Franchi, Massimo

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u n i ve r s i t y  o f  co pe n h ag e n  Københavns UniversitetThe Integration Order of Vector Autoregressive ProcessesFranchi, MassimoPublication date:2006Document versionPublisher's PDF, also known as Version of recordCitation for published version (APA):Franchi, M. (2006). The Integration Order of Vector Autoregressive Processes. Cph.: Department of Economics,University of Copenhagen.Download date: 02. Feb. 2020DISCUSSION PAPERS Department of Economics University of Copenhagen       The Integrat  AutoregrMassStudiestræde 6, DK-Tel. +45 35 32 3http:/  06-05 ion Order of Vector essive Processes   imo Franchi      1455 Copenhagen K., Denmark 0 82 - Fax +45 35 32 30 00 /www.econ.ku.dk THE INTEGRATION ORDER OF VECTORAUTOREGRESSIVE PROCESSESMASSIMO FRANCHIUNIVERSITY OF COPENHAGENANDUNIVERSITY OF ROME “LA SAPIENZA”Abstract. We show that the order of integration of a vector au-toregressive process is equal to the difference between the mul-tiplicity of the unit root in the characteristic equation and themultiplicity of the unit root in the adjoint matrix polynomial. Theequivalence with the standard I(1) and I(2) conditions (Johansen,1996) is proved and polynomial cointegration discussed in the gen-eral setup.1. IntroductionAn autoregressive process is integrated of order d, if its characteristicequation has d roots at z = 1 and the remaining lie outside the unitcircle. This is not true in the multivariate case, because the order ofintegration of a vector autoregressive processes is not established bythe multiplicity of the unit root in the characteristic equation. For thisreason, some extra conditions are needed in order to write down themoving average representation. Johansen (1988, 1992) imposes neces-sary and sufficient conditions on the parameters of the autoregressiveprocess and derives the corresponding moving average representationfor I(1) and I(2) processes. His work is related to Engle and Granger(1987), who start from the moving average representation of an I(1)process which exhibits cointegration and derive the corresponding errorcorrection model; unfortunately the proof of the Granger Representa-tion Theorem is not correct (see Johansen (2005a) for a counterexampleto Lemma 1). Other proofs of the same theorem are based on the SmithDate: February 17, 2006. I am very grateful to Søren Johansen for his preciousinsights and his continuous help throughout the development of the paper. E-mailaddress: massimo.franchi@econ.ku.dk.12form of a matrix polynomial (see Engle and Yoo (1991), Ahn and Rein-sel (1990) and Haldrup and Salomon (1998)) and on the Jordan rep-resentation of the companion form (see Archontakis (1998) and Bauerand Wagner (2003)). Other relevant papers in this area are Gregoireand Laroque (1993) and Gregoire (1999), who discuss polynomial coin-tegration in a very general setup and Neusser (2000), who points outsome interesting algebraic properties of the I(1) model. An attemptto characterize explicitly the polynomial cointegration properties of anI(d) process from its autoregressive representation is la Cour (1998).See Johansen (2005a) for an exhaustive survey of the mathematical re-sults concerning the representation theory and Johansen (2005b) for anapplication of similar ideas to fractional integration and cofractionality.In this paper we study the I(d) multivariate case and show thatone can determine the order of integration of a vector autoregressiveprocess as the difference between the multiplicity of the unit root inthe characteristic equation and the multiplicity of the unit root in theadjoint matrix polynomial. This result arises from observing that thereduced rank of the characteristic polynomial at z = 1 translates intoa zero versus non zero statement about the adjoint matrix polynomial.This then allows to write the inverse in such a way that the order ofthe pole at the unit root becomes explicit, resembling what happens inthe univariate case.The paper is organized as follows: in section 2 we introduce theVAR(k) model and the standard definitions of integration and cointe-gration and in section 3 we prove the main Theorem on I(d) processes.In section 4 we show the equivalence with the standard I(1) and I(2)conditions (Johansen, 1996) and in section 5 we discuss polynomialcointegration. The last section contains some concluding remarks.2. VAR(k) model and standard definitionsConsider the p−dimensional autoregressive model(2.1) Xt = Π1Xt−1 +Π2Xt−2 + · · ·+ΠkXt−k + ²t,or Π(L)Xt = ²t and ²t is an i.i.d. process.Definition 2.1. The process Xt = C(L)²t is stationary if C(z) =∑∞i=0Cizi converges for |z| < 1 + δ for some δ > 0; it is I(0) when itis stationary and C(1) 6= 0; it is I(d), d > 0, if ∆dXt is I(0).3Cointegration and polynomial cointegration are defined as followsDefinition 2.2. The I(d) process Xt is cointegrated if there exists βsuch that β′Xt is I(b) with b < d. It is polynomially cointegrated if thereexists βk for k = 0, · · · , d− 1, such that∑d−1k=0 β′k∆kXt is stationary.3. Poles, order of integration and multiplicitiesThe characteristic polynomial of (2.1) is the p× p matrix functionΠ(z) = Ip −k∑i=1Πizi,and the characteristic equation is defined as |Π(z)| = 0, where |Π(z)| =det(Π(z)) is a polynomial of degree n ≤ kp, |Π(z)| =∑ni=0 dizi. From|Π(0)| = 1 it follows that zero is not a root of the characteristic equa-tion. Let nr be the number of distinct roots zi, each with multiplicitymi; the determinant can thus be written as(3.1) |Π(z)| = dnnr∏i=1(z − zi)mi = (z − 1)mg(z),where g(1) 6= 0 and 1 ≤ m ≤ n.Assumption 3.1. The only unstable root is at z = 1; that is |Π(z)| = 0implies zi = 1 or |zi| > 1.Evaluating the characteristic polynomial at the roots of the char-acteristic equation we get reduced rank matrices; at the unit root wewrite Π(1) = −αβ′, where α and β are p×r matrices of full rank r < p.The inverse is defined as the adjoint matrix Πa(z) = Adj(Π(z)) di-vided by the determinant(3.2) Π(z)−1 =Πa(z)|Π(z)| , z 6= {1, · · · , znr}.Since Π(z) has reduced rank at the roots of the characteristic equation,(3.2) is not defined at z = {1, · · · , znr}. These singularities are knownto be poles but at the moment nothing can be said about their order.Proposition 3.2. If Assumption 3.1 holds, then Xt is I(d) if and onlyif Π(z)−1 has a pole of order d at z = 1.Proof. By definition Xt is I(d) if ∆dXt = C(L)²t with C(z) =∑∞i=0Cizi convergent for |z| < 1+ δ for some δ > 0 and C(1) 6= 0; thenΠ(z)−1 = C(z)/(1− z)d has a pole of order d at z = 1.4Now consider the numerator in (3.2) at z = 1: when we evaluate theadjoint matrix polynomial at the unit root we have that1Πa(1) = 0p when r < p− 1,because the determinant of any p−1×p−1 minor extracted from Π(1)is zero andΠa(1) 6= 0p when r = p− 1,because there is at least one non singular p− 1× p− 1 minor in Π(1).It follows that z−1 can be factored out a times from Πa(z), for somea > 0 when r < p− 1 and for a = 0 when r = p− 1; consequently wehave thatΠa(z) = (z − 1)aH(z),where H(1) 6= 0p and a ≥ 0.The only way of having a pole at z = 1 in (3.2) is if Πa(z) goes tozero at a slower rate than |Π(z)|. Equivalently, we could say that itmust be the case that a < m.This is exactly what Theorem 3.3 below makes precise.Theorem 3.3. If Assumption 3.1 holds, the order of integration of Xtis equal tod = m− a,where H(1) 6= 0p in Πa(z) = (z − 1)aH(z) and m is the multiplicity ofthe unit root in the characteristic equation.Proof. Assume that a = m− d and H(1) 6= 0p; thenΠ(z)−1 =Πa(z)|Π(z)| =H(z)(z − 1)dg(z) ,from which we see that Π(z)−1 has a pole of order d at z = 1; thus Xtis I(d) by Proposition 3.2. Assume now that Xt is I(d); by Proposition3.2 it follows that Π(z)−1 has a pole of order d at z = 1. Then (z −1)(d−1)Π(z)−1 is not defined at z = 1 and G(z) = (z−1)dΠ(z)−1 is suchthat G(1) 6= 0p. This defines H(z) = Πa(z)/(z − 1)(m−d) which impliesa = m− d and H(1) = g(1)G(1) 6= 0p.Thus the order of integration of the process is simply equal to themultiplicity of the unit root in the characteristic equation minus themultiplicity of the unit root in the adjoint matrix polynomial. We knowhow to calculate the roots of a polynomial; then it is clear that we can1We use 0p for the p× p zero matrix and Ip for the identity matrix of the samedimension. For non square matrices we write both the row and column indexes.5find m, the number of roots at z = 1 in (3.1) and mij, the numberof roots at z = 1 in entry i, j of Πa(z). Then a = minij mij and theprocess is integrated of order d = m− a.Example 1 (Johansen, 1992): Consider the model−(1 22 4)Xt +(0 12122 + δ)∆Xt +(0 −1212−1)∆2Xt = ²t,with characteristic polynomialΠ(z) =(−1 −2 + z2(1− z)−2 + z2(1− z) −3 + δ(1− z)− z2),and characteristic equation|Π(z)| = −(1− z)(δ + 1− z + z24(1− z)).Assumption 3.1 is satisfied if δ = 0 or δ ≥ 3. When δ ≥ 3, m = 1 andg(z) = δ + 1− z + z24(1− z) is such that g(1) = δ. Since Πa(1) 6= 0 wehave that a = 0 and d = m = 1. When δ = 0, m = 2 and g(z) = 1+ 14z2is such that g(1) = 54; then Πa(1) 6= 0 implies d = 2.Example 2 (Paruolo, 1996): Consider the modelXt = 0 0 120 1 0120 2Xt−1 + 0 0 −120 0 0−120 −1Xt−2 + ²t,with characteristic polynomialΠ(z) = 1 0 − z2(1− z)0 1− z 0− z2(1− z) 0 (1− z)2 ,and characteristic equation|Π(z)| = (1− z)3(1− z24).Then Assumption 3.1 is satisfied, m = 3 and g(z) = 1− z24is such thatg(1) = 34; the adjoint matrix polynomial isΠa(z) = (1− z)3 0 z2(1− z)20 (1− z)2(1− z24 ) 0z2(1− z)2 0 1− z ,from which it is easily seen that a = 1 and thus that d = m − a = 2;thus the process is integrated of order 2.64. Equivalence with the standard I(1) and I(2) conditionsWe want to prove the equivalence with the standard conditions inJohansen (1996) and derive the explicit expression of H(1). We intro-duce the following notation: let A⊥ be the orthogonal complement ofan m × n matrix A of rank n < m, let A¯ = A(A′A)−1 and write theTaylor expansion of Π(z) at z = 1 asΠ(z) = Π(1) + Π˙(1)(z − 1) + Π¨(1)2(z − 1)2 + (z − 1)3Π3(z).The order of integration is established (Johansen, 1996) by some re-duced and full rank conditions on specific matrices: Xt is I(1) if andonly if |Π(1)| = 0 and |α′⊥Π˙(1)β⊥| 6= 0; similarly, the I(2) conditionstates that the order of integration is two if and only if |Π(1)| = 0,|α′⊥Π˙(1)β⊥| = 0 and |α′2θβ2| 6= 0 where θ = Π¨(1)2 + Π˙(1)β¯α¯′Π˙(1),α2 = α⊥ξ⊥, β2 = β⊥η⊥, α′⊥Π˙(1)β⊥ = ξη′ and ξ, η are p− r× s matricesof full rank s < p − r. Using (3.1) and Theorem 3.3 we rewrite theidentity Π(z)Πa(z) = Πa(z)Π(z) = |Π(z)|Ip as(4.1) Π(z)H(z) = (z − 1)dg(z)Ipand H(z)Π(z) = (z − 1)dg(z)Ip. At z = 1 they read αβ′H(1) =H(1)αβ′ = 0p and mean thatH(1) = β⊥ζdα′⊥,for some p− r × p− r matrix ζd of rank 0 < rd ≤ p− r.Proposition 4.1 (I(1) case). A necessary and sufficient condition for|α′⊥Π˙(1)β⊥| 6= 0 is thata = m− 1in Πa(z) = (z − 1)aH(z) and H(1) 6= 0p. The explicit expression forH(1) isH(1) = g(1)β⊥(α′⊥Π˙(1)β⊥)−1α′⊥.Proof. Assume a = m−1 so that d = 1; differentiate (4.1) at z = 1to get Π˙(1)H(1)− αβ′H˙(1) = g(1)Ip and thusα′⊥Π˙(1)β⊥ζ1 = g(1)Ip−r.Then g(1) 6= 0 implies |ζ1| 6= 0, |α′⊥Π˙(1)β⊥| 6= 0 and ζ1 = g(1)(α′⊥Π˙(1)β⊥)−1,and thus the I(1) condition is satisfied.Assume now the I(1) condition holds and suppose d = m − a > 1;differentiating (4.1) at z = 1 we get α′⊥Π˙(1)β⊥ζd = 0p−r; since ζd 6= 0p−rthis contradicts |α′⊥Π˙(1)β⊥| 6= 0 and implies m− a = 1.7In the next proposition we consider the I(2) case:Proposition 4.2 (I(2) case). A necessary and sufficient condition forα′⊥Π˙(1)β⊥ = ξη′ and |α′2θβ2| 6= 0, where ξ and η are p−r×s matrices offull rank s < p−r, θ = Π¨(1)2+Π˙(1)β¯α¯′Π˙(1), α2 = α⊥ξ⊥ and β2 = β⊥η⊥is thata = m− 2in Πa(z) = (z − 1)aH(z) and H(1) 6= 0p. The explicit expression forH(1) isH(1) = g(1)β2(α′2θβ2)−1α′2.Proof. Assume a = m − 2 so that d = 2; the first derivativeof Π(z)H(z) = H(z)Π(z) = (z − 1)2g(z)Ip implies α′⊥Π˙(1)β⊥ζ2 =ζ2α′⊥Π˙(1)β⊥ = 0p−r and thus |ζ2||α′⊥Π˙(1)β⊥| = 0; if |α′⊥Π˙(1)β⊥| 6= 0then ζ2 = 0p−r contradictsH(1) 6= 0p; thus |α′⊥Π˙(1)β⊥| = 0, α′⊥Π˙(1)β⊥ =ξη′ where ξ and η are p − r × s matrices of full rank s < p − r andζ2 = η⊥ψξ′⊥ for some p−r−s×p−r−smatrix ψ of rank 0 < t ≤ p−r−s;then H(1) becomesH(1) = β⊥η⊥ψξ′⊥α′⊥ = β2ψα′2.Note that the first derivative of (4.1) provides the equality(4.2) β′H˙(1) = α¯′Π˙(1)H(1).The the second derivative of (4.1) implies(4.3) α′⊥Π¨(1)H(1) + 2α′⊥Π˙(1)H˙(1) = 2g(1)α′⊥.Using Ip = β⊥β¯′⊥ + β¯β′ we see that α′⊥Π˙(1)H˙(1) = ξη′β¯′⊥H˙(1) +α′⊥Π˙(1)β¯β′H˙(1) = ξη′β¯′⊥H˙(1) + α′⊥Π˙(1)β¯α¯′Π˙(1)H(1) by (4.2). Thus(4.3) becomes(4.4) α′⊥[Π¨(1)2+ Π˙(1)β¯α¯′Π˙(1)]H(1) + ξη′β¯′⊥H˙(1) = g(1)α′⊥.Pre and post multiplying (4.4) respectively by ξ′⊥ and α¯2, we see that(4.5) α′2θβ2ψ = g(1)Ip−r−s.Then |ψ| 6= 0, |α′2θβ2| 6= 0 and ψ = g(1)(α′2θβ2)−1 follow from g(1) 6=0 and the I(2) condition is satisfied.Assume now the I(2) condition holds and suppose d = m − a > 2;(4.5) becomesα′2θβ2ψ = 0p−r−sand ψ 6= 0p−r−s contradicts |α′2θβ2| 6= 0 and implies m− a = 2.8These two equivalences allow us to understand the standard I(1) andI(2) conditions as imposing a particular shape to the adjoint matrixpolynomial, which in turn ensures that the pole at the unit root hasorder one or two; in these cases the principal part of the Laurent expan-sion of (3.2) around z = 1 consists of one or two terms and translatesinto a moving average representation which involves the cumulation(or the double cumulation) of the stationary process that generates thestochastic trends.5. Polynomial cointegrationThe results of the previous section can thus be interpreted as al-ternative proofs of the Granger Representation Theorem in the I(1)and I(2) cases: the order of integration is established by Theorem3.3, the explicit expression for H(1) indicates the directions in whichcointegration is to be found, and the restrictions implied by (4.1) andits derivatives define the (polynomial) cointegration properties of theprocess. In the I(2) case, for example, we write the inverse of Π(z) asΠ(z)−1 =H(z)(z − 1)2g(z) , z 6= {1, · · · , znr},where H(1) 6= 0p and g(1) 6= 0, and H(z) asH(z) = H(1) + H˙(1)(z − 1) + (z − 1)2H2(z).From the calculations in the proof of Proposition 4.2, we have thatH(1) = β⊥η⊥ψξ′⊥α′⊥, |ψ| 6= 0,(5.1)β′H˙(1) = α¯′Π˙(1)H(1).(5.2)Thus the polynomial cointegration properties of the process are thefollowing:Proposition 5.1. Let Xt be I(2) and β1 = β¯⊥η; then β′Xt and β′1Xtare I(1), and β′Xt + α¯′Π˙(1)∆Xt is I(0).Proof. From (5.1) we have that β′H(1) = 0r×p and β′1H(1) = 0s×p;thus the functions β′Π(z)−1 and β′1Π(z)−1 have a pole of order one atz = 1 and correspond to β′Xt and β′1Xt being I(1). Using (5.2) it iseasy to see that the function{β′ + (1− z)α¯′Π˙(1)}Π(z)−1 has no poleat z = 1 and corresponds to β′Xt + α¯′Π˙(1)∆Xt being I(0).9The only polynomial cointegrating relation that involves the levelsand reduces the order of integration from two to zero is:β′Xt + α¯′Π˙(1)∆Xt.Note that (5.1) and Ip = β¯β′+ β¯1β′1+ β¯2β′2 imply H(1) = β¯2β′2H(1);thus the coefficient of the pole in (1−z)Π(z) is actually β¯2β′2H(1). Thismeans that there are terms in α¯′Π˙(1)∆Xt which do not help eliminatethe non stationarity of β′Xt and thus are not needed. Thus the minimalchoice is to takeβ′Xt + α¯′Π˙(1)β¯2β′2∆Xtas in Johansen (1992).Now consider a process integrated of order d. The Taylor expansionof Π(z) isΠ(z) =d−1∑n=0Π(n)(1)n!(z − 1)n + (1− z)dΠd(z)and translates intod−1∑n=0Π(n)(1)(−1)nn!∆nXt +Πd(L)∆dXt = ²tin terms of the stochastic process. Since ∆dXt is I(0) and Πd(L) is afinite order polynomial, Πd(L)∆dXt is stationary.Then∑d−1n=0Π(n)(1) (−1)nn!∆nXt is also stationary. The polynomialcointegrating relation that involves the levels and reduces the orderof integration from d to zero is simplyβ′Xt + α¯′Π˙(1)∆Xt − α¯′ Π¨(1)2∆2Xt + · · · − (−1)d−1α¯′Π(d−1)(1)(d− 1)! ∆d−1Xt.The difficulties arise when we want to find the minimal representa-tion (see la Cour, 1998, for the I(3) case). Further investigation is stillneeded to find a tractable solution in the general case.6. ConclusionThis paper has extended the way we understand the order of inte-gration in the univariate case to vector autoregressive processes. It hasshown that there exists a very natural representation of the inverse ofthe characteristic polynomial, in which the order of the pole at theunit root is explicit. This result significantly simplifies the proof of theGranger Representation Theorem in the I(1) and I(2) cases.10ReferencesAhn, S. & C. Reinsel (1990) Estimation for partially nonstationarymultivariate autoregressive models. Journal of the American Statis-tical Association 85, 813–23.Archontakis, F. (1998) An alternative proof of Granger’s represen-tation theorem for I(1) systems through Jordan matrices. Journal ofthe Italian Statistical Society 7, 111–127.Bauer, D. & M. Wagner (2003) A Canonical Form for Unit RootProcesses in the State Space Framework. Discussion paper, Univer-sity of Bern.Engle, R. & C. Granger (1987) Co-integration and Error Cor-rection: Representation, Estimation, and Testing. Econometrica 55,251–276.Engle, R. & S. Yoo (1991) Cointegrated economic time series: anoverview with new results. In Engle and Granger (ed.) Long-run Eco-nomic Relations: Readings in Cointegration. Oxford: Oxford Univer-sity Press.Gregoire, S. (1999) Multivariate time series with various hidden unitroots, Part I, Integral operator algebra and representation theorem.Econometric Theory 15, 435–468.Gregoire, S. & G. Laroque (1993) Multivariate integrated timeseries: a polynomial error correction representation theorem. Econo-metric Theory 9, 329–342.Haldrup, N. & M. Salomon (1998) Representation of I(2) cointe-grated systems using the Smith-McMillan form. Journal of Econo-metrics 44, 303–325.Johansen, S. (1988) The mathematical structure of error correctionmodels. Contemporary Mathematics 80, 359–386.Johansen, S. (1992) A representation of vector autoregressive pro-cesses integrated of order 2. Econometric Theory 8, 188–202.Johansen, S. (1996) Likelihood-based Inference in Cointegrated Vec-tor Auto-Regressive Models. Oxford: Oxford University Press.Johansen, S. (2005a) Representation of cointegrated autoregressiveprocess with application to fractional process. Manuscript, Univer-sity of Copenhagen.Johansen, S. (2005b) A representation theory for fractional autore-gressive processes. Manuscript, University of Copenhagen.la Cour, L. (1998) A parametric characterization of integrated vectorautoregressive (VAR) processes. Econometric Theory 14, 187–199.Neusser, K. (2000) An algebraic interpretation of cointegration. Eco-nomics Letters 67, 273–281.Paruolo, P. (1996) On the determination of integration indices inI(2) systems. Journal of Econometrics 72, 313–356.

The Integration Order of Vector Autoregressive Processes

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The Integration Order of Vector Autoregressive Processes

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