptions of Proposition 2, weak form, hold. Let yt be a solution of Equation (10), so that (1−L)yt=S(L)−1B(L)ut, (A2) and suppose that ythas the autoregressive representation ˜ A(L)yt=˜ B˜ut, (A3) where ˜ A(L) is a rational matrix with poles outside the unit circle, ˜ A( 0 ) = Ir , ˜ut is a nonsingular q-dimensional white noise, ˜ Bis a full rank r×qmatrix5. We have ˜ A(L)[(1−L)yt]=(1−L)˜ B˜ut. (A4) The assumption that ˜ B is full rank and the argument used e.g., in Brockwell and Davis (1991), p. 111, Problem 3.8, imply that ˜ut is fundamental for ( 1 −L)yt . Thus ˜ut=Qut , where Q is a nonsingular q×qmatrix (see Rozanov (1967), p. 57), and ˜ B˜ut= [ ˜ BQ]ut. On the other hand, from (A2) and (A4): ˜ A(L)S(L)−1B(L)ut= (1−L)[ ˜ BQ]ut. (A5) As ut is nonsingular, ˜ A(L)S(L)−1B(L)=( 1 −L)[ ˜ BQ] . Setting L= 0 we have ˜ BQ =B( 0 ) , so that (A3) becomes ˜ A(L)yt=B(0)ut(A6) while (A5) becomes ˜ A(L)S(L)−1B(L)ut= (1−L)B(0)ut. (A7) The impulse-response function of yt to ut resulting from (A6) is H(L)B( 0 ) , where H(L)˜ A(L) = Ir . Multiplying both sides of (A7) by H(L)we obtain S(L)−1B(L) = (1−L)H(L)B(0), so that H(L)B(0)is obtained by cumulating S(L)−1B(L)and is therefore independent of ˜ A(L). Appendix C. Data Generating Process for the Simulations The simulation results of Section 3.4 are obtained using the following specification of (14): A(L)yt=A∗(L)(1−L)yt+αβ0yt−1=C(0)ut=GHut, where r= 4, q= 3, c= 3, the degree of A(L) is 2, so that the degree of A∗(L) is 1. A(L) is generated using the factorization A(L) = U(L)M(L)V(L), where U(L)and V(L)are r×rmatrix polynomials with all their roots outside the unit circle, and M(L) = (1−L)Ir−c0 0 Ic! 5 Multiplying both sides of (A3) by ( 1 −L) and using (A2) , we obtain ˜ A(L)S(L)−1B(L)ut= ( 1 −L)˜ B˜ut . Comparing the spectral densities of right- and left-hand terms, it is easy to prove that ˜ut must be a q -dimensional, nonsingular white noise and the rank of ˜ Bmust be q. Econometrics 2020,8, 3 22 of 23 (see Watson 1994). To get a VAR(2) we set U(L) = Ir−U1L , and V(L) = Ir , and then, by rewriting M(L) = Ir−M1L, we get A1=M1+U1, and A2=−M1U1. Regarding the generation of the data, the diagonal entries of the matrix U1 are drawn from a uniform distribution between 0.5 and 0.8, while the extra–diagonal entries are drawn from a uniform distribution between 0 and 0.3. U1 is then multiplied by a scalar so that its largest eigenvalue is 0.6. The matrix G is generated as in Bai and Ng (2007): (1) ˜ G is an r×r diagonal matrix of rank q where ˜ gii is drawn from the uniform distribution between 0.8 and 1.2, (2) ˇ G is obtained by orthogonalizing an r×r uniform random matrix, (3) G is equal to the first q columns of the matrix ˇ G˜ G1/2 . Lastly, the orthogonal matrix H is such that the upper 3 × 3 submatrix of GH is lower triangular. The results are based on 1000 replications. 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